1998 Vol. 1 - ABCM

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Associayao Brasileira de Ciencias Mecanicas- ABCM Instituto Alberto Luiz Coimbra de P6s-Graduayao e Pesquisa de Engenharia - COPPE!UFRJ Instituto Militar de Engenharia - IME/RJ

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21 a 25 de Setembro de 1998,

Rio de Janeiro, Brasil.

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Transi<;iio e Turbulencia I Escola de Primavera Associac;ao Brasileira de Ciencias Mecanicas - ABCM ABCM Carlos Alberto de Almeida, Presidente Hans Ingo Weber, Vice-Presidente Paulo Batista Gonr,:alves, Secretario Geral Felipe Bastos de F. Rachid, Diretor de Patrim6nio Nestor Alberto Zouain Pereira, Secretario Comite de Ciencias Termicas da ABCM Antonio Cesar P. Brasil Jr. Leonardo Goldstein Jr. Jose Alberto do Reis Parise Jurandir Itizo Yanagihara Atila P. Silva Freire Joao Luis F. Azevedo Silvia Azucena Nebra de Perez

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Conselho Cientifico da Escola de Alvaro T. Prata Atila P. Silva Freire Daniel Onofre A. Cruz Leonardo Goldstein Jr. Luis Fernando A. Azevedo

Primavera UFSC COPPE/UFRJ UFPA UNICAMP PUCIRJ

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Palestrantes Convidados lnternacionais Robert A. Antonia University of Newcastle, Australia Plymouth University, UK Leslie J. S. Bradbury Marcel Lesieur Institut National Polytechnique, Franr,:a Michael Gaster Queen Mary College, UK Palestrantes Convidados Nacionais Angela 0. Nieckele Pontificia Universidade Cat6lica!RJ Ant6nio Cesar P. Brasil Jr. Universidade de Bras!lia Aristeu da Silveira Neto Universidade Federal de Uberlandia Atila P. Silva Freire COPPEIUFRJ Daniel Onofre A. Cruz Universidade Federal do Para Cesar J. Deschamps Universidade Federal de Santa Catarina Luis Fernando A. Azevedo Pontificia Universidade Cat6lica/RJ Philippe Patrick M. Menut COPPE/UFRJ

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Editores Atila P. Silva Freire Philippe Patrick M. Menut Su Jian

COPPE/UFRJ COPPEIUFRJ COPPE/UFRJ

Secrettirios Mila R. Avelino Claudio C. Pellegrini Patricia Chedier

UERJ FUN REI COPPEIUFRJ

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eooeooe oeoeoeo ooeeeoo ooeeeoo oeoeoeo eooeoo•

••••••• Till' B11ll'h Council

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AMBASSADE DE FRANCE Service Culture!, Scientifique et de Cooperation

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C A P E S

Cooodeno<;OO de Apellelc;ocrnento de Pessoal de Nlvel 5..,...,.,.

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Funda~io Unlveraltarla Jose Bonifacio

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CNPq CONSEUfO No\CIONAL DE

DESENVOLVIMEHTO CIENTIFICO E Tr:CNOU)GICO

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FIJNilA(iAollEAMI'IIRO A PESOUISA DO ESTADO

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A I Escola de Primavera em Transit;iio e Turbulencia e uma iniciativa do Comite de Ciencias Termicas da Associat;lio Brasileira de Ciencias Mecanicas (ABCM). Sonho antigo da comunidade de meciinica dos fluidos, ela agora se toma realidade grat;as ao entusiasmo de alguns pesquisadores e o apoio generoso de certas entidades. 0 Conselho Britanico, a Embaixada da Frant;a e a CAPES financiaram a vinda dos palestrantes intemacionais. 0 apoio decisivo da Fundat;iio Jose Bonifacio permitiu que os Anais com as contribuit;oes tecnicas e as notas dos mini-cursos estivessem disponiveis aos participantes em tempo habil. A Reitoria da UFRJ e a COPPEIUFRJ garantiram a acomodat;lio de todos os palestrantes convidados, nacionais e intemacionais. A FADESP, Federat;ao de Amparo ao Desenvolvimento do Estado do Para, financiou a viagem de alguns palestrantes nacionais. 0 CNPq, a FAPERJ e a FINEP colaboraram com a impressao do livro sobre Turbulencia e de urn volume especial da Revista Brasileira de Ciencias Mecanicas dedicado ao evento, os quais deverao estar a disposit;iio dos leitores a partir do comet;o do ano que vern. 0 lnstituto Militar de Engenharia cedeu graciosamente suas excelentes instalat;oes para estagiar o evento. Seu corpo de professores e funcionarios foi ~empre muito solicito. Finalmente, reitero o enorme apoio recebido dl\ UFRJ, em todos os niveis, para o pleno sucesso da Escola de Primavera em Transit;iio e Turbulencia. Em tempos dificeis, de criticas contundentes e dilacerantes, e sempre urn conforto ter a certeza de pertencer a uma instituit;iio seria, com excelente quadro funcional, discente e docente.

APSF

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Transi<;ao e Turbulencia I Escola de Primavera Associayllo Brasileira de CiSncias MecAnicas - ABCM

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Effect of different surface roughness on a turbulent boundary layer R. A. ANTONIA e P-A.·KROGSTAD Modeling of turbulent flow through radial diffuser Cesar J. DESCHAMPS

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Transition to turbulence of low amplitude three-dimensional disturbances in flat plate boundary layers Marcello A. F. de MEDEIROS

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Uma formulayllo de parede para escoamentos turbulentos com recirculayllo D. 0. A. CRUZ, F. N. BATISTA eM. BORTOLUS

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Mass injection in a wake of a fixed and rotating cylinder Jose Antonio G. CROCE, Fernando M. CATALANO

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Application of a non isotropic turbulence model to stable atmospheric flows and dispersion over 3D topography Fernando T. BO<;:ON, Cl6vis R. MALISKA

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Explorando o nllo-fechamento no equacionamento da turbulSncia Harry E. SCHULZ

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Laminar-turbulent transition: the nonlinear evolution of three-dimensional wavetrains in a laminar boundary layer Marcello A. F. de MEDEIROS

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Investigayllo experimental da transiyllo de escoamentos num sistema pulverizador jato-placa ' Marcelo B. da SILVA, Leonardo M. AMORIM, Aristeu da SILVEIRA NETO

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On Kaplun limits and the multi-layered asymptotic structure of the turbulent boundary layer Atila P. SILVA FREIRE

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Nonlinear evolution of a three-dimensional wavetrain in a flat plate boundary layer Marcello A. F. de MEDEIROS

157

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Transiyao e Turbulencia I Escola de Pdmavera

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Associa~iio

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Brasileira de Ciencias Mecfulicas - ABCM

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Analise de transh;:ao da camada limite sobre a pa de urn modelo de turbina eolica de eixo horizontal Jaqueline B. do NASCIMENTO, Fernando M. CATALANO Uma solu~iio para turbulencia gerada por grades oscilantes Harry E. SCHULZ Comparison of some models of turbulent Prandtl number for low and very low-Prandtl-number fluids Marcelo C. SILVA, Ricardo F. MIRANDA e Lutero C. de LIMA Dynamics of coherent vortices im mixing layers using direct numerical and large-eddy simulations Jorge H. SIL VESTRINI

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181

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195

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( Turbulent shallow-water model for orographic subgrid-scale perturbations Norberto MANGIAV ACCHI, Alvaro L. G. A. COUTINHO e Nelson F. F. EBECKEN

243

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( Instabilidade e turbulencia: uma forma de niio-linearidade encontrada no caos Harry E. SCHULZ Effect of wave frequency on the nonlinear interaction between Gl:lrtler vortices and three-dimensional Tollmien-Schlichting waves Marcio T. MENDON<;A, Laura L. PAULEY and Philip MORRIS em prot6tipo de flutua~oes de pressiio na bacia de dissipa~iio da usina de Porto ColOmbia Jayme P. ORTIZ, Fatima M. de ALMEIDA, Erton CARVALHO e Ricardo D. BORSARI

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Medi~oes

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EFFECT OF DIFFERENT SURFACE

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ROUGHNESSES ON A TURBULENT BOUNDARY LAYER

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R. A. Antonia.* and

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Krogstad+

*Department of Mechanical Engineering University of Newcastle, N.S.W., 2308, Australia +Department of Mechanics, Thermo and Fluid Dynamics Norwegian University of Science & Technology, N-7034 Trondheim, Norway

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ABSTRACT The classical treatment of rough wall turbulent boundary layers consists in determining the effect the roughness has on the mean velocity profile. This effect is usually described in terms of the roughness function t:::.U+. The general implication is that different roughness geometries with the same nu+ will have similar turbulence characteristics, at least at a sufficient distance from the roughness elements. Measurements over two different surface geometries (a mesh roughness and spanwise circular rods regularly spaced in the streamwise direction) with nominally the same nu+ indicate significant differences in the Reynolds stresses, especially those involving the wall-normal velocity fluctuation, over the outer region. The differences are such that the Reynolds stress anisotropy is smaller over the mesh roughness than the rod roughness. The Reynolds stress anisotropy is largest for a. smooth wall. The small-scale anisotropy and intermittency exhibit much smaller differences when the Taylor microscale Reynolds number and the Kolmogorov-norrnalized mean shear are nominally the same. There is nonetheless evidence that the small-scale structure over the three-dimensional mesh roughness conforms more closely with isotropy than that over the rod-roughened and smooth walls.

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1

Introduction

The technological importance of wall-bounded turbulent flows is well accepted. In many situations, turbulent boundary layers develop over surfaces that are hydrodynamically rough over at least some part of their length. The major impact of surface roughness is to perturb the wall layer in such a. way as to lead, in general, to an increase in the wall shear stress. This has obvious implications to both the shipbuilding and aviation industries (e.g. Schlichting, 1968). It also adversely affects the overall performance of turbines, compressors and other bladed turboma.chinery (e.g. Acharya et al., 1986).

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Figure 1: Typical mean velocity distributions, normalized using wall variables, over smooth and · rough walls. The Clauser roughness function t::.U+ is shown.

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The increase in the wall shear stress is almost invariably accompanied by an increase in the wall heat or mass transfer rate. This has major implications in engineering, e.g. in terms of improving the efficiency of heat exchangers or in meteorology, in the context of the atmospheric surface layer over vegetated surfaces. Pimenta et al. (1979) mentioned a number of applications, including nose-tips on re-entry vehicles and transpirationcooled turbine blades where there is heat transfer to or from pervious rough walls. Clauser (1954,1956} presented a method of analysing the effects of surface roughness on the mean velocity distribution; the scheme has proved to be robust and continues to be used. He argued that the inner portion for rough walls must have a logarithmic region with the same slope as for a smooth surface. According to this now "classical" scheme, the sole effect of the roughness is to shift the log-law intercept Cas a function of the roughness Reynolds number k+ Urkfv. With the exception of the "roughness sublayer", the inner mean velocity distribution on a rough wall is then described by

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where t::.u+ represents the roughness-caused shift, as illustrated in Figure 1. (Note that the shift is generally downward although certain surfaces, e.g. longitudinal riblets, can, under certain conditions, produce a positive shift). The outer velocity, when expressed in defect form, viz. (2} ut- u+ = J (~)

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(c5 is the boundary layer thickness) is identical for both smooth and rough walls. Although (1) and (2} have received widespread experimental verification, a few remarks are necessary, especially with respect to t::.U+. The significance of this quantity cannot be overstated since, as Ha.ma (1954} showed, t::.u+ =

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for the same magnitude of the Reynolds number U1 li* / v ( 6* is the displacement thickness). The magnitude of f::.U+ is not uniquely determined by k, even for the so-called k-type roughness, for which

t::.u+ = ~~:- 1 ln k+

(3)

However, experimental support for (4) is lacking, at least for a boundary layer. While there is mild support for a correlation based on E+ (E is the error in surface origin), e.g. Perry et al. (1969), Wood and Antonia (1975), Osaka et al. (1984), Bandyopadhyay (1987), such a proposal is only tenable over a limited range of x (Raupach et a!., 1991). Nonetheless, a surface comprising two-dimensional transverse bars with narrow cavities, which is generally associated with a "d-type" roughness, is of interest since it appears to satisfy the conditions for exact self-preservation, as set out by Rotta (1962), i.e. both Ur/U1 and d6fdx are independent of x. Evidence in support of this was provided by Tani (1986,1987), Osaka et a!. (1982), Osaka and Mochizuki (1988) and Djenidi and Antonia (1998). While this evidence is hard to refute, cogent physical reasons as to why this surface is closer to equilibrium than either a smooth wall or other rough walls have not been formulated; some work is being done in this direction (Djenidi and Antonia, 1998), especially in the context of self-sustaining energy production mechanisms. Although we subscribe to Raupach et al.'s assessment that there are difficulties with the division of roughness into k and (especially) d type classes, we see no reason why Clauser's proposal, encaptured by (1), (2) and (3), shoud not continue to be useful. We emphasise however that the scheme only addresses the mean velocity distribution. According to the Reynolds number similarity hypothesis (Townsend, 1976) or the wall similarity hypothesis (Perry and Abell, 1977), turbulent motions outside the roughness sublayer are independent of the wall roughness at sufficiently large Reynolds number. This is of course consistent with the universality of f(y/6) in (2). The validity of this hypothesis has recently been challenged by Krogstad et a!. (1992), also Krogstad and Antonia (1994). The experimental evidence presented in the last two papers indicates that the outer layer distributions of the wall-normal turbulence intensity and the Reynold's shear stress are markedly different between a mesh roughness and a smooth wall. There was also evidence of major differences in the characteristics of the large scale motion between the two surfaces. One implication of these results is that the communication between the wall and the outer region is more important than has hitherto been thought. Another, possibly more serious, inference is that there may be a fundamental difference in the momentumtransport process contrary to what the equality of the log-law slope, Eq. (1), between smooth and rough surfaces may imply. As a consequence, classical mixing length

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The additive constant C 1 depends, inter alia, on the roughness density, e.g. Dvorak (1969) and Raupach et al. (1991). The latter authors showed that (3) is very well supported by both laboratory and atmospheric surface layer data over an impressive range of k+. A distinction has been made between k-type roughness and d-type roughness for which t::.u+ = ~~:- 1 lno+ + C2 . (4)

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( calculations, albeit allowing for a shift in origin, are unlikely to explain the differences in the Reynolds shear stress -(uv) (angular brackets denote time averaging), in the outer region of the layer. This paper continues to examine possible differences between smooth and rough walls. A particular strategy we adopt is to consider different types of roughness geometries but with the important requirement that flU+ is kept constant. We assess the influence of different surface conditions on the larger-scale (shear-stress carrying) as well.as the smaller scale motions. Although the generally accepted wisdom is that the latter are less likely to be affected by the nature of the surface than the former, especially if the Taylor microscale Reynolds number R>. is kept constant, the possibility that a reduced anisotropy of the large-scale motion (e.g. Krogstad and Antonia, 1994; Shafi and Antonia, 1995) could impact on the anisotropy of the small-scale motion should not be dismissed. Following a brief description (Section 2) of the surfaces we consider, we address mainly the anisotropy of both the large scale {Sections 3 and 4) and the small scale motion (Section 5). We do not consider here the effect of the roughness on the heat transfer characteristics of the boundary layer nor do we treat the implications that the present observations may have on calculation methods. These topics merit to be addressed separately, at a future date.

2

Experimental Conditions

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Two rough surfaces are considered. One is essentialy three-dimensional, consisting of a woven stainless steel mesh screen. The other is basically two-dimensional, consisting of circular rods placed in a spanwise direction at regular streamwise intervals. Dimensions for these two geometries are given in Table 1, where details on experimental conditions are also included. Both the screen and the rods were glued, in separate experiments, on to the aluminium wall of the wind tunnel working section (5.4 m long, inlet area 0.9 m x 0.15 m). The ceiling of the working section was adjusted to set the pressure gradient to zero. The mesh screen covered a streamwise distance of 3.5 m; for the rods, the distance was 3.2 m. Further details for the mesh screen and rod roughness experiments can be found in Krogstad et al. (1992) and Krogstad and Antonia (1998) respectively. For reference, measurements were also made on a smooth wall, though in a different wind tunnel; only a few relevant experimental details are shown in Table 1. Whenever possible, the results obtained by Spalart (1988) for a smooth wall boundary layer direct numerical simulation are shown; only results at the highest value of R9 ( = 1410) have been used. U was measured both with a Pitot tube (0.81 mm outer dia.) and single and X-hot wires (Pt-10% Rh). Most of the wires used has a diameter (dw) 2.5 Jlm and a length lw of 0.5 mm; the measured frequency response was 12.5 kHz at U = 7 mfs. For the small-scale experiments, smaller diameter wires (dw = l.2Ji.m and ew = 0.22 mm) were used; the frequency response was about 23 kHz at U = 7 mfs. The X-wires had an included angle of no•.

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In-house constant temperature circuits were used to operate the hot wires. The output signals from the circuits were filtered (cut off frequency fc) amplified to optimise the input range of the 12 bit (16 channel sample and hold) A/D converter and sampled (sampling frequency J.). For most measurements, fc '::::! 5 kHz, J. '::::! 10 kHz and the record duration was about 30 s. For the small-scale measurements significantly larger values of fc and J. were used, with fc set close to the Kolmogorov frequency fi( (e.g. Table 2); record durations up to 180 s were used.

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Table 1. Characteristics of the Different Surfaces and Basic Experimental Conditions Surface mesh screen (wire dia. = 0.69 mm) circular rods (dia. = 1.60 mm) smo<>th

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3

Mean Velocity and Velocity Fluctuation Moments

Mean velocity distributions are shown in Figure 2 for the three surfaces, with wall variable normalization. Both the present smooth wall results at Rs = 12570 and the DNS smooth wall results of Spalart (1988) for Rs = 1410 are shown. These two distributions overlap, in agreement with the law of the wall. The two rough wall distributions also overlap in the inner region indicating that nominally the same value of f:l.U+ (see Table 1) was indeed for each of the two rough surfaces and the experimental conditions chosen. As originally anticipated by Clauser [1954,1956; see Eq. (2)] and subsequently verified by many investigators, the velocity defect (Ui - u+) [Figure 3], is the same for rough and smooth walls. Reynolds stresses are plotted in Figures 4 ((u+ 2 )), 5 ((v+ 2 )) and 6 (-(u+v+)) in terms of yfo. If we focus our attention primarily on the outer layer, several salient observations can be made 2 ) than either (v+ 2 ) or -(u+v+). Even in the case of (u+ 2 ) (Figure 4), there are some differences between the three surfaces, (u+ 2 ) tending to be larger over the rod

1. There is much closer agreement between the different surfaces for (u+

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roughness. The differences are believed to be genuine, since they fall outside the range of experimental uncertainty. 2. The most pronounced differences seem to occur on (v+ 2 ) (Figure 5) and -(u+v+) (Figure 6), implying that the wall-normal motion is the most affected by the type 2 of surface. Note that (v+ ) and -(u+v+) are also largest for the rod roughness, implying a much stronger momentum transport for this particular surface. Relative to the smooth surface, there is clearly much more activity associated with the wall-normal velocity fluctuation over the rough surface. 3. Although t;;.U+ is the same for the two roughnesses, the Reynolds stress distributions are different. This alone considerably limits the generality of t;;.U+ as a descriptor of the effect that different surface conditions have on the momentum transport. Outer layer differences for the turbulent kinetic energy k+

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[Figure 7] between the rough surfaces are less pronounced than those for (v+ 2 ) or -(u+v+), reflecting the dominant contribution froq1 (u+ 2 ) to (k+). Both rough wall distributions lie significantly above the smooth wall distribution. A comment with regard to the smooth wall DNS distribution shown in this and previous figures seems appropriate. The agreement between the DNS distribution a.t R6 = 1410 and the present smoth wall measurements at R9 = 12570 is generally quite good in the outer region, implying that the moderately low ~ DNS results of Spa.la.rt can serve as a. reliable smooth wall reference against which the effect of the rough wall can be assessed. Velocity triple products a.re expected to be a more sensitive indicator of the effect of surface condition than second-order moments. Andreopoulos and Bradshaw (1981) noted that triple products were spectacularly altered for a. distance up to 10 roughness heights above a surface covered with floor-sanding paper. Bandyopadhya.y and Watson (l988) reported that instantaneous motions involved in the shear stress flux near the

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rods - thus representing a gain of turbulent energy - but positive over the mesh screen. 2 The skewness (Sa= (a 3 )/(a 2 ) 312 ) and flatness factor (Fa= {a4 )/{a2 ) ) of u and v are shown in Figures 9 and 10 respectively. While Su changes sign, irrespectively of the type of surface, near the wall, Sv changes sign only over the rods. In contrast to Sv, Fv is, like Fu, practically unaffected by the &urface up to y /6 ~ 0.6. There are nevertheless differences between the three surfaces in both Fu and Fv a.s the edge of the turbulent/non-turbulent interface is approached.

4

Anisotropy of Reynolds Stresses

Ratios such as {v 2 )/{u2 ), {v 2 )/{w 2 ) or -(uv)/(v 2 ) provide a rough indication of the departure of the Reynolds stresses from isotropy. Figure lla shows that, while there may not be large differences in {v 2 ) / {u2 ) between the two rough surfaces, the magnitude of this ratio is significantly smaller for a smooth wall (the DNS data of Spalart, 1988, is used here and in subsequent figures). The inference is that the anisotropy- at least tor the Reynolds normal stresses- is reduced over a rough wall. Note that {v2 )/(u 2 )

( (

( ( (

9

(

( ( (

v o~B~ Su -1

1-

t:l

0

~aj

Figure 9: Skewnesses of u and v. (a) u; (b) v. Symbois as in Figure 2.

B~ ~ ~

0

(

t-

0

.

v 0

-3 2.0.

1.0

{ 0 0

( (

'v

0

1.5 ~ Sv

~(

v 0

-2

(

1'!1

I

0

(b)

0

0

v

( 0

11 0

~

(

0

0

rg 0

v

o.oF;v I

-0.5 0.0

{

(

I

0.5

'

1.0

1.5

y/o

(

is somewhat larger, in the region y/6 < 0.4, for the mesh tha.n for the rod roughness. This closer tendency towards isotropy for the mesh roughness is better illustrated by the ratio (v 2 ) / (w 2 ) in Figure llb over a. significant fraction of the layer. The ratio -(uv)/(v 2 ), Figure llc, is also smaller for the mesh than the rods. The smooth wall values of -(uv}/(v 2 } are largest, reflecting the greater anisotropy for this surface. A better measure of the anisotropy of the Reynolds stresses is provided by the second (II) and third (I II) invariants of the Reynolds stress anisotropy tensor b;; defined by b·. _ (u;u;} _ 6;; 11 (u;u;} 3

=

where (u;u;} (q 2 ) is the mean turbulent energy and 6;; is the Kronecker delta tensor. The states that characterise b;; can be identified on a plot of -II (= b;;b;;) vs III (1/3b;;b;kb~c;}, as originally proposed by Lumley and Newman (1976}. The limiting values of the invariants delineate an anisotropy invariant map or partially curvilinear triangle as shown in Figure 12. The upper linear boundary of the triangle characterises 2-component turbulence, such as might be expected in the vicinity of a smooth wall; the DNS boundary layer data of Spalart (1988) shown in the figure and the DNS channel flow data of Kim et a!. (1987}, shown in Antonia et a!. (1991 }, confirm this behaviour.

(

( (

( (

(

( (

(

(

\ ( (

( 10

(

t

., (

(

(

40~------~------,-------~

0

( 30

( (

v

v

Fu 20

0

10

0

(

'1

Figure 10: Flatness factors of u and v. (a) u; (b) v. Symbols as in Figure 2.

0

(

~

( (

-

25r-------~------~----(b) 0

(

(

0

20

( (

ljJ

v

15 0

Fv 10

v 0

~~~"ret~

(

( (

( ( ( ( (

( ( ( (

0

0

v

tg

0.0

0.5

1.0

1.5

yto

Figure 12 clearly highlights the greater tendency towards isotropy of the rough wall layers relative to the smooth wall layer. The majority of the rough wall data are much closer to the bottom cusp (i.e. isotropy) of the triangle. In particular, this behaviour is better approximated (inset in Figure 12) by the mesh data than the rod data, thus corroborating previous inferences from Figure 11. A few data points lie just outside the right axisymmetric boundary of the triangle; this is most likely due to the uncertainties in measuring the Reynolds stresses. The information in Figure 12, which is presented solely in terms of invariants, is, unlike that in Figure 11, independent of the choice of coordinate axes. This feature of Figure 12 would be worth exploiting when better quality data - for example with adequately resolved LDV - become available in the vicinity of different rough surfaces. A further measure of the anisotropy of the Reynolds stresses is provided by the parameter F ( 1 + 27 I II + 9II) which is proportional to the product of the three eigen values of (u;uj} / (u;u;}. F is bounded between 0 and 1. It is zero along the linear boundary of the triangle, which describes a 2-component state of turbulence. It is 1 when both I I and I I I are zero, i.e. for isotropic turbulence. Figure 13 clearly shows that, almost everywhere in the layer, F is largest for the mesh roughness and smallest for the smooth wall.

=

( ( (

(

~

' 0'-------...L.-----

(

(

0

[J

( (

0"

-

11

( ( (

2.0

(a)

i

1.51 II

"!:s

v

II

1.?

"!.

I

Figure 11: Reynolds stress ratios. (a) (v 2 )/(u 2 }; (b). {v 2 )/(w 2 ); (c) -(uv)/(v 2 }. Symbols as in Figure 2.

(

'{

{

(

v /0

v

0.5

I (

0.0 2.0'

(b)

/~

~~OonO~.Jl

- 1.0~ II

'\v II

f (

0

(

(

1

0.5

( 0.0 1.0

I

0.8

1II

0.6

>

:1

y

( (c)

\

(

(

vv v

v

v

11'1~

v ~

v

( 0

0

0

0

0

0

0

0 0

<

0.4

( 0.2 0.0

0.2·

0.4

0.6

0.8

1.0

y/8

5

( (

Small-Scale Anisotropy and Intermittency

(

( Ideally, the anisotropy of the small scale-motion should be quantified by evaluating the anisotropy invariants of the energy dissipation rate tensor, in similar manner to the way the Reynolds stress invariants were obtained in the previous section. Anisotropy invariants of E;j require all velocity derivatives to be known. This is not yet feasible

( ( ( (

l 12

( (

\ ( (

( (

0.4 r---,---,------,,-------,,-------,

( 0.3

(

(

-n

·-

0.2

(

0.05

0.1

(

o.oo

Figure 12: Invariant map for the Reynolds stress anisotropy tensor. Symbols as in Figure 2.

M

'=";-~;;--~~-~L=~~=~ 0.04 0.00 0.02 0.06 0.08

(

o.o -0.02

(

Ill

( 1.0r---.---~--,----,-----~

(

(

Figure 13: Invariant function F. Symbols as in Figure 2.

( (

F

( (

( (

0.2

0.6

0.4

0.8

1.0

y/8

( ( (

( ( (

( ( (

(

experimentally (it is possible from direct numerical simulations, e.g. Kim and Antonia, 1993, but these have yet to be performed for.surface geometries of comparable complexity to those considered here). We therefore limit ourselves here to considering relatively simple measures of small-scale anisotropy, as provided for example by the ratios ((8v) 2 )c/((8v) 2 ))m and r/J~(kt)f¢':,'(k1 ), where the superscripts c and m denoted calculated (using isotropy) and measured values. As noted in the introduction, the small-scale anisotropy is expected to depend on a number of factors, such as R>., the mean shear, the proximity to the wall or the presence of a turbulent/non-turbulent interface. Here we consider data at y/8 ~ 0.2, a location where the influence of the last two factors should not be significant. Experimental conditions were chosen so that the magnitude of R>. is about the same for the three surfaces considered. Also, the magnitude of s•::: S(v/(e)) 112 , where S::: 8(U)/8y is the mean shear, is nominally t.he same for the three surfaces. The experimental conditions are summarised in Table 2.

(

( ( (

13'

(

1

103

Figure 14: Secondorder moments of longitu-. dina} and transverse veloc- . ity increments. The normalization is by Kolmogorov variables. Also shown is the ratio ((8v) 2n((8v)2)m, where the superscripts c and m refer to calculated (using isotropy) and mea1mred values. Rod roughness : 0,

<(liu*)2>m <(liv*)2>m

e

"t

<(liv•)•>•

t

o

-

--%~-"""-''-·~=j ~

........_.•. .,,.-.;:=""'·-

;;;

"!;;10

.n

I

I

I I

!fill

1

1

t

10 1

10°

1 pu!

I

I

I I

qui

I

I

1 1

!Ill!

103

102

~

10•

r*

((8u*)2)m; +, ((8v*)2)m; -, ((8v*j2) ((8v) 2 ) 0 /((8v) 2)m 0



: - -, smooth wall; - - -,mesh roughness;---, rod roughness. Table 2. Experimental Conditions for Different Surfaces Yielding Approximately the Same R~ at y/8 ~ 0.2 Surface

y/8

Mesh Rods Smooth

0.22 0.20 0.19

Uoo

(U)

(m/s)

(m/s)

10 10 23

6.66 5.34 18.4

Jgt

~

Rs

R~

77t (mm)

(kHz)

0.49 0.43 0.35

8000 12000 12570

240 248 230

0.126 0.109 0.077

8.41 7.90 38.1

( (

(

( (

( (

( )



(5)

As expected, the ratio ((8v*) ) /((8v*) 2)m, also shown in the figure for the other two surfaces, is close to 1 at small r* and increases as r* increases. As r* --+ oo (or, to a reasonable approximation r* ;:::: L*, where Lis the integral length scale), ((8u*) 2 ) --+ 2(u* 1 ) and ((8v*) 2 ) = 2(v* 2 ). Since ((8u*Jl)m was used as input in Eq. (5), ((8v*) 2 ) 0 = ((8u*) 2)m when r* is large enough for 8((8u*) 2 )m j8r* to be zero. The different levels of the ratio ((8v) 2 )" / ((8v) 2 )m in Figure 14 reflect the different values of the ratio (v 2 )m /(u 2 )m for the three different surfaces (Table 2). 0

(

(

0

2

(

(

0

2

(

( 0.09 0.11 0.12

Distributions of ((8u*) 2)m, ((8v*) 2 )m and ((8v*) 2) are shown in Figure 14 for the rod roughness. The values of ((8v~) 2 ) were calculated using the isotropic relation

:r) ((8u)

(

(



t77 is the Kolmogorov length scale v314 j(f) 114 i /K is the Kolmogorov frequency Uj21r71

2 ((8v) ) = ( 1 + ~

(

( (

(

( (

( ( (

14

( (

'

\ ( (

(

(

(

4>:m(k~)

(

4>;"'(k1'l 103

(

4>:'(k1) 10'

( (

~

~ ::::.

¢~(kt)!¢':(kt):

105 r~~.......,.--~~..,..,-~~.....-~~~-

( (

Figure 15: Kolmogorovnormalized spectra of u and v. The ratio ¢~/ ¢': is also shown. The superscripts c and m refer to calculated (using isotropy) and measured distributions. Rod roughness : - -, ¢:m(kj);

102 ~

~ t

=""'""*

htnh

I

!!!p!!l

10"3

I

!!!p!!l

I

1



111!1111

10"2

I

e;

1

100

III!IIQ

10"1

10

10°

~

.2 . .

- - -,

¢~m(ki);;

-, ¢:c(kj).

--,smooth wall; ----,mesh roughness; - - -, rod roughness.

k1

{ ( ( (

( ( (

( (

( ( (

( ( (

( ( (

(

Distributions of ¢:m(ki), ¢:m(ki) and ¢:c(ki) are shown in Figure 15 as a function of ki, where k1 is the one-dimensional wavenumber. ¢: 0 (ki) was evaluated from the isotropic relation

¢.(kt) =

~ (1- k1 0~J ¢u(kt)

with ~::'(k 1 ) as input. The departures of ¢~(kt)f¢l::(kt) from 1 at high wavenumbers of course reflects that of (( ov )2) c f (( ov )2) m from 1 at small separations (Figure 14). It is difficult to select unambiguously, from ratios in Figures 14 and 15, a surface for which the anisotropy is smallest. One could conclude that all three surfaces satisfy isotropy equally well to a rough approximation. Alternately, it may be argued that isotropy is somewhat better satisfied, especia.lly in terms of the extent of the range- in either r• or ki - for the mesh roughness. A more detailed examination of the smallscale structure for this roughness supports this argument, in particular, transverse vorticity spectra were found (Antonia. a.nd Shafi, 1998) to satisfy isotropy a.t least as well as the boundary layer measurements (Ong a.nd Wallace, 1995) in the NASA-Ames wind tunnel at R>. ~ 1400. Speculatively, the small-scale structure over three-dimensional surfaces such a.s the mesh screen may satisfy isotropy more closely than that over twodimensional roughnesses such as the rods. Although more work is needed, for example, vorticity measurements have yet to be made over the rods or, for that matter, over the present smooth wall, the experimental conditions (in particular the magnitudes of Tf and fk) in Table 2 clearly indicate that small-scale statistics should be measured more accurately over rough surfaces than smooth walls. A measure of small-scale intermittency is provided by the departure of the IR exponents (u(P) where

((ou)P)

~

r'•(Pl

from the corresponding Kolmogorov (1941) values. According to Kolmogorov (1941),

((ou)P) ~ rP/ 3

( (

( (

IS

(

{ 3.------.-------.-------.------~

/;;u(P)

~-------~

2

_o-----

0

I

I

I

j

2

4

6

8

/;;v(P)

0

.

~

~

9

0 ~

0

Figure 16: Scaling range exponents for (louiP) and (loviP) at nominally the same R>. (~ 240) and yfo (~ 0.2). (u(P) : 0, smooth wall; D, mesh roughness; 'V, rod roughness. (.(p) : 0, smooth wall; D, mesh roughness; 'V, rod roughness .

( (

(

(

I (

(

(

p

( 3,------,-------.-------.------~

;§k1

2 /;;u(P)

/;;v(P)

Iy

Iy

•• y

•• y

••

:')4 !

Figure 17: Scaling range exponents for (louiP) and (loviP) at different R>. but the same yfo (~ 0.2). Open and solid symbols represent (u(P) and (.(p) respectively. <:J, R>. = 240; D, 330; 0, 375.

y

( (

(

( (

( (

2

4

6

8

p

(

when r lies within theIR. This departure, which is usually described as the "anomalous scaling", increases asp increases (e.g. Anselmet eta!., 1984) but is detectable even for small values of p (e.g. Stolovitzky_ and Sreenivasan, 1993). The magnitude of (u(P) was estimated using the extended self-similarity (ESS) method of Benzi et, a!. {1993); the scaling range was however restricted to that over which (loul 3)r•-l was approximately constant (this range is loosely identified here with theIR). For consistency, the same range was used to determine (.(p), theIR exponent of (lov!P), viz.

(lovjP)

~ r(.(p)

.

It is worth emphasising that ESS only yields relative, rather than absolute, estimates of (u(P) and (.(p). The distributions of (u(P) over the three different surfaces at yfo ~ 0.2 (Figure 16) are nearly the same, implying that the intermittency affects each flow in similar fashion. For comparison, the prediction of the lognormal model (Kolmogorov,

( (

(

( (

(

( (

( (

( 16

( (

'-

(

(

( ( ( (

( ( ( ( (

( (

( ( ( (

( ( ( (

( (

( ( ( (

( ( (

1962), viz.

1'( ) = ~ - pp(p- 3) ~ p

3

18

where p (~ 0.2) is the intermittency factor, has been included in Figure 16. While it is in rea.!lonable agreement with the experimental estimates of (u(p), it does not differentiate between (u(p) and (.,(p). The experimental magnitudes of (.,(p) are significantly smaller than those of (u(p), possibly suggesting that transverse velocity fluctuations are more intermittent than longitudinal velocity fluctuations. Another possibility may be that v is more sensitive to anisotropy tha.n u. In this context, we note that (.,(p) is generally bigger for the mesh roughness than the other two surfaces, in support of the earlier suggestion that the small-scale turbulence over three-dimensional surfaces satisfies isotropy more closely than that over either a smooth wall or a two-dimensional surface roughness. As the Reynolds number (Re or R~) increases, the anisotropy is expected to decrease, thus bringing (.,(p) into closer alignment with (u(p). This tendency is illustrated in Figure 17 for the rod-roughness. Note that (u(P) is hardly affected by the increase in R~. It should be mentioned that the increase in (.,(p) with R~ in Figure 17 and the difference between the three (.,(p) distributions in Figure 16 are statistically significant, allowing for experimental uncertainty in estimating (.,(p).

6

Conclusions

The Clauser roughness function t::.U+ is a useful descriptor of the effect that the surface roughness has on the mean velocity distribution in the inner region of a boundary layer. Also, the mean velocity distribution in the outer region is, to a good approximation, unaffected by the roughness. However, there is as yet no adequate scheme which describes the effect the roughness has on the Reynolds stresses in the outer region of the boundary layer. In particular, we have noted that for two different roughness geometries for which t::.u+ is approximately the same, the outer layer distributions of the Reynolds stresses, especially those involving the wall-normal velocity distribution, are discernibly different. They also differ with respect to the sipooth wall. The Reynolds stress differences are such that the Reynolds stress anisotropy is smallest over the mesh screen and largest for the smooth wall. We have also shown that when R~ and a Kolmogorov-normalized value of the mean shear are kept constant, the differences in the small scale anisotropy are only small. Nonetheless, there is evidence indicating that, for the three-dimensional roughness, the small scales are more closely isotropic than for either the two-dimensional roughness or the smooth wall. Consistently, the difference between IR power-law exponents of longitudinal and transverse velocity structure functions appears to be smallest for the mesh screen roughness. A useful feature of rough walls is that a particular value of R,. can be attained at a particular value of yf ofor a significant smaller free stream velocity relative to a smooth wall. There are consequently less severe experimental constraints fn rough wall layers in the context of adequately resolving the small-scale motion.

( ( ( (

17

(

t

Acknowledgements

(

RAA acknowledges the continued support from the Australian Research Council.· We would like to acknowledge the contribution from Mr. R. Smalley to the rod rough- · ness experiment and many useful discussions with Dr L. Djenidi in the context of a turbulent boundary layer over the so-called "d-type" surface.

(

(

(

References ACHARYA, M., BORNSTEIN, J. and ESCUDIER, M.P. 1986. Turbulent boundary layers on rough surfaces, Expts. in Fluids, 4, 33-47. ANDREOPOULOS, J. and BRADSHAW, P. 1981. Measurements of turbulence structure in the boundary layer on a rough surface, Boundary-Layer Meteorol., 20, 201-213. ANSELMET, F., GAGNE, Y., HOPFINGER, E. J. and ANTONIA, R. A. 1984. High order velocity structure functions in turbulent shear flows, J. Fluid Mech., 140, 63-89. ANTONIA, R. A., BROWNE, L. W. B. and KIM, J. 1991. Some characteristics of small scale turbulence in a turbulent duct flow, J. Fluid Mech., 233, 369-388. ANTONIA, R. A. and SHAFI, H. S. 1998. Small scale intermittency in a rough wall !turbulent boundary layer, Expts. in Fluids (to appear) BANDYOPADHYAY, P.R. 1987. Rough-wall turbulent boundary layers in the transition regime, J. Fluid Mech., 180, 231-266. BANDYOPADHYAY, P. R. and WATSON, R. D. 1988. Structure of rough-wall turbulent boundary layers, Phys. Fluids, 31, 1877-1883. BENZ!, R., CILIBERTO, S., TR.IPICCIONE, R., BAUDET, C., MASSAIOLI, F. and SUCCI, S. 1993. Extended self-similarity in turbulent flows, Phys. Rev. E, 48, R29-R32. CLAUSER, F. H. 1954. Turbulent boundary layers in adverse pressure gradient, J. Aeronaut. Sci., 21, 91-108. CLAUSER, F. H. 1956. Turbulent boundary layer, Adv. Appl. Mech., 4, 1-51. DJENIDI, L. and ANTONIA, R.. A. 1997. Reynolds stress producing motions in smooth and rough wall tm:bulent boundary layer, in R. Panton (ed.) SelfSustaining Mechanisms of Wall Turbulence, Southampton, Computational Mech. Pubs., 181-199. DVORAK, F. A. 1969. Calculation of turbulent boundary layers on rough surfaces in pressure gradient, AIAA Jnl., 7, 1752-1759. ERM, L. 1988. Low Reynolds number turbulent boundary layers, Ph.D. Thesis, University of Melbourne. HAMA, F. R.. 1954. Boundary layer characteristics for smooth and rough surfaces, Trans. Soc. Naval Arch. Marine Engrs., 62, 333-358. KIM, J. and ANTONIA, R.. A. 1993. Isotropy of the small scales of turbulence at low Reynolds number, J. Fluid Mech., 251, 219-238.

(

( ( (

( (

( (

( ( ( ( (

(

(

(

( ( (

( (

( 18

(

(

(

(

(

( ( {

( (

( ( (

( (

(

( ( (

(

( {

( (

( ( ( ( (

( (

(

KIM, J., MOIN, P. and MOSER, R. D. 1987. Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 177, 133-166. KOLMOGOROV, A. N. 1941. Energy dissipation in locally isotropic turbulence, D"okl. Akad. Nauk. SSSR, 32, 19-21. KOLMOGOROV, A. N. 1962. A refinement of previous hypothesis concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech., 13, 82-85. KRbGSTAD, P.-A. and ANTONIA, R. A. 1994. Structure of turbulent boundary layers on smooth and rough walls, J. Fluid Mech., 277, 1-21. KROGSTAD, P.-A. and ANTONIA, R. A. 1998. Surface roughnes effects in turbulent boundary layers (in preparation) · KROGSTAD, P.-A., ANTONIA, R. A. and BROWNE, L. W. B. 1992. Comparison between rough- and smooth-wall turbulent boundary layers, J. Fluid Mech., 245, 599-617. LUMLEY, J. L. and NEWMAN, G. R. 1976. The return to isotropy of homogeneous turbulence, J. Fluid Mech., 82, 161-178. ONG, L. and WALLACE, J. M. 1995. Local isotropy of t.he vorticity field in a boundary layer at high Reynolds number, in R. Benzi (ed.) Advances in Turbulence V, Dordrecht, Kluwer Academic Pub., 392-397. OSAKA, H. and MOCHIZUKI, S. 1988. Coherent structure of a d-type rough wall boundary layer, in M. Hirata and N. Kasagi (eds.) Transport Phenomena in Turbulent Flows : Theory, Experiment and Numerical Simulation, New York, Hemisphere, 199-211. OSAKA, H., NAKAMURA, I. and KAGEYAMA, Y. 1984. Time averaged properties of a turbulent boundary layer over a d-type rough surface, Trans. Japan Soc. Mech. Eng., 50, 2299.2306 OSAKA, H., NISHINO, T., OYAMA, S. and KAGEYAMA, Y. 1982. Self-preservation for a turbulent boundary layer over ad-type rough surface, Memoirs of the Faculty of Engineering, Yamaguchi University, 33, 9-16. PERRY, A. E. and ABELL, C. J. 1977. Asymptotic similarity of turbulence structures in smooth- and rough-walled pipes, J. Fluid Mech., 79, 785-799. PERRY, A. E., SCHOFIELD, W. H. and JOUBERT, P. N. 1969. Rough-wall turbulent boundary layers, J. Fluid Mech., 37, 383-413. PIMENTA, M. M., MOFFAT, R. J. and KAYS, W. M. 1979. The structure of a boundary layer on a rough wall with blowing and heat transfer, J. Heat Transfer, 101, 193-1'98. RAUPACH, M. R., ANTONIA, R. A. and RAJAGOPALAN, S. 1991. Rough-wall turbulent boundary layers, Appl. Mech. Rev., 44, 1-25. ROTTA, J. C. 1962. Turbulent boundary layers in incompressible flow, in A. Feric, D. Kucheman and L. H. G. Stone (eds.) Progress in Aeronautical Science, Oxford, Pergamon, 1-220. SCHLICHTING, H. 1968. Boundary Layer Theory, 6th ed., New YUork, McGrawHill. SHAFI, H.S. and ANTONIA, R. A. 1995. Anisotropy of the Reynolds stresses in a turbulent boundary layer on a rough wall, Expts. in Fluids, 18, 213-215.

(

( (

(

19

(

(

SPALART, P. R. 1988. Direct simulation of a turbulent boundary layer up to R 9 = 1410, J. Fluid Mech., 187, 61-98. STOLOVITZKY, G. and SREENIVASAN, K. R. 1993. Scaling of structure functions, Phys. Rev. E, 48, R33-R36. TAN!, I. 1986. Some equilibrium turbulent obundary layers, Fluid Dyn. Res., 1, 49-58. TAN!, I. 1987. Equilibrium, or nonequilibrium, of turbulent boundary layer flows, Proc. Japan Academy, 63, 96-100. TOWNSEND, A. A. 1976. The Structure of Turbulent Shear Flow, CUP. WOOD, D. H. and ANTONIA, R. A. 1975. Measurements of a turbulent boundary layer over a d-type surface roughness, J. Appl. Mech., 42, 591-597.

{

( (

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( 20

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\

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MODELING OF TURBULENT FLOW THROUGH RADIAL DIFFUSER

(

(

(

( (

Cesar J. DESCHAMPS Departamento de Engenharia Meclinica Universidade Federal de Santa Catarina S8040-900 Florian6polis - SC, Brazil e-mail: [email protected]

( (

( ( ( ( (

( (

( ( ( (

(

21

(

(

( (

MODELING OF TURBULENT FLOW THROUGH RADIAL DIFFUSER

(

Keywords: Radial Diffuser, Renormalization Group k-e Model (

(

ABSTRACT

(

The work considers the modeling of turbulent flow in radial diffuser with axial feeding. The flow through the diffuser is characterized by the Reynolds number Re based on the feeding orifice

(

(

diameter d. Due to its claimed capability to predict flow including features such as recirculation

(

region, curvature and adverse pressure gradient (all of them existing in the flow considered here),

(

the RNG k-e model of Orzag et a!. (1993) has been applied in the present numerical analysis. The

( (

governing equations are numerically solved using the finite volume methodology, and the (

approximation of convective transport was performed using the higher-order accurate QUICK

(

scheme. Experimental results for different values of displacement between the disks and Reynolds

(

number when compared to the numerical solution showed that the RNG k-r. model can provide a

(

good prediction of the flow.

(

(

( (

( ( (

( 22

( (

(

(

(

NOMENCLATURE

(

Acl

Entrance cross sectional area of the diffuser, [= 1tds]

Af

Cross sectional area of the feeding orifice, [= 1td 2 /4]

d

Orifice diameter

D

Front disk diameter

(

I

Turbulence intensity, [=Cutuinl

(

e

Orifice length

{

m

Mass flow rate

p

Pressure, also step height in Fig. l

(

p*

Dimensionless pressure,[= 2p/pUfnl

(

Patm

Atmospheric pressure

r,x

Cylindrical coordinates

Re

Reynolds number based on the orifice diameter, [= pUind/f.l]

(

s

Displacement between the disks

(

t

Step width

(

u,v

Velocity components

Din

Average velocity in the orifice

(

y+

Dimensionless distance from the wall, [= pu• y /f.l]

(

u•

Wall-friction velocity [=.F:/Pl

(

( (

( (

(

(

(

( (

( (

( ( (

( ( ( (

(

Greek Svmbols

1-l

Air absolute viscosity

p

Air density

"'.

Dimensionless stream-function[= 'I' I rn] 2j

( (

INTRODUCTION

A three dimensional schematic view of a radial diffuser and the geometric parameters that govern Ia

( (

and Ib, respectively. The fluid enters the diffuser flowing axially through the feeding orifice, hits

(

the front disk, and after being deflected by it a radial flow is established. The impact of the flow on

(

the flow, including a backward facing radial step (not shown in the 3-D view), are shown in

Fi~s.

the front disk produces a bell-shape pressure distribution and, depending on the gap between the (

disks and on the flow Reynolds number, negative pressure regions can occur. The radial diffuser geometry of Fig. I represents a basic configuration for many engineering

(

( flows. Examples include reed type valves of reciprocating compressors (Prata and Ferreira, 1990), aerostatic bearings (Hamrock, 1994), vertical take off and landing aircrafts with centrally-located

(

downward pointing jets (Moller, 1963 ), electro-discharge machining (Osenbruggen, 1969), aerosol

(

impactors used as inertial separation device for collecting air borne particles (Marple et al. 1974),

(

injection molds used for polymer processing (Pearson, 1966), pilot valves employed in hydraulic and pneumatic components (Hayashi et al., 1975), and prosthetic valves used to replace diseased

I (

natural heart valves in humans (Mazumdar and Thalassoudis, 1983 ). (

Despite the numerous works related to laminar radial flows, very little attention has been given to

(

(

L

1

¢>DI

-~--~--

··v

( (

lD '!· tJ·

(

I

(

Feeding orifice

~~-~cl· --~

1_j- Air flow (a)

Front disk

(b)

(

(

(

( (

Figure 1 - Geometry of radial diffuser with axial feeding. 24

(

(

(

(

(

(

turbulent flows. For references on laminar radial flow the interested reader is referred to Hayashi et

(

al. (1975), Wark and Foss (1984), Ferreira et al. (1989), Gasche et al. (1992) Prata et al. (1995),

(

Possamai et al. (1996), and the literature cited therein.

(

The few works dealing with radial turbulent flow focused on pure radial flow between parallel

(

disks _without considering the inlet (Ervin et al., 1989 and Tabatabai and Pollard, 1987).

(

Apparently, the first attempt to solve the turbulent flow in axially feeding radial diffusers was made

(

(

by Deschamps et al. ( 1988). There it was found that the high Reynolds numbet k-e model used to

(

close the averaged Navier-Stokes equations was unable to predict the flow, even with the inclusion

(

of correction terms to take into account effects such as flow curvature. The poor quality of the

(

numerical solution was attributed to the wall-functions needed in the model. This was confirmed

(

later when a low Reynolds number k-e model, which does not use wall-functions, produced better

( {

flow predictions (Deschamps et al. 1989). Nevertheless, even for this model there were significant

(

differences between experiments and computations. In reality, the k-E model is known to produce

(

excessive turbulence in the presence of adverse pressure gradient, as is the case at the entrance of

(

the radial diffuser. This leads ·to an overprediction of turbulence intensity and to delay, or even

(

suppress, any eventual flow separation hinted at by the laboratory measurements.

( ( ( (

The main goal of the present paper is to perform a numerical simulation with experimental validation of the turbulent flow through the geometry depicted in Fig. 1.

Due to the claimed

capability to predict flows that include features such as stagnation and recirculation regions,

(

curvature and adverse pressure gradients (all of them present in the flow considered here) the RNG

(

k-£ model ofOrzag et al. (1993) was adopted in this work.

( ( (

EXPERIMENTAL SETUP AND PROCEDURE A careful experimental setup was built to measure the pressure distribution on the front disk as a

( ( ( ( (

function of the Reynolds number, Re, and the displacement between the disks, s. Of paramount importance in the experiments was the correct adjustment of the displacement to the desired value 25

(

( due to the strong influence of this parameter on the flow field. The uncertainty associated to the

(

experimental data is less than 5%. A description of both the experimental setup and procedure will

(

not be presented here due to space limitations, but can be found in Possamai et al. (1995).

( (

(

TURBULENCE MODELING

( The foundation of most turbulence models is the time/ensemble averaging of the flow transport equations, where the turbulent component of any property is defined as the departure from its time/ensemble averaged value. For an isothermal and incompressible flow under the effect of no body force the equations of motion (here written in Cartesian tensor notation) are:

(

Mass conservation,

( (

(

aui

=D

(1)

axi

( (

(

Momentum conservation for the U; component of velocity,

( (

ui auj

axi

=-.!_(~)+__£_[v auj -uiu·J p

axj

axi

axi

(

.

(2)

J

(

( (

The term

UjU j

appearing in the above equation is the Reynolds stress tensor and is never

(

negligible in any turbulent Oow. Equations (1) and (2) can only be solved if the Reynolds stresses are known, a problem referred to as the 'closure problem' since the number of unknowns is greater than the number of equations. A brief account of some common techniques used to close Eq. (2) is

(

(

( given below. A good review of several other methods is provided by Markatos ( 1986).

(

( (

26

(

( (

(

(

( (

Reynolds Stress Models

f

By manipulating the Navier-Stokes equation for instantaneous velocity it is possible to obtain a

(

transport equation for the Reynolds stress uiuj. However, the resulting set of equations is not

(

(

closed since they include higher order correlations, such as the third order moments uiujuk, as

(

well as correlations between fluctuating velocities and pressure. Any attempt to provide transport

( (

equations for these higher order correlations leads to the appearance of even. higher correlations

( (

and, consequently, to a dramatic increase in the computing time required to calculate flows. For this reason, most works using this level of modeling have used transport equations for second order correlations and simple algebraic expressions to approximate the triple-moments and the

(

(

correlations between fluctuating velocities and pressure. Models for closing the Reynolds stress

(

transport equations, following the aforementioned procedure, were proposed quite early (see for

(

instance Rotta, 1951 ).

(

A considerable obstacle for the use of Second Moment closures is the modeling of the near-wall

(

region and, despite much effort have been directed to the solution of the problem, progress has not

(

c (

reached the point for the full benefit of flow in complex geometries as the one considered here. On the other hand, the employment of wall-functions, to bridge the whole of the near-wall sublayer

( (

where viscous effects are significant, is unsuitable for the present flow situation, as hinted at by

(

by the condition ofQ1inimum turbulence level that must be observed when using wall-functions (for

( (

practical reasons usually fixed as / ,., 11.6). Naturally, in flows where important features occur

previous works (Deschamps et a!., .1989). This is mainly related to the numerical aspect and caused

close to the walls it is quite difficult to balance the needed grid resolution against the minimum

(

( (

(

value for /. Given the foregoing reasons and in order to predict the flow in radial diffusers it seems to be essential to avoid the use of wall-functions and to include the near-wall region in the calculations.

(

f ( (

27

(

(

Eddy Viscosity Models

The assumption that turbulence is proportional to the velocity gradient, acting like the viscous stresses, was first made by Boussinesq, who introduced the concept of a 'turbulent' or 'eddy'

( ( (

viscosity v 1. A generalized form of the Boussinesq's hypothesis, proposed by Kolmogorov, is as follows:

~

( (

-

UjU j = -Vt

[8U· 8Uj] + Jliijk 2 Ox~ + Oxj

( (3)

,

(

(

where liij is the Kronecker delta and the kinetic energy of the turbulent motion, k, is defined as

(

k = (uiui)/2. Substitution of Eq. (3) into Eq. (2) results in the averaged Navier-Stokes equations with the Reynolds stresses modeled via the viscosity concept: (

(

8Ui 18( 2 ) 8[ (8Ui 8Uj}]

U · - = - - - p+-pk + - Veff - + J Oxj p Oxj 3 Oxj Oxj Oxj

(4)

(

( (

where Veff =

v + Vt. By far the most common choice for calculating of v1 has been that in terms of

the turbulence kinetic energy k and its rate of dissipation, e, i.e.

( ( (

( (

k2

(5)

Vt=CJ.l-;-

(

( Models of this kind were originally proposed by Harlow and Nakayama (1968) and subsequently

( (

refined by Launder and Spalding (1972). Later, Jones and Launder (1972) included low-Reynoldsnumber effects into the k-e model (by making certain coefficients dependent on the turbulent

( (

(

28

( (

(

( (

(

(

Reynolds number) so that it can be used to compute near-wall flows as well as those where wall effects are not present.

(

Due to its robustness, economy and acceptable results for a considerable amount of flows the ·k-e

(

model has been the most used model for numerical predictions of industrial flows. However, it is

I

known. to have deficiencies in some situations involving streamline curvature, acceleration and

(

separation; all of them are present in the case of flow through radial diffusers.

For instance,

(

(

turbulence is very sensitive to small amounts of curvature of the streamlines; see Bradshaw(1973).

(

The effects of curvature tend to increase the magnitude of the turbulence shear stress where the

(

angular momentum of the flow decreases in the direction of the radius of curvature, and to decrease

(

when the angular momentum increases with the radius.

(

turbomachine blade the skin friction may be reduced by the curvature on the convex surface by as

( (

much as 20 % and increased on the concave surface by a comparable amount. Such effects cannot

(

Hence, for example, on a typical

be accounted for in turbulence models based on the simple eddy-viscosity hypothesis unless some

(

ad hoc extra terms be introduced into the equations (it is opportune to state that the Reynolds stress

(

equations, in the case of curved streamline flows, contain in exact form extra-strain production

(

terms that account for the curvature). Moreover, in the presence of adverse pressure gradient

(

regions, the equation for E is known not to be capable of responding to the surge of kinetic energy,

( ( (

returning an excessive level of turbulence that can lead to a delay of an eventual flow separation or even to a total suppression of it, as pointed out by Rodi and Scheuerer ( 1986). Finally, in the case of

( (

separated flows Sirripson et al. (1981) found that the Reynolds shearing stresses must be modelled

{

A new form of k-E model has been proposed by Orzag et al. (1993) and was derived from the

(

original governing equations for the fluid motion using Renormalization Group (RNG) methods.

(

by relating them to the turbulence structure and not to the local mean velocity gradients.

The novelty of the so called RNG k-E model, compared to the standard k-e model, is that constants

(

(

and functions are evaluated by the theory and not by empiricism and that the model can be applied

(

to the near-wall region without recourse to wall-functions or ad-hoc function in the transport

(

29

(

~

(

(

equations of the turbulence quantities. Due to this mathematical foundation, compared to the semi-

(

empirical approaches adopted in the standard k-E, Orzag and his colleagues argue that the RNG k-E

( model offers a wider range of applicability. Some examples of flows where the RNG k-E model has (

been seen to return better predictions than the standard k-E are those including flow separation,

(

streamline curvature and flow stagnation. As pointed out before, all these flow features are present in the case of radial diffusers and, therefore, it seemed natural to adopt the RNG k-e model in the

(

present work.

(

The effective viscosity in the RNG k-E model is given by

(

v,ff

(

-{t•F.},r

( (6)

( (

which is valid across the full range of flow conditions from low to high Reynolds numbers. The

(

( turbulence kinetic energy k and its dissipation e appearing in Eq. (3) are obtained from the following transport equations:

( ( (

ak

ak]

a[

U j - = - a.Veff- +v 1S2 iJx j iJx j iJx j

( -E

(7)

,

( ( (

a[

8£- = - a.v U· Jjjx. J

Ox·

J

2

e

8£-] +C 1-v E 2 E rr 1S -C 2--R iJx· E k E k '

(8)

J

( ( (

where the values of C 81 e C 8 2 are equal to 1.42 and 1.68; respectively. The inverse Prandtl

(

( number a. for turbulent transport is given by the following relationship:

( (

( 30

( (

' ( (

( (

a -1.392910.63211 a+ 2.392910.3679

I

a 0 -1.3929

a 0 + 2.3929

v

(9)

Veff

(

(

with a 0 = 1.0. The rate of strain term, R, is given by

(

( (

R =ell113 (!- TJ I TJo )e2

1+~, 3

(

(10)

k

( (

where TJ =Ski E, TJo,., 4.38, ~ =0.012 and S = 2SijSij in which Sij is the rate of strain tensor. In 2

( (

regions of small strain rate, the term R has a trend to increase v eff somewhat, but even in this case

(

Veff still is typically smaller than its value returned by the standard k-E

(

elevated strain rate the sign of R becomes negative and veir is considerably reduced. This feature

( ( (

model. In regions of

of the RNG k-E is responsible for substantial improvements verified in the prediction of large separation flow regions.

(

Also the reduced value of Ca 2 in the RNG theory , compared to the value of 1.9 used in the

(

standard k-E turbulence model, acts to decrease the rate of dissipation of e, leading to smaller values

( (

ofveff·

(

(

Boundary conditions

(

Boundary conditions at inlet, walls, axis of symmetry and outlet are required to solve Eqs. (4),(6)

(

and (7). For the inlet boundary it was recognized by Ferreira et at. ( 1989) that, as the flows exits

( ( (

the feeding orifice of area Ar and enters the diffuser of area Ad , the strong reduction of the passage area given by the ratio Ad I Ar = 4sld brings about a strong flow acceleration next to the orifice wall for small values of sld. Due to this phenomenon the inflow velocity profile at the feeding orifice

( (

plays no role in the solution of the flow field in the diffuser; therefore, the inlet boundary condition

(

31

(

t ( was specified as U=U;n and V=O. Although no information is available for the turbulence kinetic energy, numerical tests indicated that when the level of the turbulence intensity was increased from

(

( (

3% to 6% no significant change was observed in the predicted flow. Therefore, a value of 3% of turbulence intensity was used in the calculation of all results shown in this work. Finally, the

(

distribution of the dissipation rate was estimated based on the assumption of equilibrium boundary (

layer, that is

( e = c/'4k3/2 !em

(II)

( (

where C 11 =0.09, and the mixing length was calculated using an empirical coefficient for turbulent

( (

pipe flow, that is,

em = 0.07d/ 2.

(

At the solid boundaries the condition of no-slip and impermeable wall boundary condition were

(

imposed for the velocity components, that is, U=V=O, with calculations being extended up to the

(

walls across the viscous sublaycr. For the turbulence quantities k and e rather than prescribing a

(

condition at the walls, they were calculated in the control volume adjacent to the wall following a

( (

two-layer based non-equilibrium wall-function. In the plane of symmetry, the normal velocity and the normal gradients of all other quantities were set to zero.

(

At the outlet boundary two different procedures had to be adopted. For Did = 3 the diffuser exit

(

is far enough downstream and a condition of parabolic flow can be assumed. Yet, for the much

(

smaller ratio, D/d = 1.45, this is not possible and, therefore, the solution domain had to be extended

(

well beyond the diffuser exit and the atmospheric pressure verified in the experiment was set to the

(

( outlet. The boundary condition for k in this case was fixed according to a turbulence intensity of

( 3% whereas the dissipation rate was estimated based on the same assumption of equilibrium

(

boundary layer used at the orifice inlet, Eq. (II). Given the wall jet characteristic of the flow

(

exiting the diffuser it is expected that any eventual inaccuracy of the above outlet conditions will

(

not have a significant impact on the numerical solution.

(

32

(

(

' ( {

(

NUMERICAL METHODOLOGY

(

The numerical solution of the governing equations was performed using the commercial

(

computational fluid dynamics code FLUENT, version 4.2 (1993). In this code the conservation

(

equations for mass, momentum and turbulence quantities are solved using the finite volume

(

discret!zation method (Patankar, 1981 ). For this practice the solution domain is divided in small

(

control volumes, using a non-staggered grid scheme, and the governing differential equations are

(

integrated over each control volume with use of Gauss' theorem. The resulting'system of algebraic

( (

equations is solved using the Gauss-Seidel method and the SIMPLE algorithm.

(

In the finite volume method, interpolation at the control volume faces of properties transported by

(

convection can be of primary importance on the accuracy of the numerical results. The classical

( (

approach of first order accurate upwind ditTerencing usually suffers from severe inaccuracies in

(

complex flow situations originated by truncation errors and streamline-to-grid skewness.

A

consequence of the first is that the only truncation-error-free problems are those whose solutions

( (

vary almost linearly with the grid index in the streamwise direction. The second source of error

(

occurs where the vector velocity is not aligned with the grid lines (as in recirculating flow regions),

(

and usually is referred to as false diffusion. Recirculating regions of course are a common feature

(

in radial diffusers and, therefore, such flow situations are susceptible to this sort of error. An

(

effective approach to reduce truncation error, while still maintaining the grid size within

(

computational resource limits, is the introduction of a more accurate differencing scheme into the

( (

numerical analysis.' In the present work, the QUICK scheme was adopted in the solution of the

(

momentum equations, yielding a second order accuracy for the interpolated values. Yet, for the

( ( ( ( (

transport equations of turbulence quantities the Power Law Differencing Scheme (PLDS) of Patankar (1980) was adopted since the unboundedness of the QUICK scheme usually introduces serious numerical instabilities, causing calculations to diverge.

evidence (Craft, 1991) that in the case of these equations the source terms are dominant, with the convective terms playing secondary role.

( (

(

Nevertheless, there is some

3~

l ( Two grid levels (70x80 and 100xl40, axial x radial) were used to assess the numerical truncation

(

( error. The refinement was mainly promoted in the entrance of the diffuser, where flow property

(

gradients are steeper. Of great help to this test was some evidence of the discretization needed for the analysis and made available b¥ Deschamps et al. ( 1989) and Possamai et al. (1995).

(

Because of the strong non-linearity of the equations, under relaxation coefficients were required.

(

For the velocity components these coefficients were 0.15, for pressure 0.25 and for the turbulence

(

quantities 0.2. At the very tirst interaction of the numerical procedure the sum of the residual of all

( (

algebraic equations (all five variables included) was on the order of one; convergence was stopped ( when this sum was less than 5xio-4.

(

RESULTS AND DISCUSSIONS

(

The flow through the radial diffuser in Fig. I was investigated for three displacements, s/d

(

(=0.05, 0.07 and 0.1 0), three Reynolds number, Re (=10,000; 20,000 and 40,000), and two diameter

( ratios, 0/d (=1.45 and 3). Additionally, for D/d=I.45, the effect of a backward facing radial step in the back disk was also considered (p/d=0.039 and t/d=O.l38, according to Fig. I).

(

(

( Numerical Solution Validation

The numerical solution was validated by means of sensitivity tests of the resul.ts with respect to grid refinement and boundary conditions. The numerical solution is expected to represent thus a

( ( (

pure prediction of the flow through the turbulence model, for which an assessment was possible by ( a comparison between numerical results and experimental data. The numerical validation and the

(

turbulence model assessment were conducted with reference to results of pressure distribution on

(

the front disk.

(

(

( ( ( 34

(

(

oc\ ( ( (

Figure 2 shows the result emerging from the sensitivity tests. In Fig. 2a results of radial pressure I

distribution on the disk surface yielded by two different levels of grid refinement (70x80 and

(

100xl40) are compared for the flow situation D/d=3 and Re=20,100. It is clear that both results are (

I

(

i

virtually the same, and hence, in order to reduce computing times the less refined grid will be used

(

I

in the remaining calculations. Another source of uncertainty that had to be investigated is related to

(

I

the inlet boundary condition for turbulence at the entrance of the feeding orifice since the

(

I

(

I

(

experimental setup used in this work cannot provide such information. Because the flow upstream of the test section follows a straight smooth pipe, it was assumed that levels of turbulence are

(

moderate and therefore the levels of turbulence intensity I tested were 0.03 and 0.06. The result of

(

the test (Fig. 2b) shows that the pressure distribution on the disk surface was not significantly

(

affected by the variation in the level of turbulence intensity. A value of!= 0.03 will be adopted for

(

the remaining calculations.

(

( (

(

Assessment of the Turbulence Model

Figures 3 to 5 show the radial pressure distribution along the front disk surface obtained from the

(

experiments and computations for a variety of flow geometries. In all situations the pressure profile

(

exhibits a plateau on the central part of the curve (r/d < 0.5), as previously verified for the laminar

,

( ( ( (

40 [

r\1..1-'ll- .. 1..a-n

n~:. D--""' ~......

40

1

P*3o

p"30 ·-

20

20 ·-

10

10

(

0

(

-10

-10 ·-

-20

-20

( (

-30

-30

(

( ( ( (

O/d=3; sld=0.05; Re=20,100

... ···-~~....

0

40 '-·----~-1.5 -1.0

-0.5

0.0

I 0.5

J

1.0 rid

1.

-40 ·-1.5

(a) Computational mesh refinement



----- ·-·----

I =0.03 I= 0.06

---~~··

I

• __ _.L_

-1.0

-0.5

0.0

0.5

1.0 rid

(b) Turbulence intensity at the inlet

Figure 2 -Sensitivity of results for pressure distribution to numerical parameters.

35

1.

' ( ( (

40 p* 30

/(11

1: ~n·c··,,

'' . .

'""<.

-10 -20

·t·,

-

-30

-40 -1.5

-1.0

10

-0.5

0.0

0.5

'

(

--~-~~.~.

(

(

5

_: 'r- ',~," .

l

-10 -15 1.0

'·:,\

,,

/

\

Experiment Computation

Re=19,800

p' 15

--"\ ~ .....•... ~

1

20

20 [O/d=3 -- -s/d=O - - -07

Experiment Computation

O/d=3; sld=O.OS; Re=20,100

\ ___ _j~

-20' -1.5

1.5

-1.0

(

\ .< -0.5

0.0

(

___ j ___ _

!____ _

0.5

1.5

1.0

r/d

r/d

(b)

(a)

(

(

( Figure 3 -Numerical and experimental results for pressure distribution; D/d=3. (

(

50

0td=1.45; sJd=o.os: Re==18~a0o

p• 40 ,;./.J

30

._,

-U

0

,,.....

\

5 0 -5

-0.25

I

I

0.00

0.25

(

( (

'

"

•,•,

I -0.50

(

10

t.

J

-10

(

15

\

10

Computation

20 \

20

EXPeriment

JO [ 0Jd=1 45, SJd-=o.oi;Rez:1a.soo p• 25 -

Expenment Computation

'

---------~-

0.50

-0.50

0.75

! -0.25

I. 0 00

0.25

r/d

0.75 (

0.50 r/d

(

(b)

(a)

(

Figure 4- Numerical and experimental results for pressure distribution; D/d=1.45. ( 50

p• 40

·6/d-=1 45-:s/d.;o.os; Re=fi.8o0

30 ( D/d=1 45, sld=O 07, Re:1s:2·oo with step

Expenmenl Computation

_with step

p• 25

20 10 0

I

20

30

-10

\nr. - '

-20

I_ \

-30 I -0.75

l -0.50

~~-----

\.

"I 10

\

CJI

I .; ,,

-0.25

-5

I0.50

.. ,,

..

-10 -0.75

0.75

--..,

\\,

·-·,_~'

-0.50

( (

0~---.~ - - -

·\:/ I --- _ _L__ 0.00 0.25

_L_

( Expenmenl Computation

/,"

-0.25

0.00

r/d

0.25

( ( (

0~0 ~ 0~5( r/d

(b)

(a)

(

Figure 5- Numerical and experimental results for pressure distribution; D/d=1.45; with step. (

( 36

(

(

' ( ( flow by Ferreira and Driessen (1986) and Ferreira et al. (1989). Also similar to the laminar flow is

(

(

the sharp pressure drop at the radial position r/d "' 0.5, which is due to the change in the flow

{

direction. For the outer part of the curve (r/d>0.5) the pressure level never recovers a positive

(

value, a situation which is also verified in the laminar flow for combinations of both large

(

displacement and high Reynolds number.

(

(

For the diameter ratio D/d=3 (Figs. 3a and b) the pressure level goes to almost zero for regions r/d > 1.25. This is not the case for the smaller diameter ratio D/d=l.45 where the pressure even at

( (

the exit of the diffuser is seen not to reach the atmospheric condition, but still remains negative.

(

The reason for this being that in the latter case the diffuser does not have sufficient length to allow

(

for complete pressure recovery, which does not defy expectation. The effect on the flow of the

(

backward facing radial step is explored in Figs. 4 (without step) and 5 (with step).

(

situations investigated here, the influence of the step is small. However, a careful examination of

(

For the

Figs. 4 and 5 suggests as a consequence of the step, a small increase in the pressure values for

( (

r/d>0.5.

(

The good agreement between experiments and computations seen in Figs. 3 to 5 provided

(

confidence in the turbulence model. Thus, the next step in the analysis was to generate numerical

(

simulations for flow situations not included in the experimental investigation. The computations

( ( (

were then conducted for D/d=l.45 without the backward facing radial step, considering three displacements (s/d=0.05, 0.07 and 0.10) and two Reynolds numbers (Re=10,000 and 40,000). The results plotted in Figs. 6a and b at first sight show no significant difference between the pressure

( (

distributions on the valve surface for the two Reynolds numbers explored.

(

distinction between the curves is that for increasing Re values, the magnitude of the negative

( (

pressure profiles decreases. To support the explanation for this feature Fig. 7 was prepared. In this

(

s/d=0.05 and 0.10 at two flow rate conditions (Re= I 0,000 and 40,000). For the smaller value of s/d

(

figure dimensionless stream-function contours are plotted at the entrance of the diffuser for

the flow is seen to separate at r/d"' 0.5 and to reattach downwards inside the diffuser. As the gap

( (

(

However, a first

37

(

( between the disks is increased to 0.10, and the flow inertia becomes stronger, the separation region is increased and the recirculating zone moves into the diffuser exit. Since the negative pressure

(

(

( values are dictated by the 11ow passage area in the diffuser, the rise in those values with increasing values ofRe is a direct consequence of the growth of the separated flow region in the diffuser.

( (

Another important detail of Fig. 6 is disclosed with the help of Fig. 8. There the pressure

(

distributions for Re = I 0,000 and 40,000, normalized by the pressure value at the center of the front

(

disk (rld=O), are presented for s/d =0.05 and 0.1 0. The figure shows that the pressure drop at r/d""

(

0.5 is more pronounced for smaller displacements. This is an expected result since as the gap

(

( between the disks increases, the change in the flow direction at r/d "" 0.5 becomes less stiff.

(

Additionally, for s/d=0.05 an increase in the Reynolds number brings about a considerable enhancement of the negative region in the pressure distribution, whereas, tor s/d=O.J 0, the Reynolds

(

number effect in the shape of the pressure distribution is much Jess prominent. This feature is

( (

related to the size of the separated flow region in the diffuser, as can be noticed from Fig. 7.

(

( 40

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p' 30

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40 p' 30 20 10

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(a)

(

\

Figure 6- Numerical results for pressure distribution on valve reed; D/d=l.45.

( 0.75

( (

( ( (

( (

( 38

(

(

\ (

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( ( (

( (

Air

fl;__..:::,_} ~ ~

Airflow

Region of zoom

1.000 egionofzoom

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( (

0.920

0.920

0.800

o.eoo 0.560

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(a) s/d=0.05; Re=lO,OOO

(b) s/d=0.05; Re=40,000

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( ( ,.0

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0.920

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0.800

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(c) s/d=O.lO; Re=IO,OOO

(d) s/d=O.l 0; Re=40,000

(

(

Figure 7- Stream-function contours at the diffuser entrance, D/d=1.45.

( ( (

(

39

(

'

( 100

(

1.00

p*iP~

0.50

·_

0.00 "\ \

-0.50

=c

,.~.-

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- - ·\--·-·

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L

L -0.50

L -0.25

I_ 0.00

I 0.25

I 0.50 rid

-100L -0.75

0.7

(

Re= 10,000 Re = 40,000

-{)50

\ .J

-100 -0.75

( Dfd"'1.45, s/d=D.10

0.00 ,_

Re = 10,000 Re .. 40,000

/

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0.50

L -0.50

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(

I

I

I

-025

0.00

0.25

.L 0.50 rid

0.7

(b)

(a)

Figure 8- Results of normalized pressure distributions on valve reed; D/d=1.45.

(

(

(

( CONCLUSIONS The present work has presented a numerical and experimental investigation of the incompressible

( (

turbulent and isothermal flow in a radial diffuser. This is the basic !low problem associated with

( several engineering flows, such as automatic valve reeds of reciprocating compressors, aerostatic

(

bearings and aerosol impactors. The !low was analyzed for different parameters such as Reynolds

(

number, diameter ratios and gap between the disks.

(

The RNG k-E turbulence model used to predict the flow was found to reproduce well the experimental results. It should be mentioned though that a complete assessment of the turbulence

(

( model would require comparisons between numerical results and experimental data of turbulence quantities, such as Reynolds stresses. This could not be addressed in the present work due to limitations of the experimental setup.

( (

One of the main features observed in all flow situations is the presence of a separated flow region

(

in the diffuser. This contributes greatly to the negative pressure region observed along the entire

(

diffuser on the front disk surface. For the cases investigated here, it seems that as the gap between

(

the disks is increased the shape of the pressure distribution on the disk surface becomes less and

( (

less dependent on the Reynolds number and the gap itself.

(

( 40

( (

\ (

( ( ( (

( ( (

REFERENCES Bradshaw, P., 1973, "Effects of streamline curvature on turbulent flow", NATO, AGARD monograph No. 169. Craft, T. J., 1991, "Second-moment modeling of turbulent scalar transport", Ph.D. Thesis, UMIST, England. Deschamps, C.J., Ferreira, R.T.S and Prata, A.T. 1988, "Application of the k-E Model to

(

( (

Turbulent Flow in Compressor Valves", Proc. 2nd Brazilian Thermal Science Meeting, Sao Paulo, pp. 259-262 (in Portuguese).

(

Deschamps, C.J., Prata, A.T. and Ferreira, R.T.S., 1989, "Turbulent Flow Modeling in Presence

(

of Stagnation, Recirculation, Acceleration and Adverse Pressure Gradient", Proc. X Brazilian

(

Congress of Mechanical Engineering, Rio de Janeiro, Vol. I, pp. 57-60 (in Portuguese).

(

( (

Ervin, J. S., Suryanarayana, N. V. and Ng, H. C., 1989, "Radial, Turbulent Flow of a Fluid Between Two Coaxial Disks", ASME J. Fluids Eng., Vol. Ill, pp. 378-383.

(

Ferreira, R. T. S. and Driessen, J. L., 1986, "Analysis of the Influence of Valve Geometric

(

Parameters on the Effective Flow and Force Areas", Proc. 9th Purdue Int. Compressors Technology

(

Conference, West Lafayette, USA, pp. 632-646.

(

Ferreira, R. T. S., Deschamps, C. J. and Prata, A. T., I 989, "Pressure Distribution Along Valve

( (

(

Reeds of Hermetic Compressors", Experimental Thermal and Fluid Sciences, Vol. 2, pp. 201-207. Gasche, J. L., Ferreira, R. T. S. and Prata, A. T., I 992, "Pressure Distributions Along Eccentric

(

Circular Valve Reeds of Hermetic Compressors", Proceedings of the International Compressor

(

Engineering Conference at Purdue, West Lafayette, USA, Vol. IV, pp. 1189-1198.

(

Hamrock, B. J., 1994, :'Fundamentals of Fluid Film Lubrication", McGraw-Hill.

(

Harlow, F. H. and Nakayama, P. I. 1968, "Transport of Turbulence Energy Decay Rate", Rep.

(

LA 3854, Los Alamos Sci. Lab.

( ( ( ( (

Hayashi, S., Matsui, T. and Ito, T., 1975, "Study of Flow and Thrust in Nozzle-Flapper Valves", ASME J. Fluids Eng., Vol. 97, pp. 39-50. 41

(

( Jones, W. P. and Launder, B. E., 1972, "The Calculation of Low-Reynolds-Number Phenomena with a Two-Equation Model of Turbulence", J. Heat Mass Transfer, Vol. 16, pp. 1119-1130.

(

( (

FLUENT, 1993, Fluent Inc., Centerra Resource Park, 10 Cavendish Court, Lebanon, NH 03766. Launder, B. E. and Spalding, D. B., 1972, "Lectures in Mathematical Models of Turbulence", Academic Press.

( (

(

Markatos, N. C., 1986, "The Mathematical Modelling of Turbulent Flows", Appl. Math.

(

(

Modelling, Vol. I 0, pp.190-220. Marple, V. A., Liu, B. Y. H. and Whitby, K. T., "Fluid Mechanics of the Laminar Flow Aerosol-

( (

Impactors", Aerosol Science, Vol. 5, pp. 1-16.

( Mazumdar, J. and Thalassoudis, K., 1983, "A Mathematical Model for the Study of Flow

(

Through Disc-Type Prosthetic Heart Valves", Medical and Biological Engineering, Vol. 21, pp.

(

400-409.

(

Moller, P. S., 1963, "Radial Flow Without Swirl Between Parallel Discs", The Aeronautical

(

(

Quartely, May, pp. 163-186. Orzag, S. A., Yakhot, V. Flannery, W. S., Boysan, F., Choudhury, D. Marusewski, .T., Patel, B.

( (

1993, "Renormalization Group Modeling and Turbulence Simulations", So, R. M. C., Speziale, C.

(

G. and Launder, B. E. (eds.), Near-wall turbulent flows. Elsevier Science Publisher, Osenbruggen, C. V., 1969, "High Precision Spark Machining", Philips Technical Review, Vol.

(

30, pp. 195-208. Pearson J. R. A., 1966, "Mechanical Principles of Polymer Melt Processing", Oxford, Pergamon

(

(

Press.

( Patankar, S. V., 1980, "Numerical Heat Transfer and Fluid Flow", Washington D. C.,

( Hemisphere Pub!. Corp.

(

Possamai, F. C., Ferreira, R. T. S. and Prata, A. T.,1995, "Pressure Distribution in Laminar

(

Radial Flow Through Inclined Valve Reeds", ASME International Mechanical Engineering

( (

( 42

(

(

' (

( (

Congress, Heat Pump and Refrigeration Systems Design, Analysis and Applications, AES Vol.34,

(

pp. 107-119.

(

Prata, A. T. and Ferreira, R. T. S, 1990, "Heat Transfer and Fluid Flow Considerations in

(

Automatic Valves. of Reciprocating Compressors", Proceedings of the 1990 International

(

Compressor Engineering Conference, West Lafayette, USA, Vol. I, pp. 512-521.

(

Prata, A. T., Pilichi, C. D. M. and Ferreira, R. T. S., 1995, "Local Heat Transfer in Axially

(

( ( (

Feeding Radial Flow Between Parallel Disks", ASME J. of Heat Transfer, Vol.ll7, pp. 47-53. Rodi, W. and Scheuerer, G., 1986, "Scrutinizing the k-E Turbulence Model under Adverse Pressure Gradient Conditions", ASME J. Fluids Engng., Vol. I 08, pp. 17 4--179.

(

Rotta, J. A 1951, "Statistische theorie nichthomogener turbulenz", Z. Phys., Vol. 129, pp. 547.

(

Simpson, R.L., Chew, Y-T and Shivaprasad, B.G., 1981, "The Structure of a Separating

(

Turbulent Boundary Layer. Part I. Mean Flow and Reynolds Stresses", J. Fluid Mechanics, Vol.

( ( ( (

( (

113, pp. 23-51. Tabatabai, M. and Pollard, A., 1987, "Turbulence in Radial Flow Between Parallel Disks at Medium and Low Reynolds Numbers", J. Fluid Mechanics, Vol. 185, pp. 483-502. Wark, C. E. and Foss, J. F., 1984, "Forces Caused by the Radial Outflow Between Parallel Disks", ASME J. Fluids Eng., Vol. 106, pp. 292-297.

( ( ( ( ( (

( ( ( ( (

( (

43

-

-

-----

\ ( (

( ( (

(

( ( (

( (

( ( (

(

( (

TRANSITION TO TURBULENCE OF LOW AMPLITUDE THREE-DIMENSIONAL DISTURBANCES IN FLAT PLATE BOUNDARY LAYERS Marcello A. F. de MEDEIROS

(

( ( (

Departamento de Engenharia Mecanica Pontiflcia Universidade cat61ica de Minas Gerais Av. Dom Jose Gaspar, 500 30535-610 Belo Horizonte- MG, Brazil e-mail: [email protected]

( ( ( ( (

( ( ( ( (

( ( (

45

t (

TRANSITION TO TURBULENCE OF LOW AMPLITUDE THREE-DIMENSIONAL DISTURBANCP,S IN FLAT PLATE BOUNDARY LAYERS Departamento de Engenha.ria Mecanica Pontiffcia Universidade Cat6lica de Minas Gerais Av. Dom Jose Gaspar, 500, Belo Horizonte, 30535-610- MG- Brazil e-mail: marcello!Omea. pucminas. br SUMMARY This paper presents results of an experimental study of the nonlinear evolution of a threedimensional Tollmien-Schlichting wavetrain excited by a harmonic point source in a flat plate boundary layer. The three-dimensional wavetrains behaved very differently from twodimensional ones. In particular, the first sign of nonlinearity to appear was not a subharmonic mode, but a mean flow distortion. This distortion had a spanwise structure consisting of regions of positive and negative mean distortion distributed like longitudinal streaks, which became more complex as the nonlinearity developed. The observations suggest that the early stages of the nonlinear interaction may be explained by a weakly nonlinear mechanism. The subsequent stages seem to involve mechanisms that are as yet not known.

(

(

( ( (

( ( ( (

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1

Introduction

Owing t.o it.s import.anc.e in fundament.al and applied fluid mechanics, t.he laminar turbulent. t.ransit.ion ha.q at.t.ract.ed t.he at.t.ent.ion of many researchers. Over the years many aspects of the phenomenon have been understood, but there are still important areas where further research is needed. Among the less researched areas is the so called natural transition. This area is concerned with the transition process that originates from randon disturbances that are always present in natural condit.ions. Recent. studies (Ga.qt.er 1978, Shaikh 1997, Medeiros &"Ga.qt.er 1997, Medeiros & Gast.er 1998) have shown that. the t.ransit.ion that. is observed in such environments display some features that are not observed when the most commonly studied case of periodic plane disturbances are used to drive the process. It apears that the streamwise modulation and the three-dimensionality that characterise the natural transition give rise to nonlinear inteactions that dp not occur for regular plane wa.vetrains. The cited previous works have studied either wavepa.ckets or white noise which display the combined effects of strea.mwise modulation and three-dimensionality. The complexity of the nonlinear int.eract.ion obsP.rvAd ha.q madP. it. difficult. to interpret. the results. At first it appeared that the important ingredient was the streamwise modulation, rather than the spanwise modulation, that is, the three-dimensionality (Gaster 1984, Medeiros & Gaster 1994, Medeiros & Gaster 1995). However numerical simulations of twodimensional wavepa.ckets by Medeiros (1996) have shown that spa.nwise modulation is essential to the process. The current paper focuses on the effect of threedimensionality alone by investigating the nonlinear evolution of a. wavetrain emanating from a point source. Preliminary results of this investigation have been present.ed by Medeiros (1996, 1997, 1998) Investigations of three-dimensional wavetrains in shear layers have also been carried out by other researchers (Kachanov 1985, Mack 1985, Seifert 1990,

Seifert & Wygnanski 1991, Wiegand, Bestek, Wagner & Fasel 1995), but these were restricted to the linear regime.

2

Experimental Results

The current experiments were conducted in the low turbulence wind tunnel of the University of Cambridge, Cambridge, UK 1 . Details of the cxperiemental set-up and precedures can be found in Medeiros (1996, 1997, 1998) In experiments with wa.vetrains the flow is usually disturbed by a continuous harmonic source. In the current series of experiments a long but. finite 200Hz wavetrain is excited from a point source. The linear evolution of a two-dimensional mode with frequency 200Hz is shown by the straight line on the instability diagram, figure l. The excitation was introduced at R 6 about 800 and measurements were taken up to R~ 2100, as indicated by the dashed lines in t.he figure. One can see that the excitation was introduced upstream of branch I of the neut.ral r.nrv~ and t.hat. measurement.s were taken beyond hranr.h II, after whir.h l.he Tollmien-Schlir.ht.ing waves decay. The waves r.ross branch II at. Rj around 1700. The evolut.ion of l.he clisturhances observed experimentally along t.he cent.reline is shown in figure 2. The mea.qurements were l.aken al. a nondimensional clist.anr.e of 0.525*. Using finite wavetra.ins, the flow is disturbed by an event that can he repeated. Therefore ensemble averages can be taken in order to get a clearer signal. The records displayed here, as well as those shown in other figures, were obtained from 64 ensembles. The first important observation is that wave amplitudes grow up to Rj about 1700 and thereafter decay, consistent with the linear theory, figure I. A mean flow distortion that is not predicted by the linear theory is also observed . Initially the distortion is negative, but further downstream 1

This tunnel is now lnr:at.P-d at. Queen Mary Md We!i.tfield Col-

lege, London University, London, UK

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46

I

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,

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(

o[)r----·-·-:O.Hi a:l. 0.14

0

;

10.12

j:~: -0Jl6

( (

·--- ..

:~+~~:::::,._~··:::·_::...

0.04 0.02

o~-~s~oo~-~,ooo~-~,~,oo=---=,ooo~-~~~~-:c:,ooo

( (

wavetrains. It could have remained undetected if a cont,inunus wavet.rains were used. It. is possible that. t.he use of continuous wavet.rains haw~ prevent.ed t.hese mean flow dist,ort.ions from being observed in previous experiments with three-dimensional wavetrains. Measurements were also taken off the centerline of the flow to provide a three-dimensional view of the mean flow distortion. Velocity records were taken at different spanwise locations 10mm apart from each other covering the entire width of the disturbance field. 1b make the streaks clearer the oscillating part of the signal was digitally filtered. Details of the signal processing can be found in Medeiros (1997, 1998). A picture of the evolution of the streak structure as it evolves downstream is shown in figure 3. Initially there is a central region

Reynold~

numbet- R,.-

X•l.lmT.,•.I19h

Figure 1: The inst.abilit.y diagram showing the path of the 200Hz Tollmien Schlicht.ing wave.

(

X•I.Om T.-.010421

R,·

(

1100

( (

1000

---~---

1946

X.0.9m T.•.01193•

-~---

1855

X..O.Rm T.•.0744s

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9900

(

"i

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'--~--

1760

f

X..0.7m T.-.o595s

-.. .

!' -:.,,. .

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. X.0.6rn T.-..0446s

1659

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j

X..(l.Sm T.-.029fh;

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600

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-2 -1 -.5

500

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111 ••m ii(l 11 1lllfl/li_OOSU:······----

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0.1

0.2

0.3

0.4

Figure 3: Dow"nstream evolution of the streaks

1312

0.5

time/"

( ( ( ( (

Figure 2: The evolution of the 200Hz three-dimensional wa.vetrain along the centerline of the plate at a. distance of 0.525' from the wall. switches to positive. It is remarkable that the change in the trend of the mean flow distortion occurs close to where the disturbance crosses branch II. The mean flow distortion is made very clear by the use of finite length

of negative mean flow r'istortion together with two lateral regions of positive uean flow distortion. This structure suggests the exiPt-~nce of a pair of counter-rotating concent.rat.ions of vor ,city which would ptL•h down high momentum fluid in t.he la.t.eral regions and lift up low momentum fluid in t.he cent.ral region. However, it. is as yet unclear whether these mechanisms are actually taking place. The concentrations of vorticity are probably too weak to be considered vortices and perhaps the lift up/ push down effect is too small to affect the flow. As the waves evolve, the structure becomes more complex. At x=lOOOmm the appearence of a region of positive mean flow distortion right at the center of the wavetrain is observed. This corresponds to the change in the sign of the mean flow distortion shown in figure 2. From station x=1000mm onwards the structure does not display remarkable changes, apart from the broadening of the central positive mean flow distortion.

( (

(

.s

47

•( [t

is interest.ing to look at the evolution in the fre-

quency domain. This has hnen carrier! oul. by Medeiros (1997). However, l.he spanwise resolut.ion of t.he experiment. is relat.ively low. Th<> results hncame difficult to interpret because alias effects could not be ruled out. Therefore, care should be taken in analyzing those results. What is clear is that initially the nonlinear mechanism generates only two region of positive mean Row distortion and one central region where the distortion is negative, while further downstream the spanwise wavenumbers become significantly larger. Also important is to investigate the structure of the mean Row distortion in the direction normal to the wall. Figures 4 and 5 show contour plots of the mean Row distortion on planes perpendicular to the Row direction at 900mm and llOOmm from the leading edge. The

!(,·-·-···-

~--·

·---.--

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(

....

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[_~ -s

-2 -1

1

2

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Figure 5: The mean Row distortion distribution on a plane perpendicular to the flow direction llOOmm from the leading edge.

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The figure shows how the disturbance field slowly evolves X=l.lm

( X:1 Om

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[~ -5 -l -1

1

2

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X=0.9m

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Figure 4: The mean flow distortion distribution on a plane perpendicular to the Row direction 900mm from the leading edge.

' f.....

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mean Row distortion is concentrated inside the boundX=07m ary layer. In the external part of the flow no sign of the mean Row distortion is observed. At x=900mm the structure is basically composed of two regions of positive mean Row distortion and a central region of negaX"'0.6m tive mean Row distortion. Whereas the positive lumps ">,·:i, :\:,:':: are fairly concentrated the negative rei;on spreads over a larger portion. Moreover, the negative region appears XE(J.!Im to be composed of several lumps. The profiles resemble t.hat. of t.he Klebanoff modes wit.h a single maximum inside t.he boundary layer, part.icularly for t.he posit.ivo X=0.4m streaks. The maximum is located between 71(= . ,.... \:·,:·.~.~:-· ~{/' 1 and 2, which is also consitent with Klebanoff modes. At x=llOOmm the negative central region splits into a [~ 34om -2 -1 -.S .!1 I 2110_. number of regions and lumps of positive mean flow distortion arise. At this stage the central part of the disFigure 6: Evolution of the mean Row distortion. turbance field is too complex for any definite conclusion concerning the location of the maxima. The evolution of the profile along the streamwise from a relatively simple structure at x=800mm into a direction may be more meaningful than the analysis of much more complex one at x=llOOmm. An overall view of the transfer of momentum from the signal at a particular streamwise station, figure 6.

0.~:<.,.>

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=:'.>'··M\:')",~'

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the low velocity streaks t.o the high velocit.y streaks is The parameter E characterizes the amplitude of the disgiven by t.he rlist.rihut.ion of the displacement, t.hkkness t.nrbancP. and since it, is small il. is nat.ural t.o at.t.empt. to variat.ion over t.hn ent.im disturbance field, figure 7. In expand t.he solnt.ion of the pert.urbed problem in powers of '· However, we not.e t.hat, it. is not. clear that. it, 1200,_-is legitimate to do that, because there is no guarantee that such series will converge. Therefore in practice it is IIIXl necessary to verify whether the solution found is really a good approximation to the flow. 1000 The r.oeffident.s of equat.ion 1 are given by t.he ha.'IP. flow solut.ion. For t.he boundary layer t.he equat.ions of motion are non-dimensionalized by the free-stream velocity U00 and the displacement thickness &•, therefore fMX! R= where v is the kinematic viscosity. With the ! 700 additional assumption that the base flow is parallel, that is, V = (U,O,O), the system of equations (1-2) permits the normal modes solution

t""

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~

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~

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Figure 7: The displacement thickness variation over the plate relative to displacement thickness of the Blasius profile.

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3

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(

ilo(11) ?o(y) wo(y) fio(11)

e•(a.z+a.z-Pt)

(3)

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where u, v, w represent the velocity components in the streamwise, normal to the wall and spanwise directions (x, y, z), respectively. In the expression /3 is the non-dimensional frequency, and O
Some Theoretical Considerations

[(U- c) (~2

[~2

-

k

2

)-

~11~]}

c)] ~

+

1!0

k4

=0

(4)

'O<•R~~

- k -; •R01,(U= iio (5) The standard procedure used to analyse the stability of flows involves the decomposition of the velocity field and with boundary conditions the pressnre field int.o a bAse part. V, P, which is a solnt.ion of t.he eqnat.ions of mot.ion, and a small distnrbance at y = 0 iio(Y) = 0, f.iio(Y) = 0 il(y) =0 a".. , part, v, p. For analysing nonlinear int.eract.ions of the disat y-+oo iio(y) '-* 0, ilYvo(Y) -+ 0, i)(y)-+ 0 turbance field it is assumed further that the velocity and (6) pressure disturbances can be written as where k2 = oi + 0<~ and c = !; is the phase velocity of 2 3 V = EVo + E Vt + E V2 .•• the mode. Equation (4) is the Orr-Sommerfeld equation 2 P = EPo + € Pt + €~P2 · • • (OSE) which governs the linear stability the flow. The Substituting into the Navier-Stokes equations, subtract- OSE together with the associated homogeneous bounding out the base flow and collecting terms of order E one ary conditions, constitutes an eigenvalue problem. Nonarrives at a linear system of equations describing the dis- trivial solutions, or modes, have to satisfy a dispersion relation turbahce field: F(01., 01., /3, R) = 0. (7)

8

lfvo

1

+ (V · 'll)vo + (vo · 'll)V = -'llPo + R'I1 2Vo

t 'll·vo

= 0.

(1) (2)

2

The dispersion relation for two-dimensional modes (01. = O) in a flat plate boundary layer is represented in figure 1. The solid curve represents neutral modes. Modes

( (

=

Po

this picture the details of the complex distribution in the direction perpendicular to the wall are lost, and an averaged view of the distortions is obtained. The figure shows a somewhat more symmetric pictures of the flow than that of figure 3. For the positive lateral regions the displacement thickness distribution indicates a structure similar to that suggested by figure 3. The central region, on the other hand, indicated that the central negative region splits into three regions separated by two newly generated positive regions.

(

l[ l

49

( ouside the loop are stable whereas those inside the loop leading edge. The fundamental waves also reach a max· are nnstahln. imum wit.hin the experirwmt.al domain, figure 2, which If now t.he I.P.rms of order F2 am col!P.r.t.P.d from is consistent. with l.he amplitude rlependenr.e of t.he nonP.quant.ion I and 2 one arives at. an OSE for t.hn li1 with a linear int.eract.ion rliscusserl ahove. non homogeneous term which is a function of vo and Po· At x= lOOOmm modes of higher span wise wavenumOwing to the quadratic nature of the nonlinear terms ne- ber are also present. Care should be taken in analysing glected at order (<) the non homogeneous terms of order these results because the span wise resolution is relatively (<2) have the form poor and alias effects can not be ruled out. It is tempting e•l±(a,x+a,.z-Ct)±(a.x+a.,z-tlt)!. (8) to think chat these modes originate from a second generation of the nonlinear interaction discussed above, but For a harmonic point source there is only one a, and this time including the first nonlinearly generated mode. However, the observations show that these modes conone {3, but a number of three-dimensional modes (a,). Therefore, the quadratic interaction produces tinue to grow even after the first nonlinear mode has demodes with streamwise wavenumbers 0 or 2a,. The ex- cayed. Other results, not shown here, indicate that these perimental observations show the appearance of modes modes do not decay within the experimental domain, wit.h st.reamwise wavenumber 0, t.he mean flow clist.or- xOOr---~----~·--~--~--~-

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a key in~redient of the mechanism of production of tur- Medeiros, M. A. F. & Gaster, M. (1995), The nonlinbulenm. It hM also been shown t.hat. modulat.ed waves ear behaviour of modulat.ed Tollmien-Sr.hlicht.ing give rise t.o l.ransit.ion ~tt. smaller amplitudes in r.omprison wavP.s, in 'IUTAM Conference on nonlinear instability and transition in tri-dimensional boundary laywit.h plane wavnt.rains. It. is possible l.h~tt the modulaP.rs', Manchester, pp. 197-206. tion of the waves provide a short cut between the early wavelike behaviour and the vortical structures observed in turbulent flows. These conjecture is currently being Medeiros, M. A. F. & Gaster, M. (1997), 'The nonlinflar evolnt.ion of wavepar.ket.s in a laminar boundary investigated. The possibility that the initial mean flow layers: Pnrt. I'. (snhmit.t.erl t.o !.he J. Fluid Mr.r.h.). distortion be generated from the self interaction of waves is also being further investigated. Medeiros, M. A. F. & Gaster, M. (1998), 'The nonlinear evolution of wavepackets in a laminar boundary layers: Part. II', J. Fluid Mech. (to he published). References Cohen, .J., Breuer, K. S. & Haritonidis, .J. H. (1991), 'On the evolution of a wave packet in a laminar boundary ilwer', .T. Fluid Mech. 225, 5754l06.

Seifert, A. (1990), On the interaction of small amplitude disturbances emanating from discrete points in a Blasius boundary layer, PhD thesis, Tel-Aviv UnivP.rsity.

Gaster, M. (1962), 'A note on the relation between temporally-increasing and spatially-increasing dis- Seifert, A. & Wygnanski, L (1991), On the interaction of wave trains emanating from point sources in a Blaturbances in hydrodynamic instability', J. Fluid sius boundary layer, in 'Proc. Con£. on Boundary Mech. 14, 222-224. Layer Transition and Cont.rol', The Royal Aeronant.ical Society, Cambridge, pp. 7.1-7.13. Gaster, M. (1965), 'On the generation of spatially growing waves in a boundary layer', ]. Fluid Mech. Shaikh, F. N. (1997), 'Investigation of transition to tur22, 433-441. bulence using white noise excitation and local analGaster, M. (1978), The physical process causing breakysis techniques', J. Fluid Mech. 348, 29-83. down t.o turhuhmce, in '12t.h Naval Hydrodynamic.s SqnirP., H. B. (1933), 'On t.he stability of t.hreeSymposium', Washington. dimensional distribution of viscous fluid between Gaster, M. (1984), A non-linear transfer function deparallel walls', Proc. Rou. Soc. London A 142, 621scription of wave growth in a boundary layer, 628. in V. V. Ko7.lov, ed., 'Laminar-turbulent transition', IUTAM Symposium, IUTAM, Springer- Wiegand, T., Bestek, H., Wagner, S. & Fasel, H. (1995), Experiments on a wave train emanating from a Verlag, pp. 107-114. point source in a laminar boundary layer, in '26th Kachanov, Y. S. (1985), Development. of spat.ial AIAA Fluid Dynamics Conference', San Diego, CA. wave packets in boundary layer, in V. V. Kozlov, ed., 'Laminar-turbulent transition', springerVerlag, pp. 115-123. Mack, L. M. (1984), Boundary-layer linear stability theory, in 'Special course on stability and transition of laminar flow', AGARD Rep. No 709. Mack, L. M. (1985), Instability wave patterns from harmonic point sources and line sources in laminar boundary layers, in V. V. Ko?.lov, ed., 'Laminarturbulent transition', springer-Verlag, pp. 125-132. Medeiros, M. A. F. (1996), The nonlinP.ar behaviour of modulat.ed Tollmien-Schlicht.ing waves, PhD t.hesis, Cambridge University- UK. Medeiros, M. A. F. (1997), Laminar-turbulent transition: the nonlinear evolution of three-dimensional wavetrains in a laminar boundary layer, in 'Proc. of the XIV Brazilian Congress of Mechanical Engineering', Bauru. (in CD-ROM). Medeiros, M. A. F. (1998), nonlinear evolution of a three-dimensional wavetrain in a flat plate boundary layer, in '21th Congress of the International Council do the Aeronautical Sciences', Melbourne. (t.o he published). Medeiros, M. A. F. & Gaster, M. (1994), The nonlinear behaviour of modulated Tollmien-Schlichting waves: experiments and computations, in 'Second EUR.OMEC Conference', Warsaw- Poland.

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51

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UMA FORMULA(:AO DE PAREDE PARA ESCOAMENTOS TURBULENTOS COM RECIRCULA(:AO D.O.A. CRUZ & F.N. BATISTA Grupo de Turbomaquinas I Departamento de Engenharia Medinica/ CT-UFPA 66075-900 Betem-PA-Brasi/- gtdem@amazon.com.br

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M.BORTOLUS Universidade Federal de Minas Gerais- Departamento de Engenharia Mecanica 31270-901 Bela Horizonte-MG-Brasi/- borta@vesper.demec.ufmg.br

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Abstract This work presents a new wall law formulation for recirculating turbulent flows. An alternative expression for the internallenght which can be applied in the separated region is also presented. The formulation was implemented in a numerical code which solves the k-& model using the finite volume method. The solution is then compared with an experimental case existing in the literature and describes the phisics of the problem more accurately than the standard k-E model.

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Palavras-chave

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Turbulencia, recircula~il.o, modelo k-E, lei da parede

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l.INTRODU(:AO 0 modelo k-E tern se tornado ao Iongo dos anos urn dos mais populares modelos de turbulencia sendo utilizado em inumeras aplica((oes pniticas. Essa popularidade se deve principalmente a uma composiyao de uma relativa simplicidade de implernenta((ao corn uma certa generalidade de aplicayao. Apesar dessas vantagens o modelo k-E ainda apresenta algumas dificuldades, principalrnente na descri((ao dos escoamento proximo a superficies s61idas. Com o objetivo de contornar essa dificuldade varias solu((oes foram propostas as quais se dividem basicamente em dois grupos. 0 primeiro chamado de modelo de alto nfunero de Reynolds turbulento que utiliza as chamadas Lei de Parede Logaritmica (CHIENG, c.c. & LAUNDER, B.E.,(l980) C!OFALO,H. & COLLINS, M.W., (1989)) e requer uma malha pouco refinada junto a parede, o segundo grupo e conhecido como modelos de baixo nfunero de Reynolds turbulento e utiliza uma malha bastante refinada junto a parede (PATEL, v.c., ROD!, W. & SHEUERER, 0.,(1985)). Ambos os grupos apresentam caracterlsticas distintas,

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53

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( sendo que o primeiro possui uma maior robustez e facilidade de convergencia, embora n1io descreva adequadamente o escoamento em muitos casos como nas regioes onde ocorre recirculav1io, por exemplo. 0 segundo grupo, embora consiga retratar melhor o escoamento, e computacionalmente mais dispendioso, alem de, em alguns casos, apresentar uma certa dificuldade de convergencia. Grande parte da dificuldade dos modelos do primeiro grupo em descrever certos tipos de escoamento esta concentrada na utilizav1io da lei da parede Jogaritmica classica a qual foi deduzida para escoamento sobre uma placa plana e n1io se aplica a escoamentos onde ocorra separay1io e recirculaviio. No presente trabalho sera apresentado uma versiio da lei da parede logaritmica a qual pode ser empregada ao Iongo de todo o escoamento, inclusive nas regiiies de separavilO e escoamento reverso. Uma definiviio alternativa do comprimento caracteristico da regiiio interna do escoamento sera utilizado (CRUZ, D.O.A. & SILVA FREIRE, A.P., (1995)). Essa formulayilO sera implementada em UITI c6digo numerico 0 qual utiJiza 0 metoda dos volumes finites, e os resultados obtidos para o escoamento em torno de urn degrau descendente (figura 1) seriio comparados com resultados experimentais assim como com a formulaviio do modelo k-e de Spalding e Launder (1974).

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Domlnio de cillculo

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..£.=

l.ll ~

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I_

~

I~~--......_

H

YL

. , Linhalde corrente di111sona

- --

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:::::;+

H

Camada

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.....

1

SH

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Ponto de recolamento

~Z~n~ ~ ~~~~~ ____"_'

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Camada limite : :-rredesenvolvida : 1

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_________ _]

20H

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Figura 1 - Geometria do problema e dominic computacional

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2. AS EQUA<;OES GOVERNANTES Na analise do escoamento foram utilizadas as equayoes medias de Reynolds para o caso incompressivel, juntamente com as equayiies da energia cinetica turbulenta (k) e da dissipayiio (e) as quais sao mostradas abaixo: Dk Dt

=~[.2~]+vT[au; axj

crK

axi

axj

+auj] au axi axj

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-e

(I)

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oe=_i__[2 ~]+c,~vT[au; +aui]au -c 2 ~ ot

ax;

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cr 1

axi el

Onde

VT

k

= cfl k;

C~=0,09,

axj

ax; axj

(2)

k

CI=1,44;C2=1,92; cr,=1,3;
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3. A FORMULA<;:AO DE PAREDE

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Na regiiio proxima a parede onde OS termos de inercia sao pequenos a equar;ao da conservar;ao da quantidade de movimento pode ser aproximada pe1a seguinte re1ar;ao(CRUZ, D.O.A. & SILVA FREIRE, A.P., ( 1995)):

au -,-, =-+--y 't'w 1 dp v-+uv ax p p dx

(3)

onde v ea viscosidade cinematica, 't'w e a tensao na parede e p e a pressao, u'v' representa a tensiio de cisalhamento turbulenta e pea massa especifica do fluido. Na equar;ao acima, o !ado esquerdo representa a tensao cisalhante total a qual e dada pela soma do termo laminar e do termo turbulento como descrito abaixo:

au ax

, ,='t

v-+uv

(4)

No calculo da tensao na parede sera utilizada aqui uma versao simplificada da lei da parede proposta por Cruz e Silva Freire (1995 ), alem de uma equar;ao alternativa para o calculo do comprimento caracterfstico da regiilo interna do escoamento. Ambas as relar;oes silo mostradas abaixo:

u=~~ /_:s.._ +.!.. dp y + ~ u_! ln(__'j_) \tw\kl/p pdx \•w\k Lc

(5)

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(

Lc

=

-~+-1(~)2 +2v dp u P

V

P

dx

R

(6)

_!_ dp

p dx

(

( ( (

55

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( onde k e a constante de Von Karman=0.41,

UT=

~

( e a velocidade de fricyaO e

UR

e

descrito pela seguinte equayao:

( {

(7)

uR=.ft

As equa96es (5) e (6) representam uma generalizayao da Lei da Parede Logaritmica classica que pode ser aplicada em varias regioes do escoamento . Na regiao Ionge i:lo ponto de descolamento na de a tensao na parede e positiva e dP/dx y <<1:w temos que as equa96es (5) e (6) assumem a seguinte forma:

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u=

~k uT + u.!k ln(..L) Lc

(8)

Lc==v/uT

(9)

ou seja , a Lei Logaritmica classica . Proximo ao ponto de descolamento onde equayao fica sendo dada por :

1:w =0 a

~

( (

u==~~y dP k pdx

(10)

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A equayao (9) e semelhante a expressao proposta por Stratford ( 1959) a qual descreve o perfil de velocidade dos escoamentos turbulentos proximo a parede na regiao de separayao onde a tensao de cizalhamento na parede tende a zero. Na regiao de recircula~j:ilO onde dP/dx y >>twas rela~j:oes (5) e (6) pode ser rescritas da seguinte forma:

U

==

-~

UT -

Lc == 2\•w I

~ In(

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(

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(

zJ

(II)

( (

(

:~1

(12)

A principal diferen9a enter as equa9oes (9) e (8) e as equa9oes (11) e (12) alem do sinal negativo na expressao que descreve a velocidade esta na definiyao do comprimento caracteristico do escoamento proximo a parede (equayao (12)) o qual difere do comprimento caracteristico classico dado pela equa9ao (9).

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Como objetivo de evitar urn procedimento iterativo adicional o qual poderia dificultar a convergencia do esquema numerico o seguinte conjunto de equayoes foi adotado no calculo da tensao na parede:

t=Cljk+ au " v-

( ( (

(13)

()y

t.,. =

u.Jtpk clj ( .Jt) j.l

(14)

In Eyv

( ( (

1 dp

t - tw

p dx

y

--=--

(

( (

2

_two+

L

P

( (

-....

V( P

c

)

+ 2 v dp

dx uR

(16)

_!_ dp

p dx

( (

(15)

ul-tpk

(17)

'•"[2£1•1{zJ]

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onde E=9.8 0 conjunto de equayoes acima representa uma lineari'Zayll.o no procedimento de obtenyll.o da tensao na parede, o que torna o calculo mais robusto alem de acelerar a convergencia do c6digo numerico. A dissipayao e entao descrita pela seguinte relayll.o:

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(

l

1dpj

f:=C~k.ft pdx j.l +--ky

(18)

k.ft

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(

57

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4. ANALISE DOS RESULT ADOS

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A formulav1lo aqui apresentada foi utilizada no calculo do escoamento em torno de urn degrau descendente sendo os resultados comparados com os experimentos de Vogel e Eaton (1985) as equayoes governantes na sua formulay1lo conservativa s1io discretizadas utilizando a formulav1io de volumes finitos acoplada com urn esquema hibrido para o tratamento dos termos convectivos e difusivos simultaneamente. 0 conjunto de equayoes diferenciais foi resolvido iterativamente utilizando uma versao robusta e extensivamente testada do codigo numerico TEACH-2E (Teaching Eliptic Axi-Symmetrical Caracteristic Heuristically) o qual incorpora o algoritmo SIMPLE especifico para acoplamento press1iovelocidade em escoamentos incompressiveis. A melhor relay1io entre precis1io e tempo de CPU foi conseguida utilizando-se uma malha de 146x I 02 pontos. Nas figuras abaixo sao mostrados os perfis de velocidade na regiao proxima a parede para varias estayoes do escoamento. Uma comparayao entre os resultados obtidos pela presente formulay1lo com o modelo k-E padr1io de Launder e Spalding (1974) e com os resultados experimentais de Vogel e Eaton ( 1985) e feito. Os resultados obtidos com a formulayiio aqui proposta se mostram superiores aos da formula9iio padr1io, descrevendo melhor o perfil de velocidade, principal mente proximo a parede, sem nenhum acrescimo no tempo de computav1io. Na verdade o calculo numerico feito com o modelo padr1io necessita de rna is iterayoes para convergir ( 1200), que o modelo aqui apresentado (II 00) para uma precis1io da ordem de SE-3. Vale ressaltar que o procedimento proposto retrata a pequena zona de recirculav1lo secundaria presente no escoamento proximo ao degrau (figura 5). Outras formulavoes os quais tambem empregam a lei logaritmica chissica nas equavoes do modelo k-E original (CHIENG, C.C. & LAUNDER, B.E.,(l980) CIOFALO,H. & COLLINS, M.W., (1989)) dividindo o escoamento proximo a parede em varias camadas, fornecem melhores resultados que a formulaviio padrao, essas formulavoes, contudo, nao reproduzem a zona de recircula!j!1iO secundaria (BORTOLUS, M. & GIOVANNINI, A (1995))

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1.2

/~-:

0.8

/

___ .,.. 0

I-,

•.)

0.4

•)

.............0"""'8"""'

•)

FOITTU~Proposta

•)

·=·

o.o

-l..':'.-~~----,-.o.2

J --0I

•) ·'

0

0.0

0.2 u/Uoo

( (

(

i

,y/-

0.4 -

'- j

0.8-

0

.<'),..,.. 0

i

r ----.·- r-

1.2

o.o 0.4

0.

-j o---~-~r-

--.---- -r - r

T ---.-

(

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-,--~,--

0.8

(

Figura 3- Comparacao do Perfil de Velocidade Junto a Parade (x-Xr)!Xr=-{).44

(

-02

0.0

0.2

0.4

0.8

u/Uoo

Figura 2- Comparacao do Perfil de Velocidade Junto a Parede (x-Xr)!Xr"-0.33

58

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J

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(

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( 1.2--

0.8

~-

1 08li

(

/

I

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---

//

/

~----

0

M)dalo Padrto Formul~ Proposta

_.-/

~

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04

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~



0.4

Eltperimento \moat & Eaton

r)

Fonnula;.lo Pfoposte M:xltlloPadrlo

------------~---;.

,-- ==--r--0.0-+--~' -0.20

,..-----,!'~-

0.00

0.0

0.20 u/Uoo

0.6

0.40

-+-----r-------,------- ------,----r=-0.04

-0.02

0.00

--,---r-r

-----------------0.02

Figura 4· Compara~lo do Perfil de Veiocidade Junto A Parade (x-Xr)/Xr=-0.55

Figura 5· Compara~lo do Perfil de Veiocidada Junto A Parade Pr6xlmo ao Degrau

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Nas figuras acima Xr representa a posi!j:iio do ponto de recolamento. Na figura abaixo e mostrada uma compara~j:ilo entre o fator de atrito na parede (Cf) fornecido pelo procedimento aqui desenvolvido com os resultados do modelo k-e padriio e com os dados experimentais. Novamente os resultados aqui obtidos reproduzem melhor o experimento que a formula~j:ilo padriio tanto antes quanto depois do ponto de descolamento. Pode-se observar na regiiio pr6ximo ao degrau (x/H 0) que a tensiio na parede (Cf) e positiva, atestando a presenlj:a da pequena zona de recircula~j:ilo secundaria como ja havia sido mencionado. A predi~j:ilo do descolamento secundario constitui urn importante teste para a capacidade do modelo k-e de descrever os escoanientos pr6ximo a superficies s61idas.

=

0.003

(

(

•)

0.002 0

( (

o.

0.001

(

0

0 o.)

0

0

(

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/'

_-·'o

0.000

( .().001

Q

fWIINI119.. P,....t•

0

Q

E•..,t...ntoV,..t•l!.toll

o oO

( (

-0.002

, - - ) - ·-r 0.00

4.00

8.00

12.00

16.00

20.00

x/H

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Figura 6 - Compara<;lio do coeflclente de frlc~;lio (Cf) para os diversos casos

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0.04

u/Uoo

59

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( Na tabela I, aparecem os valores do ponto de recolamento fomecido pelas duas teorias juntamente com o valor obtido experimentalmente. 0 calculo feito com a teoria proposta fornece urn comprimento da zona de recircula<;:ao mais proximo do valor experimental quando comparado com o valor obtido pelo modelo k-E padrao. Isto ocorre porque a equa<;:ao (18) fornece valores maiores para a taxa de dissipa<;:ao junto a parede causando urna diminui<;:ao ·da viscosidade turbulenta nessa regiiio o que causa urn aumento da zona de recircula<;:ao.

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CALCULO DO PONTO DE RECOLAMENTO EXPERIMENTAL PRESENTE FORMULACAO MODELO PADRAO

6.6 6.0

5.5

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Tabela 1 - Pontos de recolamento calculados e experimental

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< 5. CONCLUS.AO

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No presente trabalho uma nova formula<;:ao para a descri<;:ao dos escoamentos turbulentos proximo a superficies s61idas foi apresentado. Essa formulayiio foi implementada em num c6digo numerico o qual resolve as equa<;:oes medias de Reynolds juntamente com as equa<;:t'ies classicas do modelo k-e de turbulencia atraves do metodo dos volumes finites. Foi mostrado que a formula<;:iio proposta reproduz melhor os experimentos de Vogel e Eaton que a formula<;:ao de Launder e Spalding com urn custo computacional equivalente. Os resultados obtidos sugerem que a capacidade de descri<;:ao dos escoamentos das equa<;:oes do modelo k-e classico, podem ser melhorados atraves da utiliza<;:ao adequada de uma formula<;:ao de parede, este incremento na precisao do modelo ocorre sem que haja alguma perda de sua robustez ou algum acrescimo de custo computacional, caracteristicas estas de grande importiincia na solu<;:ao de problemas praticos de engenharia.

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6. REFERENCIAS BIBLIOGRAFICAS

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( I. BORTOLUS, M. & GIOVANNINI, A., "Numerical Prediction of Wall Heat Transfer in Complex Turbulent Flows"- Turbulence-Heat and Mass Transfer 1, pp.228-232, BEGELL HOUSE, 1995. 2. CHIENG, C.C. & LAUNDER, B.E., "On the Calculation of Turbulent Transport Downstream for an Abrupt pipe expansion", Numerical Heat Transfer, Vol.3, pp189-207, 1980.

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( 60

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3. CIOFALO,H. & COLLINS, M.W., "k-E Predictions of Heat Transfer in Turbulent Recirculating Flows Using na Improved Wall Treatment", Numerical Heat Transfer, Part B, Vol.l5, pp21-47, 1989.

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4. CRUZ, D.O.A. & SILVA FREIRE, A.P., "The Assimptotic Structure of the Thermal Boundary Layer Near a Separation Point" - Turbulence-Heat and Mass Transfer I, pp.5762, BEGELL HOUSE, 1995.

5. LAUNDER, B.E. & SPALDING, D.B., "The Numerical Computation of Turbulent Flows", Computer Methods in Applied Mechanics, Vol.3, pp269-289, 1974.

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6. PATEL, V.C., RODI, W. & SHEUERER, G., "Turbulence Models for Near Wall and Low-Reynolds Number Flows: a Review", AIAA Journal, Vol.23, n.9, pp 1308-1319, Sept. 1985.

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7. STRATFORD,B.S., "A n Experimental Flow With Zero Skin Friction Throughout its Region of Pressure Rise ", Journal of Fluid Mechanics, Vol.S, n.l7,. 1959.

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8. VOGEL, J.C. & EATON, J.K., "Combined Heat Transfer and Fluid Dynamic Measurements Downstream of a Backward Facing Step", Journal of Heat Transfer Transactions of ASME vol.107, pp 922-929, 1985.

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61

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Mass Injection in a Wake of a Fixed and Rotating Cylinder

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Jose Antonio G. Croce, MSc Student

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Fernando Martini Catalano, Ph.D. MRAeS, MAIAA

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Universidade de Siio Paulo - USP Escola de Engenharia de Siio Carlos- EESC

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Departamento de Engenharia Mecanica

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Laborat6rio de Aeronaves

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Av. Dr. Carlos Botelho, 1465

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CEP 13560-250- Siio Carlos- SP- Brasil e-mail - crocc@sc.usp.br

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Abstract

An experimental work was carried out in order to measure the effect of a mass injection in a wake of a fixed and a rotating two-

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dimensional cylinder. A jet made the injection of mass from a small cylinder located just behind the main cylinder. The jet

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position could be changed in order to assure that the blowing was direct into the wake. A very low turbulence wind tunnel was

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used for flow visualization of both fixed and rotating cylinder. The velocity distribution behind de cylinder was measured by a constant temperature hot wire anemometer in a grid of I00 points and at two downstream positions. A small open circuit wind was used for drag and lift measurements through a special two component balance. Results showed that small jet flows could reduce

significantly the wake of both fixed and rotating cylinder with the inherent reduction of the base drag. A significant increase on the lift to drag ratio could be achieved on the rotating cylinder due to the reduction of drag. The jet and wake mixing measured in detail with a fme grid at the downstream positions showed that the jet mixing layer immediately smoothes out the velocity gradient between jet and wake. As the jet is blowing outside the main cylinder this system could be used to control the vortex shedding of structures such as bridges, chimney and towers.

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1.

lntrodu~io

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Qualquer metodo que possibilite uma reduc;iio significativa na forc;a de arrasto de urn corpo sujeito

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a ac;iio de

ventos alem do carater cientifico, produz avanc;os nas rnais diversas areas da engenharia. Estruturas altas como

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( ( edificios, chamines, ou pontes estao a todo tempo sujeitas a a'tiio de escoamentos atmosfericos os quais produzem o ( aparecimento de for'
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referencia o sentido do escoamento. Alem das for'
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de pressiio entre duas superficies. Urn exemplo cllissico de indu'tiiO de sustentac;:iio e o cilindro de efeito Magnus. Urn ( cilindro rodando e imerso em urn escoamento, gera altos valores de for'
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bastante estudado e utilizado como no caso do uso de cilindros com rota'tiiO na embarca'tiiO conhecida como "Backau", urn navio construido nos anos trinta por FLETTNER (1925), o qual usava no Iugar de velas dois cilindros girando.

Outra aplica'tiio apresenta urna aeronave com cilindros rotativos no Iugar de asas desenvolvida tambem na decada de ( trinta. Varias outras aplica'tiies foram testadas; como uso de cilindros rotativos para a confec'tiio de bombas hidraulicas ( e turbinas e61icas, estudados por CAMARGO VIERA (1965); em dirigiveis em forma de uma grande esfera, a qual girava lentamente enquanto em voo para gerar sustenta'tiio pelo efeito Magnus, desenvolvidos pelos lNSTITUTOS \ CANADENSE DE PESQillSA AERONAUTICA (1988).

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Metodos para a reduc;:iio de arrasto de perfil em corpos rombudos abrangem urn grande niunero de ( possibilidades. Dentre eles pode-se citar a succ;:iio da carnada limite com a intenc;:iio de atrasar a separac;:iio e ( 1

consequentemente reduzir o tamanho da esteira, PRANDTL ( 1927). Outros processos visam aumentar a intensidade da turbulencia do escoamento atraves do posicionamento de pequenos cilindros

a frente

do corpo para provocar a (

dirninui'tiio do arrasto. Recentemente, IGARASHI (1997) obteve bons resultados de redu'tiio de arrasto em prismas (

1

quadrados atraves do posicionamento a frente do modelo de urn pequeno cilindro. Ainda pode-se citar o uso de ( rugosidade na superficie de corpos imersos em escoamento, para induzirem a transic;:ao da carnada limite rna is cedo , ( atrasando o ponto de separac;:ao com conseqiiente reduc;:iio da forc;:a de arrasto. Com o metodo aqui proposto pretende-se determinar as caracteristicas aerodinamicas de urn processo de ( reduc;:iio do arrasto em cilindros parados e rotativos. Este metodo consiste em se colocar na esteira dos modelos urn ( pequeno cilindro contendo urna serie de furos no sentido do escoamento. Isto e feito com a intenc;:iio de se injetar ( massa na esteira e desta forma reduzir seu tarnanho e alterar tambem seu comportamento. Sendo o equipamento de ( inje'tiio de massa montado extemamente a estes corpos, o processo em quase nada altera a forma destes. Isto

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vantajoso pois permite adaptac;:iies ao contrario de modificac;:iies estruturais. Para determinar as caracteristicas do ( escoamento foram realizados urna seqiiencia de ensaios de visualizac;:ao, anemometria a fio quente dos perfis de ( velocidade da esteira, e de medidas das forc;:as de arrasto geradas.

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2. Descri ..iio do Experimento

2.1. Experimento de Visua/izar,:iio

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0 experimento de

visualiza~ao

com

fuma~a

foi montado em urn tlinel de vento de circuito aberto, de baixa

velocidade, e com caracteristicas de baixo indicie de turbulencia. 0 modelo utilizado consistia de urn cilindro de diametro igual a 0,059 metros montado, atraves de urn eixo intemo, em urn mancal de rolamento fixado na parede da camara de ensaio. Ao redor do mancal foi montado urn anel onde foi fixada uma placa circular que continha o pequeno cilindro para o jato. 0 cilindro de jato e composto de pequenos furos, com diametro igual a 0,0016 metros, em todo seu comprimento para a inje91io de massa. Para o caso do cilindro rotativo, urn sistema composto de motor eletrico de corrente continua e todo o cinematismo necessario foi montado na parte traseira da camara de ensaio. A Figura I detalha melhor o aparato usado.

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Clm•r• cleo Enuio

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CilindtocomfondtJ

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SmidorJo E$COOmonfO

( Figura I - Experimento de visualiza~ao.

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2.2. Anemometria a flo quente.

( 0 levantamento dos pertis de velocidade da esteira do cilindro rodando e parado foi montado em outro !Unel de ( vento de circuito aberto, com rnaiores velocidades de escoamento. 0 modelo tinha mesmo diametro, igual a 0,059

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metros, mas com a diferen9a de ter dois eixos em suas extremidades para a fiXac;iio nos mancais de rolamentos (Figura 2). As caracteristicas do cilindro de inje9iio de massa e a placa foram mantidas iguais ao do modelo usado na ( visualizac;iio. Para as medidas dos pertis de velocidades usou-se

urn anemometro de tio quente de temperatura {

constante. Foram levantadas curvas dos perfis de velocidades , tanto para o cilindro fixo quanto para o girando, a

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distancia constante a montante em urn 'grid' de 100 estac;oes.

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Mancal Supenor

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l"arl'ld8Gupetk'lldaCAmaf&rleEn~alo

\ Cilmdro Rotan ...o

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Cihndro d& de Massa

lnJ9~~o

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Meuqtllnhmor

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ParwdelnkKrordaCamilradaEnnlo

:ssa

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f ln]a~~~a:: Figura 2 - Experimento de Anemometria.

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( 3.3. Medidas de Fort;:as Aerodimimicas.

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Para as medidas das fon;as aerodinamicas foi utilizada urna balanc;a aerodinamica de dois graus de liberdad/ para a medi9iio da forc;a arrasto. Esta balanc;a foi projetada e construida exclusivamente para este tim. 0 modelo foi mesmo utilizado nos ensaios de anemometria montado em urn suporte em forma de "lY' conectado

J.

a balanc;a(

aerodinamica. A leitura dos valores de arrasto foi medido por urn ampliticador instrumental de precisiio e lido em u ' microcomputador PC atraves de urn aplicativo capaz de ler pequenas variac;oes de forc;a da ordem de 0,001 N. Desta (

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forma obteve-se alta precisao para pequenas variar;oes de fort;a de arrasto. A Figura 3 apresenta urna vista geral da balanr;a sem o modelo no suporte.

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Figura 3 - Vista Geral da Balanr;a Aerodin8mica.

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3. Resultados 3.1.

Visualiza~tio

do Escoamento

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Os resultados obtidos a partir dos ensaios de visualizat;iio sao apresentados a seguir. Sao apresentados as curvas

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dos perf!S de velocidade da esteira basics e de duas esteiras com injer;iio de massa tanto para o cilindro parado quanto

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para o rodando. A velocidade do escoamento foi mantida constante no valor de 1,4 metros por segundo, resultando em urn nu.mero de Reynolds de 5506,7 baseado no diAmetro do cilindro. Sao apresentados duas situar;oes de injer;ao de massa com o jato I e jato 2 com valores de quantidade de movirnento iguais a 0,0 129 N e 0,0395 N respectivamente.

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0 escoamento ao redor do cilindro parado e rodando pode ser visto na Figura 4. Os dois modelos encontram-se

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nas mesmas condir;oes de escoamento, e o cilindro rotativo esta girando a uma rotar;iio constante de 950 rpm. Pode-se

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notar, como esperado, a esteira do cilindro parado maior do que a do cilindro rotativo. Como pode ser visto na Figura 5, o efeito do jato na esteira do cilindro parado e bastante significativo quanto a redur;iio de seu tamanho. Pode-se tambem constatar que a esteira niio apresenta de forma evidente a formar;ao de

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67

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( grandes vortices descolados e peri6dicos. lsto pode ser tornado como a evidencia da influencia da inje~iio massa na ( altera~iio

da

forma~iio

do 'Vortex Shedding'. Na Figura 6 observa-se tambem a

diminui~iio

do tamanho da esteira do (

cilindro parado. No entanto agora e possivel observar tambem uma redu91io da esteira do cilindro com rota~iio.

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( ( Figura 6 - Cilindro parado e rodando com jato 3

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3.2. Levantamento dos perfis de velocidade. A seguir

e apresentado os resultados dos levantamentos dos perfis de velocidades da esteira do cilindro para e a uma velocidade media na carnara de ensaio de 2,3 metros por segundo.

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rodando. Estes levantamentos foram feitos

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Manteve-se a velocidade relativamente baixa devido ao fato de que nestas condic;iies o efeito dos jatos na reduc;iio da

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esteira foi rnais pronunciado. Isto niio impede de extrapolar os resultados para outras condic;iies de escoamento, devido principalmente a similaridade de perfis garantir o comportamento basico igual ern urn vasta faixa de condic;iies de escoamento.

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Nas figuras 8 e 9 pode ser visto que a reduc;iio da esteira e significativa com o aumento do valor da injec;iio de

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rnassa. A figura 7 apresenta os jatos que foram utilizados nesta parte do experimento. A Figura 8 apresenta os resultados para a esteira de urn cilindro parado sern jato e com urna serie de jatos. A Figura 9 apresenta os mesmos

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resultados mas para o cilindro rotativo. A seguir emostrado a relac;iio dos jatos usados com seu valor de quantidade de

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movirnento.

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Jato 40- 0,02041 N

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Jato 60 - 0,0547 N Jato 80- 0,08715 N

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Jato 100- 0,15475 N

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Jato 120- o. 19794 N

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~jato4D

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-+-jato60 --.!>-jato SO -a-jato1DD

24,5

~jato120

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19,5

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c:

14,5

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9,5

4,5

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0,1

0,2

0,3

0,4

0,5

0,6

0,7

y/Ymn.

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Figura 7- Perfis de velocidade dos jatos usados.

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0,8

0,9

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'{ ---e-EstM/Umd --e-JV120/Umd 1 ,-&-JV100/Umd

1,4

1,3

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0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

PosiCiiiO de Medlda (Adlmensional)

Figura 8- Pertis de velocidade das esteiras sem e comjatos (Cilindro Parado).

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1,2

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1,15 1,1 1,05 0

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:iE 0,95

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0,9 0,85

-e- Esteira(Ad) I

0,8

EV120(Ad) ---tr- EV1 OO(Ad) --G-

0,75 1

0,7 0

0,2

0,4 Esta~;io

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0,8

de Medida (Adimensional)

Figura 9- Pertis de velocidade das esteiras sem e com jatos (Cilindro Rodando).

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3.3. Medidas de Forfa

( A seguir siio apresentados os resultados das medidas de forfi:a utilizando a balanfi:a aerodinamica. Foram, ate ~ momenta, realizados apenas as medidas como cilindro parado, isto devido a complicarroes tecnicas nas medirroes co"( o cilindro rodando. As velocidades usadas variaram entre 5 e 20 metros por segundo. Foi usada esta faixa

dl (

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velocidades pois foi a melhor que se adequou as

condi~,:oes

de

medi~,:iio

da

balan~,:a

e limitada pela velocidade maxima

controlavel do time) de vento.

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-&-Cd(O)

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1,4

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-&-Cd(60) -.!>-Cd(80) -+-Cd(100) -&-Cd(120)

1,6

1,2

.,0

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~-==~~

0,8

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0,4 20000

40000

30000

80000

50000

70000

No. Reynold•

Figura I 0- Curvas de Cd pelo nUm.ero de Reynolds para varios tipos jatos.

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1,1

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1

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...,..,

0,8

0

0,6

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0,4

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::: I 0,3

0,4

0,5

0,6

0,7

Figura II -

Diferen~,:a

0,9

entre Cd provocado por cada jato e Cd da esteira b&sica.

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0,8

Vtvmax

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-.!>- delloCd(-1 00). -+-dolloCd(120)

'-/ 0,7

0,5

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::&:.Cieitiicd(&o)-~ -G-dellaCd(80)

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4. Discussio (

4.1. Visualizar;ao do escoamento

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Atraves das imagens obtidas pelo ensaio de visualiza9iio, fica claro que a redu9iio do tamanho da esteira tanto para o cilindro parado quanto para o rodando e bastante grande. Na Figura 4 pode-se ver a esteira formada pelos dois ( cilindros, fixo e rodando. Logo ap6s na Figura 5 a esteira tern suas caracterfsticas modificadas pela inje9iio de massa.

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rna is nobivel aqui e a aparente elimina9iio dos vortices peri6dicos no cilindro fixo. Apenas uma pequena modifica9iio ( no comportamento do cilindro rodando e notada. No entanto assim que o valor da vaziio do jato e aumentada, uma ( notavel diminui9iio no tamanho da esteira para ambos cilindros e observada. Pode-se aqui concluir que a inje9iio de massa provoca uma diminui9iio na pressiio estatica no escoamento

a montante, fazendo com que o gradiente de pressao \

na superficie dos cilindros fique menos adverso.

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4.2. Levantamento dos perjis de velocidade

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De acordo com o que pode ser visto pelos resultados da anemometria, fica claro que a ideia de reduzir distribui9iio de velocidades em esteiras atraves da inje9iio de massa,

e urn

objetivo realizado. Os resultados(

apresentados na Figura 8, para o cilindro parado e na Figura 9 para o cilindro rodando, comprovam que a inje9iio de( massa a uma determinada velocidade, a esteira sofre uma grande modifica9iio. Esta modifica9iio pode ser uma redu9iio( em seu tarnanho como pode levar ate a uma esteira invertida com acrescimo de velocidade e niio de perda. lsto pode ~ significar o surgimento de uma for9a de empuxo.

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Os resultados das medidas de for9a de arrasto apresentados nas Figura I 0 e II mostram que o fenomeno redw;:ao da esteira pela inje9iio de massa niio comporta-se forma seqiiencial. Em outras palavras fica claro que

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grande redu9iio no valor do arrasto e obtido. No entanto esta redu9iio niio e muito mais aumentada mesmo com acrescirno na inje9iio de massa. Isto, aparentemente, leva

a uma conclusiio de que o fenomeno

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esta mais relacionadt

com a injeyiio de urn alto nivel de turbulencia no escoamento. Assirn sendo, a carnada limite na superficie do modelo (

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tende a transicionar rnais cedo do que o esperado, elirninado a separao;Bo que e em alguns casos laminar para uma

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separaviio, bern rnais a frente, turbulenta. Isto representa uma grande diminuiviio na forva de arrasto entre os nfuneros

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de Reynolds 20000 a 30000 e tendendo a urn patamar logo ap6s.

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5. Conclusoes

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Foram observados uma serie de provas experimentais que caracterizam a injeyiio de massa em urna esteira como sendo urn metodo eficiente na reduviio do arrasto em corpos rombudos. Tanto os experimentos de visualizaviio

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quanta' o de mediviio dos perfiS de velocidades mostraram uma reduviio grande nas dimensoes da esteira e uma

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mudanc;a de suas caracteristicas. No entanto o experimento de medida das forvas de arrasto mostrou que o fenomeno

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para certos numeros de Reynolds ser inerente

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a camada limite do que a esteira. Para os casos de separaviio laminar o

efeito rnais significativo para a reduviio do arrasto esta na trarJSiviio prematura da camada limite, devido ao aurnento do indice de turbulencia provocada pela presenc;a do jato. lsto leva a urn atraso na separaviio da camada limite com a

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conseqiiente reduviio do arrasto de perfil. Mas todos estes resultados comprovam que houve de fato uma reduc;iio no

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valor do arrasto de perfil atras dos modelos estudados.

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Este metoda

e, do ponto de vista de montagem, muito aconselhavel para o uso em aplicavoes praticas. Dentre

elas pode-se citar a reduviio do arrasto e o controle do 'Vortex Shedding' de pontes, torres altas, charnines e dernais estruturas sujeitas a aviio de ventos. Outra aplicaviio

e a implementaviio do sistema de injeviio de rnassa para a reduviio

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do arrasto em aparelhos que pretendem o uso de cilindros rodando, efeito Magnus, melhorando assim a performance

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atualmente prejudicada pelo alto valor da forva de arrasto encontrada nestes casos. Todas estas aplicayoes praticas iriio

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necessitar de pouca adaptayiio para a montagem do sistema estudado.

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Referl!ncias

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BATCHELOR, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press, Londres.

BIRKHOFF, G., ZARANTONELLO, E. H. (1957). Jets, Wakes, and Cavities. Academic Press Inc, Nova Iorque.

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( ( CAMARGO VIEIRA, R. C. (1964). Contribui{:iio aoestudo das aplica{:oes dos cilindros rotativos. Sao Carlos. 82p.

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Tese ( Livre Docencia ) - Escola de Engenharia de Sao Carlos, Universidade de Sao Paulo.

( DEPARTMENT OF EXTERNAL AFFAIRS (1988). Tecnologia dos transportes: A experiencia Canadense. Ottawa. ( , Ontario. Canada. /folder/

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FLETTNER, A. (1925). The Flettner Rotor Ship. Engieering. P. 117-20,jan.

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IGARASill, T. (1997). Drag reduction of a square prism by flow control using a small rod. Journal of Wind Engineering and Industrial Aerodynamics. 69-71 ,p.l41-53.

JOHNSON, W. (I 986). The Magnus effect- Early investigations and a question of priority. lllternational Journal of (

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Mechanics Sciences. V. 28, n. 12, p.859-72.

( PRANDTL, L. (1927). The generation of vortices in fluids of small viscosity. Aeronautical Reprints, n.20.

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SCHLICHfiNG, H. (1968). Boundary Layer Theory. McGraw-Hill Book Company, Nova lorque, 6' Edi~tao.

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SWANSON, W. M. (1961). The Magnus Effect: A Summary of Investigations to Date. Journal of Basics Engineering. ( ASME 830, p. 461-70, set.

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TOLLMIEN, W. (1945). Berechnung turbu1enter Ausbreitungsvorgiinge. NACA TM 1085.

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APPLICATION OF A NON ISOTROPIC TURBULENCE MODEL TO STABLE ATMOSPHERIC FLOWS AND DISPERSION OVER 3D TOPOGRAPHY

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Fernando T. BO<;ON, Clovis R. MALISKA

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Departamento de Engenharia Mecanica Universidade Federal do Parana C.P. 19011 81531-990 Curitiba- PR, Brazil e-mail: bocon@demec.ufpr.br

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APPLICATION OF A NON ISOTROPIC TURBULENCE MODEL TO STABLE ATMOSPHERIC FLOWS AND

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DISPERSION OVER 30 TOPOGRAPHY

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ABSTRACT A 110n isotropic turbulence model is ex/ended and applied /o three dimensional stably strat(fied

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flows and di!>]Jersion calculations. The model is derived.fhml the algebraic s/ress model (including wall proximity e.ffecl.\), but il retains the simp/icily of the ·'eddy viscosily" concept ()f./irs/ order models. The "modified k-6" is implemenled in a three dimensional numerical code. Once the flow is resolved, the predicted velocity and turbulence .field~ are inle1polated info a second grid and used

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to solve the concentration equation To evaluate the model, various steady slate numerical solutions

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are compared with small scale dispersion experiments which were conducted at Ihe wind twmel of

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distinct idealized complex lopographies (flat and hilly lerrain) are sludied. Verlical prcifiles of

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velocity and pollutant concentration are shown and discussed. Also, comparisom are made against

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Milsubishi Heavy Industries, in Japan. Stably stratified flows and plume dispersion over lhree

the results obtained with I he s/andard k-6 model.

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Keywords: Atmospheric di.\persion, flow over hills, anisotropic k-6, numerical simulalion

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INTRODUCTION

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Atmospheric boundary layer flows are object of intense study over the last years. A more

(

comprehensive understanding of the complex phenomena involved in this particular type of flow is

( (

I

(

i

being sought, aiming the analysis of structural implications due to strong winds (neutral atmosphere), the pollutant dispersion under neutral or stable conditions and also for meteorological purposes. The

( (

76

( (

!

' (

( (

phenomenal increase in computer power over the last two decades has led to the possibility of

(

computing such flows by the integration of the (modeled, time-averaged) Navier-Stokes equations.

( (

Raithby eta/ (1987) employed the k-e model (with modification in the

C~

value) to calculate

(

the neutrally buoyant flow over the Askervein hill, and compared their numerical results with the

(

experiment made over the real terrain in Scotland. ·Dawson eta/ (1991) also used the k-e model

(

(with some modification in the constants of the dissipation equation) to simulate the flow and

(

dispersion over Steptoe Butte (Washington, USA) under neutrally and stably stratified atmosphere.

( (

Their results were favorably compared with experimental data, indicating that mathematical models

(

using the eddy viscosity assumption in the turbulence closure could be used to predict the flow and

(

pollutant dispersion over complex terrain. In Brazil, Dihlmann (1989) studied numerically the

(

thermal discharge (from chimneys) into neutral and stable stratified environments. Santos et a/

(

(1992) applied the standard k-e model to simulate the discharge of a chimney and the correspondent

( ( (

( (

plume dispersion over a flat terrain. Queiroz eta/ (1994) applied the standard k-e model to study (in two dimensions) the effect of heat islands in the atmospheric diffusive capacity. Koo (1993) developed a non isotropic modified k-eto account for different eddy diffusivities in the lateral and vertical directions in the atmosphere. His model is derived from the algebraic stress

( (

model and was applied in one dimensional problems to predict the vertical profiles of velocity,

(

potential temperature and turbulence variables for the horizontal flow in a homogeneous atmospheric

(

boundary layer. Also, the model was applied in two dimensional problems to simulate the sea breeze

(

circulation and the manipulation of the atmospheric boundary-layer by a thermal fence. Koo' s model

(

(

(

is similar to the level 2.5 model of Mellor and Yamada (1982). Recently, Castro and Apsley (1997) compared numerical (using a "dissipation modification" k-e model, as named by the authors) and laboratory data for two dimensional flow and dispersion over topography. Also, Bo"on and Maliska

( (

(1997a, 1997b) extended the non isotropic k-e model ofKoo (1993) to numerically simulate the flow

(

and pollutant dispersion over complex idealized topography, under neutral stratification.

( (

Ti

t ( ( Computational results were compared with experimental data obtained from a wind tunnel simulation.

( (

More sophisticated models, like the Reynolds stress model, were also applied to pre,dict environmental flows and pollutant dispersion, for instance the work of Andrdn (1990). Sykes and

( ( (

Henn (1992) applied the Large Eddy Simulation technique to simulate plume dispersion. Our view is

I

(

that for the time being, because of limitations in computer resources, those more complex turbulence

(

models (like Reynolds stress and LES) are not suitable for most engineering problems, due to large

(

f

CPU time and memory required.

( In the present work we extend the application of Koo's modified k-e model to predict three

(

dimensional stably stratified flows and pollutant dispersion over complex terrain. The prediction of

(

the plume dispersion downwind from a pollutant source is obtained from the solution of the

(

concentration equation. To do so, it's necessary firstly to calculate the velocity tield and eddy

(

viscosities in the region of interest. Thus, convection and turbulent diffusion of the plume may be (

calculated.

I

(

(

<

MATHEMATICAL MODEL

<

The task of computing the concentration field downstream from a pollutant source is divided

(

into two decoupled steps. Firstly we calculate the flow (velocity, temperature and turbulence

(

variables) in the region of interest. Secondly, we use the computed velocity field and eddy

( (

diffusivities to solve the concentration equation. This separation can be done as we consider that the

( pollutant release does not disturb the flow. In fact, in the wind tunnel experiment, against which we compare our results, the tracer gas was released with practically no momentum nor buoyancy force.

( ( (

I

(

78

(

I

(

)

(

-~

(

( (

Flow and Dispersion Modelling

(

The governing equations for the stratified flow are the conservation of mass, momentum and

(

energy, written below in the usual tensor notation. Dispersion of a pollutant is computed from the

(

concentration equation, after the flow is resolved.

( (

OU;

=0

(1)

OX;

(

(

au; Ot

+u-~=-..!_~+v~+~(-u~u'.) J

Pax;

axj

axjaxj

axj

I

(2)

J

(

( ( ( (

( (

aT+ U aT _ ~(-u~J 1 ·--

at

()xj

T')

(3)

axj

ac +u.~=~(-u~c')

ot

J

ox;

ox;

(4)

I

where p is the pressure deviation with respect to the hydrostatic pressure. Primed variables denote turbulent fluctuations. As we are simulating wind tunnel flows, the Coriolis effect is neglected.

( (

Modelling of fluctuation terms are described in the next section.

{ (

( (

Turbulence Modelling In environmental flows the non isotropic character of turbulence is notable, specially in the

(

case of dispersion of a scalar (pollutant) in the flow. For the

(

instance, vertical fluctuations ate much inhibited due to buoyancy forces (arising from the positive

(

~ase

of stably stratified flows, for

vertical temperature gradient), while horizontal fluctuations are not. Even neutrally stratified flows

( (

feature some anisotropy. So, it is not expected that isotropic turbulence models may well reproduce

(

the non isotropic turbulent diffusion. However, standard k-e is successfully applied for environmental

(

flows calculation where horizontal gradients (of velocity, temperature and turbulence variables) are

(

smaller than the vertical gradients. In these situations, turbulent diffusion is significant only in the

(

( (

vertical direction, and an isotropic model can handle it appropriately. On the contrary, in the problem

79

c (

(

of pollutant dispersion from a point source, both vertical and horizontal concentration gradients are

( significant, so are the corresponding turbulent diffusion. For this situations, a better description of

( the anisotropy in turbulent exchanges is necessary

(

(

In his Ph.D. thesis, Koo (1993) proposed a modification on the classic k-& model, through

(

use of algebraic stress model including wall proximity effects. The resulting model was compared to

I

( data and higher order (turbulence closure) simulations tbr one and two dimensional atmospheric flows. The modified k-E reproduced well the observed behaviors.

(

In our work we extend the application of the Koo's modified k-E model to three dimensional

( flow and to dispersion problems. A description of the turbulence model is given below. Detailed (

description of derivation of the model can be seen in Koo (1993). Following the Boussinesq's eddy viscosity concept, Reynolds stresses are related to the gradient of the velocity components as

I

J

--'

axj

axj

3

(

(

.(au. +-au;) --ko 2 -u! u'. =K 1 m

(

(S) IJ

(

'

( where

K~

is the turbulent eddy viscosity in the j direction. Analogously, turbulent heat exchange

( (

and mass dispersion are expressed, respectively, by

I

< -u~T'=Ki J h8T --

(6)

ax;

( (

(

-ujc'=KJ~

(7)

• axj

(

where Ki, and K~ are the eddy diffusivity in the j direction, respectively for heat and concentration.

(

(

Eddy viscosities (for momentum) and eddy diffusivities (for energy and concentration) are expressed

(

as functions of turbulent kinetic energy and its dissipation rate. For the vertical direction:

( (

Kzk2 m-Cm-

(8)

E

80

(

'

(

I

( (

7 (

(

(

K'h

=Chkl-

( (

K;=c

(

(

(9)

E

'

~

(10)

E

And for the horizontal directions:

(

(

~ E

K'm = KYm =C J-1

(11)

( (

(

K'h = KhY =~ K' Pr,

( (

K;=KY =

(12)

K~

(13)

'sz.-

(

(

Cm is the proportionality coefficient for eddy viscosity in the vertical direction.

c.. and C. are

(

the proportionality coefficients for eddy diffusivities in the vertical direction, respectively, for heat

(

and mass concentration. These three coefficient are defined by functions of flow structure (from the

(

algebraic stress model). Pr, is the turbulent Prandlt number (=O.S) and Sc, (=O.S) is the turbulent

(

Schmidt number.

( ( (

C =2 (c, -1)(E 7 -AGu) m 3 E4 Ea G G E 4 + - - u-E,E 7 GM+E,A uGM

(14)

CIT

( ( (

C =C = h

c

2

(c. -I)+E5 GM em

3

(2E 4 E 9 ) (crr+c;Tf)E 4 + - - - +E 6 G" Eto

(IS)

( ( ( ( (

I C k'12 f=-=-·kvz kvZE

(16)

Except for GJ..., Gn and f, the coefficients in equations ( 14) and (IS) are model constants, which can be found in Koo ( 1993 ). f is the wall function which reflects the effect of the ground

( ( (

81

4 (

(

proximity on the Reynolds stresses and turbulent heat and mass fluxes, I is the turbulence length

( scale, kv is the von Karman constant (=04), z is the distance from the ground and c.= 0.13.

( The C'", Ch and C. proportionality coefficients are functions of

G~~
the productioll of

(

turbulent kinetic energy by mean velocity shear, and 0 11, the production (or destruction) of turbulent

(

kinetic energy by buoyancy effects

(

I

( GM

=(~r[(~~r +(~:rJ

(17)

(

(

(

k)' ae GH =gJl (-; oz

( 18)

< (

Turbulent kinetic energy and its dissipation rate are computed from their well known prognostic equations:

ok

ot

(

(

+u-~=__i_(K~, ~)+P+G-e J

oxj

(

axj

(Jk

(19)

axj

( (

J

OE+ uOE OE +C. (P+G)--C E i - =o- (K~ -.. 2 -e 1 8t

axj

i)xj

cr. axj

k

(

2

k

(20) (

I

( P is the production term due to mean velocity gradients

ou, _

(au, ou;)

( (

_ - ,-, K; ou, --+-- P --u.u.--• J OX; m OX; ox, axj

(21)

( (

G is the production (or destruction) term due to buoyancy

-

G =gJl w'T'

aT

=-gJl K~­

( (22)

oz

( (

Constants in equations (II), (19) and (20) are those from the standard k-e model, and can be

( (

seen in table 1.

( 82

(

I

(

I

(

( ( (

(

( (

( (

1

-r +.: : ~:

--~--

: I

1:,

I

Table 1 -Constants of the turbulence model

(

(

(

NUMERICAL METHOD

(

(

The finite volume method is employed to solve the governing equations, in a non orthogonal,

(

generalized curvilinear coordinate system. Co-located arrangement is used for variables storage in

(

the grid, and the QUICK interpolation scheme with source deferred correction term Lien ( 1994) is

(

applied on the convection terms, except for turbulence variables where a hybrid scheme (WUDS of

(

(

( ( (

Raithby and Torrance, 1967) is adopted. Our own codes NAVIER (\99\) and SMOKE (1997) are used to solve the governing equations, respectively, for the flow and concentration. As the grid used for computing the flow is not adequate for the concentration calculation, a second grid (refined near the source) is used for the last purpose. Velocities and eddy diffusivities

( (

obtained from the flow solution are interpolated into the second grid for the concentration

(

calculation. Also, in order to verifY grid dependent errors, the computations are made in a coarse and

(

in a fine grid. Figures I and 2 illustrate some of the coarse

(

(inflow boundary at left). Fine grids are 9Sx41x41 and 128x64x64 for flow and concentration,

( (

(

grids. Only half domain is resolved, because of symmetry.

( ( ( (

used for flow and concentration

respectively. Coarse grids have half the number of volumes in each direction, with respect to the fine

(

(

gri~s

83

f (

1000-

~r~,-

--·

.,

-~·~-,---

( (

( (

500

(

~~+--

( (

:':~Jld

1/l/f.-- ~-

I

-500

500

0

1000

(

(

(

I

(

1500

(

(

(

=m .-----

1000

( r-r-r--r--r-

~+--~+----

--

-

500-

-

·-f-~-·-

--·

---!----

-~-=+----

~ t--

t-~-

0 -1000

I

-500

0

500

1000

(

(

(

--··t==t==

t-- 1--

(

(

--

-

(

.-.--r-r-r-r--,-.,------,-r---1--------,---~-----r--·------

1500

(

(

( (

( Figure I -Vertical (at the xz symmetry plane, above) and horizontal (below) views of the coarse grid for hill height 200mm (42xl8xl8 volumes)- vertical dimensions (z direction) exaggerated

( ( (

( 84

(

( (

'

( (

(

500

f ( ( ( (

(

{

0 -500

(

0

500

(

1000

1500

1000

1500

(

(

500

(

(

250

( (

0 -500

( (

0

500

( Figure 2- Vertical (at the xz symmetry plane, above) and horizontal (below) views of the coarse

(

grid for hill height 200mm (64x32x32 volumes)- vertical dimensions (z direction) exaggerated

( To verify the model performance, in a first step, the above described modified k-e is applied

( (

to simulate wind tunnel experiments.

( ( (

THE WIND TUNNEL EXPERIMENT

( Pollutant dispersion wind tunnel experiments were conducted at the Mitsubishi Heavy

(

(

Industries, in Nagasaki, Japan, 1991. A report containing the results was obtained directly from that

(

company. Wind tunnel test section is 2.Sm wide, lm high and 10m long. Axisymetric hills of

( ( (

'

different heights (0, 100 and 200mm), were positioned with the top located at (x,y)=(O,O). Hill shape

85

c (

can be seen in tigures I and 2. Streamwise direction is x, lateral is y and vertical is z. Source of tracer gas was positioned at (x, y, z)=(-500 mm, 0, 50mm) for hill heights 0 and IOOmm, and at (x, y, z) = ( -SOOmm, 0, 1OOmm) for hill height 200mm. Cases of neutral

(~ T=O,

( (

( Pasquill class D) and stable (

atmosphere (~T=20°C, Pasquill class E) were performed. Streamwise velocity; velocity fluctuations, temperature and concentration were measured at various locations.

(

(

)

( (

NUMERICAL EXPERIMENTS AND BOUNDARY CONDITIONS Three different wind tunnel experiments were computationally simulated. They are designated

.

(

( (

with a letter - indicating stability class - followed by a number indicating hill height in mm. Hill heights of 0, 100 and 200mm were simulated. Neutral flows (Pasquill class D) and pollutant

( (

and Maliska (1997a,

(

1997b). At this time, stably stratified flows (Pasquill class E) are considered. At the inflow boundary,

(

dispersion over these topographies were already numerically studied by

Bo~on

velocity, temperature and turbulent kinetic energy are specified according to experimental measured

( (

values. As the dissipation rate of turbulent kinetic energy was not measured during the experiment,

I

(

its inflow profile is calculated according to a prescribed turbulence length scale. For neutral boundary

(

layer atmospheric flows, this length scale increases linearly with the distance from the wall (height

(

above the surtace, in the present problems). However, in the case of stably stratified flows the

(

turbulence length scale does not increase linearly with the height, but it is limited to a maximum

.

'

( (

value (Castro and Apsley, 1997)

(

I=~

(23)

k. z I+-0.08SL

(

(

( where z is the distance from the ground, k. is the von Karman constant (=0.4) and Lis the Monin-

(

Obukhov length (=0.13m), which was calculated from the experimental values of velocity and

(

temperature near the ground.

(

( 86

( (

)

( (

( {

Outflow conditions are that of zero gradient for all variables. For velocity, lateral and upper

(

boundaries are impermeable, with zero tangential stresses. For all other variables, lateral and upper

(

boundary conditions are of null fluxes. Wall functions are invoked to apply boundary conditions

(

appropriate to a rough wall (Zo = l.Se-4m) at the ground. Symmetry conditions are applied at the

(

boundary coincident with the plane of symmetry (y ':"' 0).

( (

(

TREATMENT OF NEAR SOURCE DIFFUSIVITY

(

(

After applying the modified model and computing concentrations, we constated that, for all

(

the cases studied (neutral and stable stratification), there was a large unrealistic plume spread near

( (

the source and, consequently, low concentrations everywhere in the domain (specially up to SOOmm downstream the source). Taking a look at the turbulence length scale near the source, we noticed

(

r

that it is larger than the plume dimensions. It means that the turbulent eddy sizes present in the flow

(

are bigger than the plume, and could not promote such a observed diffusion in the numerical

(

simulations. Therefore, we speculate that the length scale to be applied in the eddy diffusivities for

(

the concentration should be appropriately reduced for the initial stages of plume spread, according to

(

local plume dimensions. Based on a Gaussian plume distribution near the source, as a first

( (

{

investigation, we decide to reduce linearly the eddy diffusivities computed from the flow solution, to

.

be applied in the concentration calculations. Using this simple procedure, the quality of the results

(

improved considerably. Reduction of eddy diffusivity near the source is made taking its value (at the

(

source location), obtained from the flow solution, and applying it in the Gaussian model for diffusion

(

from a point source, to calculate how far from the emission point the plume width is about five times

(

the local turbulence length scale. This value (five) was empirically determined from the analysis of

( (

the results. It was found, however, that the value is roughly the same for all the cases studied. At a

(

given distance from the source, plume width is defined as the distance from the plume center line to

(

the point where the concentration is 10% of its peak value. Indeed, further work is needed to better

(

s7

(

(

(

model the initial stages of plume spread, where its dimensions are smaller than the characteristic

(

turbulence length scale of the flow. (

I

< (

RESULTS AND DISCUSSION

( In this section, some results of the flow and concentration calculations are presented, for the

(

three stably stratified cases which were computationaly simulated. Figures 3, 4 and 5 show vertical

(

profiles of concentration on the symmetry plane (y=O) for the cases EO, E I 00 and E200. The graphs

( in each figure refer to different positions downstream the source. In figure 3, for the case of flat (

terrain (EO), it can be seen a good agreement between numerical and experimental values. At the

(

position x=200mm (third graph in fig. 3), we believe that there possibly was a mistake with respect

(

to the report of the experimental results, which are inconsistently underestimated (as it can be noticed by comparison between the two peak concentration values corresponding to the positions

(

I

( (

x=O and x=500mm).

( Figures 4 and 5 show, respectively, the concentration profiles for the cases of hilly terrain EIOO and E200 (hill heights 100 and 200 mm). From the view of a numerical analist, these are the most critical cases, due to the characteristics of stable flow and complex topography. Although the

( (

I

(

( peak concentration values are fair well predicted, their locations are not. For the problem of

( pollutant dispersion over complex terrain, the plume path is dictated by the deviations in the mean

(

flow caused by the irregular topography. A correct description of the plume path requires a

(

sufficiently accurate prediction of the flow field (which "drives" the plume).

(

Figures 6 and 7 show vertical profiles of the streamwise component (u) of velocity on the

(

( symmetry plane (y = 0) for the cases EIOO and E200. For both cases, the modified and the standard

(

k-e model produced nearly the same velocity profiles, and the recirculation zone in the lee side of the

(

hill was underestimated. Different inflow turbulent length scales were tested at the inflow to verify a

( (

88

( (

I

( (

( (

possible influence, but it was noticed that the flow after the hill top is essentially determined by local

(

conditions. A possible explanation tbr this model defect would be that the pronounced velocity

(

gradients in this region, due to the three dimensional open recirculation zone (see figure 8), increase

(

the production of turbulent kinetic energy and consequently enhance the eddy viscosities there, thus

(

diminishing the size of the recirculation. However, a. comparison between numerical and wind tunnel

(

(

measured values did not reveal that the turbulent kinetic energy level has been overestimated by the

(

mathematical model. Thus, regarding to the above cited problem of high eddy viscosities in the

(

recirculation zone, the model drawback should be attributed to the dissipation equation, which is

(

underestimating e, and not to an overestimation of the production of turbulent kinetic energy (P).

(

( (

The underestimation in the size of the open recirculating three dimensional zone by the flow model leads to an incorrect determination of the plume path. As the numerical m()del foresees a

(

smaller recirculation, the flow, after passing over the hill top, brings (convectively) the plume down

(

to the ground. Thus, in the numerical simulation, the center line of the plume (the place of the peak

(

concentration values along the plume path) is nearer the ground than the experimental plume, and the

(

(

ground level concentration numerical values result higher than the measured ones.

( (

(

For the case E200, figure 5 also shows the concentration results produced by the standard ke

~ode!

(isotropic), which clearly are more diffusive than those from the modified model. In this

sense, it is demonstrated that the non isotropic character of the turbulent diffusion under stably

( ( (

stratified flow conditions is relevant, and that the modified dispersion of a plume in such conditions.

(

( ( (

( ( ( (

89

k-e

has better ability to predict the

( (



k-e anisotr6pico (grosseira)

(

k-e anisotr6pico (fina)

(

Tunal de vento

( 200-

200-

x = -200 mm

( 150-

150

e.s

(

\

x=O

. ', .. __.._ ____ _,__________ _____

100

N

~

-........:._:

·--. -

50 r-;-;---.~----

0

100

---

200

600

400

c

(

' '·......_. . -·---.._____

-

(

--------~

50-

.----;---~--.J_--- ____•__ _:} (

OT

..,..-----,--

r··T----1-'j

200

100

0

800

(

.

----..:..______

,_

.--~r--r-r--r--,---,-.-----T---~

0

(

300

c

(

400

(

( ( 200

150

e .s

200

x= 200 mm

x = 500 mm

(

(

l~

'~ ·--.._

100

.

..... __ ~

N

100

......_____

. ·~---

50

rr-1

0

50

1

1

1 ,

1

1 t

100

r-r

c

150

"''-~....... ,.._

(

. '.·

(



~.,_)

~-----'

0

(

"'----~ , __

200

50-

,

.. '),

....

(

_.J--~= __.../!

----~­

0

~

,-~~--r-T'-;-r-···

0

250

(

50

100

c

150

200

( (

(

( Fig. 3 - Case EO - Concentration at the symmetry plane - flat terrain

(

( ( ( (' 90

( (

t

(

( k-e anisotr6pico (grosseira)

( -

(

•J

(

(

200

(

150



• (

k-e anisotr6pico (fina) Tune! de vento

250 I

x=O

x = -200 mm 200

1: E

s

100

150

.

N

--------

50

(

100

( 0

(

~---·..- --,---~-..,

0

200

'- .-T~,-~r-·r--r---,.-~~~ 50 ~---,------,-- r400 600 800 0

( 250

( (

( ( (

E

x =200 mm

1

250

200 .

150 ~'

150

50

(

500

c

x =500 mm

100

J·.

100

...

)'

50

o I , ,,-,---. 0

'---r

250

..

(

(

I

200 -)

~ 1001

.----1----T

c

( (

-

-~..,.---1"!1

200

c

300

400

.

..

i

0 j 0

\ J.

.. '

--,---,~

100

··-,.

.

c

. ~~-T-~

200

300

( (

Fig. 4 - Case E I 00 - Concentration at the symmetry plane - hill height I 00 mm

(

( ( ( ( ( ( (

91

(

(

(

k-e anisotr6pico (grosseira)

-

e

k-e anisotr6pico (tina)

(

k-e classico

(

Tune! de vento

300--,

300l x

250~

e

2oo

§. N

150

i\_. _

1

·-)

--

(

x=O

=-200 mm

(

,,\ \.

250

-~ ::---

(

\

(

-~~,-~

.

( ~-

----~

\

;-~....-~-~--·----~

~-~:;

-~-

200

(

y,

(

,/"

100

( ...,---.--.....,..,--~-.-!

50 200

0

400

600

c

150

800

,---.-.-T -,-,'1

0

100

200

300

c

400

500

(

(

( 350300

e §.

(

350

x =200 mm

x = 500 mm

(

300-



250

250- ·"........__,

.. .· . .

'·------:--... ·

N

150

(



-, ___ -.__.

~-......... <:-...

200

~~ \:_

200



~-

')

( ..... .,."--'.~

'·'::::.:.:...

.

150

~

100

100-

)

(

\ /

'

(

50

50

~----.------.--------.-1

0

(

--....

100

c

200

0

300

·.0

50

·r~

100

c

150

200

( ( (

( Fig. S - Case E200 - Concentration at the symmetry plane - hill height 200mm

(

( (

( (

( 92

( (

'

I (

( (

modified k-e (coarse)

(

modified k-e (fine)

(

standard k-e



(

( (

5QQ

I

400

I

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Wind Tunnel



50~ -1

~

x = -200 mm

j

I

400

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./

300

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200 100 ~~

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-

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0.5 U (m/s)

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x=O

~-~~

-,.--,-_, ....,.

-~

..



r

1.0

0.0

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x = 500 mm



(

(

500

J

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x = 250 mm

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(

400

J

300 -1

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J 200 .. 100-

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300-

)

i

200

~

1.0

1.5

0

0.0

0.5 1.0 U (m/s)

1.5

( ( ( (

Figure 6- Case EIOO- vertical profiles ofstreamwise component of velocity (u) at the symmetry plane (y

=

0) for different positions upstream and downstream the hill top (x = 0)

( ( 93

(

( (

(

k-e modif. (grosseira)

(

k-e classico



500

(

Tunel de vento

x



=-200 mm

500

~

400

e§.

(

k-e modif. (tina)

N /

200

"/ •

••

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,,

'

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400-

j

300

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x=O

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;/

100 -\--,-- ~r--r' ·.--.---r T--..------.-- 200 0.0 0.5 1.0 0.0 U (m/s)

-~

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r---l--,.-------,---r-----~~1~--1---.,---

0.5 U (m/s)

1.0

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500-,

-

!

31JO

x

l •

-1

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• • ••./ / /

.

..

x =500 mm

5001

.

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l.

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0•

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./'

200

1/

'I

100

~

-,-1

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'

300

i' •

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400

(: ',

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=250 mm

I

I

.•

•• •

f

•I

(

'!;/

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/

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0

I

1.5

0.5 1.0 U (m/s)

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Figure 7- Case E200- vertical profiles ofstreamwise component of velocity (u) at the symmetry

(

plane (y =0) for different positions upstream and downstream the hill top (x = 0)

(

-0.5

0.0

0.5 U (m/s)

1.0

0.0

(

( (

94

( (

(

(

~-v/~

(

~ ~~~~ ~ ~~O:o:::~ ,,,,,,/;/

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,

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c ( (

v- v / I ; ; 11 1 I

.

.

/

/

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.. .___.-.,. . /,../ ' ,"',. /, // .....1 ' /,// /

:///

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Figure 8 - Top view of velocity vectors I 0 mm above the ground - case E200

(

(

(

CONCLUSIONS

( (

A modified non isotropic k-e model is applied to simulate three dimensional stably stratified

(

atmospheric flows (Pasquill class E) and dispersion over an idealized complex terrain. The results for

(

the velocity field are similar to those obtained with the standard model, because the vertical gradients

(

of flow variables are not great (when compared with the concentration field), resulting nearly the

(

same turbulent diffusion for both the standard (isotropic) and the modified (non isotropic) k-E

( (

models. However, for the concentration results, the differences between the numerical solutions

(

obtained with the standard and the modified k-e models are quite distinct. The agreement of the

(

concentration values - against the wind tunnel results - is very satisfactory for the case of flat terrain.

(

Over hilly terrain, the concentration peak values are fair well predicted, but their locations (height

(

above the ground) are not. The problem is attributed to a model fault in predicting the open

( ( (

(

recirculating three dimensional zone, which occurs in the lee side of the hills. In the recirculation

95

(

( zones, the eddy viscosities are overestimated and thus the size of the recirculation is underpredicted. Consequently, the plume path over the top and the lee side of the hill is not well predicted, with

( (

respect to its height above the ground, which is underestimated. Despite of these drawbacks, the

(

modified non isotropic k-e produces better results than the conventional isotropic model.

(

( (

ACKNOWLEDGMENTS

(

( We are grateful for the financial support provided by CNPq and CAPES.

( ( (

REFERENCES Andren, A., 1990, "A Meso-Scale Plume Dispersion Model. Preliminary Evaluation m a Heterogeneous Area", Atmospheric Environment, v. 24A, n. 4, p. 883-896. Bo~n,

F. T. and Maliska, C. R., 1997a, "Numerical Modelling of Flow Over Complex Terrain",

(

( (

Proceedings, XVIII Iberian Latin American Congress on Computational Methods in

(

Engineering, Brasilia- DF, published in CD-ROM.

(

Boc;;on, F. T. and Maliska, C. R., 1997b, "Numerical Modelling of Flow and Dispersion Over Complex Terrain", Proceedings, XIV Brazilian Congress of Mechanical Engineering, ABCM, Bauru-SP, p. 211-218. Castro, I.P. and Apsley, D.D., 1997, "Flow and Dispersion Over Topography: A Comparison Between Numerical and Laboratory Data for Two-Dimensional Flows", Atmospheric

( ( (

( (

Environment, vol. 31, no 6, pp. 839-850. Dawson, D., Stock, D.E. and Lamb, B., 1991, "The Numerical Simulation of Airflow and

(

Dispersion in Three-Dimensional Atmospheric Recirculation Zones", J. Applied Meteorology,

(

vol. 30, pp. 1005-1024.

(

Dihlmann, A., Maliska, C. R., Silva, A. F. C., 1989, "Soluc;;ao Numerica da Descarga de Jatos

(

Poluentes em Meio Estratificado", Anais do X Congresso Brasileiro de Engen haria Mecanica,

(

ABCM, Rio de Janeiro-RJ, p. 101-104.

(

Koo, Y.S., 1993, "Pollutant Transport in Buoyancy Driven Atmospheric Flows", Ph.D. Thesis, The Louisiana State University and Agricultural and Mechanical Col.

96

(

( (

(

I

4.

( ( Lien, F.S. and Leschziner, M.A., 1993, "Upstream Monotonic Interpolation for Scalar Transport

(

{

( (

With Application to Complex Turbulent Flows", Int. J. For Numerical Methods in Fluids, vol. 19, pp 527-548. Mellor, G.L. and Yamada, T., 1982, "Development of a Turbulence Closure Model for Geophysical Fluid Problems", Reviews of Geophysics and Space Physics, vol. 20, no 4, pp. 851-875.

(

NAVTER, 1991, "Desenvolvimento de C6digos Computacionais para Solw;:iio de Problemas de

(

Escoamentos de Alta Velocidade", Relat6rio preparado para o Instituto de Atividades Espaciais

(

do Centro Tecnico Aeroespacial - UFSC - Dep. Eng. Mecanica- Parte VII.

( (

(

Panofsky, H. A, Dutton, J A, 1984, "Atmospheric Turbulence - Models and Methods for Engineering Applications", John Wiley & Sons, New York. Raithby, G.D. and Torrance, K.E., 1967, "Upstream-Weighted Differencing Schemes and Their Application to Elliptic Problems Involving Fluid Flow", Computer and Fluids, vol. 2, pp. 12-26.

(

Raithby, G. D, Stubley, G. D , and Taylor, P A, 1987, "The Askervein Hill Project: A Finite

f

Control Volume Prediction of Three-Dimensional Flows over the Hill", Boundary-Layer

( (

( (

Meteorology, v. 39 SMOKE, 1997, "C6digo Computacional para a Solw;:iio da Dispersiio de Escalares em Geometrias Tridimensionais", Laborat6rio de Simulac;:iio Numerica em Mecanica dos Fluidos e Transferencia de Calor do Departamento de Engenharia Mecanica da Universidade Federal de Santa Catarina SINMEC.

( (

( (

(

( ( ( ( (

( ( ( (

( (

97

-

--

/

{ (

(

Explorando o Nao-fechamento no Equacionamento da Turbulencia

(

( (

Exploring the Nonclosure Question in Turbulence Equations

(



(

(

{

Harry Edmar Schulz Laborat6rio de Hidraulica Ambientai-CRHEA Departamento de Hidraulica e Saneamento Escola de Engenharia de Sao Carlos-Universidade de Sao Paulo C.P.359, 13560-270, Sao Cartes, S.P., Brasil (Trabalho desenvolvido no lnst~ut fOr Hydromechanlcs, Unlversitat Kartsruhe, Alemenha)

( ( ( ( (

(

(

Abstract

In this study it is shown that the use of the turbulent kinetic energy dissipation rate as a relevant parameter in the search of a closure equation for turbulence. evokes the concept of entropy generation. The Gouy-Stodola theorem is used to show this connection. Further, an alternative formulation for turbulence is presented here. This formulation is very similar to the entropy equation for thermal radiation. It is shown that simple relations exist between the parameters defined in this formulation and the entropy generation rate. Keywords: entropy generation rate, Gouy-Stodola theorem, nonclosure in turbulence, turbulence equations.

( (

( (

(

( ( (

Resumo

Uma discussiio e conduzida no sentido de mostrar que o uso da taxa de dissipa~iio de energia cinetica turbulenta, como variirvel relevante na busca de uma equa~iio de fechamento para o prbblema de turbulencia, evoca o conceito de gera~iio de entropia. Isto e feito recordando o teorema de Gouy-Stodola. Posteriormente, um equacionamento alternativo para turbulencia e explorado visando verificar a identidade passive! entre varic:iveis deste equacionamento e a taxa de gerar;iio de entropia. Este equacionamento possui uma sentelhan~a formal muito forte com a e(jlwr;iio de entropia em radia~iio termica. Palavras-chave: taxa de gera~iio de entropia, teorema de Gouy-Stodola, niio-fechamento em turbutencia, equar;oes de turbulencia.

(

(

( ( (

lntroduc;:ao

(

No estudo da turbulencia com bastante frequencia nos concentramos no conjunto de equar,:oes que governam o movimento de fluidos newtonianos, isto e, fluidos que, se adrnitidos como compondo urn continuo, seguem uma proporcionalidade entre tensao e gradiente de velocidade. Neste caso, as equar,:oes de Navier-Stokes e a equar,:ao da continuidade compoem urn conjunto fechado e representam convenientemente a segunda lei de Newton e o principia de conservar,:ao de massa. Entretanto, utilizar essas equar,:oes para representar parametres flutuantes, ainda submetidas

(

99

(

( (

I

( ao operador de media, acrescenta a dificuldade de ser necessaria estabelecer equar;:oes adicionais para exprimir as funr;:oes de correlayiio entre flutuar;:i:ies de velocidades, ou de outras grandezas, que decorrem dos procedimentos tradicionalmente aceitos para efetuar as medias. Surge, portanto, o problema do fechamento, no qual se impi:ie que sejam introduzidas equar;:i:ies complementares com informayi:ies novas acerca do fenomeno," porem sem introduzir variaveis novas. 0 problema e antigo e tern conduzido a diferentes aproximar;:i:ies. A experiencia acumulada ao Iongo deste seculo conduziu gradativamente ao uso da taxa de dissipar;:iio de energia (s) como urn parfunetro relevante na modelayiio da equayiio faltante. Urn exemplo de sucesso bastante expressive e o modelo k-s, que liga a taxa de dissipayiio de energia ao conceito de viscosidade turbulenta, este ultimo uma simplificayiio conveniente para calcular escoamentos turbulentos. Neste modelo, sao utilizadas as equar;:i:ies de Reynolds (segunda lei de Newton), da continuidade (conservar;:i!.o de massa) e da energia (transporte da energia cinetica turbulenta). Nesta ultima surge a taxa de dissipayiio de energia e, na btisca de maiores informayi:ies, gera-se uma nova equayiio para esta taxa (a qual geralmente e apresentada desvinculada, de forma mais evidente, de urn principia fisico aparente) e, finalmente, conecta-se esta variavel a viscosidade turbulentaja mencionada. 0 sucesso em utilizar a taxa de dissipaylio de energia como variavel relevante induz apergunta elementar: se ja foi utilizado o conceito de conservaylio de energia, no qual parte e dissipada e no qual se define a taxa de dissipayiio, como justificar uma nova equar;:iio para esta dissipaylio sem evidenciar urn conceito fisico adicional? Evidentemente o desenvolvimento da equar;:lio adicional fundamenta-se na propria fisica dos escoamentos turbulentos, na qual o principia utilizado esta imbutido. Mas podemos ainda caminhar no sentido de explorar as possibilidades decorrentes do uso da taxa de dissipar;:lio de energia e indicar os conceitos que podem estar sendo evocados ao fomecermos uma aproximayiio adicional para a mesma.

( ( (

( (

( ( (

(

( ( ( (

( (

Os Principios Fisicos Usuais em Turbulencia Como em qualquer problema em Mecanica dos Fluidos, a formulayiio em turbulencia utiliza, conforme ja foi mencionado, os seguintes principios:

( ( (

- Conservar;:iio de massa. - Conservar;:lio (transporte) da quantidade de movimento. 0 uso das equayoes de Navier-Stokes para descrever o escoamento de fluidos em condiyoes especiais de proporcionalidade entre tensiio e taxa de deformar;:i!.o gerou uma comunidade de especialistas neste tipo de escoamento e permitiu o avanyo no calculo de solur;:oes numericas explicitas para estas equayoes. 0 estudo da turbulencia se enquadra nesta comunidade, uma vez que as equayoes utilizadas sao uma aplicar;:lio das equar;:oes de Navier-Stokes. - Conservar;:lio (transporte) de energia. Em turbulencia utiliza-se a "energia cinetica turbulenta", que e desenvolvida tambem a partir das equayoes de Navier-Stokes. 0 uso das flutuar;:oes de velocidade garante a interpretaylio que vincula esta energia apenas a parcela turbulenta do escoamento. A equaylio de conservar;:iio ou transporte resultante apresenta todos os termos deste tipo de equacionamento, ou seja, de forma resurnida: variar;:iio temporal, advecyiio, difusiio, produyiio e dissipar;:iio. Os principios mencionados, que conduzem cada urn a sua propria equar;:iio, sao de facil assimilar;:iio. A "equar;:lio adicional", que deve possibilitar o fechamento do problema de turbulencia, tern dispendido urn esforyo razoavel por parte dos pesquisadores, os quais apontam para o uso de uma equa<(lio que envolva a taxa de dissipar;:lio de energia, ou que a descreva. Isto nao implica em dizer que a taxa de dissipar;:iio de energia represente o unico problema a ser resolvido no fechamento das equayoes. Entretanto, os termos desconhecidos slio representados, comumente, como dependentes da taxa de dissipar;:lio de energia.

100

(

(

(

(

( ( (

( (

(

(

t ( ( I(

Como exemplo de uso da taxa de dissipayil.O de energia para obter uma soluyil.o particular podemos mencionar a soluyil.o de Kolmogoroff para a distribuiyil.o da energia sabre os diferentes numeros de ondas. A lei dos 5/3 que recebe o seu nome e desenvolvida no contexto da turbulencia isotropica, utilizando analise dimensional, na qual urn procedimento de "inclusil.o-exclusil.o de variavel relevante" e utilizado para a viscosidade cinematica (Schulz, 1997). Este procedimento, embora conduza ao resultado correto, mostra que talvez a melhor forma de analisar o problema ainda niio foi atingida. Na analise dimensional efetuada, a taxa de dissipayil.o de energia permite obter a forma do espectro, bern como definir escalas de velocidade e comprimento caracteristicas para os maiores numeros de ondas.

(

(

( (

(

( ( (

( (

( (

( (

(

f (

No sentido de obter urn equacionamento mais geral, o uso das equayoes de Navier-Stokes para gerar uma equayil.o de transporte (conservayil.o) da vorticidade e, conseqiientemente, uma equa9ii0 de transporte (conservayil.O) da taxa de dissipayil.O da energia para altos numeros de Reynolds, e talvez o procedimento reconhecido como aquele mais vinculado a propria fisica do escoamento. Este procedimento tambem inclui uma nova variavel, a vorticidade, para a qual se propoe a sua propria lei de variayil.o. Em uma primeira analise, portanto, conseguiu-se associar uma nova equayil.O a urn novo conceito (vorticidade). Contudo, esta nova relayiio traz consigo os problemas ja mencionados decorrentes do uso constante das equayoes de Navier-Stokes para a gera9iio de novas equayoes de conservayil.o ou transporte: parcelas componentes apenas estao definidas na propria equayiio e niio possuem comportamento determinado por outras rela9oes conhecidas. Assim, essas parcelas necessitam ser deterrninadas a partir de novas considera9oes. Conseqiientemente, a conservayil.o (transporte) da vorticidade, embora tenha perrnitido estabelecer uma relayiio para a taxa de dissipayil.o de energia, transfere o problema do fechamento para outras equayoes adicionais e, evidentemente, para outros principios fisicos. Tendo em vista este aspecto, comenta-se, no presente trabalho, uma abordagem que conduz o problema da taxa de dissipayil.o de energia para urn ponto de vista no qual a taxa de gerayiio de entropia e evidenciada. Niio se pretende, evidentemente, resolver o problema do fechamento, mas mostrar as possibilidades decorrentes deste ponto de vista.

Taxa de Dissipat:;:ao de Energia e o Teorema de Gouy-Stodola

( ( {

( ( ( ( ( ( ( ( ( ( (

Uma verificayii.o da possibilidade de vinculo entre a taxa de dissipayii.o de energia e outros con.ceitos nil.o aventados na abordagem tradicional de turbulencia e interessante. A entropia e urn principia fisico de uso corrente, ainda niio utilizado na formulayiio ate o momen~o apresentada. Uma forma passive! de abordagem pode talvez ser melhor analisada a partir de urn exemplo ilustrativo. Se se considerar urn agitador qualquer que mantem urn fluido em UJV estado de turbulencia estacionaria, podemos dizer que estamos continuamente alterando a ordem deste sistema (fluido), ou continuamente gerando entropia no mesmo. Assim, esta-se utilizando o estado turbulento como urn gerador de entropia. Como conseqiiencia, o equacionamento de entropia deve ser convenientmente adaptado para representar a gerayiio da mesma. Note-se ainda que a manutenyiio de urn estado turbulento tambem exige urn fornecimento continuo de energia, que e dissipada, o que indica que uma relayiio entre as duas grandezas deve ser passive!. Esta relayiio ja esta establecida pelo teorema de Gouy-Stodola (ver Bejan, 1982, ou Schulz, 1997) 0 seu desenvolvimento, embora tradicional na literatura, e pouco utilizado no desenvolvimento de soluyoes para os escoamentos turbulentos, motivo pelo qual e incluido no presente texto. Uma breve apresentayiio do teorema de Gouy-Stodola e aqui feita, onde inicialmente sao colocadas as equayoes basicas com as quais se trabalha. 0 desenvoJvimento na forma integral aqui apresentada e conveniente para a rapida assirni!ayii.O do conteudo do teorema.

101

( (

1 - Equar;:iio de Conservar;:iio de Massa, na sua forma integral usual:

(

a I pdVol=- I pV.dA --

~

C/11'\:

(1)

sc

.

( (

Ou:

(

oM =L:m- L:m at

u

(2)

(

E.....

( p e. a massa especifica do fluido, Vole o volume de controle, a massa e

V e a velocidade, A e a area, Me

m representa o fluxo de massa atraves das paredes do volume de controle.

( (

2- Primeira Lei da Termodinamica, na sua forma integral usual:

( Q-W=: IepdVol+ I(e+E.)pv.tiA

(3)

p

sc

C/tl't:

(

vl

E= fepdVo/

h=u+E.

e=z-+gz+u

p

1'1:

V V L ~· h+-+gz - L~ · h+-+gz 2 2 2

a

-E = Of

)

+Q-W•

(

( (5)

(

(

Sal

Q e a potencia termica, acelera~io

I

(4)

2

)

Brvra

(

w

e a taxa de trabalho, e e a energia especifica, p e a pressao, g e a da gravidade e z e a cota.

3 - Segunda Lei da Termodinamica, na sua forma integral usual:

I

I -- I

(

(o C!)

a atl't:spdVol+ scspV.dA?. seTl AdA

(6)

(

s e a entropia por unidade de massa. Define-se a entropia S como:

= fspdVo/

S

.(

( (7)

l't"

( Tem-se, entilo:

(

as Q - ? . L:ms- L:ms+at

Enlt"a

u

(8)

T

(

Portanto, verificando-se sempre a desiguladade 8, define-sea entropia gerada como sendo:

S. Oor

(

as " " Q =--L.ms+L.ms--?.0 at Entra Sal T

( (9)

( (

102

( (

t ( ( (

4 - Teorema de Gouy-Stodola:

( Da equac;:ao 5 e da desigualdade 8 resulta:

(

2

(

. W<>

)

Em'a

(

( ( ( ( ( ( (

2

V V Lrh( h+-+gz-I;,S - Lrh h+-+gz-I;,S 2 2 (

(10)

Q(

0= wm.. -W

(11)

Utilizando as relac;:oes 5, 10 e 11, obtem-se:

. (as "' "' . Q)

U= I;, - - L..rils+ L..ms-y; iJl Entra Sai 0

(

Das equac;:oes 9 e 12 tem-se, imediatamente:

(

8 --(E-I;,S)

To e a temperatura do ·ambiente onde esta imersa a parte do volume de controle por onde atravessa o calor, sendo tambem a temperatura daquela parte da superficie de controle. Alterac;:oes de temperatura sao consideradas ocorrerem no interior do volume. A desigualdade 10 mostra que a potencia utilizavel do sistema possui urn limite superior, quando se adota a igualdade. A diferenc;:a entre este limite superior e a potencia utilizavel e a potencia utilizavel perdida (potencia dissipada).

(

(

)

Sai

(12)

0 =I;, Sa.,

(13)

(

Esta e uma forma conveniente de apresentac;:ao do teorema de Gouy-Stodola para a Medl.nica dos Fluidos (Bejan, 1982). Nota-se que a taxa de dissipac;:ao de energia (potencia dissipada) no volume de controle e diretamente proporcional a taxa de gerac;:ao de entropia oeste mesmo volume. A equac;:ao 13 nao permite encaminhar no memento qualquer proposta de fechamento, mas mostra que ao buscarmos soluc;:oes que envolvem a taxa de dissipac;:iio de energia nos problemas de turbulencia, estamos apenas seguindo o caminho natural que evoca o conceito de entropia. Assim, o equacionamento em turbulencia envolve de fato os conceitos de massa, quantidade de movimento, energia e entropia. A identidade imediata entre as duas grandezas sugere que a questiio da taxa de dissipac;:iio de energia deva ser observada a partir do ponte de vista da taxa de gerac;:ao de entropia.

( (

Uma Formulayao que Usa a Analogia com Equayoes de Entropia

( ( (

( (

( ( ( ( ( (

( (

0 uso de analogias com formulac;oes de Entropia na area de Fenomenos de Transporte, vinculados ou niio a escoamentos turbulentos, e bastante marginal, havendo poucos textos especificos, como Bejan (1982). Como a entropia tern as suas raizes hist6ricas no desenvolvimento da termodinfunica, as equac;:oes de radiac;:ao termica, que envolvem freqilentemente a entropia, passam desapercebidas pelo estudioso em mecanica dos fluidos. Entretanto ha urn potencial de analogia que deve ser explorado, tendo em vista a gama de resultados positives que a area de radiac;:ao termica gerou e tern gerado ao Iongo de sua existencia. Urn exemplo de analogia com equacionamento de entropia vinculado a radiac;ao termica e detalhadamente discutido em Schulz (1997), de onde foi adaptada a tabela 1, utilizando a simbologia do presente texto. Este equacionamento foi estudado no ambito do tratamento conjunto de 103

( resultados esparsos em turbulencia. Esses resultados esparsos foram coletados niio da aplicar;:iio numerica da teoria estatistica da turbulencia, mas dos estudos fenomenol6gicos da area, ou seja, daqueles estudos que originaram solur;:oes independentes a partir de pontes de vista independentes e fundamentados na experimentar;:li.o. A proposta basica e oferecer uma formular;:ao que contenha uma raiz comum para os diferentes resultados obtidos. A busca dessa raiz comum e motivada pela crenr;:a de que fenomenos decorrentes da ar;:ao turbulenta dos fluidos possam ser descritos por' uma "mecanica da turbulencia", conforme ja sugerido em Mdnin e Yaglom (1979, 1981). Os resultados explorados nesta formular;:iio sao as equar;:oes de velocidade de escoamentos (Chezy e equar;:ilo universal de perda de carga), equar;:oes de transferencia de grandezas escalares (coeficiente de transferencia proporcional a taxa de dissipayao de energia elevada ao expoente '14), a definir;:ao da taxa de dissipar;:ao de energia como proporcional a razao entre a velocidade caracteristica ao cuba e a escala do turbilhiio, a definir;:ao do numero de Reynolds com as grandezas usuais de turbulencia, as relar;:Oes"para escalas de velocidade e comprimento, os resultados classicos de escoamentos ajusante de grelhas e a lei dos 5/3 de de Kolmogoroff Note-se que esses sao resultados bern conhecidos e comprovados, porem possuem diferentes formas de abordagem para a sua justificativa te6rica. Alguns se fundamentam no uso de analises dimensionais convenientemente conduzidas, porem sem relar;:oes aparentes entre elas. Uma caracterisitica adicional desta formular;:li.o eque a mesma se ocupa com resultados vinculados a grandezas "macrosc6picas" do escoamento, como o volume de urn turbilhao, a sua velocidade caracteristica, a taxa de dissipar;:iio de energia e coeficientes de transferencia. A discussli.o acerca da aplicar;:iio desta formular;:ao direciona-a para a tubulencia isotr6pica. Detalhes referentes a estrutura da turbulencia (distribuir;:5es espectrais, por exemplo) nilo aparecem como variaveis do equacionamento apresentado. Embora a tabela 1 mostre uma forte identidade entre a formular;:ao aqui discutida e a equar;:ao para radiar;:ao termica, esta identidade e casual. Ela foi verificada em urn estagio mais avanr;:ado do desenvolvimento da formular;:ao, tendo servido como apoio para indicar o valor do expoente da grandeza x (Schulz, 1997).

( ( (

( (

( ( (

(

( (

(

(

('

(

Tabela 1: Semelhanr;:a formal entre os equacionamentos de Radiar;:ao Termica e de Turbulencia Equacionamento de Radia~iio Termica 4u Vol dS = -dVol + -du 3T T

(

(

Equacionamento Alternativo de Turbulencia 4u Vol dr/J= -dVol+-du 3x x

( (

"(

u

i li

(

T

X

u=a T 4

li=a x 4

s

(

I

( Na primeira coluna tem-se que S e a entropia, u e a densidade de energia e T e a temperatura absoluta. Na segunda coluna r/J e uma forma de representar o numero de Reynolds do turbilhilo, iJ e densidade de potencia e X e uma variavel vinculada a transferencia de propriedades fisicas. Nos trabalhos iniciais de Schulz (1990), x representou o coeficiente de transferencia de massa. Verificouse, ainda, que para situar;:5es nas quais r/J e constante (iso-1/J), x representa a escala de velocidade turbulenta. Vale lembrar que a densidade de potencia e a taxa de dissipar;:iio de energia por unidade de massa estao simplesmente relacionadas atraves da massa especifica. Assim:

( ( ( (

(

li e=P

(14)

( (

104

(

(

t ( ( Em outras palavras, as relayoes obtidas envolvendo a densidade de potencia podem igualmente ser utilizadas com a taxa de dissipayao de energia. A identidade verificada na tabela 1 ainda pode ser mais valorizada, considerando as seguintes equayoes:

( (

(

Para a entropia:

( (

dS= dQ T

(

Para o equacionamento alternative:

( 15)

(

JV'dVol)

0 d(ln

(

d¢ =

(

( ( ( ( ( (

( ( ( (

i( ( ( ( ( ( ( ( ( (

X

( (

= u Vol

(16)

Q e a quantidade de calor transferida, 0 e a potencia total dissipada em urn volume de fluido Vol e V e a velocidade caracteristica neste volume. As equayoes 15 e 16 mostram formas semelhantes e relacionam S e ¢com outras grandezas expressas na forma diferencial. Do equacionamento alternative, ve-se que a variayao da grandeza ¢ esta relacionada com a taxa de dissipayao de energia e, conseqiientemente, com a taxa de gerayao de entropia. Porem uma relayao direta entre as duas variaveis ainda nao esta fornecida. A formulayao permite, todavia, estabelecer esta relayao, de modo bastante simples. Alguns resultados parciais, entretanto, necessitam ser demonstrados. Nos procedimentos que seguem, De a escala dos turbilhoes. 1 - Coeficiente de transferencia de grandeza escalar: Partindo da equayao proposta na tabela 1, do fato·conhecido de que x = x(u), e reconhecendo a regra da cadeia ( expressa par vezes como derivar em relayao a uma variavel enquanto a outra e mantida constante), obtem-se:



4u!

oVol = 3x ,, (17)

;~ = v:l~ Efetuando a derivada segunda com a variavel faltante:

o2r/J 4 oVolou = 3x

4ti dx

-37 dti (18)

o2r/J olio Vol

X

Lembrando da igualdade de Schwarz (derivadas segundas iguais) e rearranjando, tem-se:

dx X

(

0

Vol

1 dli

4

(19)

1i

105

(

( (

A integrayiio produz: X=

a 1 Ji 114

x

ou

=al' &

( (20)

114

a 1 e a 1 ' sao constantes de integrayiio (possuem dimensoes). Assim, obtem-se o resulta~o de que 0 coeficiente de tran.sferencia de grandezas escalares e proporcional a taxa de dissipayiib de energia elevada ao expoente '!..Para situayoes iso-¢, Schulz (1997) mostra que x representa a escala de velocidade caracteristica do turbilhao, valendo para esta escala a mesma relaviio de dependencia mostrada nas equayoes 20.

( (

(

( (

2 - Ntimero de Reynolds associado ao turbilhiio

( Igualando a equayiio da tabela 1 e a equaviio 16, tem-se:

=~dVo/ + du

d(lnfVJdVo/1 vo1

(

3 Vol

)

(21)

u

(

Exprimindo o diferencial do logaritmo de forma extensa e introduzindo a viscosidade cinematica (constante):

j

Y

3

4dVo/

f V d;ol) - 3 Vol (

Vol

du

----+-

3

(22)

u

(

( (

V

(

Integrando, para velocidade caracteristica constante no turbilhiio:

Vl_Vol--al Vo/4/J zi

( (23)

vl

( (

Reconhecendo, em turbulencia isotr6pica, a proporcionalidade entre o volume do turbilhiio e sua escala de comprimento (D) elevada ao cuba, e que o primeiro membra da equayiio 23 representa uma potencia do numero de Reynolds, tem-se: Re =a 3 D

(

(

3

fV dVol)

a\_vo1

(

4 3 '

u

113

ou

Re = a /

n•'l & Ill

( (

( (24)

(

a1, aJ e a3 • sao constantes (com dimensao e que podem canter a viscosidade cinematica) e Re e o numero de Reynolds do turbilhiio. A equayiio 24 e utilizada em diferentes abordagens, na literatura (ver Monin e Yaglom, 1979, por exemplo), sempre fundamentada em argumentos dimensionais. Na presente abordagem esta relaviio surge de operayoes elementares efetuadas com a equacionamento alternativo apresentado. 3- Rela~iio entre ¢eo numero de Reynolds associado ao turbilhiio Utilizando a equayiio 16 e seguindo uma argumentayao semelhante aquela conduzida para obter uma expressao para a numero de Reynolds do turbilhiio, tem-se:

(

( (

(

( ( (

106

( (

' (

( (

dr/> ~ Q_ dRe

3

(25)

x Rel

( (

Utilizando a definic;:ao de tem-se:

0

apresentada na equac;:ao 16 e o resultado geral da equac;:ao 20,

(

( (

(

(

dt/J =a

4

ul'•D 3 dRe

(26)

Re

Considerando, agora, o resultado geral da equac;:ao 24, resulta:

dt/J

= a 5 Re

514

d Re

(27)

(

A integrac;:ao desta equac;:ao produz:

(

tP = a

(

f (

s' Re9'4 +as''

(28)

e

a4, as, as' e as'' sao constantes (com dimens5es). Ve-se que ¢ uma forma de representac;:iio do numero de Reynolds do turbilhao. Para grandes numeros de Reynolds, a equac;:iio 28 e simplificada para:

(

t/J =a /Re9'•

(

4- Rela,:lio entre ¢ e a taxa de gera,:lio de entropia

(

(29)

Os resultados intermedh'lrios apresentados anteriormente sao agora utilizados para exprimir a

(

relac;:iio entre ¢ e

(

Do teorema de Gouy-Stodola, expresso na equac;:iio 13, e da definic;:iio de equac;:iio 16, define-sea taxa de gerac;:ao de entropia por unidade de volume, sa,:

Sa" .

0

apresentada na

( ( (

(

it= To

sa..

Assim, todos os resultados envolvendo a densidade de potencia podem ser expresses como func;:iio da taxa de gerac;:iio de entropia por unidade de volume. Das equac;:oes 24 e 30 tem-se:

( ( (

(' (

( ( (

( (

(30)

1

Sa, =(a

Re 3

(31)

~7;,) 1)4

Este resultado mostra que a taxa de gerac;:iio de entropia cresce em aumentando o numero de Reynolds do turbilhao. 0 numero de Reynolds e a taxa de gerac;:ao de entropia, portanto, tern uma forte identidade entre si, havendo ainda, na relac;:iio entre as duas grandezas, urn fator multiplicativo encvolvendo uma potencia do volume considerado. Unindo a equac;:iio 29 (para grandes numeros de Reynolds) e a equac;:ao 31 resulta:

s aor

314 -

-(a~

1

a •;13 7;, r/4

(

¢)

(32)

Dl 107

( ( Reconhecendo que o volume do turbilhiio finalmente:

sa./14

=a

e proporcional a sua

escala ao cubo, resulta,

( (

6!

\

{33)

(

e

a1 e uma constante j representa a a grandeza tP por unidade de volume. Obteve-se, portanto, uma lei de potencia simples entre as duas variaveis, podendo-se dizer, portanto, que j (ou tP /Vol) e uma forma conveniente de representar a taxa de gera~iio de entropia por unidade de volume (na situa~iio de grandes numeros de Reynolds). Assim, demonstra-se que o equacionamento alternative de turbulencia apresentado na tabela I envolve, como grandeza relevante, a taxa de gera~ao de entropia. A discussiio conduzida por Schulz (1997) mostra a busca deste vinculo e apresenta sugestoes que apontam para a necessidade de maiores estudos, visando verificar as reais possibilidades desta opr;:iio de abordagem do problema da turbulencia e localizar as suas limita~oes. 0 campo de trabalho, todavia, e aberto, uma vez que as equa~oes que govemam os movimentos turbulentos continuam nao-fechadas. No item seguinte e apresentada uma aplica~ao para a obten~ao da equa~iio de Chezy, que mostra a forma direta como alguns resultados estao agregados a esta formula~iio.

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( ( ( (

Equac;ao de Chezy a partir da Formulac;ao Alternativa Apresentada

(

Em termos de equacionamento de escoamentos turbulentos, pode-se dizer que qualquer expressiio empirica para avaliar a vaziio desses escoamentos e uma equar;:ao para turbulencia. 0 problema pratico desta avalia~ao com certeza acompanha o homem desde os seus prim6rdios. Talvez, na hist6ria das civiliza~oes, o exemplo mais ilustrativo seja os sistemas hidniulicos elaborados pelos engenheiros romanos. Aquedutos extensos foram construidos para suprir cidades, com suas termas e fontes. Alguns desses aquedutos representam solur;:oes arquitetonicas e de engenharia altamente criativas, considerando os recursos da epoca. Muito provavelmente nao se empreendia uma constur~ao desse porte sem haver alguma avaliar;:ao, pelo menos empirica, da vazao a ser transportada para suprir as necessidades previstas. A Jiteratura, entretanto, niio faz menr;:ao de equa~oes que tivessem sido utilizadas por esses engenheiros. Talvez o primeiro equacionamento empirico que produziu alguma repercussao tenha sido a equar;:ao de Chezy, proposta em 1775 (citado em Netto, 1977), que relaciona velocidade media do escoamento, o raio hidraulico da ser;:ao trasnversal e a declividade da linha de energia (em escoamentos livres., adotada como sendo a declividade do canal). Nota-se que a equa~ao de Chezy e anterior a qualquer questionamento acerca das equa~oes de Navier-Stokes ou acerca de turbulencia. Entretanto, a sua simplicidade, associada ao fato de envolver variaveis de facil medida (sem a introdur;:ao de conceitos fisicos novas) garantiram o seu sucesso. No caso da determina~iio da velocidade media em urn duto ou canal, convem utilizar uma escala de comprimento associada a se~ao transversal do mesmo. No presente estudo considera-se o dillmetro equivalente da ser;:iio considerada. Na figura 1 mostra-se urn esquema desta situar;:ao de estudo. 0 volume na figura 1 e dado pela expressao:

(

A=;rDl 4

(

Vol= A.D

{34)

Assim verfica-se que se mantem a proporcionalidade entre o volume e a escala de comprimento ao cubo. Partindo da equar;:ao apresentada na tabela 1 e da equar;:ao 16 tem-se:

( ( ( (

( (

( (

( ( (

(

( ( ( (

108

(

(

' (

( (

( ( (

Vol= A.D

( ( (

( (

(

A=;rD2 4

( (

( (

Figura 1: Volume considerado para a analise efetuada. A escala de comprimento D representa o diametro equivalente da seylio transversal

( (

J

d(ln V 3 d(AD)J

=~dAD+ du 3 AD

AD

( (

(35)

u

Como se procura uma relaylio para a velocidade media no volume considerado (portanto constante neste volume), tem-se, ap6s a integraylio da equaylio 35:

(

( (

(

( ( ( ( (

( (

V3 AD= a 7 (AD)

413

u

(36)

a 1 e uma constante de integraylio. A equaylio 3 permite obter a segunda expresslio que envolve a potencia dissipada no escoamento, necessaria para a resoluylio do problema, ou seja: (37)

U=rQh,

Q ea vaziio e hp e a perda de carga no escoamento. A unilio das equayi5es 36 e 37, lembrando que Q=VA, produz:

V =(a 7)"2 Jl/2 (AD) 1/6

J

=h, I D

J ea declividade da linha de energia. Considerando dutos de seyliO circular, tem-se: 116 V={a7)112; ( )

J112Dl'2

(39)

(

( (

(38)

109

t (

( Ou, representando a equa9iio 39 de forma mais simplificada, resulta:

( (

V=

a,p75

(40)

( (

0 diametro utilizado representa tambem o diametro hidniulico, uma vez que a constante multiplicativa necessaria pode ser inserida sem problemas na constante de proporcionalidade fX8. A equa9iio 40 e a equa9iio de Chezy, que. surge, nesta an{llise, como decorrente da aplica9iio da formula9iio alternativa para turbulencia. E interessante observar que a equa9iio de Chezy aparece aqui como resultado de uma abordagem que visa tratar o problema especifico da turbulencia, no sentido de unificar diferentes resultados. Entretanto, em 1775 este problema niio estava colocado. As equa9oes consideradas como aquelas que governam os fenomenos ligados a turbulencia niio eram conhecidas e talvez apenas homens como Leonardo da Vinci pudessem ter observado e levantado questoes associadas ao movimento turbulento da agua (recordando 0 seu celebre desenho dos turbilhoes ajusante do despejo provindo de urn duto de agua). Boussinesq e Reynolds passariam a se interessar pela questiio apenas no seculo seguinte. A motiva9iio de Chezy muito provavelmente foi aplicativa, mas as variaveis envolvidas e os expoentes entiio sugeridos mostram uma sensibilidade aguda na observa9iio e explica9iio da realidade. Embora sem procurar vincular o resultado final obtido com a motivayiio original de Chezy, pode-se dizer que o seu equacionamento constitui de fato urn primeiro modelo para escoamentos turbulentos, · quando apenas as grandezas medias sao consideradas.

( ( ( (

(

(

(

( (

Conclusoes

(

Neste trabalho desenvolveu-se, inicialmente, uma discussiio visando mostrar que os esforyos para buscar uma equa9iio adicional para a turbulencia conduzem ao conceito de taxa de gerayiio de entropia. Esta conclusiio se baseia na concentraviio de resultados envolvendo a taxa de dissipayiio de energia e no teorema de Gouy-Stodola. Urn desenvolvimento resumido do teorema de Gouy-Stodola foi inserido no texto, visando demonstrar a sua simplicidade e a conveniencia de se falar em entropia no equacionamento de turbulencia. Embora niio se tenha tentado fornecer uma equa9iio de fechamento para o problema de turbulencia, conseguiu-se demonstrar que a entropia e o conceito evocado quando se utilizam equa9oes empiricas e semi-empiricas para a taxa de dissipa9iio de energia. Vale lembrar que problemas complexos que concentram a sua resolu9iio no uso de uma variavel, como e o caso dos escoamentos turbulentos e da taxa de dissipa9iio de energia, podem conduzir a situa9iio de se utilizar inadvertidamente o mesmo principia de forma multipla, ora rigorosamente, ora de forma simplificada (o que pode ocorrer atraves das informayoes empiricas). Isto pode gerar urn desafio conceitual e confusoes sobre a validade das aproximayoes feitas. A inser9iio da taxa de gerayiio de entropia, com sua identidade imediata com a taxa de dissipa9iio de energia, deve perrnitir redirecionar as discussoes e oferecer urn campo mais amplo para a apresentayiio de novas aproximayaes para o problema de turbulencia. Adicionalmente a discussiio acima mencionada, foi apresentada uma formulaviio alternativa para turbulencia (extraida de Schulz, 1997), cuja caracteristica mais marcante e a forte semelhanva existente com a equayiio de entropia para radia9iio terrnica. Esta semelhan9a induziu abusca de uma identidade entre as grandezas utilizadas nesta formulaviio e a entropia, mostrando-se que ha uma lei de potencia relacionando a grandeza rp da formulaviio mencionada e a taxa de geraviio de entropia, ambas por unidade de volume. A utilidade deste equacionamento alternative e demonstrada a partir de urn exemplo no qual se obtem a equayiio de Chezy para escoamentos em dutos ou canais. A 110

( ( (

( ( (

( (

(

(

( ( (

( (

(

'

\ ( ( ( (

generalidade da validade deste tipo de formulayao, ou as suas restriy5es de aplicavao, nao foram discutidas, uma vez que o objetivo do trabalho foi o de valorizar o ponto de vista da utilizavao da taxa de geravao de entropia nos problemas de turbulencia.

( ( ( (

Agradecimentos 0 au tor agradece aFAPESP, pelo apoio obtido atraves do processo 1997/11743-0 para execu9iio de . pesquisa no exterior, na qual o presente trabalho se insere, e ao Prof. Gerhard Jirka, anfitriiio no Institut fur Hydromechanik, Universitat Karlsruhe, Alemanha.

( (

Referencias Bibliograficas

(

Bejan, A, 1982, "Entropy Generation Through Heat and Fluid Flow", A Willey-Interscience Publication, John Willey & Sons, New York.

( (

Monin, AS. e Yaglom, A.M., 1979, "Statistical Fluid Mechanics-Mechanics of Turbulence", Vol t, the MIT Press, Massachussets.

( ( (

(

:c I

(

Monin, A.S. e Yaglom, A.M., 1981, "Statistical Fluid Mechanics-Mechanics of Turbulence", Vol 2, the MIT Press, Massachussets. Netto, J.M.A.,1977, "Manual de Hidniulica", 3a. reimpressiio da 6a. edi9iio, Volumes 1 e 2, Editora Edgard Blucher Ltda, Sao Paulo. Schulz, H.E., 1990, "Investiga9ao do Mecanismo de Reoxigenaviio da Agua em Escoamento e sua Correlaviio com cr Nivel de Turbulencia Junto a Superficie", Tese apresentada a Escola de Engenharia de Sao Carlos, Universidade de Sao Paulo, 896 p.

(

( I

Schulz, H.E., 1997, "Teste de uma Formulaviio Alternativa em Turbulencia", Tese apresentada Escola de Engenharia de Siio Carlos, Universidade de Sao Paulo, 86 p.

I(

( (

( ( ( ( (

( (

(

( (

111

a

---

--

------

/

\ ( ( ( ( (

(

( LAMINAR-TURBULENT TRANSITION: THE NONLINEAR EVOLUTION OF THREE-DIMENSIONAL WAVETRAINS IN A LAMINAR BOUNDARY LAYER 1

( ( (

A. FARACO DE MEDEIROS 2 Departamento de Engenharia Mecanica Escola de Engenhariade Sii.o Carlos - Uni11ersidade de Sii.o Paulo Rua Dr. Carlos Botelho, 1465, Sii.o Carlos, 13560-250 - SP - Brazil MARCELLO

(

( ( ( ( (

( (

( (

Abstract This paper presents results of an experimental study of the transition in boundary layers. The experiments were conducted in a low-turbulence wind tunnel. The process was triggered by a three-dimensional TollmienSchlichting wavetra.in excited by a harmonic point source in the plate. Hot-wire anemometry was used to measure the signal and investigate the nonlinear regime of these waves. It was observed that the three-dimensional wavetrain behaved very differently from two-dimensional ones. In particular, it did not involve the growth of subharmonics or higher harmonics. The first nonlinear signal to appear was a mean flow distortion. This had a spanwise structure consisting of regions of positive and negative mean distortion distributed like streaks, which became more complex as the nonlinearity developed. Elsewhere studies have revealed the existence of streak-structures in turbulent flow. It is conjectured that the current experiments may provide a link between early wave-like instabilities and some coherent structures of turbulent boundary layers.

( (

1

Introduction

( ( ( ( (

( (

( ( (

( (

Laminar-turbulent transition in boundary layers is a subject in fluid mechanics that has gained increasing interest in recent years. It is known that the process usually involves waves of small amplitude, the so called Tollmien-Schlichting (TS) waves, which amplify as they travel downstream. These waves are excited by disturbances in the flow such as wall vibrations, acoustic waves, free-stream turbulence or wall roughness. When the waves reach some finite size they cause the breakdown of the laminar flow structure creating a turbulent flow. For small enough amplitudes the evolution of these waves can be described by a linear version of the Navier-Stokes equation, the Orr-Sommerfeld equation (Lin 1955). However, experiments show that prior to transition the TS waves behave nonlinearly (Klebanoff, Tidstrom & Sargent 1962). Moreover, the amplification rates are often considerably larger in the nonlinear regime and therefore the transition point is ultimately determined by this stage. However, scientists 1 This paper has been published in the proceedings of the 14'h Brazilian Congress of Mechanical Engineering, in CD-Rom 2 Current Address: Departamento de Engenharia Meciinica- Pontiffcia Universidade Cat6lica de Minas Gerais - Av. Dom Jose Gaspar, 500, Belo Horizonte, 30535-610- MG- Brazil. E-mail:marcello@mea.pucminas.br

113

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( have not been able to explain entirely the nonlinear regime of the TS waves, and this remains a very active field of research. Most of the research on nonlinear TS waves has concentrated on the evolution of plane twodimensional wavetrains. This effort has been able to established that when a two-dimensional TS wave reaches some threshold amplitude a secondary instability sets is causing strong amplification of three-dimensional modes (Kachanov 1987, Herbert 1988, Corke & Mangano 1~89). The mechanism is of a parametric resonant nature and saturates in the form of a staggered pattern of >. vortices tliat, for some yet unclear reason, breakdown into turbulence. If the primary TS wave is very large, the process involves the generation of harmonics producing an aligned arrangement of>. vortices (Kachanov 1994). The situation of more practical interest, however, involves highly three-dimensional modulated waves. Moreover, experiments have shown that these waves cause transition in a way that is remarkably different from that of two-dimensional regular wavetrains, and often the appearance of turbulence spots is observed (Shaikh 1997). As an example of this more generic type of waves, three-dimensional wavepackets have been studied a number of times (Gaster & Grant 1975, Gaster 1975, Cohen, Breuer & Haritonidis 1991, Konzelmann 1990, Medeiros & Gaster 1995, Medeiros 1996, Medeiros & Gaster 1997, Medeiros & Gaster 1998). These studies however have not been able to explain the complicated nonlinear behaviour observed. Research is now concentrating on a simpler three-dimensional wave, namely the three-dimensional wavetrain (Mack 1985, Kachanov 1985, Seifert 1990, Seifert & Wygnanski 1991, Wiegand, Bestek, Wagner & Fasel 1995) in the hope that this could bring some insight into the more complex cases. These works were mainly concentrated on the linear evolution. The work by Wiegand included the nonlinear regime, but the investigation was restricted to flow visualization. The objective of our current work is to investigate the nonlinear regime of these waves in a more detailed and quantitative way. This paper presents some preliminary findings.

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Experimental set-up

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Experiments on transition in boundary layers are usually carried out in wind tunnels in which the levels of disturbances of the flow are kept to a minimum. This in turn excites very small amplitude TS waves which take a long downstream distance to reach the amplitudes that cause breakdown to turbulence. It is then possible to disturb the flow with some controlled wavemaker and excite TS waves artificially. If the artificially excited waves are substantially larger than the naturally arising ones, the transition process can be controlled and repeated to a considerable degree even at the highly nonlinear stages. This provides a fundamental tool for research in this subject. The experiments here presented were carried out in a 0.9m x 0.9m low-turbulence wind tunnel in which the free-stream velocity was 16.7mfs. The RMS free-stream turbulence level was of about .008%. The boundary layer studied developed on a 2m long elliptic nosed plate placed vertically at the centre of the tunnel. The plate was slightly inclined to the tunnel wall on the working side in order to compensate for the boundary layer growth so as to ensure a constant pressure in the streamwise direction. Fine adjustment of the pressure gradient was achieved by flaps at the trailing edge of the plate. The artificial excitations were produced by a loudspeaker embedded in the plate and coupled to the flow via a .3mm hole located on the centre line of the plate 203mm from the leading edge. A more detailed description of the set up is given by Medeiros (1996). First the loudspeaker was driven continuously by a 200Hz sine wave and the disturbance flow field was studied. The streamwise velocity fluctuations were detected by a constant temperature hot-wire anemometer. The anti-aliases filter was adjusted at 600Hz. Measurements were taken at a constant nondimensional distance from the wall, namely yfo* = .58 (o* = displacement thickness), which is close to the inner peak of the 114

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streamwise velocity eigenfunction. Figure 2 shows velocity records taken along the centreline of the How at different distances from the leading edge of the plate. These results can best be interpreted with the help of figure 1. This figure shows the stability diagram of the flat plate boundary layer obtained by solving the Orr-Sommerfeld equation. It indicates in a Reynolds number(R)xnondimensional frequency ((J = 27rflj*/Uoo) plane the region of flow instability. The curve indicates the neutrally unstable waves and separate the unstable region (the inner part) from the stable region. The picture also displays, inside the loop, curves of constant amplification rates. Waves traveling downstream in the boundary layer follow straight lines that irradiate from the origin of the coordinate system. The line displayed in figure 1 corresponds to the wave in figure 2. It is observed that the first measuring station (R = 1312) is located within the unstable region. This is confirmed by the observation that the wave amplifies from station x=500 to x=600mm. From stations x=800 to 900mm the signal starts to decays. That corresponds to the region where the waves cross the upper part of the neutral curve and return to the stable region.

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The nonlinear three-dimensional wavetrain

A more detailed study of the linear evolution the wavetrain was carried out, but this paper focuses on the more interesting nonlinear regime. Previous studies have used a continuous wavetrain to excite the flow. Here the excitation used to study the nonlinear regime was a finite wavetrain. However, the finite wavetrain was made long enough so as to behave like a continuous one. Also, care was taken that the ends of the wavetrain were very smooth in order that the modulation did not affect the results. In this way, the hot-wire records obtained were composed of two parts, a disturbed part and an undisturbed one, figure 3. The background noise in the experiment was very low, but the random part of the signal was further reduced by ensemble averaging a set of 64 records generated by identical excitations. Analysis of the signals revealed no sign of subharmonic or higher harmonic. In fact, the first sign of nonlinearity was clearly a mean flow distortion also indicated in the picture. A somewhat surprising behaviour was that the mean flow distortion changed from negative to positive somewhere along the evolution. Under this experimental conditions, the positive mean How distortion occurred after the disturbances had crossed the second branch of the instability loop. Although the fundamental disturbances were decaying in this region, the positive mean flow distortion was larger then the negative one. To shed more light into the phenomenon, 115

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the experiments were extended to include measurements off the centreline of the flow, in order to produce a three-dimensional view of the disturbance field. Figure 4 shows contour plots of the wave field as it passes a downstream station, namely x=llOOmm. It is important to note that this view was constructed from 41 spanwise equally spaced time series measured by hot-wire anemometer and it should not be taken as a snap-shot of the flow at a particular time. Figure 4 shows some streak structures which developed nonlinearly in the flow. A clearer view of streak structures is obtained by filtering out the oscillating part of the signal, figure 5. Figure 6 shows the streamwise evolution of the nonlinearly generated mean flow distortion. At x=.7m the mean flow distortion is already apparent. It is observed that the distortion is 116

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(; not negative everywhere, but has some spanwise structure. At x=.8m the distortion is larger, but remains similar in structure. At x=.9m a more complex structure is forming. It is very difficult to obtain spanwise symmetry in experiments at so large streamwise distances, and this affects the interpretation of the results. However, the picture conveys the idea that the positive lumps of mean distortion on the edge of the disturbance field are splitting. At x=l.Om the central lump of negative distortion is split in two by a localized positive mean distortion. It is remarkable that th.e phenomenon occurs exactly at the centreline of the flow, although the disturbance field appears to be asymmetrical. After x=l.Om the spanwise distribution of the streak structures remains essentially the same with a widening of the positive distortion generated at the centre of the wave field. From figure 6 it appeared that the spanwise wavelength at station x=1.3m is approximately half of that at x=. 7m. To try and gain more insight into the phenomenon the signals were also studied in Fourier space. Two dimensional discrete Fourier transforms of the signals were taken mapping the spectra onto a nondimensional frequency (F = 104 .8/R)xspanwise wavenumber (a,)plane, figure 7. In the figure the frequency coordinate was stretched and only the frequencies close to zero were shown. Initially the mean flow distortion appears as a double peak of symmetrical spanwise wavenumbers. As the nonlinearity developed the spanwise distribution became more complex and more peaks appear in the spectra. The change of sign of the mean flow distortion on the centreline at x=l.Om manifests itself in the appearance of spanwise modes of even higher wavenumbers.

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Discussion and conclusions

The current experiments have shown that the nonlinear evolution of three-dimensional wavetrains does not seem to be linked with the appearance of sub or higher harmonics as occurs with two-dimensional ones. In fact, the first nonlinear signal to appear was a mean flow distortion. The experiment was not designed for a detailed investigation of the origin of this distortion, but some conjectures can be made. It is possible that the nonlinearity arises from the Reynolds stresses terms (u'v') that are neglected in the linear approximation. It is known that this terms can produce both harmonics and mean flow distortion (Stuart 1960). In a three-dimensional case this mechanism would produce both spanwise and time harmonics, but it is possible that the time harmonics have been highly dumped because of their high frequency and only the mean distortion remained. The appearance of the higher spanwise wavenumber number modes might be connect with a secondary Reynolds stress interaction which wtmld produce even higher harmonics. A similar mechanism has been found in the so called oblique transition (Elofsson & Alfredsson 1997). Our experiment, however, did not have enough spanwise resolution for a definitive conclusion. In any case, the results appear to provide a link between the early wave-like instability and streak structures which seem to be a key ingredient of boundary layer transition (Monkewitz 1997).

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References

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Cohen, J., Breuer, K. S. & Haritonidis, J. H. (1991), 'On the evolution of a wave packet in a laminar boundary layer', J. Fluid Mech. 225, 575-606.

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Corke, T. C. & Mangano, R. A. (1989), 'Resonant growth of three-dimensional modes in transitioning Blasius boundary layers', J. Fluid Mech. 209, 93-150. Elofsson, P. A. & Alfredsson, P. H. (1997), 'An experimental study of oblique transition in plane poiseuille flow'. (submitted to the J. Fluid Mech.). 118

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X=0.7m T0 =.0595s

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X=0.6m T0 =.0446s

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X=0.5m T0 =.0298s

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X=0.4m T 0 =.0149s

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-2 -1 -.5

.5 l

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Figure 6: Evolution of the mean fiow distortion. Gaster, M. (1975), 'A theoretical model of a wave packet in the boundary layer on a flat plate', Proc. R. Soc. London A 347, 271-289. Gaster, M. & Grant, I. (1975), 'An experimental investigation of the formation and development of a wavepacket in a laminar boundary layer', Proc. Royal Soc. of London A 347, 253-269. 119

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( Figure 7: Evolution of the mean flow distortion in Fourier space. Herbert, T. (1988), 'Secondary instability of boundary layers', Ann. Rev. Fluid Mech. 20, 487526.

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Kachanov, Y. S. (1985), Development of spatial wave packets in boundary layer, in V. V. Kozlov, ed., 'Laminar-turbulent transition', springer-Verlag, pp. 115-123. 120

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Kachanov, Y. S. (1987), 'On the resonant nature of the breakdown of a laminar boundary layer', .!. Fluid Mech. 184, 43-74. Kachanov, Y. S. (1994), 'Physical mechanisms of laminar boundary layer transition', Ann. Rev. Fluid Mech. 26, 411-482. Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. (1962), 'The three-dimensional nature of boundary layer instability', J. Fluid Mech. 12, l-34. Konzelmann, U. (1990), Numerische Untersuchungen zur rii.umlichen Entwicklung dreidimensionaler Wellenpakete in einer Plattengrenzschichtstromung, PhD thesis, Universitii.t Stuttgart. Lin, C. C. (1955), The Theory of Hydrodynamic stability, Cambridge University Press. Mack, L. M. (1985), Instability wave patterns from harmonic point sources and line sources in laminar boundary layers, in V. V. Kozlov, ed., 'Laminar-turbulent transition', springerVerlag, pp. 125-132. Medeiros, M. A. F. (1996), The nonlinear behaviour of modulated Tollmien-Schlichting waves, PhD thesis, Cambridge University - UK. Medeiros, M. A. F. & Gaster, M. (1995), The nonlinear behaviour of modulated TollmienSchlichting waves, in 'IUTAM Conference on nonlinear instability and transition in tridimensional boundary layers', Manchester, pp. 197-206. Medeiros, M. A. F. & Gaster, M. (1997), 'The nonlinear evolution of wavepackets in a laminar boundary layers: Part I'. (submitted to the J. Fluid Mech.). Medeiros, M. A. F. & Gaster, M. (1998), 'The nonlinear evolution of wavepackets in a laminar boundary layers: Part II', J. Fluid Mech. (to be published).

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Monkewitz, P. (1997), personal communication.

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Seifert, A. (1990), On the interaction of small amplitude disturbances emanating from discrete points in a Blasius boundary layer, PhD thesis, Tel-Aviv University.

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Seifert, A. & Wygnanski, I. (1991), On the interaction of wave trains emanating from point sources in a Blasius boundary layer, in 'Proc. Conf. on Boundary Layer Transition and Control', The Royat' Aeronautical Society, Cambridge, pp. 7.1-7.13. Shaikh, F. N. (1997), 'Investigation of transition to turbulence using white noise excitation and local analysis techniques', J. Fluid Mech. 348, 29-83. Stuart, J. T. (1960), 'On the nonlinear mechanisms of wave disturbances in stable and unstable · parallel flows', J. Fluid Mech. 9, 1-21. Wiegand, T., Bestek, H., Wagner, S. & Fasel, H. (1995), Experiments on a wave train emanating from a point source in a laminar boundary layer, in '26th AIAA Fluid Dynamics Conference', San Diego, CA.

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121

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Investiga~ao

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Transi~ao

de Escoamentos num

Sistema Pulverizador Jato-Placa

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Experimental da

Marcelo

Ba~ci

da Silva, Leonardo Machado Amorim e Aristeu da Silveira Neto

Departamento de Engenharia Meclinica da Universidade Federal

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de Uberllindia- 38400-206- Uberllindia- M.G.

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Neste artigo apresenta-se os resultados obtidos com uma tecnica de fotografia a alta velocidade, que serve para congelar fenomenos dinfunicos de altas frequencias envolvidos no problema de gerac;:ilo de gotas por meio de urn sistema pulverizador jato-placa. Foram analisadas as instabilidades que caracterizam a transic;:ao do jato e observou-se importantes fenomenos fisicos, como por exemplo as instabilidades dinfunicas e os buracos formados sobre o filme de liquido ap6s a placa aspersora.

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lntrodu~ao

Foi conduzida urna investigac;:ilo experimental dos mecanismos fisicos envolvidos na transic;:ao de escoamento de urn sistema pulverizador jato-placa. 0 principio deste tipo de pulverizador e comurnente usado para sistemas de irrigac;:ilo na agricultura, mas encontra-se tambem aplicac;:ao em outras areas, como na injec;:ilo de combustive! nos equipamentos de combustilo interna e nos sistemas de protec;:ilo contra incendios. A investigac;:ilo experimental contribui para a compreensilo dos mecanismos de transferencia de massa, momento e energia em escoamentos com duas fases dispersas. 0 sistema montado para a investigac;:ao experimental e composto de urn bico injetor que projeta urn jato de agua contra urna placa conica. 0 jato e concentrico com a placa, como ilustra a figural. Urn filme de liquido se forma ap6s a placa. A espessura deste filme diminui devido a abertura radial. As forc;:as de tensoes viscosas atuantes provocam uma diminuic;:ilo da quantidade de movimento do fluido e ele se expande para fora da placa conica causando uma reduc;:ilo adicional na sua espessura. 0 filme escoa na direc;:ao radial para fora da placa, e se quebra em gotas. Para urn alto nfunero de Reynolds, a interac;:ao do filme com o ar gera ondas tridimensionais de grande amplitude, e com o aurnento dessas amplitudes, perfurac;:Bes surgem ao Iongo do filme de liquido.

Trabalhos Preliminares Urn numero de trabalhos importantes tern sido direcionado para este problema. 0 interesse primario de tais trabalhos e o entendimento dos mecanismos fisicos que aumentam as taxas de transporte de calor e massa envolvendo jatos livres: Stevens e Webb (1992), Buyevich e Ustinov (1994), Elison e Webb (1994) e Mansour e Chigier (1994). Uma revisao bibliografica mais detalhada sobre os mecanismos de formac;:ao de gotas foi desenvolvido por Kolev (1993). Um primeiro modelo para a transformac;:ilo de urn filme lfquido em gotas foi proposto por Dombrowski e Jones (1963). Eles acreditam que as instabilidades que aparecem sobre o filme liquido favorece a acelerac;:ao do processo de formac;:ao das gotas. Urn modelo flsico similar foi proposto por Chin eta/. (1991), usando urn formalismo de entropia maxima. Spielbauer e Aidum (1994) propuseram que a quebra em 123

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Aparato Experimental

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0 equipamento experimental usado para investiga~j:ao dos mecanismos fisicos envolvidos na transi~j:iio do esc.oamen~o ~e urn sistem~ pulverizador jato-?laca e apresenta~o na figura I. A se~j:aO. de testes e constitutda de uma ca1xa coletora a qual e eqmpada com tres janelas de vidro (W). 0 equipamento e composto por I bomba (P), urn reservat6rio de agua (R), tres rotametros (FM) que controlam o fluxo de agua, tubos PVC de 25 mm de diiimetro, urn bico de inje~j:iio e uma placa conica (NP). A pressiio de opera~j:iio foi monitorada por um manometro de Bourdon (M) localizado antes do bico de injer,;iio.

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( Figura t. Esquema da Montagem Experimental. As fotografias foram tiradas usando uma camera Pentax 35 mm e uma lampada estrobosc6pica de alta frequencia. Esta Hlmpada foi control ada por urn circuito eletronico para fomecer urn flash com durar,;iio da ordem de 4 flS (quatro microsegundos). A iluminar,;ao de curta durar,;iio permite congelar os fenomenos de altas frequencias como as instabilidades do jato, ondas, perfurar,;oes no filme liquido e formar,;iio de gotas. Instabilidades sobre o jato e sobre o filme liquido livre assim como perfurar,;oes localizadas e acumulo de massa nas bordas do filme liquido foram tambem claramente evidenciadas. Varios tipos de tecnicas de iluminar,;iio sao possiveis, mas ap6s varios testes, a tecnica de iluminar,;ao "back-lighting" fo.i adotada, como ilustrado na figura I. Foi usado urn filme ISO 400. Para mais detalhes veja Tarqui (1996).

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Resultados As fotografias mostram o comportamento global do processo fisico envolvido na transir,;ao do escoamento sobre o sistema jato-placa. Na figura 2 pode-se visualizar a estrutura turbulenta do jato livre, tendo como parametro de controle o nfunero de Reynolds, aqui definido como Red =Ud I v , onde U e a velocidade media, d e o diiimetro do bico injetor e v e a viscosidade cinematica. Na figura 2(a) mostra-se urn regime laminar na saida do jato e na figura 2(b), continuar,;ao da anterior, mostra-se a quebra do jato, promovida pelas instabilidades de Rayleigh. Pode-se observar dois tipos de gotas: as maiores que aparecem primeiro e que geram filamentos de liquido entre elas, sobre os quais aparecem as 124

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instabilidades de Rayleigh favorecendo o aumento das gotas menores, comumente chamadas gotas satelites. Sob esse baixo numero de Reynolds, a superflcie liquida tern uma aparencia muito suave, sem instabilidades.

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Figura 2. Visualizar;iio das instabilidades de Rayleigh no jato livre de agua; Re£F5.000; (a) saida do bico injetor; (b) sequencia espacial da imagem precedente.

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Figura 3. Visualizar;iio de instabilidades tridimensionais no jato livre de agua; Rect=l20.000; (a) saida do bico injetor; (b) a 40 em do bico injetor (break-up do jato).

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125

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( 0 escoamento relacionado com alto numero de Reynolds, Re£Fl20.000, pode ser visualizado na figura 3. As imagens 3(a) e 3(b), mostram o escoamento numa sequencia espacial. Visualiza-se nesta figura os disrurbios complexos tridimensionais sobre a superficie do jato (imagem (a)). Estes disturbios tornam-se mais pronunciados com o desenvolvimento do escoamento e o jato perde sua regularidade (imagem (b)), mostrando instabilidades peri6dicas muito fortes. Eles iniciam o desprendimento de ligamentos e de pequenas gotas que caracterizam o inicio da quebra. As gotas sao criadas pelo arraste aerodiniimico sobre as instabilidades. Estes resultados qualitativos comparam-se muito bern com aqueles fomecidos por outros autores: Taylor e Hoyt (1981), Eroglu eta/. (1991), Chigier (1991), e Mccreery e Stoots ( 1996)

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Figura 4. Visiio geral do escoamento; visualizaf,:iio das instabilidades senoidais e quebra do filme liquido em gotas; Rect=32.000; diiimetro da placa e 44 mm; diiimetro do bico injetor e 2,8mm.

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f ( ( Figura 5. Visiio superior do filme liquido; quebra em perfuraf,:5es e em gotas; Rect=32.000; diiimetro da placa e 44 mm; diiimetro do bico injetor e2,8 mm.

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A figura 4 mostra uma vista geral do escoamento, onde sao observadas ondas senoidais sobre a napa de fluido as quais sao amplificadas para finalmente degenerarem em gotas. Na figura 5 mostra-se uma vista superior da napa liquida referente a Rea=32.000. 0 cisalhamento do ar sobre a napa cria oscila~oes e varia~oes na espessura da mesma. A a~ao do cizalhamento e de ondas resulta em for~as localizadas de tensoes altas. Quando a espessura local do filamento cai abaixo de uma valor critico, perfura~oes ocorrem, causando rupturas antes da regiao principal de quebra em gotas. A napa toma-se pontuada (perfurada) em buracos isolados. Este fenomeno foi tambem observado por Chigier ( 1991 ), Mccreery e Stoots (1996) e Reise Silveira Neto (1991).

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Figura 6. Visao superior do filme liquido; quebra em perfura~oes e em gotas; Rea=36.000; diametro da placa e 44 mm; diiimetro do bico injetor e 2,8 mm.

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Figura 7. Visao superior do filme liquido; quebra em perfuractoes e em gotas; Rea=SO.OOO; diametro da placa e 44 mm; diametro do bico injetor e 2,8 mm.

Nas figuras 6 e 7 mostra-se o escoamento referente a Rea=36.000 e 50.000 respectivamente. Com o aumento do nfunero de Reynolds, a frequencia dos buracos tambem aumenta. Pode-se observar que o processo de forrnactao das gotas toma-se altamente 127

c ( ( complexo. Entretanto a natureza nao homogenea da regiao frontal da formar;:ao de gotas permanece. Na figura 8 mostra-se uma vista lateral do escoamento referente a Rea=50.000, ilustrando as oscilar;:i'ies de altas amplitudes. Comparando as figuras 4 e 8 verifica-se que essas oscilar;:i'ies aumentam com o numero de Reynolds. Jsto e muito importante para aplicar;:i'ies pniticas do tipo sistemas de irrigar,:ao na agricultura, pais estas oscilar,:i'ies favorecem o aumento na largura da faixa de solo irrigada. Logo, urn aumento no n(Jmero de Reynolds permite maximizar a area irrigada. Altemativamente estas oscilar,:i'ies podem ser amplificadas por excitar,:ao meciinica ou por modificar,:i'ies geometricas.

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Figura 8. Visao lateral do filme liquido, mostrando as oscilar;:oes de grande amplitude; Rea=50.000; diiimetro da placa e 44 mm; diiimetro do bico injetor e 2,8 mm.

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Conclusoes

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A tecnica de fotografia de alta velocidade foi usada para capturar e congelar os fenomenos de altas frequencias que ocorrem na transiyao do escoamento sabre urn sistema pulverizador jato-placa. Com as fotograflas mostra-se a estrutura do escoamento: as instabilidades caracteristicas do jato; as oscilat,:Qes sabre o filme liquido ap6s sair da placa e as perfurar,:oes sabre o filme liquido; o processo de acumulo de massa na borda do filme liquido e nas bordas dos buracos. Tambem visfveis sao a emissao de gotas e ligamentos de fluido no jato livre. Parece que o mecanismo fisico mais importante para a formar;:ao de gotas sao as perfurar,:oes que aparecem sabre o filme Iiquido. Etas sao transportadas convectivamente na direr,:ao radial e determinam a natureza nao homogenea das frentes de gotas. Outro importante fenomeno observado foram as oscilar,:oes no tilme liquido. Quando amplificadas, estas oscilar,:oes causam aumento na frente de gotas. Esta caracteristica e muito importante com relar,:ao ao desempenho dos sistemas de irrigar,:ao porque elas deterrninam a area molhada do solo.

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Agradecimentos- Os autores agradecem ao CNPQ e a Fapemig pelo suporte financeiro e ao curso de P6s-Graduar,:ao da Engenharia Meciinica da Universidade Federal de Uberliindia.

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Referencias

Buyevich, Y. A. and Ustinov, V. A. 1994. Hydrodynamic condition of transfer process through a radial jet spreading over a flat surface. Int. Journal Heat and Mass Transfer, 37, 165-173 ..

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Chigier, N. 1991. Optical imaging of sprays. Pro g. Energy Combust. Sci., 17, 211-262.

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Chin, L. P., Larose, P. G., Tank.in, R. S., Jackson, T., Stumd, J. and Switzer, G. 1991. Droplet distributions from the breakup of a cylindrical liquid jet. Physical of Fluids A, 3, 18971906.

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Dombrovski, N. and Jones, W. R. 1993. The aerodynamic instability and disintegration of viscous liquid sheets. Chern. Eng. Science, 18, 203-214. Elison, B. and Webb, B. W. 1994. Local heat transfer to impinging liquid jets in the initially laminar, transitional and turbulent regimes. Int. Journal Heat and Mass Transfer, 37, 1207-1216. Eroglu, H., Chigier, N. and Fraga, G. 199L.Coaxial atomizer liquid intact lengths. Phisics of Fluids A, 3, 303-308. Kolev, N. I. 1993. Fragmentation and coalescence dynamics in multiphase flows. Experimental Thermal and Fluid Science, 6, 211-251. Mansour, A. and Chigier, N. 1994. Turbulence characteristics of cylindrical liquid jets. Phys. Fluids, 6, 3380-3391.

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Mccreery, G. E. and Stoots, C. M. 1996. Drop formation mechanisms and size distributions for spray plate nozzles. Int. J. Multiphase Flow, 22, 431-452. Reis, W. and Silveira Neto, A. 1993. Comportamento dinfunico de urn jato incidente sabre placas aspersoras-cenario da transh,:ao a turublencia, Proceedings of the XII Brazilian Mech. Eng. Congress, 2. 1033-1036. Spielbauer, T. M. and Aidun, C. K. 1994. The wave thining and breakup of liquid sheets. ASME .Journal of Fluids Engineering, 116, 728-734.

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Stevens, J. and Webb, B. W. 1992. Measurements of the free surface flow structure under an impinging, free liquid jet. Journal of Heat Transfer, 114, 79-84. Taylor, J. J. and Hoyt, J. W. 1981, Water jet photography- techniques and methods. Exp. Fluids, l, 113-120. Tarqui, J. L. Z. 1996. Desenvolvimento de urn metoda de fotografia para o estudo de fenomenos de altas frequencias em aspersores jato-placa, Master of Science Dissertation, Federal University ofUberlandia.

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( ( On Kaplun Limits and the Multi-Layered Asymptotic Structure of the Turbulent Boundary Layer

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( by Atila P. Silva Freire Mechanical Engineering Program (PEM/COPPE/UFRJ), C.P. 68503, 21945-970 - Rio de Janeiro - Brazil.

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Abstract In the present work, some formal propertie.• of singular perturbation equations are studied through the concept of "equivalent in the limit" of Kaplun, so that a proposition for the principal equations is derived. The proposition shows that if there is a principal equation at a point ('I, 1) of the (3 x E) product space, 3 space of all positive continuous functions in (0, 1), E = (0, 1], then there is also a principal equation at a point (T/,•) of (3, xE), first critical order. The converse is also trne. The proposition is of great implication for it ensures that the asymptotic structure of a singular perturbation problem can be determined by a first order analysis of the formal domains of validity. The turbulent boundary layer asymptotic st.ructure is then studied by application of Kaplun limits to three test case..: the zero-pressure boundary layer, the separating boundary layer and the shock-wave interacting boundary layer. As it tun•• out, different asymptotic structure.• are found depending on the test cases considered.

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In physics and mathematics many phenomena are modelled through intricate equations that present no analytical solution. As such, engineers, physicists and applied mathematicians are forced do develop techniques that yield approximated solutions to the problems they are faced with. These techniques often resort to sophisticated procedures which only in a very few cases are restrained to fully analytical frameworks. In most cases, analytical procedures have to be combined with numerical procedures to produce an approximated solution. Perturbation methods have evolved along the past forty years into a powerful tool for solving a large class of complex problems. They have, therefore, become a basic working tool of many engineers and applied mathematicians. In fact, a large number of papers can be found in literature which use perturbation methods as their primary solution procedure. The purpose of this work is twofold: i) to consider more thoroughly some fundamental concepts and ideas used in solving perturbation problems, and ii) to study the turbulent boundary layer asymptotic structure by applying Kaplun limits to the Navier-Stokes equations. While some precise definitions can be enunciated, and exact results obtained to find uniform approximations and to perform the matching of functions, the determination of the domain of validity of an approximation is always difficult. Two important results in perturbation theory are the intermediate matching lemma and the extension theor~m of Kaplun. These results are of fundamental importance for the construction of matched asymptotic expansions, but say nothing about the domain of validity of the approximations. To circumvent this difficulty, Kaplun(1967) applied the concept of limit-processes directly to the equations rather than to the solutions and enunciated an Ansatz, the Ansatz about domains of validity, which relates t,he domain of validity of solutions with the formal domain of validity of equations (a concept which is easily defined). Examples are known where Kaplun's ideas fail; however,

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Introduction

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131

c (1 ( for some difficult problems, e.g., the Stokes paradox of fluid mechanics, only consideration of these ideas can clarify the conceptual structure of the problem. Here, we study some formal properties of equations yielded by the definition of "equivalent in the limit" of Kaplun, and relate them to the actual problems of determining the overlap domain and of matching asymptotic expansions. The concept of "richer than" of Kaplun and Lagerstrom(1957) is given a more elaborated interpretation which leads to the derivation of a theorem for the principal equations. The theorem shows that if there is a principal equation at a point (1J, 1) of the (::::. x E) product space, :=: = space of all positive continuous functions on (0, 1], E = (0, 1], then there is also a principal equation at a point (1), c) of (3 x E), € =first critical order. The converse is also true. The consequence of this theorem is that, no matter to what order of magnitude we want an approximation to be accurate, it is always possible to find high-order solutions at points (1J, 1) of the (::::. x E) space (17 = point of the :::: space obtained th;ough passage of Kaplun's limit process, where a principal equation is located) which satisfy the required degree of accuracy, overlap and cover the entire domain. Since the basis of perturbation methods comes from heuristic ideas and intuitive concepts rather than some general theory, basic questions are normally answered by physical arguments and an acquired experience. Texts with an applied-oriented nature, such as the books of Cole(1968), Nayfeh(1973), Van Dyke(1975) and Kevorkian and Cole(1981), show through examples how these heuristic ideas can be put into practice, and how some simple rules can be devised for the analysis of perturbation problems. These rules have become very popular over the past years, but are known to fail in many situations. To formalise such rules and show the conditions under which they work, Eckhaus(1969, 1972, 1973) and Fraenkel(1969) carried out more mathematically-oriented analysis which had to be, necessarily, of a more limited scope. Further important works on perturbation methods with mathematically-oriented approaches are the books of O'Malley(1974) and of Wasow(1976) where some exact results are derived for some types of ordinary differential equations. In an attempt to make clearer Kaplun's ideas, Lagerstrom and Casten(1972) published a work where a survey of some ideas on perturbation methods was presented. Again, a heuristic approach was used. The work, however, presented some new definitions and results which were known to work for leading-order approximate solutions. Some of these results have recently been revisited in publications by Lagerstrom(l988) and by Silva Freire and Hirata(1990). Most of Kaplun's ideas have been developed in literature in connection with boundary value problems. Nipp(l988), however, used the concepts presented in Kaplun( 1967) and Lagerstrom and Casten(1972) to work out a systematic approach to solve a large class of singularly perturbed initial value problems. His analysis considers both the formal and the rigorous aspects of the problem, yielding a procedure to find formal approximations of order unity. Although many of Nipp's ideas carry over·to boundary value problems, much work still needs to be done concerning the determination of higher order approximate solutions for these problem. The formal properties of equations here studied are aimed at boundary layer problems. The theorem of the principal equations formalises the notion of distinguished limit so often used in literature, allowing Kaplun's ideas to be used in a systematic manner. The matched asymptotic expansions method, for example, depends on two crucial guesses for the determination of approximate solutions: the choice of the stretching function and the choice of the asymptotic expansions. These choices are normally guided by physical arguments, but are in the end always made by trial-and-error. In fact, the determination of the stretching function and of the asymptotic expansions has always been seen as an art. With the theorem of the

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principal equations, the stretching function is immediately found, whereas the appropriate gauge functions for the asymptotic expansions are obtained from Kaplun's concept of critical orders. The asympt.otic structure of the turbulent boundary layer has been extensively investigated by a number of authors in the past twenty years. Unlikely the laminar flow case, whose solution has been known since the sixties, the turbulent problem poses some questions which still have to be understood and answered. Of course, all difficulties stem from the introduction of the time-averaged equations. To make these equations a determined system, closure conditions must be introduced to relate the Reynolds stresses to the mean flow velocities. The Reynolds stresses, the time averages of the fluctuating velocities, describe the effect of turbulent fluctuations on the mean flow; if they could be determined, the mean flow equations could solved and the asymptotic structure unveiled. Many closure conditions have been proposed in literature but, unfortunately, none of them are generally valid. Using only the hypothesis that the order of magnitude of the Reynolds stresses do not change throughout the boundary layer, some authors (Yajnik(1970), Mellor(l972)) have found the turbulent boundary layer to have a two-deck structure consisting of a wall region and a defect region. Other authors using closure conditions in terms of eddy-viscosity (Bush and Fendell(l972)) or"- € (Deriat and Guiraut(1986)) models have reached the same conclusion, making the two-deck asymptotic structure of the turbulent boundary layer the basis of most subsequent work. Recently, however, there has been a claim that the turbulent boundary has instead a threelayered structure (Long and Chen(l981), Sychev and Sychev(1987), Melnik(1989)) and that this is the only structure that can possibly handle flows subject to pressure gradients. In this work, the asymptotic structure of the turbulent boundary layer is investigated by applying Kaplun limits to the Navier-Stokes equation. Three cases will be investigated here: i) turbulent flow over a fiat surface with zero-pressure gradient, ii) separating turbulent boundary layer flow, and iii) the interaction of a normal shock wave with a turbulent boundary layer.

2

The Fundamentals of the Theory

We shall consider perturbation methods to find approximate solutions to differential equations • of the form



E1(x, y, ... , y(n))

+ E2(x, y, ... , y
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(1)

that is, equations where the small parameter € multiplies the highest derivative term. E; is a given function of the variables x,y, ... ,y
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E--+ 0,

exists and is -:f 0}.

(2)

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A partial ordering is constructed on these functions by defining

{ ord 'TJ! < ord

lim

:!.!. = 0,

0.

(3)

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A set D of order classes is said to be convex if ord OJ. ord 62 E D and ord c5 1 < ord () < ord 62 together imply ord B E D. A set D is said to be open if it is converse and if ord B E D implies the existence of functions 'Y, {j such that ord B > ord 'Y E D and ord () < o1·d 6 ~ D. A set D, on the other hand, is said to be closed if it is convex and has particular elements ord 81 , ord 62 such that o1·d li,::; ord B::; 01·d 62 for every ord ()ED. Two order sets, D and D' are said adjacent if: i) D' > D and ii) 7J < D' and 7} 1 > D --+ 7J 1 > 17. We may refer to D' as

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'1]2

<=?

E --+

'T/2

being the upper adjacent region of D. Analogously, D is said to be the lower adjacent region of D'. Definition (Lagerstrom, 1988). We say that J(x, E) is an approximation to g(x, E) uniformly valid to order li( €) in a convex set D (J is a li-approximation to g), if

E --+

0, uniformly for x in D.

(4)

The function ti(E) is called a gauge function. The essential idea of 7}-limit process is to study the limit. as E -+ 0 not for fixed x near the singularity point Xd, but for x tending to Xd in a definite relationship to E specified by a stretching function rJ(E). Taking without any loss of generality xd = 0, we define X

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. J(x,y)-g(x,y) _ l zm li(E) - 0'

Xry= 'TJ(E)'

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G(xry;E) = F(x;E),

(5)

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with ry(E) a function defined in::::.

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Definition (Meyer, 1967). If the function G(xry;+O) = limG(x, 1;E), E uniformly on {xry/lxryl > 0}; then we define !imry F(x; E)= G(xry; +0).

--+

0, exists

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Thus, if 7J -+ 0 as E -+ 0, then, in the limit process, x -+ 0 also with the same speed of ry, so that x/ry tends to a non-zero limit. value. One of the central results of Kaplun 's work is the extension theorem, which is here presented in the following version (Meyer, 1967).

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Kaplun's extension theorem. If j(x; E) is a ~(E)-approximation to g(x; E) uniformly in a closed interval Do, then it is so also in an open set D :) Do.

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The above theorem was firstly published in Kaplun and Lagerstrom(l957) in connection ( with the Stokes paradox for flow at low Reynolds number. It needs to be complemented by an Axiom and by an Ansatz to relate the formal domain of validity of an equation with the (

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actual domain of validity of its solution. The idea of Kaplun was to shift the emphasis to applying limit-processes directly to the equations rather than to the solutions, establishing some rules to determine the domain of validity of solutions from the formal domain of validity of an equation. The set of equations that will result from passage of the limit is referred to by Kaplun as the "splitting" ofthe differential equations. The splitting must be seen as a formal property of the equation obtained through a "formal passage of the ?)-limit process". To every order of 17 a correspondence is induced, lim~ _, associated equation, on that subset of 3 for which the associated equation exists. Definition. The formal limit domain of an associated equation E is the set of orders 17 such that the 17-limit process applied to the original equation yields E. Passage of the ?)-limit will give equations that are distinguished in two ways: i) they are determined by specific choices of 1), and ii) they are more complete, or in Kaplun's words, "richer" than the others, in the sense that, application of the 1)-limit process to them will result in other associated equations, but. neither of them can be obtained from any of the other equations. Limit-processes which yield "rich" equations are called principal limit-processes. The significance of principal limit-processes is that the resulting equations are expected to be satisfied by the corresponding limits of the exact solution. The notion of principal equation will be formalised below. The above concepts and ideas can be given a more rigorous interpretation if we introduce Kaplun's concept. of equivalent in the limit for a given set of equations for a given point (?J, 6) of the (3, E) product space. Given any two associated equations E 1 and E2, we define the remainder of E 1 with relation to E2 as

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IR(x~; e) = Et(x~; e)- E2(x~; e),

where e denotes a small parameter. According to Kaplun(1967), R should be interpreted as an operator giving the "apparent force" that must be added to E2 to yield E 1 . Definition (of equivalence in the limit) (Kaplun, 1967). Two equations E1 and E 2 are said to be equivalent in the limit for a given limit-process, lim~, and to a given order, 6, if

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(7)

Proposition 1: If E ~ E' for the point (17', 6') of the product space 3 x E, then E ~ E' for all points (11, 6) such that 1) = 1)1 and 6 » 6'. Conversely, if E rf E' for the point (17', 6'), then E rf E' for all points (?J, 6) such that 1) = 1) 1 and ord 6 «: ord 6'.

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The following propositions are important; they can be found in Kaplun(1967). The symbol is used to indicate equivalent in the limit whereas rf indicates not equivalent in the limit.

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Proposition 2: If E ~ E' for the point (ry, 8) of the product space :=: x E, and if associated equations for that point exist for E, then they exist also for E' and are identical for both. Proposition 3: If associated equations exist for E and E' respectively, corresponding to 17 = r/ and the sequence 8 = 80, 8~, ... , 8~, 8' where 8~ > 6' > 6~+ 1 , and are identical for'both, then E ~ E' for the point (ry', 61). We can make the following definition.

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Definition (of formal domain of validity). The formal domain of validity to order {J of an equation E of formal limit domain D is the set D. = D U D:s, where D:s are the formal limit domains of all equations E; such that E and E; are equivalent in D; to order 6. Definition (of principal equation}. An equation E of formal limit domain D, is said to be principal to order 6 if: i} one can find another equation E', of formal limit domain D', such that E and E' are equivalent in D' to order 8; ii) E is not equivalent to order 6 to any other equation in D. An equation which is not principal is said to be intermediate. To relate the formal properties of equations to the actual problem of determining the uniform domain of validity of solutions, Kaplun(1967) advanced two assertions, the Axiom of Existence and the Ansatz about domains of validity. These assertions constitute primitive and unverifiable assumptions of perturbation theory.

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Axiom (of existence) (Kaplun, 1967}. If equations E and E' are equivalent in the limit to the order 6 for a certain region, then given a solution S of E which lies in the region of equivalence of E and E', there exists a solution S' of E' such that as E -+ 0, IS- S'l/6-+ 0, in the region of equivalence of E and E'.

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In other words, the axiom states that there exists a solutionS' of E' such that the ''distance" between S and S' is of the same order of magnitude of that between E and E'. In using perturbation methods, the common approach is to consider the existence of certain limits of the exact solution or expansions of a certain form. This is normally a sufficient condition to find the associated equations and to assure that the axiom is satisfied (Kaplun(1967). Equivalence in the limit, however, is a necessary condition as shown by propositions (1) to (3). To the axiom of existence there corresponds an Ansatz; namely that there exists a solution S of E which lies in the region of equivalence of E and E'. More explicitly, we write. Ansatz (about domains of validity) (Kaplun, 1967). An equation with a given formal domain of validity D has a solution whose actual domain of validity corresponds to D. The word "corresponds to" in the Ansatz was assumed by Kaplun to actually mean "is equal to"; this establishes the link we needed between the "formal" properties of the equation and the actual properties of the solution.

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The Ansatz can always be subjected to a canonical test which consists in exhibiting a solution S' of E' which lies in the region of equivalence of E and E' and is determined by the boundary conditions that correspond to S. Because the heuristic nature of the Axiom and of the Ansatz, comparison to experiments will always be important for validation purposes. The theory, however, as implemented through the abov~ procedure, is always helpful in understanding the matching process and in constructing the appropriate asymptotic expansions.

3

The Proposition of the Principal Equations

The "splitting" of the equations obtained through the definition of equivalent in the limit may be extended to higher orders by introducing a fictitious perturbation of an arbitrary order 6. Thus, according to Kaplun( 1967), for higher orders the splitting of the equat.ions corresponding to arbitrary limit processes becomes more complicated and less significant; the operation of splitting is then merely reduced to exhibit some of the typical associated equations and some of the sufficient conditions under which they are associated. In fact, Kaplun lists three reasons why the splitting for higher orders should not be considered in detail: i) the equations associated with a given point (17, 6) depend on the choice of the 6~ for the corresponding limit process and may depend on the amount of information used in connection with the preceding terms, ii) the 6~ depend to greater extent on boundary conditions and hence are difficult to determine a priori, and iii) many trivial splitting of the associated equations arise, corresponding to expansions of the preceding terms by different limit processes. Here, we want. to further extend the above notions. In what follows we will show that, for certain points of the (S, I:) product space, the determination of the associated equations will depend on the choice of some discrete values of 6~. It results that the order of validity of an approximation is defined by open intervals determined by the discrete 6~'s. Furthermore, no trivial splitting will result in these certain points. To extend the previous results to higher orders, we consider solutions of the form

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f = fo

Definition (of critical order) (Kaplun(1967)). An order ord 6.(e) is said to be critical if: i) the corrections to fo to any order (in D,D = {(/ord6.(e) < ord( < 1}, are trivial;

( (

(8)

where t..(e) E S. Some questions are now in order. Which function is t..(e) foi a given differential equation? Is t..(e) the same for all regions of the domain? The first question is complex and involves speculating on the existence and uniqueness of solutions. Of course, uniqueness of 6.(e) can never be assured since given any t..(e), one can always present another t..'(e) such that t..'(e) is exponentially close to t..(e) . Thus, according to Kaplun, there will always be a "question of choice" for the determination of the appropriate asympt.otic expansions which must be solved relying on intuition and physical insight. An adequate t..(e) can however be determined in a very natural way. We require t..(e) to be such that the resulting equation for h does not provide a trivial solution. A 6.(e) satisfying this condition is said to be a criticalt..(e). Analogously, its order, ord6.(e), is called critical order. More precisely:

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ii) the corrections to fo to any order ( in the complement of D are not trivial.

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The above definition suggests that approximate solutions for different regions of the domain should not in general have the same A(e). Of course, equal A's might happen as a mere coincidence; however, it is important to give emphasis to that, normally this is not the case. To find the several order approximate equations we substitute Eq.(8) into the original equations and perform elementary operations such as addition, multiplication, subtraction, differentiation and s<;> on. If these operations are justified, that is, if they do not lead to any non-uniformity, we then collect the terms of same order of magnitude and construct a set of approximate equations. Thus, it is clear that in the process of collecting terms, to each term E 1 of order, say v there will always correspond another term E2 of order vll.(e).

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Consider now an equation E where E 1 and E2 denote the first two critical order terms. We call the operator TI 1(E) = E 1 the first order projection of E onto E 1. Analogously, the operator TI 2 (E) = E 2 is called the second order projection of E onto E 2 . We can then enunciate the following proposition.

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Proposition (of the principal equations). If there is a principal equation, E 1 , at a point (ry, 1) of the (2, E) product space, then there is also a principal equation, E, at a point (7J, e) of (2, E) with E1 = IT 1(E).

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Proof: Suppose E 1 is a principal equation at a point (17, 1) of the (2, E) product space. Then one can find a term Ru such that Ru is order unity in D (the domain of EJ) but o1·d 1 < o!"dRu < ord e in Du, the upper adjacent domain of D. Here £ denotes the first critical order. Define E~ = E1 - Ru. Let now E2 and E2 denote the first order associated equations in D and Du respectively. Then, there is a term R21 such that R21 is order e in D but ord R21 < 01·d e in Du. Define E2 = E2 - R21lt results that the structure of the lower adjacent region is

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ord Ru < ord E; < 01·d R21 < ord E~

(9)

This yields that no other equation is equivalent to order e to equation E = E; + Ru + E2+ R 21 in D. However, E and E'(= E; + Ru + E2) are equivalent in Du to o1·de. We conclude E is a principal equation at a point (7J, e) of the (2, E) product space. The converse of the above proposition is obviously true, that is: if E is a principal equation at a point (ry, e) of the (2, E) space, where 7J denotes the formal limit domain of E and e the first critical order, then TI 1 (E) is a principal equation at a point (7J, 1) of the (2, E) space. What the above proposition clearly states is that the position in the (2,E) product space where the principal equations are located can se searched by looking only at the lowest order associated equations. Furthermore, it says that these lowest order approximations are good up to the first critical order and that.no trivial splitting will arise. This fact is only valid for the particular point in (2,E) space where the principal equation holds. In the upper and lower adjacent domains trivial splitting will occur. It results that higher order splitting should not, in fact, be considered. The principal equations of the problem, those that retain most of the information about the problem solution,

( (

( (

(

(

(

< (

( (

( 138

'

( (

\

l, (

( (

(

can have their position determined only through an analysis of the lowest. order terms. Then the concept of critical order can be applied to the solution to find the appropriate asymptotic expansions for the problem.

( (

4

(

Boundary layer problems are historically important in the development of singular perturbation methods. In fact, the basic ideas of singular pert.urbation methods can be traced back to Prandtl's boundary layer theory of a laminar flow. Prandtl's matching principle for laminar boundary layers was systematically discussed and generalised in the fifties yielding well established procedures and solutions which have rendered the laminar flow problem solved. In the seventies, the interest shifted to turbulent flow. Two approaches were used: in the first, asymptotic techniques were applied to the averaged equations without appealing to any closure model (Yajnik(1970), Mellor(1972)); in the second, eddy-viscosity (Bush and Fendell(1972)) or "· - e (Deriat and Guiraud(1986)) models were used to find high order approximations. The theories divided the turbulent boundary layer into two regions, becoming the basis of most subsequent work. Other authors, Long and Chen(1981), Sychev and Sychev(1987), lvlelnik(1989), however, have recently claimed that the turbulent boundary layer has instead a three-layered structure. This structure considers a new region in which a balance of inertia forces, and pressure and turbulent frict.ion forces occurs. The formulation of Melnik is based on a two-parameter expansion of the boundary layer equations, the new additional small parameter resulting from the particular turbulence closure model he uses. The discussions that have led to the development of the three-layered asymptotic model for the turbulent boundary layer result from the recognition that two-layered models cannot deal with large flow disturbances in the stream-wise direction. When a turbulent boundary layer is subjected to a large longitudinal adverse pressure gradient, the velocity deficit is large and the mean momentum equation is non-linear; this makes the classical matching arguments which result in a log-law and in a two deck structure, not valid anymore. The classical wall characteristic velocity, the friction velocity, may become an inappropriate scaling parameter so that new formulations will have to be developed for the problem at hand. Here, we will investigate the turbulent boundary layer from the point of view of Kaplun 's single limits. The purpose is to formally arrive at a three-layered structure which is compatible with the class of problems to be studied: the zero-pressure gradiel}t turbulent boundary layer, the separating turbulent boundary layer and the shock-wave interacting t.urbulent boundary layer.

( ( (

( ( (

( (

( ( (

( (

( ( (

The Asymptotic Structure of the Turbulent Boundary Layer.

( ( ( ( ( (

4.1

The zero-pressure gradient turbulent boundary layer

For an incompressible two-dimensional turbulent flow over a smooth surface in a prescribed pressure distribution, the time-averaged motion equations; i.e., the continuity equation and the Navier-Stokes equation can be written as

8u.; = 0, 8Xj

( ( (

( (

139

(10)

(

{

( 11-j

ou; = _ !!..!:_ OXJ OXj

_ _!!__ €2

OXj

(uju.:) + ~R 8oxj1~;,

'

l

2

(11)

where the notation is classical. Thus (x 1,x2) = (x,y) stand for the co-ordinates, (u 1,u2) = (u, v) for the velocities, p for pressure and R for the Reynolds number. The dashes are used to indicate a fluctuating quantity. In the fluctuation terms, an overbar is used to indicate a time-average. All mean variables are referred to some characteristic quantity of the external flow. The velocity fluctuations, on the other hand, are referred to a characteristic velocity 1LR, firstly introduced in Cruz and Silva Freire(1998). The correct assessment of the characteristic velocity is fundamental for the determination of the boundary layer asymptotic structure. For unseparated flows the characteristic velocity is known to be the friction velocity; for separating flows it reduces to (v(dpfdx)fp) 113 . For the moment, we will consider attached flow so that we can write

(

I

(

'

( ( (

( ( (

( ord (u;) = ord (u.T).

(12)

(

This result is valid for incompressible flows as well as for compressible flows (see, e.g., Kistler(l956) and Kistler and Chen(l956)). The small parameter E is, therefore, defined by

(

'IJ.R

t/.T

Uoo

Uoo

t=---

( (13)

The asymptotic expansions for the flow parameters are written as u(:r, y) = u 1 (x, y)

+ w2(x, y),

(

' (

' I

( (14)

( v(x, y) = ry[v1(x, y)

+ tv2(x, y)],

(

I

(15)

( p(x,

y) = PI(x, y)

+ ep2(x, y),

{ \ (16)

(

w; 1 (x,y) +€ 2 u; 2 (x,y).

u;(x,y) =

(

(17) (

u'v'(x, y) =



2

u'v~(x, y)

+ t 3 u.'v2(x, y).

( (18)

( To find the asymptotic structure of the boundary layer we consider the following stretching transformation

( (

( ( 140

( (

' ( ( ( ( fj =

( (

y~ = TJr€)'

!t.;(x, y~) = u;(x, y).

(19)

(

with rJ(E) defined on 2. Upon substituti~n of Eq.(19) into Eqs.(14) to 17) and upon passage of the ry-limit process onto the resulting equation we get:

(

x-momentum equation:

( (

D12

Du = P1.

D12

= €:

Du = Pt,

1)

(

= 1:

Du

€2 <

(

1)

< €:

Du = Pt,

T/

= €2:

Du

(

( ( (

( (

( ( ( (

D12

(u'v1) 0 = 0, (u'v!) 1i = 0,

Du

1/E R
3

(u'v!z)v

:

(1i."vf)o =

o,

ry=1/E 2 R: (n'v1) 1j =0, 1/ER <

17

< 1jE 2R: (u'vi) 1i = 0,

rJ = 1/ER:

(u'vDv

(1L2)yy = 0,

= (il.2)!ifi,

(22) (23)

= P2

- (u.'v!z)v,

(24)

= 0,

(25) (26)

(u'v!z)u = 0,

(27)

(u.'u2)u = (u2)!i!i•

(28)

(u.'u!z)v

= 0,

(29)

(~)o

= (il.a)vv·

(30)

where t.he following operators were used

au.. D;i = ii.;

_ auj

a: + v; ay~'

P; =

_.!:_ap; pax'

(31)

The above equations were arranged in three columns according to their respective order of approximation. The first column corresponds to the first order of approximation; the third one to the second order of approximation. The middle column corresponds to orders between the first and the second critical order. The extreme left of the lines indicates the point in the domain where the ry-limit process was applied. Passage of the ry-limit process onto the y-momentum equation does not give any relevant information ..In £act, we will find that for ord rJ < ord E the first and second order pressure terms will dominate all the other terms. All information regarding the asymptotic structure of the boundary layer is, therefore, contained in the x-momentum equation.

(

(

= Pt,

= P2,

(20) (21)

Du = P1

( (

+ D21 + D21

(u'v~)y,

- (u'v2)o, 2

(

(

D12

- (u'vi)y, €3 < T/ < €2 : I)= €3:

(

(n'v!) 1i = 0,

= Pt

( (

+ D21 = P2, + D21 = P2, D12 + D21 = P2

= Pt,

€ < T/ < 1:

1)

( (

ord(6) = ord(E)

ord(6) = 1

(

141

(

( ( '! The term u. 1 (x, y 11 ) is missing from equations (29) and (30) since from the no-slip condition = 0 near the wall. Equations (24) and (30) are distinguished in two ways: i) they are determined by specific choices of 1], and ii) they are "richer" than the others in the sense that, application of the limit process to them yields some of the other equations, but neither of them can be obtained from passage of the limit process to any of the other equations. Thus, according to the definitions introduced in the previous sections, these equations are the principal equations. We have seen that principal equations are important since they are expected to be satisfied by the corresponding limits of the exact solution. A complete solution to the problem should then according to the Axiom of Existence and Kaplun's Ansatz, be obtained from the principal equations located at points ord I) = m·d € 2 and ord 1J = ord (1/€R). The formal doma:ins of validity of these equations cover the entire domain and overlap in a region determined according to the definition of equivalent in limit. To find the overlap region of equations (24) and (30), we must show these equations to have a common domain where they are equivalent. A direct application of the definition of equivalence in the limit to equations (24) and (30) yields

(

'11. 1

lR = D(!I.J)- P(pJ)

+ D(u2)-

P(p2)- (u2)uu- (ii.3) 1i!i.

€"'

(32)

I

(

'

( (

(

(

(

f (

Noting that the leading order term in region ord (1/€R) < ord 1) < ord €2 is the turbulent term, of ord (€ 2 /1]), we normalise the above equation to order unity to find -

(

1)

(

(

(33)

(

The overlap domain is the set of orders such that the 7)-limit process applied to lR tends to zero for a given a. Then since ord (8f8y) = € and ord (8/8x) = 1, the formal overlap domain is given by

(

lR= 2 R €

( (

= {1J/

1

ord (€l+"'R)- < ord

2

< ord (€ +"')}.

(34)

(

According to Kaplun's Ansatz about domains of validity, the approximate equations, Eqs. (24) and (25), only overlap if set (34) is a non-empty set, that is, if

(

Doverlap

1)

(

1 (In R 3). O
(35)

The implication is that the two-deck turbulent boundary layer structure given by the two principal equations, equations (24) and (30), provides approximate solutions which are accurate to the order of €"~··, where O
( ( (

( ( (

( (

( 142

(

\

I

'( (

(

(

(

D~

n Dl

=

{oTdryf

2

oTd(l/ER) < oTd(17) < 01·d(€ )}

(36)

3 ord(l/€ 2 R) < oTd(ry) < 01·d(€ )}.

(37)

and,

( D~ n D; = {ord17/

( (

(

f ( (

( (

( ( ( (

We conclude that the turbulent boundary layer has a two-deck structure very much like the one derived by Sychev and Sychev. This structure, however, must change as a separation point is approached. We shall see this next. Before we move forward, however, some comments about the intermediate equations will de made. For the formal limit domains which are not adjacent. to the principal equations two approximated equations rue always defined, separated by the first two critical orders. In this case the interpretation is simple and the local approximated equations and solutions well defined. For the regions adjacent to the principal equations, however, a correction with order between the first two critical orders is found. The interpretation of these equations is more complex and must be made in an individual basis. For example, in t.he turbulent boundary layer problem under consideration, the solution in the upper adjacent region must take into consideration, as the first two order of approximation equations, the leading order equation and the intermediate order equation; these equations will provide non-trivial solutions with physical information. For the lower adjacent region, however, the intermediate order equation provides a trivial solution; thus, no extra information is obtained from this equation except that the overlap domain for the first two order of approximation is not given by equation (37) but by

(

D~ n D; = {ordryf

ord(l/€ 2R) < ord(ry) < ord(€ 2)}.

(38)

(

( (

( ( (

( (

( ( (

4.2

The separating turbulent boundary layer

The above asymptotic structure must undergo some modifications if flows subjected to adverse pressure gradients are to be considered (Cruz and Silva Freire, 1998): A major difficulty· found for a direct translat.ion of the classical models into models that apply for separating flows is the characteristic velocity used in the former approach. When the friction velocity, u.., is used to develop the asymptotic structure of the boundary layer, a non-uniformity will occur near a separation point where 1l.r = 0. These difficuJties force into' the adverse pressure gradient problem a new small parameter of the order of R- 113 , which is used to scale a power-y layer that replaces the logarithmic layer. This new intermediate layer defines a third characteristic scale which must be considered together with the wall and defect layer characteristic scales. Thus, t.hree sets of characteristic scales are needed for the asymptotic description of adverse pressure gradient turbulent boundary layers (see Durbin and Belcher(1992)). The result is that any theory advanced for the problem should explain in asymptotic terms how the far upstream two-deck structure reduces to a three-deck structure near a separation point. Equivalently, any theory should show how the logarithmic layer vanishes as separation is approached, and how the y 112 -layer is formed.

( ( ( (

143

(

I

( ( In Cruz and Silva Freire( 1998) a new scaling procedure was introduced through asymptotic arguments that resulted in an algebraic equation for un that yielded a changeable asymptotic structure for the boundary layer consistent with the experimental data. The theory followed the approaches of Yajnik(1970) and of Mellor(1972), not imposing any functional relation~hip between quantities determined by the Reynolds stress field and by the velocity field. Here we will repeat part of t.he theory to illustrate how the results of the previous section can be extended to separated flows. In the region defined by the principal equation, Eq.(30), a balance between the turbulent and viscous stresses exists so that we may write

a - - +. J.L8 2u

- ( -pu'v') fJy

fJp = -. fJy2 fJx

(39)

In this region, the characteristic length is given by vfun. Then, considering that the turbulent fluctuations are of the order of the reference velocity, un, and that the viscous term can be approximated by fJu2 ord(wa:y) = ord(rw),

(40)

{ (

'

(

(

(

< t ( (

it results from simple order of magnitude arguments that the characteristic velocity can be estimated from the algebraic equation

( (

3 Tw V fJp ·u.R- -un- - - = 0. p p fJx

( (41)

( (

Passing the limit as

Tw

tends to zero onto the above equation,

( (

un---+

~ fJp) 1/3

( p8x

'

(42)

so that the characteristic velocity for the near separation point region proposed by Stratford(1959) and by Townsend(1976)is recovered. The characteristic velocity un is determined by the highest real root of (41). It follows that, close to the separation point., ord (E 2) = ord (1/ER), and the two principal equations merge giving origin to a one deck structure. This merging provokes the disappearance of the log-region, reducing the flow structure to a wake region and a viscous region. To find the asymptotic structure of the separating boundary layer we apply the following stretching transformation to the equations of the previous section

( (

( (

(

( (

( 144

( (

I

-( ( ( (

X

( (

with t.(e) defined on 3. The resulting flow structure is given by:

(

x-momentum equation:

( ord C. = ord 1 :

(

. av.2 . aii.2 a·h + 112- + -axll. ayry axe:,.

11.2--

f m·d e2 < ord

(

( (

2

A

ord e = ord u:

a·P2 a. • a· 11.2 +v2- + -axll. ayry axll. ord C. < m·d e2

(

a2 11.2 a211.2

a~ a2· a2· u 1v 1 + - U22- + - 11.2 -, ayry axll. ayry2

(46)

=O;

(47)

. av2 . av2 a·h u.2-a +v2-a +-a X
ord C. = ord 1 :

(

ord

2 €



= ord t.:

=

ord C.

< ord

e

2

a2v2 :

(

(

(

(49)

(50)

a2v2 -

~-+--2-

ax~

ay'l

o

.

(51)

Note that in region (t.,71) = (e 2 ,e 2 ) the full Navier-Stokes averaged equation is recovered. The leading order equations for •1 1 together with the no-slip condition at the wall gives ii. 1 = 0. In Cruz and Silva Freire(1998) the asymptotic structure of the thermal turbulent boundary layer is also studied through Kaplun limits. The procedure is basically the same and the

( (

o,

11

(

c

=

• fJv2 • av2 av2 a~ a1;.'1 ~'1 a 2 v2 a 2 v2 11.2-a +v2-a +-a =--a --{)--+-a 2 +-a X
{ ( (

. (48)

0,

. av2 . av2 av2 11.2-a + v2-a X
ord e2 < ord 1 < ord C. :

( (

(

(45)

,

= - - - ·- - - -

-a 2 +-a 2 xll. Yry

:

= 0

y-momentum equation:

(

(

a~ u 1 axll.

(44)

= 0,

. au.2 . au.2 a·h v.2-- + v2- + - axll. ayry axll.

m·d 1 :

• 11.2 11.2--

( ( ( (

t. <

(43)

= t.(e)'

X
145

t '

( ( results similar to those derived for the velocity field. For more details concerning this problem the reader is referred to the original work. There, new formulations are advanced for the law of the wall for the velocity and the temperature fields for separating flows. To these formulations, experimental and numerical validation are given based on the works of Vogel and Eaton{l985) and Driver and Seegmiller(1995). The resulting asymptotic structure for both the velocity and the temperature bound'ary layers is shown in Figure 1, which was taken from Cruz and Silva Freire(1998). This figure incorporates the dependence of the structure on the Prandtl number.

( (

( {

1 (

4.3

The shock interacting boundary layer

(

For a compressible flow, the two-dimensional Navier-Stokes equations of mean motion can be cast in terms of a mass-weighted-averaging procedure. The continuity and the momentum equations can then be written in the following non-dimensional form:

(

,.

a

ax. (pi'i-j) = 0 )

(52)

'

(

(

a (-- _ l

OXj

ap

pu;u'i = - 8x;

a ( -,-, 1 ) + ax -pu;ui + [ t i j '

( {53)

1

(

( where the stress tensor

T;j

is given by

( (

Tij

=

8u1 + Jl. (au; + -au.1 ) .

)..{jij-

ax,

OXj

(54)

8x;

( (

These equations are complemented by the energy and the state equations. It follows that

2

a =

~('y + 1)- ~('y-

l)ii.;U.j,

p= pt.

a=

~

v~·

(

( (55)

(

(56)

(

( In the above equations, x, ·u, p, t and p have their classical meaning. >.is the bulk viscosity (= -2/3Jl.), Jl. is the viscosity and 6;; the Kronecker delta. The non-dimensional velocities, pressure, temperature and density are all referred to their critical values just outside the boundary layer and ahead of the shock. R denotes the Reynolds number. The dashes denote turbulent fluctuations; the bars and the tildes denote respectively conventional time averaging

(

(

( (

( 146

( (

t (

( (

( ( (

(

and mass-weighted averaging. In what follows, for the sake of simplicity, the bars and the tildes will be omitted. The order of magnitude of the turbulent terms in the equations of motion can be estimated through t.he experimental results of Kistler(1959), of Kistler and Chen(1963) and of Morkovin(l962). These author have shown that: (a) u.', p', and t' have the same order of magnitude and (b) the root square-mean value of p' is proportional to u'. Thus, in view of the above remarks, the scales of fluctuation can be written as

(

ord(n')

(

= ord(v') = ord(p') = ord(t') = ord(ur),

(57)

( (

ord(p') = ord(u.~).

(58)

(

(

( (

To find the asymptotic structure of the flow in the interaction region, we consider the same stretching transformation used in the two previous sections of the paper, that is,

( (

(

y

X

xc. = Ll(e)'

(

ii.;(xc., YrJ)

YrJ = 7J(e)'

=

u;(x, y),

(59)

with Ll(e) and 7J(e) defined on 3. Following previous studies of the problem by Messiter(1980) and by Silva Freire(1988), in the interaction region we separate the asymptotic expansions for the solution into a rotational and an irrotational part. Thus, we introduce here the two small-parameters

(

( (

Uoo

( (

( ( ( (

11 ·•

1~ = a*

u = 1 + W 01 (X, y)

+ U.rup(y),

v = e312 v01 (x,y),

( (

VPw,

(61)

where a• is the critical sound speed in the external flow just ahead of the shock, Tw is the laminar stress at the wall, and Pw is the density at the wall. From now on, the reader is asked not to confuse the new definition of e with its previous definition in the previous sections. As a result, the velocity profiles can be written as

( (

(60)

and

( (

1

E=~-'

147

(62)

(63)

(

( (

where u"' and '11.(3 represent respectively the irrotational and the rotational parts of the flow. Upon substitution of Eqs.(62) and (63) into the equations of motion, and passage of the r1-limit process onto the resulting equations, we get for the x-momentum equation:

(

( a ( _ _ ) a ( ~ _) afia ord 1J = 1 : - a puaua +-a puava = - - a , X6 YQ X6

{64)

(

a ( • _ ) a ( _ _) afia ord uT2 < ord 1J < 1: - a puaua +-a PUaVa =--a , X6 YQ X6

(65)

(

a ( _ _ ) a ( _ _) ofia 2 ord 17 = ord v.T: - - PUaU- 0 +-a pu.aVa =--a ax6 YQ X6

ord 1/ RuT < ord 1J < ord

--::-;-:;,) +a- ( -puava , ayQ

u; : _aaYQ ( -pii~v~) = 0,

( (66)

(67)

(

(

2

a ( -pu' ----) a u.f3 = 0 ord '7 = ord 1/R-u"T · v' +J.L-" ayQ "Ct Ct ay~ a2 uf3

ord 1J < ord 1/ RuT : J.L7}2 = 0. YQ

(

>

(68)

( (

(69)

Since we are considering the flow in the interaction region, in passing the 17-limit we have taken ord(~) = ord(E). The other equations, continuity, energy and state, do not give any contribution to the asymptotic structure. In fact, Silva Freire{1989) has shown that if the full energy equation is considered, and the concepts of section two are applied to the full set of equations, then the overlap domains of the velocity field and of the temperature field will coincide. The continuity equation simply implies that

( (

( ( (

( 1J

ord(v) = .L\ord{u).

(70)

The classical two-deck structure of the turbulent boundary layer is then clearly seen from equations {64) to {69). Note that Eqs.{66) and (68) are the principal equations; their overlap domain is identical to the overlap domain determined for the incompressible flow case. In the vicinity of the shock wave, however, the asymptotic structure above deduced must change. The strong pressure gradient imparted to the boundary layer by the shock wave alters the balance of terms in the equations of motion, giving rise to a new structur~ where for most of the boundary layer the problem becomes an inviscid one. The need for the establishment of an inviscid rotational flow model for the description of the interaction has been recognised since Lighthill{1953) proposed his linearized solution for the laminar problem. The result is that all recent theories advanced for the turbulent problem must somehow accommodate the inviscid rotational interaction model without contradicting the features of the turbulent flow.

(

(

( ( (

(

(

( 148

(

(

t

(

( ( (

(

f (

( (

To surmount this difficulty, the theories of Melnik and Grossmann(1974), of Adamson and Feo(1975), of Messiter(1980) and of Liou and Adamson(1980) consider the introduction of a blending region in the interaction region. The blending layer is, in fact, nothing more than the turbulent region defined by the overlap domain and derived by our asymptotic analysis of the problem. The absence of an equation similar to equation (67) in the matched asymptotic expansions method is the main reason for the difficulties this method presents. Likewise, this is the reason why the structure depicted by Eqs.(66) to (69) can deal with the interaction problem. · To take into account for the presence of the shock wave, we pass the ~-limit. process onto Eqs.(64) to (69). The result is:

( (

ord 1:::. = ord e:

a (Puouo) . • +-a a (pu<>v<> . • ) =--a afio , -a Xt. y,., Xt.

(71)

( (

( ( (

( ( ( ( (

( ( ( ( (

( ( ( (

o1·d

€/ Rv. ,.3 < ord

1:::.

< ord ery/u,.2 :

a (pu..,u.<> . •) 01·d t::. = ord eI Ru,.3 : -a Xt.

a (pu.,u., • . ) + -a a (pv.o.vo . . ) -_ --a afio , -a Xt. y,., Xt.

a (pv.ov., • . ) + -a y,.,

2

0

ord !:::.) < ord

3.

a2u.p

€/ Ru.,. : J.L{i'i'

= 0.

(73)

0

(74)

Y,.,

The change in the asymptotic structure of the flow in the interaction region is noticeable from the above equations. In particular, we note that as the shock is approached, that is, as the order of magnitude of 1:::. increases, the validity domain of the outer principal equation changes position until the two principal equations merge at (t::.,ry) = (e/(u~R), 1/(u,.R)). Indeed, as shown by the calculations, at the beginning of the interaction the outer principal equation is positioned at (t::.,ry) = (e,u~). However, as the order of magnitude of 17 varies from u~ to 1/u.,.R, this equation moves along the path (ery/u~,ry) until reaching the point (€/u~R, 1/u.,.R). The flow structure is then shown to reduce from a classical two deck structure to a one deck structure near to the foot of the shock wave. According to these results, there is a region at the foot of the shock where the full boundary layer equations are recovered. The results of this section are shown in Figure 2; they will be compared with the experimental data of Sawyer and Long(1982). Figures 3 and 4 reproduce, from the experimental data, a map which indicates the dominant region of every term in the equations of motion. Both cases, Mach numbers of 1.27 and of 1.37 are shown. The meaning of the shades in gray is clear. Thus, the farthest from the wall tone corresponds to the inertia and pressure gradient terms, the intermediate tone to the turbulent terms and the remaining tone to the viscous terms. The shock wave is located at x = 0.

(

(

afio --a Xt.

a (-pu ---::-;--,-) + J.L{i'i', a U.p +-a v y,., Y,.,

(

(

=

(72)

149

t (

( Observe, as predicted by the asymptotic theory, the complete dominance of the inertia and pressure terms in the vicinity of the shock. This feature is particularly striking for the 1.37 Mach number case where the influence of the shock extends down to the viscous layer. The consequence is that the phenomenon is, for most of the interaction region, and, to a leading order, governed by inviscid equations.

5

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Final Remarks

In the first part of the paper, some ideas of Kaplun concerning limit processes have been extended to higher orders through the proposition of the principal equations. This result is central "to our work, for it ensures that the asymptotic structure of a singular perturbation problem can be uniquely determined by a first order analysis of the formal domains of validity. The resulting principal equations are expected to be satisfied by the corresponding limits of the exact solution, so providing approximate solutions that overlap and cover the entire domain of validity. In the second part of the paper, application of Kaplun limits to the equations of motion has shown the zero-pressure turbulent boundary layer to have a two deck structure, the principal equations being located at points (e 2 , 1} and (1/eR, 1} of the product space (2 x L:). The present results are very much in accordance with the earlier works of Yajnik, of Mellor and of Bush and Fendell. They seem to corroborate the idea that a one-parameter theory can correctly describe the flow structure and, furthermore, do not give any evidence to suggest the contrary. The present analysis has also shown how the two-deck turbulent boundary layer structure develops into a one-deck structure near a separation point. This results seems, at first, contradictory to the three-layer structure found by other authors (Melnik( 1989}, Durbin and Belcher(1992)}. However, we point out that all local equations derived by these authors are intermediate equations, in the sense of Kaplun, being therefore, contained in the domain of validity of the principal equations here derived. In other words we may say that those theories are "contained" in the present theory. Of course, the principal equations are of difficult solution, do not providing closed analytical solutions; however, only these equations give fundamental insight to understand how the viscous and defect layers merge as a separation point is approached. In what concerns the problem of interaction between a shock-wave and a turbulent boundary layer, the application of Kaplun limits to the equations of motion has shown the flow to attain an one deck structure, which is distinct from those of other authors but consistent with the experimental data of Sawyer and Long(1982} and with the general knowledge of the problem we have. The theory, as presented here, can formally explain how an inviscid flow region is formed at the foot of the shock wave, resulting from the disappearance of the fully turbulent region. Acknowledgements. Many people have decisively contributed to the completion of this work. Drs. S.L.V. Coelho e N. Chokani read and commented an early draft of the first part of the paper. The author also benefited from useful discussions with Prof. R. Narasimha and Dr. J. Su. Many of the analytical difficulties were discussed with Dr. D.O.A. Cruz. This work was financially supported by the CNPq (Ministry of Sciences and Technology) through grant No 350183/93-7.

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References

Adamson, T. C.Jr. and Feo, A.; Interaction Between a Shock Wave and a Turbulent Boundary Layer at Transonic Speeds, SIAM J Appl. Math., vol. 29, pp. 121-145, 1975. Bush, W. B. and Pendell, F. E.; Asymptotic Analysis of Turbulent Channel and Boundary Layer Flows, .J. Fluid Mechanics, vol. 56, pp. 657-681, 1972. Cole, J. D.; Perturbation Methods in Applied Mathematics, Blaisdell, Massachussets, 1968. Cruz, D. 0. A. and Silva Freire, A. P.; On Single Limits and the Asymptotic Behaviour of Separating Turbulent Boundary Layers, Int. J. Heat Mass Transfer, vol. 41, pp. 2097-2111, 1998. Deriat, E. and Guiraud, J.P.; On the Asymptotic Description of Turbulent Boundary Layers, J. Theor. Appl. Mech., Special issue, pp. 109-140, 1986.

(

Driver, D. M. and Seegmiller, H. L.; Features of a Reattaching Turbulent Shear Layer in Divergent Channel Flow, AIAA J., vol. 23, pp. 163-171, 1995.

(

Durbin, P. A. and Belcher, S. E.; Scaling of Adverse-Pressure-Gradient Turbulent Boundary Layers, J. Fluid Mechanics; vol. 238, pp. 699-722, 1992.

( ( (

(

Eckhaus, W.; On the Foundations of the Method of Matched Asymptotic Expansions, J. de Mecanique, vol. 8, pp. 265-300, 1969. Eckhaus, W.; Boundary Layer in Linear Elliptic Singular Perturbation Problems, SIAM Review, vol. 14, pp. 225-270, 1972.

(

Eckhaus, W.; Matched Asymptotic Expansions and Singular Perturbation; North-Holland, Amsterdam, 1973.

(

Fraenkel, L. E.; On the Method of Matched Asymptotic Expansions, Proc. Camb. Phil. Soc., vol. 65, pp. 209-284, 1969.

( (

( ( ( ( (

Kaplun, S.; Fluid Mechanics and Singular Perturbations, Academic Press, 1967. Kaplun, S. and Lagerstrom, P. A.; Asymptotic Expansions of Navier-Stokes Solutions for Small Reynolds Numbers, J. Math. Mech., vol. 6, pp. 585-593, 1957. Kevorkian, .J. and Cole, J.D.; Perturbation Methods in Applied Mathematics, Springer Verlag, Heidelberg, 1981. Kistler, A. L., Fluctuation Measurements in a Supersonic Turbulent Boundary Layer. Phys. Fluids, vol. 2, pp. 290-296, 1959. Kistler, A. L. and Chen, W. S., A Fluctuating Pressure Field in a Supersonic Turbulent Boundary Layer, J. Fluid Mechanics, vol. 16, pp. 41-64, 1963.

(

Lagerstrom, P. A. and Casten, R. G.; Basic Concepts Underlying Singular Perturbation Techniques, SIAM Review, vol. 14, pp. 63-120, 1972.

(

Lagerstrom, P. A.; Matched Asymptotic Expansions; Springer Verlag, Heidelberg, 1988.

( ( (

Lighthill, M. J.; On Boundary Layer Upstream Influence. Part II: Supersonic Flow without Separation. Proc R Soc London A, vol. 217, pp. 478-507, 1953.

( ( (

151

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( Liou, M. S. and Adamson, T. C., Interaction Between a Normal Shock Wave and a Turbulent Boundary Layer at High Transonic Speeds. Part II: Wall Shear Stress. ZaMP, vol. 31, pp. 227-246, 1980. Long, R. R. and Chen, J.-C.; Experimental Evidence for the Existence of the "Mesolayer" in Turbulent Systems, J. Fluid Mechanics, vol. 105, pp . .19-59, 1981. Mellor, G. L.; The Large Reynolds Number Asymptotic Theory of Turbulent Boundary Layers, Int. J. Engng. Sci., vol. 10, pp. 851-873, 1972. Melnik, R. E.; An Asymptotic Theory of Turbulent Separation, Compt. & Fluids, vol. 17, pp. 165-184, 1989. Melnik, R. E. and Grossmann, B.; Analysis of the Interaction of a Weak Normal Shock Wave with a Turbulent Boundary Layer. AIAA paper No. 74-598, 1974. Messiter, A. F.; Interaction Between a Normal Shock Wave and a Turbulent Boundary Layer at High Transonic Speeds. Part I: Pressure Distribution. ZaMP vol 31, pp. 204-227, 1980 Meyer, R. E.; On the Approximation of Double Limits by Single Limits and the Kaplun Extension Theorem, J. Inst. Matl1s. Applies., vol. 3, pp. 245-249, 1967.

(

{ ~

( (

(

( ( (

( (

Morkovin, M. V.; Effects of Compressibility on Turbulent Flows, Int Sym on "Mecanique de Ia turbulence", pp. 367-380, 1962.

(

Nayfeh, A. H.; Perturbation Methods, John Wiley, N.Y., 1973.

(

Nipp, K.; An Algorithmic Approach for Solving Singularly Perturbed Initial Value Problems, Dynamic Reported 1, John Wiley, N.Y., 1988.

(

O'Malley, R. E. Jr.; Introduction to Singular Perturbations, Academic Press, New York, 1974.

( Sawyer, W. G. and Long, C. J.; A Study of Normal Shock-Wave Turbulent Boundary-Layer Interactions at Mach Numbers of 1.3, 1.4 and 1.5. Royal Aircraft Establishment, Technical Report No 82099, 1982.

(

Silva Freire, A. P.; An Asymptotic Approach for Shock-Wave/Thanspired Turbulent Boundary Layer Interaction. ZaMP, vol. 39, pp. 478-503, 1988.

(

Silva Freire, A. P.; On the Matching Conditions for a Two-Deck Compressible Turbulent. Boundary Layer Model. ZaMM, vol. 69, pp. 100-104, 1989. Silva Freire, A. P. and Hirata, M. H.; Approximate Solutions to Singular Perturbation Problems: the Intermediate Variable Technique, J. Math. Analysis and Appl., vol. 145, pp. 241-253, 1990. Stratford, B. S.; An Experimental Flow with Zero Skin-Friction throughout its Region of Pressure Rise, J. Fluid Mechanics, vol. 5, pp. 17-35, 1959. Sychev, V. V. and Sychev, V. V.; On Turbulent Boundary Layer Structure, P.M.M. U.S.S.R., vol. 51, pp. 462-467, 1987. Townsend, A. A.; The Structure of Turbulent Shear Flow, Cambridge University Press, 1976. Van Dyke, M.; Perturbation Methods in Fluid Mechanics, Parabolic Press, 1975.

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4.

( ( ( Vogel, J. C. and Eaton, J. K.; Combined Heat Transfer and Fluid Dynamic Measurements Downstream of a Backward Facing Step, Journal of Heat Transfer, vol. 107, pp. 922-929,

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1985.

(

WMow, W ; Asymptotic Expansions for Ordinary Differential Equations, lnterscience, 1965.

(

Yajnik, K. S.; Asymptotic Theory of Turbulent Shear Flow, J. Fluid Mechanics, vol. 42, pp. 411-427, 1970.

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NONLINEAR EVOLUTION OF A THREE-DIMENSIONAL WAVETRAIN IN A FLAT PLATE BOUNDARY LAYER

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( Marcello A. Fa.raco de Medeiros

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Departamento de Engenharia Medi.nica

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Pontiflcia Universidade Cat6lica de Minas Gerais Av. Dom

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500, Belo Horizont.e, 30535-610 - MG - Brazil

F.rmail:marcello@mea. pucminas. br

Abstract

Introduction

1

This paprr presents results of an experimen-. The laminar-turbulent transition point is often tal study of the transition in boundary lay- a crucial element in the design of wings. In ers. The experiment.s were conducted in spite of it., the prediction of the transition point a. low-turbulence wind tunnel. The process is still a challenge in fluid mechanics. Transiwas triggered by a. three-dimensional Tollmien- tion is affected by a large number of parameter, Schlichting wa.vetrnin excit.ed by a. harmonic such as free-stream turbulence, pressure gradipoint source in the plate. Hot-wire a.nemome- ent, conditions of the leading edge and degree try was used to measure the signal and investi- of isotropy and homogeneity of the free-stream gate the nonlinear regime of these waves. It was turbulence. It lias now been established that there observed that the three-dimensional wavetra.in behaved very differently from two-dimensional are a number of different routes to transition. ones. In particular, it did not involve the Among t.hem the most. studied is t.he To\lmiengrowth of subharmonir.'l or higher harmonics. Schlichting route. This route involves the expoThe first. nonlinear signal t.o appear was a mean nent.ial amplification of two dimensional waVP.'I, flow distortion. This had a spanwise struc- which, if the amplitude is large enough, give ture consisting of regions of positive and neg- rise to three dimensional waves via a secondary ative mean distortion distributed like longitu- instability -mechanism (Herbert 1988). The dinal streaks, which became more complex as three dimensional waves can be of two types the nonlinearity developed. The structure in depending on the kind of resonant interacthe direction perpendicular to the wall has also tion that occurs. With primary resonance the been studied. Initially the distortion profiles three dimensional waves have frequency equal resembled Klebanoff modes, but further down- to the fundamental two dimensional waves, stream it also became more complex. Else- whereas with parametric resonance the three where studies have revealed the existence of dimensional waves are subharmonics of the streak-structures in turbulent flows. It is con- fundamental. In both cases the wave syRjectured that t.he current experiments may pro- t.em that arisP.'I, saturate!! in the form lambda vide a link bet.ween P.arly wave-like instabili- vortices (Corke & Mangano 1989, Kachanov ties and some coherent structures of turbulent & Levchenko 1984, Kachanov 1987, Kachanov boundary layers. 1994). With subahrmonic resonance the vor-

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bance, namely, t.hat generated by a white noise nat.ing from a point source. This paper presents excitations. Shaikh compared the evolution some results of these invest.igations. Prelimiof disturbances generated from different white nary results have been presented by Medeiros noise sequences. The sequences had the same (19!)7 d) and Medeiros (1997 a) spectral content, only the phase of the comInvestigation of three-dimensional waveponent.s relat.ive t.n each other were different.. trains has also been carried out by other reHe observed that. the transition point. was very searchers (Kachanov 1985, Mack 1985, Seifert sensitive t.o t.he phase composition of the distur- 1990, Seifert & Wygnanski 1991, Wiegand, bance, and concluded that the amplitude was BP~c;t.ek, Wagner & Fa.'lel 1995), hut these were not the only parameter affecting' the process. n~st.rict.ed to the linear st.age. The works of Gaster and Shaikh pointed to the fact that. the nonlinear mechanisms in Experimental Results the evolution of modulated waves were different 2 from those observed in plane wavetrains. However the wavepackets of Gaster were too restric- The experiments were condnc:t.ed in t.he low t.urUniversity of Camtive while the white noise sequences of Shaikh hnlence wind tunnel of the 1 bridge, Cambridge, UK • The boundary layer were too generic, making it. difficult. to int.erpret the results. By allowing the amplitude developed over a flat plate. The pressure graof the wavepackets to be complex, Medeiros dient was controlled with the help of a flap at (1996, 1996) generated experimentally a set the t.railing edge of t.he plate. With this set up of wavepackets with identical envelopes but a fairly small pressure gradient was obtained, wit.h different. pha.'le relative to t.he envelope. figure 1. Alw, t.he profiles mea.'lured were close With this he was able to show the influence to the theoretical Blasius profile, figure 2. The on the phase of the evolution of modulated measurements covered the region ± 200mm in waves (Medeiros 1997b). Later he also showed the spanwise direction from the centreline with experimentally that although the subharmonic profiles 2cm apart.. resonance appeared to be present in the process, it alone could not explain the observations (Medeiros 1997c). The nonlinear mechaM>•··••••• T ; nism observed involved the production of subharmonic waves. The fact that not only amplitude but also phase affects the transition process is oflarge practical importance because the 0 0.2 0.4 0.6 0.8 I 1.2 1.4 distance from leading edgelm transition prediction met.hods so far used have only acconnt.ed for the amplitude of the waves. Figure 1: The static pressure distribution over The wavepackets are modulated both in t.he plat.e streamwise and spanwise directions. At first it appeared that the important ingredient was The disturbances were introduced in the the streamwise modulation, rather than the spanwise modulation (Gaster 1984, Medeiros & flow via a loudspeaker embedded in the plate Gaster 1994, Medeiros & Gaster 1995), how- and communicated to the How through a .3mm ever numerical simulations of two-dimensional hole, located on the centreline of the plate wavepadret.s by Medeiros (1996) have shown 203mm from the leading edge. The velocthat. spanwise modulation is essent.ial t.o t.he ity records were measured with a 2.5~m gold process. This ha.'l led t.o t.he invest.igat.ion of plate tungsten hot-wire connected to a constant 1 This tunnel is now located at Queen Mary and the nonlinear evolution of waves that are only spanwise modulated, namely, a wavetrain ema- Westfield College, London University, London, UK

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Figure 2: The mean flow. Lines represent the Tollmien-Schlichting waves decay. The waves theoretical Blasius profile. Asterisks represent cross branch II at. Ri around 1700. the measured values The evolution of the disturbances observed experimentally along the centreline is shown in temperature anemometer. The hot-wire was figure 4. The measurements were taken at a mounted on a three-dimensional traverse gear. nondimensional distance of 0.528•. Using finite The streamwise and spanwise positions of the wavetrains, the disturbance is an event that can wire could be directly measured from the tra- be repeated. Therefore ensemble averages can verse mechanism. The distance from the wall be taken in order the get a clearer signal. The had to be obtained indirectly by measuring t.he records displayed her,,, as well as those shown velocity profile and comparing to the theoret- in other figures, were oht.ainecl from 64 ensemically obtained. The procedure has been used bles. The first important. observation is that. in similar experiments in this tunnel (Shaikh wave amplit.ucle.~ grow up to R& about. 1700 1993) and has proved to be fairly accurate. and thereafter decay, consistent with the linMore details of the experimental set up are ear theory, figure 3. A mean flow distortion is also observed that is not predicted by the lingiven by Medeiros (1996). In experiments with wavetrains the flow ear theory. Initially the distortion is negative, is usually disturbed by a continuous harmonic hut. further downstream switches to posit.ive. It. source. In the current series of experiments is remarkable that the change in the trend of a long but finite 200Hz wavetrain is excited the mean flow distortion occurs close to where from 11. point. source. The linear evolution 11. the disturbance crosses branch II. The mean two dimensional mode wit.h freqnenr.y 200Hz flow distortion is made very clear by the use is shown by the straight line on the instabil- of finite length wavetrains. It could have reity diagram, figure 3. The excitation was in- mained undetected if a continuous wavetrains troduced at R, about 800 and measurements were used. It is possible that the use of continuwere taken up t.o Ri 2100, as indicat.t>.d by the ous wavetrains have prevented these mean flow dashed lines in the figure. One can see that. the distortions from being observed in previous exexcitation was introduced upstream of branch periments with three-dimensional wavet.rains. I of the neutral curve and that measurements Figure 5 shows the evolution of a wavewere taken beyond branch II, after which the train of considerably smaller amplitude. Here

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Figure 7: SpMwise structure of the disturbance Figure 9: Spanwise structure of the mean flow field 900mm from t.he leading edge and 0.52!5* distortion (streaks) 900mm from the leading from the wall. edge and 0.52!5" from the wall. Row distortion. This structure suggests the existence of a pair of counter-rotating concentrationR of vort.icit.y which would push down high momentum fluid in t.he la.t.eral regions and lift. up low momP.nt.um fluid in t.he cent.ra.l region. However, it is a.s yet unclear whether these mechanisms are actually taking place. The concentrations of vorticity are probably too weak to be considered vortices and perhaps the lift up/push down effect is too small to affect the flow. As the waves evolve, the structure becomes more r.omplex. At x=lOOOmm

the appearence of a region of positive mean flow distortion right at the center of the wavetrain is observed. This corresponds to the change in t.he Rign of the me11.11 How dist.ort.ion ohservP.d in figure 4. From station x=lOOOmm onwards the struct.ure doesn't. experienr.e remarkable changes, apart from the broadening of the central positive mean flow distortion. It is interesting to look at the evolution in the frequency domain which was carried out by Medeiros (1997a). However, the spanwise resolution of the experiment is relatively low. The

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Figure 10: Downstream evolution of the streaks results became difficult to interpret because alias effects could not be ruled out. Therefore, care should be taken in analyzing those results. What. is dear is that initially the nonlinear mechanism generates only t.wo region of positive mean flow distortion and one central region where the distortion is negative, while further downstream the span wise wavenumbers become significantly larger. Also important is to investigate the structure of the mean !low distortion in the direction normal to the wall. Figures 11 and 12 show r.ontom plots of the mean How distortion on planes perpendicular to the How direction 900mm and llOOmm. The mean How clist.ortion is concentrated inside the boundary layer. In the external part of the How no sign of the mean How distortion is observed. At x=900mm the st.mcture is ha..<>ically composed of two regions of positive mean flow distort.ion and a central region of negative mean How distortion. Whereas the positive lumps are fairly concen-

~.:~ Figure 12: The mean flow distortion distri!Jution on a plane perpendicular to the ftow direction llOOmm from the leading edge. trated the negative region spreads over a larger portion. Moreover, the negative region appears to be composed of several lumps. The profile

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( resembles that of a Klebanolf mode with a sin- the displacement thickness variation over the gle maximum inside the boundary layer, partic- entire disturbance field, figure 14. In this picularly for the posit.ive streak.'!. The maximum ·~·r-------~--~--~--~-----is located between q(= II•~) 1 and 2, which 11001 is also similar to that of ki~banoff modes. At x= llOOmm the negative central region splits into a number of regions and lumps of positive '""' mean flow dist.ort.ion arillf!. At t.hiR st.age the I <'.f.ntral part of the dist.urb11.11ce t.he signal is too r.omplex for any definite r.onclnsion concerning J ... the location of the maximum. The evolution of the profile along the atreamwise direction may be more meaningful "" than the analysis of the signal at a particular streainwise station, figure 13. The figure shows

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ture the details of the complex distribution in the direction perpendicular to the wall are lost, and an averaged view of the distortions rl'.sults. The figure shows a somewhat more symmetric pictures of the Row than that of figure 10. For the positive lateral regions the displacement thickness distribution indicates a structure similar to that suggested by figure 10. The central region, on the other hand, indicated that the cent.ral negat.ive region split.s into t.hree regions separP.t.M by two newly generat.P.d posit.ive regions.

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Figure 13: F:volnt.ion of t.he mean flow distortion. how the disturbance field slowly evolves from a relatively simple structure into a much more complex one at. x= llOOmm. An overall view of the transfer of momentum from the low velocity streaks to the high velocity streaks is given by the distribution of

3

Conclusion and Discussion

The work studies the nonlinear evolution or wavP.trains em11.11ating from a point sourcP. in a flat. plat.e boundary layer. The first int.ereRt.ing result. wa.o; that the first indication of nonlinear behaviour was not a subharmonic signal, but a mean flow distortion that formed longi-

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tuuinal streaks. This mean How distortion may wavetraius. It is possible: that. l.lw modulahave arisen from self interaction of the modes tion of the waves provide a short cut between via the Reynolds stresses term, but this con- the early wavelike behaviour and the vortical jecture is yet unproven. Initially the mean flow structures observed in turbulent llows. These distortion displayed a relativelly simple span- conjecture is currently being investigo.tc•l. The wise stmr.t.nre with a c:entral region of negative possibility that. the initial mean flow dist.ort.ion mean flow distortion and two lateral regions be generated from the self interaction of waves of positive mean flow distortion. It. appeared is a.lso heing investigated. that the strength of the nonlinear interaction at this stage was stronger off the centerline of the flow. This might be linked to the fact that for References some frequencies the three-dimensional wavetrains also display amplitude maxima off the Breuer, K. S. & Harit.onidis, J. H. (1990}, 'The evolution of a localised dist.urhance in a centerline in the linear regime (Wiegand et al. laminar boundary layer - part 1 - weak 1995}. This would be consistent with these bedisturbances', .!. FlUid M.:ch. 220, 569ing generated from the Reynolds stresses. 594. The st.rur.ture of these dist.ort.ions in the direr.t.ion perpendicular t.o the wall was also Breuer, K. S., Cohen, .1. & Uaritonidis, .1. H. investigated. Initially they resemble klebanolf ( 1997}, 'The late stages of transition inmodes with one amplitude peak at a position 1) duced by a low-amplitude wavepacket in between 1 and 2. a laminar boundary layer', .1. Fluid Mech. 340, 395-·411. The relativelly simple structure gives rise to a fairly complicated flow field further downstream. The more complicated field appears Cohen, .1. (1993}, 'The initial evolution of a wave packet in a boundary layer', Phvs. to originate at a position close to the second Fluids 6(3}, 1133-11.13. branch. The more complicated structure arises in the central portion of the wavetrain, where Cohen, J., Breuer, K. S. & llaritonidhs, .1. H. the initial negative mean flow distortion was (1991}, 'On t.he evolution of a wave packet formed. There, regions of positive mean flow in a laminar boundary layer', ./. Fluid distortion arises and the profiles of the distorMech. 225, 575-606. tions do not display the Klebanoff mode shape. Despite the complexity, an overall view shown Corke, T. C. & Mangano, R. A. (1989), ·Resby t.he displacement. t.hickenes.'l variat.ion over onant growth of three-dimensional modes the plate, suggests that. the central negative rein t.ransitioning Blasius boundary layers', gion splits into three. The positive mean flow .1. Fluid Mech. 209, 93-150. distortion regions do not change considerably Elofsson, P. A. & Alfredsson, P. II. (1997), 'An along the process, except in amplitude. experimental study of oblique transition in Streaks have also been observed in by pass plane poiseuille flow'. (submitted to the .1. transition. However, there the streaks tend to Fluid Mech. ). keep their spanwise spacing, as oppose to what is observed in the current experiment. Longi- Ga.~ter, M. (1975), 'A theoretical model of a tudinal streaks have been observed in turbuwave paeket. in the boundary layer on a flat. lent. flow. In same cases they appear t.o he plat.e', Pror.. R. Sor.. Londm1 A 347, 271a key ingredient. of the mechanism of produc289. tion of turbulence. It. has also been shown that modulated waves give rise to transition Gaster, M. (1978), The physical process at smaller amplitudes in comprison with plane causing breakdown to turbulence, in

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165

I

t ( '12th Naval Hydrodynamics Symposium', Mack, L. M. (1985), Instability wave pat.terns from harmonic point sources and WMhington. line sources in laminar boundary layers, Ga..'lter, M. (1984), A non-linear transrer in V. V. Kozlov, ed., 'Laminar-turbulent function description of wnvP growth in transition', springer-Verlag, pp. 125-132. a boundary layer, in V. V. Kozlov, P.d., 'LIUllinar-t.nrhnlent. t.ranslt.ion', IU- Medeiros,. M. A. F. (1996), The nonlinTAM Symposium, JtJTAM, Springerear behaviour of modulated TollmienVerlag, pp. 107 -114. Schlir.hting waves, PhD thesis, Cambridge University - UK. G&..'ltl'!r, M. (1993), Theorigins~fturbulence, in 'Proc. ConLon New Approaches and ConMe1lP.iros, M.A. F. (1997a), Laminar-t.nrhulent cepL'I iu 1'nrhulenc:P.', Birkhauser Rase!, transition: the nonlinear P.volnt.ion of Switz1•rland. three-dimensional wavetrains in a laminar boundary layer, in 'Proc. of the XIV Gaster, M. & Grant, I. (1975), 'An experimenBrA.?.ilian Congress of Mechanical Engi' t.al invest.igat.ion or t.lu~ formation and dP.neering', Bauru. in CD-ROM. VPlopmtml. of a wavepacket in a laminar boundary layer', Pror:. Royal Sor.. of l,onMedeiros, M.A. F. (1997b), 'The nonlinear evorlon A 347, 253-269. lution of wavepackets in a laminar boundary layers: Part i'. (submitted to the .J. Henningson, D. S., Lundbla.dh, A. & .JohansFluid Mech.}. son, A. V. (1993}, 'A mechanism for hypRs.CI t.rllnsit.ion from lor.alizP.d clisturhances in wall-bounded shear flows', .!. Pluid Medeiro.c;, M.A. F. (1997r:), 'ThP nonlinear evolution of wavepackets in a laminar boundMrr:h. 250, 169-207. ary layers: Part ii'. (submitted to the J. Herbert, T. ( 1988}, 'Secondary instability of Fluid Mech.). boundary layers', Ann. Re11. Fluid Mech. 20, 487-526. Medeiros, M. A. F. (1997d}, Nonlinear meanflow distortion caused by a. wavetrain emKachanov, Y. S. (1985), Development ofspat.ial anating from a harmonic point source in a wave packets in boundary layer, in V. V. flat-plate boundary layer, in 'EURDMEC Kozlov, ed., 'Laminar-t.nrhulent. t.mnsicolloquium on stability and transition of t.ion', ~pringPr-Verlag, pp. 115-123. boundary layer flows', Stuttgart. Kachanov, Y. S. (1987), 'On t.he resonant nature of the breakdown of a laminar bound- Medeiros, M. A. F. & Gaster, M. (1994), ary layer', .T. Pluid Mer.h. 184, 43-74. The nonlinear behaviour of modulated Toll mien-Schlichting waves: experiments and computations, in 'Second EUROMEC , Kachanov, Y. S. (1994), 'Physical mechaConference', Warsaw - Poland. nisms of laminar boundary layer transition', Ann. Re11. Fluid Mech. 26, 411-482. Medeiros, M. A. F. & Gaster, M. (1995), The nonlinear behaviour of modulated Kar.hMov, Y. S. & Levehenko, V. Y. (1984), 'The resonant. int.erar.t.ion of disTollmien-Schlir.ht.ing waves, in 'IUTAM Conference on nonlinear inst.ahilit.y and t.urhanres at. laminar-t.nrhnlent t.ransit.ion in a boundary layer', J. Pluid Mech. transition in tri-dimensional boundary layers', Manchester, pp. 197-206. 138, 209-247.

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166

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Medeiros, tv!. A. F. & Gaster, M. (199G), 3d structures of nonlinear wavepackets generated from different excitations in a boundary layer, in 'EUROMEC colloquium on dynamics of localised disturbances in engineering flows', Kalrsruhe. Seifert, A. (1990), On the interaction of small amplitude disturbances emanating from discrete points in a Blasius boundary layer, PhD thesis, Tel-Aviv University. Seifert, A. & Wygnanski, l. (1991), On the interaction of wave trains emanating from point sources in a Blasius boundary layer, in 'Proc. Con f. on Boundary Layer Transition and Control', The Royal Aeronautical Societ.y, Camhriclge, pp. 7.1-7.13. Shaikh, F. N. (1993), Turbulent spot in a transitional boundary layer, PhD thesis, Cambridge University. Shaikh, F. N. (1997), 'Investigation of t.ransit.ion l.o t.nrhulenr:e using white noise excitation and local analysis techniques', J. Fluid Mech. 348, 29-83. Shaikh, F. N. & Gaster, M. (1994), 'The nonlinear evolution of modulated waves in a boundary layer', Journal of Engineering Mathematics 28, 55-71. Wiegand, T., Bestek, H., Wagner, S. & Fasel, H. (1995), Experiments on a wave train emanating from a point source in a laminar boundary layP.r, in '26th AIAA Fluid Dynamics Conference', San Diego, CA.

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ANALISE DE TRANSICAO DA CAMADA LIMITE SOBRE A PA DE UM

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MODELO DE TURBINA EOLICA DE EIXO HORIZONTAL

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ANALYSIS OF BOUNDARY LAYER TRANSITION ON A BLADE OF HORIZONTAL AxiS WIND TURBINE MODEL

( (

Jaqueline B. do Nascimento

(

Fernando Martini Catalano

(

Universidade de Sao Paulo - USP

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Escola de Engenharia de Sao Carlos - EESC

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Departamento de Engenharia Meciinica

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Laborat6rio de Aeronaves

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Av. Dr. Carlos Botelho, 1465 CEP 13560-250 - Sao Carlos - SP - Brasil e-mail:jaquelin@sc.u;-p.br

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Abstract

(

11tis work pres ellis all aerodynamic analysis of boundaiJI. layer tra11sitio11 011 blades of a horizo11tal axis wi11d turbine model.

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The techniques used were a mixture of 11aphthaline a11d trichloride sprayed 011 swfaces of blade. The tests were pet:formed

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with a blade ill a11d out of the wind twmel (stopped a11d rotatillg, respectively), in the Aircraft Laborat01y of the L{11iversity

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of Silo Paulo. The visualizatioll tests present qualitative and quantitative informatiolls about boundary layer tratlfilion

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phenome11011 upo11 the blade. The results obtained show that the boumlary layer tra11sition took place before the inner of the

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regio11s blade, where the qfter part of the aitfoi/s are co11cavet, showi11g the i11jluence of aitfoil type on tra11sition position.

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Key-words: bou11dmy layer tra11silio11, aitfoil, wind turbines, visua/izatio11.

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169

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I Resumo

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Este trabalho apresenta

ID118

anilise acrodinilmica do fcnomeno de

transi~o

de camada limite, sobre as pas de

de turbina e6lica de eixo horizontal. A metodologia adotada utiliza mna tecnica de

visuali7.a~ilo

por

sublima~ilo

ID11

modelo

(

(mistura de (

naltaleno com wn diluente), Cl\ia mistura

e pulvcrizada

sobre todas as ruperficies das pas (inlradorso e exlradorso). Os

ensaios foram realizados com a pa parada, dentro da cfunara de ensaio do time! de vento do Laborat6rio de Acronaves da Universidade ~ Silo Paulo; bern como com o rotor girando livrcrncnte. 0 ensaio de

visualiza~i!o

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apresenta infonna~oes ni!o (

apenas qualitativas, mas tambCm quantitativas sobre o fcnomeno de obtidos mostnun que a

tnnsi~i!o

transi~i!o

da camada limite ocorrcu mais cedo na regiilo da p3 proxima a rai7, onde os pcrfis

transi~o

da camada limite, acrof61ios, turbinas e6licas,

I

f (

aerodinilmicos possuem concavidade na parte traseira, comprovando a influencia do tipo de perfil na posi~o da transi~o. Palavras-chave:

I

do cscoamento sobre a pa. Os resultados

visualiza~o.

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Introdu~iio

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0 estudo aerodinamico de turbinas e61icas tern, nos Ultimos tempos, despertado o interesse por

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parte dos pesquisadores do mundo inteiro. A analise aerodinamica dessas maquinas envolve parametros

(

variados. Entre estes, a escolha do perfil adequado para o uso em turbinas e61icas e de grande

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importlincia. Com o desenvolvimento de perfis especfficos para este tipo de uso, que surgiu com o

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( conhecimento adquirido na area aeronautica, varios metodos de analise tf!m sido desenvolvidos,

OS

(

quais tern contribuido muito para o desenvolvimento cientffico e tecno16gico das turbinas e6licas de

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eixo horizontal.

(

Bjiirck ( 1989) apresenta urn estudo sobre o desenvolvimento de novos aerof6lios no Instituto de ( Pesquisa Aeronautica da Suecia, e como ponto de partida do estudo, foram colocadas as caracteristicas ( I

desejaveis de urn aerof6lio, as quais podem ser divididas em propriedades estruturais e propriedades ( ' aerodinlimicas. Estruturalmente, a espessura e a geometria dos aerof6lios !!do os parlimetros mais { importantes. Ja as propriedades aeodinamicas consideradas sao:

rela~iio sustenta~iio-arrasto,

(· coeficiente (

( 170

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de sustenta<;iio de projeto, coeficiente de sustenta<;iio maximo e comportamento do estol, baixa sensibilidade a diminui<;iio do desempenho devido a rugosidade do bordo de ataque e momenta de 'pitching'. I

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Este trabalho, desenvolvido no Laborat6rio de Aeronaves da Universidade de Sao Paulo,

(

apresenta urn metoda de analise aerodiniimica sobre a pa parada (dentro da camera de ensaio do tune!

(

de vento) e com a pa girando (rotor girando livremente na saida do tlmel), cujo objetivo foi estudar o

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fenomeno de transi<;ao da camada limite. Os perfis adotados na constru<;iio da pa foram FFA-W 1-xxx (com espessura variada de 27.1% a 12.8%c e sao perfis especificos para turbinas e61icas.

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0 Estudo Aerodinamico e a Visualiza~io da Camada Limite

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0 estudo aerodiniimico das pas de urn rotor e61ico concentra-se basicamente no comportamento da camada limite sobre estas, ou seja, no comportamento do fluxo que escoa em regiao limitada ao

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redor das piis. Isto, de urn modo geral, envolve esfor<;os para que o desempenho aerodiniimico seja o maior possivel, obtendo-se urn aumento no desempenho total do sistema. Os metodos de visualiza<;ao de escoamento sao importantes nao apenas para apresentar uma

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visao qualitativa clara do fenomeno do escoamento, mas tambem, em muitos casos, produzir

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informa<;ao quantitativa. Existem uma serie d!! tecnicas para indica<;ao da posi<;ao da transi<;ao de

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escoamento laminar para turbulento na camada limite do corpo, e estes dependem do comportamento

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fisico ou quimico do deposito localizado na superficie do corpo (Pankhust & Holder, 1968). Existe uma serie de metodos para indica<;ao da posi<;ao de transi<;ao de fluxo laminar para

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turbulento na camada limite de urn corpo, dentre os quais destaca-se o metoda de sublima<;ao, que foi.

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adotado por Nascimento (1998a). Esta consiste, basicamente, na pulveriza<;ao de uma mistura de nafataleno (por exemplo) com urn liquido voliitil, sobre a superficie da pii, de modo que o dep6sito

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171

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resultante toma-se sensivelmente aparente e podem, desta forma, serem registrados por meio de

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imagens fotognificas (Figura I).

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I

Os perfls adotados e suas caracteristicas aerodiniimicas

Os perfis adotados napa estudada neste trabalho (WI-271, Wl-242, Wl-211, Wl-182, Wl-152



(

e Wl-128) foram desenvolvidos pelo Institute de Pesquisa Aeronautica da Suecia (The Aeronautical

(,

Research Institute of Sweden, FFA)( Bjorck, 1989).

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Segundo (Bjorck, 1989), o .desenvolvimento de novas perfis na FFA foi uma tentativa de se

( obter aerof6lios com boas caracterfsticas tanto em fluxo turbulento, como em fluxo laminar. Este, (

I

realizando ensaio de visualizar;iio em tilnel de vento, atraves de tufas de Iii, sobre o aerof6lio Wl-152 e

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\

Wl-128 (espessura de 15.2% e 12.8% da corda, respectivamente), verificou que, na lateral de pressiio,

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apresentaram um comportamento completamente laminar. Ja na lateral de sucr;iio apresentaram urn

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comportamento laminar numa faixa de 40% do fluxo sobre a supericie.Uma outra observaciio

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importante quanto a estes aerof61ios eo seu desempenho em alto Nfunero de Reynolds, uma vez que,

a (! (

medida que o mimero de Reynolds aumenta, aumenta tambem a margem de separar;iio.

< ( M~todo

de VisuaHza~io de Escoamento por meio da

T~cnica

de SubHma~io

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Foi realizado um ensaio de visualizar;iio de escoamento sobre a pa de um modelo de turbina ( e6lica, projetado e construido no Laborat6rio de Aeronaves da Universidade de Sao Paulo. A pa ( ensaiada tern 0.9m de raio (considerando o difimetro total do rotor), mas o comprimento de raio (

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adotado no ensaio foi de 0.84m, o qual corresponde o comprimento entre a primeira estar;iio (aerof61io ( 1

W!-271) e a ultima (aerof61io Wl-128).

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Figura I - lmagem da pa dentro da climara de ensaio do time! de vento

(lingula de ataque de 16

graus), ap6s o registro da transir;iio da camada limite por meio do metoda de sublimar;iio com naftaleno.

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Figura 2- Imagem (aproximada) da pa na saida do tlinel de vento (angulo de passo de 5 graus), ap6s o registro da transir;ao da camada limite par meio da tecnica de sublimar;iio , com o registro da transir;ao da camada limite.

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173

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< ( ( 0 ensaio foi dividido basicamente em duas etapas, ambas com Numero de Reynolds de 200000.

(

A primeira etapa consistiu no ensaio da pa parada dentro da cftmara de ensaio do tUne! de vento, com ( Angulo de ataque (a) variando de 0 a 16 graus. A segunda etapa consistiu no ensaio com o rotor

(

girando, utilizando o fluxo na' saida do tUne! de vento, na regiiio proximo ao ventilador. Para este

(

ensaio, o tUne! de vento recebeu alguns cuidados tecnicos quanto a uniformiza~ao de seu fluxo, uma

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vez que este apresentava-se bastante turbulento e rotacional. Desta forma, foi colocada uma tela (

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proxima ao ventilador e uma colmeia na saida do time! como meio de uniformizar o fluxo de saida I

'

naquela

(

se~ao.

Quanto a faixa de angulo de ataque (a) do ensaio, este foi calculado considerando o

Angulo de passo

(

(p - angulo de montagem da pa no cubo do rotor). Uma vez fixado o angulo de passo (

da pa, calculou-se o angulo de ataque equivalente. Assim, foi passive! estabelecer uma compara~ao (

1

entre os dados levantados com a pa parada e com as pas do rotor em funcionamento. Deve-se ressaltar ( que este Angulo de ataque, quando o rotor esta em funcionamento, varia de acordo com a

posi~ao

( ( ·'

radial.

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0 material utilizado para visualiza~iio do escoamento foi uma mistura de naftaleno com urn (

I

.I

solvente (tricloroetano). Esta mistura foi pulverizada sobre toda a superflcie da pa, tanto no intradorso ( como no extradorso, e ap6s a sublima~iio desta, considerando o efeito do fluxo sobre a pa, registrou-se ( imagens (Figuras I e 2) que possibilitaram a

visualiza~iio

da regiiio de ocorrencia da

transi~ao,

da (

camada limite laminar para turbulenta. As imagens registradas foram transpostas em curvas a tim de (

( fomecerem a posi~iio (em percentual da corda, x/c), da

transi~iio

envergadura ( percentuais do raio, r/R), em cada angulo de ataque.

da camada limite, ao Iongo de todo a ( (

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( 174

.

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Resultados e Discussiio

( (

Conforme citado anteriormente, o ensaio de visualizayiio de escoamento com o rotor girando, na

(

salda do tlmel de vente, foi feito variando-se o angulo de passe e calculando-se o angulo de ataque

(

equivalente. Deste modo, foi passive! analisar o comportamento do fluxo (transiyiio da camada limite),

(

escoando sabre a superficie da pa, comparando-o como caso parade.

( Enquanto a faixa de angulos de ataque no ensaio com a pa parada foi de 0 a 16 graus, no caso

( (

girando, esta faixa foi mais ampla, de -6 graus (na ponta) a 14 graus (na raiz), o que comprova que que

(

as turbinas e6licas de eixo horizontal trabalham em angulos de ataques altos nas regioes pr6ximas

{

raiz.

( ( (

a

Os graficos apresentados nas Figuras a seguir apresentam as curvas representativas do caso parade, enquanto que para o caso do rotor girando sao apresentados pontes indicatives do angulo de ataque, corresponde a situayiio da pa parada como forma de manter mna relayiio para analise.

{ (

Considerando inicialmente a posiyiio a I 0% do raio da pa, conforme apresentado na FIGURA 3,

(

onde o angulo de ataque da pa girando variou entre II e 14 graus, observou-se que a transiyao ocorreu

( (

antes para o caso parade do que no caso como rotor. A mesma tendencia pode ser observada para o I

caso com o rotor girando, a 20% do raio, conforme apresentado na FIGURA 4.

( Translejio da camada limite a 10'Ao do raiO

( ( (

( (

1-+-lllc(O 1)tunel

-' xlc(0.10)

I

~~:~ . ·=:::==:I 0

5

10

15

angulo de ataque

( (

Figura 3 - - Local da transiyiio da camada limite em relayiio

( (

(

175

a corda, a I 0% do raio da pa.

t (• (

( ( Trans~llo

da cam ada limite a 20%do ralo

O.S~· E•/c~02)~ne·l.

x/c(02~)J

(

........ ·-

(

0.6

~

(

.~

0.4

0.2

..........___

0

(

... -

----····~

10

5

0

15

(

angulo de ataque

(

Figura 4 - Local da

transi~ao

da camada limite em rela~iio

acorda, a 20% do raio da pa.

' j

I ' (

rota~iio

Apesar de os efeitos da

serem maiores na ponta da pa, observa·se de resultados

anteriores (Wood, 1991) que o estol e amenizado na parte da pa proxima

a raiz.

( (

0 resultado das

(

'

(

j

FIGURAS 3 e 4 apontam para uma extensiio maior de camada limite laminar nesta regi1io, o que pode confrrmar o atraso na

separa~iio

da parte turbulenta, pois a camada limite laminar s6 se sustentara se o

(

gradiente de pressiio for favoravel. Este gradiente favoravel de pressao pode ter sido resultado da

(

i

rota~ao.

(

'

(

I

Deve·se ressaltar que os testes foram restritos (pequena

nao foram utilizadas tecnicas de

visualiza~ao

seria necessaria o levantamento da

distribui~ao

da

separa~ao

varia~ao

de angulo de ataque) e que

da camada limite turbulenta. Alem disso

( de pressao ao Iongo da corda para observar o gradiente

de pressao. Resultados anteriores confirmam que, com a pa girando, o gradiente de pressao

e aliviado

na raiz.

~ (

( Ate 20% do raio, conforme observado anteriormente, a pa, no caso em que o rotor estava (

girando, apresentou uma faixa mais ampla de camada limite laminar a que pode ser, explicado pelo fato ( de os efeitos tridimensionais e de rota~ao sao menores na raiz do que na ponta. Prosseguindo com a analise da

transi~ao ao Iongo de todo o raio da pa, observou-se que em

(

(

( tomo de 30 a 40% do raio a tendencia anterior

come~ou

FIGURAS 5 e 6.

a se inverter, conforme pode ser visto nas

(> (

176

(

0

(

l

(

\ ( ( ( (

Trans ioda cam ada limite a JO"~doralo

o.7 c~~o~~ ... x/c(o.~cn.mllN>i ·:~·

(

0.6

(

0.5

. -...:.......

u 0.4

(

)I

( (



.......__.._

~ . ~

0.3 0.2 0.1 0

~-------··

0

5

angulo de

atlque

15

10

(

(

Figura 5 -- Local da transi~iio da camada limite em rela~iio il corda, a 30% do raio da pa.

( (

Trans~ioda camada

J

.il £""'-""'~~. ' """~l

(

{

)I

(

0.3 0.2 01

0+-------

(

0

5

10

15

Angub de atlque

(

(

limite a 40%do raio

~--~--·

Figura 6 - Local da transi~ao da camada limite em relar;:iio

a corda, a 40% do raio da pa.

(

(

Para r/R maiores, com a pa girando, em iingulos de ataques menores, a

transi~iio ocorre~

em

( (

( ( (

posi~oes

da corda inferiores, comparados ao caso da pa parada dentro do time! (FIGURA 6). Em I •

posir;:oes medias da pa, em tomo de 40 e 50% do raio (FIGURAS 6 e 7),esta inversiio .e mais evidente, uma vez que, a partir desta regiiio, os efeitos siio mais intensos, promovendo desta forma, transir;:iio da I

camada limite laminar para turbulenta, mais cedo.

(

( (

( (

(

{ (

177

t (• (

(

I ---

( llmslc;iio ca cam:talirme a !D'Io ooralo

(

L--;~~~___~_-~

o.sl ---- I o,a[~I o,4 -. ~I

~

02

I

'j) ~5

- --r--

----

----~----_J

.5

0

irgjode!ta:!Je

acorda, a 50% do raio da pa.

Trans~da camada limite a 60%1%> ralo

1:--------.---AAJ

L~l xJ_c:_(06)1~ _:_x!•:(03-

--

·l_~_;rr-==~

I

I

-5

5

0

~rguo

( ( (

1

(

j

(

)

15

10

Figura 7- Local da transir;:iio da camada limite em relar;:iio

(

10

de ataque

Figura R - Local da transir;:iio da camada limite em relar;:ao

(

( (

(

'

(

( 15

a corda, a 60% do raio da pa.

;

( ( (

Alem da regiiio media da pa, quando os angulos de ataque passam a ser negativos, a tendencia

(

de transir;:ao ocorrer mais cedo do que no caso parado continuou, conforme mostram as Figuras 8, 9 e
( Trans~oda

(

camada limite a 70%1%> ralo

F~(o:7)t.-;;;--::-xlc(O.nil

r----- ~-- -::r==~---------J ~I

0.41 _

~

(

(

(

0.2

~-~--'l+---~-~-10 -5 0 5 10 Argulo de ataque

---~-"-

(

~--··

15

------~

Figura 9- Local da transir;:ao da camada limite em relar;:iio

acorda, a 70% do raio da pa.

(

( (

( 178

( (

'

\ ( ( (

(

Tr~da camada.limite a 80"/odo raio

~~oB)tln!l

(

____xk
~-----------~c:::-=-._. ______

(

(

------·

-,~

~:+--~, ~~ 1

~

. \

1---~-1---.----~-------1

(

{

-10

-5

0

5

argw de 21aq..1e

10

15

( (

Figura I0- Local da transiyao da camada limite em relayao

acorda, a 80% do raio da pa.

( ( (

Observou-se tambem, ao Iongo do experimento, o efeito de alguns graos maiores de natataleno que induziram a transiyao da camada limite mais cedo, con forme mostra a Figura 2.

{ ( (

Conclusoes

(

Observou-se que o escoamento sabre o perfil apresenta uma extensao laminar bastante

{

considenivel. Com relayao ao perfil aerodinamico, nota-se que na regiao onde este possui concavidade

(

na parte traseira, devido ao gradiente de pressao adverso nesta regiao, a transiyao ocorreu mais cedo.

(

( (

(

Este efeito e ainda maior nesta regiao de concavidade, para pequenos angulos de ataque, podendo, inclusive ocorrer separay~o. Por esta razao e que que se utilizam, geralmente, estes tipos de perfis em regioes da pa que operam a altos angulos de ataque e baixa velocidade. Nota-se aiD?a que o tipo de

(

perfil influencia sobremaneira, na posiyao da transiyao; uma asa com urn mesmo perfil ao Iongo da

(

envergadura tenderia a apresentar uma faixa constante de escoamento laminar.

(

( (

Referencias Bibliogrificas Anderson Jr., J.D. (1991). Fundamentals of Aerodynamics. 2"ediyao, McGraw-Hill, New York.

( (

( ( (

179

(

'

(I (

,, (

Bjorck, A. (1989). Airfoil design for variable RPM horizontal axis wind turbines. Proceedings of AMSTERDAM EWEC'89.

( (

Bjorck, A. (1990). Coordinates and Calculations for the FFA-Wl-xxx, FFA-W2-xxx and FFA-W3-xxx series of airfoils for horizontal axis wind turbines. FFA TN. Stockholm.

( 1

(

Hill; D. C. & Garrad, A. D. ( 1989). Design of airfoils for wind turbine use. Wind energy: Technology ( '

·(

'

Visualiza~iio de Fluxo sobre as I

1

and Implementation. Proceedings of Amsterdam EWEC'89. Garrad Hassan and Partners Bristol. Nascimento, J. B. and Catalano, F. M. (1998 4 ) (submetido). "Ensaio de

(

Pas de urn Modelo de Turbina E61ica de Eixo Horizontal". Octavo Congreso Chileno de Ingenierif ( Mecanica, Santiago, Chile.

(

Nascimento, J. B. and Catalano, F. M. (1998). "Estudo Aerodiniimico do Efeito da Rugosidade no (

1

Desempenho de urn Modelo de Turbina E6lica de Eixo Horizontal". Tese de Doutorado, EESC- (

(

USP, Sao Carlos.

Pankhurst, R. S. and Holder, D.W. (1968). Wind-tunnel technique (an account of experimental methods ( in low- and high speed wind tunnels). Sir Isaac Pitman&Sons Ltd., London. Schlichting, H., "Boundary Layer Theory", 7th. ed., McGraw-Hill Book Company, New York, 1974.

' 1

( (

)

(

I

'

Tangier, J. L. & Somers, D. M. (1985). Advanced airfoils for HA WTs. Proceedings Wing Power'85 ( Conference, SERJ/CP-217-2902, Washington, DC. American Wind Energy Association, pp. 45-51. ( Wood, D. H. (1991). "A three-dimensjonal analysis of stall-delay on a horizontal axis turbine". Journal ( of Wind Engineering and Industrial Aerodynamics, 37, 1-14, Amsterdam.

( ' (

I

(

( (

( (

I

( 180

(

t

.1

\ ( (

Uma Solugao para Turbulencia Gerada par Grades Oscilantes

( (

(

A Theoretical Solution for Turbulence Generated by Oscillating Grids

(

( Harry Edmar Schulze Fazal Hussain Chaudhry Laborat6rio de Hidraullca Ambientai-CRHEA Departamento de Hidniulica e Saneamento Escola de Engen haria de Sao Carlos-Universidade de Sao Paulo C.P. 359, 13560-270, Sao Carlos, S.P., Brasil.

( ( (

(Trabalho desenvolvido no lnslitut fUr Hydromechanik, Universitat Karlsrul1e, Alemanha)

(

Abstract

( (

The k-6 model is used to quantify the turbulent field generated by oscillating grids for the situation of stationary turbulence. A theoretical solution for a single oscillating grid is obtained, which superimposes well with experimental data from other sources. Further, a general solution is furnished, which is applied to the case of two oscillating grids. The relevant phenomena m this kind offlow are the diffusion and the dissipation of the turbulent kinetic energy. 1:-xact solutions for the power consumption are furnished, as well as series solutions for the spatial behaviour of the turbulent kinetic energy and the energy dissipation rate. The series solutions show to be adequate to study the situation of two oscillating grids. Keywords: grid turbulence, diffusion-dissipation 111 turbulence, k-c model, isotmpic turbulence.

(

( ( ( (

(

Resume

( (

( (

( ( (

(

0 problema do equacionamento do campo turbulento gerado par grades ou grelhas asci/antes e abordado a partir do usa do modelo k-6. As simplificac;oes possiveis neste tipo de escoamento permitem propor uma soluc;iio te6rica simples para o caso de uma zlnica grelha osci!ante irnersa em urn jluido. Esta soluc;iio sobrepoe-se bern a dados experimentat:s encontrados na literatura da area. A turbu!encia para regioes niio-pr6ximas das grelhas envolve as processos de difusiio e dissipar,:iio da energia cim!tica turbulenta, sern haver advecc;ao ou produr,:iio des/a energia. ·Esta caracteristica e utilizada para explorar o uso do equacionarnento basico k-t: tarnbem para uma situac;iio geral, apresentando-se equac;oes que perrnitem averiguar mais imediatamente a forma esperada para os perfis da energia cinetica turbulenta, a/em de permitir calcular a potencia dissipada como fimc;iio de va/ores limites da energia cim!tica turbulenta. 0 equacionamento e utilizado, entiio, para obter a soluc;iio para o caso de duas grelhas asci/antes. Palavras-chave: turbulencia gerada par grelhas, difusiio-dissipac;iio em turbulencia, mode/a k-6, turbulencia isotr6pica.

( (

( ( (

( (

lntrodu9ao Textos classicos de turbulencia, como Hinze (1959), apresentam formas de abordagem para resolver o problema de turbulilncia gerada em urn escoamento a jusante de uma grelha, esta colocada 181

1

t

( ' perpendicularmente :l direyiio preferencial do escoamento. 0 uso da hipotese de congelamento de ( Taylor e da teoria da semelhanya de von Karman conduzem a resultados observados em laborat6rio, ( sendo, portanto, urn case chissico de soluyiiO para escoamentos turbulentos que podem ser denominados de "simples". Este case cliflssico e resolvido no contexte da turbulencia isotr6pica. E, ( contudo, interessante observar que outro escoamento de geometria razoavelmente simples envolvendo grelhas, nao e tile largamente discutido na literatura da area, no sentido de divulgar ( amplamente uma soluyao te6rica. E o case do escoamento gerado por grelhas ou grades oscilantes. ( Evidentemente o termo"escoamento", no case de grelhas oscilantes, esta vinculado ja ao campo turbulento em si, uma vez que nao existe urn escoamento medic que venha a se estabelecer em uma ( direyao preferencial,. mesmo considerando diferentes "compartimentos" ou regioes menores no interior do fluido em estudo. Todas as escalas de velocidade sao decadentes na direyiio normal a ( grade, nao havendo uma velocidade de referenda conhecida a priori. 0 problema e difusivo- ( dissipative e ai reside a dificuldade de sua resoluyiio. Para os termos difusivos devem ser ·feitas hip6teses simplificadoras que conduzam a uma soluyao viavel. ( , Algumas propostas de soluyao, visando descrever o comportamento de diferentes variaveis, sao encontradas na literatura, sempre associadas a hip6teses que buscam vinculo com a realidade { 1 fisica (Bouvard e Dumas, 1967, Thompson e Turner, 1975, Hopfinger e Toly, 1976, Nokes, 1988, ( De Silva e Fernando, 1994, Voropayev e Fernando, 1996, Srdic et al., 1996). No presente trabalho tambem e apresentada uma proposta de soluyiio, a qual foi elaborada ainda no contexte da ( 1 turbulencia isotr6pica, utilizando aproximayoes vinculadas a hip6tese de Boussinesq e ao modele k6. Esta proposta reproduz de forma satisfat6ria o comportamento observado para a energia cinetica ( turbulenta e para a taxa de dissipayiio de energia em uma regiao do escoamento gerado por uma { , grelha oscilante. Para o case de duas grelhas oscilantes, o equacionamento permite obter uma expressilo para o citlculo da potencia dissipada na regiiio entre as grelhas e ainda sugerir soluyoes por ( expansiio em serie de potencias, para as variaveis k e 6. ( '

(

0 Caso da Adveq:ao da Turbulemcia

(

Visando justificar algumas hip6teses adotadas no case da difusao-dissipayiio, apresenta-se aqui, inicialmente, a soluyiio usual do problema de advecyilo da turbulencia a partir da equayiio da energia cinetica turbulenta. A equayiio de conservayiio da energia cinetica turbulenta, em sua forma completa, e geralmente apresentada como:

ok +U ok =-_!!__[u.(u,u 1 +f!_)]-.,-;u-t3U, -vou, ou, t3 t

' t3 x,

t3 x,

'

2

p

1

'

t3 u, t3 u, 6=v----

t3 x,

t3 xi t3 xi

(I)

( (

(

( (2)

t3x1 oxj

I

(

'

(

'

k e a energia cinetica turbu]enta, 6 e a taxa de dissipayiiO desta energia, U, e a ve]ocidade media ( na direyiiO i, u, e a flutUayiiO de ve]ocidade na direyiiO i, V e a viscosidade cinematica e p e a \ flutuayiio de pressilo. Sem variayoes temporais e havendo apenas transferencia por advecyiio, esta ( equayao assume a forma:

(

_ ak

U-=-6 0 XI

(3)

I

(

'

( 182

(

(

I

'

( (

( (

( (

As observa<;:oes mostram que o numero de Reynolds da turbulencia (escala de velocidade turbulenta multiplicada pela macro-escala de turbulencia, sendo este produto dividido pela viscosidade cinematica) e constante para uma regiao do escoamento a jusante de grelhas colocadas transversalmente ao mesmo. Desta forma, uma vez que a viscosidade cinematica turbulenta tambem e definida como o produto entre uma esc ala de velocidade da turbulencia (em turbulencia isotr6pica, a raiz quadrada da media quadratica das flutuat;:6es de velocidade, aqui brevemente referida como intensidade turbuleqta) e uma escala de turbulencia (a macro-escala), vemos que esta viscosidade deve ser constante. Utilizando o modelo k-6 para descrever a viscosidade turbulenta, resulta:

( (

e

v, =C"

(4)

6

( ( (

(

!!._!:_=- c" k' 8x

c" e uma constante de proporcionalidade. 0 termo multiplicativo no segundo membra e constante, se considerarmos a velocidade constante. A soluc;ao para a equac;ao diferencial 1.5 e:

( (

l

k=-c--1-

C,, I v, 6

= (

{

{

( ( ( (

( (

ou

a k=---

ou

6

x+xo

_!'_X+kv, U, o

(

( (

(S)

v,U,

c,,

b =

l )'

(x+xoY

(6a)

(6b)

v,U,x+k;

a, b e x 0 sao constantes. As equac;oes 6a e 6b sao confirmadas experimentalmente, sendo as constantes relacionadas, ainda, com caracteristicas geometricas das grelhas utilizadas. (Hinze 1959, Monin e Yaglom, 1979, 1981). A equac;ao 6b satisfaz tambem a equat;:ao diferencial para 6, geralmente apresentada no modelo k-6. Para o caso de advecc;ao, esta equac;ao assume a forma: 06

ox,

c,~~ tJ. k

(1)

Utilizando as equa<;:oes 6b e 7 obtem-se o resultado C2=2, que e bastante proximo do resultado, 1,92 geralmente utilizado (ver Eiger e Shen., 1997, par exemplo).

( (

0 Caso de Difusao-Dissipac;ao da Turbulencia para uma Grelha Oscilante

(

Para o estudo da evolu<;:ao da energia cinetica turbulenta em processos difusivos, unidimensionais, novamente parte-se das equa<;:5es 1 e 2. Para escoamento estacionario e admitindo a situac;ao de grelha oscilante em fluido em repouso, a equa<;:iio se reduz a:

(

( (

p)]

[ (u,u 1 -8 u -+ 8 x, ' 2 p

ou,v -ou,=8 x1 8 x1

(8)

( ( (

183

(

{· (, 0 termo entre colchetes, sem equacionamento definitive, e usualmente substituido pelo produto do coeficiente de Boussinesq (viscosidade turbulenta) multiplicado pelo gradiente da energia ci~etica turbulenta e por uma constante de proporcionalidade ( O"k). Tem-se, entao:

I (! (

8(v 88xk) =e 8x t

(9)

~

(

0 sucesso em r~presentar o campo turbulento para o transporte advectivo com numero de Reynolds constante (viscosidade turbulenta constante) induz que o mesmo procedimento seja adotado para o transporte difusivo. Assim, a equac;:iio 9 passa ser escrita como:

f

( { (

8(8k)

vt ;;:- 8x 8x = e

(10)

(

'

f ' Com a definic;:iio 4 para a viscosidade turbulenta tem-se:

r:~(~:)

=

e

c,, O"k

I

k'

e = --, VI I

( ( (II)

(

Uma soluc;:iio explicita para a equac;:ao nao-linear 11 e:

(:J

( (12)

- (X+eJ

B 2 e uma constante de integrac;:iio. Para a taxa de dissipac;:iio de energia obtem-se, a partir das equac;:oes 4 e 12:

c,(2~} 8

e

= (x +B

j

(

k--····,

v,

(

( (

i

( (

J

(

!

(

'

1

,r

(13)

Para grandes distiincias da origem (centro de oscilac;:iio da grelha), a energia cinetica turbulenta segue uma lei de decaimento com a· potencia -2 da distiincia, enquanto que a taxa de dissipayiio de energia segue uma lei de decaimento com a potencia -4 da distancia. As figuras 1 e 2, adaptadas de • Matsunaga et al. (1991 ), mostram resultados experimentais (convenientemente normalizados e indicados, devido a esta normalizac;:ao, com "*") que confirmam as tendencias previstas para uma regiao do escoamento. A linha tina na figura 2 foi construida utilizando a equayiio 14, com as constantes nela indicadas.

( ( ( (

( e * = -,-----'-0,_14_6

(14)

(x * +0,550)

Convem frisar que as constantes utilizadas niio pretendem ser constantes universals. A equac;:iio 13 satisfaz tambem a equac;:iio diferencial parae, geralmente apresentada no modelo k-e. Para o caso de difusiio-dissipac;:iio, esta equac;:iio assume a forma:

184

(

'

(

)

( (

(

(

\ (

(

1 10 ~~~~~~~~::~1·r1-rll~-----,---,--~,-~rT1l------,---,-~-----i I I II I I I I I II II I I I I I I I

~

( (

I I

.&

( (

(

(

Re 0

(

• A ...

(

[J

• •

(

<>

(

-11

(

10

10"

( {

(

I

SIM

4,24.10~

0,8 0,8 1,6 8,8 1,6

5,82.102 7,44.10 2 9,18.103 1,10.10 3 4,01.10 3 9,60.10 4 1,28.10

I

I

a,a 0,8 1,6

I I II

3

!I

I

1!11!!1

I

I

•n"2

1n"

I

I

II

I I I

1

10°

Figura 1: Dados de energia cinetica turbulenta (normalizada k*) de Matsunaga et al. (1991) em fun<;:ilo da distancia ao centro de oscila<;:ao da grelha (normalizado x*). Observa-se a regiao com o expoente -2 da distancia. S c! a amplitude, de oscila<;:ilo, Me a largura da malha quadrada, Re e o numero de Reynolds calculado como Re={sJJ)!v, onde f c! a freqilencia de oscila<;:ao.

( (

(

1 ~~~.-.-'T"TIT---.-.-.~~~~n---.--.~~~

...

(

0

(

• A • 0 •

Para grandes distancias: sa x·•

( (

SIM

4,24.10~

0,1 1,1

5,82.102

1,44.10 1,11.1032

1,1

1,?1.10

1,1 0,1 0,1

D,l

4,01.10~



1,11.10

0

1,21.11

4

1,1

( ( (

Equa~tao

( (

( (

(

Simula~tllo

101 I

10-3

I

I

I

14

numerlca de Matsunaga et al. (1991) I

I

I

I I I

I

I

I

(

I

I

1 1 I

I

I

I

I

I

W

I

I

HP

Figura 2: Dados de taxa de dissipa<;:ao de energia (normalizada &*). de Matsunaga et al. (1991) em fun<;:ao da distilncia ao centro de oscila<;:ao da gre1ha. Silo comparados os dados experimentais, a tendencia para grandes distAncias ('!",.-.), a , equa<;:ilo 14 eo resu1tado numerico de Matsunaga el al. (1991).

( (

I

1 10

2 10

185

t (

'

(

( _!!_ (;:__,__ 8

ox

a,

e.) = c

ox

2

~

(Is)

k

t

'

a , e C1 sao constantes. 0 uso desta equat;:iio, em conjunto com a soluyiio 13, mostra a validade desta ultima e produz ainda uma relat;:iio interessante entre as diferentes "constantes" do<( modelo k-e. Tem-se:

I

10

C, a' .

(16) (

--=---3 a*

(

A equat;:iio 16 deve ser satisfeita para os va[ores "universais" das constantes envolvidas q'ue sao ( ' encontrados na literatura. Utilizando os val ores de Matsunaga et a!. (1991 ), tem-se, para o segundo f membra, o valor 2,50. Ja os valores apresentados por Eiger e Shen (1997), produzem o resultado ' 1.92. Ambos os resultados nao coincidem com o valor 3,33 sugerido pelo primeiro membra, mas ( pod em ser admitidos satisfat6rios, se considerarmos a sua ordem de grandeza.

( 0 Caso Geral de Difusao-Dissipac;:ao da Turbulencia

( (

a

I

Uma equat;:iio governante semelhante equat;:iio 10 ja foi apresentada na literatura da area, para grelhas oscilantes, por Bouvard e Dumas ( 1967), seguindo argumentos diferentes daqueles aqui ( apresentados. E interessante mencionar que esses autores nao fornecem a solut;:iio explicita, mas ( ' comentam que a equayiio tern soluyiio, que pode ser obtida com a aplicayiio dos procedimentos usuais para as equayoes elipticas. Posteriormente, Thompson e Turner (1975) apresentam urn ( modelo de decaimento espacial que se fundamenta no decaimento temporal proposto por Batchelor e obtem uma equat;:iio diferencial de primeira ordem, que conduz tambem a uma expressiio na forma de { ) potencia da distiincia, porem sem a present;:a da origem virtual representada pela constante no ( _; denominador. Os procedimentos seguidos parecem suficientemente saudaveis para interpretar as caracteristicas turbulentas de escoamentos gerados por uma unica grelha oscilante. Neste caso, tanto ( I o modelo de primeira ordem de Thompson e Turner (1975) como o modelo de segunda ordem aqui apresentado ou aquele de Bouvard e Dumas ( 1967) poderiam ser utilizados. Isto porque o modo ( I como os modelos foram desenvolvidos conduz sempre a uma soluyiio na forma de uma lei de ( potencia. No caso aqui apresentado esta lei de potenciaja contem o expoente -2. No caso do modelo de Thompson e Turner, o expoente permanece inc6gnito, o que permite ajusta-lo a diferentes dados. ( Embora esta caracteristica pareya apontar para uma maior generalidade do modelo de primeira ordem, e preciso lembrar que est~ modelo decorre de uma aproximat;:iio temporal (sempre de ( primeira ordem em qualquer formulat;:iio de turbulencia), que uma equayiio diferencial de primeira ordem admite apenas urn contorno e que o modelo de Thompson e Turner, com coeficientes ( ' constantes, portanto, fica restrito aquela situayiio de turbulencia gerada por apenas uma grade ( oscilante. Em outras palavras, se duas grades oscilantes forem introduzidas em urn escoamento de modo que fiquem paralelas e com urn espat;:amento entre as mesmas, o modelo de primeira ordem ( nao permite prever com acerto a evolut;:iio da energia cinetica turbulenta no espat;:o entre as grades ou mesmo no espat;:o externo proximo as grades. Isto e evidente porque a soiuyiio da equayii.o ( diferencial com coeficientes constantes proposta sera sempre uma lei de potencia que admite urn ( unico expoente, nii.o importando o seu valor. Como no caso de duas grades oscilantes tem-se duas regioes com alta agitayiio (proximo as grades) e uma regiiio de mlnimo para a agitat;:iio turbulenta ( i (meia distiincia entre as grades), nenhuma variat;:iio exponencial com urn unico expoente pode preencher esses requisites. A questiio ainda mais relevante, entretanto, talvez seja a o fato de ter sido ( gerada uma equayiio governante para o fenomeno (bastante bern aceita na literatura da area) que nii.o ( I 186

(

\ ( (

parece relletir corretamente a realidade fisica do fenomeno. A equa<;ao diferencial de primeira ordem, como e normalmente apresentada, nao representa bern o processo difusivo em urn ' escoamento generico, o que englobaria, por exemplo, o caso das duas grades oscilantes. E passive! que considera<;6es adicionais acerca das variaveis envolvidas permitam adequar o modelo de forma que possibilite avaliar escoamentos difusivos mais abrangentes. Este estudo, contudo, nao foi encontrado na literatura consultada. A equa<;ao pr_oposta por Bouvard e Dumas, por outro !ado, parte ja da discussao de que em um experimento com grelhas oscilantes a difusao e a dissipa<;iio da energia cinetica sao os fenomenos mais relevantes.A forma da equa<;il.o finale identica a equa<;ao II, sendo talvez esta a razao porque k:l seu uso nao se generalizou (equa<;ao diferencial de segunda ordem nao-linear). Admitindo alguns pariimetros como constantes, para chegar a equa<;iio final, os autores deram margem ao surgimento' de criticas ao equacionamento devido a essas hip6teses simplificadoras. Entretanto, o merito da formula<;iio e que a mesma se fundamenta em uma argumenta<;iio que mantem as principais caracteristicas fisicas do problema e parece nao impor resultados fisicamente inviaveis para o problema em questiio. 0 desenvolvimento da tbrmula<;iio que aqui fbi seguido e semelhante aquele apresentado em Schulz (1997). Outros autores, como Matsunaga et a!. (1991) utilizam este desenvolvimento para estudar o escoamento atraves de simula<;6es numericas com o modelo k-e. Todavia, apesar das demonstra<;6es de viabilidade desta formula<;il.o, a mesma ainda nao e utilizada como uma forma de explora<;iio de caracteristicas basicas dos escoamentos difusivo-dissipativos. A equa<;il.o 9 apresenta urn grau de liberdade que merece ainda ser explorado, que e a presen<;a da viscosidade turbulenta. Utilizando a defini<;ao 4 result a:

( (

( ( (

( {

(

( ( ( ( (

t3 (

ox

( (

k k) 2

c,, t3 8 ~-;-a; =

(17)

A integra<;iio desta equa<;iio produz:

( i

( (

( ( ( ( ( (

a< k 3=3 -

z c~

{J c·dx+B }2 +B

(18)

2

I

.\

B1 e B2 sao constantes de integra<;iio. Note-se que e passive! representar diretamente uma variavel (na caso a energia cinetica turbulenta) como fun<;iio de opera<;oes efetuadas apenas sabre a segunda variavel (no caso a taxa de dissipa<;ao de energia). Esta forma de representa<;ao e uma simplifica<;ao aguda do problema nao-linear existente e permite ·explorar, por exemplo, como solu<;5es empiricas para uma variavel interferem no desenvolvimento da segunda variavel. Evidentemente, uma vez havendo uma rela<;iio direta, e passive! obter a rela<;iio inversa, de forma a explicitar a taxa de dissipa<;iio de energia. Neste caso, tem-se:

J2 c

c= ±d- - - " (k dx 3ak

3

-B)

(19)

2

( ( (

( (

A equa<;iio 19 envolve apenas uma deriva<;iio para a energia cinetica turbulenta, o que representa uma vantagem substancial em rela<;ii.o a equa<;iio correspondente 17. As equa~toes 18 e 19 , siio caracteristicas para escoamentos difusivo-dissipativos. A partir da equa<;ao 19 pode-se calcular a. potencia consumida em urn espa<;o no qual apenas existem processos difusivos e dissipativos. Assim, admitindo que em uma posi<;ao =0 a energia cinetica turbulenta assume o valor k 0 e que em uma posi<;iio generica x a mesma assume urn valor maximo para o espa<;o considerado(k=kmax), obtem-se:

(

( (

187

'

(

. ( f2c~ ( 2 c~ ( ~) W-=2pA f3~k k"',.-B,- Ja• k -B2)) 3

3

)

0

(

(20)

I

(

( p e a massa especifica do fluido e A e a area transversal a direcao X, que define 0 volume do ( espaco de trabalho. A constante de integracao remanescente deve ser resolvida para cada caso particular de estudo. Como exemplo, a integracao da equacao 19 para urn escoamento entre duas ( grelhas oscilantes e apresentada no item seguinte. A equacao 18 permite, por outro !ado, verificar tendencias para a variaciio de k a partir de variacoes conhecidas de e. 0 caso mais simples e a ( situacao de e constante, que conduz a seguinte relacao para a energia cinetica turbulenta:

k "

3 [

a • ( e 'X 2 c~

2

+ 2B, eX +

sn

I {

1/J

+ B,

I

. (21) ]

\

( 1

Para pequenos valores de x, tem-se que a energia cinetica assume urn valor constante. Para { grandes val ores de x, por outro !ado, verifica-se que ha a tendencia a seguinte proporcionalidade: (

k-

3

a•

(

)1/3 (e xr'

( 2C,,

3

(22)

( (

I

Esta forma de dependencia pode ser extraida da literatura, por exemplo no estudo das condicoes de contorno em superficies livres e superficies s61idas para a resolucao numerica de ( problemas de escoamentos em meciinica dos fluidos (ver Eiger e Shen, 1997, e Demuren e Rodi, ( ' 1984, por exemplo). Esses estudos estao fundamentados basicamente em proposicoes empiricas e argumentos dimensionais para a definicao das varhiveis relevantes. Os autores citados apresentam ( uma equacao que, se utilizada para explicitar k com a representa9ao de variaveis aqui definida, ( resulta em: 213

k-- c~/4 K ( )

(ex)

(23)

213

(

I

(

I

(

Ke a constante de von Karman, com valor da ordem de 0,41. Valores encontrados na literatura ( (Matsunaga et al. 1991) para as constantes envolvidas nos coeficientes de ( e x)m conduzem ao coeficiente 2,55 para a equaciio 22 e ao coeficiente 1,85 para a equa9ao 23. Embora nao haja ( igualdade nos resultados obtidos, mais uma vez a comparaviio da ordem de grandeza de ambos e satisfat6ria. Resultados como a equa9iio 22 sao animadores e mostram que a formulavao apresentada ( pode ser aplicada em situayoes nas quais se espera que processes difusivo-dissipativos sejam os mais ( relevantes na descriyao de urn fenomeno. Assim, junto a superficies livres, por exemplo, espera-se que a produ9ao de energia cinetica turbulenta seja reduzida, uma vez que os gradientes de ( velocidade sao suprirnidos. Restam, entao, a difusao e a dissipacao como mecanismos que \ determinam as caracteristicas turbulentas do escoamento nessas regioes. Para a obtenviio de uma avalia9ao dos perfis de k e e optou-se, neste trabalho, utilizar ( expansoes em series de potencias para as duas variaveis e calcular os val ores dos coeficientes atraves de operavoes sucessivas. A equaclio 18 serve de base para o primeiro conjunto de resultados. ( As vari{weis k e e sao expressas da seguinte maneira: (

( 188

(

\

\

I

I

I

'

(

(

s~2:s,x'

k"' Lk

r=O

( (

J

X J

(24)

p•O

A utilizayao dessas rela~t5es na equa~tao 18 conduz a uma igualdade entre series de potencias, cujos coeficientes de igual ordem devem ser iguais. Isto produz o seguinte conjunto de igualdades (aqui apresentado apenas ate o termo de quarta ordem, mas que pode ser expandido ate qualquer ordem):

( ( (

k = \js.B: + B

Ordem zero:

s. =

2

0

(

f ( (

(

!J t

k;

Ordem3: k

Ordem4: k

s. (2B =31:2 --3t:- + & 1

3

4

2

o

s. =--, 3 k0

(2Sa)

(25b)

( ') Ordem 2: k 2 = -s. k 2 B 1& 1+& 0 --k 3 0 0

(

(

3

Ordem 1: k 1 = 2S.B1e 0 3k~

( (

zc:-

) 0 & 1

(25c)

-

k~ 3

k

2

2k 1 k 2 --k-

0

(B- - +t:~- + - -s,) - --1t: 3

2

4

(25d)

0

2& 0 3

k;k, k{ 2k 1 k 3 -----k 20 k0 k0

(2Se)

( (

( (

( ( (

( ( ( ( ( (

( ( (

( (

As constantes B1 e B2 devem ser obtidas a partir dos contornos. Ve-se que cada novo coeficiente k, e dependente dos coeficientes anteriores da energia cinetica turbulenta, que as operayoes envolvidas no calculo dos novos coeticientes conduzem sempre a ni1meros reais (nii.o existem operayoes que geram numeros complexos) e que ha, tambem, uma dependencia para com os coeticientes da taxa de dissipayiio de energia. As duas primeiras caracteristicas sao positivas, no que tange a obten~tao de urn perfil. Porem a terceira caracteristica exige que se utilize uma equayiio adicional para a taxa de dissipal(iio de energia. Aqui convem utilizar a equayiio I 5, que e equayiio diferencial de s para o caso de difusiio-dissipal(iio. Efetuando as deriva~toes indicadas e rearranjlmdo os termos, obtem-se:

dkdc dcd& d'c 2k 2 c - - - k 3 - - + k 3 c--=S c dzdz dzdz dz 2 •

4

(26)

s. == c, u, c,.

Verifica-se que a equayiiO 26 e linear para /C, 0 que implica que uma equayaO equivalente a equar;:iio 18 pode ser tambem obtida, se isto for desejado. Entretanto, no presente texto, interessa utilizar as definil(oes 24 para obter novas relar;:oes para os coeficientes de k e &. Mais uma vez, a utilizar;:iio dessas relal(5es na equar;:ao 26 conduz a uma igualdade entre series de potencias, cujos coeficientes de igual ordem devem ser iguais. Isto produz, agora, o seguinte conjunto de igualdades

189

(

( 1

(apresentado apenas ate o termo de terceira ordem devido ao espa90 requerido, mas que pode ser I expandido ate qualquer ordem): ( Ordem zero: 2k~e 0 e 1 k 1 -k~e;+2e 0 e 2 k~=S,e~

Ordem 1:

I

I

(27a) (

'

(

I

(

z[(zkok,e o+kje ,)e\k, +2kge o(e ,k, +[2F. ,e 1+6& 0 e ,]kg +(3ko'k 1)2e 0 £

+e

2=

,kl)]-[(3kgk,)e ;+4kge le,]+

4S,e ~e 1

(27b) .(

i

(

'

(

Ordem 2:

f

z{[(k,' + 2kok,)e o+ 2kok,e ,+ ko' e 2 )e ,k, + (2kok,e o+ ko' e ,)( 2e ,k, + 2e ,k ,)} + + 2(ko' e o)(3e 1k3 + 4e2 k 2 + 3e 3 k1 ) - [(k,' + 2k0 k,)k0 +2k 0 k~ + ko' k, }e;-12ko'k,e ,e 2 - k~(6e ,e 3 + 4e ~) + [ze i+ 6e ,e 3+ 12e 0 e 4 ]kg +

+[2e 1e 2 +6e 0 e 3K3kgk,) + 2e 0 e ,[(k,' + 2k0 k,)k 0 +2k0 k 12 +ko'k,] = = s,(6e ~e

~+ 4e ~e ,)

I

( (

I

(27c) (

(

(

Ordem 3·

('

z[z(kok3 + k,k, )so+ (k,' + 2kok,)e ,+ 2kok, e ,+ ko' e 3 )e ,k, + +2[(k: +2k0 k 2 )e 0 +2k 0 k 1e ,+ko'e ,](ze ,k, +2e ,k,)+

,)(3e 1k3 + 4e,k 2 + 3e ,k,) + + 4ko'e 0 (2e ,k4 + 3e ,k3 + 3e ,k, + 2e 4 k,)-

+ 2(2k 0 k 1e 0 + kge

-[z(k0 k3 +k,k,)k0 +(k~ +2k0 k,)k, +2k0 k 1k 2 +kgk,]e;-12(ko'k, + k 0 k,')e ,e 2 - 3k; k,( 6e 1e 3 + 4e.;)- kg(se ,e 4 + 12e2e ,) +

+[8e ,e 3 + 12e ,e 4 + 20e 5 e ~]kg+ [ze ;+ 6e ,e 3 + 12e 0 e .](3ko'k,) + 2 1

( (

I

( (

I

(

I

(27d) (

(

+[ze 1e 2 +6e 0 e ,l(k,' + 2k0 k,)k 0 + 2k0 k + ko' k2 ] +

(

+2e 0 e,[z{k0 k3 +k,k,)k 0 +(k: +2k0 k 2 )k, +2k0 k,k 2 +ko'k,J=

(

=

)

zs.[ze ~(e oe ,+ e ,e ,) + 2e oe ,(e :+ 2e o6 ,)]

(

\ ( A apresentayao algo rica em caracteres nao deve impressionar, porque tambem aqui os coeficientes sao sucessivamente calculados a partir dos valores ja conhecidos, de menor ordem. ( Assim, ve-se que na equa9ao 27a pode-se isolar o coeficiente e2, na equa9ao 27b pode-se isolar o ( coeficiente e3 , na equayao 27c pode-se isolar o coeficiente e,, na equa9ao 27d pode-se isolar o coeficiente e5, e assim sucessivamente. As opera96es envolvidas para a obten9iio de cada coeficiente ( 190

(

'

l

(

( produzem apenas valores reais, o que viabiliza este calculo Por outro !ado, a obten<;:ao das expressoes 25 e 27 segue padroes bern definidos, ditados pelas equa<;:6es diferencias e integrais iniciais, que podem ser facilmente reproduzidos em uma rotina de calculo. Como ocorreu com as equa<;:oes 25, tambem aqui os coefientes dependem de forma interligada com aqueles para a energia cinetica turbulenta. E preciso notar ainda que se tern coeficientes que dependem de quatro . constantes que devem ser difinidas a partir dos contornos. Essas constantes siio aqui apresentadas · como ko, eo, e1 e B; ou B2. Note-se que, devido a equa<;:iio 25a, que relaciona tres das constantes mencionadas, uma delas e superflua (quando duas forem conhecidas, a terceira e imediatamente conhecida). Essas quatro constantes siio evidentemente esperadas, uma vez que se tern urn problema composto de duas equar;;oes diferenciais de segunda ordem, cada qual envolvendo, portanto, duas constantes de integrar;;iio. A obten<;:iio sucessiva dos coeficientes pode ser assim conduzida: - ko, eo e e1 siio considerados conhecidos, assim como B1, que aparece nas equar;;oes 25. - k1 e k2 siio obtidos das equar;;oes 25b e c. - e2 e e1 siio obtidos das equar;;oes 27a e b. - k1 e k. siio obtidos das equar;;oes 2Sd e e. - t:4 e t:5 siio obtidos das equar;;oes 27c e d.

( ( (

( ( (

( (

( I

( (

Dois a dais, os coeficientes podem ser calculados alternando-se o conjunto de equar;;oes para a energia cinetica turbulenta e para a taxa de dissipar;;iio de energia. Apesar da simplicidade aparente, preciso mencionar que a determinar;;iio das constantes de integra<;:iio pode ser complexa (primeiro passo acima arrolado) e que o uso pratico das expressoes aqui obtidas provavelmente exige o truncamento em ordens inferiores das series infinitas apresentadas. Niio obstante, trata-se de uma solur;iio te6rica e geral, a qual pode ser utilizada para estudar o comportamento de casas particulares de escoamentos turbulentos difusivo-dissipativos. No presente trabalho, o caso de duas grelhas oscilantes e abordado a seguir

(

e ,

(

(

( ( (

0 Caso da Difusao-Dissipac;ao da Turbulencia para Duas Grelhas Oscilantes

( ( (

( (

(

(

No problema de duas grelhas oscilantes, as propriedades turbulentas (como a energia cinetica turbulenta) devem passar por pontes criticos (minimos ou maximos) no centro do espar;amento entre as grelhas, se a gerar;;iio de turbulencia for igualmente intensa em ambas. Assim, o plano central age como urn espelho e qualquer fun<;:iio que descreva uma propriedade deste escoamento dev~ ser for<;:osamente uma fun<;:iio par em torno da origem do sistema de coordenadas, localizado convenientemente neste plano central. As grelhas ficam entiio posicionadas em x=+I-L, sendo L o valor da meia-distancia. Como tanto a energia cinetica turbulenta como a taxa de dissipar;;iio de energia sao fun<;:5es pares de x tem-se que a constante B1 da equar;;iio 1. 18 deve ser zero, o que acarreta o valor k/ para a constante B2, sen do k0 o valor de k na origem. A equar;;iio 1.19 pass a a ser representada, entiio, por:

(

(

dt:=± dx

J2--''(k3-k3) c 3crk

(28)

o

( ( (

A partir da equar;;iio 28 pode-se calcular a potencia consumida no espar;;o entre as grelhas ( W), considerando que em x=+I-L tem-se k=kmax, que representa o maior valor da energia cinetica turbulenta. Tem-se, entiio:

{

( (

(

191

4 (

W=2pA

1o

.(

)~C"(k' -k') 3 ak 0

(29)

ma..'(

t

p e a massa especifica do fluido e A e a area das grelhas oscilantes. Pode-se avaliar ( aproximadamente o coeficiente da equar;iio 29 a partir de valores de Iiteratura para as constantes ( envolvidas. Tem-se, entiio:

'

(

W"" 0,44 p A ( ~ k ~"'- k ~)

1

(30) ( )

Como avaliar;iio adicional, o valor media da taxa de dissipar;iio de energia por unidade de .( massa na regiiio entre as grades oscilantes pode ser obtido diretamente das equar;oes 29 ou 30, ( fornecendo, para esta ultima, a previsiio: .

(

(3J)

(

I

( Finalmente, os perfis de energia cinetica turbulenta e de taxa de dissipar;ii.o de energia podem ( ser obtidos das equar;oes 25 e 27, para as quais os termos de ordem impar devem ser anulados. Desta forma, tem-se, a partir das equar;oes 25: ( Ordem zero: k 0 =

s;''

ou B, = kg

Ordem 1: k ,"' 0

(32a)

{

(32b)

(

( Ordem 2: k

3

sk (fi o') ,= )"k"2

C! k

sk = 2 c,

0

Ordem 3: k 3 = 0

(32c)

(32d)

3..L(2&---fi,) -~' k'

Ordem 4: k 4 = 3k' 0

0

3

k

(

J

(

I

( (

I

(

J

(

I

(32e)

0

(32f)

Ordem 5: k 5 = 0

(

( Das equar;oes 27 obtem-se:

t

Ordem zero:

S,& ~

fi ,= 2k ~

( C, a,

s =-c t

(33a)

(

(

"

(

Ordem 1:

fi ,= 0

(33b)

(

( 192

(

\

I

\ ( (

Ordem 2.

( 64

(

(

=

2S>· ~6 2 -7kgk 2 6 6k 03 6

06 2

+ k~6;

(33c)

0

Ordem 3.

(

=0

(33d) I

6 5

(

Ordem 4:

( (

6

( (

6

=

s,( 3 6 ~6; + 26 ~6

4)-

llkok~ 6 o6 2- 1 Jkg k46 o6 ,- 20k~ k 2 6 o6 4 + k~ t: 2 8 15kgt:

4

(33e)

0

Ordem 5:

(

t:

7=

(33t)

0

( (

( (

( (

(

(

Observa-se, portanto, no caso de duas grades oscilantes, uma sensivel simplitica'
(

( ( (

( ( (

( ( (

(

Conclusoes Os escoamentos turbulentos gerados por grelhas ou grades oscilantes foram analisados a partir das equa'
(

( (

193

t ' ( (

Agradecimentos

(

0 autor agradece a FAPESP, pelo apoio obtido atraves do processo 1997/11743-0 para execuc;:ao de pesquisa no exterior, na qual o presente trabalho se insere, e ao Prof. Gerhard Jirka, anfitriao no Institut fur Hydromechanik, Universitat Karlsruhe, Alemanha.

( {' (

Referencias Bibliograficas

(

Bouvard , M. e Dumas, H., 1967, "Application de Ia Methode du Fil Chaud a Ia Mesure de Ia ( , Turbulence dans I'Eau", La Houille Blanche, n2 7, pp. 723-734. "

( Demuren, A.O. e Rodi, W., 1984, "Calculation of Turbulence-driven Secondary Motion in ·Noncircular Ducts", Journal ofFluid Mechanics, Vol.l40, pp.189-222. ( De Silva, I.P.D. e Fernando, H.J.S., 1994, "Oscillating Grids as a Source of Nearly Isotropic ( Turbulence", Physics of Fluids, Vol6, n° 7, pp. 2455-2464. (

J

Eiger, S. e Shen, H. W., 1997, "An Analysis of the Free Surface Boundary Condition of the ( Dissipation Rate of Turbulent Kinetic Energy", trabalho submetido para publicac;:iio, cedido ( como informac;:iio pessoal.. (

Hinze, J.O., 1959, "Turbulence", McGraw-Hill, New York.

I

(

Hopfinger, E.J e Toly, JA, 1976, "Spatially Dt:caying Turbulence and ist Relation to Mixing ( Across Density Interfaces", Journal of Fluid Mechanics, Vol. 78, pp. 155-175.

( Matsunaga, N.; Sugihara, Y. e Komatsu, T., 1991, "A Numerical Simulation of Oscilating-Grid Turbulence by using the k-e Model", in Lee, J.H.W. e Cheung, Y.K., (editors) "Environmental ( Hydraulics", Vol.!, AA Balkema, Rotterdam, the Netherlands, pp.427-432. ( Monin, AS. e Yaglom, A.M., 1979, "Statistical Fluid Mechanics-Mechanics of Turbulence", Vol 1, the MlT Press, Massachusets. Monin, A.S. e Yaglom, A.M., 1981, "Statistical Fluid Mechanics-Mechanics of Turbulence", Vol2, the MIT Press, Massachusets.

(

I

(

I

( (

Nokes, R.I., 1988, "On the Entrainment Rate Across a Density Interface", Journal of Fluid Mechanics, Vol188, pp.185-204. ( Schulz, H.E., 1997, "Teste de uma Formulac;:ao Altemativa em Turbulencia", Tese apresentada Escola de Engenharia de Sao Carlos, Universidade de Siio Paulo, 86 p.

a (

Srdic, A.; Fernando, H.JS. e Montenegro, L., 1996, "Generation of Nearly Isotropic Turbulence using Two Oscilating Grids", Experiments in Fluids, 20, pp.395-397.

(

( (

Thompson, S.M. e Turner, J.S., 1975, "Mixing Across an Interface due to Turbulence Generated by ( an Oscilating Grid", Journal ofFluid Mechanics, Vol67, pp. 349-368.

( Voropayev, S.I. e Fernando, H.J.S:, 1996, "Propagation of Grid Turbulence in Homogeneous Fluids", Physics ofFiuids, Vol.8, n2 9, pp. 2435-2440. 194

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COMPARASIONS OF SOME MODELS OF TURBULENT PRANDTL NUMBER FOR LOW AND VERY LOW-PRANDTL-NUMBER FLUIDS

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Marcelo C. Silva* Ricardo F. Miranda Lutero C. De Lima

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(*) Department of Energy UNICAMP - Campinas SP - Brazil Department of Mechanical Engineering Universidade Federal de Uberlandia 38400 089 Uberlandia MG Brazil

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ABSTRACT

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The present article studies some models of turbulent Prandtl number for low and very low-Prandtl-number fluids. It was investigated model of Cebeci, two models of Kays, and model of Wassel and Catton. Three different low and very lowPrandtl-number fluids and aire were studiedand compared with experimental data found in the literature. It was verified the behaviour of the turbulent Prandtl number close to the wall as function of dimensionless wall distance and eddy diffusivity of heat. It was observed that near the wall Prt shows asymptotic behaviour for all models and fluids, except for model of Cebeci and air. For the wall distance, y+, greater than 700 all models and fluids show values of Pr1 around 0.9-1.0.

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Keywords: Turbulence; Turbulent Prandtl Number; Boundary Layer.

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195

t (

I

f NOMENCLATURE A+ 8+

c

c1. c2. c3, c4

k I Pr Pr1 T

T. Tw u ~·



uo u v v'T' X

y

/ a 8 EH

EM '1

ll y v

~ p tw

e

l

Van Driest Constant function of Eq. (15) specific heat at constant pressure, J/kg° C constants (equation 18) molecular thermal conductivity, W/m°C Prandtl mixing length, m · Prandtl number = v/a turbulent Prandtl number = eMf EH time-averaged temperature, °C temperature of free stream, °C temperature at wall surface, °C time-averaged velocity in x direction, m/s 2 2 turbulent shear stress, m /s mean velocity in wall coordinates = u/u • velocity at outer edge of boundary layer, m/s

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m/s shear velocity = time-averaged velocity in y direction, m/s turbulent heat flux, m°C/s distance measured in direction of mean flow, m distance measured in direction normal to mean flow, m distance from wall in wall coordinates= yu"tv thermal diffusivity, m% boundary layer thickness, m 2 eddy diffusivity of heat, m /s 2 eddy diffusivity of momentum, m /s transformed y-coordinate dynamic viscosity coefficient, kg/ms intermittency factor kinematic viscosity coefficient, m% transformed x-coordinate 3 density, kg/m . shear stress at wall surface, N/m 2 dimensionless temperature = (T - T.)/(Tw - T.)

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INTRODUCTION Prediction of heat transfer in turbulent flows demands the solution of the energy equation, which in its turn depend on solution of the momentum equations. The fact is that such equations have fluctuation terms and those terms usually are expressed as function of the mean velocity and temperature gradients. For the case of the momentum equation, the concept of eddy diffusivity of momentum Em has been introduced and, together with the theory of the Prandtl's mixing-length, it was reached satisfactory numerical prediction in comparison to experimental data. Similarly for the case of the energy equation, the concept of eddy diflusivity of heat ~>h is used although it is not yet completely consensus. As stressed by Cebeci (1973), various assumptions have been made about eddy dilfusivity of heat, and several expressions have been proposed in attempts to predict the mean temperature distribution within the boundary layer. The solution of the governing equations of a turbulent flow needs the application of turbulence models which are based on the ratio of the eddy dilfusivity of momentum Em to the eddy diffusivity of heat gh that is the turbulent Prandtl number (Bremhorst and Krebs, 1993). One assumption that has been used extensively is the one due to Reynolds. According to his assumption, heat and momentum are transferred by the same process, which means that both eddy diffusivities are the same. This assumption leads to a turbulent Prandtl number of unity (Cebeci, 1973). More than lour decades, the literature discusses the behaviour of the turbulent Prandtl number mostly tor air boundary layer and until now no definite conclusions has been reached. Reynolds (1975), in a review, has examined more than 30 different ways of predicting the turbulent Prandtl number and affirmed that the existing procedures range from purely empirical to formal analyses based on the Reynolds stress equation. Kays (1994) examined available experimental data on Pr1 lor two-dimensional turbulent boundary layer and for fully developed flow in a circular duct or a flat duct. More recently, De Lima, Silva and Miranda (1998) made , a comparative analysis of different models lor the turbulent Prandtl number for air boundary layer. The influence on the calculation of various thermal parameters such as dimensionless temperature profile, turbulent heat flux, eddy conductivity of heat and Stanton number were also investigated. They observed that the behaviour of the turbulent Prandtl number is relatively constant along the boundary layer. Discrepancies in Pr1 shown by almost all models and experimental data at mainly to the near wall region and in the ''wake" region of the boundary layer had little effect both on the calculation of Stanton number and on the calculation of such other thermal parameters. If on one hand there are extensive data on turbulent Prandtl number tor fluids such as air and water, on the other hand there are few publications on turbulent Prandtl number for low and very low Prandtl number fluids. The scarcity of publications on Pr1 for low and very low Prandtl number fluids make difficult the evaluation of models and experimental data. Notwithstanding these aspects are very important then considerable interest does exist in applications of such fluids as, tor example, cooling of nuclear reactors and valves of internal combustion engines. While for air and fluids with molecular Prandtl number greater than 0.7 the turbulent Prandt number can be assumed constant (for example 0.9) along the boundary layer, for the case of fluids with low and very low Prandtl that

( ( (

197

( ( assumption can not be considered. As it will be shown in this article close to the wall the majority of models of Pr1 for such class of fluid points to values much higher than 0.9. However higher values of y+ (say y+ > 700) all models of Pr1, independently of fluid, its values goes to 0.9.

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GOVERNING EQUATIONS

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For a steady: two-dimensional, constant property turbulent air boundary layer over a flat plate with negligible body force, negligible viscous dissipation and no pressure gradient, the governing equations can be expressed as follows.

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Continuity equation

(

au av -+-=0 ax ay

(1)

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Momentum equation

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a.]

a.- =a- [(v+eM)ua.- + v a 0> 0• o/

(2)

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I

( Energy equation

(

a [(a+eH)i!l' J

rJI- = urJI -+v

(3)

0>

o/o/

&

( (

and the definition of the turbulent Prandtl number is: Pr = 1

~ _ U';(oT I By) EH - v'T(oU I By)

Substituting for equation:

EH

( (4)

in Eq. (3) and ·after rearranging, thus it has, for energy

u = 0;

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()y

Pr

Pr,

8y

(5)

y ~ oo; u = U»

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The conservation of mass and momentum equations (1) and (2) require specification of the velocity components at the wall and at the free stream. That is,

y = O;

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uOTax +vaT =~{v[l._+~]OT} ()y

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(6)

Doing the same for the thermal energy equation requires specification of the temperature at the wall and in the free stream:

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198

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( y

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f (

= 0;

T = Tw; y ~ oo; T =T.

(7)

In order to solve equations (1 ), (2), (3) and (5) with their corresponding boundary conditions, a turbulence model has to be introduced to evaluate the eddy quantities EM and EH and consequently the turbulent Prandtl number Pr1. After Chyou (1991 ), the theory of turbulent wall shear layers is still in a state of intense study, and new breakthroughs are continually in sight. But the simplest of all the schemes proposed remains the very old Prandtl mixing-length model, and with new information available on the very important behaviour of the viscous sublayer, the mixing-layer model provides a remarkably adequate basis for many engineering applications especially for some simple flow patterns. The following calculations were based on this turbulence model:

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c:, = e'la~t 0JI

(8)

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( ( ( (

To evaluate the mixing length e, the outer region of the boundary layer and the near wall region must be considered separately. Still after Chyou (1991 ), for flows remote from walls, e is usually taken as uniform across the layer and proportional to the thickness of the layer. For a boundary layer on a wall, the variation of e in the outer part is similar to that in free turbulent flows, but e is proportional to the distance from the wall for the near wall region. The coefficient of proportionality between the Prandtl mixing-length e and the thickness of the layer is normally the Von Karman constant K = 0.4, that is e= 0.4y. However, for the region close to the wall, the viscous sublayer, equation (8) needs to be modified. The Van Direst's hypothesis was applied to the wall region as

( e=

{ (

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0.4y[t-exp(- y' I A+)]

(9)

where A+= 26 (the Van Driest's constant) and/ a dimensionless distance defined 112 as / = y(p rw) /fl, in which rw is the shear stress on the wall. By that way the eddy-viscosity expression based on Prandtl's mixing length modified by Van Driest for the inner viscous sublayer will be

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1

E'"' = 0.16y [1-

exp( -y• I A+ )]\m I Oyj

(10)

( For the outer region the eddy-viscosity expression is given by

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Emo =

being y the Kleban off's intermittence factor under the formula

y

( (

(11)

O.OJ68UJ)y

and

=[1 + 5.5(y I t5)

6

r

(12)

c/ the boundary layer displacement thickness defined as

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199

L (

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8'=J:lt-~~}y

(13)

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( For the turbulent Prandtl number Cebeci (1973) proposed a model which is dependent upon the distance of the wall, given as follows:

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( Pr = K(l-exp(-y+IA+)] ' K H [1- exp(- y +I B+)]

(14)

(

(

where K = 0.4, KH = 0.44, A+= 26 and B+ is given by the following equation.

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s· = J~r L:~=l c, (1og 10 Prt

(15)

(

(

being Ct = 34.96, C2 = 28.79, C3 = 33.95, C4 = 6.3 and Cs = -1.186. A very simple model was proposed by Kays (1994) who considered Pr1 as a function· of Pr and e,, lv which is the turbulent Pech~t number (Pe1). As highlighted by him his model is similar to one suggested by Reynolds (1975). The Kays' model is in the form:

Pr, =[0.7/Pr·(cM/v)] + 0.85

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(16)

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Kays (1994) has proposed another equation for Pr1, modifying equation (16) to the form:

( (

Equation (17) was proposed by Kays with the intention of including experimental data on liquid metals. The step of introducing an analogous variation for. e" 1v as cited by Reynolds (1975) was early taken by Wassel and Catton (1973) in the form:

C 3

Pr' = C1· Pr·

[I -

c~ /v))

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[1-exp(~ )] -C ex{ Pr· (

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(17)

Pr, =[2.0/Pr·(c"/v)] + 0.85

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(18)

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( where Ct = 0.21, C2 = 5.25, C3 = 0.20 and C4 = 5.0 The limit between both regions is determined by the condition where

( Emi (

Emo·

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200

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t

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The following dimensionless parameters and variables were adopted in Eqs. (1)- (13):

( u=~.

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(

s(x)= r•U.., dx J. "

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u~

T- Too T,..

-T~

(19)

and independent variables as

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(}=

v=3_ and

u~

and

u.,

'l(x,y)= Y v(2s)"

(20)

being 11 function 1; and 8. After Schlichting (1979) 11 varies between 0.5 and 0.8. In a study of validation of this turbulence model made by the present author (De Lima and Pereira, 1983) n = 0.5 was used because good results were reached in comparison with experimental data. Therefore equations (1 ), (2) and (5) with boundary conditions will be:

·

6V

(2st ~~ + iJrJ = 0

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at]

2s ) 2" -Cit+ Vov- =t3- [
(22)

88 v8e _8 [v(-1 + r."'/vJ8eJ --(2 .,~)2" u-+ 81; Of] Of] Pr Pr, Of]

(23)

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(21)

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being 1

v(2E,) " u 8rJ + (21:,)" V

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V = Uco(x)

(24)

Ox

The boundary conditions for equations (16-19) are

o: u = o, v = o. e = o

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~; =

(

11 = o: u = o, v = o, e = 1 11 --t co: u = 1, v = o, e = o

(25)

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Equations (21-24) were numerically discretized and solved by an tridiagonal implicit finite-difference method as outlined by De Lima and Pereira (1983). The finite-difference equations use a variable grid in the 11 direction which permits shorter steps close to the wall and longer steps away from the wall (Cebeci, 1970). The grid has the property that the ratio of lengths of any two adjacent

( ( (

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201

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I intervals is a constant, that is,

/j.77 i

= 8fj.77 i-l

.

The grid has five points in the ~­

(

direction and three points in the T]-direction. The computer program used in this study was used a grid wich 300 points in the T]-direction. The choice of 8 = 1.03 and /j.Tlt = 0.035 shown a satisfactory combination which permitted a computation time per station of 3 seconds under the convergence criterion of 10-4. Concerning ~-direction there is no restriction on the number of stations, and the starting position was at x = 0.0914 m. The calculations were started with an initial velocity profile based on the concept of friction velocity and the logarithmic laws of the wall. Since the velocity and temperature profiles are decoupled for non-buoyant flows, velocity fields are solved first. Thereafter temperature field was solved with an initial one-seventh law profile.

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{ RESULTS AND DISCUSSION

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For the present comparative study of different models of Pr1 for low and very low Prandtl number fluids, were selected liquid sodium (Pr = 0.0058), mercury (Pr = 0.025), a fluid with P, = 0.1 which corresponds to a gas mixtures as for example hydrogen-xenon and air (Pr = 0. 71 ). Due to the few data in the literature, with the purpose of validation and comparison, this selection of fluids was intentionally made as similar to the selection of Bremhorst and Krebs (1993).

Pr

,,_~

z

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Cebeci

16

i

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=0.71

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Pr= O.l Pr = 0.025

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Pr = 0.0058

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'-~

4

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'-,

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"' "' "' ---

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Q L _______ L-.....,__L

tE+t

1E+2

'

1E+3

y+

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( Figure 1. Turbulent Prandtl Number Calculated from Model of Cebeci.

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( 202

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'

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Figure 1 shows the variation of the turbulent Prandtl number as a function of the wall distance y+. In this case the model used was the one of Cebeci (eq. 14). The effect for liquid sodium and mercury is very remarkable and it is apparent that Pr1 is a strong function of the molecular Prandtl number close to the wall and constant away from the wall. It can also be noted that increasing the molecular Prandtl number will result in the decreasing of the turbulent Prandtl number near the wall. As expected air did not show such behaviour. 30~ I

(

~

( (

( (

~ ll

~

2o

Pr=O.?l

- - . Pr = 0.1

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-

1

(

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~1- - - - - - - - - - - - -

Pr = 0.025 Pr = 0.0058

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-'0

£ 5

J

10

\ \

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"'

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(

'-.-.

---

----.

~----==-= -

--=---

-:oo---

OL-----~--~-L~_L~LLL------L---L~--~-L~~

(

1E+1

1E+2 y+

1E+3

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t ( (

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(

Figure 2. Turbulent Prandtl Number Calculated from Model of Kays (I). Figures 2 and 3 show the variation of Pr1 as a function of y+ and fluids the models of Kays (Kays 1 - eq. 16 and Kays 2 - eq. 17). Firstly Kays (1994) used equation 16 for Pr, observing that its calculation were very close to experimental data and DNS results published in the literature. Subsequently Kays proposed equation 17 for the turbulent Prandtl number for low and very low Prandtl fluids. He explained that although equation 16 is well adjusted to experimental and DNS results, when temperature profiles and Nusselt numbers were measured directly equation 17 is more consistent than equation 16. With exception of air, the turbulent Prandtl number calculated by models of Kays (eq. 16 and eq. 17) present a very dramatic behaviour close to the wall. Some investigators argue that the region close to the wall is unimportant (Me Eligot and Taylor, 1995) since at such location and for low and very low Prandtl number fluids the molecular transport of energy or momentum is expected to be high than the turbulent transport. Notwithstanding the molecular Prandtl number has a strong repr~senting

( ( (

203

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I ,

effect on Pr1 close to the wall as shown by the present investigation. As highlighted by Me Eligot and Taylor (1995) it is difficult to measure Pr1 accurately near the wall. As the wall is approached, the experimental uncertainties grow to the point where the measurements can not be used with confidence to discriminate between hypothesized models. In particular, both the turbulent shear stress and the turbulent heat flux go to zero as the wall is approached, so Pr1 becomes -01-0 and its limiting value is bound to be uncertain. On the other hand the same Me Eligot and Taylor affirm that for gas mixtures with Pr ranging from about 0.18 to 0.7, and where in our opinion could be included liquid metals, predicted heat transfer parameters are expected to be strongly dependent on the representation of thermal energy transport in the viscous layer, y+ < 30. This expectation evolves since, in a typical high Reynolds number flow, about 40% of the thermal resistance can be concentrated in the region 5 < y+ < 30, which covers only about 0.2% of the radius or boundary layer thickeness. Kays2

30~--

1

Pr= 0 71 Pr=O.l

( ( ( (

( (

( (

l

{ ( (

Pr = 0.025 Pr = 0 0058

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,

(

20 ·-

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J

(

~ ~

(

\ 10

(

\ \..

'

( ""-

(

....___

'-..

--= -: ---

(

,...-;___~ ---=-:- ..:::.

OL-----L-~--L-Li-LLLL---~---L~~-L~~

181

1E+2

(

1E+3

y+

(

Figure 3. Turbulent Prandtl Number Calculated from Model of Kays (II).

( The model of Wassel and Catton (eq. 18) was applied to the same fluids treated in this study and curves of Pr1 are shown in the Figure 4. Again as occurred with the model of Cebeci and models of Kays, the model of Wassel and Catton show dramatic behaviour of the turbulent Prandtl number of liquid metals (sodium and mercury). In fact, with the exception of air, the model of Wassel and Catton present valued of Pr, greater than value present by the other models studied here. If one takes a closer look on the behaviour of liquid metals as shown in Figures 5 and 6 for mercury and sodium, respectively, it is observed that most of the models point to a value of infinity to the turbulent Prandtl number when the wall is approached.

( (

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{

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( 204

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The model of Cebeci is the only one which points a finite value to Pr1 close to the wall.

(

Wassel & Canon

-·--\- -----;r = 0 7-;--- -~

30

(

Pr = 0.1

(

Pr = 0.025

( (

( ( (

iz

Pr = 0.0058 20

I

j

\ \ tO

\

""

( (

(

\

~-

=------------

Qk---L-~_J-~_L~~----~--L~~-L~~

1E+1

1E+2 y+

(

( (

'-..

1E+3

Figure 4. Turbulent Prandtl Number Calculated from Model of Wassel and Catton.

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( (

( ( (

( ( ( (

( ( (

Figure 5. Turbulent Prandtl Number Calculated from various models and liquid mercury.

( (

(

205

'

(

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For the case of liquid mercury flow (Fig. 5) model of Wassel and Catton and models of Kays converge to infinite value of Pr1 when the wall is approached. The model of Cebeci points to a value 8 for Pr1 at the wall. All models converge to a constant value of Pr1 (around 0.9) for y+ > 400. The same behaviour will be seen for the case of liquid sodium flow (Fig. 6). However estimation of all four models is higher than for mercury and the convergence to about 0.9 will be displaced toy+> 1000.

(

( (

( 30 ,

Pr 1

0.0058 ---

Cebeci

(

Kays I

(

Kays 2

(

Wassel & Catton

i

\

J

(

10

~-

(

\

\

~

~

(

\

20

(

\

\

(

\

" "0

1E+1

....._

L-L-LJ 1E+2

"

( '-

( (

1E+3

y•

(

Figure 6. Turbulent Prandtl Number Calculated from various models and liquid sodium.

(

( In general, it is noticeable that the lower the molecular Prandtl number the higher will be the turbulent Prandtl number mainly for situatio.n of flows close to the wall. Although many researchers agree that the turbulent Prandtl number goes to values very high for a low and very low molecular Prandtl number fluid it is very difficult through the knowledge of the relationship Pr, x y+ discriminate what would be the value of Pr1 close to the wall of a certain fluid flow. Attempts of scaling Pr, as a function of Reynolds number proved fruitless. However Bremhorst and Krebs (1992) found that for liquid sodium their data collapsed with experimental data of many researchers and with a curve given by Pr1 = 1.8 exp {-1.5 ~:Hfa.) + 0.9 for the range of 0.053 < EH/a. < 3.0. Figures 7 to 11 will show calculated Pr, using all models studied here as function of ~:HI'a. for this referenced range. Additionally will be used liquid sodium experimental data of Bremhorst and Krebs, Fuchs and of Sheriff and O'Kane, as a representative data and as a tool for the comparative analysis of the models here studied. Fuchs {referenced by Bremhorst and Krebs, 1993) measured temperature profiles in fully developed pipe flow with constant wall heat flux at a nominal temperature of 220°C. These gave EH

(

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( 206

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f (

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in the core region and &M was obtained by use of published velocity field data. Sheriff and O'Kane (1981) reported &Hand Pr, data for point source Injection placed on the centerline of fully developed pipe flow at Pr = 0.0071 - 0.0072 .. Bremhorst et al. (1989) performed measurements in water and liquid sodium with a point source using a multibore jet block. As no cross-stream velocity gradients existed, direct calculation of a turbulent Prandtl number was not readily possible, although eddy diffusivities of heat were determinable. Subsequently Bremhorst and Krebs (1992) extended the latter experiment by significantly reducing the ambient flow surrounding the point source flow. The resultant flow was similar to a free jet for which Pr, could be calculated from measured velocity and temperature profiles however comparison with boundary layer measurements were difficult. It was firstly observed by Bremhorst and Krebs (1992) that experimental results on Pr, from different researchers show good consistency when Pr, is considered as a function of etla. than other parameters such as Uoo or Reynolds , number directly even that this procedures excludes the case of ewa. = 0, as seen in Figures 7 - 11. However, as pointed by Sheriff and O'Kane (1973) omission this limit is not of practical significance, since turbulent heat transfer becomes negligible close to the wall. The procedure of studying Pr1 against ewa. as put henceforth will be a useful tool for the comparative analysis of different models of Pr, for flows of low and very low molecular Prandtl fluids.

:(

(

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( ( ( (

( (

( ( (

( ( (

Figure 7. Turbulent Prandtl Number Calculated from Model of Cebeci as function of EH/fJ..

( ( (

(

207

(

(

( {

The model of Cebeci (eq. 14) as represented in Figure 1 for various fluids, when put in terms of eddy diffusivities of heat £H/a will present behaviour as shown in Figure 7. For fluids such as air (Pr = 0.71) and gas-mixture (Pr = 0.1) the model of Cebeci show finite values of Pr1 at the wall. The same does not happen to very low Prandtl number fluids such as liquid mercury (Pr = 0.025) and liquid sodium (Pr = 0.0058). The fact is that for the region ewa < 0.5, for liquid metals, the turbulent Prandtl, calculated by model of Cebeci, increases significantly, being apparent that this model is not able to predict Pr1 of flow of very low Prandtl number fluids in the range eHfa < 2. For the case of liquid sodium the model of Cebeci follow very closely experimental data at the region ewa > 0.5. The model Kays, represented by equation 16, was calculated for all fluids studied here and is present in Figure 8. Excluding air the calculated curves of all other fluids collapsed in one curve which follow closely experimental data for liquid sodium (Pr = 0.0058). It is also interesting to observe that other expression proposed by Kays (Eq. 17) presented the same behaviour as the one presented by equation 16, but with higher calculated values of the turbulent Prandtl number, when compared with experimental results. The calculated curves of Prt. for all fluids, using equation 17, are shown in Figure 9. It seems that both expressions proposed by Kays for the turbulent Prandtl number for low and very low Prandtl number fluids do not discriminate between one and another fluid collapsing all to one single curve for each expression.

( (

(

(

( (

( ( ( (

( (

3.5 r--~-~-~----~-

_ _o:K::a,y_,s__,l'------------

(

Pr=0.71 Pr= 0.1

(

Pr = 0.025

(

Pr= 0.0058

I

2.5 ."'h

I

I

"'~

(

+

Bremhorst & Krebs A

0

Bremhorst & Krebs B

(

6

Fuchs

+

(

Sheriff & O'Kane

(

6'

(

1.5

(

--------0

0

+ 0.5

(

+

(

L--~.L....__ ___L_ _ _ ___i._ _ ___J_ _ _......l_ _ __j

2

0

3

e.H/a.

Figure 8. Turbulent Prandtl Number calculated from various and Kays (1) as function of eH/a.

( (

( (

( 208

( ' (

\ ( (

(

3.5

Ka s2

Pr=0.71

(

Pr= 0.1

.

(

Pr = 0.025

(

Pr= O.OOSB

(

( (

f (

i:z

l"

-:: c

1.5 -

Bremhorst &. Krebs B

6

Fuchs

+

·~

., 0

0

+

( o.5 L _

(

o

f

Sherif&. O'Kane

,I CJ.

-¢qo

(

I

Bremhorst &. Krebs A

0

.,.

""' £ 5

~1-

+

I~

2.5

(

( (

~

___L_ ___L_

+

___,_ _ _ _-I ; ; -

2

0

--

3

&H/a

Figure 9. Turbulent Prandtl Number calculated from various and Kays (II) as function of eH/a.. Wassel & Catton

3.5

----, Pr = 0.71

(

Pr= 0.1

( ( (

( (

( (

Pr = 0.025 \~

f

2.51~

\

+

~

J 1.5

·~

- - - - - - - -

-~

- -+ - - -0 - - - -0 - - -lf:qo

+

( (

(

(

0

+

0.5 '------'------1----L..-----L---.J._----1 0 2 3 sH/a

Figure 10. Turbulent Prandtl Number calculated from various and Wassel and Catton (II) as function of eHfa..

(

(

Sheriff & O'Kane

CJ.

(

(

Fuchs



'-""'

.,.

Bremhorst & Krebs B

6.

~

CJ.

Bremhorst & Krebs A

D

·,'\,

z

i

Pr = 0.0058 ~

209

(

( (

(

The model of Wassel and Catton for Prt represented by equation 18 is shown in Figure 10. The model of Wassel and Catton did show the usual behaviour of Pr1 for the fluid air. However for the case of low and very low Prandtl number fluids, as can be seen in that figure, calculated value of Prandt are higher than experimental data of liquid sodium, and practically all curves collapse in one curve showing the same aspect of the expressions of Kays. Considering that the most available experimental data of low and very low Prandtl number fluids is for liquid sodium, it is interesting to make comparative analysis of all proposed models relative to such fluid. In Figure 11 there are experimental data on liquid sodium taken from Bremhorst and Krebs, from Fuchs and from Sheriff and O'Kane and calculated Pr1 using call models studied here.

( ( (

(

( ( (

3.5

r---

_

__,_P_,_r-=~0~.0~05~8~--::-:---:----~ Cebeci

(

Kays I

(

Kays 2

t;

..Q

E ;::1

z

2.5

~~~ ' \

Q

~

0..

~

''

"

'-

'

~

c :; "'

-e

'

~ .\

"::I

""

(

Wassel & Catton

\

+

Bremhorst & Krebs A

0

Bremhorst & Krebs B

6

Fuchs



(

(

1

I

(

Sheriff & O'Kane

( (

~

1.5 -

(

( + 0.5 0L

__

(

+

-·---;----__!__~~ 2

(

(

3

EH/a

Figure 11 . Turbulent Prandtl Number calculated from various models with experimental data of liquid sodium.

( ( (

Experimental results of Fuchs and of Sheriff and O'Kane cover the range r....,/a < 1. For the range 0.2 < r....,/a < 1 results of Fuchs and Sheriff and O'Kane are practically the same. Fuchs present some experimental results for the region r....,/a < 0.2 indicating that Pr1 approach a limit of 2.6, as seen in Figure 11. For the region r....,/a > 1 available experimental results for liquid sodium are due to Bremhorst and Krebs. Exclusively for the case of liquid sodium experimental results of many researchers evidently indicate that when r....,/a < 0.2 the turbulent Prandtl number

(

( (

( ( (

210

( (

(

( ( (

( (

( ( (

increases significantly but to a limited valued of 2.6 Pr1 will decrease to 1.3 in the range 0.2 :5 ewa. :5 1 and Prt will be approximately 0.9 - 1.0 when ewa. > 3. As seen in Figure 11 one model of Kays (equation 17) and model of Wassel and Catton overpredict the turbulent Prandtl number relative to experimental data of liquid sodium. For the region ewa. > 0.2 the model of Cebeci and the other model of Kays (equation 16) follow experimental results very closely though the model of Kays could not discriminate among fluids, practically collapsing to one single curve. For the situation where ewa. < 0.2 all models point to values going to infinity although experimental results point to limited value for Prt.

(

f (

(

f f (

f (

f

CONCLUSIONS Various models of turbulent Prandtl number for low and very low Prandtl number were studied in this article. There were investigated model of Cebeci, models of Kays, and model of Wassel and Catton. It was verified that different from studying the relationship Prt against dimensionless wall distance y+ the relationship Pr1 against eddy diffusivities of heat ewa. provides a better tool which make it easier · to compare models. Experimental data for liquid sodium pointed that Prt is 2.6 when ef'!a. 4 0, Pr1 goes to 1.3, when ef'ia. 4 1 and Prt goes to 0.9 - 1.0 when ewa > 3; The wall distance based model of Cebeci was the only one which followed closely experimental data and which discriminated better among various low and very low Prandtl number fluid. One model of Kays followed very closely experimental however could not discriminate among fluid, collapsing all to one single curve. None of the models could follow experimental data on the region ewa. < 0.2 but in that region the molecular conduction dominates and Prt is of little importance.

( ( (

( ( (

( ( (

( ( ( (

REFERENCES "Experimental Determined Turbulent Prandtl Bremhorst, K. and Krebs, L. Numbers in Liquid Sodium at Low Reynolds Numbers", International Journal of Heat and Mass Transfer, vol. 35, n. 2, pp. 351-359, 1992. Bremhorst, K. and Krebs, L. "Eddy Diffusivity Based Comparisons of Turbulent, Pran'dtl Number for Boundary Layer and Free Jet Flows With References to Fluids of Very Low Prandtl Number", Journal of Heat Transfer, vol. 115, pp. 549552, 1993. Cebeci, T. "Laminar and Turbulent Incompressible Blilundary Layer on Slender Bodies of Revolution in Axial Flow", ASME Journal of Basic Eng., vol. 92, pp. 545554, 1970. Cebeci, T. "A Model for Eddy Conductivity and Turbulent Prandtl Number", ASME Journal of Heat Transfer, vol. 95C, pp. 227-234, 1973.

( ( (

211

\ ( ( De Lima, L.C.; Silva, M.G. and Miranda, R.F. "Comparative Analysis of Different Models for the Turbulent Prandtl Number", accepted to the Journal of the Brazilian Society of Mechanical Sciences, 1988.

( (

(

De Lima, L.C. and Pereira Filho, H.V. "Turbulent Boundary Layer with Heat Transfer on Curved Surfaces" (in Portuguese), VII Brazilian Congress of Mechanical Engineering, pp. 107-114, Uberlandia MG, Brazil, 1983.

(

( Jischa, M. and Rie~e. H.B. "About the Prediction of Turbulent Prandtl and Schmidt Numbers from Modeled Transport Equations", International Journal of Heat and Mass Transfer, vol. 22, pp. 1547-1555, 1979.

(

(

Kays, W. M., "Turbulent Prandtl Number - Where Are We?", ASME Journal of Heat Transfer, Vol. 116, pp. 284-295, 1994.

(

{ Na, T.Y. and Habib, I.S. "Heat Transfer in Turbulent Pipe Flow Based on a New Mixing Lenght Model", Appl. Sci. Res., vol. 28, pp. 415, 1973.

( (

Reynolds, A. J., "The Prediction of Turbulent Prandtl Schmidt Numbers", International Journal of Heat and Mass Transfer, Vol. 18, pp. 1055-1069, 1975.

(

(

Sheriff, N. and O'Kane, D.T. "Sodium Eddy Diffusivity of Heat Measurements in a Circular Duct", International Journal of Heat and Mass Transfer, vol. 24, pp. 205211,1981.

(

(

Wassel, A. T., and Catton, 1., "Calculation of Turbulent Boundary Layer Over Flat Plates with Different Phenomenological Theories of Turbulence and Variable Turbulent Prandtl Number", International Journal of Heat and Mass Transfer, Vol. 16, pp. 1547-1563, 1973.

(

( ( ( ( (

(

( (

(

)

(

(

( (

( 212

(

(

)

(

( ( (

( (

( ( (

( ( ( ( ( (

( (

DYNAMICS OF COHERENT VORTICES IN MIXING LAYERS USING DIRECT NUMERICAL AND LARGEEDDY SIMULATIONS

(

Jorge H. SILVESTRINI

(

Departamento de Matematica Pura e Aplicada Universidade federal do Rio Grande do Sui Av. Bento Gonyalves 9500 91501-970 Porto Alegre- RS, Brasil e-mail: silvestr@mat.ufrgs.br

( (

( (

( ( (

( ( ( (

( ( ( ( (

213

' I

l

(

( (

DYNAMICS OF COHERENT VORTICES IN

(

MIXING LAYERS USING DIRECT NUMERICAL

( {

AND LARGE-EDDY SIMULATIONS

(

' (

Abstract.

(

Coherent vortices in turbulent mixing layers are investigated by means of Direct Numer-

( (

ical Simulation (DNS) and Large-Eddy Simulation (LES). Subgrid-scale models defined in

( spectral and physical spaces are reviewed. The new "spectral-dynamic viscosity model",

(

that allows to account for non-developed turbulence in the subgrid-scales, is discussed.

(

Pseudo-spectral methods, combined with sixth-order compact finite differences schemes

( (

{when periodic boundary conditions cannot be established}, are used to solve the Navier-

(

·1

Stokes equations. Simulations in temporal and spatial mixing layers show two types of

( pairing of primary Kelvin-Helmholtz (Kif} vortices depending on initial conditions {or

(

upstream conditions}: quasi-2D and helical pairings. In both cases, secondary stream~ise

(

vortices are stretched in between the [(If vortices at an angle of 45° with the horizontal

( (

plane.

These streamwise vortices are not only identified in the early transitional stage

( of the mixing layer but also in self-similar turbulence conditions. "The Re dependence of

(

the "diameter" of these vortices is analyzed. Results obtained in spatial growing mixing

(

layers show some evidences of pairing of secondary vortices; after a pairing of the primary

(

/(elvin-Helmholtz (KH) vortices, the streamwise vortices are less numerous and their dia-

(

meter has increased than before the pairing of [(If vortices.

(

( Key words: Coherent Vortices, Mixing Layer, Direct Numerical Simulation, Large-Eddy Simulation, Subgrid Scales Models.

(

( (

( 214

(

(

4, (

( (

Introduction

(

(

Since coherent vortices play a crucial role in mass, heat and momentwn transport in

(

geophysical and industrial. turbulent flows, their identification has been one of the main

(

objectives of research in turbulence theory in the last years. To be characterized as

(

coherent three conditions are required (Lesieur, 1997):

( (

i} a concentration of vorticity w enough so that fluid trajectories can wind around,

( ( ( (

( ( (

ii) with a life time longer than their local turnover time scale w- 1 and, iii} that has the property of unpredictability, in the sense of the sensibility to initial or

boundary conditions. These coherent vortices are normally called by the name of the hydrodynamic instability which originated them (Kelvin-Helmholtz vortices, Gortler vortices), or by their orientation

( (streamwise vortices), or by their form (hairpins, lambda vortices). Here is presented some

( (

numerical evidences of streamwise vortices, their origin and evolution, in transitional and

(

turbulent mixing layers.

(

(

Two numerical techniques for the simulation of turbulent flows were used: Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES). In DNS, all turbulent

( (

(

( ( ( ( ( ( (

(

scales should be simulated explicitly, in three dimensions of space, from the integral scale

1 I down to the Kolmogorov scale (v /E) / 4 for free shear flows (in wall shear flows the dissipative scale is the viscous thickness v/u.). This implies high-order schemes, small time steps, very fine 3D grids and, in practice, low Reynolds nwnbers, since it can be proved that the total number of degrees of freedom to represent a turbulent flow is of the

215

'I'

( ( order of R13 . This is the main restriction of DNS to simulate turbulent flows of practical

( (

interest.

(

Since, in general, one is interested in the large

sc~les

of the flow, which contains most of

< the information about momentum and heat transfer, LES strategy consists in simulating, explicitly and in three dimensions, all motion larger than a certain cut-off scale. The smaller scales are modeled through a sub-grid model.

( (

Full pseudo-spectral methods (for temporal mixing layers) and pseudo-spectral methods combined with high order compact finite differences methods (for spatial developing

'

mixing layers) are used to solve the incompressible Navier-Stokes equations. Within this

{

context, it is firstly presented, the general formalism of LES carried out in spectral space

(

I



for the case where periodicity may be assumed in the three directions i.e. the temporal ( case. Extension to flows where only two directions may be assumed as periodic i.e. the

(

spatial mixing layer, is then briefly described. The subgrid-scale modelisation strategy

(

is explained and some subgrid-scale models defined in spectral and physical space are

(

described. In particular, the new spectral-dynamic viscosity model, is presented. Finally

( (

some DNS and LES results of temporal and spatial mixing layers, denoting the origin and ( evolution of streamwise vortices, are discussed.

( (

(

Large-Eddy Simulation

( In this section, the LES formalism for incompressible flows in spectral space is con-

(

sidered. The detailed description of this formalism may be found elsewhere (Lesieur and

(

Metais, 1996; Lesieur, 1997). For compressible flows, the LES formalism may be found in

(

( Comte et.al. (1994), Ducros et.al. (1996) and Silvestrini (1996), among others.

( (

216

f (

(

( (

( ( (

Full Periodic Problem Let u(k, t) and O(k, t) be the spatial Fourier transform of the velocity u(z, t) and the scalar

(

8(z, t) fields of an incompressible flow. Assuming periodicity in the three spatial directions,

(

and using pseudo-spectral·methods (Canuto et.al., 1988), the conservation equations of

( (

mass, momentum and scalar read in spectral space as :

(

(

ii(k, t). k =

(:t

+ vk 2 )

.I

o,

(1)

ii(k,t) = F[F- 1(u(k,t))

X

F- 1 (w(k,t)))- ikF,

(2)

(

(

(:t + ~~:k )

2 B(k, t) = -ik.F[r 1(B(k, t))r 1 ( u(k, t))),

(3)

(

( ( ( (

( ( (

where k

= (k,., k~, k,) stands for the wave number vector, F denotes the Fourier transform

operator, w is the vorticity vector and P

respect the incompressibility condition. The LES formalism introduces the filtering operation :

f(k) = G(k)/(k),

(4)

where

(

filter in Fourier space, defined as :

(

G( k)

is the Fourier transform of the filter function G( z). Here is used the cut-off

G(k) = { 1• if

( (

( (

(

lkl ~ kc,

(5)

1

0, otherwise

(

( (

The

pressure P is eliminated in eq. (2) by projection on the plane orthogonal to k, in order to

( ( (

= pf p +u · u/2, is the generalized pressure.

I

where kc =

1r /

t::. is the cutoff wave number associated to the grid mesh D.. Introducing the

operation (5) in Eq. (1), (2) and (3), the LES equations read :

a vk)2 u(k,t) " . k.(k,t), 217

(6)

\

t ( (

a

2 ·( 9 (at+ "k) o k,t ) = t 1 kJ
(

+ t 19kJ>dk,t.)

(7)

( (

The terms on the r.h.s of Eqs. (6) and (7) denote the supergrid-scale and sub-grid scale

(

transfers due to nonlinear terms involved in Navier-Stokes equations in Fourier space. The supergrid-scale transfers need no modelling since they can be explicitly calculated in the

( large-eddy simulation as :

(

t 1kJ
tfkJ
~<~

1

JkJ
(8)

(9)

where IT is the projector on the plane normal to the wave number vector k. The unknown

(

subgrid-scale transfers tJkJ>ko{k, t) and tfkJ>dk, t), should be modelled. Following Kntich(

nan ideas (Kraichnan, 1976), it was proposed to model these transfers with the aid of the

(

spectral eddy viscosity and diffusivity (Chollet and Lesieur, 1981), as :

(

IJkJ>ko{k,t) = -vt(k,kc,t) k 2 li(k,t), 9

(10)

(

2"

(11)

tJkJ>dk, t) = -Kt(k, kc, t) k fJ(k, t), where models to calculate v 1(k, kc, t) and

(

K1

(k, kc, t) should be introduced.

(

( (

(

Partial Periodic Problem

( Now we want to take into account the streamwise developing character of the mixing layer

(

and therefore we need to change the temporal problem to a spatial developing problem,

(

where no periodic conditions may be assumed in the streamwise direction.

( (

Let now U.(x, kw, t) and O(x, kw, t) be the spatial bidimensional Fourier transform of the velocity and passive scalar fields, where we assume that x is the streamwise no

( (

( 218

( (

'

(

( ( (

(

(

periodic direction and k2o = (kv, k,) is the wave vector defined in yz plane. With this decomposition the mass conservation equation read now as

au ax+ i(kyv + k.w) = 0,

(12)

( ( ( ( ( ( (

( (

and the momentum and passive scalar conservation equations as :

{:t {:t -~ (::2 v ( ::2

-

k

2

k

)}

2 )}

where V = (ajax,iky,ik.)

u = u x w - v p,

(13)

8 = - v . ~,

(14)

and~

denotes the Fourier transform.

The main difference between the system of Eqs. (12,13,14) with reference to the system Eqs. (1,2,3), is that now the pressure can no longer be eliminated by projection and then

(

( (

we should solve the Poisson equation that arrives when we take V· of Eq. 13, that is :

V·VP =V·uxw.

(15)

( (

Since we have now two directions where the variables are described in spectral space

(

and one in physical space, two filters should be introduced: the cutoff in spectral space

( (

(the 2D counterpart of Eq. (5)) and a. top-hat filter for the physical direction, which m~y

(

be defined as :

( (

( ( (

G(x)

~

l

1, if

lxl -< ~2'

(16)

0, otherwise

The filtering operation applied to Eqs. (13,14, 15) give us the LES equations to be used in the spatial developing mixing layer case.

(

( ( (

(

219

(

( (

(

Subgrid-Scale Models

( Here will be discussed some subgrid-scale models developed at the Grenoble turbulence

(

school (for a complete description of these models see Lesieur & Metais, 1996). Others

(

models developed elsewhere may be found in Smagorinsky {1963), Germano et al. {1991)

(

( and Ghosal et a/. {1995). Comparison of some of these subgrid-scale models in academic

( tests cases were reported in Comte et a/. {1994) and Comte et al. (1995).

(

(

·. In Spectral Space

(

Assuming a k- 5 / 3 inertial range at wave numbers greater than kc, it was proposed

( (

(Chollet & Lesieur, 1981) to renormalize the eddy viscosity with the aid of [E(kc, t)/kc]t,

(

where E(k, t) is the three-dimensional kinetic-energy spectrum. More precisely, the eddy

(

viscosity in spectral space writes

( (

llt(k, kc, t) = T<(k/kc) v;"'(kc, t)

(17)

~t;"'(kc, t)

(18)

( (

ll:t{k, kc, t) = C(k/kc)

( (

with

(

ll;"'(kc, t) = 0.267 (

E{~:• t)] t

(19)

( (

I<(k/kc) = 1 + 34.5

(20)

e-3.03(k
(

( The constant 0.267 was obtained with the aid of the EDQNM (Eddy-Damped Quasi-

(

Normal Markovian) non-local interactions theory (Lesieur, 1997), using leading-order

(

expansions in powers of the small parameter k/kc, and assuming that E(k) follows a

(

(

( 220

( (

•( ( ( (

Kolmogorov law extending above the cutoff. In eq. (20), K(k/ kc) displays a strong over-

(

shoot (cusp-behaviour) in the vicinity of k/kc = 1 (Kraichnan, 1976). This is due to local

(

or semi-local interactions in the neighborhood of k0 • If one goes back to physical space, the

(

plateau part of the spectral eddy viscosity corresponds to a classical eddy-viscosity formu-

(

lation, which assumes, in fact, a separation of scales between supergrid and subgrid scales.

( (

This is of course wrong, and fixes the limits of the eddy-viscosity formulation. Therefore,

(

the cusp part of the spectral eddy viscosity is important since it contains effects beyond

(

the classical eddy-viscosity concept. The eddy-diffusivity was found to have, qualitat-

(

ively, the same behaviour, with a corresponding turbulent Prandtl number

P; = v;"' / ~~:f

( (

approximately constant and taken equal to 0.6 (Lesieur, 1997).

(

The major drawback of the eddy viscosity described by Eq. (17) is that it assumes a

(

Kolmogorov spectrum at the cutoff. This condition is obviously not satisfied in transitional

(

regions, or close to a wall, even at high Reynolds numbers. To avoid this problem, eddy

(

coefficients may now be evaluated in a less restrictive context than previously. Assuming

( (

( ( ( (

that the kinetic energy spectrum follows a power law E(k) ex k-m instead of a Kolmogorov law, it is found (Metais & Lesieur, 1992) :

v;"'(k 0 ,t)=0.31

c;l

5 -m ( -m)t (E(kc,t)]! 3 m +1 kc

(21) I

where the Kolmogorov constant Ck = 1.4 and the associated turbulent Prandtl number

( ( (

P; = 0.18 (5- m) .

(22)

The model defined by equations (17), (21) and (22) was called the spectral-dynamic model

(

( (

(SDM). The model was used by Lamballais (1996) for LES of turbulent channel flow with excellent results. A full presentation of the model may be found in Silvestrini et al., 1998.

(

(

(

221

I

I

I

(

( ( Note finally that eq. (21) is valid only for m ::; 3. Form > 3, the choice was to set

( (

the eddy-viscosity equal to zero. From a practical viewpoint, this may be justified by

( considering that if the kinetic energy spectrum is steep enough, there

i~

no energy pile-up

at high wave numbers, so that no subgrid-scale modelling is actually necessary.

( ( ( (

In Physical Space.

( To determine eddy viscosities in physical space, the kinetic energy at the smallest resolved

scale~=

rr /kc should be measured. One of these local spectra is Fu.(:z:, t), the

(

(

second-order structure function of the resolved velocity field, defined as :

Fu,(:z:, t) = (llu(:z:

+ r, t))- u(:z:, t)ll 2) 11 rii=Ll,

(23)

( (

and related to the three dimensional kinetic energy spectrum in isotropic turbulence through Batchelor's formula (Batchelor, 1953) :

FM(:z:, t) = 4

l

0

k,

E(k, t)

(

sink~) l- - dk. k~

(

( ( (24)

In the case of infinite Kolmogorov spectra, energy-conservation arguments yield the

( ( (

structure-function model (SF model) (Metais & Lesieur, 1992), defined by

vfF(:z:,t) = 0.105 Cf(312 ~ VF2,;(:z:,t),

(

(25) (

As it involves velocity increments instead of derivatives, the SF model has the advantage

(

of being defined independently of the numerical scheme used. It is nevertheless not much

(

better for transitional flows than the Smagorinsky model: low wave number velocity flue-

(

tuations corresponding to unstable modes yield v,'s so large that affects the growth rate

(

( of weak instabilities, like Tollmien-Schlichting waves (Ducros, 1995).

(

( 222

( (

t ( (

( (

( (

One way of remedying this is to apply a high-pass filter onto the resolved velocity field before computing its structure function (Ducros et al. 1996). With a triply-iterated secondorder finite-difference Laplacian filter denoted - , one finds E(k )/ E(k) ::::: 40 3 (kf kc) 9 for all

(

k, almost independently of the grid mesh and the velocity field. With the same formalism

(

used for the structure-function model, this yields the filtered structure-function model (FSF

(

model), defined by :

( ( (

Iv[5F(:c,t) = 0.0014 Ci(3/2 A yF2.o.(:c,t).

(26)

With this model it was possible to perform a LES of a spatially-growing boundary

(

( (

layer (a.t Mach 0.5) between Re., = 3.3 10 5 and 1.14 106 , which widely encompasses the transition region (Ducros et al.l996).

( (

( ( (

Mixing Layers Simulations In this section simulations of temporal and spatial growing mixing layers a.re presented. The temporal approximation is obtained by ta.king a. reference frame moving with the

( (

average velocity (U1 +U2)/2 (where U1 and U2 a.re the velocity of the two parallel streams).

( (

(

Temporally Growing Mixing Layer A DNS and a. LES of temporal mixing layers differing in the Reynolds number will be

( (

presented at first. In both cases, the temporal mixing layers a.re initiated by a. hyperbolic-

(

tangent velocity profile, U tanh 2y / 6; to which is superposed a. small quasi-2D random

(

perturbation. 6, denotes the initial vorticity thickness. Then we will analyze the influence

(

of the initial conditions by showing coherent vortices formed in two LES which differ only

(

( (

(

223

'I

(

( ( i;1 th" initial perturbation added to the base profile.

( (

For all the simulations the computational domain is cubic of side L, = Ly = L, == 4 .\.,

(

where .\. == 7 O; == 2rr / k. is the wavelength of the most amplified stream wise mode

(

predicted by the inviscid linear-stability theory (Michalke, 1964). Such a domain allows

(

two successive pairings of Kelvin-Helmholtz (KH) vortices during a simulation. Periodic

(

boundary conditions are imposed in the streamwise (x) and spanwise (z) directions, while

( (

free-slip boundary conditions are employed for y == ±Ly/2, by means of pure sine or cosine expansions. The time derivative is approximated by a third-order low-storage Runge-Kutta

(

(

scheme (Williamson, 1980). Aliasing errors (Canuto et a/. 1988) are minimized by taking more collocation points in physical space (120 3 ) than Fourier modes (96 3 ).

( (

The DNS with an initial Reynolds number of Res, == Uo;fv == 100, is presented first.

( This simulation is called DNSQ2DT. Fig. 1 denotes the vorticity structures of the mixing

(

layer by visualization of vorticity lines. The threshold value of the vorticity norm is w;/3

(

(w; being the initial maximal vorticity modulus equal to 2U jo;). At t = 350;/U, the first

(

pairing of Kelvin-Helmholtz (KH) vortices is complete. At this time, two KH vortices

( (

can be seen, with stretching of vorticity lines in between. These streamwise vortices are called hairpins and are characterized (as we will see) by pairs of vortices of different signs. At t

= 700;/U,

the end of the simulation, only one KH vortex remains. The side views

show the stretching of vorticity lines at an angle near of 45° with respect to a. horizontal

( (

( ( (

plane. The origin of these streamwise vortices, which has been observed in laboratory

( experiments for a long time (Konrad 1976, Bernal & Roshko 1986), may be explained by

(

the intense deformation rate imposed by the KH vortices in the stagnation zones. The KH

(

vortices strain the vorticity lines, which are originally oriented in the spanwise direction,

( (

( 224

( (

t (

( ( (

and align them in the streamwise direction. The stretching of vorticity lines may be analyzed considering the vorticity equation for

( (

(

a perfect fluid : Dw;

Dt =w;S;;,

(27)

(

(llw;ll <
(

and assuming that the vorticity in the stagnation region is weak

(

gives the main direction of straining of streamwise vorticity of 45° with respect to a

(

horizontal plane.

( (

Eq. (27)

Now, results from aLES using the spectral-dynamic model (DM) with an initial Reyn-

(

olds number of Re6, = 2000, are presented. This simulation is called LESQ2DT. In this

(

LES, the spectral-dynamic model is used in its "standard" version, defined by equations

(

(17),(20) and (21). The spectrum slope miscalculated at each time step (and at each

(

sub-step of the Runge-Kutta method), from the three-dimensional kinetic energy spectrum,

(

( (

using a least-square method applied to wave numbers ranging between kc/2 < k < kc. The time evolution of the slope of the three-dimensional kinetic energy spectrum close

(

to the cutoff (m), and of E(kc) is presented on Fig. 2. Until t

(

fUOdel is applied since m

(

:5 3. Between t

= 10 ~;/Uno eddy-viscosity

= 10 ~;/ U (tim~ of KH vortices roll-up) and

20 ~;/U (beginning of the first pairing), the slope of the 3D kinetic energy spectrum takes,

( ( ( (

(

a rather constant value of m = 2.5. After that, and untii the end of the simulation, m tends asymptotically to 2. At this time, the Reynolds number" based on the local vorticity thickness

c5

(defined as 2U/I (w.) (y = 0)1) is Re6

= 24000.

In Fig. 3 can be seen the time evolution of the non-normalized three-dimensional kinetic

(

( (

energy spectrum, for the simulation LESQ2DT. Mixing-layer experiments at this Reynolds number do possess a very good k- 513 Kolmogorov law over a quite long range at large wave

( ( (

225

i

(

( ( numbPrs. The figurf> shows a Kolmogorov law only over a short range, whereas the slope

( (

is stPeper near kc, (close to -2, in agreement with Fig. 2). This is the main disagreement between the spectral dynamic model and experiments in mixing layers:

(

(

Fig. 4 shows the vorticity-modulus isosurfaces of the quasi 2D mixing layer at t = 35

(

and 75/i;/U with a threshold value of w;. The figure shows the moment of the two pairing

(

of KH vortices and the intense stretching of streamwise vortices. Note that the isovalue

(

( has increased three times with reference to the previous DNS. At the end of the simulation, there is only one KH vortex. Note finally the presence of intense small-scale vortices.

(

(

For the two simulations presented, DNSQ2DT and LESQ2DT, we analyze now the evolution of the "diameter" of the stream wise vortices with viscosity. Fig. 5 shows isolines

( of vorticity modulus in the same transversal plane and at the same time, for the DNS at

( Re=lOO and for LES at Re=2000. In the DNS we can measured:::::; 2/i; while in the LES

(

d:::::; rl;. These values may be compared with an experimental one, d:::::; 1.3/i;, obtained at

(

1400 (Huang & Ho, 1990). The agreement is fairly good and the tendency is correct.

(

Re:::::;

This shows the strength of LES to reproduce features of turbulent flows. Now, we present results of aLES (also with the SDM model) where the initial conditions

(

(

are defined hy the same base profile hut the perturbation is a three-dimensional isotropic

(

one. The initial Reynolds number is also Re 0, = 2000. The simulation was stopped at

(

t = 60 li;/U and is called LES3DT.

(

l"igure () shows the temporal evolution of m and E(kc) for the whole simulation. The

(

( spectrum slope decreases initially from the high initial value (m :::::; 9). At t = 10 li;fU,

(

which correspond to the time of the vortex roll-up, we have m :::::; 3. It means that the eddy-

(

viscosity was inactive (see equation 21) up to this instant, and that all the dissipation was

( (

( 226

( (

)

(

( (

(

due to molecular viscosity. Hence instabilities are allowed to grow without any influence

(

of the eddy viscosity, which is certainly desirable. Between t = 10 and 30 o;fU (moment

(

of the first pairing), the slope m decreases from 3 to 2. After that, m remains very close

( (

to 2 up to the end of the simulation. The temporal evolution of E(kc), which reaches

(

its maximum at t = 25 o;/U and then decreases slowly, might indicates that a. "quasi-

(

equilibrium state" characteristic of the self-similar regime was attained.

( (

Statistics of the recorded velocity profiles were used to determine the temporal evolution of the local vorticity thickness, and compared with experimental data. of spatially-growing

(

(

mixing layers carried out by Bell and Mehta (1990). The l.h.s. of figure 7 shows o(t). A

(

fairly good linear growth is established very early at a rate of U- 1 dofdt = 0.19. During the

(

first pairing (t ~ 30), the spreading slows down, and then it starts rising again at the sa.me

(

linear rate. In spite of the differences in the spatial growth of mixing layers reported in

( (

several works (Silvestrini, 1996), and also between the spatial and the temporal problem,

(

the growth rate found here is very close the traditionally accepted mixing layer spatial

(

growth of 0.18 reported in Brown and Roshko's experiments (Brown and Roshko, 1974).

(

The r .h.s. of figure 7 and figure 8 show, respectively, the mean strea.mwise velocity and

( (

·velocity components variances at the end of the simulation (t = 60 o;fU). The agreement see~s

(

between numerical and experimental data is good and

( (

state has been established a.t the end of the simulation, as far as mean and variances of

( ( (

(

velocity are concerned. To confirm this point, normalized three-dimensional kinetic energy spectra. are presented on figure 9 (the normalization is made by U and the local vorticity thickness .5). The good collapse of the different spectra for t

( (

= 50,

55 and 60 o;fU is

another good indicator that a self-similar regime is attained. Note also that Bell & Metha.

( (

to indicate that a self-similar

227

(

( { (

have considered that a self-similar regime was established at a streamwise distance of

( abont 250 cl; from the splitter plate, with the velocity ratio >.

= uu, - uu2 = 0.25. I+

If this

(

2

distanc.e is transformed in a corresponding elapsed time for a temporal r!J-ixing layer, using the convection velocity Uc = (U1 x •• ,, >. x •• ,, t,.,, = -u: = u the value found t.,.,, = 62.5 o,f

+ U2 )/2

( (

and writing

(

(28) [J

( (

is very close to the time c.onsidered for present statistics.

(

Let us look now at the three-dimensional vortical structure. Figure 10 presents a per-

( spective view of vorticity-modulus isosurfaces (threshold wi), at t = 14, 26, 40 and 60 o;JU. At t = 14 8;/U, one can see a dislocated array of four rolling-up Kelvin-Helmholtz vortices,

(

similar to the configuration found in previous DNS of Comte et a/. (1992) and laborat-

(

ory experiments of Chandrsuda et a/. ( 1978), and called "helical pairing". Secondaries

(

( streamwise vortices are also stretched by the deformation field induced between the big

( vortices. At t = 26 o;/U large structures pair. The subsequent pairing is more difficult.

(

to identify from the vorticity isosurfaces, mainly because of a rapid growth of small-scale

(

structures. At the end of the simulation (t = 60 o;/U), the vorticity field displays only

(

the presence of intense small-scale vortices, with no obvious preponderant orientation.

(

( By contrast, the low-pressure field (see figure 11) indicates the presence of one big quasi

(

two-dimensional vortex, stretching thinner longitudinal vortices. Note however that the

(

computational domain is too small at this instant, with regard to the vortex size.

(

(

Spatially Developing Mixing Layer

(

( Here, results from a LES using the filtered structure-function (26) of a spatial growing

{

mixing layer are presented. The simulation is called FSFQ2DS. The numerical code used,

(

( 228

( (

'

' (

( (

(

that solves Eqs. (13,15), combines pseudo-spectral methods in the spanwise and transverse directions with compact finite-difference of sixth order (Lele, 1992) in the streamwise dir-

( (

ection. Free-slip conditions are still imposed upon the boundaries. Non-reflective outflow

(

boundary conditions are approximated by a multi-dimensional extension of Orlansky's

(

discretization scheme. The temporal integration is performed by means of a low-storage

(

3rd order Runge-Kutta scheme, with a fractional step procedure for the pressure-gradient

(

correction.

(

( (

The profile prescribed at the inlet is :

_( )

U1

+ u2 ul -

U2

u y = - 2 - + - 2 - tan

h 2y 8,·

(29)

( (

plus small-amplitude random perturbations. The velocity ratio is chosen as: R = (U1

(

U2 )/(U1 + U2 ) = 0.5. The domain's dimension are L., = 1128;, Lu

( (

(

-

= 288; and L. = 148;,

and the grid mesh is cubic with 384 x 96 x 48 collocation points. The last record from a

I

previous DNS run a Re = 100 was used to initialize the FSFQ2DS run (Silvestrini, 1996).

(

Fig. 12 shows an isosurface of vorticity modulus at the end of the simulation, with

(

a. threshold value of 2/3w; In the figure ma.y be ideptified, from left to right, two KH

( (

. vortices undergoing a first pairing, after that a cluster of three KH vortices undergoing also a first pairing, and, a.t the end, a billow made of 4 fundamental KH vortices, whose

1

(

( (

second pairing is in progress. Between the KH vortices, streamwise vortices are stretched by the sa.me mechanism observed in temporal mixing layers.

(

Fig. 13 shows a.n isosurface of stream wise vorticity. The black and grey colors, denotin'

(

pairs of vortex of different sign, identify the "legs" of the hairpins. The figure enables us to

(

( (

analyze the evolution of the diameter of the stream wise vortices with the pairing of the KH vortices. After the pairing of the three KH vortices, strearnwise vortices are less numerpus

( ( (

229

(

( ( ~1it

bigger than before the pairing.

More quantitatively, if two transversal planes are

( (

fixed, befor" (x = 588;), and after this pairing (:r = 888;), the diameter of the streamwise vortices can be measured. A loose estimation gives dbef :::::: 0.58; and

da~t :::::: 8;,

(

which may

(

be transformed to the local vorticity thickness d&e/ :::::: 0.138 and daft :::::: 0.148 with the

(

aid of the streamwise evolution of the vorticity thickness (Silvestrini, 1996). These values

(

seem to reinforce the idea that a pairing of KH vortices induces a merging of streamwise

( (

vortices, with their diameter being scaled with the local vorticity thickness 8. But this calculation has not reached self-similarity: kinetic-energy spectra in the down-

(

(

stream region are in k- 5 12 and r.m.s. velocity fluctuations have a departure of about 20%

(

from P.xperiments. Therefore simulations in longer domain are necessary to understand

(

the downstream evolution of coherent vortices to self-similar conditions. (

Results from a LES using the filtered structure function model but where the upstream conditions are now perturbed by an isotropic noise is full reported in Comte et al. (1998).

(

(

( (

Conclusions

( The LES formalism in spectral space for incompressible turbulence was reviewed. Some subgrid-scale models defined in spectral or physical space were discussed and their limit-

( ( (

ations discussed. LES in temporal and spatial mixing layers show the main characteristic

(

of these flows: the formation and pairing of KH vortices with'intense stretching of stream-

(

wise vortices in between. These stream wise vortices may also merge when the KH vortices

(

pair. The diameter of these vortices seems to be scaled with the local vorticity thickness

(

suggesting that both kind of merging are related. The strength of LES to reproduce the

(

( coherence of the vortex organization in a turbulent flows is to be remarked.

( (

230

( (

t ( ( (

Acknowledgments

( (

This work was developed at the MOST /LEG! team in Grenoble, France. The a.uthor is

(

grateful to Prof M. Lesieur, E. La.mba.llais and P. Comte for useful discussions, a.nd to P.

(

Begou for computational assistance. Calculations were carried out at the IDRIS (Institut

(

du Developpement et des Ressources en Informatique Scientifique, Paris). The a.uthor is

(

(

supported by a Research Fellowship from FAPERGS/RS.

(

(

References

( ( ( (

G. K. BATCHELOR., The Theory of Homogeneous Turbulence, Cambridge Univ. Press., 1953.

J. BELL and R. MEHTA, Development of a two-stream mixing layer from tripped a.nd

(

( ( (

( ( ( ( ( ( (

untripped boundary layers, AIAA Journal, 28, 2034-2042, 1990. L. BERNAL and A. RoSKHO, Streamwise vortex structure in plane mixing layer, J. Fluid Mech., 170, 499-525, 1986.

G. BROWN and A. RosKHO, On density effects a.nd large structure in turbulent mixing "layers, J. Fluid Mech., 64, 775-816, 1986. C. CANUTO, M. Y. HUSSAINI, A. QUAR.TERONI, and T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988.

J. P. CHOLLET and M. LESIEUR., Parameterization of sma.ll sca.les of the threedimensional isotropic turbulence utilizing spectral closures, J. Atmos. Sci., 38, 2747-2757,

( (

(

1981.

J.P. CHOLLET a.nd M. LESIEUR, Modelisation sous maille des flux de qua.ntite de

( ( ( (

231

(

( ( mouvement et de chaleur en turbulence tridimensionnelle isotrope, La Meteorologie, 29-

( (

30, 183-191, 1982.

(

P. CoMTE, F. DUCROS, J. SILVESTRINI, E. LAMBALLAIS, . 0. METAlS, and

(

M. LESIEUR, Simulation des grandes echelles d'ecoulements transitionnels, In Proc. 74th

(

Fluid Dynamics AGARD Symposium on "application of direct and large eddy simulation

(

to transition and turbulence", Chania, Crete, 1994.

( (

P. COMTE, J.H. SILVESTRINI, and E. LAMBALLAIS, A straightforward 3d multi-block

(

unsteady Navier-Stokes solver for direct and large-eddy simulations of transitional and

(

turbulent compressible flows, In Proc.

(

77th Fluid Dynamics AGARD Symposium on

"progress and challenges in CFD methods and algorithm", Seville, Spain, 1995.

( (

P. COMTE, J.H. SILVESTRINI, and P. BEGOU, Streamwise vortices in large-eddy simulations of mixing layers, European Journal of Mech. B/ Fluids, 17(3), 1998. F. DUCROS, Simulations numeriques directes et des grandes echelles de couches limites compressibles, These, Institut National Polytechnique de Grenoble, 1995.

F. DUCROS, P. CoMTE, and M. LESIEUR, Large-eddy simulation of transition to tur-

(

(

( ( ( (

bulence in a weakly compressible boundary layer over a flat plate, J. Fluid Mech., 326, 1-36, 1996.

( (

M. GERMANO, U. PIOMELLI, P. MOIN, and W. H. CABOT, A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A, 3(7), 1760-1765, 1991.

( ( (

S. GHOSAL, T. LUND, P. MOIN, and K. AKSELVOLL, A dynamic localization model

( for large-eddy simulation of turbulent flows, J. Fluid Mech., 286, 229-255, 1995. L. HUANG and C. Ho, Small-scale transition in a plane mixing layer, J. Fluid Mech.,

(

(

(

210, 475-500, 1990.

( ( 232

(

(

' (

( (

( (

( ( ( (

(

J. KONRAD, An experimental investigation of mixing in two-dimensional turbulent shear flows with applications to diffusion-limited chemical reactions, PhD thesis, California

Institute of Technology, 1976. R. H. KRAICHNAN, Eddy viscosity in two and three dimensions, J. Atmos. Sci., 33, 1521-1536, 1976.

E.

LAMBALLAIS, Simulations numeriques de Ia turbulence dans un canal plan tournant,

These, lnstitut National Polytechnique de Grenoble, 1996.

( ( (

S. K. LELE, Compact finite difference schemes with spectral-like resolution. J. Comp. Phys., 103, 16-42, 1992.

(

M. LESIEUR, Thrbulencl! in fluids, (Third Edition), Kluwer Academic Publishers, 1997.

(

M. LESIEUR and 0. METAlS, New trends in large-eddy simulations of turbulence,

( ( (

( ( (

Annu. Rev. Fluid Mech., 28, 1996.

0. METAlS and M. LESIEUR. Spectral large-eddy simulation of isotropic and stably stratified turbulence. J. Fluid Mech., 239, 157-194, 1992. A. MICHALKE. On the inviscid instability of the hyperbolic tangent velocity profile. J. Fluid Mech., 19, 543-556, 1964.

( (

J.H. SILVESTRINI. Simulations des grandes echelles des zones de melange; application

(

d Ia propulsion solide des lanceurs spatiaux. These,

( (

Grenoble, 1996.

(

ln~titut

National Polytechnique de

J .H. SILVESTRINI, E. LAMBALLAIS, and M. LESIEUR. Spectral dynamic model for LES

of free and wall shear flows. in press Int. Journal of Heat and Fluid Flow, 1998.

( (

(

J. SMAGORINSKY. General circulation experiments with the primitive equations. Mon. Weath. Rev., 91(3), 99-164, 1963.

( ( (

(

233

I

(

( J. H.

WILLIAMSON.

(

Low-storage runge-kutta schemes. J. Comp. Phys., 35, 48, 1980.

(

(

(

I

( I

FIGURES

( ( (

( (

(

( (

(

( ( (

( (

(

I Figure 1: Vorticity lines at t = 35 and 70 .S;/U (top views above and side views below) - Simulation DNSQ2DT.

( (

(

(

( ( ( 234

( (

' (

(

( (

lOr---------------------------------------------~

(

(

., .,

(

,

10-l

-------------------- ~·~

10-· 10-·

(

10->

(

10-·

(

10-·

(

QL---~---L--~~--~--~--~----~--~--~

0

20

( (

10-1(

80

60

40

tt5;/U Figure 2: Time evolution of m (-) and of E(kc) (---)-Simulation LESQ2DT.

( ( ( (

10-1

E

I

I

I

I

I I I II

I

I

I

I

a

( 10- 2

( (

( (

10-3 ,-...

:!J:!J........ w

10-4

( ( ( (

( ( (

k Figure 3: Time evolution of the non-normalized three-dimensional kinetic energy spectrum from t = 10 to 85 6;/U- Simulation LESQ2DT.

(

( ( ( (

235

(

(

( ( (

(

( ( (

( (

( ( (

(

(

( (

Figure 4: 1110811rfacea of vorticity modulus

llwll

=

w1

at t

=35 and 75 51/U (top views a.bove and side

views below)- Simulation LESQ2DT.

( (

( l

i

l.:::r::·

10

'ht:·

( ( (

.. L !

( (

:1

I

(

+

·+··· ! -10

.,.

-10

10

0

(

(

10

''"

(

KH vortices, for the DSNQ2DT (left) and LESQ2DT (right), at t = 35 5;fU.

(

Figure 5: Isolines

of vorticity modulus at the same transversal plane in the stagnation region between

( ( 236

(

(

'

(

( ( (

10,------------------------------------------,

( 8

(

( ( (

( 0~~--~--~--~~--~--~--~~~~--~--~~

(

0

20

40

ti5;/U

( (

Figure 6: Time evolution of m (-)and of E(kc) (---)-Simulation LES3DT.

( (

i

\

( (

.

I

-T

( ( 10

(

(

()/();

yj()

(

0

(

(

-1

( (

( (

(

40

20

-1

60

tU/15;

(

0.~

Figure 7: Left, time evolution of the local vorticity thickness ; right, comparison of mean strea.mwise velocity (straight line) with experimental data of Bell and Mehta (1990) (circles) - Simulation LES3DT.

(

(

0

/U

( (

-o.~

237

( (

I

(

I (

(

( ( ( {

(

0 0

\

0 0 0 0

(

0 0

(

-1

(

-1

/U 0

0.05

0.1

2

/U

0.15

0

2

0.05

( ' 0.1

( (

{ 0

yj~

0

( (

0

0

(

0

-1

-1

/U 0

0.05

0.1

/U 0.02

0.04

2

(

0.08

(

Figure 8: Comparison of present velocity fluctuations variances (linea). with experimental data (symbols) of Bell and Mehta (1990)- Simulation LES3DT.

f (

(

( (

(

238

(

'

(

'

(

\

(

( ( ( ( ( (

( ( ( 10-2

(

}::

I

I

I

I I I I I I

I

I

I

I i I II I

....

!i

( ( 10-l

(

( ( (

(

,...... N

'0

::::>

::::::: ,......_ ~­

~-

......... w

( ( ( (

10-·~--L-~~~~~~~L-~~~~

101

1

ko

( (

(

Figure 9: Normalized three-dimensional kinetic energy spectra at t =50(-), 55(_:--) and 60 - -) - Simulation LES3DT.

(

( (

( ( (

( ( ( (

102

239

oi/U (-

( (

J

( (

.< (

( (

(

( ( (

(

( (

( (

{ ( ( (

( (

( Figure 10: Flom left to right and top to bottom, perspective views of.the mixing layer at t =14, 26, 40 and 60 6;/U showing isosurfaces of the vorticity modulus at a threshold of w;- Simulation LES3DT.

( (

(

(

( (

I

( ( 240

(

(

I

-( ( ( ( ( (

(

( ( (

( (

( ( (

( (

( ( (

( ( ( (

( ( (

TURBULENT SHALLOW-WATER MODEL FOR OROGRAPHIC SUBGRID-SCALE PERTURBATIONS Norberto Mangiavacchi, Alvaro L. G. A. Coutinho, Nelson F. F. Ebecken Center for Parallel Computations

COPPE/Federal University of Rio de .Janeiro PO Box 68506, R.J 21945-970, Rio de .Janeiro, Brazil E-mail: norberto,alvaro@coc.ufrj. br, nelson@ntt. ufrj. br Web page: http:/ /www.nacad.ufrj.br · Key Words: 'TUrbulent flows, shallow-water equations, subgrid-scale models, orographic perturbations, parallel processing, spectral methods

Abstract. A parallel pseudo-spectral method for the simulation in distributed memory computers of the shallow-water equations in primitive form was developed and used on the study of turbulent shallow-waters LES models for orographic subgrid-scale perturbations. The main characteristics of the code are: momentum equations integrated in time using an accurate pseudo-spectral technique; Eulerian treatment of advective terms; and parallelization of the code based on a domain decomposition technique. The parallel pseudo-spectral code is efficient on various architectures. It gives high performance on vector computers and good speedup on distributed memonJ systems. The code is being used for the study of the interaction mechanisms in shallow-water flows with regular· as well as random orography with a prescribed spectrum of elevations. Simulations show the evolution of small scale vortical motions from the interaction of the ' large scale flow and the small-scale orographic perturbatiqns. These interactions transfer energy from the large-scale motions to the small (usually unresolved} scales. The possibility of including the parametrization of this effects in turbulent LES subgrid-stress models for the shallow-water equations is addressed.

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INTRODUCTION

The shallow-water model is used· as a test bench for understanding many fundamental dynamical problems, such as atmospheric flows, tides, storm surges, river and coastal flows. In many such applications, the flow is mostly a two-dimensionai turbulent flow. Numerical simulations usually are performed using Large-Eddies Simulation (LES) techniques, which require the parametrization of scales smaller than the computational grid, which is done by subgrid-scale models. The best known subgrid-scale model for LES of turbulent flows is the Smagorinsky model [1). Later models, such as Germano [2), and Lilly [3) overcome some of Smagorinsky model limitations, in particular the need to specify a parameter, and the tendency of beeing over-dissipative. None of these models, however, address the effects of the perturbations introduced by an irregular orography with scales smaller than the computational grid. Small scale orographic perturbations play an important role in the dynamic cascade in turbulent shallow-water flows. Neglecting these effect can result in degraded forecast capabilities in atmospheric simulations. Therefore, better subgrid-scale models, including subgrid-scale orographic effects, may result in significantly improved weather forecasts. Numerical simulations of the detailed behaviour of turbulent shallow-water flows in the presence of small scale orographic perturbations can provide a fundamental understanding of the mechanisms involved in such flows, and provide valuable data for the development and validation of new sub-grid models. Such simulations require the space and time accurate integration of the turbulent flow field and additionally model accurately the effects due to the orography. Since such simulations are very computationally intensive, higher resolution shallowwater flow simulations could profit from the scalable performance available on parallel arquitectures. Pseudo-spectral methods, due to their high accuracy and performance when used in simple domains, and their intrinsic parallelism, can be applied successfully for the simulation of shallow-water flows in distributed memory computers. In this work we describe a new parallel pseudo-spectral code designed to perform high resolution, space and time accurate simulation of shallow-water flows on various current distributed memory architectures. The parallel algorithm explores the intrinsic parallelism of the pseudo-spectral method and it is based on a domain decomposition approach. The remainder of this paper is organized as follows. In the next section we briefly review the governing equations of shallow-water flows. The section that follows details our pseudospectral method, based on Fourier expansions. The next section presents the parallel implementation,.designed to achieve high performance, while retaining portability across different platforms. The following section shows the numerical results for a test case and discusses the parallel performance of the code on several computers, and results of several simulations of the shallow-water equations with various orographic perturbations are discussed. Finally, the paper ends with a summary of the main conclusions of this work.

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GOVERNING EQUATIONS

The governing equations are the shallow-water equations, also known as Barre de SaintVenant equations. The derivation is given, for example, by Lesieur (4]. Here we will restate the main assumptions. We start from the Navier-Stokes equations locally on a sphere, with a fluid of constant uniform density p. The fluid is assumed to have a free surface with elevation H(x, y, t) above a reference plane, and to lie above a topography of height h,(x, y). The depth of the fluid layer is h(x, y, t), and the surface elevation is H = h + h, (see figure 1).

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The pressure at the free surface is uniform and equal to p0 • The pressure is hydrostatically distributed along the vertical, and the horizontal velocity field v, depends only on the horizontal space variables x and y, and on the time. Integrating the equations along the vertical, one obtains

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THE NUMERICAL METHOD

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We use a Fourier· Pseudo-Spectral Method in order to discretize the system of partial differential equations into ordinary differential equations. The pseudo-spectral appt·oach avoids the high cost of computing quadratic and higher terms in Fourier representation, which require convolutions, but still preserving the Spectral Convergence properties of standard spectral methods. The Pseudo-spectral approach gives the best computational cost/benefit for simple geometries, in particular for periodic domains. We proceed using the expansion of the flow variables in Fourier series in the (x) and (y) horizontal directions, u(x) =

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To be able to perform high resolution simulations, which are required to perform direct numerical simulations, the code was parallelized using a domain decomposition approach, which is described next.

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PARALLEL ALGORITHM

The parallel algorithm explores intrinsic parallelism of the pseudo-spectral technique. It is based on a domain decomposition approach in latu sensu. Three kinds of operations are involved in the parallel pseudo-spectral method: (1) computation of products in physical space, (2) inversion of the Poisson operators, computation of derivatives, and filtering operations in Fourier space, and (3) computation of the discrete two-dimensional Fourier transforms. The first two kinds of operations can be performed without any communication, when the domain is partitioned among the processors, in physical space and in

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Fourier space respectively. The only part that requires communication is the computation of the the discrete two-dimensional Fourier transforms. Here also, for portability, we have chosen to use the transposition approach, that allows to use highly optimized single processor one-dimensional FFT routines, that are normally found in most architectures, and a transposition algorithm that can be easily ported to different distributed memory architectures. The transposition algorithm used in the FFT is the so called dir·ect transposition algorithm where at each stage of the communication algorithm each node sends to his pair all the data that has that node as final destination. At each stage the processor pairs are defined using a mapping of the processors onto a hypercube, and a relative addressing strategy. The number of stages is p - 1 where p is the number of processors. The transposition algorithm can be summarized as follows: • For i=l, 2, ... ,p- 1 do: Each node m collects and sends ton = XOR(m, i) all blocks that have node n as final destination, and replaces them with the blocks that receives from n. • Unshuffle the resulting data in each node. Here XOR stands fo the exclusive or boolean operation. The Poisson operator is diagonal in Fourier space. After decomposition of the domain among the nodes, the Poisson operator will require 0( ~·) computations per node, at each time step, and no communications. Here N is the resolution in each dimension. The two-dimensional FFT's will require 0( 2 N' l~g.,(N)) computations and (p-1) bidirectional communications of length ~: when using the described transposition algorithm. Since the latency time is much shorter than the communication time for large problem sizes, the total communication time is essentially 0( ~' ). Therefore the ratio of communications to computations is 0(-1-) log 2 (N) · Except for a few global operations, all the communication arc lumped in the transposition algorithm making it easily portable to other distributed memory computers. The implementation using PVM, as the one given in [5], can run on a CRAY T3D and an IBM SP2 without changes in the transposition algorithms, only requiring to load the appropriate machine dependent single-processor one-dimensional FFT routines. To achieve better performance on the SP2, versions employing the MPI and MPL libraries were also implemented with minor additional effort, and considerable improvement in the performance.

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SINGLE NODE AND PARALLEL PERFORMANCE I

To evaluate the performance of the code in parallel machines, two-dimensional computations were performed with resolutions ranging from 256 x 256 to 1024 x 1024.

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Table 1: Single-processor performance (256 x 256 resolution)

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CPU IBM SP2 wide IBM SP2 thin IBM SP2 wide CRAY T3D CRAY J90 CRAY T90

FFT TIME(s/100 iters) Fortran 246 ESSL 90 ESSL 49 scalar library 156 vector library 24.7. vector library 3.38

Mftopjs 10 28 50 16 100 731

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High single-processor performance is obtained across various platforms using the provided optimized high performance libraries for the FFTs. The first and third cases in table 1 (SP2 wide using FORTRAN and ESSL library respectively) show that single-processor performance in the IBM SP2 can be improved by a factor of 5 by using the ESSL library. On CRAY T3D implementation, the scalar FFT library subroutines CCFFT, SCFFT, and CSFFT were used. On the CRAY J90 and T90 version, the vector library FFT subroutines CFFTMLT, SCFFTM, and CSFFTM where used. Parallel performance was measured on a 4-processor IBM SP2. and on a 32-processor CRAY T3D. When using the PVM libraries on the SP2, the speed-up curves show that the parallel efficiency drops to about 75% for two processors. This is caused by the transposition algorithm, which is not present in the single processor case, and that introduces a significant overhead even when using only 2 processors. However efficiency continues above 50% even for quite larger numbers of nodes, as long as the problem size is adequate. When using the dedicated MPL library on the SP2 we obtain some improvement in the efficiency (on two processors about 80% for 256 x 256), showing that the communication cost is reduced by using the MPL library. In fact, the efficiency is about 50% even when using 32 T3D nodes for the case 1024 x 1024. Hence, a CRAY T3D with 32 nodes outperforms a single CPU of a CRAY J90 running in vector mode. The scalability of the parallel code clearly indicates its potential to beat more powerful vector processors, as the T90, if more processors were added to the parallel machine.

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RESULTS

For the scope of this work we will analize the case of negligible rotation (J = 0). The following simulation were performed using a physical space resolution of 64 by 64 points, unless otherwise spcci ficd.

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6.1

Test problem

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A number of runs were performed to assess the correctness of the code. One usual test for shallow-water solvers is the dam-break problem. In this problem, there is sharp variation in the surface elevation If, in otherwise quiescent. initial conditions. The elevation is H = H 0 at the left, and H = H 1 at the right of the dam. From this initial sharp front, two waves propagate in opposite directions, at two different speeds. The wave travelling in the shallower side is a shock wave, while the wave propagating on the deep side is a rarefaction wave. In this test it is assessed the ability to accurately simulate the propagation of sharp waves on the surface elevation. A typical result from this test is shown in figure 3. A two-dimensional extension to the dam-break problem was also simulated, in order to verify if the code accurately accounts for more complex two-dimensional wave interactions. Results are shown in figure 4.

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Some simulations were performed using regular orographic perturbations. In such simulations the orography is given by sinusoidal perturbations of the form

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Regular Orography

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Two different kinds of initial conditions were analised. On the first ease, shown in figure 5, a single vortex with a scale larger than the orographic perturbation is left to evolve.

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Figure 5: Simulation of shallow-water flow over regular orography. Single vortex. Vorticity field for (a) flat orography, and (b) sinusoidal orography (single 2-d mode). Interactions with the orography cause the development of six small-scale peaks close lo the center of the vortex. (c) Elevation of the orographic perturbation.

In the abscence of the orographic perturbations, the vortex undergoes a slow decay, since the backgroung viscosity is very small. In the presence of the orographic perturbation the vortex develops small-scale peaks, which are originated from interactions between the vorticity field and the orography. The occurrence of the peaks can be explained simply by conservation of potential vorticity (w +f)/h. When passing over deeps, h increases causing the increase of vorticity by vortex stretching, and the opposite occours over the hights. On a second case, the initial conditions are given by a turbulent fiow, as shown in figure 6. In this case, the interaction cause a faster rate of decay of the turbulent motions with scales close to the scale of the orographic perturbation than the ease without perturbation. The effect of the orographic perturbation can be better visualized looking at the energy spectra E(k) of the two flows, shown in figure 7. Two kinds of effects can be seen. The first most ostensive effect is the forcing of modes dose to the perturbation scale (8J2) and its harmonics. The forcing causes the peaks observed in the spectrum at discrete points. The other effect is the reduction of energy at scales away from the forcing frequencies, in particular for the large scales (small k). This is the most important effect as far as parametrization of subrid-scale perturbations is concerned.

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f Figure 6: Simulation of shallow-water turbulent flow decay over a flat orography and over regular orographic perturbations. (a) Vorticity field of the initial r.ondition. (b) Vorticity field for t = 1 x 105 with flat orography, and (c) with sinusoidal orographic perturbation, showing vortical structures with scales larger than in (b). (d) Elevation of the sinusoidal orography (single 2-d mode). Interactions with the perturbed orography in (c) cause a much faster rate of decay of the turbulent motions with scales close to the scale of the orographic perturbation than the previous case (b). The decay of the large scales is also affected.

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6.3

Irregular Orography

A more complex behaviour can be expected in the case that the urography has a continuou~ spectrum of perturbations. Two particular cases are of interest. In the first case the spectrum of perturbation is similar to the spectrum of the velocity, while in the second case the spectrum of the orographic perturbation is peaked at higher modes. The elevation spectra for this two cases are shown in figure 8, and the resulting spatial distributions are shown in figure 9.

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Figure 9: Terrain for irregular orography. Elevation of the orographic perturbation used in the simulations. a) with spectrum similar to the velocity spectra. b) with spectrum with more energy at higher modes.

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6.4

Considerations on Parametrization of Orographic Subgrid-scale Perturbat ions

From the results of the previous simulations it has been shown that the effect of the subgrid orographic perturbations is not a simple dissipative mechanism, acting instead in the full range of dynamic scales. The transfer of energy among scales is enhanced, and, in the case of of orography with a elevation spectrum containing perturbations with high energy at small scales, there is strong transfer of kinetic energy from large to small scales. No simple dissipative model can reproduce this kind of mechanism with accuracy. In order to take into account these effects in large eddies simulations, one crude approach is to add a perturbation at the smallest resolved scales with the same energy than the subgrid scale perturbations. A very promising approach, however is to use upscaling technique.s to accurately compute the energy transfer due to the smallest scales, and to use this information in the large scale model.

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A method to solve two-dimensional shallow-water !lows using pseudo-spudral methocb was developed. Resuls of simulations using the method show physically consistent. results. Based on this method, a high performance parallel pseudo-spt~ctral method for the simulation of two-dimensional shallow-water flows was developed. The code is designed to perform high resolution, space and time accmate simulations of shallow-water flows on various distributed memory architectures. The parallel Pseudo-Spectral code is efficient on various architectures. It gives good speedup on distributed memory systems (IBM SP2 and T3D). Simulations performed with the code show that the interaction mechanisms between orography and the vorticity field lead to the development of small scale vortical motions that have a faster decay rate, at the expense of the energy of the large scales. This kind of interaction cannot be accurately represented by a simple dissipative uwdel, requiring more refined techniques such as upscaling.

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We gratefully acknowledge the Center of Parallel Computations of the Federal University of Rio de Janeiro (NACAD-COPPE) for providing us time on t.he IBM SP2. and CRAY ]90 machines. We are also indebted to Silicon Graphics/Cray Research Division by the computer time in their CRAY T3D and T90 machines at Eagan, MN, USA. The first author is currently supported by a Research Grant from FAPERJ (E-26/151.(118/97).

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CONCLUSIONS

ACKNOWLEDGEMENTS

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REFERENCES

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[1] J. Smagorinsky. General circulation experiments with the primitive equations. Mon. Weath. Rev., 91, 99-164 (1963). [2] M. Germano, U. Piomelli, P. Moin, and W. H. Cabot. A dynamical subgrid-scale eddy viscosity model. Phys. Fluids .4, 3, 1760-1765 (1991). [3] D. K. Lilly. A proposed modification to the germano subgrid scale closme method. Phys. Fluids A, 4, 663 (1993). [4] M. Lesiem. Turbulence in Fluids. 1\luwer Academic Publishers, 2nd edition, (I !)90). [5] N. Mangiavacchi, A.L.G.A. Coutinho, and N.F.F. Ebecken. Parallel pseudo-spectral computations at NACAD. Proc. of SBAC-PAD, 1 (1997).

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lnstabilidade e Turbulencia: Uma Forma de Nao-Linearidade Encontrada no Caos

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Instability and Turbulence: A Kind of Nonlinearity found in Chaos Harry Edmar Schulz Laborat6rio de Hidraulica Ambiental - CRHEA Departamento de Hidn!ulica e Saneamento Escola de Engenharia de Sao Carlos-Universidade de Sao Paulo C.P.359, 13560-270, Sao Carlos, S.P., Brasil (Trabalho desenvolvido no lnstttul fOr Hydromechanlk, Unlversil~t Karlsruhe, Alemanha)

Abstract

The Navier-Stokes equations and the Reynolds equations are used to generate a simplified onedimensional model, which maintains the nonlinear characteristic of the original equations. The discretization of this simplified model leads to an equation similar to the population growth equation used in the earlier studies of chaos. As a consequence, the behaviour of data obtained using the growth equation must also be observed in mathematical ::,ystems which use the NavierStokes or the Reynolds equations. As these models are believed to be representative of a physical reality, also physical ::,ystems found in the nahtre may follow the same behaviour. The example of flow in shallow water layers is used in this study. Keyworcls: chaos, nonlinear systems, transition flows, period doubling. Resumo

As equa{:oes de Nm 1ier-Stokes e de Reynolds siio utilizadas para gerar um modelo simplificadp, unidimensional, que mantem a caracteristica niio-linear das equa{:oes originais. Atraves da discretiza{:iio deste mode to simplificado mostra-se que a equa{:iio final obtida reproduz a equa{:iio logistica de crescimento populacional, que foi utilizada nos estudos pioneiros acerca do caos. Desta forma, o comportamento descrito pela equa{:iio logistica de crescimento populacional tambem pode ser esperado em sistemas matemciticos descritos pelas equa9oes de Navier-Stokes e de Reynolds discretizadas. Sendo esses modelos matematicos representativos da realidade que pretendem descrever, epassive/ esperar o mesmo comportamento nos sistemas fisicos encontrados na natureza. No presente estudo utiliza-se como exemplo o escoamento em aguas rasas. Palavras-chave: caos, sistemas niio-lineares, escoamentos de transi{:iio, duplica{:iio de periodos.

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0 estudo de turbulencia recebeu urn impulse algo inovador a partir das decadas 70-80 com os trabalhos referentes ao caos, que mostraram que esquemas numericos simples podem conduzir a evoluyoes complexas das grandezas envolvidas. Uma quantidade razmivel de trabalhos foi conduzida desde entao, mostrando as coincidencias comportamentais entre os sistemas fisicos naturais e os resultados qualitativos desses esquemas numericos simples. Mesmo resultados numericos, relacionados com a obtenvao de constantes vinculadas ao modelo de duplica<;;ao de periodo de May e Feigenbaum (ver Feigenbaum, 1978) foram tambem observados experimentalmente com aproximavao razoavel. Neste sentido, o modelo de duplica<;;ao de periodo permitiu efetuar uma conexao, ainda que mnemonica, entre a evoluviio prevista para as variaveis matematicas e a evoluyao estabilidadeinstabilidade-turbulencia observada em muitas situayoes em mecanica dos fluidos. A grande vantagem didatica em visualizar o processo de instabilidade e evolu<;;ao para a turbulencia, baseada

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( no modelo de duplica~iio de periodo, como resultante das intera~iies niio-lineares existentes na natureza e reproduzidas nos modelos matematicos, sugere que se utilize este modelo de forma mais intensa como ferramenta de suporte ao estudo da mecanica dos fluidos. Entretanto, tambem e preciso que uma conexiio mais formal seja elaborada, vinculando a expressiio de recorrencia basica dos estudos de caos via duplica~iio de periodo, com expressiies numericas decorrentes do uso das equacoes de Navier-Stokes ou das equa~iies de Reynolds (no caso de escoamentos turbulentos). Evidentemente as equa~oes de Navier-Stokes podem ser discretizadas de diferentes maneiras, mas ~ostra-se, neste trabalho, que e possivel seguir uma forma de discretizayiio em uma situayiio de escoamento simplificada, a qual conduz a uma expressiio similar a equayiio basica adotada para o estudo da duplicayiio de periodos. Desta forma, a niio-linearidade presente nas equacoes de NavierStokes ou nas equaciies de Reynolds mostra ser do mesmo tipo daquela existente na equayiio de recorrencia dos estudos de caos, sendo que evolucoes comportamentais obtidas para esta equayiio podem ser esperadas tambem para as equacoes de Navier-Stokes ou de Reynolds, bern como para os sistemas reais que elas pretendem representar. No presente trabalho considera-se uma situayiio uni-dimensional e pretende-se que o mesmo seja fundamentalmente urn texto didatico. 0 objetivo basico e mostrar que a niio-linearidade presente nas equacoes estudadas pode conduzir a instabilidade e, posteriormente, aturbulencia. Esta forma de abordagem e mais adequada a uma introducao acerca de instabilidade em mecanica dos fluidos do que o usual tratamento formal, que sobrecarrega matematicamente o primeiro contato como tema.

Situac;:ao de Aguas Rasas e Simplificac;:ao Conveniente das Equac;:oes Governantes 0 escoamento em aguas rasas, tanto no estudo de casos larninares como no estudo de casos turbulentos, gera esteiras de VOrtices que, a despeito da grande diferenya entre OS numeros de Reynolds correspondentes a cada caso, mantem uma semelhanya plastica notavel entre si. Este fato desperta novamente a curiosidade acerca das causas deste tipo de movimento e de como podemos descreve-lo da melhor forma. A questiio torna-se ainda mais interessante porque acreditamos que ja dispomos da equacao governante para este fenomeno, isto e, as equacoes de Navier-Stokes devem ser v{llidas para escoamentos larninares ou turbulentos. Possuimos tambem equacoes (construidas a partir das equacoes de Navier-Stokes) que, acredita-se, descrevam corretamente os escoamentos turbulentos (equayoes para grandezas medias, como as equacoes de Reynolds). Assim, temos as equayoes, ou mesmo urn conjunto delas, que sao de diftcil tratamento, niio perrnitindo uma visualizayao imediata das caracteristicas dos escoamentos que pretendemos entender. No caso dos escoamentos de aguas rasas, as caracteristicas dos mesmos perrnitem que uma serie de simplificacoes seja feita nas equacoes de Navier-Stokes ou nas equayoes de Reynolds, que conduz a uma forma final que mantem a caracteristica niio-linear do equacionamento original. Esta niio-linearidade e aqui explorada atraves de uma aproximacao numerica, nos moldes dos trabalhos de May e Feigenbaum (ver Feigenbaum, 1978), procurarando reproduzir as propriedades que os autores mencionados enfatizaram. A situacao fisica que aqui se explora e o escoamento superficial ( ou o escoamento da superficie) em urn corpo de agua raso, com direyiio do escoamento paralela ao eixo x. 0 equacionamento basico para a direyiio x, para as equacoes de Navier-Stokes e para as equacoes de Reynolds, e dado por:

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Equacoes de Navier-Stokes:

au au au au 1 op (o U o U o U) - + U - + V - + W - = - - - + v - - + - - + - - +B 01 OX oy 0Z , p OX ox oy 0Z 2

2

(

2

2

2

2

(!) X

(

( 258

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U, V e W sao as componentes da velocidade do escoamento nas dire96es x, y e z, respectivamente. p e a pressao, p e a massa especifica do fluido, v e a viscosidade cinematica do fluido e Bx e a forya de campo na direyao x.

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Equa96es de Reynolds (com as tensoes de Reynolds decorrentes dos produtos entre flutuay6es de velocidades):

(

rJ [J __ rJ[J _rJ(] _rJ(J --+U-+V--+W-= rJt rJx oy rJz

(

( rJ- ( v--uu au -~ +rJ- ( v---uv au -) +a- ( v---uw ou -)) +B_ -1-oJ5 -+

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( ( ( (

p OX

OX

OX

oy

0y

0Z

0Z

(2)

X

A barra superior indica a operayao de media temporal efetuada sobre as grandezas ja definidas nas equa<;:oes de Navier-Stokes. u, v e w sao as flutua<;:oes de velocidade nas dire<;:oes x, y e z. As simplifica<;:6es que podemos fazer, para o estudo do escoamento da superficie em urn experimento em tanque para aguas rasas, siio as seguintes: l - 0 escoamento campo (Bx=O).

e horizontaL

2 - 0 escoamento preferencial velocidades nas dire<;:oes y e z.

Assim, niio ha componente na dire<;:iio x para as for9as de

e na dire<;:iio paralela ax.

Assim, no momento, desprezam-se as

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3 - Por se tratar de urn escoamento de superficie, niio existe varia<;:iio de pressiio ao Iongo de x. 4 - Efeitos viscosos siio desprezados junto a superficie livre. 5 - As flutua<;:6es de velocidade mantem caracteristicas constantes ao Iongo de toda a superficie. Convem frisar que a simplifica<;:iio 2 da ao sistema a caracteristica unidimensional procurada. Mas esta simplifica<;:iio tambem faz com que, em urn escoamento de caracteristicas bi-dimensionais, niio se possa evocar o principio da conserva9iio de massa, o que implica que os procedimentos aqui seguidos devam ser encarados validos quanto ao aspecto qualitativo do comportamento da velocidade, e niio quanto ao valor numerico desta ultima. Esta ressalva e pertinente porque o problema fisico real precisa da segunda dimensao (sem a qual niio se pode descrever a forma~tiio de vortices).

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As simplifica<;:oes apresentadas conduzem as seguintes equa<;:6es:

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Equa<;:oes de Navier-Stokes:

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-+U-=0

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Equa<;:oes de Reynolds:

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au ot

au ox

(3)

u _au

rJ -+U-=0

ot

(4)

ox

259

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( 0 que se observa e que a caracteristica nao-linear das equa96es originais foi mantida. No casu das equa96es de Reynolds, a hip6tese de homogeneidade da turbulencia na superficie elimina o problema de ser necessaria considerar os produtos de flutuay5es de velocidade. Evidentemente as equa96es 3 e 4 possuem soluy5es explicitas, que podem ser obtidas, por exemplo, por separat;:ao de variaveis. Esta solw;ao e mostrada aqui na equa9ao 5, mas a mesma nao e utilizada neste texto no estudo dos aspectos referentes a nao-linearidade das equa96es 3 e 4. Alem disso, a solut;:ao carrega consigo a questao referente ao principia da conservat;:iio de massa anteriormente mencionado.

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(

u

C -ax -1.--

011

u = c2 -- at

(5)

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C1, C1 e a sao constantes. Como o escoamento em aguas rasas com a format;:ao de esteira de vortices ocorre tanto em pequenos numeros de Reynolds (o caso chissico da instabilidade em torno de urn cilindro, por exemplo) como em gran des numeros de Reynolds (is to e, em escoamentos seguramente turbulentos, como aqueles descritos por Dracos eta!., 1992), optou-se por utilizar aqui a situa91io de grandes numeros de Reynolds para o desenvolvimento do trabalho. Assim, a notat;:ao da velocidade envolve a media temporal associada com a equat;:iio 4. Esta escolha visa conduzir a discussiio no sentido de discutir as instabilidades que surgem tanto em escoametnos laminares como em escoamentos turbulentos com caracteristicas medias bern definidas. Na figura I e mostrada a situat;:iio de trabalho. Apresenta-se, nesta figura, urn escoamento preferencial na diret;:ao x no qual uma fonte de perturbayiio (cilindro) foi colocado. Os pontos que correspondem a malha de discretizat;:iio espacial (diret;:ao x) sao tambem indicados.

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Discretizacao das Equacoes Simplificadas

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' Para efetuar a analise numerica nos moldes dos modelos simples de caos, utilizou-se aqui uma discretizat;:iio por diferent;:as finitas progressiva no tempo e regressiva no espat;:o, ou seja:

(

au - u (i,j + 1)- u (i,j) ot

(6)

t:.t

( (

au - '{] (i,j)- '{] (i -1,)) ox

(

(7)

t:.x

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Neste caso, i representa a posi9iio espacial na malha de discretizat;:iio e j representa a posit;:ao temporal nesta malha. A uniiio da!r equat;:oes 4, 6 e 7 produz:

( (

[ 6.t 6.t ] U(i,j+1)=U(i,)) 1+ t:.xU(i-1,))- t:.xU(i,j)

(8)

(

( Pela figura I ve-se que a velocidade na posit;:ao i-1 sempre sera a velocidade do escoamento imperturbado, isto e, Uo. Assim, a equat;:iio 8 passa a ser representada por:

(

( U(i,J +I)= U(i,J)[1 +a U 0

-

(9)

a U(i,J))

t:.t

(

(

a= t:.x

( 260

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'

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Dire<;ao do escoamento

(

Obstaculo

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Figura !a: Situa~ilo considerada para ilustrar o presente trabalho. 0 escoantento de aguas rasas em regime turbulento, gerando a esteira de v6rtices para as grandezas medias.

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lm-,W//$7/////h'§'fi'#'#'$/W
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Direyao do escoamento

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-----~

Obstaculo

-7

Uo

----·------~------·--(i,j)

(l-1,j)

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(

r~/#$$##$$,?#7,0%

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Figura 1 b: N6s da malha de discretiza~i!o na dire~ao x. 0 indice i representa a posi~ilo no espa~o e o indice j representa a dimensl!o temporal

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(

261

I , ( (

(

Uma simplificac;:ao substancial da equac;:ao 9 e obtida aplicando a transformac;:ao:

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y,=-a l+aUo U(i,j)

( (10)

\

A

=~a Uo

(

(

4

(

Resulta, entao:

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Y,.,=4Ay,(1-y,)

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t representa a discretizac;:iio do tempo para a nova variavel y. Assim como a equac;:iio 5, esta ·.equar;:ao conduz a valores absolutos que nii.o devem ser aplicados diretamente como solur;:ii.o do escoamento. Convem lembrar, mais uma vez, que interessa verificar o comportamento esperado (qualitativo) decorrente da nii.o-linearidade das equar;:oes governantes. 0 fato notavel e que a equar;:ii.o II e exatamente a equar;:ii.o discretizada de crescimento populacional que foi utilizada nos trabalhos de May e Feigenbaum (ver Feigenbaum, 1978) para o estudo das propriedades inerentes a sistemas nii.o-lineares simples que apresentam caracteristicas comportamentais complexas. Entretanto, esta equar;:iio deriva aqui de simpliflcac;:oes e discretizar;:oes feitas sobre as equac;:oes de Navier-Stokes e de Reynolds. A equac;:iio II possui urn parametro de "controle", representado pelo coeficiente A ao qual podemos arbitrar valores e estudar o comportamento temporal do sistema. Os trabalhos classicos de Feigenbaum (1978) foram reproduzidos em uma grande quantidade de artigos e livros (ver Briggs & Peat, 1989, por exemplo}, adicionando informac;:oes relevantes e dando base para a teoria do caos. Esses resultados podem ser aqui utilizados, no ambito do problema flsico em estudo. As transformar;:oes que foram efetuadas permitiram ainda apresentar a equar;:iio II em uma forma normalizada, de modo que y varia entre 0,0 e 1,0 se o coeficiente A. tambem variar entre 0,0 e 1,0. Na presente analise os valores dos intervalos de tempo e espar;:o sllo mantidos constantes, de forma que o valor de A. passa a ser dependente apenas do valor da velocidade do escoamento. Ve-se, pela equar;:iio I 0, que quanta maior a velocidade, maior A.

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Caracteristicas das Respostas de acordo com o valor de A. 0 experimento numerico efetuado para entender o "caminho para o caos" e notavelmente simples. Os ca!culos sao efetuados de forma a se obter a evoluc;:ao temporal e o assim denominado "comportamento eventual" da velocidade no n6 a jusante do cilindro (fonte de perturbac;:iio) indicado na figura I. Cada calculo deste comportamento eventual e efetuado para urn valor fixo de A. Em outras palavras, impomos valores para a velocidade de montante e veriflcamos, experimentalmente, o comportamento da velocidade de jusante. 0 valor de montante e refletido no valor constante de A para cada experimento. As seguintes caracteristicas podem ser observadas: I - Valores da velocidade de montante que mantem 0 ~A.< I I 4 conduzem a urn valor final de y=O (comportamento eventual). Evidentemente niio se espera urn valor final da velocidade igual a zero, mas sabemos que, para velocidades baixas, a velocidade media de entrada e a velocidade media no ponto (iJ) da figura I sao iguais, isto e, sua diferenr;:a e nula. A figura 2a mostra urn esquema desta situar;:iio.

262

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i

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2 - Valores de velocidade de montante que mantem l I 4
- Estagio de duplica.,:ii.o de periodos Neste estagio, o comportamento eventual que era, para baixas velocidades, constante, passa a apresentar oscilat;:oes periodicas, similares aquelas observadas para a componente longitudinal da velocidade na formayao da esteira de vortices a jusante de urn cilindro (figura 2c). As oscilat;:oes apresentam-se, inicialmente, com estrutura interna simples, lembrando uma repeti9iio senoidal (valores altos e baixos repetidos alternadamente). Neste caso, tem-se urn fenorneno que se repete apos 2 intervalos de tempo (periodo 2). Posteriormente, aumentando cada vez mais a velocidade, a estrutura interna das oscilay5es torna-se cada vez mais complexa. Assim, passa-se par urn periodo 4 (isto e, os valores se repetem apos quatro intervalos de tempo), por urn periodo 16, seguido de um periodo 32, e assim par diante. Tem-se, entao, o que se denominou de estagio de duplicayiio de periodo, que segue a relayiio 2", onde n indica em qual duplicat;:iio nos encontramos. Esta forma de evolut;:ao, na qual a estrutura interna do escoamento mostra-se cada vez mais complexa em aumentando a velocidade (ou o numero de Reynolds, se quizermos utilizar a terminologia usual para este tipo de observat;:iio experimental) e amplamente conhecida em meciinica dos fluidos.

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(

- Estagio de caos

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Neste estagio, o comportamento eventual da velocidade de jusante, que era periodico, passa a apresentar urn "periodo infinite", ou, o que e o mesmo, os seus valqres passam a niio se repetir mais. Tem-se urn comportamento dito caotico. Em meciinica dos fluidos, apos as passagens de escoamentos laminares estaveis para escoamentos periodicos, tem-se, com o aumento do numero de Reynolds, (da velocidade, no presente exemplo) a transit;:iio para a turbulencia (figura 2d). No presente equacionamento esta transit;:iio e-nos apresentada como caos (muito embora nao se pretenda aqui discorrer sobre as semelhant;:as ou discrepancias entre caos e turbulencia). 0 valor limite de A. para o estagio de duplicat;:ao de periodos e, conforme indicado no item anterior, algo em torno de 0,892498. Para o caso dos escoamentos em aguas rasas que aqui utilizamos como exemplo, sabemos que os mesmos ja se encontram na situayao de escoamento turbulento. 0 que e interessante observar e que, em termos de propriedades medias desses escoamentos turbulentos, novamente ha a passagem por urn processo de instabilizat;:iio e gerat;:ao de movimentos periodicos, semelhante instabilidade que ocorre nos movimentos laminares. Este tato esta bern exemplificado nos trabalhos experimentais de Jirka (1992), Dracos et al. (1992) e Chene Jirka (1995), por exemplo. A questao natural que entao surge por que o processo de instabilizayao das propriedades medias em escoamentos turbulentos segue os padroes existentes no processo de instabilizayiio de escoamentos laminares? A resposta esta contida nas equat;:oes governantes para ambos os fenomenos. No caso dos escoamentos laminares, utilizamos as equat;:oes de Navier-Stokes, sem

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263

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a

e:

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(

:. .

--,~

,;l= 7 u

(

( Th

--

( 00

~-"!-·-

"

"

-

..

0.0~. ~10

4.0"";

"

"

(

""''""

(

/

:1AAMAM~A,

(

(

( 2c

2d

. c=

(

10

"

"

..

Figura 2: Evolu9tles temporais dey para diferentes valores de A.. Todas as evolw;:tles foram obtidas com urn valor inicial de 0,5 paray. (a) Evolu~o dey para 2=0,2. (b) Evolu9i!o dey para 2=0,6. (c) Evolu9~0 dey para 2=0,8. (d) Evolu91io dey para 2>0,892498.

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( 0,75

1,0

Figura 3 · Bifurca9tles observadas para os val ores dey em fun9i!o do parfunetro de controle A..

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264

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restric;oes No caso dos escoamentos turbulentos utilizamos as equa<;:oes de Reynolds para as grandezas medias, as quais mantem as mesmas caracteristicas niio-lineares do equacionamento original (Navier-Stokes). As equa<;:oes 3 e 4 mostram isto e permitem concluir que os comportamentos observados em baixos numeros de Reynolds, dependentes da niio-linearidade das equa<;:6es (instabilidades), tambem podem ocorrer em altos numeros de Reynolds para as grandezas medias.

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Resultado Quantitativa Universal

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A determina<;:iio do encadeamento de duplica<;:iio de periodo no caminho ao caos (ou ao estado de turbulencia, no presente trabalho) e uma caracteristica do tipo de nao-linearidade encontrado nas equac;oes governantes da medinica dos fluidos. Quando o escoamento observado passa de urn estagio com urn periodo T para urn estagio com periodo 2T. ocorre o que e denominado de bifurcac;ao, que pode ser vista talvez com mais propriedade no grafico do comportamento eventual em func;ao de X A figura 3 apresenta urn esquema das principais caracteristicas deste grafico, onde os valores de possiveis dey para cada A. duplicam a partir de val ores bern definidos de A.. Uma forma de visualiza<;:ao experimental da duplicac;ao de periodos e efetuar uma analise espectral sabre os dados coletados, localizando as freqi.iencias dominantes. Assim, para valores de A. no intervale 3 I 4
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limn-~"'

A n+l- A. n :::4 6692 ' -A. n+l - '

(12)

/L n+2

0 numero de Feigenbaum e irracional. Assim, a validade do valor apresentado restringe-se as casa decimais utilizadas. A pergunta natural que segue a esta proposta numerica e: esta constante pode ser observada em problemas de mecanica dos fluidos? Como ja foi dito, esta e uma expressi!o que tern caracteristicas universais. Assim, resultados experimentais obtidos em mecanica dos fluidos 265

t ( ( ( (

(

I

(

i

~

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(

{ 4a fo

fo/2

f

fo

4b

(

f

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i

~

(

l

(

( ( ( 4c

4d

....___. fo/4

fo/2

fo

f

fo/4

fo/2

Figura 4: A duplica<;ilo de perfodo vista a partir de uma analise espectral. Cada duplica<;Ao de perlodo implica em uma freqilencia adicional com metade do valor da freqilencia anterior. A Ultima figura mostra a situa<;ilo de tuibul'encia, onde o espectro apresenta-se sem picos preponderantes.

fo

f

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\ ( conduzem ao numero de Feigenbaum quando analisados com respeito a duplicac,:ao de p<:modos A analise seguida pode ser considerada elegante, porque permitiu, atraves de urn exemplo numerico simples, no qual simplificac,:oes fortes como unidimensionalidade e desconsiderac,:ao dos processos difusivos foram feitas, obter urn resultado de caracteristicas universais

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Conclusoes

(

Mostrou-se, 'neste trabalho, uma forma de obter a equac,:ao de recorrencia normalmente utilizada no estudo do caos, a partir de simplificac,:5es e de discretizac,:oes convenientemente conduzidas nas equac,:oes usuais da meciinica dos fluidos. Mostrou-se que a caracteristica relevante a ser analisada, neste tipo de estudo, e o tipo de naolinearidade que a equac,:ao original contem. No presente caso, as equac,:5es analisadas conduzem a urn processo de duplical(iio de periodos e ao caos, que foi associado a instabilizal(iio com geral(iio de vortices e ao estado de movimento turbulento do escoamento utilizado como exemplo. No presente texto foi mantido o aspecto didatico. Quest5es referentes a associac,:ao entre caos e turbulencia niio tbram levantadas. Mostrou-se a simplicidade existente no entendimento da instabilidade de escoamentos utilizando este ponto de vista, o qual minirniza a carga matematica necessaria ao iniciante. Evidentemente frisa-se que os metodos tradicionais devem ser tambem analisados, porem em urn estagio posterior de formal(iio do pesquisador. Mostrou-se resultados classicos da teoria do caos, como o numero de Feigenbaum, que demonstram a universalidade das conclus5es obtidas. Mostrou-se que as equac,:oes de Navier-Stokes e as equal(5es de Reynolds, para as grandezas medias em escoamentos turbulentos, apresentam o mesmo tipo de niio-linearidade, o que faz com que se esperem comportamentos semelhantes para as variaveis nos escoamentos laminares e as variaveis medias nos escoamentos turbulentos. Utilizou-se como exemplo o escoamento turbulento em aguas rasas.

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Agradecimentos 0 au tor agradece aF APESP, pelo apoio obtido atraves do processo 1997/11743-0 para execuc,:iio de pesquisa no exterior, na qual o presente trabalho se insere, e ao Prof Gerhard Jirka, anfitriiio no lnstitut fur Hydromechanik, Universitiit Karlsruhe, Alemanha.

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Referemcias Bibliograficas

(

Briggs, J. e Peat, D, 1990, "Die Entdeckung des Chaos", Carl Hanser Verlag, Munchen, Wien.

( (

Chen, D. e Jirka, G H, 1995, "Experimental Study of Plane Turbulent Wakes in a Shallow Water Layer", Fluid Dynamics Research. Vol.l6, pp. 11-41.

( ( (

( (

Dracos, T., Giger, M. e Jirka, G.H, 1992, "Plane Turbulent Jets in a Bounded Fluid Layer", Journal ofFluid Mechanics, VoL241, pp.587-614. Feigenbaum, MJ. 1978, "Quantitative Universality for a Class of Nonlinear Transformations", Journal of Statistical Physics, Vol.l9, nQ1, pp.25-52. Jirka, GH, 1992, "In Support of Experimental Hydraulics: Three Examples from Environmental Fluid Mechanics", Journal ofHy
( ( (

267

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Effect of Wave Frequency on the Nonlinear Interaction Between Gortler Vortices and Three-Dimensional Tollmien-Schlichting Waves

(

(

Marcio T. MENDON«;A, Laura L. PAULEY 1 & Philip J. Morris

(

Centro Tecnico Aeroespacial Pc. Mal. Eduardo Gomes, 50- CT AIIAE/ ASA-P 12228-904 Sao Jose dos Campos- SP, Brazil e-mail: mendonca@valley-bbs.com. br

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1

1 The Pennsylvania State University

(

Dept of Aerospace Engineering University Park, PA 16802, USA

( ( (

(

Abstract

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(

The nonlinear interaction between Gortler vortices (GV) and three-dimensional Tollmien-Schlichting (TS) waves is studied with a spatial, nonparallel model based on the Parabolized Stability Equa.-

(

tions (PSE). In this investigation the effect of TS wave frequency on the nonlinear interaction is

(

studied. As verified in previous investigations using the same numerical model, the relative ampli-

(

tudes and growth rates of GV and TS waves is one of the dominant parameters in GV /TS wave

( ( ( ( (

interaction. In this sense, the wave frequency influence is important in defining the streamwise distance traveled by the disturbances in the unstable region of the stability diagram. For 'threedimensional TS waves, in the range of frequencies that result in significant disturbance gro~th, there is little change in the total growth of the disturbances for different wave frequencies, and so

(

the influence of frequency on the nonlinear interaction is small.

(

Keywords: Gortler vortices, Tollmien-Sch.lichting waves, boundary layer stability, instability,

(

transition.

( (

269

(

( M. T. Mendont;a et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

( ( (

Introduction

(

( Due to centrifugal effects, laminar boundary layer flows over concave surfaces may develop

(

, counter rotating longitudinal vortices called Gi:irtler vortices (GV). These vortices develop

(

inflectional velocity profiles that are sensitive to other types of instabilities leading t9 tran-

( '

( sition to turbulence. The transition may be undesirable since it increases skin friction and

(

heat transfer rates and, if inevitable, it must be predicted with accuracy to allow, for exam-

(

pie, the correct design of cooling systems. Among the Aerospace Engineering applications

(

where laminar flow over concave surfaces is important, one can highlight the flow over the

(

pressure side of turbine blades and the flow inside supersonic converging diverging nozzles.

(

( Besides the presence of the vortices, other types of instabilities may also be present and the

(

nonlinear interaction between the GV (Gi:irtler vortices) and these other instabilities may

(

anticipate the transition to turbulence. More specifically, when the curvature of the wall is

(

small the flow may become unstable also to Tollmien-Schlichting (TS) waves which interact

( (

nonlinearly with the vortices.

( Tani and Aihara (1969) presented experimental results for the interaction between GV

(

developing on a concave wall and TS (Tollmien-Schlichting) waves generated by a vibrating ribbon. They concluded that the main effect of the vortices on the TS waves is through the spanwise change in boundary layer thickness. Nayfeh (1981) used the method of multiple scales to study the effect of GV on the

(

(

( development of TS waves. He found that the vortices strongly destabilize TS waves having

(

spanwise wavelength twice the wavelength of the vortices. His results were not confirmed

(

by Malik (1986) who used a temporal, parallel model and found an inconsistent length scale

(

in Nayfeh's formulation. Malik (1986) found that TS waves with spanwise wavelength half

(

270

( (

'

(

( (

M. T. Mendow;a et a/.: Effect of Wave Frequency on tlle Nonlinear Interaction ...

(

( ( (

the wavelength of the vortices are destabilized by the nonlinear interaction. Srivastava and Dallmann (1987) used the method of multiple scales to study the same problem, but also allowed for TS wave amplitudes of the same order of magnitude as the

i

( (

(

vortices. Their results showed good agreement with Nayfeh's results despite the fact that Nayfeh's formulation was incorrect. This result raises doubts about their other findings.

(

To correct the problem in his previous paper, Nayfeh reworked his formulation and

(

presented new results in Nayfeh and Al-Maaitah (1988). They solved the stability equations

(

using both Floquet theory and the method of multiple scales. This time, their results agreed

( (

with Malik (1986) in the sense that resonance occurs when the spanwise wavelength of the

(

oblique wave is half that of the wavelength of the vortices. They also presented some

(

parametric studies on the effect of Reynolds number and frequency.

( (

(

Malik and Hussaini (1990) extended Malik's (1986) temporal, parallel formulation to allow TS wave amplitudes of the same order of magnitude as the vortices. They studied

(

the interaction between GV and two-dimensional TS waves and concluded that the growth

(

rate is larger than the growth of the unperturbed wave. Their results agree with Nayfeh

(

and Al-Maaitah's (1988) results in the sense that interactions take place at a relatively large

(

amplitude of the vortices. Although the model could be used for amplitudes of the waves of

( ( (

the same order of magnitude as the amplitude of the vortices, they only presented results for small amplitude waves.

(

Malik and Godil (1990) presented another paper using the same formulation used by

( (

Malik (1986). They showed that the nonlinear interaction between GV and two-dimensional TS waves leads to the development of oblique waves with a spanwise wavelength equal to

( (

( ( (

that of the vortices. Again, they limited their study to small amplitude TS waves. Their results indicate that the upper branch TS waves are excited while the lower branch waves 271

'

(

( M. T. Mendonr;a et a/.: Effect of Wave Frequency on the Nonlinear Interaction ...

( ( (

are relatively insensitive to the vortices. All these investigations have used local models or temporal, parallel models. Local

(

( models are not suitable to study the development of GV which are governed by parabolic equations that,· rigorously, can not be simplified to ordinary differential equations, except

< (

at large wavenumbers. In this way, local models have been used to study the development

(

of TS waves in boundary layer flows with embedded streamwise vortices. Temporal models

(

are not the most appropriate to describe the physics of spatially developing vortices, and

r (

nonparallel effects are important both for low spanwise wavenumber vortices and for threedimensional TS waves. Only results forTS waves with small amplitudes have been presented

(

in previous works.

(

Mendonc;a, Morris and Pauley (1997) used a spatial, nonparallel model to verify the conclusions obtained in previous investigations that used local or temporal, parallel models.

(

(

( Their model was based on the Parabolized Stability Equations (PSE) (Bertolotti, 1991).

(

They showed that the conclusions obtained in previous investigations are valid, but the

(

assumption of parallel mean flow does influence the results. They also presented results for

(

TS wave amplitudes of the same order of magnitude of the vortices which result in significant nonlinear interaction. In this case the bre~down to turbulence may be anticipated. Their

( ( (

results show the importance of growth rates and initial amplitudes as controlling parameters in GV /TS wave interaction.

( (

In a second paper Mendonc;a, Morris and Pauley (1998a) used the same spatial model based on the PSE equations to investigate the effect of Gortler number and spanwise

(

(

( wavenumber on the nonlinear interaction between GV and two-dimensional TS waves. They

( showed that it is not possible to isolate the effects of initial amplitude, growth rate, Gortler

(

number and wavenumber. This controlling parameters are interrelated and the nonlinear

(

272

(

(

)

(

( (

M. T.

Mendon~a

et aJ.: Effect of Wave Frequency on tl1e Nonlinear Interaction ...

(

(

interaction is strongly dependent on the relative amplitudes of the vortices and TS waves.

(

In this sense two types of interactions have been identified. If the TS wave amplitude is of

(

the same order of magnitude of the vortices, the development of the mean How distortion

( (

and of the vortices higher harmonics are strongly destabilized. If the vortices are stronger

(

than the TS wave, the vortices damp the development of the TS wave. These two different

(

types of nonlinear interaction has been called "Type I" and "Type II" interactions.

( (

( (

The effect of wave frequency on the interaction between GV and two-dimensional waves has been studied by Mendonr;a, Morris and Pauley (1999). They concluded that the longer the path of the TS wave under the unstable region of the TS wave stability diagram, the

(

stronger the disturbance and the higher the nonlinear interaction with the vortices. As ob-

(

served in previous studies, when the TS wave amplitudes are of the same order of magnitude

(

as the vortices, very strong nonlinear interaction takes place resulting in earlier breakdown

(

(

to turbulence or strong destabilization of the vortices.

(

The present investigation expands the results presented by Mendon<;a, Morris and

(

Pauley (1999). It uses the same spatial, nonparallel model to study the influence of fre-:

(

quency on the nonlinear interaction between GV and three.dimensional TS waves. Again,

(

the model allows TS wave amplitudes of the same order of magnitude as the vortices so that

(

( (

the influence of the TS waves on the development of the vortices can be accounted for.

(

(

Formulation

( (

The coordinate system used in the present work is the same coordinate system presented

(

by Floryan (1980). It is based on the streamlines (1/1") and potential lines (.P") of the

(

inviscid How over a constant radius of curvature waiL

(

(

273

This coordinate system has the

(

( ( M. T. Mendonca et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

( (

advantage of producing a decay of the curvature away from the wall; at the wall it is surface oriented, but away from the wall it approaches a Cartesian system. In the normal direction a transformation is applied in order to cluster grid points close to the wall.

(

'

'

{

The Navier-Stokes equations for an incompressible flow of a Newtonian fluid are simplified by assuming that the dependent variables are decomposed into a mean coml?onent and a fluctuating component as follows: (

...

...

u• = u•

.... ,

+ u•'

and

p• = p• +p•'.

(1)

( (

where u• = [u•,v•,w•f is the velocity vector and p• is the pressure. The superscript •

(

indicates dimensional variables.

(

The equations are nondimensionalized using scaling parameters, where

u;.,

o0 =

o0 and

U;.,

as the length and velocity

(v•¢ 0/U;.,)' 12 is the boundary layer thickness parameter,

is the free stream velocity, ¢0 is a reference length taken as the streamwise location

(

( (

( (

where initial conditions are applied, and v• is the kinematic viscosity. Floryan (1980) derived the equations for the zeroth order and first order approximations for the mean flow and for the perturbation quantities. He concluded that for the zeroth order

( (

( approximation the mean flow equations reduce to the Prandtl boundary layer equations for

(

the flow over a flat plate. The only remaining curvature term for the perturbation equations

(

zeroth order approximation is the term in the momentum equation in the normal direction

(

given by:

(

( Go2 (2Uu' + u'2) ' 2 Re

where

Go=

Re(k·o~)(l/2),

Re =

u;,A) v•

(2)

Go is the Gortler number, k• is the curvature of the wall, and Re is the Reynolds number. 274

< (

(

( (

(

( (

M. T. Mendonc;a et al.: Effect of Wave Frequency on tl!e Nonlinear Interactiou

(

(

The resulting momentum and continuity equations are written in vector form:

( (· 1

(

8~ 8~ Aat +Bt 84>

1

8~

1

8~

1

+C, 8tjJ +Dtaz +E

(82 ~ 8¢2

2

1

18 ~

1

1

82 ~ )

+ g2 8tjJ2 + 8z2

+F,~~ = G,

(3)

( ( ( (

where~~=

[u 1,v1 ,w1,p1jT, and the expressions for the coefficient matrices can be found in

Mendon11a (1997). The boundary conditions are given by:

(

( (

u1 = v 1 = w 1 = 0

tjJ = 0,

at

(4)

( ( (

(

( (

8u1 8v 1 8w 1 1 8tjJ' 8tjJ' 8tjJ ' P --+ 0

as

tjJ --+ oo.

(5)

The boundary condition for pressure at the wall is given by t.he momentum equation in the normal direction applied at tjJ = 0.

(

{

Parabolized Stability Equations

( (

The governing equations for the perturbation variables are simplified, leading to the Parab-

( (

olized Stability Equations (PSE) developed by Herbert and Bertolotti (Bertolotti, 1991).

(

nonparallel, nonlinear effects to be accounted for without the heavy demands of a direct

(

The resulting set of equations describes the spatial evolution of disturbances, and allows

numerical simulation. The simplifications leading to the PSE are presented below.

(

(

The set of equations represented by Eq. (3) are elliptic and the perturbations propar-

(

gate in the flow field as wave structures. The governing equations can be simplified if the

(

wavelike nature of the perturbations are represented by their frequency w, wavenumbers a

( (

275

(

( M. T. Mendont;a et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

(

( and {3, and growth rate -y. The perturbation ' is assumed to be composed of a slowly varying shape function n,m and an exponential oscillatory wave term Xn,m· It is represented

(

( (

,,

mathematically as a Fourier expansion truncated to a finite number of modes:

N

<~>'=

M

:E L

(6)

n,m(IP, t/>) Xn,m(IP, Z, t),

n=-N m=-M

where n,m (¢>, t/>) = [un,mo Vn,m• Wn,m, Pn,m]T is the complex shape function vector, and

(

( Xn,m(IP, z, t) = exp

[£:

an,m(e)de

+ im{Jz- inwt]

,

(7)

( (

( an,m(¢>) = 'Yn,m(IP)

+ ina(IP).

(8)

( (

This procedure is similar to a normal mode analysis but, in this case, the shape function

n,m is a function of both 1P and t/>.

( (

The streamwise growth rate 'Yn,m• the streamwise wavenumber a, and the spanwise

(

wavenumber {J were nondimensionalized using the boundary layer thickness parameter 80.

(

The frequency w was nondimensionalized using the free stream velocity

u;,.,

( and the bound-

( ary layer thickness parameter 80 .

(

For linear problems only the fundamental mode is significant. With the growth of the

(

amplitude of the fundamental, higher harmonics become significant as well as the mean flow

(

distortion (MFD) n

= 0,

m

= 0.

As the nonlinearities become stronger, higher harmonics

are considered by increasing the number of modes N, M in the truncated Fourier expansion. The form of an,m (Eq. 8) reflects the fact that the phase speed of higher harmonics should

( (

(

(

be the same as the phase speed of the fundamental mode to avoid dispersion of the wave

(

structure.

( 276

(

1

'

4 ( (

M. T. Mendom;a et al.: Effect of Wave Frequency on tl1e Nonlinear Interactiuu ...

(

( (

The perturbation variable <1"1', as defined in Eq. (6), is substituted in Eq. (3). The equation is then simplified by assuming that the shape function, wavelength, and growth

(

( ( ( ( (

rate vary slowly in the strearnwise direction. In this way, second order derivatives and

i

products of first order derivatives can be neglected in the strearnwise direction. (3) and performing a harmonic balance df

After substituting these terms into Eq.

the frequency, a set of coupled nonlinear equations is obtained. For each mode (n, m) the equation is given in vector form by:

(

( (

( ( ( (

_ An,m <1"1n,m

a .._

-

awn m

"'n,m + C • + + -Bn,m ---a;pn,m ~

-D n,m

a2
(9)

where the coefficient matrices can be found in Mendom;a {1997) The resulting equations are parabolic in


and the solution can be marched downstream

(

given initial conditions at a starting position ¢ 0 . This is true as long as the instabilities are

(

convected instabilities such that they propagate in the direction of the mean How and do

( ( ( (

not affect the How field upstream. The pressure gradient in the streamwise momentum equation also makes the system of equations nonparabolic. For incompressible How Malik and Li {1993) suggest that suffi-

(

ciently large steps in the strearnwise direction will avoid the elliptic behavior of the problem.

(

They also show that dropping the pressure gradient term altogether does not change the

(

results for the level of approximation given by the PSE. In the present model the pressure

( (

gradient term is not included.

(

The boundary conditions for Eq. {9) are derived from Eq. {5). At the wall, homoge-

(

neous Dirichlet no-slip conditions are used. In the far field, Neumann boundary conditions

(

are used for the velocity components and a homogeneous Dirichlet condition is used for

( (

277

(

1

( ( M. T. Mendonca et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

(

<

pressure.

(

For the parabolic formulation, it is necessary to specify initial conditions at a starting

(

position tf>o downstream of the stagnation point at the leading edge of the curved plate. For

(

,TS waves the initial conditions are obtained from the solution of the eigenvalue problem

(

posed by the Orr-Sommerfeld equation. For GV the initial conditions are also give!! by a

(

I

local normal mode analysis. (

(

Normalization Condition

(

The splitting of ~'(t/>, t/J, z, t) into two functions,

~n,m(t/>,

t/J) and Xn,m(t/>, t/J, z, t), is ambiguous,

( (

since both are functions of the streamwise coordinate tf>. It is necessary to define how

l

t/J), and how much will

(

be represented by the exponential function Xn,m(tf>, t/J, z, t). This definition has to guarantee

(

that rapid changes in the streamwise direction are avoided so that the hypothesis of slowly

(

much variation will be represented by the shape function

~n,m{t/>,

changing variables is not violated. To do this, it is necessary to transfer fast variations of

( (

~n,m(t/>,

t/J) in the streamwise direction to the streamwise complex wavenumber an,m(tl>) =

/n,m(tP)

+ ino:(tf>).

If this variation is represented by bn,m 1 for each step in the streamwise

( (

direction it is necessary to iterate on an,m(tf>) until bn,m is smaller than a given threshold. (

At each iteration k, an,m(tf>) is updated according to:

( (

(an,mh+L

= (an,m)k + (bn,mh

(10)

( (

The variation bn,m of the shape function can be monitored in different ways. In the present implementation the following is used: 278

(

( ( ~

I

(

( (

M. T. Mendonqa et al.: Effect of Wave Frequency on tile Nonlinear InteractioH ...

(

(

( (

( ( (

-

1

{

00

bn,m- fooo llun,mll2d,P Jo

(-tun,m .8iin,m) 8

(ll)

where uJ,m is the complex conjugate of Un,m· The integral of llun,m\1 2 was used to assure i that the variation is independent of the magnitude of Un,m·

( (

Numerical Method

(

(

The system of parabolic nonlinear coupled equations given by Eq. (9) is solved numerically

(

using finite differences. The partial differential equation is discretized implicitly using a

(

second order backward differencing in the streamwise direction, and fourth order central

(

differencing in the normal direction. The resulting coupled. algebraic equations form a block

( (

pentadiagonal system which is solved by LU decomposition.

(

To start the computation a first order backward differencing is used. The first order

(

approximation is also used in a few subsequent steps downstream in order to damp transients

(

more efficiently. For the points neighboring the boundaries, second order central differenciJlg

(

in the normal direction is applied.

( ( ( (

( ( ( ( ( ( (

(

The nonlinear terms are evaluated iteratively at each step in the strearnwise direction. The iterative process is used both to enforce the normalization condition and to enforce the convergence of the nonlinear terms. A Gauss-Siedel iteration with successive overrelaxation is implemented. The nonlinear products are evaluated in the time domain. To do this, the dependent variables in the frequency domain are converted to the time domain by an inverse Fast Fourier Transform subroutine. The nonlinear products are evaluated and the results are transformed back to the frequency domain. The complex wavenumber is updated at each iteration according to Eq. (10), and the 279

t ( M. T. Mendom;a et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

( ( (

variation in the shape function is monitored through Eq. (11). The iteration is considered converged when the normalization condition is no larger than a given small threshold. In the present implementation this threshold is

' (

w-s. (

(

Code Validation

( {

A comparison between PSE results and the experimental results from Kachanov and Levchenko ( (1984) for subharmonic breakdown is presented. The subharmonic breakdown is charac-

(

terized by the nonlinear interaction between a finite amplitude two-dimensional TS wave

(

and small amplitude three-dimensional waves with half the frequency of the 20 TS wave. (

The starting conditions are: Re = ber (3 1, 1 = €t,l

400, frequency w2 ,0 = 0.0496, spanwise wavenum-

0.1333, frequency w 1, 1 =

0.0248, initial amplitudes

€ 2 ,0

=

0.439%, and

( (

= 0.0039%.

(

Figure 1 presents a comparison between the PSE results and the experimental results

(

from Kachanov and Levchenko {1984) for the amplitude of different harmonics. It shows

(

( that the PSE is able to reproduce the development of all harmonics with good accuracy. According to Joslin, Street and Chang (1993), the small differences between experimental

'

and computational results observed for higher harmonics can be attributed to small differ-

(

ences between the experimental conditions reported by Kachanov and Levchenko and the

(

(

actual experimental conditions. Those differences were due to a small streamwise pressure

( gradient and a larger frequency.

(

Good comparisons were also obtained with numerical results from Bertolotti (1991) for

(

K-type breakdown and with numerical and experimental results from Malik and Li {1993)

(

and Swearingen and Blackwelder (1987) respectively for nonlinear GV development.

(

280

(

f

(

(

(

M. T.

Mendon~a

et al.: Effect of Wave Frequency on tile Nonlinear luteractiou ...

(

( (

:}

10' 1

(

10'2

(

10·3

(

1o·•

(

1o·•

(

r--....--......-.......--~-

·-···

10·• ·--

·--

---

/

550

(5.1)

600

650 Re

( (

Figure 1: Subharmonic breakdown. Comparison between the PSE results and experimental

( (

results from Kachanov and Levchenko (1984).

( ( ( (

Results

( (

( ( (

(

In this section the effect of TS wave frequency on the nonlinear interaction between G V and three-dimensional TS waves is studied. The following computational parameters are used in the calculations: the number of grid points used in the normal direction is 250 with 200 grid points clustered inside the boundary layer region, the step size dx is 10, the number

(

of Fourier modes N in the strea.rnwise direction is 6 and in the spanwise direction M is 5

(

(given the symmetry conditions, a total of 143 modes are considered, but only 42 modes

(

are stored). For a typical case, 180 steps in the strea.rnwise direction takes 133.1 minutes of

(

( ( ( (

( (

CPU time, with 8.5 seconds per iteration on an IBM RS6000 workstation Model 560. The following test cases consider vortices specified by Go

=

5, b

=

{J/re 103

=

0.1,

with an initial amplitude €av = 1%, interacting with TS waves of different frequencies. The TS waves initial amplitude is €rs = 0.5%. Four different frequencies are considered: 281

' (

M. T. Mendonca et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

(

~

F = wfre 106 = 30, 50, 75 and 100. The starting streamwise positions are defined by the lower branch of the neutral curve and are given, with respect to the frequencies above, by

(

(

Re = 700, 575, 465, and 400. The vortices and the TS waves are followed to a streamwise {

position past tlie upper branch of the neutral curve. Both fundamental resonance and

(

subharmonic resonance are investigated.

(

Before investigating the effect of wave frequency on GV /TS waves interaction it is helpful to look at the general conclusions about GV /30-TS wave interaction. A study on the

( ( (

nonlinear interaction between GV and three-dimensional TS waves was presented by Mendon<;a, Morris and Pauley (1998b). For fundamental resonance the nonlinear interaction

(

results in the following: the nonlinear interaction has little effect on the development of the

(

fundamental modes; a Fourier spectrum broadening is observed, resulting in the develop-

(

ment of Fourier modes that would not grow without interaction (e.g. modes (1,0), (1,2),

( (

(2,0), (2,2), etc.); the development of mode (0,2) is governed by the development of the

(

TS wave; the development of the MFO (mean flow distortion, mode (0,0)) is governed by

(

the development of either the vortices or the TS waves, depending on which one results in

(

the stronger MFO. These nonlinear effects can be observed in figures 2 through 9 for the

(

four levels of wave frequency investigated i~ the present study. For subhannonic resonance,

( (

with the spanwise wavenumber of the vortices two times the spanwise wavenumber of the

(

TS waves, the nonlinear interaction results in a strong effect on the development of the GV.

(

This is due to the development of longitudinal vortices associated with the development of

(

the TS waves. Again, the development of the MFO is governed by either the vortices or the TS waves, depending on which one results in the stronger MFO. These nonlinear effects can be observed in figures 10 through 17. The results for the effect of wave frequency on GV /30-TS waves interaction for fun-

282

(

(

( ( (

( (

\ ( (

M. T. Mendonc;a et al.: Effect of Wave Frequency on t1Je Nonlinear Interaction ...

( (

damental resonance are presented in figures 2 through 9. Figures 2 through 5 show the

(

development of the Fourier modes (0,0), (0,1), and (0,2) due to the nonlinear interaction

( (

(symbols) and due to the development of the vortices and TS waves without interaction

(

(solid lines and dashed lines respectively). It can be observed that the variation of the TS

(

wave frequency does not significantly change the results of the nonlinear interaction. The

(

only noticeable difference is on the development of the MFD which depends on the strength

( (

of the MFD due to the vortices and TS waves: for F = 50 the MFD is governed mostly by

(

(

the development of the MFD due to the TS waves, while for F = 100 the influence of the TS waves on the development of the MFD is delayed until a position farther downstream.

(

Figures 6 through 9 show the development of the Fourier modes ( 1,1), ( 1,0), ( 1,3), and

(

(1,2). The symbols are the results due to the nonlinear interaction and the dashed lines

(

due to the development of the TS waves without interaction. Again, for the four levels

( (

of wave frequency investigated there is no significant change in the nonlinear interaction

(

characteristics. The growth of the TS waves is not strongly affected by the change in the

(

wave frequency. The only exception is for F = 100 where the TS waves reach a lower final

(

amplitude. These weaker TS waves are more strongly affected by the nonlinear interaction

(

( ( ( ( (

through the stronger development of additional harmonics (e.g. the Fourier modes (1,0) and (1,2)) that reach final amplitudes of the same order of magnitude of the fundamental mode (1,1). The results for subharmonic resonance are presented in figures 10 through 17. The development of modes (0,2), (0,0) and (0,4) are presented in figures 10 through 13, and

(

(

the development of modes (1,1), (1,3), (2,0), and (2,2) are presented in figures 14

throug~

(

17. The results show that the nonlinear interaction characteristics are not significantly

(

dependent on the wave frequency. Again, the only exception is for F = 100, where the

( (

283

(

( M. T. Mendonca et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

(

(

(

'::J~

'::J~

..

10' 1

l ~-------:::--·- /

10''

10'' .. -~--

(

(0,1~:-~~

. '_..1---~·

2

10

1 ••

10''

•'\0,0)

(0 •.

10'4 '

10"5

(0,2)

/

10'3

~

2_)-~.:··· . ··~~: •• ,)/:_ (0,0)



. •

/·/~

[

•·

10.1 700(":800

..---

.. ··

(0,2) • / ,/

10~

(

,:

~,.

(

(O,~)'

;'

.··'

/ / (0,0)

~;::

( (

'

10''

.-----···

( 10~~~----~------~----~

900 1000 11 00 1200 1300

sao

Re

750

1ooo

1250 Re

(

( Figure

2:

=

Re

Fundamental

700, F

=

Figure

resonance.

=

30, f3rs

0.07.

O.Q7.

f3av

Fundamental

Re

=

f3av

= 0.0575.

575, F

=

resonance.

=

50, f3rs

0.0575.

1

10 2

'::J

..--..---------.-""'"'~.--,

_______.. (0,1)

_.........._~-~



10"'

~ !

10"2

.~

(

,___,..__,...--.---....-(0,1) J~--~__..........·· ..... •

.....

(

(0,2

(

•. ···''(0,01 ..... /

10''

10''

10 4

10''

10"5

1o·•

10"8 . . _ _ . . _ _ . . __ _.__...._,_.....__...... 400 500 soo 700 8oo 900 1000 Re

800

ooo

Re

(

(

10'

7oo

(

(

g

'::J! 10

3:

,,J:// ///

(0,2) •.•. •·

.········ /.0-t>-/

,

(

(0,0)

(

~-

( ( (

Figure Re

=

4: 465, F

Fundamental resonance.

=

75, f3rs

=

0.0465.

f3av = 0.0465.

Figure Re

=

5: 400, F

Fundamental =

100, f3rs

resonance. =

0.04.

f3av = 0.04.

(

(

( (

Note: solid lines represent results due to the GV without interaction, dashed lines represent

( (

results due to TS waves without interaction, symbols represent results due to GV /TS waves interaction.

( (

284

( (

·-(

( (

M. T. Mendom;a et al.: Effect of Wave Frequency 011 the Nonlinear I11teraction

(

( (

~ E

;::)2

;;:)

10'

1

... . ..... .. ....

10' 2

(

( ( (

(

10''

,~~,...--.---~-

( 10''

,._..__

..

10 2

... ~-· ;·

(1 ,1).

····· .... -•

... ,. ··

(1 ,0)•••.:: ••• (1 ,2)

10'4 I

10' 5

I : ! ' •

.... ·

(1,3\.-

1o·•

(1 '1) ••••• -~.

10

.....

.(13) ..... ·:

10 3

'

'

....

(1 ,0).; ·: ••• (1 ,2)

4

: t • • .:· ••

:·:·

.·-·

10'5

/

106 1000

750

1250

500

Re

750

1000

Re

( (

Figure

(

Re

( ( ( ( ( ( (

( (

=

6:

=

30, f3rs

0.07.

~

=

Re

Fundamental

575, F

resonance.

50, firs

=

0.0575.

=

~

10'~

10' 1

.~·=~ 1

•.•.•.• -

.

(1,0) • ;

t

10''

(1~1!.··:~:~:·':

. . . ..: ·.·• ...

10''

• (1,2)

10' 4

. • I: • ._r ~

10'.

Figure

=

600

8: 465, F

700

800

Fundamental

=

(1 ·~>

.

.

10'' '

•(1.,3)

75, f3rs

900

500

Re

.

600

resonance.

Figure

9:

== 0.0465.

Re

400, F

fJav = 0.0465.

fJav

.. .

............... --

10''

.

.

..

···.~·-·;·

.::::' .. .. . ... : •.••

(1,1) •.•

10'6

Re

7:

;;:)

:::I 10'4

Figure

fJav == 0.0575.

;;:)

( (

resonance.

f3av = 0.07.

500

(

=

700, F

( (

Fundamental

(1,2)

(1,3) 700

800

Fundamental 100, firs

900

Re

resonance. 0.04.

0.04.

(

(

Note: dashed lines represent results due toTS waves without interaction, symbols represent

(

results due to GV /TS waves interaction.

( (

(

285

•( M. T. Mendom;a et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

( (

strength of the TS waves is not enough to disturb the development of the GV as strongly

( (

as in the first three cases (F = 30, 50, 75).

( (

Discussion

(

The results show that for three-dimensional TS waves the wave frequency does not have

(

the same strong influence on the nonlinear interaction as it does for two-dimensional TS

( (

waves, at least for the range of frequencies investigated. For two-dimensional TS waves

( the results from Mendoll(;a Morris and Pauley {1998a) indicate that the most important

(

controlling parameters in GV /TS wave interaction is the relative amplitudes of the vortices

(

and TS waves. In the instability diagram presented in Figure 18 a given TS wave follows

(

a line of constant frequency F as it travels downstream. The higher the frequency F of

(

( the two-dimensional TS waves the weaker the TS wave. The TS wave travels a shorter

(

streamwise distance in the unstable region in the stability diagram and is subject to lower

(

growth rates. The weaker the TS waves, the stronger the dominance of the vortices in the

(

nonlinear interaction. Lower frequencies result in stronger two-dimensional waves which

(

may grow to amplitudes on the same order of magnitude as the vortices, -resulting in a TS

( (

wave dominance over the nonlinear interaction.

(

For three-dimensional TS waves, the stability diagram shows that the unstable region

(

may define a closed region as seen in Figure 18. In these cases low frequencies may result

(

in little or no disturbance growth. In a certain range of TS wave frequencies there is little

(

variation on the strearnwise distance traveled by the disturbances under unstable condition

(

( and TS waves with different frequencies have similar growth along the streamwise direction.

(

Since the growth rate of the TS waves does not vary much with frequency in that range

(

of frequencies, the effect of frequency is lower in that range. When the frequency F is

(

286

( (

-( (

(

M. T. Mendonqa et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

( (

(

i

::::>i

::::>

10" 1

10"'

( (0,2)

(

~ ..... _.....

( .. --··

(00)

1o·•

.---~

10-3

'.

10".

r---...-____,____,..___,..,._

10~

( ··--···(0,0)

(

750

(

1250

1000

eoo

Re

800

100

900

1000

11 oo

Re

( (

Figure

(

Re

(

/Jav

=

10: 700, F

Subharmonic

=

30,

resonance.

/Jrs = 0.07,

0.14.

Figure

11:

=

Re

Subharmonic

575, F

=

50,

resonance.

/Jrs = 0.0575,

0.115.

/Jav

(

( (

i

::::>i 10" 1

::::>

!--.----.----.-__,.........,-

10"2

( 10"3

( (

(0,~_)

10"4 10"5

(

1o·•

(

800

-10,0)

'-'-_._______.._____.____. 500

900 Re

750

1000

he

( ( ( (

Figure

12:

Re

=

465, F

/Jav

=

0.093.

Subharmonic

=

75,

resonance.

/Jrs = 0.0465,

Figure

13:

Re

400, F

/Jav

Subharmonic

=

100,

/Jrs

resonance.

=

0.04,

0.08.

( (

Note: solid lines represent results due to the GV without interaction, dashed lines represent

( ( (

( (

results due toTS waves without interaction, symbols represent results due to GV /TS waves interaction.

287

( ( M. T. Mendon~ et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

(

(

(

-:JI 10' 1

1o·•

,...--.-__,....._...,.-..

':::J~

-·' ... :2:0~-~:::.;·::·:·.~_::~...-

··--·-····---·-·

10''

(1,3).:. ·.'

100 800

(

·._-:•:··· (2,2)

(

.···~··::::-·

~

• 5

(20) _,: ·.•.,.. ,..·· I

,',:·

10'"



900 1ooo 11 oo 1200 1300 Re

.:--r·r.:·!.·_..



10

~.::-···.:: ~....... :~'·

1o·•

(

.·f

(1,3) .'

10'"

_.- .. ,.,._,_. (2,2)

1o·•

. ---· -·. -·~~ ~~! .. -·· -·· -· _.:.:: :·,>j

1o·•

.

.-

.-::.

10'2

.• -~--~ / ;:.

(1,1)

(



..~;

...... --..

1o·•

(

---..-........,__--.----

10'1

500

... •

(

• .. •

/ 600

100

800

ooo

1ooo

Re

(

( Figure

14:

Subharmonic

700, F

Re

=

f3cv

= 0.14.

30,

resonance.

f3-rs

0.07,

':::JI

f3cv

= 0.115.

10'

3

·--·--·--··...

,.·· (1,3)/ ,•'

10" 10'1 10-·

l

.-;:-1

10' 10"

Figure

16:

~

1o·•

600

Subharmonic

(

(

10 900

(

(1.1)4 •.• -•-:-·'i·;~"

4

'···--·•·•·~ ;: ,3)_::<·<.:.~.~::: I •

( (

I

~2 0)··--··. • • • .:• _,- (2,2) •... :~·:.-· '

.. ,

Re

,.

(

• I ,• ,

1:' ..... "~·····

t

(

(

,4~-4i·;-•:. •

.

700

0.0575,

=

(

(

10'4

(2,0~·//':;-.

600

50, f3-rs

575, F =

,....------..---

~!'·

... -:.·

>-'(2 2) . . _,..)~:A·.:. ·" ' :--·-·:.~:·......

L-..!.....:.: 400 500

resonance.

2

··#···-·/

.•..•.• -~-··::-·.--·

(1,1)

Subharmonic

=

10'1

,,-

2

15:

Re

':::Jft

10'1 1-.---~~

10'

Figure

(

-;··. : 500

resonance.

Figure

=

Re

=

fJcv

= 0.08.

17:

750

Subharmonic

1000

Re

resonance.

(

( (

Re

=

465, F

f3cv

=

0.093.

=

75, /Jrs

0.0465,

400, F

=

100, /Jrs

=

0.04,

(

( (

Note: dashed lines represent results due to TS waves without interaction, symbols represent

(

results due to GV/TS waves interaction.

(

( 288

( (

I

' (

(

M. T. Mendom;a et al.: Effect of Wave Frequency on tl1e Nonlinear Interaction

( (

9 0.100 ,.........,_....,.._...,..._...,.._...,__..,...._., F-100

( (

0.075

( ( ( (

2so 5oo 750 1ooo 1250 15oo 1750 Re

(

(

Figure 18: Neutral curves for oblique waves for different spanwise wavenumbers.

( (

close to the boundaries of the closed region defined by the stability diagram, and at certaiD

(

frequencies and spanwise wavenumbers that result in small growth of the disturbances, the

( (

.

effect of frequency on the nonlinear interaction is lower.

(

( (

Conclusions

( ( (

AB observed in previous investigations the relative amplitude of the vortices and TS waves

is one of the most important parameters in the GV /TS wave nonlinear interaction. Three-

( (

dimensional TS waves are characterized by little variation in the total growth of the dis-

(

turbance for different frequencies in a certain range of frequencies. In this way, since the

(

change in frequency in this range does not result in strong change in the total growth of the

(

TS waves, no significant change in the nonlinear interaction was be observed. At frequencies

(

(

that are close to the limits that define the closed unstable region, and for wave frequencies

(

that result in little growth of the fundamental waves, a weaker influence of the TS waves

(

on the nonlinear interaction can be observed.

( (

289

I

c ( M. T. Mendom;a et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

( (

(

References

( Bertolotti, F. P. 1991. Line'!r and Nonlinear Stability of Boundary Layers With Streamwise

Varying Properties. Ph.D. thesis, The Ohio State University, Columbus, Ohio.

( ( (

Floryan, J. M. 1980. Stability of boundary layer flows over curved walls. Ph.D. thesis, (

Virginia Polytechnic Institute and State University. Joslin, R. D., Street, C. L., and Chang, C.-L. 1993. Spatial Direct Numerical Simulation

( (

( of Boundary-Layer Transition Mechanics: validation of PSE Theory. Theoretical and

Computational Fluid Dynamics, 4(6), 271-288. Kachanov, Y. K., and Levchenko, V. Y. 1984. The Resonant Interaction of Disturbances at

{ (

( (

Laminar-Turbulent Transition in a Boundary Layer. J. Fluid Mechanics, 138,209-247. Malik, M. R. 1986. Wave Interaction in Three-Dimensional Boundary Layers. AIAA Paper

I

( (

( 86-1129.

(

Malik, M. R., and Godil, A. A. 1990. Nonlinear Development of Gi.irtler and Crossflow Vortices and Gi.irtler/ Tollmien-Schlichting Wave Interaction. NTIS,.AD A 221 107.

( ( (

Malik, M. R., and Hussaini, M. Y. 1990. Numerical Simulation of Interactions Between Gortler Vortices and Tollmien-Schlichting Waves. J. Fluid Mechanics, 210, 183-199.

( ( (

Malik, M. R., and Li, F. 1993. Transition Studies for Swept Wing flows Using PSE. AIAA

Paper, 99-0077.

( ( (

Mendon~a,

M. T. 1997. Numerical Analysis of the Interaction between Gortler Vortices

(

and Tollmien-Schlichting Waves using a Spatial Nonparallel Model. Ph.D. thesis, The

(

Pennsylvania State University.

( 290

i

( (

I

(

( (

M. T. Mendow;a et al.: Effect of Wave Frequency on tl1e Nonlinear Interaction ...

( (

Mendon'
M. T., Morris, P. J.,

and Pauley,

L. L. 1997.

Gortler Vortices

(

Tollmien-Schlichting waves interaction: reassessment of previous results with a spa-

(

tial/nonparallel model. In: XIV Brazilian Congress of Mechanical Engineering.

( ( (

Mendon'
(

( ( (

of Fluids.

Mendon'
(

Vortices and tw<>-dimensional Tollmien-Schlichting waves: Effect of Gortler number

(

and spanwise wavenumber. Submited to The Physics of Fluids.

( ( (

(

Mendon'
(

( ( ( ( ( (

Nayfeh, A. H. 1981. Effect of Streamwise Vortices on Tollmien-Schlichting Waves. J. Fluid Mechanics, 107, 441-453.

Nayfeh, A. H., and Al-Maaitah, A. 1988. Influence of Streamwise Vortices on TollmienSchlichting Waves. Physics of Fluids, 31(12), 3543-3549.

( ( ( ( ( ( (

(

Srivastava, K. M., and Dallmann, U. 1987. Effect of Streamwise Vortices on TollmienSchlichting Waves in Growing Boundary Layers. Physics of Fluids, 30(4), 1005-1016.

Swearingen, J.D., and Blackwelder, R. F. 1987. The Growth and Breakdown of Streamwise Vortices in the Presence of a Wall. J. Fluid Mechanics, 182, 255-290. 291

(

\

( M. T. Mendonca et al.: Effect of Wave Frequency on the Nonlinear Interaction ...

( (

Tani, 1., and Aihara, Y. 1969. Gortler Vortices and Boundary Layer Transition. ZAMP, 20,

(

( 609-618.

( (

(

( ( ( (

( (

(

)

( (

( ( (

( ( ( ( (

( ( (

( (

( 292

( (

I

~(

(

( (

(

( ( (

( ( ( (

{ (

(

( ( ( (

( (

( ( (

( ( ( ( ( (

( (

MEDI(:OES EM PROTOTIPO DE FLUTUA<;:OES DE PRESSAO NA BACIA DE DISSIPACAO DA USINA DE PORTO COU)MBIA Jayme Pinto Ortiz Universidade de Sao Paulo Escola Politecnica -Departamento de Engenharia Mecinica Sao Paulo - SP Escola de Engenharia Maua - IMT Email: jportiz@usp.br Fatima Moraes de Almeida Furnas Centrais Eletricas S.A. Rio de Janeiro - RJ Email: fatimama@fumas.com.br Erton Carvalho Furnas Centrais Eletricas S.A. Rio de Janeiro- RJ Ricardo Daruiz Borsari Centro Tecnol6gico de Hidrautica e Recursos Hidricos- CTH Universidade de Sao Paulo Escola Politecnica - Departamento de Hidriwlica e Sanitaria Abstract The hydraulic jump stilling basin of the Porto Colombia Hydroelectric Power Plant was severely eroded due the formation of horse-shoe vortex in the chute-blocks region. The problem ocurred for normal operation flow rate, which were only of order of36% (5700 m3/s) ofthe maximum flow rate (16000 m3/s). A Subcommission was stablished inside the Commission of Hydraulic and Fluid Mechanics of the "Associafi:llo Brasileira de Recursos Hidricos - ABR.ff' to study the problem, which involved the participation of the three major hydraulic laboratory of Brazil (LAHEIFURNAS - Rio de Janeiro,' CTH!FCTH- Silo Paulo and CEHPAR- Curitiba) This Subcomission developed a work that was presented in the last biannual symposium of the ABRH, which took place in Vit6ria!ES, in november, 1997. As a recomendation of the Consulting Board, FURNAS decided to remove all the chute blocks and to design a new end-sill for the stilling basin; in the other hand, following the recommendations of the Subcommission, decided too for the instrumentation of the stilling basin. using pressure transducers. In the article here presented, it is showed the instrumentation work developed in the prototype to install seven pressure transducer in one vain of the spillway and the stilling basin, the proceedings for the calibration process and data acquisition, and, the results of the pressure fluctuations measurements. The simultaneous data acquisition results from the seven pressure transducers aligned on the spillway and stilling basin permit a criterious analysis of the behavior of the pressure fluctuations in the hydraulic jump. The prototype results, which are rare in the literature, together with the hydraulic models studies, which are been developed, give a unique possibility for the development not only of applied research to solve hydraulic structures problems, but also, of basic research in turbulence. Introdu~io

Em novembro de 1990 foi criada uma Subcornissio de Pesquisa no ambito da Comissio de Hidraulica e Mecinica dos Fluidos da Associ&fi:llo Brasileira de Recursos Hldricos - ABRH. Esta Subcomissao foi composta pelos Laborat6rios de Hidraulica de Furnas (LAHE)!Rio de Janeiro,

( (

(

293

(

I

( CTH/FCTH/Sio Paulo e CEHP AR/Curitiba, que se comprometeram a so mar esforc;:os ern tomo de urn tema para pesquisa conjunta. Em novembro de 1993, no X Simp6sio Brasileiro de Recursos Hidricos da ABRH, realizado na · cidade de Gramado - RS, o Laborat6rio de FURNAS (LAHE) propos a escolha do tema " Estudo de Flutuac;:io de Pressao ern Bacia de Dissipac;:ao". Tal proposta foi vinculada a oportunidade surgida com o ensecamento da Bacia de Dissipac;:l[o do Verdedouro da UHE de Porto Colombia, de propriedade de FURNAS Centrais Eletricas S.A., programado para o periodo de estiagern do anode · 1995. 0 ensecamento da bacia permitiria o desenvolvimento dos trabalhos preliminares necessaries a sua instrumentac;:ao. No periodo de 17 a 23 de maio de 1996, realizou-se na UHE de Porto Colombia, urna campanha de medic;:io de flutuac;:oes de pressao instantaneas na bacia de dissipac;:!o do vertedouro. Para as vazoes vertidas de 500, 1000, 2000, 3000 e 4000 m3 Is forarn aquisitados dados instantaneos de pressao ern sete pontos da bacia de dissipac;:llo, distribuidos ao Iongo do vllo do vertedouro extremo direito e distantes 7,20 m do muro lateral da bacia (ver figura 1). Os dados forarn aquisitados em conjunto pelas equipes e sistemas de aquisic;:ao das seguintes instituic;:Oes: - Laborat6rio de Hidniulica de Furnas - LAHE/FURNAS; - Centro Tecnol6gico de Hidraulica e Recursos Hldricos/Fundac;:l[o Centro Tecnol6gico de Hidraulica - CTH/FCTH. Urn relato sobre os trabalhos desenvolvidos pela Subcomissio acirna referida foi apresentado no XII Simp6sio da Associac;:Ao Brasileira de Recursos Hidricos - ABRH, ern Vit6ria/ES, em novembro de 1997 (ABRH/FURNAS, 1997). 0 trabalho aqui apresentado mostra alguns resultados de flutuac;:oes de pressio rnedidas no prot6tipo a partir dos referidos sistemas de aquisic;:ao de dados. Os resultados obtidos a partir da aquisic;:io simultinea de sinais de sete transdutores de pressao, alinhados na bacia, perrnitem uma analise criteriosa do comportamento das flutuac;:oes de pressao no ressalto, no dominio do tempo e da frequencia. 1

ldentlftca~lo

1

(

( ( (

( (

'

'

( (

( (

(

( (

( (

I

(

(

Historico

)

'

( do Problema

(

A Usina Hidreletrica de Porto Colombia, que iniciou sua operac;:io no anode 1973, situa-se no Rio Grande, na divisa dos Estados de Minas Gerais e Sao Paulo. Seu vertedouro possuia uma bacia de dissipac;:io convencional tipo II do USBR, cujos parametros de projeto haviam sido alterados em func;:io dos estudos em modelo realizados naquela epoca. Uma inspec;:io subaquatica realizada em outubro de 1983 revelou o pessimo estado de conservac;:ao das estruturas componentes da bacia. Observou-se que, ao Iongo dos, ate entia, 10 anos de operac;:io da Usina, urn par de erosoes laterais e simetricas havia surgido ern cada um dos seus 36 blocos de queda (" chute blocks"). Na laje da bacia, imediatamente a jusante dos " chute blocks" , crateras de erosio que atingiam as dimensoes medias de 2,00 m de comprirnento por 1,65 m de targura e 0, 70 m de profundidade, possuiam sinais evidentes de arrancamento das ferragens do concreto. No restante da laje apenas duas irregularidades superficiais foram encontradas. Ern alguns blocos da soleira terminal ("end sill") tambem foi observada a existencia de pequenas erosoes. Pesquisando-se o hist6rico das vaz5es vertidas na Usina, verificou-se que, ao Iongo de todo o seu ocorrida de 5700 m3/s, registrada em fevereiro de 1983, periodo de operac;:llo, a descarga ' nio havia atingido 36% de sua capacidade plena de vazio (16000 rn3/s). Nesta ocasilio mediu-se sabre a bacia a carga total de 20, 80 m. Desta forma, as eros5es encontradas na bacia ocorreram para condic;:Qes de operac;:!o bern inferiores Aquela que seria sua solicitac;:io maxima (q = 98 m3/s.m., para I= 163,oo m).

maxima

( ( ( ( ( (

( ( (

( ( (

294

'

( (

J

(

( ( (

Diagnostico Preliminar

(

Todos os dados coletados sabre o vertedouro foram submetidos a aprecia~ito de consultores que forneceram a FURNAS seus pareceres tecnicos acerca do assunto. A an8lise destes documentos, juntamente com os registros de casos semelhantes ao de Porto Colombia, possibilitou o diagn6stico preliminar de que os danos ocorridos na bacia tiveram origem em um processo tipico de cavi~o por v6rtice. Este processo estaria instalado apenas na regiito dos "chute blocks" e na laje da bacia logo a jusante dos mesmo~ atribuindo-se as erosoes ocorridas no restante da laje e no "end sill", a urn processo de desgaste devido a circula~ao de material erodido dentro da bacia. Este diagn6stico foi reavaliado ao Iongo do tempo.

( ( ( ( ( (

Pesquisa em Modelo

(

Estudos em modelo fisico realizados no LAHEIFVRNAS, que contaram com a participayio do CTHIFCTH (ABRHIFURNAS, 1997) comprovaram o diagn6stico preliminar. Em que pesem as limital,)oes de analise do fenomeno de formal,)ito de vortices ferradura ("horse-shoe vortex") em modelos em escala Froude, os resultados destes modelos sao extremamente uteis na indica~oes de tendencias cavitantes em estruturas hidraulicas. Os estudos em modelo conduziram a formul~o da hip6tese de que, as erosoes encontradas na inspel,)ilo subaquatica de outubro de 1983, na regiilo do "end sill", tambem eram resultado de um processo de cavital,)ito por v6rtice ferradura. A realiza~ao de uma segunda inspe~ilo sub-aquatica na bacia, em outubro de 1990, confirrnou as suspeitas levantadas no estudo em modelo. Os estudos ern modelo permitiram ainda estudar algumas alternativas de bacia a partir da remo~ito completa dos "chute blocks", otimizando-se portanto a soleira terminal de modo a minimizar o risco de cavita~ito por v6rtice, mesmo que comprometendo o melhor desempenho hidraulico do ressalto hidraulico dentro da bacia. Como ensecamento da bacia de dissipa~ito do vertedouro da UHE de Porto Colombia (ver fotos 1, 2 e 3) foi impJantada a bacia otimizada em modelo, removendo-se os "chute blocks" e construindose uma nova soleira terminal ("end sill").

( ( ( ( ( ( (

( ( (

( ( (

( ( (

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Objetivos do Estudo 0 estudo sistematico pas flutua~es de pressio em bacias de dissipa~ito tern sido conduzido por diversos investigadores, entretanto, a obten~ito destes dados, em sua grande maioria, e efetuada ern modelos hidraulicos reduzidos e dificilmente, confrontados com medi~oes em prot6tipo. Com o ensecamento da bacia de dissip~ito do vertedouro da UHE de Porto Colombia, para realiza1,1ilo das obras de recuper~ito a que a mesma foi submetida, surgiu, para a Subcomisslo "Desenvolvimento de Pesquisa" da ABRH, a oportunidade de instrument~ito dessa bacia visando, nao s6 a uma analise comparativa entre os dados obtidos no prot6tipo e ern modelos, como tambem a determin~o da real eficiencia hidraulica da bacia ap6s as modifica~es introduzidas em sua geometria, a deterrninal,)itO dos esfor~s hidrodinamicos que a mesma estara submetida ern diversas situacoes de oper~ito e o inicio da form~ito de urn banco de dados que, certamente, em muito contribuini para o projeto de novas bacias. Tendo em vista a importancia desses dados que, se nilo ineditos, silo raros dentro da literatura nacional e internacional, no que se refere a medi~ito em prot6tipo, pOde-se contar com o apoio de FURNAS que teve especial interesse na realiza~ilo desse trabalho. 295

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( Portanto o principal objetivo do trabalho aqul apresentado e relatar alguns resultado de flutua~oes de pressio medidas na bacia de dissipa~ilo do prot6tipo da UHE de Porto Colombia a partir dos sistemas de aquisi~ilo de dados mencionados anteriormente. A anAlise dos graticos de espectros de pot6ncia, varia\)io dos sinais no tempo, distribui\)io de probabilidade etc, dilo urna ideia clara do caminhamento dos picos (maiores esca1as de turbuiSncia) • associados as estruturas de coer&lcia do ressalto, que e urn fenomeno hidniulico caracterizado por baixas frequencias e grandes amplitudes de flutua~toes de pressio.

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Instrumenta~Ao

do Prototipo

A escolha dos pontos de medi~tio de pressilo no prot6tipo foi norteada a partir de estudos pr6vios em modelo hidraulico (ABRHIFURNAS, 1997). Decidiu-se pela instalaQio em prot6tipo de sete transdutores alinhados longitudinalmente ao Iongo do eixo da bacia de dissipa\)io relativa ao vilo n°l da extrernidade direita do vertedouro. Os pontos de medi~ de pressilo estio identificados na figura I por: DA, DB, DC, DI, 02, 03, OS e 07. Os pontos DA,DC e DB estilo Jocalizados, respectivamente, nos pontos extremos de tangencia e no centro da curva de concordftncia existente entre o perfil vertente e a laje horizontal da bacia de dissipa\)io. Os demais pontos estilo localizados todos na bacia de dissipa~tlo.

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Projeto dos Embutidos

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Para a instalaQio dos transdutores de pressio no prot6tipo adotaram-se os seguintes criterios de projeto: • Durante a obra de modifica\)io e reparos na bacia, seriam embutidas, na mesma, tubulatrOes e dispositivos de espera para a futura instala~ dos transdutores (ver figura 2); • A installlQio dos transdutores s6 ocorreria por ocasiio da realiza~tio da campanha de meditrio; • Ap6s o termino da campanha os transdutores seriam removidos; • Ao )ado de cada ponto de medi~tio de pressio instantiinea seria instalada uma tomada piezometrica que serviria de testemunho para as mediQoes. Maiores detalhes sobre o projeto e instal~o dos embutidos podem ser encontrados em ABRHIFURNAS (1997) e Carvalho et alii (1997).

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Transdutores

( Para a realizatr~o das medi~toes, foram especificados transdutores cuja faixa de trabalho englobasse o intervalo de valores de pressilo de possivel ocorr&lcia. Como precautr~o quanto ao surgimento de pressaes negativas, os transdutores foram ajustados pelo fabricante, para operarem na faixa de O,S a S bar ( -5,1 a 51 m.c.a.). Esta faixa corresponde a uma varia\)io de 4 a 20 rnA que, convertida, fomece urn sinal de tensilo de 1 a S Volts. Este sinal de tensio foi monitorado e aquisitado pelos sistemas de aquisi~ de dados do LAHE e do CTHIFCTH. Foram utilizados sete transdutores com as seguintes caracteristicas: • Fabricante: Hytronic; • Modelo: H-2S; • tipo de sensor: Piezoresistivo; • Faixa de opera~tilo: -O,S a S bar; • Repetibilidade: ± 0,2S% FE; • Histerese: ± 0,2S% FE; • Linearidade: ± 0,2S% FE; • Salda: 2 fios: 4 a 20 rnA; • Tempo de resposta: 2 rns; • Alimenta~tilo: IS a 30 Vee.

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Externamente os transdutores foram identificados segundo o codigo do fabricante como segue: IK112 ( DA); IK-121 (DB); IK-116 (DC); JG-19 (D l); IK-127 (D3); IK-122 (DS); IK-133 (D7).

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Adapta~oes

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Efetuadas

Os transdutores foram rosqueados em flanges metalicas, juntamente com capsulas metaJicas cilindricas (ver foto 4). Tais flanges tern as mesmas dimensOes dos flanges cegos, usados para tamponar as caixas metalicas embutidas no piso da bacia. Os fios do transdutor (sinal de corrente) foram emendados isoladamente e vedados, e o cabo de extensio foi introduzido em uma mangueira ("poli-flow") de diametro 3/8 poL Nas extremidades foram usadas conexOes anilhadas, para fixa~io das mangueiras e veda~tiio das capsulas e cabos eletricos. Na outra extremidade da mangueira foi providenciado urn bujao com uma presilha, para enla~ar o cabo-de-a~to guia, colocado no interior da mangueira nervurada usada como conduite. Com o auxllio destes cabos, as mangueiras forarn puxadas ate a superficie. Detalhes sabre as adapta~oes efetuadas e a metodologia de instala~ao dos transdutores podem ser vistos em Carvalho et alii {1997). Originalmente os transdutores eram de pressao relativa, mas esta op~iio mostrou-se inadequada, tendo em vista a possibilidade de penetra~iio de Agua no interior das capsulas, a despeito de todas as precau~oes tomadas para que tal niio ocorresse. Caso houvesse entrada de agua nas capsulas e, consequentemente, no interior dos inv6lucros dos transdutores, os componentes eletronicos e os elementos sensores dos transdutores sofreriam danos irrepaniveis, impossibilitando qualquer tipo de medida. Desta forma, em fun~tiio das responsabilidades e riscos envolvidos e, conforme sugestlo do fabricante, os transdutores foram ajustados e lacrados, sendo que no interior dos involucros e pressio atuando na membrana sensora interna, havia uma pressao de 914 mbar, referente atmosferica na data em que os transdutores foram lacrados. Sendo assim, quando sem carga, porem submetido a uma pressao atmosferica diferente de 914 mbar, os mesmos indicavam uma saida em corrente correspondente a uma carga ficticia igual diferen~ entre a pressiio atmosferica e 914 mbar. Portanto, uma vez conhecida a pressao atmosfenca local, ajustava-se a curva do fabricante a nova situa~tiio. Ao Iongo de todos os ensaios foi efetuada a mediciio da pressao atmosferica.

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Especifica~tt'ies

tecnicas relativas a conversores, fontes de alimenta~iio, placas de aquisi~io de dados, micocomputadores, softwares de aquisi~io encontrarn-se em ABRHIFURNAS (1997). Conforme salientado anteriormente, os dados foram aquisitados a partir de dais sistemas de aquisioio independentes, operando simultaneamente. Instrumenta~iio

de Apoio

Como instrumenta~tiio de apoio foram utilizados: • Multimetro digital Fluke de 5 \Ia digitos, modelo 45 (COPPE); • Multimetro digital Minipa 4 \Ia digitos modelo ET2700 (FURNAS); • Oscilosc6pio Tektronic duplo tra~o, SO MHz, modelo 2205 (COPPE); • Analisador de espectro HP, modelo 3582A (FCTH); • Filtro ativo programilvel (FCTH); • Barometro Bruel & Kjaer, faixa de trabalho de 790 a 1041 mbar e sensibilidade de S mbar (FURNAS); • Bomba manual F. Maskina P. Tube RP-60 (p = 120 kglcm2, Q = 7,5 I Vmin ); • Medidores de nivel (''PIO" - fabric~iio FURNAS).

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dos Transdutores, Sistemas de Apoio, Procedimentos de Calibra~ilo

Ap6s a instala~o. montagem e verifica~iio de todo o sistema de apoio (microcomputadores, conversores, fontes, multimetro etc) os transdutores foram conectados aos m6dulos, realizando-se o primeiro ensaio de calibra~iio. Para tanto, foram feitas oito marca~oes metro a metro nas mangueiras "poly-flo" , mar~oes estas que se constituiram em referencia de profundidade a que os transdutores ~riam submetidos mergulhando-os 'll partir da superficie. A cada metro realizou-se uma aquisi~iio de 10 s. tanto na imerslo, quanto na emersiio. Realizado o primeiro ensaio de calibra~iio dos transdutores, partiu-se para a sua instala~iio nos pontos determinados. Este trab~o ficou a cargo da equipe de mergulhadores da CESP, ja que toda a etapa de instala~io propriamente dita dos sete transdutores, foi subaquatica. A opera~iio de instah~~o dos transdutores foi monitorada da superficie, atraves de filmagem subaquatica. Inicialmente f&-se uma avalia~io da quantidade de sedimentos depositados no fundo da bacia e da eventual necessidade de urn vertimento de .limpeza Embora a quantidade de residuos no interior das caixas metalicas fosse pequena, em alguns casos foi necessilrio usar-se uma bomba manual, para desobstruir a passagem da mangueira do transdutor. Concomitantemente foram retirados os plugs dos piez0metros e aplicada a mesma bomba para desobstrui-ios. Os mergulhadores localizaram os pontos de ins~io dos transdutores. verificando que os flanges cegos e parafusos que os fixavam estavam em born estado de conserva~iio. Em seguida tais flanges foram retirados e os transdutores instalados urn a urn. As presilhas dos bujoes foram presas aos cabos-guia das mangueiras nervuradas correspondentes e na oportunidade, o mergulhador que executou a opera~o refor~ou tal enlace com uma amarr~ de cordiio. Todas as mangueiras com os cabos de extensio dos transdutores passaram livres atraves das mangueiras nervuradas. Com as mangueiras jA na superficie, fora.m retirados os bujoes de veda~ e emendados os cabos, para que chegassem ate a cabine de medi~ao. Tais emendas foram identicas as que foram feitas entre os transdutores e os cabos de extensio.

Campanha de

Medi~oes

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( . Calibra~llo dos Transdutores Durante a campanha de medi~io de pressaes, foram realizados sete ensaios de calibra~iio estatica dos transdutores, distribuidos da seguinte forma: • CALO 1 - Ensaio inicial com todos os transdutores, antes de sua inst~io; • CAL02- Ensaio com os transdutores: IK-122, IK-127, IK-133, JF-105, JF-103-, JG-19, IK112, 1K121, IK116; • CAL03 - Ensaio como transdutor JF-105; • CAL04, OS, 06, 07- Ensaio final com todos os transdutores, exceto JF-103, JF-105. Todos os dados foram aquisitados em Volts, tendo sido necessilrio, atraves das curvas de calibra~o dos tansdutores. transforma-los para m.c.a.

Procedimentos de Ensaios As calibra~Oes estaticas foram feitas para niveis de ilgua medios na bacia de 8,3 m. Sendo assim, todos os transdutores foram imersos ate a profundidade de 8 m, de metro em metro, e depois emersos ate a superficie, de igual forma. Adotou-se intervalo de aquisi~o de 10 s e frequencia de amostragem de 100 Hz como val ores padroes para a calib~ estatica. A primeira aquisi~ era feita sempre com o sensor na atmosfera, depois a uma profundidade de S em e, posteriormente, para cada metro irnerso ou emerso. Para os varios ensaios de calibra~io realizados foi verificada a pressio atmosferica local, como se segue: • CALO I - 973 mbar; 298

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• CAL02 - 972 mbar; • CAL04 ate 07 - 973 mbar. Como a disperslio maxima entre os valores de presslio foi de I mbar e a precislio do barometro utilizado era de 5 mbar, adotou-se para todos os ensaios de calibr~io, a presslio atmosfenc~ de 973 mbar. A curva do fabricante foi entao ajustada para a nova pressiio atmosferica de 973 mbar. Maiores detalhes sobre o procedimento de calibra.;:ao podem ser encontrados em ABRHIFURNAS (1997). A tabela I apresenta, para cada urn dos ensaios realizados, as equa.;:oes medias, por transdu~or, utilizadas na conversiio volt/m.c.a.

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Tabela 1- Equa~oes de conversio volts/m.c.a. para os ensaios realizados (mJ/s)

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TRANSDUTOR lK- 112 IK- 121 IK- 116 JG-19 IK- 127 lK- 122F IK- 133F

500/3000 p = 14 652xT-19,907 p = 14,348xT-19 753 p = 14,556xT-19 510 p = 14 546xT-20 229 p = 14 761xT-19 603 p = 14 451xT-19,177 p = 14,399xT-20,074

500 p = 14 652xT-19 882 p = 14,348xT-19 728

p- 14 556xT-19,485 p = 14 546xT-20 204 p = 14,761xT-19,578 p = 14,451xT-19 152 p = 14,399xT -20,049

1000/2000/3000/4000 = 14,652xT-19 892 = 14 348xT-19 738 _!>_- 14 556xT-19,495 p = 14 546xT-20 214 p = 14 761xT-19 588 p = 14 451xT-19 162 p = 14,399xT-20,Q59 p p

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Vertimento de Limpeza

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Para avalia.;:ao das condi.;:oes de trabalho, realizou-se no dia 17/05/96 uma inspe.;:ilo subaquatica da bacia. Tal inspe..ao teve como principal objetivo, alem da verific~io das con~oes de dep6sito de sedimentos na bacia como urn todo, o reconhecimento dos locais de insta1a.;:ao dos transdutores e dos piezometros. Apesar da pequena espessura da camada de sedimento encontrada no fundo da bacia, essa insp~ao indicou a necessidade de se promover urn vertimento de limpeza da mesma, pois, se a camada de sedimento nlio chegava a impedir a realiza.;:iio dos servi.;:os de instala.;:iio, prejudicava, enormemente, a visibilidade dentro d'agua, dificultando o trabalho de monitoramento que se fazia da superficie das imagens das filmagens feitas pela equipe de mergulhadores. Solicitou-se portanto, a Opera.;:io da Usina, para o dia 19/05, o vertimento das vaz<>es de 500 e 3000 m3/s, ambas por urn periodo de aproximadamente uma hora. Os transdutores relativos aos pontos DA, DB, DC, situados no vertedor foram instalados antes do vertimento (dia 18/05), ja que prJltiCIIlllente nao foram afetados pe1o dep6sito de sedimentos. Durante a opera.;:iio de vertimento para limpeza, foi possivel o estabelecimento da seguinte metodologia de ensaio; • A equipe de opera.;:ao da Usina anotou com intervalos de 10 em 10 minutos todas as condi90es 1 de contomo dos ensaios; . • Antes e ap6s o periodo total de vertimento, foram obtidos os valores das pressoes estaticas nos piezometros e aquisitados os mesmos valor.es atraves dos transdutores, para efeito de verificayio da calibra.;:ilo dos mesmos; • Durante o vertimento de cada urna das vazoes ensaiadas, foram medidos de 5 em 5 minutos os niveis d'agua nos dois postos limnimetricos instalados na ilha a jusante. Estes dados eram anotados e transmitidos, via radio, para a equipe responsavel pelas medi.;:oes, que procedia a analise da tendencia a estab~o da vJIZAo em trinsito. • Uma vez estabilizado o nivel d'agua de jusante, para a vaziio vertida, os valores de pressaes instantaneas foram aquisitados e, concomitantemente, mediram-se os valores das presslies medias em cada urn dos pontos monitorados (DA,DB e DC). Ap6s o vertimento de limpeza, foram concluidos os trabalhos de instal~ao dos demais transdutores. 299

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Ensaios Realizados

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Os ensaios completos foram realizados para as vazoes vertidas de 500 m3/s (no dia 20/05), de 1000 e 2000 m3/s (no dia 21/05) e de 3000 e 4000 m3/s (no dia 22/05). Devido as condifVoes operacionais da Usina nli.o foi possivel a realizafVli.O do ensaio com a vaz!o vertida de 6000 m3/s, conforme previsto na programa!Vli.O original. As fotos 5 e 6 ilustram as condifVoes de escoamento no interior da bacia de dissipafVli.o.

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Analise dos Resultados do Prototipo

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As figuras 3 a 7 mostram a varia~o no tempo das pressoes instantineas medidas nos sete transdutores de pressli.o instalados, para vaz5es de 500 m3/s (figura 3) a 4000 m3/s (figura 7). Observa-se claramente nestas figuras o processo de produfVli.O e de ampliafVli.O de turbulencia, particularmente para as vaz5es mais altas. Para cada figura sli.o apresentados valores de press5es instantineas maxima, media e minima, a diferen!Va entre os valores maximo e minima e a cota do transdutor. Os resultados destes graficos conduzem aos valores da tabela 2 onde sli.o apresentados, para cada vazio, resultados de amplitude de flutua~ de pressli.o para os transdutores situados no vertedouro e no inicio da bacia (DA, DB, DC, D I).

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Tabela 2- Resultados de amplitude de DA, DB,DC,Dl Vazlo (m3/s)

lsoo 1000 2000 3000

4000

DA(m) 4,17 7,32 5,51 6,22 4,06

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Transdutores de Pressio DB(m) DC(m) 3,87 3,84 5,53. 4,98 7,51 8,03 6,62 9,03 8,13 11,16

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5,15 5,06 7,46 10,10 12,93

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As figuras N°s 3 a 7 mostram urn aumento gradativo da energia de turbulencia na bacia com o aumento da vazio, com amplitudes de flutU8fVoes atingindo valores da ordem de 13 m na entrada do ressalto para a va.zli.o de 4000 m3/s. A anAlise destas figuras mostra ainda, atraves do registro simultineo nos sete transdutores de pressli.o, o processo de fo~ de picos de flutua~es de pressio que sli.o convecionados parajusante, sendo atenuados no final da bacia (ver figuras 5, 6 e 7). As figuras 8 a 14 mostram os resultados da analise espectral que evidenciam que o ressalto e urn fenomeno hidrautico turbulento de baixa frequencia.A figura 8 mostra que o transdutor DA (canal I), instalado no vertedor nli.o e afetado pela energia de turbulencia do ressalto para a vaz!o maxima vertida de 4000 m3/s, o que nli.o acontece no transdutor DB (figura 9- canal 2), que ja mostra o efeito desta energia para a va.zli.o mencionada. Os picos de energia de turbulencia, para a vaz!o mAxima ensaiada, ocorrem sobre os transdutores DC (canal3) e Dl (amal4), ja localizados no inicio da bacia de dissipa~o (figuras 10 e 11 ), observando-se que a energia de turbulencia das maiores ·escalas conceatram-se na faixa de 0 a 5 Hz, como era de se esperar. .As .figuras 12, 13 e 14 (transdutores OJ - canal 5, 05 - canal 6 e 07 - canal 7) mostram a atenua!Vli.O gradativa do pico de energia de turbulencia. As figuras 15, 16 e 17 mostram os resultados de flutua~Voes de pressli.o no dominio do tempo que evidenciam que o ressalto e urn fen&neno bidrautico que se caracteriza por grandes amplitudes de flutu~es. A figura 15 apresenta os resultados de amplitude de pressli.o (~>...- Pnun) ao Iongo do rtssalto para vazoes variando entre 500 e 4000 m3/s. Nota-se claramente nesta figura o aumento gradativo das amplitudes de flutuafVli.o de pressilo com a va.zli.o, atingindo-se valores da ordem de 20

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m, notando-se ainda que a zona critica do ressalto para a vazlio maxima vertida se situa entre os transdutores DC (canal 3) e Dl (canal4), como ja observado na amilise espectral. As figuras 16 e 17 apresentam as linhas de pressao maxima, media e minima ao Iongo do ressalto para as vazoes de 2000 e 4000 m%, notando-se que para a vaziio de 4000 m3/s ocorrem pressoes negativas da ordem de 4 m.c.a. na zona critica do ressalto (regiao dos transdutores DC e Dl). ·

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Importincia do Tema para o Desenvolvimento de Pesquisa

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Pesquis11. Aplicada em Engenharia Hidraulica

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A literatura tern reportado a ocorrencia de cavi~iio e erosio por vortices ferradura em estruturas de dissipa~iio para vazoes operacionais bern abaixo das vazCSes de projeto dos vertedores ( BORSARI & ORTIZ - 1987). No caso da UHE Porto Colombia, que tinha urna bacia tlpo D do USBR, erosCSes significativas foram observadas para descargas mllximas inferiores a 36% (5700 m3/s) da capacidade plena de vazio(l6000 m3/s). lsto esta associado ao fato que para vazCSes mais baixas o v6rtice ocorre proximo as fronteiras s6lidas, acarretando danos as estruturas. A decisiio de engenharia, mencionada neste relat6rio, de se remover os blocos de queda da bacia de Porto Colombia, praticamente elirnina a possibilidade de cavita~io nesta regiio, ja que os vortices ferradura niio devem mais se formar, apesar de ainda ocorrerem pressCSes negativas, conforme mencionado no item anterior. Todavia, como as medi'roes de flutua~oes de pressao no prot6tipo foram feitas ap6s a elimina~io dos blocos, os dados disponiveis nio perrnitem um estudo direto da forma~ilo dos referidos vortices. Por outro lado, embora nio exista a possibilidade de se estudar a cavita'rio no prototipo, os resultados permitirio estudar com detalhes os efeitos de escala em rnodelos em semelhanQ& de Froude, ja que sio raras as possibilidades de instrumenta'rilo e medi~o em prot6tipo. A analise dos resultados dos estudos em modelos hidriulicos que estilo sendo conduzidos nos laborat6rios de hidraulica, assim como a interpreta~io dos efeitos de escala, nilo serio tratados no trabalho aqui apresentado. Todavia, e interessante frizar que a interpretayio correta destes efeitos de escala nos estudos de semelhan'ra de Froude deve perrnitir a utiliza~ilo com seguran~a dos chamados ''blocos supercavitantes" em estruturas de dissip~ por ressalto. 0 banco de dados disponivel a partir das medi'roes de Porto Colombia sera extremamente uti! para OS pr6ximos projetos de estruturas de dissipa~ilo. Conforme salientado anteriormente, 0 ressalto e urn fenomeno hidraulico de baixa frequencia e de grande amplitude de flutu~Cies, sendo que a analise de seu comportamento em prototipo, nos dominios do tempo e da frequ&lcia, permitira a deterrnina~io, com seguranQ&, das maiores escalas de turbulencia e das solicita~Cies hidrodinimicas das estrutQras. E interessante salientar que as medi~es feitas com os sete transdutores perrnitirilo uma analise criteriosa dos momentos de diversas ordens (media, desvio-padrio, variincia, assimetria, curtose etc), das fun'rCies de a'utocorrela~io, das correla~Cies espaciais entre transdutores, dos espectros de potencia, dos espectros cruzados entre transdutores etc. Todas estas inform~es devem pennitir visualizar o carninhamento do " roller" na bacia, o desenvolvimento e amortecimento de picos, a dura~iio e o tamanho das maiores escalas de turbilh5es, a possivel dis¥ibui'rilO de probabilidade de pressCSes flutuantes, que supCSe-se nio gaussiana na zona critica do ressalto, a velocidade de convec'rilO do ressalto, que supoe-se menor que a velocidade media do escoamento etc. Estas informa~es estilo sendo trabalhadas, atraves do banco de dados disponivel, e serio motivo de uma outra publica~ilo, devendo ser utilizadas para o aprimoramento dos criterios de projeto de bacias de

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( Pesquisa Basica em Turbulencia

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A modelagem da turbulencia e urn assunto em franca evoluciio que depende, para o seu desenvolvimento, do convivio balanceado entre pesquisas experimentais em 1aborat6rio, utilizac;:ilo de instrumentacilo e tecnicas modernas de medicilo, informacOes de campo e desenvolvimento de novas tecnicas nurnericas. Em publicac;:ilo recente da revista da ABCM, Freire et alii (1998) apresentam a evolucilo do estado da arte na modelagem em turbulencia no Brasil. A publicacao embora bastante detalhada e ex:tensa, e ainda parcial, pois se restringe apenas aos trabalhos publicados na referida revista, o que mostra a dificuldade do dominio do assunto com aplicacOes em diversas areas do conhecimento cientifico. Nos ultimos dez anos o estudo da turbulencia em escoamentos tern sido conduzido a partir de dois pontos de vista ate certo ponto antagonicos. Por um lado, sustenta-se que a turhulencia urn fenomeno esencialmente randornico, de forma que deve ser modelada a partir de ferramentas da estatlstica pesquisando-se os valores med.ios das quantidades turbulentas. Seguindo a linha estabelecida pela teoria de Kolmogorov de 1941 da cascata de turbilhOes (apud Frisch, 1996), a turbul!ncia se origina a partir da deformacilo do escoamento med.io pelas instabilidades hidrodinimicas e fronteiras dos escoamentos, gerando-se as maiores escalas de turbilMes (escalas integrais) que vilo se decompondo em cascata ate as menores escalas (escalas dissipativas). Por outro lado ha urn outro ponto de vista que sustenta qua a turbuiSncia e composta por estruturas de coerEncia de modo a poder ser tratada como urn fenomeno deterministico. Uma anAlise mais aprofundada do tema indica que cada um destes pontos de vista nilo pode ser tratado isoladamente. Confonne friza Lesieur (1990), deve-se considerar as estruturas de coerencia como parte integrante do escoamento turbulento. Em outras palavras, quando se analisa um sinal turbulento de um escoamento qualquer e se identificam estruturas de coerencia, observa-se que estas estruturas podem conservar a forma geometrica por algum tempo, mas silo imprevislveis em termos de posicilo no espaco e em tais situaQOes, a anAlise randOmica do sinal turbulento, utilizando as ferramentas da estatlstica, continua sendo fundamental para o estudo do fenomeno. Dentro desta filosofia portanto a turbulencia poderia ser interpretada como: " urn processo de geracilo
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para registro Iongo, nao sendo todavia conclusivos. Os resultados disponiveis do prot6tipo de Porto Colombia, onde foram feitos registros de flutuar,:oes de pressao atraves de dois sistemas de aquisir,:ao em paralelo, com intervalos de amostragem entre 5 minutos (sistema CTH) e 45 minutos (sistema FURNAS), alem da possibilidade de se fazerem registros longos nos modelos, abrem a perspectiva de se conhecer a verdadeira distribuir,:Ao de probabilidade de pressoes no ressalto bidraulico, o que por si s6 seria urn trabalho inedito. Com rela~o a simular,:ilo numenca da turbulencia, sabe-se que para a simul~ direta de escoamentos turbulentos, a partir da equa~o de Navier-Stokes, ha necessidade de se explicitar todas as escalas de turbulencia, desde as maiores (escalas integrais) ate as menores (escalas dissipativas de Kolmogorov). Considerando-se que a distancia entre as maiores e as menores escalas aumenta com o numero de Reynolds, torna-se impossivel nos dias de hoje a simular,:ilo numerica direta dos escoamentos turbulentos encontrados na natureza com numeros de Reynolds superiores a 10', sendo que o ressalto se enquadra nesta ~o. No sentido de se suprir esta dificuldade, a utilizar,:ilo dos chamados ''LES - Large Eddy Simulation Turbulence Model" tern sido objeto de discussoes em diversos simp6sios internacionais e publicar,:Oes recentes (ver Wilcox, 1993 ). Os ''LES" tern como principal interesse descrever o comportamento das maiores escalas do escoamento, que normalmente, contem as infonnar,:oes desejadas sobre a estrutura de coerencia da turbulencia e os processes de transferencia de energia para as menores escalas. Estes modelos requerem menos capacidade computacional, pois nilo tern o objetivo de simular as escalas dissipativas. que sio introduzidas no ca!culo a partir do estabelecimento de uma bip6tese. A simulayilo via "LES", em principia, deve permitir a previsilo das propriedades estatisticas do escoamento turbulento pesquisado (momentos de diversas ordens, distribuir,:ilo espectral, coeficientes de transferencia para as menores escalas etc), alem de possibilitar a previsao da forma e da topologia da estrutura organizada das maiores escalas do escoamento, embora nao tenha condir,:io de reproduzir corretamente a fase destas estruturas. considerando-se que os escoamentos turbulentos sao imprevisiveis no espayo. Considerando que a base de dados disponivel consiste em dados de pressoes instantaneas, isto traz uma certa limitar,:ao ao desenvolvimento da modelagem computacional. No entanto, Song & Zhou em trabalho recentemente concluido, apresentam resultados de tlutuayoes de pressilo de escoamento de superficie livre em vertedores, a partir da simular,:ao via ''LES", utilizando o supercomputador CRAY C90 da University of Minnesota/USA. Alem disso, os valores de velocidades instantaneas sempre poderilo ser medidos em modelo, deste que se disponha de medidores instantaneos de velocidade. . Pretende-se portanto, a partir do banco de dados de Porto Colombia estudar a possibilidade dei aplicar,:ao dos chamados "LES" na simular,:ao numerica do ressalto hidraulico em bacias de dissipayao.Finalmente e importante enfatizar a contribuir,:ao pesquisa na de instrument89io aplicada a medir,:ao de fenomenos bidraulicos turbulentos. Na pratica da Engenharia Hidraulica aplicada ao estudo das estruturas de dissipar,:ao tern sido mais comum a utiliza~o de transdutores piezoresistivos para a medir,:ilo de flutuar,:oes de pressao. Dois cuidados devem ser tornados quando da utilizar,:ao destes transdutores. Em primeiro Iugar, os transdutores devem ter wna membrana suficientemente pequena, de modo a registrar as menores escalas de turbulencia. Em segundo Iugar, preferencialmente, devem, ser instalados faceados aos pontos de mediyao, de modo a impedir a atenuayao ou amplificar,:ao das respostas em frequencia introduzidas por sistemas de mangueiras e adaptadores instalados entre o ponto da tomada de pressiio e o ponto de instal~ do transdutor (ver Ortiz & Barbosa- 1995). No caso de Porto Colombia, estes cuidados foram tornados, faceandose os transdutores, o que torna, .mais uma vez, o trabalbo medito em termos de instrum~ de prot6tipo. Nos estudos em modelo, pesquisas poderilo se conduzidas com transdutores faceados ou nao, de modo a se estabelecer wna modelagem maternatica de correr,:ilo dos sinais de presslo, quando da necessidade de utilizar,:ilo de sistemas de mangueiras e adaptadores entre a tomada de pressao e o transdutor.

a

303

area

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E interessante

salientar que pelo fato dos numeros de Reynolds de prot6tipo e modelo serem diferentes, diferenQaS deverilo ocorrer nos respectivos espectros de potencia, mesmo no caso de utilizaQilo de transdutores faceados (tanto no prototipo, como no modelo). Sendo assim, os dados de Porto Col&nbia pe.rmitirilo contribuir com as pesquisas relativas aos estudos de resoluQilo espacial e em frequencia das escalas de turbutancia, a partir de mediQ5es de tlutuaQ(ies de presslo ·com transdutores piezoresistivos. 1

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ConclusiJes

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Pelo exposto, conclue-se que o trabalho aqui apresentado sobre o tema "mediooes em prototipo de tlutua~es de pressilo na bacia de dissip~ da UHE de Porto Colombia", ultr.apassou todas as expectativas de sucesso e se reveste de uma importincia muito grande, pois o banco de dados disponivel content info~es que devem trazer beneficios nilo s6 A pesquisa aplicaila a engenharia de projeto de estruturas hidraulicas, mas tambem pesquisa basica do estudo da turbulencia, que permanece ainda, neste final de seculo, como urn dos t6picos nlo resolvidos da ciencia

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Rela~o

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dos Participantes dos Ensaios no Prototipo

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Coordenador Geral

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Erton Carvalho - FURNAS

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Particlpantes

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CTIIJFCTHIEPUSP:

COPPEIUFRJ:

CEilPAR:

Fatima Moraes de Almeida Marcos da Rocha Botelho Rogerio Sales G6z Jose Zanini Filho Edson Ricardo Holanda Andre Luiz Venincio Cleober Michel Tosta Zanini Rosselini Ranieri Agostini Alcior Novaes de Faria Jayme Pinto Ortiz Flavio Spipola Barbosa Claudio Menegatte Filho Fabio Nascimento de Carvalho Igor Afonso Fragoso Sinildo Hermes Neidert Edie Roberto Taniguchi Jose Jungi Ota

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CESP:

Mauro Henn6genes L. Cove Roberto Ferreira de Alvarenga Rogerio de Oliveira Francisco Antonio A. Gouveia Jose Pereira da Silva

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Referencias Bibliogaficas

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ABRH/FURNAS (1997)" Relato do Trabalho da Subcomissao Desenvolvimento de PesquisaEstudo de Flutua~ao de Pressao em Bacia de Dissipa~ilo - Caso Usina de Porto Colombia" 64 pags., novembro, 1997. Borsari, R.D. & Ortiz,J.P. (1985) "A Modela~ao de Fenomenos de Cavita~ao e sua . Preven~ao em Modelos Segundo Criterio de Froude''. Anais do VI Simp6sio Brasileiro de Hidrologia e Recursos Hidricos, vol. 2, pag. 120-13 I Carvalho,F.N.; Almeida,F.M. and Fragoso,I.A. (1997) "Aquisi~io de Dados no Prot6tipo da UHE de Porto Colombia". Anais do XII Congresso da Associa~ilo Brasileira de Recursos Hldrlcos ABRH, vol. 1, novembro, 1997. Freire,A.P.S.; Avelino,M.R. and Santos,L.C.C. (1998) "The State of the Art in Turbulence Modelling in Brazil". Revista Brasileira de Ciencias Mecanicas- ABCM, vol.XX, No.l,pp.l38, march, 1998. ' Frisch,U. (1996) "Turbulence". Cambridge University Press, Reprinted, 1996, 296 pages. Lesieur,M. (1990) "Turbulence in Fluids". Kluwer Academic Publishers, second revision edition, 412 pag. Ortiz,J.P. and Barbosa,F.S. (1995) "Criterios de Escolha de Transdutor Eletrico de Pressilo como Padrao de Medida para Laboratories de Hidraulica". Anais do Segundo Simp6sio Brasileiro de Medi~ao de Vazao, IPT/SP, p.41 5-424. Toso,J.W. and Bowers,C.E. (1988) "Extreme Pressures in Hydraulic Jump Stilling Basins". Journal ofHydraulic Engineering, vol. 114, No.8, pp.829-843, august, 1988. Wilcox,D.C. (1993) "Turbulence Modeling for CFD... DCW Industries, Inc. La Canada, <;:alifornia,' i 460 pag. Song,C.C.S. and Zhou,F. "Simulation of Free Surface Flow over a Spillway". 33 pags.

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( ( Figura 16 • PRESSOES MAxiMAS, MEDIAS E MiNIMAS, AO LONGO DO VERTEDOR Q '" 2000 m31s

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Figura rJ • PRESsCES MAx.IMAS, MEDIAS E MINIMAS, AO LONGO DOVERTEDOR Q • 4000 m31s

PRESs0ES MAxiMAS, MEDIAS E MINIMAS, AO LONGO DO VERTEDOR • Q:o4000 M31S 20.000

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• L__ ___ _

(' Laborat6rio de Meciinica da Turbulencia, PEM/COPPE/UFRJ C.P. 68503, 21945-970, Rio de Janeiro

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1998 Vol. 1 - ABCM

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