modeling the pressure-strain correlation of turbulence - an invariant dynamical systems approach

8/12/2019 Modeling the Pressure-strain Correlation of Turbulence - An Invariant Dynamical Systems Approach

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MODELING THE PRESSURE-STRAIN CORRELATIONOF TURBULENCE - AN INVARIANT DYNAMICAL

SYSTEMS APPROACH. lon For

Charles G. Speziale and Sutanu Sarkar* ,,ICASE, NASA Langley Research Center , ed jJ

Hampton, VA 23665 -tioThomas B. Gatski . t i

NASA Langley Research Center - . .i CodesHampton, VA 23665 a,nd/or

pocial

ABSTRACT ..... j___The modeling of the pressure-strain correlation of turbulence is examined from a basic

theoretical standpoint with a view toward developing improved second-order closure models.Invariance considerations along with elementary dynamical systems theory are used in theanalysis of the standard hierarchy of closure models. In these commonly used models, thepressure-strain correlation is assumed to be a linear function of the mean velocity gradientswith coefficients that depend algebraically on the anisotropy tensor. It is proven that forplane homogeneous turbulent flows the equilibrium structure of this hierarchy of modelsis encapsulated by a relatively simple model which is only quadratically nonlinear in theanisotropy tensor. This new quadratic model - the SSG model - is shown to outperform theLaunder, Reece, and Rodi model (as well as more recent models that have a considerablymore complex nonlinear structure) in a variety of homogeneous turbulent flows. However,some deficiencies still remain for the description of rotating turbulent shear flows that areintrinsic to this general hierarchy of models and, hence, cannot be overcome by the mereintroduction of more complex nonlinearities. It is thus argued that the recent trend ofadding substantially more complex nonlinear terms containing the anisotropy tensor may beof questionable value in the modeling of the pressure-strain correlation. Possible alternativeapproaches are discussed briefly.

*This research was supported by the National Aeronautics and Space Administration under NASA Con-tract No. NAS1-18605 while the first and second authors were in residence at the Institute for ComputerApplications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665.

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1. INTRODUCTIONThe pressure-strain correlation plays a pivotal role in determining the structure of a wide

variety of turbulent flows. Consequently, the proper modeling of this term is essential forthe development of second-order closure models that have reliable predictive capabilities.Rotta (1951) developed the first simple model for the slow pressure-strain correlation (i.e.,the part that is independent of the mean velocity gradients) which describes the returnto isotropy behavior of turbulence within the framework of a full Reynolds stress closure.This model has served as a cornerstone for the representation of the slow pressure-straincorrelation in a variety of the commonly used second-order closures such as the Launder,Reece, and Rodi (1975) model. Subsequent to this work, Lumley (1978) demonstrated theneed for nonlinear terms in models for the slow pressure-strain correlation and derived anonlinear representation theorem for the slow pressure-strain correlation based on isotropictensor function theory. In the high-Reynolds-number and small anisotropy limit, the Lumley(1978) model reduces to the Rotta model.

The simplest model for the rapid pressure-strain correlation that is used in second-orderclosure modeling is based on the assumption of isotropy of the coefficients of the meanvelocity gradients; this gives rise to a rapid pressure-strain model with a single term thatis proportional to the mean rate of strain tensor (see Rotta 1972 and Mellor and Herring1973). Starting with the work of Launder, Reece, and Rodi (1975), anisotropic models forthe rapid pressure-strain correlation have been formulated wherein the coefficients of themean velocity gradients are taken to be functions of the anisotropy tensor. In the Launder,Reece, and Rodi model, the fourth-rank tensor of coefficients of the mean velocity gradienttensor is linear in the anisotropy tensor whereas most of the newer models developed duringthe last decade are nonlinear (see Shih and Lumley 1985, Haworth and Pope 1986, Speziale1987, Reynolds 1987, and Fu, Launder, and Tselepidakis 1987). The nonlinear models ofLumley and co-workers have been primarily developed by the use of realizability constraints(see Lumley 1978). In contrast to this approach, the nonlinear model of Speziale (1987) wasderived using a geostrophic flow constraint (i.e., material frame-indifference in the limit oftwo-dimensional turbulence) whereas Reynolds (1988) has attempted to develop models thatare consistent with Rapid Distortion Theory (RDT).In this paper, the general hieiarchy of closure models for the pressure-strain correlationwill be considered which are linear in the mean velocity gradients, with coefficients that arefunctions of the anisotropy tensor. This hierarchy of models, which was motivated by analy-ses of homogeneous turbulence, encompasses all of the closure models for the pressure-straincorrelation that have been used in connection with second-order closures. A general repre-sentation for this hierarchy of closure models will be derived by means of invariant tensor

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function theory. This general representation for the pressure-strain correlation will then beapplied to plane homogeneous turbulent flows - the class of flows that have long played a piv-otal role in the screening and calibration of such models. However, there will be one notabledifference with previous work on this subject: the simplest generic form of this hierarchy ofmodels will be sought that has the same equilibrium structure in the phase space of planehomogeneous turbulent flows as the general model. This generic form - which will be termedthe SSG model - is only quadratically nonlinear in the anisotropy tensor. It has the advan-tage of being topologically equivalent to the general model in plane homogeneous turbulencewith the simplicity of structure that allows for the determination of all empirical constantsbased on calibrations with pertinent RDT results and two well documented physical exper-iments (i.e., homogeneous turbulent shear flow and the decay of isotropic turbulence). Thisnew SSG model will be shown to perform better than the commonly used Launder, Reece,and Rodi model for a variety of homogeneous turbulent flows which include plane strain,rotating plane shear, and the axisymmetric expansion/contraction. However, there are stillsome remaining deficiencies in the new model, particularly for rotating shear flow. Basedon an analysis of the bifurcation diagram for rotating shear flow, it will be shown that thesedeficiencies are intrinsic to this general hierarchy of pressure-strain models and cannot beeliminated by the addition of more complex nonlinear terms. The implications that theseresults have on turbulence modeling will be discussed in detail along with suggested futuredirections of research.2. THE GENERAL PRESSURE-STRAIN MODEL

We will consider the turbulent flow of a viscous, incompressible fluid governed by theNavier-Stokes and continuity equations

Ovi Ovi OP+-7 + Ov o, (1)Ovi= 0. (2)

In (1) - (2), vi is the velocity vector, P is the modified pressure, and v s the kinematicviscosity of the fluid. The velocity and pressure are decomposed into ensemble mean andfluctuating parts, respectively, as follows:

v=Ui,+u, P= P+ p. (3)Here, the mean and fluctuating velocity are solutions of the transport equations

+ - o_ + 2 O-- (4)

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ax- 0(5)+ j- = 7,J7, + V oU (6)

at Ox1 Oxi Ox Ox, Ox1auia-= 0 (7)where rij -- u- s the Reynolds stress tensor.

The Reynolds stress tensor -rij is a solution of the transport equationOr11 T I V - V~i 0L-T - - T I- + Cij + vvr, 1 (8)

whereC,,k =_ U 3 Uk + PU76tk + P-i86ik 9)

P Ox+ U~ (10)Ou1 Ou ,eq = 2vu- --- 3 11 )are, respectively, the third-order diffusion correlation, the pressure-strain correlation, andthe dissipation rate correlation. Equation (8) is obtained by taking the symmetric part ofthe ensemble mean of the product of the fluctuating velocity u3 with equation (6). For homo-geneous turbulent flows, at high Reynolds numbers where the dissipation is approximatelyisotropic, the Reynolds stress transport equation (8) simplifies to

i- -T -z - r-i - - c6 (12)where

S=i---- (13)Ox,. Ox,.is the scalar dissipation rate. Equation (12) becomes a closed system for the determ;nationof rij in terms of 9U i/0x1 once closure models for Hi and e are provided. Sinc, 1li is theonly unknown correlation that contains directional nformation, it is clear th.t it will playa pivotal role in determining the structure of the Reynolds stress tensor for a given meanvelocity field. This dominant influence of II, n the evolution of the Rcynolds stress tensorin (12) has motivated researchers to rely on homogeneous turbulent flows for the testing andcalibration of pressure-strain models.

The fluctuating pressure p is a solution of the Poisson eq,.AtionV OupOut u i 14)=-Oxi O xO Ox, Oxj Oxi

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which is obtained by subtracting the divergence of (4) from the divergence of (1). In theabsence of boundaries, equation (14) has the general solution

1 00 1 a2u; 0v + u au . au au .udV'. 15)ax 3 1 3 1)For homogeneous turbulent flows (where the mean velocity gradients are spatially uni-

form) the pressure-strain oorrelation takes the formI1,, = A,, + M,;kL- (16)

where 1f (a , + j) (2 u VAdj= (17)47r x Oxax Ix-X

1 a(ui + uji au t dV'M, kt =jd (18) ~ o 8x4x -xlIt has been shown that A13 and M 3kj are functionals - over time and wavenumber space - of

the energy spectrum tensor; see Weinstock (1981, 1982) and Reynolds (1987). In a one pointclosure, this dependence would suggest models for Aj and Mikt that depend on the historyof the Reynolds stress tensor and dissipation rate. The simplest such models are algebraicin form:

Aij = eA-j(b) (19)Mijkl= KMdikl(b) (20)

where1

bij 3t -5'k i 21)Ta

K ITkk (22)are the anisotropy tensor and turbulent kinetic energy. In (19) - (20), Aj and M,kj canonly depend on rij through bij since they are dimensionless tensors that vanish in the limit ofisotropic turbulence. Virtually all models for the pressure-strain correlation that have beenused in connection with second-order closure models are of the general form of (19) - (20).The use of this hierarchy of models for general inhomogeneous turbulent flows is based onthe assumption of a local homogeneous structure.

The mean velocity gradient tensor can be decomposed into symmetric and antisymmetricparts as follows:

am , + Oij (23)

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where2 Ox, 1Oxi

W12 i~xj iaxi 25are the mean rate of strain tensor and mean vorticity tensor, respectively. Then, the modelfor the pressure-strain correlation specified by (19) - (20) can be written in the equivalentform

Ij= cf, ( (b, S', '') (26)where -, K, . K_Sij -s ,= w, ,= 27)C Care the dimensionless mean strain rate and vorticity tensor whereas f(L) denotes the part ofthe function fi that is linear n the mean velocity gradients and traceless. Form invarianceunder a change of coordinates requires that fj transform as

Qf(b,, ,)Q T = f(QbQ T , QSg Q T ,QW-4Q T (28)where Q is the rotation tensor (and QT is its transpose) which characterizes a change inorientation of the coordinate axes. In mathematical terms, (28) requires that fj be anisotropic tensor function of its arguments. By using known representation theorems forisotropic tensor functions (see Smith 1971) to construct fj - and then by taking the linearand traceless part of fj - the following model for Ij is obtained:

II, = fPleb i + f2e(bjkbkj - i bmn8i) + / 3Kg,,2b+ 134K (bik3 jk + b 1k3 k - 2 , r, ,6,)3 (29)2 b - j jj+3 5 K(bjkbkeS1j + bjkbk13Sz - bjmbmn, n 6, )

+ 6K(bWozj, + bjk'Zk) + # 7K(bikbk1,Ztj + bbkjZ)where

A, =+o(I )+Oi(II, III)-tr(b . )+ (1,III)-tr(b.), =1,2 (30)C C3, =/ (IIIII), j = 3,4,...,7 (31)

II = bjb,j, III = b,jbjkbk, (32)

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and tr(.) denotes the trace (see Appendix A for more details). Equation (29) represents themost general form of the hierarchy of models (19) - (20) for the pressure-strain correlationthat is consistent with the crucial physical constraint of invariance under coordinate trans-formations. It will be shown later that the Launder, Reece, and Rodi model is the linearlimit of (29) wherein 031,#3,,34, and 06 are constants while 012,135, and 07., are zero.Finally, in regard to the general model, a few comments should be made concerning non-inertial frames of reference. In a non-inertial frame, the mean vorticity tensor U, must bereplaced by the intrinsic (i.e., absolute) mean vorticity tensor defined by

WO = a7i, + e, if, (33)where Ql,, is the rotation rate of the non-inertial frame relative to an inertial framing and em3is the permutation tensor (see Launder, Tselepidakis, and Younis 1987 and Speziale 1989a).Furthermore, Coriolis terms must be added to the Reynolds stress transport equation whichthen takes the form

= j ai 2 6 2(rijemjQ2m 'remkiSm)in an arbitrary non-inertial reference frame.

3. PLANE HOMOGENEOUS TURBULENCEWe will consider the general class of plane homogeneous turbulent flows for which the

mean velocity gradient tensor can be written in the forma-vj 0 S+W 0 -- S - 0 0 . 35)5Xj 0 0 0 )

Since the mean continuity equation (5) requires that 6vi-]/Dx = -&c 2/8X 2 in plane homoge-neous turbulent flows, equation (35) results by simply aligning the coordinates at a 450 anglerelative to the principal directions of the symmetric part of oui/xj. Of course, in order tomaintain the homogeneity and two-dimensionality of the mean flow, S and w are constantswhile Qj is given by = (0,0, (36)(hence, the rotation is in the plane where the mean velocity gradients are applied). Equations(35) - (36) encompass, as special cases, plane shear, plane strain, rotating plane shear, androtating plane strain turbulence.

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The Reynolds stress transport equation (34) for plane homogeneous turbulence can bewritten in terms of the anisotropy tensor b1j(t*) (given that t - St is the dimensionlesstime) as follows: dbii _ - - bik 3 -k + Wyk

+2Qe3kj) - bjk(3ik + w,*k + 2-Qe3ki) (37)

where 010 1 0 0100 , 0 ) 38)0 0 0 0 0 0

II .= 1 1/2KS (39)and P - rjijv3/Oj x is the turbulence production. Equation (37) must be supplementedwith a transport model for the turbulent dissipation rate in order to obtain a closed system.We will consider the most standard form of the modeled dissipation rate transport equationgiven by

Gel-.1 T'i 3 - C. 2 - (40)K Oxi Kwhere Cj1 and C,2 can be functions of the invariants II and III of the anisotropy tensor(in the most commonly used form of this model, Cj1 and C,2 are constants that assume thevalues of 1.44 and 1.92, respectively; see Hanjalic and Launder 1972). It will be argued laterthat some of the crucial conclusions to be drawn concerning the limitations of this hierarchyof closure models for the pressure-strain correlation are independent of the specific form of(40). Furthermore, it should be noted that virtually all of the alterations to (40) that havebeen proposed during the last decade are highly ill-behaved (see Speziale 1989b).

The modeled dissipation rate equation (40) can be written in the dimensionless form[(C 1-_1) -(C.2-1)]. (41)

dt SK) SKIEquation (41) is obtained by combining (40) with the transport equation for the turbulentkinetic energy

/k = P - c (42)which is exact for homogeneous turbulence. When the general model for the pressure-straincorrelation (given by (29), with O0 replaced by W1i) is substituted into (37), a closed system

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for the determination of bij and c/SK is obtained. This system of equations has equilibriumsolutions of the form (b=) g, W,) 43)where ( )o denotes the solution in the limit as time t -- o; these solutions were shown bySpeziale and Mac Giolla Mhuiris (1989a) to attract all initial conditions. The equilibriumsolutions (43) - (44) are obtained by solving the nonlinear algebraic equations that resultwhen the time derivatives on the left-hand-sides of (37) and (41) are set to zero. It is asimple matter to show that there is a trivial equilibrium solution where

=(45)which exists for all w/S and 02/S. Non-trivial equilibrium solutions where (C/SK).. 0exist for intermediate ranges of w/S and P/S wherein the trivial equilibrium solution (45)typically becomes unstable.

We will now show that the non-trivial equilibrium values of IIoIIIo,(b 33 ) , and(P/c)o are universal (i.e., completely independent of wIS and QI2S) for this hierarchy ofmodels in plane homogeneous turbulent flows. A closed system of equations for the deter-mination of the temporal evolution of II, III, b33 and c/SK are as follows:

dlI 2 ( P\ 2 b C 2P edt* SK 1- II- b33C-K + 3 SK

+ 1 I + 111 I (46)SK SK 2 cSK

+04b-K S 1 - I2II- 7 SK ESbaa P 3 P C P'Cc 1 P ctSK 2K cSK

P c 3 1- b I3 + -IIS - +-0211SK 2 SK 4 SK (47)3 P 1CP+4b3CS-K 4 41eSK

1 Pc 1 P C+4 Ib3CS3K 20I c SK

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dA SK 6 3 3 P SK(48)+-/3----33 + -/3-- b ] 3--I )

2 SK 2 S K \33 1dT = F C, 1 - (Cr - 1)1. (49)

Equations (46) - (47) are obtained by multiplying (37) with b and b 2 , respectively, and thentaking the trace after the model (29) for IIij is substituted. The non-trivial equilibriumsolutions are then obtained from the nonlinear algebraic equations

[- ('-).1 II, - 2(b )o.(1-).+2 ( )

+1II.+ 02111. - -93 ( + 4 (b33)-( (50)2 0--2 ~ =0

-(b 3 3 )c.(P). +31IlW + 1 21I(51)3 1+ 03(b o o '41I1-4 41

+ - 5 11.b3 )6) -053~11oo 0 -+1 (b3

- P)j b33)0 - (52)11 79

+1f22 [(b 33 )- 3II] = 0

Cei -1) -(C 2 -1) = 0 (53)which are derived by setting the time derivatives on the left-hand-sides of (46) - (49) tozero and dividing by c/SK. The system of equations (50) - (53) will have solutions for

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II., III , (6 33 ). and (P/e)o that are completely independent of w/S and fE/S - and henceuniversal - for plane homogeneous turbulent flows.

This universal equilibrium struoture of IIo, IIIoo, (b33). and (P/e) will now be utilizedto obtain the simplest generic form of (29) that has the same equilibrium ztructure as thegeneral model in the phase space of plane homogeneous turbulent flows. Due to these fouruniversal invariants, the quadratic terms in the rapid pressure strain correlation are notlinearly independentfor plane homogeneous turbulent flows. This quadratic part In ) of (29)is defined as follows:

-l( 3)(b3kbktX.t + bakbkt - T di) + fl7K(bjkbktVWt + bykbklWjt). (54)For plane homogeneous turbulent flows, a straightforward, although somewhat tedious, cal-culation yields the relationships

2 2 111 (55)jkbkI3~j + bjAkbktSit bbSk = -b(b3jA + bj~ - - 2111- (55)3 3b33bikbWfl + bjkbkjV 11 -b 33(bikWjk +~b2,kWiA) (56)

where we have made use of (38) and the fact that the anistropy tensor is of the formbil b1 2 0bi b12 b22 0 (57)

0 0 b33in such flows. Due to (55) - (56), and the fact that IIo, I1Io,(b 3 3)o and (P/e)o areuniversal invariants for all plane homogeneous turbulent flows, it follows that the quadraticterms in the rapid pressure-strain are directly related to the linear terms in such flows.Consequently, the equilibrium structure of (29) in plane homogeneous turbulent flows willbe indistinguishable from that of the substantially simpler model

1Hlij= CIebi3 + C26z(bikbk3 - ,n.8j3+C3 hi + c4K(b k3 j/, + bjk3 k (58)2b- - b,.,+ :5K(bjA:WjA + bjAkWik)3

where c1, C2 , ... , c5 are dimensionless constants and we have mi de use of the fact ttiattr(b 1P (59)

tr(b2 b3 3 (60)

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which was also used in the derivation of (46) - (49). In alternative terms, (58) is topolog-ically equivalent to the general model (29) in so far as the equilibrium structure of planehomogeneous turbulent flows is concerned.

It is rather striking that an analysis of the equilibrium states of arbitrary plane ho-mogeneous turbulence - coupled with the crucial physical constraint of invariance undercoordinate transformations - collapses the general pressure-strain model

= -Aj(b) + KMjk (b)- (61)(which can have as many as forty-five independent functions of b) to the substantially sim-plified model (58) that has only five undetermined constants. In the next section, a newmodel for the pressure-strain correlation will be developed.

4. THE SSG MODEL: ITS ASYMPTOTIC ANALYSIS AND CALIBRATIONNow, a new model for the pressure-strain correlation - which we will call the SSG model

- will be developed based on the previous invariant dynamical systems analysis coupled withthe following additional constraints:

(i) Asymptotic consistency in the limit of small anisotropies(ii) Consistency with Rapid Distortion Theory for homogeneously strained turbulent flows

that are initially isotropic(iii) Consistency with the equilibrium values for homogeneous shear flow obtained from the

physical experiments of Tavoularis and Corrsin (1981)(iv) Consistency with the RDT results of Bertoglio (1982), for rotating shear flows, which

predict that the most unstable flow occurs when the ratio of the rotation rate to theshear rate f2/S = 0.25 and that a flow restabilization occurs when Q/S > 0.5

(v) Consistency with the results of physical experiments on the decay of isotropic turbu-lence and the return to isotropy of an initially anisotropic, homogeneous turbulence.

Since the magnitude of the anisotropy tensor is relatively small (HbH < 0.20 for most turbu-lent flows of engineering and scientific interest), we feel that terms which are of a comparableorder of magnitude in bij should be maintained unless there is some overriding physical rea-son not to do so. In this fashion, the model can then be thought of as an asymptoticallyconsistent truncation of a Taylor series expansion of Aj and Mijk in the variable bj. Since

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the simplified model for the rapid pressure-strain correlation in (58) is of 0(b), this suggeststhat c3 - which in general can be a function of the invariants of b - should be replaced by

C3 - C3II(where C3 and Cj are constants) for asymptotic consistency. Furthermore, since the modelfor the slow pressure-strain correlation is of 0(b ), and since most engineering flows havesignificant regions where P > e, e will replace the constant cl with the coefficient

-(C + C P/E)where C and Cr* are constants. This yields the following model for the pressure-straincorrelation: Iiiy = -(Ce + C*P)b 3 + C2 (bkbk,

-bmb&j) + (C 3- CII )KSi3 (62)2+C4 K(bkSjk + bjkSk - 2bmn~mnbij)+CK(b kWjk + bjkWi).

Although (62) is topologically equivalent to (58) in so far as the equilibrium states areconcerned, it will give rise to different temporal evolutions. We feel that it is better touse (62) as our final model for the pressure-strain correlation since all terms that are of acomparable order in b13 have been maintained for asymptotic consistency.Before using constraints (ii) - (v) to calibrate the SSG model given by (62), a few com-ments are in order concerning the relationship between this new model and previously pro-posed models. The SSG model is not significantly more complicated than the commonlyused Launder, Reece, and Rodi model which can be obtained from (62) in the linear limit asC , 2 and C3 go to zero. In fact, the SSG model is substantially simpler than the recentlyproposed nonlinear models of Shih and Lumley (1985) and Fu, Launder, and Tselepidakis(1987) (see Appendix B).

The coefficients C1 and C2 have been calibrated by considering the return to isotropyproblem (see Sarkar and Speziale 1989). Of course, for the return to isotropy problem, onlythe terms containing the coefficients C, and C2 in the pressure-strain correlation survive (i.e.,the rapid pressure-strain correlation vanishes). Based upon realizability, dynamical systemsconsiderations, and the phase space portrait of return to isotropy experiments, the followingvalues of C, and C2 were arrived at by Sarkar and Speziale (1989):

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C2 = 3 Cl - 2) = 4.2. (64)Interestingly enough, the value of C, = 3.4 is quite close to the value of 3.6 for the Rotta

coefficient that is currently used in the basic model of Launder and his co-workers. However,as demonstrated by Sarkar and Speziale (1989), the quadratic term containing C2 is crucialto properly capture the experimental trends. In Figure 1, the predictions of the SSG modeland the Launder, Reece, and Rodi (LRR) model are compared in the -r7 phase spacewith the experimental data of Choi and Lumley (1984) for the return to isotropy from planestrain. The SSG model exhibits a curved trajectory that is well within the range of theexperimental data; the LRR model - as well as any model for which C2 = 0 - erroneouslypredicts a straight line trajectory. In Figures 2(a) - (b), the predictions of the SSG modeland the LRR model for the temporal evolution of the anisotropy tensor are compared withexperimental data for the relaxation from plane strain experiment of Choi and Lumley (1984)and plane contraction experiment of Le Penven, Gence, and Comte-Bellot (1985). The SSGmodel, on balance, yields improved predictions over the LRR model. For more detaileddiscussions and comparisons, the reader is referred to the paper by Sarkar and Speziale(1989) where this quadratic model for the slow pressure-strain correlation was comparedwith data from four independent experiments on the return to isotropy.

Constraint (ii), which requires consistency with RDT for a homogeneously strained tur-bulence that is initially isotropic, is commonly enforced in the turbulence modeling literature.While the dynamical systems analysis presented in Section 3 can guarantee proper long-timebehavior, this RDT constraint can be of considerable assistance in ensuring proper short-time behavior; if a model properly captures both the short and long-time behavior, it standsan excellent chance of performing well for all times. This RDT result requires that (see Crow1968) 4lim Ilij = KSij (65)

b 0 3and, hence, that

C3 = 4 (66)5for the SSG model. We found that models which deviated significantly from (66) performedpoorly in homogeneous shear flows (e.g., such models yielded spurious points of inflection inthe time evolution of the turbulent kinetic energy).

Constraints (iii) - (iv) were used to calibrate the remaining constants in the model,namely, C*, C*, C4, and C5 as well as the constant C,1 in the modeled -- transport equation.This was done using a value of C,2 = 1.83 (as opposed to the more commonly adoptedvalue of 1.92) which yields a power law decay in isotropic turbulence with an exponent of1.2 - a value which is in better agreement with available experimental data as discussed by

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with large anisotropies. This results from the fact that the finite difference form of themodeled equations usually do not exactly satisfy realizability (see Speziale and Mac GiollaMhuiris 1989a). No such problems are encountered by the weak form of realizability since it issatisfied exactly by most standard numerical formulations of the model. Finally, it should bementioned that the SSG model was not forced to satisfy material frame indifference (MFI) inthe limit of two-dimensional turbulence (Speziale 1983) which constitutes another extremeconstraint that is a rigorous consequence of the Navier-Stokes equations. It has recentlybecome apparent to us that when such extreme constraints as MFI and strong realizability(correct as they may be for general flows) are applied to highly idealized models, there is astrong possibility that the model will become overly biased so that it performs poorly in themore commonly encountered turbulent flows.5. PERFORMANCE OF THE SSG MODEL IN HOMOGENEOUS FLOWS

The SSG model given by equation (62) will now be tested in four independent homo-geneous turbulent flows. For the purpose of clarity, we will summarize the values of theconstants that were arrived at in the previous section:

C1 = 3.4, C; = 1.80, C2 = 4.2 (67)43 =, C = 1.30, C4 = 1.25 (68)

C5 = 0.40, Cj1 = 1.44, C 2 = 1.83. (69)The problem of homogeneous turbulent shear flow in a rotating frame will be consideredfirst. For this case, the mean velocity gradients and the rotation rate of the reference frameare given by

O 0 0 70) 0 0 (0Ozj 0 0 0= (0,0, f2), 71 )

respectively, in matrix form. The initial conditions correspond to a state of isotropic turbu-lence where

b =0, K = Ko, =co (72)at time t = 0. It was shown by Speziale and Mac Giolla Mhuiris (1989a) that the solutiononly depends on the initial conditions through the dimensionless parameter Co/SKo; thedependence of the solution on the shear rate and rotation rate is exclusively through thedimensionless parameter [/S. Two types of equilibrium solutions have been established

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for this problem (Speziale and Mac Giolla Mhuiris 1989a): one where (e/SK),, = 0 whichexists for all Q2/S and one where (e/SK)oo > 0 which exists only for a small intermediateband of values for l/S. The zero equilibrium value is associated predominantly with stableflow wherein K and e undergo a power-law time decay; the nonzero equilibrium values areassociated with unstable flow wherein K and e undergo an exponential time growth. The twosolutions undergo an exchange of stabilities for intermediate values of Q/S (which includesthe case of pure shear where Q2/S = 0). In this fashion, the second-order closures are ableto account for both the shear instability - with its exponential time growth of disturbancekinetic energy - and the stabilizing (or destabilizing) effect of rotations on shear flow.

In Figures 3(a) - (c), the predictions of the SSG model for the time evolution of turbulentkinetic energy are compared with those of the LRR model and the results of the large-eddysimulations of Bardina, Ferziger, and Reynolds (1983) for rotating shear flow. From thesefigures, it is clear that the SSG model does a much better overall job of capturing the trendsof the large-eddy simulations. Several observations are noteworthy: (a) the LRR modelexhibits too strong a growth rate for pure shear (Q/S = 0) in comparison to the SSG modeland large-eddy simulations, (b) both the SSG and LRR models yield too weak a growthrate for the f2/S = 0.25 case, however, the SSG model is substantially better, and (c) theSSG model properly captures the weak growth rate that occurs for fP/S = 0.5, whereas theLRR model erroneously predicts a flow restabilization. The premature flow restabilizationpredicted by the LRR model at fD/S ;, 0.37 is somewhat serious since, in addition to theresults of large-eddy simulations, linear stability theory and RDT predict that there shouldbe unstable flow for the entire range of 0 < Q/S < 0.5 (see Lezius and Johnston 1976 andBertoglio 1982). As mentioned earlier, the SSG model does not predict a flow restabilizationuntil O/S > 0.53.

It would be useful at this point to compare the performance of the SSG model in rotatingshear flow with that of some newer models that have been recently proposed. Three suchmodels - the model of Shih and Lumley (1985), the model of Fu, Launder, and Tselepidakis(1987), and the RNG model of Yakhot and Orszag (1988) - were compared in a recent studyof Speziale, Gatski, and Mac Giolla Mhuiris (1989). It was established in that study thatthe Fu, Launder, and Tselepidakis (FLT) model performed the best among these modelsin rotating shear flow. Hence, for simplicity, we will only compare the SSG model withthe FLT model. In Figures 4(a) - (c), the results for the turbulent kinetic energy obtainedfrom the SSG model and the FLT model for the rotation rates of Q/S = O,Q2/S = 0.25,and Q2/S = 0.50 are compared with the large-eddy simulations of Bardina, Ferziger, andReynolds (1983) for rotating shear flow. It is clear from these results that the SSG modelproperly captures the trends of the large-eddy simulations which indicate that all three cases

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are unstable and that the 2/S = 0.25 case has the strongest growth rate. On the otherhand,the FLT model erroneously predicts that the 2/S = 0 and /S = 0.25 cases are equallyenergetic and that the 2/S = 0.5 case has undergone a restabilization. Comparable to theLRR model, the FLT model erroneously predicts a premature restabilization at 2/S ;, 0.39.TL may be of concern that a heavy emphasis has been placed on comparisons with large-eddy simulations for rotating shear flow (unfortunately, no direct simulations or physicalexperiments have been conducted for this problem). However, it must be emphasized thatthe critical evaluations have been based on which states should be more energetic - resultswhich have been confirmed independently by RDT and linear stability theory.

A bifurcation diagram for the general hierarchy of closure models (61) is shown in Figure5 for rotating shear flow. Here, the equilibrium value of (e/SK)oo is plotted as a function of2/S. The SSG model as well as the other commonly used models have the same topologicalstructure in rotating shear flow as indicated in Figure 5. There are two equilibrium solutions:the solution where (e/SK)oo = 0 exists for all 2/S but becomes unstable in the interval AB;the nonzero solution for (e/SK)oo, which lies on the semi-ellipse ACB, exchanges stabilitieswith the trivial solution (e/SK)o, = 0 in the interval A < 2/S < B. For 2/S < A - 8A and

/S > B + 6B (where 8A and 6B represent a small increment that depends on the model)the trivial equilibrium value of (,/SK)oo = 0 is associated with solutions where the kineticenergy undergoes a power law decay with time; for A - 6A < 2/S < B + 8B, this trivialsolution is associated with solutions where the kinetic energy undergoes a power law growthwith time. The nonzero equilibrium values (e/SK)oo > 0 (on the semi-ellipse ACB) areassociated with solutions where the kinetic energy grows exponentially with time. It can beshown (see Speziale and Mac Giolla Mhuiris 1989a) that the growth rate Afor A < 2/S < Bis given by

A 1)EK(73)where a =(C 2 - l)/(Ci - 1). Hence, point C - which corresponds to the maximum valueof (e/SK)oo - represents the most energetic state with the largest growth rate of kineticenergy.

The coordinates [/S, (e/SK)oo] of points A, B, and C (in Figure 5) for the Launder,Reece, and Rodi model and the SSG model are given below:LRR Model

A = [-0.09,0], B = [0.37,0], C = [0.14,0.167]SSG Model

A = [-0.09, 0], B = [0.53, 0], C = [0.22,0.254].

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The improved performance of the SSG model in rotating shear flow is largely due to thefact that its most energetic state (point C on the bifurcation diagram shown in Figure 5) islocated in close proximity to Q/S = 0.25 - the value predicted by rapid distortion theory.However, it needs to be mentioned at this point that the reason we were not able to satisfythis RDT result exactly is due to a defect in the general hierarchy of models (61). Due to(73) and the fact that the bifurcation diagram is symmetric about its most energetic state(point C in Figure 5), the general hierarchy of models (61) erroneously predicts Richardsonnumber similarity if point C is located at Q/S = 0.25. Such models will yield solutions forK and e that scale with the Richardson number

Ri = -2(Q/S)(1 - 2Q/S) (74)and, thus, erroneously predict that the P/S = 0 and P/S = 0.5 cases are identical. Large-eddy simulations, RDT (Bertoglio 1982), and independent mathematical analyses of theNavier-Stokes equations (Speziale and Mac Giolla Mhuiris 1989b) indicate that the P/S = 0and 2/S = 0.5 cases are distinct. By moving the most energetic state a small distance tothe left of P/S = 0.25 - as is done with the SSG model - the proper growth rates obtainedfrom large-eddy simulations for Q/S = 0 and Q/S = 0.5 can be reproduced (see Figures 3(a)and 3(c)). However, the substantially larger growth rate for Q/S = 0.25 shown in Figure3(b) (which has independent support in the RDT calculations of Bertoglio 1982), cannot bereproduced by the SSG model. This is a defect in the general hierarchy of models (61) that isintimately tied to their prediction of universal equilibrium values for IIo, III , (b33), and(o/e)oo in plane homogeneous turbulent flows - an oversimplification that is not supportedby physical or numerical experiments.t Nonetheless, despite this deficiency, the SSG modelperforms reasonably well - and is superior to other existing second-order closures - forrotating shear flow as evidenced by Table 1, Figures 3(a) - (c), and Figures 4(a) - (c).

Now, we will examine the performance of the SSG model in homogeneous plane strainturbulence for which the mean velocity gradients are given by

0 -S 0 75 )0 0 0and the turbulence evolves from an initial state of isotropy. Comparisons of the model predic-tions will be made with the direct numerical simulations oi Lee and Reynolds (1985) on planestrain. Such comparisons must be made with caution due to the low turbulence Reynoldsnumbers of the direct simulations. However, comparisons with physical experiments (e.g.,

tit is not possible to tie this deficiency to the modeled c-transport equation since all dependence on ccan be eliminated in the RDT limit.18


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Tucker and Reynolds 1968 and Gence and Mathieu 1980) are equally problematical due tothe uncertainty in the initial conditions for c/SK and possible large-scale contaminationfrom the walls of the test apparatus.

In Figure 6, the time evolution of the turbulent kinetic energy for the LRR model and SSGmodel are compared with the direct simulations of Lee and Reynolds (1985) for plane straincorresponding to the initial condition co/SKo = 2.0. It is clear from this figure that bothmodels perform extremely well. In Figure 7, the time evolution of the non-zero componentsof the anisotropy tensor re shown. Although the quantitative accuracy of the models isnot extremely good, it is clear that the SSG model does better than the LRR model andreproduces the crucial trends of the direct simulations. In Figures 8 - 9, the time evolutionof the turbulent kinetic energy and non-zero components of the anisotropy tensor are showncorresponding to the initial condition of eo/SKo = 1.0. The same conclusions can be drawnfrom these results: the SSG model yields improved predictions over the LRR model and,on balance, compares reasonably well with the direct simulations which would be expectedto have somewhat elevated anisotropies due to the lower turbulence Reynolds number. Wewill not make more extensive comparisons with the predictions of other turbulence modelssince our main purpose here was to simply establish that the alterations made in the LRRmodel - to yield the SSG model with its improved behavior in rotating shear flows - do notcompromise its performance in plane strain.

Finally, we will examine the performance of the SSG model for the axisymmetric contrac-tion and expansion in homogeneous turbulence. Since the SSG model (like virtually all otherexisting models for the pressure-strain correlation) was calibrated based on plane homoge-neous turbulent flows, it would be desirable to assess its performance in a three-dimensionalflow. For the axisymmetric contraction, the mean velocity gradients are given by

O- I 0 -iS 0 (76)o9x (60 0whereas in the axisymmetric expansion they take the form

_-v ( - S 0 0 )i ) 77 )o 2The time evolution of each of these turbulent flows - from an initially isotropic state - willbe considered. Hence, as with plane shear and plane strain, the solutions will only dependon the initial conditions through the parameter eo/SKo. Comparisons will be made withthe predictions of the LRR model and the direct numerical simulations of Lee and Reynolds(1985) for the same reasons as cited above.

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In Figure 10, the time evolution of the turbulent kinetic energy for the axisymmetriccontraction is shown corresponding to the initial condition ro/SKo = 0.179 which was takenfrom the direct simulations of Lee and Reynolds (1985). From these results, it is clear thatthe SSG model yields noticeably improved predictions over the LRR model; however, bothmodels predict growth rates that are smaller than those in the direct simulations. In Figure11, the time evolution of the nonzero components of the anisotropy tensor are shown for theaxisyrmnetric contraction where eo/SKo = 0.179. While the differences between the SSGand LRR models is small, it is clear that the SSG model yields results that are more in linewith the direct simulations.

In Figure 12, the time evolution of the turbulent kinetic energy for the axisymmetricexpansion is shown for the initial condition Eo/SKo = 2.45 taken from the direct numericalsimulations of Lee and Reynolds (1985). It is clear from this figure that both the SSGand LRR models yield results that are in excellent agreement with the direct simulations.However, the time evolution of the nonzero components of the anisotropy tensor shown inFigure 13 show more significant differences. Here, the predictions of the SSG model appearto be substantially better than those of the LRR model.

6. CONCLUSIONSThe modeling of the pressure-strain correlation of turbulence has been considered based

on invariance arguments and a dynamical systems approach. Several important conclusionshave been drawn about the standard hierarchy of closures (61) which led to the developmentof a new model - the SSG model. A summary of these findings can be given as follows:

(i) It was proven that the standard hierarchy of models yields non-trivial values for theequilibrium states Il., III. (b33)., and (P/e) that are universal (i.e., that do notdepend on w/S, Q/S or the initial conditions) for plane homogeneous turbulent flows.As a direct consequence of these universal invariants, it was shown that, for planehomogeneous turbulent flows, the general model (61) for the pressure-strain correlationis topologically equivalent to a substantially simpler model - the SSG model - whichis only quadratically nonlinear in the anisotropy tensor.

(ii) The SSG model was calibrated by using existing data from isotropic decay experiments,return to isotropy experiments, and homogeneous shear flow experiments along withthe RDT results of Crow (1968) and Bertoglio (1982). By means of this more systematicmethod of calibration, the SSG model was demonstrated to perform better than theLaunder, Reece, and Rodi model - as well as the newer models of Shih and Lumley

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and Fu, Launder, and Tselepidakis - for a variety of homogeneous turbulent flows.The flows that were examined included the challenging test case of rotating shear flow(where rotations can either stabilize or destabilize the flow) and the axisymmetricexpansion/contraction which constitutes a three-dimensional mean turbulent flow.

(iii) Although the SSG model performs reasonably well for a variety of homogeneous tur-bulent flows, there are still major deficiencies with it that are intrinsic to this generalhierarchy of models. These deficiencies, emanate from the prediction of universal equi-librium values for Ioo, III , (b33 )oo and (P/e).. in plane homogeneous turbulent flows- an obvious oversimplification that is not supported by physical experiments. As aresult of this deficiency, the general model (61) erroneously predicts that rotating shearflow has growth rates that are symmetric about their most energetic value. Hence, inorder to satisfy the RDT constraint of Bertoglio (1982) - which puts the most energeticstate at Q/S = 0.25 - the models must exhibit Richardson number similarity. This isinconsistent with the Navier-Stokes equations as proven by Speziale and Mac GiollaMhuiris (1989b) and illustrated by Bardina, Ferziger, and Reynolds (1983).

(iv) Since the general model (61) for the pressure-strain correlation gives an incompletepicture of plane homogeneous turbulent flows no matter what form is taken for A-j(b)and MAijt(b), we feel that the process of adding highly complex nonlinear terms in b13is somewhat questionable. Such complex nonlinear terms in the rapid pressure-straincorrelation have been largely motivated by the desire to satisfy the strong form ofrealizability. However, it must be remembered that the strong form of this constraintonly constitutes a sufficient condition for the satisfaction of realizability in homoge-neous turbulent flows. Due to the relatively simple topological structure of the generalmodel in rotating shear flow - which is in no way altered by the addition of morecomplex nonlinearities in bij - the application of the strong form of realizability eitherremoves the degrees of freedom necessary to properly calibrate the model or leads tostiff behavior.

Despite the deficiencies discussed above, the SSG model seems to perform moderatelywell in a variety of homogeneous turbulent flows as documented in this paper. While furtherimprovements would be most welcome, we feel that it is unlikely that they will come from thestandard hierarchy of models (61). Fundamentally new approaches will be needed. Futureresearch will be directed on two fronts. The SSG model will be implemented in a fullsecond-order closure for the computation of a variety of complex aerodynamic flows that areof technological interest. We believe that when the SSG model is used within the framework

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of a sound second-order closure, it may be possible to obtain acceptable engineering answersfor a range of turbulent shear flows with streamline curvature. In parallel with this effort,entirely new directions in modeling the pressure-strain correlation will be pursued. Thesewill involve the introduction of a tensor length scale - to better account for anisotropies -and the possible solution of a transport equation for Mijkl to account for history effects inthe rapid pressure-strain. A closer examination of these issues will be the subject of a futurepaper.

ACKNOWLEDGEMENTThe authors would like to thank Dr. Nessan Mac Giolla Mhuiris for some helpful com-

ments on the dynamical systems aspects of this paper.

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ReferencesBardina, J., Ferziger, J. H., and Reynolds, W. C. 1983 Improved turbulence models based

on large-eddy simulation of homogeneous, incompressible turbulent flows. Stanford Uni-versity Technical Report TF-19.

Bertoglio, J. P. 1982 Homogeneous turbulent field within a rotating frame. AIAA J. 20,1175.

Choi, K. S. and Lumley, J. L. 1984 Return to isotropy of homogeneous turbulence revisited.Turbulence and Chaotic Phenomena in Fluids (T. Tatsumi, ed.), p. 267, North Holland.

Crow, S. C. 1968 Viscoelastic properties of fine-grained incompressible turbulence. J. FluidMech. 41, 81.

Fu, S., Launder, B. E., and Tselepidakis, D. P. 1987 Accommodating the effects of highstrain rates in modelling the pressure-strain correlation. UMIST MechanicalEngineeringDepartment Report TFD/87/5.

Gence, J. N. and Mathieu, J. 1980 The return to isotropy of a homogeneous turbulencehaving been submitted to two successive plane strains. J. Fluid Mech. 101, 555.

Hanjalic, K. and Launder, B. E. 1972 A Reynolds stress model of turbulence and its appli-cation to thin shear flows. J. Fluid Mech. 52, 609.

Haworth, D. C. and Pope, S. B. 1986 A generalized Langevin model for turbulent flows.Phys. Fluids 29, 387.

Launder, B. E., Reece, G., and Rodi, W. 1975 Progress in the development of a Reynoldsstress turbulence closure. J. Fluid Mech., 68, 537.

Launder, B. E., Tselepidakis, D. P., and Younis, B. A. 1987 A second-moment closure studyof rotating channel flow. J. Fluid Mech. 183, 63.

Lee, M. J. and Reynolds, W. C. 1985 Numerical experiments on the structure of homogeneousturbulence. Stanford University Technical Report TF-24.

Le Penven, L., Gence, J. N., and Comte-Bellot, G. 1985 On the approach to isotropy ofhomogeneous turbulence: Effect of the partition of kinetic energy amo.g the velocitycomponents. Frontiers in Fluid Mechanics (S. H. Davis and J. L. Lumley, eds.), p. 1,Springer-Verlag.

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Lezius, D. K. and Johnston, J. P. 1976 Roll-cell instabilities in rotating laminar and turbulentchannel flow. J. Fluid Mech. 77, 153.

Lumley, J. L. 1978 Turbulence modeling. Adv. Appl. Mech. 18, 123.Mellor, G. L. and Herring, H. J. 1973 A survey of mean turbulent field closure models. AIAA

J. 11, 590.Reynolds, W. C. 1987 Fundamentals of turbulence for turbulence modeling and simulation.

Lecture Notes for Von Karman Institute, A GARD Lecture Series No. 86, North AtlanticTreaty Organization.

Reynolds, W. C. 1988 Private Communication.Rotta, J. C. 1951 Statistische Theorie Nichthomogener Turbulenz. Z. Phys. 129, 547.Rotta, J. C. 1972 Recent attempts to develop a generally applicable calculation method forturbulent shear flow layers. AGARD-CP-93, North Atlantic Treaty Organization.Sarkar, S. and Speziale, C. G. 1989 A simple nonlinear model for the return to isotropy in

turbulence. Phys. Fluids A (in press).Shih, T-H. and Lumley, J. L. 1985 Modeling of pressure correlation terms in Reynolds stress

and scalar flux equations. Technical Report No. FDA-85-3, Cornell University.Smith, G. F. 1971 On isotropic functions of symmetric tensors, skew-symmetric tensors and

vectors. Int. J. Engng. Sci. 9, 899.Speziale, C. G. 1983 Closure models for rotating two-dimensional turbulence. Geophys.

Astrophys. Fluid Dynamics 23, 69.Speziale, C. G. 1987 Second-order closure models for rotating turbulent flows. Quart. Appl.

Math. 45, 721.Speziale, C. G. 1989a Turbulence modeling in non-inertial frames of reference. Theoret.

Comput. Fluid Dynamics 1, 3.Speziale, C. G. 1989b Discussion of turbulence modeling: Past and future. Lecture Notes inPhysics (in press).Speziale, C. G., Gatski, T. B., and Mac Giolla Mhuiris, N. 1989 A critical comparison of

turbulence models for homogeneous shear flows in a rotating frame. Proceedings of theSeventh Symposium on Turbulent Shear Flows, Vol. 2, p. 27.3.1, Stanford UniversityPress.

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Speziale, C. G. and Mac Giolla Mhuiris, N. 1989a On the prediction of equilibrium states inhomogeneous turbulence. J. Fluid Mech. 209, 591.

Speziale, C. G. and Mac Giolla Mhuiris, N. 1989b Scaling laws for homogeneous turbulentshear flows in a rotating frame. Phys. Fluids A 1, 294.

Tavoularis, S. and Corrsin, S. 1981 Experiments in nearly homogeneous turbulent shear flowswith a uniform mean temperature gradient. Part I. J. Fluid Mech. 104, 311.

Tucker, H. J. and Reynolds, A. J. 1968 The distortion of turbulence by irrotational planestrain. J. Fluid Mech. 32, 657.

Weinstock, J. 1981 Theory of the pressure-strain rate correlation for Reynolds stress turbu-lence closures. Part 1. J. Fluid Mech. 105, 369.

Weinstock, J. 1982 Theory of the pressure-strain rate. Part 2. J. Fluid Mech. 116, 1.Yakhot, V. and Orszag, S. A. 1988 Private Communication.

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APPENDIX A

Consider the tensor functionF,3= F1j(b, S , ). (Al)

Form invariance under coordinate transformations (28) requires that Fj be of the form (seeSmith 1971):

F = aoI + alb+ a 2b 2 + a 39 + Q4S 2

+a 5 (bS + 9 b) + a 6 (b 2S ' + S b2)+a7(bS '2 + g 2b) + as(b 2g, 2 + 9'2b2)+a9 b --- - 4b) + ajolb- (A2)+all(b 2 ' - -b 2 ) + a 12(- b 2 - -W2 b)+ai 3(sw - '4') + i4Z S.=, /_+ ,--, ,--,2 - a,21 7 -,+a5g2 -- 4-g/ 2 ) + ai6-W -4231-1) + -- 42

where a depend on all possible invariants, i.e.,a, = a,(II, II, r - 2 , tr -g12,tr ,tr bS , trbS

tr~~~~~~~Sr2 tr b9S r ',$rb , r A3)

tr W 2 tr W,1,r w S w , tr b-S - , tr bStr bw tr bS U; I= 0, 1),2,...) 17.

Taking the linear part of Fiin S ,3nd zj yields the equationF(L) = oI + Olb+02b 2 + 031 + 04(b9'

+31b) + 05(b2g + 9 b2) (A4)+3 (b- - -b) + 037b 2 ' - --b

where1o /oo(II, III) + /Ol(II, III)tr(b 9') + o2(II, III)tr(b -S') (A5)01 = io(II, III) + ,311(II, III)tr(b .-') 2(II, III)tr(b2 S') (A6)

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,62 = 320(11, III) + 0321(11, III)tr(b. S') + 022(I1, III)tr(b2 (A7)/3, = (II, III), i = 3,4,... ,7. (a8)

Then, by taking the deviatoric part of F(j ) (since HIij is traceless) and multiplying by e weobtain Hi. :

-F(L)Si.) (a)~j= (Fij kc 3.Equation (29) results when (A4)- (A8) are substituted into (A) and the identities inequation (27) are made use of.

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APPENDIX B

The pressure-strain models of Shih and Lumnley (1985) and Fu, Launder, and Tselepidakis(1987) take the following form:Shih-Lumley Model

Hi = ebi + 4+ 8a 5) K39i,

7 em1 2i ~_ ek~3~ /M3 n 13

+ (2 + 16) [rik a-v- + emjkfi2m)(B1)

+Tk(L + emil~m) 2Tmn~n~]6- 4+ 5 mn3mnb ... 1 fiA:kWk + TrikWikJ

+ik[pi + emejim)

73 k 7 + embilm) ] TkL 9whr -- (1 + .8V'F), F = 1 + 911 + 27111 (B2)10 111 = -- bijbi, III = -1ibAbi(B3)23

=3 2 + 1-F{80.1n[1 + 62.4(-11 + 2.3111)]} (B4)9

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Fu, Launder, and Tselepidakis Model=-C, rvFi [2b1, + 4-y (blkbk3 bmnbmn~iJ

-0.6 7*iic( a- . + emkilZm)

2 Ovm

+1.2Tmav ij (B5)i~iScax, 2rkrjkTtk Tik v mli3m) + k vi+ em i

+rfl611(7ikWjk + 7TjkWik)-2 4bmibnl(TrmkWnk; + rnTnk)l

whereCj*= -6011, -y=0.6, r-0.7. (B6)

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LIST OF FIGURES

Figure 1. Phase space of the return to isotropy problem: Comparison of the predic-tions of the SSG model and LRR model with the plane strain experimentof Choi and Lumley (1984).Figure 2. Time evolution of the anisotropy tensor during the return to isotropy. Com-

parison of the predictions of the SSG model and LRR model with experi-ments: (a) the plane strain experiment of Choi and Lumley (1984), and (b)the plane contraction experiment of Le Penven, Gence, and Comte-Bellot(1985).

Figure 3. Time evolution of the turbulent kinetic energy in rotating shear flows.Comparison of the predictions of the SSG model and- LRR model withthe large-eddy simulation of Bardina, Ferziger, and Reynolds (1983) forco/SKo = 0.296: (a) P/S = 0, (b) Pl/S = 0.25, and (c) P/S = 0.5.

Figure 4. Comparison of the models with large-eddy simulations (LES) of rotatingshear flow for P/S = 0,0.25 and 0.5: (a) LES of Bardina, Ferziger, an dReynolds (1983), (b) FLT model, and (c) SSG model.

Figure 5. Bifurcation diagram for rotating shear flow.Figure 6. Time evolution of the turbulent kinetic energy in plane strain for E0 /SK 02.0. Comparison of the predictions of the LRR model and SSG model with

the direct simulations of Lee and Reynolds (1985).Figure 7. Time evolution of the anisotropy tensor in plane strain for eo/SKo = 2.0.

Comparison of the predictions of the LRR model and SSG model with thedirect simulations of Lee and Reynolds (1985).Figure 8. Time evolution of the turbulent kinetic energy in plane strain for -o/SKo =

1.0. Comparison of the predictions of the LRR model and SSG model withthe direct simulations of Lee and Reynolds (1985).

Figure 9. Time evolution of the anisotropy tensor in plane strain for co/SKo = 1.0.Comparison of the predictions of the LRR model and SSG model with thedirect simulations of Lee and Reynolds (1985).

Figure 10. Time evolution of the turbulent kinetic energy in the axisymmetric con-traction for eo/SKo = 0.179. Comparison of the predictions of the LR Rmodel and SSG model with the direct simulations of Lee and Reynolds(1985).

Figure 11. Time evolution of the anisotropy tensor in the axisymmetric contractionfor o/SKo = 0.179. Comparison of the predictions of the LRR model andSSG model with the direct simulations of Lee and Reynolds (1985).

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Figure 12. Time evolution of the turbulent kinetic energy in the axisymmetric expan-sion for eo/SKo = 2.45. Comparison of the predictions of the LRR modeland SSG model with the direct simulations of Lee and Reynolds (1985).Figure 13. Time evolution of the anisotropy tensor in the axisymmetric expansion for

eo/SKo = 2.45. Comparison of the predictions of the LRR model and SSGmodel with the direct simulations of Lee and Reynolds (1985).

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EquilibriumValues LRR Model SSG Model Experiments(bll)oo 0.193 0.204 0.201(b22)_ -0.096 -0.148 -0.147(bl2) o -0.185 -0.156 -0.150

(SK/E)oo 5.65 5.98 6.08

Table 1. Comparison of the predictions of the Launder, Reece andRodi (LRR) model and the Speziale, Sarkar and Gatski (SSG) modelwith the experiments of Tavoularis and Corrsin (1981) on homogene-ous turbulent shear flow.

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LRR Model - - SSG Model -Experimental data 0

RETURN TO ISOTROPY0.20- =IIIJ/3 /

//

0.15 - 0l / /

0

0.10-

0.0 0.02 0.04 0.06 0.08B 0.10

Figure 1. Phase space of the return to isotropy problem: Comparison of the predic-tionftheSSGmodel and LRR model with the plane strain experimentof oiandLumley (1984).

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LRR Model - - SC Model -(a) Experimental data 0

RETURN TO ISOTROPY0.20

0.15,0

000 0

b~0.10 _ bilo~o5- -- z0.05-

0.100 b22

-0.15

-0.20 00.0 0.2 0.4 0.6 0.8 1.0--

Figure 2. Time evolution of the anisotropy tensor during the return to isotropy. Com-parison of the predictions of the SSG model and LRR model with experi-ments: (a) the plane strain experiment of Choi and Lumley (1984), and (b)the plane contraction experiment of Le Penven, Gence, and Comte-Bellot(1985).

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LRR Model SSG Model -(b) Experimental data 0RETURN TO ISOTROPY0.20

0 0 615 0 00.10

0 05

-0.2220 00 0 0 0 0 0 0

_ b33-0.15

-0.20 -LI0.00 0. 0 0.20 0.30 0.40T

Figure 2. Time evolution of the anisotropy tensor during the return to isotropy. Com-parison of the predictions of the SSG model and LRR model with experi-ments: (a) the plane strain experiment of Choi and Lumley (1984), and (b)the plane contraction experiment of Le Penven, Gence, and Comte-Bellot(1985).

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LRR Model SSG Modela) Simulations 0

4.0 ROTATING SHEAR FLOW0 S 0. 03.0-2.5 //

2.0 ///2.5 / /

1.5-1.0

0.5

0.0 I0 2 4 6 8 10t

Figure 3. Time evolution of the turbulent kinetic energy in rotating shear flows.Comparison of the predictions of the SSG model and LRR model withthe large-eddy simulation of Bardina, Ferziger, and Reynolds (1983) foreo/SKo = 0.296: (a) Q/S = 0, b) 2/S = 0.25, and (c) 2/S = 0.5.

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LRR Model -- SSC Model

(b) SI'mulations 04.0 ROTATING SHEAR FLOW

3.5- ~ ~ DS =0 24.0 /

3.5 0 /3.0 - 0

02.5 - //

001-.0 /

0 j /

1.0 0Q 0

0.50.0 I0 2 4 6 8 10t*

Figure 3. Time evolution of the turbulent kinetic energy in rotating shear flows.Comparison of the predictions of the SSG model and LRR model withthe large-eddy simulation of Bardina, Ferziger, and Reynolds (1983) forEo/SKo = 0.296: (a) IS = 0, (b) PIS = 0.25, and (c) 2/S = 0.5.

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LRR Model SSG Model

c) S irulatl*ons 04.0 ROTATING SHEAR FLOWo/ S 0.503.53.02.5

12.01.5

.00 0000000.5 -0.0 1 I

0 2 4 6 8 10I*Figure 3. Time evolution of the turbulent kinetic energy in rotating shear flows.Comparison of the predictions of the SSG model and LRR model with

the large-eddy simulation of Bardina, Ferziger, and Reynolds (1983) foreo/SKo = 0.296: (a) i/S = 0, (b) 2/S = 0.25, and (c) i/S = 0.5.

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ROTATING SHEAR FLOW0 Q/S = 0.00a 21S = 0.25A IS 0.50

4.03.5 LES

3.02.5 -

0 0000UO

000. 0 , o00.50.0

0 2 4 6 8 10t

Figure 4. Comparison of the models with large-eddy simulations (LES) of rotatingshear flnw for Q/S = 0, 0.25 and 0.5: (a) LES of Bardina, Ferziger, an dReynolds (1983), (b) FLT model, and (c) SSG model

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ROTATING SHEAR FLOW/S - 0.0(b) 2502o~5 S 0.504.03.5 FLT Model

3.02.5

1.5-1.0 -0.5-

0.0-0 2 4 6 8 10t*

F guz 4. Comparison of the models with large-eddy simulations (LES) of rotatingshear flow for ?/S = 0, 0.25 and 0.5: (a) LES of Bardina, Ferziger, andReynolds (1983), (b) FLT model, and (c) SSG model.

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ROTATING SHEAR FLOW(c) - 0.00(c) -- /F2S .25D/S = 0.50

4.0 /3.5 SSG Model //3.0 ///2.5 //

/,--2.0-

1.5-/.............

1 .0 ..... ...... .. ...1.0

0.50.0

0 2 4 6 8 10tFigure 4. Comparison of the models with large-eddy simulations (LES) of rotatingshear flow for 02/S = 0,0.25 and 0.5: (a) LES of Bardina, Ferziger, and

Reynolds (1983), (b) FLT model, and (c) SSG model.

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Bifutrccationu Diatg ram

ROTATING SHEAR FLOW

8

Q/S

Figure 5. Bifurcation diagram for rotating shear flow.

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LRR Model -- SSC ModelSrimulcatiorts 0

4.0 PLANE S TRAIN

3.53.02.5

,-2.01.5

1.0.0.5 0 . . .__-0.0 I I I0.0 0.4 0.8 1.2 1.6 2.0*

Figure 6. Time evolution of the turbulent kinetic energy in plane strain for Co/SKo =2.0. Comparison of the predictions of the LRR model and SSG model withthe direct simulations of Lee and Reynolds (1985).

43

EWE|EEEE


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L .R Model - - SSU, ModelS'mulat io nas 0

0.4 PLANE STRAIN

0.3 0

0.20 - -sz_ 0 o -0

0.10 .01

C

-0.30.4 I

0.0 0.4 0.8 1.2 1.6 2.0tFigure 7. Time evolution of the anisotropy tensor in plane strain for eo/SKo = 2.0.Comparison of the predictions of the LRR model and SSG model with thedirect simulations of Lee and Reynolds (1985).

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1,RR AM d[ SY'C Model,,Si'ru L or 0

4.0 SITRANI'ItAIN2.53.02.5

,-2.0-

1.51. 00.50.0

0.0 0.4 0.8 1.2 1.6 2.0

Figure 8. Time evolution of the turbulent kinetic energy in plane strain for co/SKo =1.0. Comparison of the predictions of the LRR model and SSG model withthe direct simulations of Lee and Reynolds (1985).

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L R RMo,1,l 1 SSG Mode[ -Siriuilations 0

0.4 PLANE STRAIN

0.3

0 .2 - ....0.1

, .. 0 .0 ----------- o b33

-0.1-0.2 ..oo-0.3-0.4 I0.0 0.4 0.8 1.2 1.6 2.0

tFigure 9. Time evolution of the anisotropy tensor in plane strain for Co/SKo = 1.0.Comparison of the predictions of the LRR model and SSG model with the

direct simulations of Lee and Reynolds (1985).

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LRR Model -- SSG ModelSimulations 0

2.0 AXISYMMETRIC CONTRACTION

1.8-

1.2

0.8-0.6-0.40.20.0

0.0 0.4 0.8 1.2 1.6 2.0t*

Figure 10. Time evolution of the turbulent kinetic energy in the axisymmetric con-traction for eo/SKo = 0.179. Comparison of the predictions of the LRRmodel and SSG model with the direct simulations of Lee and Reynolds(1985).

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0.4 AXISYMMETRIC CONTRACTION

0.3

0.2-0 b22, b33

0.10.0(

-0.2 -

0

-0.3 - 00.42 1

0.0 0.4 0.8 1.2 1.6 2.0t

Figure 11. Time evolution of the anisotropy tensor in the axisymmetric contractionfor co/SKo = 0.179. Comparison of the predictions of the LRR model andSSG model with the direct simulations of Lee and Reynolds (1985).

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LRR Model SSG ModelSimulations 0

2.0 AXISYMMETRIC EXPANSION1.81.6

1.41.21.00.80.6-0.4-0.200 I I0 I I

0.0 0.4 0.8 1.2 1.6 2.0t*

Figure 12. Time evolution of the turbulent kinetic energy in the axisynmetric expan-sion for co/SKo = 2.45. Comparison of the predictions of the LRR modeland SSG model with the direct simulations of Lee and Reynolds (1985).

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0.4 AXISYMMETRIC EXPANSION

0.3-0.2

0.1-0.1

~-0.0(

-0.1 b2 3-0.2-0.3

0.0 0.4 0.8 1.2 1.6 2.0t

Figure 13. Time evolution of the anisotropy tensor in the axisymmetric expansion foreo/SKo = 2.45. Comparison of the predictions of the LRR model and SSGmodel with the direct simulations of Lee and Reynolds (1985).

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NASA Report Documentation Page1. eport No. I 2. Government Accesson No. 3.Recipient's Catalog NoNASA CR-181979ICASE Report No. 90-5 _

4. Titlend Subtitle Report Date

MODELING THE PRESSURE-STRAIN CORRELATION OF January 1990TURBULENCE -- AN INVARIANT DYNAMICAL SYSTEMS 6. Performing Organization codeAPPROACH

7.Authoris) 8. Performing Organization Report No.Charles G. Speziale 90-5Sutanu SarkarThomas B. Gatski 10. Work Unit No.

505-90-21-019. erforming Organization Name and AddressInstitute for Computer Applications in Science 11. ontract or Grant No.and Engineering IMail Stop 132C, NASA Langley Research Center NASl-18605

Hampton, VA 23665-5225 13. Type of eport and Period Covered12. Sponsoring Agency Name and Address Contractor ReportNational Aeronautics and Space AdministrationLangley Research Center 14 Sponsoring Agency Code

Hampton, VA 23665-522515. Supplementary Notes

Langley Technical Monitor: To be submitted to the JournalRichard W. Barnwell of Fluid Mechanics

Final Report16. bstract

The modeling of the pressure-strain correlation of turbulence is examined from a basic theoreti-cal standpoint with a view toward developing improved second-order closure models. Invariance con-siderations along with elementary dynamical systems theory are used in the analysis of the standardI ierarchy of closure models. In these commonly used models, the pressurer.strain correlation isassumed to be a linear function of the mean velocity gradients with coefficients that depend alge-braically on the anisotropy tensor. It is proven that for plane homogeneous turbulent flows theequilibrium structure of this hierarchy of models is encapsulated by a relatively simple modelwhich is only quadratically nonlinear in the anisotropy tensor. This new quadratic model - theSSG model - is shown to outperform the Launder, Reece, and Rodi model (as well as rcore recentmodels that have a considerably more complex nonlinear structure in a variety of homogeneous tur-bulent flows. However, some deficiencies still remain for the description of rotating turbulentshear flows that are intrinsic to this general hierarchy of models and, hence, cannot be overcome bythe mere introduction of more complex nonlinearities. It is thus argued that the recent trend ofadding substantially more complex nonlinear terms containing the anisotropy tensor may by of ques-tionable value in the modeling of the pressure-strain correlation. Possible alternative approachesare discussed briefly.

17 Key Words (Suggested by Authoris)) 18. Distnbution StatementTurbulence Modeling; Second-Order Closure

Models; Pressure-Strain Correlation; 34 - Fluid Mechanics and Heat TransferHomogeneous Turbulence, .

modeling the pressure-strain correlation of turbulence - an invariant dynamical systems approach

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