A Galilean invariant explicit algebraic Reynolds stress model for turbulent curved flowsSharath S. Girimaji Citation: Physics of Fluids (1994-present) 9, 1067 (1997); doi: 10.1063/1.869200 View online: http://dx.doi.org/10.1063/1.869200 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/9/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A realizable explicit algebraic Reynolds stress model for compressible turbulent flow with significant meandilatation Phys. Fluids 25, 105112 (2013); 10.1063/1.4825282 Nonlocal analysis of the Reynolds stress in turbulent shear flow Phys. Fluids 17, 115102 (2005); 10.1063/1.2130749 Laminar and turbulent dissipation in shear flow with suction AIP Conf. Proc. 502, 497 (2000); 10.1063/1.1302427 A dynamical model for turbulence. IX. Reynolds stresses for shear-driven flows Phys. Fluids 11, 678 (1999); 10.1063/1.869939 A dynamical model for turbulence. VIII. IR and UV Reynolds stress spectra for shear-driven flows Phys. Fluids 11, 665 (1999); 10.1063/1.869938

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A Galilean invariant explicit algebraic Reynolds stress model for turbulentcurved flows

Sharath S. GirimajiInstitute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton,Virginia 23681

~Received 1 May 1996; accepted 19 November 1996!

A Galilean invariant weak-equilibrium turbulence hypothesis that is sensitive to streamlinecurvature is proposed. The hypothesis leads to a fully explicit algebraic expression for Reynoldsstress in terms of the mean velocity field and kinetic energy and dissipation of turbulence. Themodel is tested in curved homogeneous shear flow which is a homogeneous idealization of thecircular streamline flow. The agreement is excellent with Reynolds stress closure model andadequate with available experimental data.@S1070-6631~97!02703-7#

I. INTRODUCTION

Many turbulent flows of practical importance are char-acterized by curved streamlines: e.g., flow through turbines,curved pipes and channels, flows over wing sections, andswirling flows. The qualitative effects of streamline curva-ture on turbulence are known from direct numerical simula-tions, experiments, and analytical studies. When angular mo-mentum decreases with radius, which is typically the case inflows with concave streamlines, Taylor–Go¨rtler instabilitysets in, causing increased velocity fluctuations and an aug-mentation of turbulence. When angular momentum increaseswith radial distance, flows with convex streamlines typicallyfall into this category, the Tollmien–Schlichting wavespresent in the flow are attenuated, resulting in a decrease ofturbulence. The quantitative effect of streamline curvature onturbulence is generally more profound than can be explainedby a simple scaling analysis.1–4 Therefore, accurate model-ing of the effect of streamline curvature on turbulence is veryimportant.

Currently, the full Reynolds stress closure approach of-fers the most accurate means of calculating all types of tur-bulent flows. Despite advances in computer capabilities, theReynolds stress closure method is still computationally tooexpensive for many engineering calculations of complexflows. Current engineering calculations employ, at best, two-equation turbulence models. At the two-equation level of tur-bulence modeling, algebraic Reynolds stress models derivedfrom the Reynolds stress transport equation offer the mostsophistication. The algebraic model is derived by subjectingthe anisotropy transport equation to the weak-equilibrium as-sumption. Therefore, the algebraic model inherits all theweaknesses of the Reynolds stress closure model as well asthe limitations imposed by the weak equilibrium assumption.The main weakness of the standard Reynolds stress closuremodel stems from inadequate modeling of the pressure–strain correlation. In addition, due to the weak-equilibriumhypothesis, the algebraic model is unsuitable for flows whereadvection by the mean flow and turbulent transport dominatethe evolution of Reynolds stress. Despite these limitations,there is a wide range of turbulent flows for which the alge-braic Reynolds stress model is quite appropriate. In fact, inslowly evolving turbulent rectilinear flows, which form an

important class of engineering flows, the algebraic Reynoldsstress model is the solution of the Reynolds stress transportequation. In these flows, the algebraic Reynolds stress modeloffers the accuracy of the Reynolds stress closure approachat a much smaller computational expense.

The extension of the algebraic modeling methodologyfor turbulent flows with streamline curvature has so far notbeen as successful. One of the reasons for the lack of successhas been the inability to formulate a good weak-equilibriumassumption appropriate for curved flows. The weak-equilibrium assumption that is used for rectilinear flows isinconsistent with the long-time behavior of the Reynoldsstress transport equation. On the other hand, a widely usedassumption that yields consistency with the transport equa-tion long-time behavior violates the Galilean invariance re-quirement. The development of a Galilean invariant, explicitalgebraic Reynolds stress model that is the formal long-timesolution to the transport equation in slowly evolving curvedflows would constitute an important contribution and such isthe intent of this paper.

A. Algebraic modeling of rectilinear flows

In homogeneous turbulence, the Reynolds stress and dis-sipation transport equations constitute an initial value prob-lem. The kinetic energy and dissipation evolve from theirinitial value, and in the case of energetic turbulence, theygrow unbounded. The Reynolds stress anisotropy

bi j[uiuj2K

22

3d i j , ~1!

on the other hand, tends to a finite-valued fixed point orequilibrium state at large times. The fixed point of the an-isotropy transport equation is described by

]bi j]t

1Uk~bi j ! ,k'0, ~2!

where,uiuj is the Reynolds stress,K [ 12uiuj is the turbulent

kinetic energy,di j is the Kronecker delta and the comma inthe subscript indicates total derivative~including Christoffelsymbol terms! with respect to the index direction.~Through-

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out this paper, an upper case symbol indicates the mean of aquantity and lower case represents the fluctuation from themean.!

In slowly evolving flows, the turbulence can be assumedto be close to its fixed point or equilibrium state. This as-sumption@Eq. ~2!#, called the weak or structural equilibriumassumption, leads to a nonlinear algebraic fixed point equa-tion of the Reynolds stress anisotropy. The objective of al-gebraic turbulence modeling methodology is to determinethe fixed point solution~or the equilibrium state! of the an-isotropy transport equation and use it as an approximationfor time-evolved value of anisotropy. The algebraic Rey-nolds stress models can be classified as implicit or explicitdepending upon how the fixed point solution is sought. Theimplicit models5–7 solve the nonlinear equations numericallyin an iterative fashion which could~i! be computationallyexpensive and~ii ! converge to non-physical real roots lead-ing to serious errors in the calculations. The explicit alge-braic model of Gatski and Speziale8 is an analytical solutionof the algebraic equations linearized about the energeticequilibrium state of homogeneous turbulence. While thismodel works quite well near energetic equilibrium, whenused away from this equilibrium~as most practical modelsinvariably are! it leads to inconsistency between the assumedequilibrium value of the production to dissipation~P/«! ratioand the derived value of the Reynolds stress anisotropy.9 Thealgebraic model of Girimaji9 is obtained by analytically solv-ing the fixed-point equations for anisotropy in their fullynonlinear form; this model is both fully explicitand self-consistent. This model expression is the exact fixed pointsolution~i.e., in the weak-equilibrium limit! of the Reynoldsstress transport equation in two-dimensional mean flows fora variety of quasilinear pressure–strain correlation models.On the whole, the algebraic Reynolds stress methodologyhas been reasonably satisfactory in flows with nearly straightstreamlines.

B. Algebraic modeling of curved flows

The first step towards developing an algebraic Reynoldsstress model for curved flows is the formulation of a weak-equilibrium assumption that exhibits~i! correct fixed pointbehavior, i.e., consistency with the long-time behavior ofReynolds stress closure model in homogeneous curved flow,and ~ii ! Galilean invariance. The second step is the deriva-tion of an algebraic model expression for the Reynolds stressfrom the fixed point equation. It will now be demonstratedthat the weak equilibrium assumptions currently used areunsuitable for curved flows.

Long-time behavior of Reynolds stress closure model. Ina homogeneous curved flow with circular streamlines, it iseasy to demonstrate that the Reynolds stress closure model,computed in polar or circular coordinates, leads to equilib-rium value for anisotropy tensor components at long times.This fact has also been observed experimentally by Hollo-way and Tavoularis.10 In this flow, the Reynolds stress anis-tropy is independent of the downstream~angular! location,but the kinetic energy increases or decreases monotonicallydepending on whether the curvature is destabilizing or oth-erwise. In the circular channel flow experiment of Eskinazi

and Yeh,11 the turbulence structure (bi j ) changes consider-ably at the onset of curvature, but gradually the rate ofchange slows down as the anisotropy becomes axisymmetricat long times. From this evidence, the long-time behavior ofthe Reynolds stress anisotropy can be summarized as fol-lows:

]bi js

]t1Uk

s]bi j

s

]xks '0, ~3!

wherexis is a circular streamline coordinate. For these flows,

it is easily seen that the rectilinear weak-equilibrium simpli-fication @Eq. ~2!# is inconsistent with the above observation@Eq. ~3!#. The inconsistency is most easily seen in thestreamline coordinate system where the rectilinear assump-tion implies

]bi js

]t1Uk

s]bi j

s

]xks 52Christoffel symbol termsÞ0. ~4!

Therefore, the rectilinear weak-equilibrium assumption~andthe ensuing algebraic Reynolds stress model! is not suited forcurved turbulent flows.

Streamline weak-equilibrium assumption for curvedflows.It appears from the preceding that, in curved flows, thecoordinate system used to express the components of theanisotropy tensor is important. One cannot expect thebi jcomponents expressed in the Cartesian coordinates to be in-variant following a streamline due to the effect of curvature.Intuitively, it would appear thatbi j components expressed inthe local streamline coordinates should be invariant follow-ing a fluid particle. In fact, experiments and computations ofReynolds stress closure model show this to be the case forcircular streamlines. Motivated by this, some authors~e.g.,Ref. 12! have hypothesized that, in curved flows, the anisot-ropy following a streamline is constant@Eq. ~3!#. Thisstreamline weak-equilibrium assumption is then used to re-duce the anisotropy transport equation to a set of nonlinearalgebraic equations. The resulting ARSM has yielded some-what improved performance for circular flows. However, thestreamline weak-equilibrium assumption@Eq. ~3!# dependson the streamline direction, which in turn depends on thefluid velocity. In general, the fluid velocity is not Galileaninvariant. As a consequence, the streamline direction and,therefore, the streamline weak-equilibrium assumption is notGalilean invariant.13–15 Fu et al. suggest that the Galileanvariance may be the reason for the poor performance ofARSM in some of their curved flow computations.

Present work.Our objective in this paper is to extend theapplicability of algebraic Reynolds stress modeling to curvedflows by including additional physics in a Galilean invariantmanner. First, we hypothesize a new weak-equilibrium as-sumption that is~i! Galilean invariant,~ii ! properly sensitiveto streamline curvature, and~iii ! equivalent to the Rodi as-sumption in flows with rectilinear streamlines. Then we de-rive an algebraic model expression for the Reynolds stressanisotropy that is~i! fully explicit, ~ii ! self-consistent,~iii !Galilean invariant, and~iv! sensitive to streamline curvature.

The remainder of the paper is organized as follows. Thenew weak-equilibrium assumption is proposed in Sec. II.

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The Reynolds stress transport equation is reduced to a set ofnonlinear algebraic equations in Sec. III and an algebraicsolution is obtained. The model validation is performed inSec. IV and we conclude in Sec. V.

II. GALILEAN INVARIANT WEAK-EQUILIBRIUMASSUMPTION

Invoking the weak-equilibrium assumption in the stream-line coordinate system (ei

s) is ill-advised because the basis ofthe streamline coordinate system is the velocity vector whichis not Galilean invariant. Unlike velocity, the accelerationvector (d/dt)U is Galilean invariant. If the weak equilibriumassumption is made in a coordinate system that is definedusing the acceleration vector, a Galilean invariant model willresult. The derivation given in this section employs threecoordinate systems: acceleration coordinate systemei

a,streamline coordinate systemei

s, and an arbitrary computa-tional coordinate systemei . The acceleration and the stream-line coordinate systems are denoted by superscriptsa ands,respectively.

Theacceleration coordinate systemeia is defined as fol-

lows. Leta be the unit vector in the direction of acceleration„(d/dt)U…. Let e1

a be alonga and lete2a be orthogonal toa on

the plane ofa. Finally, definee3a normal to the plane of

a-field ~unit binormal vector! completing an orthogonalright-handed system. In general, the streamline and the ac-celeration coordinate systems are different. However, whenthe mean fluid trajectory is circular the acceleration~radialdirection! is perpendicular to the velocity~tangential direc-tion!: the unit vectors of the acceleration and the streamlinecoordinate systems coincide. Thecomputational coordinatesystemis an arbitrary orthogonal reference system in whichthe model calculations are performed. Our objective is toderive the final form of the model in the computational co-ordinate system. If a flow with circular streamlines is com-puted in a cylindrical coordinate system, the acceleration,streamline, and computational coordinate systems will coin-cide.

A. The new weak-equilibrium hypothesis

If the streamlines are perfectly circular, then theeia co-

ordinate system will coincide with the streamline coordinatesystem, leading to

]bi ja

]t1Ul

a]bi j

a

]xla 5

]bi js

]t1Uk

s]bi j

s

]xks 50. ~5!

When the fluid motion is noncircular in the computationalcoordinate systemei ~i.e., velocity and acceleration are notorthogonal!, we can perform a Galilean transformation to anew moving coordinate frame in which the fluid accelerationis unchanged~since it is Galilean invariant! while the direc-tion of the fluid velocity changes. If the reference frame ve-locity is chosen appropriately, the fluid velocity and accel-eration can be made orthogonal to one another. Clearly, thisframe transformation is a function of space and time. In thisnew coordinate frame, the fluid motion is locally circular.Locally, the radial, tangential, and axial directions of thiscircular flow are thee1

a, e2a, ande3

a directions, respectively.

Based upon our knowledge of circular flows@Eq. ~5!#, wenow hypothesize that along these coordinate directions theanisotropy of Reynolds stress is invariant, leading to

]bi ja

]t1Ul

a]bi j

a

]xla '0. ~6!

Let us investigate how the above hypothesis satisfies thethree required criteria listed in the Introduction.

~1! The hypothesis depends only upon the direction of theacceleration vector and, hence, is Galilean invariant.

~2! For circular flows, it is clear@Eq. ~5!# that anisotropy isinvariant following the streamline.

~3! In the absence of streamline curvature, the Lagrangianvelocity is unidirectional; the direction of acceleration isinvariant. The acceleration coordinate system can betaken to be a Cartesian coordinate system. Then, thepresent simplification@Eq. ~6!# reduces to the Rodiweak-equilibrium assumption~Criterion 3!.

For these reasons, the hypothesis given in Eq.~6! appears tobe a reasonably sound foundation for building an algebraicReynolds stress model for general curved flows.

Hypothesis in the computational coordinate system.Computations are seldom performed in the acceleration co-ordinate systemei

a. For the new weak-equilibrium assump-tion to be useful, it is necessary to express Eq.~6! in thearbitrary orthogonal computational coordinate system~x,t!whose unit vectors areei . If Tip is the transformation matrixbetween the acceleration and the computational coordinatesystems, we have the following relationships:

eia5Tipep , ep5Tipei

a , Uia5TipUp ,

~7!bi ja5TipTjqbpq , TipTjp5d i j .

The elements of the transformation matrix are known oncethe acceleration vector is known in theei coordinate system.If the mean flow is two-dimensional~many flows of practicalinterest fall into this category!, Tip can be easily calculated.Letting e1 and e2 be the coordinates along which the flowvaries ande3 be the homogeneous direction, we have

e1a5a5a1e11a2e2 ,

e2a52a2e11a1e2 , ~8!

e3a5e3 ,

wherea is the unit vector in the direction of acceleration. Forthis case,T is given by

Ti j5S a1 a2 0

2a2 a1 0

0 0 1D .

The anisotropy tensor can now be expressed as

b[bi jaei

aeja[bi eiej . ~9!

Invoking the new weak-equilibrium assumption in Eq.~6!,we can write

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]b

]t1Ul

a ]b

]xla 5F ]bi j

a

]t1Ul

a]bi j

a

]xla Geiaeja

1bi ja F ]ei

aeja

]t1Ul

a]ei

aeja

]xla G

'bi ja F ]ei

aeja

]t1Ul

a]ei

aeja

]xla G

5bi ja F ]ei

aeja

]t1Ul

]eiaej

a

]xlG . ~10!

The invariance property ofUl(]/]xl) is used to write the lastequality in the above equation.

Using Eq.~7! we can write

eiaej

a5TipTjqepeq . ~11!

Define the temporal and spatial derivatives of the computa-tional coordinate system as follows:

Fpq5]ep]t

~• !eq , Gplq5]ep]xl

~• !eq , ~12!

where~•! denotes inner product. Like the Christoffel symbol,G is a coupling function. Now we can write

]eiaej

a

]t1Ul

]eiaej

a

]xl

5TipTjqF]epeq]t1Ul

]epeq]xl

G1epeqF ]

]t~TipTjq!

1Ul

]

]xl~TipTjq!G

5epeqFTirTjq~Frp1UlG rlp !1TipTjr ~Frq1UlG rlq !

1d

dt~TipTjq!G , ~13!

whered/dt5(]/]t)1Ul(]/]xl).By employing Eq.~13! and the following identities,

bi ja Tir Tjq5brq , bi j

a TipTjr5bpr , bi ja5brsTir Tjs ,

~14!

the weak-equilibrium approximation@Eq. ~10!# can now bewritten in the computational coordinate system:

]b

]t1Ul

]b

]xl5epeqFbrq~Frp1UlG rlp !1bpr~Frq1UlG rlq !

1brsTir TjsS Tjq

dTipdt

1TipdTjqdt D G . ~15!

In component form, the weak-equilibrium approximation inthe computational coordinate system is

]bpq]t

1Ul~bpq! ,l

5brq~Frp1UlTrlp !1bpr~Frq1UlG rlq !

1brsTir TjsS Tjq

dTipdt

1TipdTjqdt D

5bprV rq8 1bqrV rp8 . ~16!

In the above equations, the axisymmetric tensorV rs8 ~whichwill be called vorticity modification due to curvature! isgiven by

V rs8 5Frs1UlG rls1TqrdTqsdt

. ~17!

In this section, we have developed a new weak equilib-rium assumption@Eq. ~16!# for curved flows. This assump-tion hypothesizes that, following a fluid particle, the anisot-ropy tensor is merely rotated due to streamline curvaturewithout any change in magnitude. The rate of rotation of theanisotropy tensor depends upon the unsteadiness of the com-putational system~via Frs!, curvature of the coordinate sys-tem ~via UlG rls!, and the changing direction of the meanacceleration vector [(d/dt)Trs]. If the computational coordi-nate system is time invariant, thenFrs50. If the computa-tional coordinate system is Cartesian, thenGrls50. If themean flow streamline is perfectly circular and the streamlinecoordinate system is used, thenTrs5d rs , leading tod/dt(Trs)50.

B. Present versus streamline weak-equilibriumassumption

Apart from the fact that the present assumption is Gal-ilean invariant whereas the streamline weak-equilibrium as-sumption is not, there are important physical differences be-tween the two models. It can be shown that the vorticitymodification due to the streamline weak-equilibrium@Eq.~3!# assumption is

V i j8 ~streamline!5UlG i l j . ~18!

For steady circular flows being computed in the cylindricalstreamline coordinate system, the two weak-equilibrium as-sumptions are equivalent. However, the streamline weak-equilibrium model cannot account for the unsteadiness orvarying direction of the acceleration. The latter feature isvery important in steady curved~noncircular! flows. For ex-ample, in the transition from a straight to a curved channelturbulent flow~or vice versa! the term containingTi j is likelyto be very important. One of the most important deficienciesof the streamline weak-equilibrium model is its inability tocapture the recovery of turbulence from imposed streamlinecurvature.12 The previous curved flow models have also beenshown to be inadequate in swirling flows. In these cases, thepresent model can be expected to perform better.

When a circular flow is computed in a Cartesian coordi-nate system,

Frs5G lrs50. ~19!

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If the present weak equilibrium assumption is used, the ef-fect of curvature is still brought to bear on the computationsvia the remaining term:

V rs8 5TqrdTqsdt

. ~20!

III. MODEL DEVELOPMENT

In this section, the new weak-equilibrium assumption isemployed to develop an explicit algebraic Reynolds stressmodel that is appropriate for curved flows.

The exact Reynolds stress transport equation in an arbi-trary inertial reference frame is given by

]uiuj]t

1Uk~uiuj ! ,k1~uiujuk! ,k5Pi j1« i j1f i j1D i j .

~21!

The terms, respectively, are the time rate of change, advec-tion, turbulent transport, production (Pi j ), dissipation~«i j !,pressure–strain correlation~fi j !, and pressure–viscous diffu-sion ~D i j ! of the Reynolds stress:

Pi j52uiukU j ,k2ujukUi ,k , « i j52nui ,kuj ,k~22!

D i j5@2puid j l2pujd i l1nuiuj ,l # ,l .

The production and dissipation rate of turbulent kinetic en-ergy are, respectively,P5 1

2Pii and«512«i i . The dissipation

rate tensor can be split into its isotropic and deviatoric parts:« i j5

23«d i j1di j . The transport equation for the anisotropy

tensor in nondimensional time is derived from Eq.~21! ~seeRef. 9!:

]bi j]t8

1Ul~bi j ! ,l1bi j S P« 21D52 2

3 Si j2~bikSk j1Sikbk j223 bmnSmnd i j !

2~bikv jk1bjkv ik!1 12 P i j* . ~23!

In the above equation the following normalizations havebeen effected using the eddy turnover time:

]t85«

K]t, Si j5

1

2

K

«~Ui , j1Uj ,i !,

~24!

v i j51

2

K

«~Ui , j2Uj ,i !, P i j*5

1

«~f i j2di j !.

Invoking the weak equilibrium assumption@Eq. ~16!#, theanisotropy transport equation in the computational coordi-nate system can be reduced to the following set of nonlinearalgebraic equations:

bi j S P« 21D522

3Si j2S bikSk j1Sikbk j2

2

3bmnSmnd i j D

2@bik~v jk2V jk!1bjk~v ik2V ik!#

1 12 P i j* , ~25!

where

V rs5K

«V rs8 . ~26!

We consider the following type of quasilinear pressure–strain model~that includes all linear models!

P i j*52SC101C1

1 P

« Dbi j1C2Si j1C3~bikSjk1bjkSik

2 23 bmnSmnd i j !1C4~bikv jk1bjkv ik!, ~27!

where theCs are numerical constants. Many of the currentpressure–strain correlation models are special cases of Eq.~27! near weak equilibrium. Using this model, the nonlinearset of simultaneous equations~25! for the anisotropy compo-nents is written in the following compact form:

bi j @L102L1

1bmnSmn#

5L2Si j1L3~bikSk j1Sikbk j223bmnSmnd i j !

1L4~bikWjk1bjkWik!, ~28!

where

L105

C10

221, L1

15C1112, L25

C2

222

3,

~29!

L35C3

221, L45

C4

221.

The total effective vorticityWij is given by

Wij5v i j12

C422V i j . ~30!

This implies that the effect of streamline curvature is tomodify the flow vorticity and, hence,Vi j is called the vor-ticity modification tensor. The effect of coordinate frame ro-tation also appears via a modified vorticity term@Ref. 8, Eq.~26!#. However, there is a major difference between the twophenomena: whereas solid body rotation of the coordinateframe modifies flow vorticity by the same amount every-where, the modification due to streamline curvature can varywith space and time.

The objective now is to solve Eq.~28! for bi j . Thesolution—in terms of the constantsL, strain rateSi j , andeffective vorticityWij—will produce the algebraic Reynoldsstress model.

A. Fully explicit solution

The set of nonlinear algebraic simultaneous equationsfor the anisotropy of Reynolds stress has the same form@Eq.~28!# with or without streamline curvature. The only differ-ence in the two cases would be the lack of vorticity modifi-cation in the rectilinear streamline case. For the case ofstraight streamlines, the solution of Eq.~28! has been derivedby Girimaji.9 The solution procedure presented in Girimaji9

departs from those previously given in the literature by treat-ing the algebraic equations in their full nonlinear form. Thefinal solution, following a brief description of the derivationprocedure, is now given.

Representation theory provides the most general tenso-rial form of the anisotropy tensor in terms of the strain and

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rotation rate tensors. For three-dimensional flows, the func-tional form is too cumbersome to be of practical value.8 It iscustomary to restrict consideration to the more tractable caseof two-dimensional mean flows and use the resultant func-tional form of the Reynolds stress expression as a model inthree-dimensional flows. For two-dimensional mean flows,the general admissible representation of the anisotropy tensoris given by

bi j5G1Si j1G2~SikWkj2WikSk j!

1G3~SikSk j213SmnSmnd i j !. ~31!

~The tensorWikWkj , which is also permitted by representa-tion theory, is not admitted for it is inconsistent with knownphysics, see Ref. 8 for details.! The unknown coefficients~eddy viscosities!, G12G3 , are functions of the constants inthe pressure–strain model and the invariants of the strain androtation rate tensors. In incompressible flows, these invari-ants are

h15Si jSi j , h25WijWi j . ~32!

It can be easily shown that the coefficientG1 is related to theturbulent eddy viscosity coefficient,Cm , commonly used inK-« modeling:

Cm52G1 . ~33!

The objective now is to determine the unknown coeffi-cients by ensuring that the representation of anisotropy inEq. ~31! satisfies Eq.~28!. Equation~28! is nonlinear and hasmultiple roots. As a result, as demonstrated in Ref. 9,G1 hasmultiple representations. The only physically meaningful so-lution of Eq.~28! is selected by requiring thatG1 be ~i! real,~ii ! a continuous function of its parametersh1 andh2, and~iii ! consistent with known physical behavior:9

G15

{L10L2 /@~L1

0!212h2~L4!2#, for h150;

L10L2 /@~L1

0!22 23h1~L3!

212h2~L4!2#, for L1

150;

2p

31S 2

b

21AD D 1/31S 2

b

22AD D 1/3, for D.0;

2p

312A2a

3cosS u

3D , for D,0 and b,0;

2p

312A2a

3cosS u

312p

3 D ,for D,0 and b.0.

~34!

The various quantities in the above equation are given by

p[22L1

0

h1L11 , r[2

L10L2

~h1L11!2

,

q[1

~h1L11!2

F ~L10!21h1L11L22

2

3h1~L3!

2

12h2~L4!2G , ~35!

a[S q2p2

3 D , b[1

27~2p329pq127r !,

D5b2

41a3

27, cos~u!5

2b/2

A2a3/27.

The other two coefficientsG2 andG3 can be calculated asfollows:

G252L4G1

L012h1L1

1G1, G35

2L3G1

L012h1L1

1G1. ~36!

The eddy viscosity coefficientCm([2G1) is a well-behaved and nonsingular function and is plotted as a functionof h1 andh2 in Fig. 1. The value ofCm typically used inK-«modeling is 0.09. It is seen from this figure that for moderatevalues ofh1 andh2 the present model predicts a value ofCm

close to 0.09. At very high values of strain or rotation rate,theCm value tends to zero. For a given strain rate~h1!, Cm

decreases with increasing rotation rate~h2!. The two othereddy viscosity coefficients are also well-behaved functionsof h1 andh2.

9

The expression for the anisotropy of Reynolds stress isfully explicit since the production to dissipation ratio@ap-pearing asbmnSmn in Eq. ~28!# is not treated quasistaticallyas done by Rodi,5 Pope,6 or Taulbee.7 Also, this expression isself-consistent even far from equilibrium since Eq.~28! issolved in its fully nonlinear form, rather than by linearizingthe equation about the equilibrium value as done by Gatskiand Speziale.8 The advantages of the current fully explicitmodel over previous models are discussed in Ref. 9.

IV. DISCUSSION AND COMPARISON WITHEXPERIMENT

In this section, we first discuss the validity of the alge-braic Reynolds stress methodology in general curved inho-mogeneous flows. Then, the explicit algebraic Reynoldsstress model is examined in the simplest of curved flows—the curved homogeneous shear flow. The model evaluation isperformed in two parts: comparison with other models and

FIG. 1. Cm as a function ofh1 andh2.

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validation against the experimental data of Holloway andTavoularis.10 The testing of the model in more complexcurved flows has been deferred to the future.

A. Validity in inhomogeneous flows

The derivation presented in the previous section is fullyvalid only in homogeneous flows. In inhomogeneous flows,upon the invocation of the weak-equilibrium assumption, theReynolds stress anisotropy evolution equation is still not al-gebraic due to the presence of the viscous and turbulenttransport terms. Therefore the algebraic Reynolds stressmethodology is formally valid only when the transport termsare negligible and the anisotropy evolution is completely de-termined by the local processes of production, pressure–strain correlation, and dissipation. In high Reynolds numberflows, the viscous transport is generally negligible. We esti-mate the importance of the turbulent transport in some typi-cal curved flows using the experimental data of Mucket al.3

and Hoffmanet al.,4 and the direct numerical simulation dataof Moser and Moin.16

Convex curvature.Turbulent kinetic energy and shearstress budgets measured in convexly curved boundary layersindicate that, due to the stabilizing effect the ‘‘turbulenttransport~by triple products and pressure fluctuations! andadvection are both considerably decreased by the applicationof curvature.’’3 Throughout the convexly curved boundarylayer, the production and dissipation are much larger in mag-nitude than the other terms and nearly balance each othercompletely. In the curved channel flow of Moser and Moin,16

in the convex portion, the production, dissipation, andpressure–strain correlation of normal stresses are muchlarger than the turbulent transport. The shear stress evolutionis predominantly determined by the balance between produc-tion and pressure–strain redistribution. It appears that theneglect of turbulent transport is a better approximation inconvex flows than in straight flows. Therefore, the algebraicReynolds stress methodology is at least as valid in convexflows as it is in straight flows.

Concave flows.The concave boundary layer data ofHoffman et al.4 and the curved channel flow data of Moserand Moin16 indicate that the turbulent transport terms arelarger in concave flows than in straight flows due to thedestabilizing effects. However, except at the very edge of theconcave boundary layer, the production, dissipation, andpressure–strain redistribution continue to be larger than theturbulent transport terms. Therefore, the algebraic Reynoldsstress methodology in concave flows is still adequate, even ifit is less valid than in convex flows.

In the near wall region, for up to 20 wall units, the struc-ture of the velocity field in convex and concave flows areidentical to that of a straight flow.3,4,16 In this region, al-though turbulent transport is large, the production and dissi-pation are extremely large and dominate the flow dynamics.This appears to indicate that the near-wall treatment incurved flow calculations can be the same as in straight flows.

B. Model validation

For the sake of simplicity, in this paper, we restrict themodel testing to curved homogeneous shear flow which isinitially isotropic. The model is validated against other mod-els and experimental data.

Description of test case.The test case is what is tradi-tionally called the curved homogeneous shear flow,10 whichreally is the homogeneous idealization of a flow with circularstreamlines. Any curved shear flow is characterized by theshear rate and the curvature factor defined, respectively, as

S85]U

]r, Cf5SUr D Y S ]U

]r D . ~37!

In these definitions,U is the tangential velocity andr is theradial coordinate. IfCf andS8 are constants, we then have ahomogeneous curved shear flow.

Consider a curved flow~in cylindrical coordinates! con-fined betweenRi ~inner radius! andRo ~outer radius!. Let thedistance between the inner and outer radius be much largerthan the turbulence length scales, but small compared to ei-ther one of the radii:

R5Ri1Ro

2@~Ro2Ri !@L, ~38!

whereL is the integral length of the turbulence. Since thewidth of the flow (Ro2Ri) is much larger than the turbu-lence length scale, the flow in the central region of the an-nulus can be considered to be unaffected by the boundariesof this flow. Let tangential velocity of this flow be given by

U5v0r1~r2R!g, ~39!

wherev0 andg are constants andr is the radial coordinate.From the above equations we have

U

r5v01S 12

R

r Dg'v05const

~40!]U

]r5v01g5const.

Clearly this flow can be approximated as a circular flowwhere the radius of curvature of the streamline isR and theshear rate and curvature factors are constants given by

S85~v01g!, Cf5v0

v01g. ~41!

The model validation is performed in this idealized flow.In the cylindrical coordinate system, if the indices 1, 2,

and 3 correspond to the radial, tangential, and axial direc-tions, the only nonzero Christoffel terms are

G12251

r, G21152

1

r. ~42!

The nondimensional mean strain rate and vorticity are givenby

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Si j51

2S~12Cf !S 0 1 0

1 0 0

0 0 0D ,

~43!

v i j51

2S~11Cf !S 0 21 0

1 0 0

0 0 0D ,

whereS5S8K/«.~As an aside let us briefly consider the case of solid body

rotation. In this case,g5O leading toS85v0 and Cf51.Then we have

Si j5S 0 0 0

0 0 0

0 0 0D , v i j5

K

« S 0 2v0 0

v0 0 0

0 0 0D . ~44!

Clearly, in a solid body rotation, a fluid particle undergoespure rotation with no straining.!

Back to the general case, the vorticity modification ten-sor is given by

V i j52

C422

U

r S 0 1 0

21 0 0

0 0 0D . ~45!

By adding Eq.~45! to the mean flow vorticity tensor in Eq.~43! we obtain the nondimensional total vorticity for the gen-eral case,

Wij5S

2@11Cf1MCf #S 0 21 0

1 0 0

0 0 0D , ~46!

whereM is the vorticity modification factor due to stream-line curvature and is given by

M524

C422. ~47!

In all of the major pressure–strain correlation models, thevalue ofC4 is less than two. Therefore, when the streamlinecurvature is convex~Cf.O!, the effect of curvature is to

augment the vorticity. As was shown in the previous section,for a given strain rate~h1!, an increase in vorticity~h2! re-sults in a decrease inCm . This decrease inCm is responsiblefor the additional inhibition of turbulence over that occurringdue to reduced production~caused by diminished strain ratedue to convex curvature!. For concave curvature~Cf,O! thetotal vorticity is lower, resulting in a higherCm , and conse-quently higher turbulence levels. In Fig. 2,Cm is plotted as afunction of the curvature factor (Cf) for a given value ofS56 which corresponds to its equilibrium value in rectilinearhomogeneous shear flow. In this calculation ofCm and all theresults presented below, the linearized version of thepressure–strain correlation model of Spezialeet al.17 is used.It is seen from Fig. 2 that the eddy viscosity is a strongerfunction of convex than concave curvature. This is consistentwith the observation of Mucket al.3 who found that turbu-lence reacts more quickly to convex than concave curvature.

Equation~47! can be used as a constraint in specifyingC4 in the pressure–strain correlation model. In order for thepressure strain model to produce physically consistent resultsin curved flows, we will need

C4,2. ~48!

If this constraint is not satisfied, the model will lead to re-duced vorticity for convex case~resulting in enhanced turbu-lence! and increased vorticity in concave flows~diminishingturbulence! inconsistent with known physics of curved flows.As mentioned earlier, all the commonly used pressure–straincorrelation models satisfy this constraint.

Comparison with other models.Algebraic Reynoldsstress models provide an inexpensive alternative to Reynoldsstress closure models~RSCM! at the cost of lesser physicalaccuracy. The closer an algebraic model calculation gets toRSCM, the better the algebraic model. Here, we evaluate thefollowing explicit algebraic models against RSCM calcula-tions: ~i! the present curved ARSM based on the new weak-equilibrium assumption, ARSM~curved!; ~ii ! the ARSM ofGirimaji9 based on the Rodi-weak equilibrium assumption,ARSM~G!; ~iii ! the ARSM of Gatski and Speziale, AR-SM~GS!; and ~iv! the standardK-« model.

The results presented in this subsection are calculated asfollows. Turbulence, which is initially isotropic~at timet50!, is suddenly subjected at timet501 to nonzero valuesof Cf andS. Calculations using the various models are per-formed starting at timet50. The kinetic energy and dissipa-tion needed as inputs to the model are calculated using thefollowing standard equations:

dK

dt5P2«,

~49!d«

dt5

«

K~C«1P2C«2«!.

The values used forC«1 andC«2 are 1.44 and 1.83, respec-tively. The initial values ofK and « are taken to be unity.The pressure strain correlation model constants used are~fol-lowing Gatski and Speziale8!

FIG. 2. Cm as a function ofCf in curved homogeneous shear flow.S5(]U/]r )(K/«)56.0.

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C1053.4, C1

151.8, C250.36,~50!

C351.25, C450.40.

In Fig. 3, we compare the computations from the variousmodels in the stabilizing curvature case~Cf50.15 and initialS52.0!. The anisotropy componentsb11 and b12 are com-pared in Fig. 3~a!. The RSCM anisotropy values, calculatedusing evolution equations, exhibit gradual growth~withslight oscillations! from zero to equilibrium values ofb1150.188 andb12520.0116. In rectilinear shear flow, theequilibrium value ofb12 is approximately20.157. There-fore, in the present case the turbulent shear stress level ismuch smaller, indicating a suppression of turbulence produc-tion. All the other models, where the anisotropy is calculatedfrom algebraic equations, exhibit a sudden jump in anisot-ropy value at the onset of curvature~t501!, followed bygradual evolution to equilibrium values. These models areincapable of calculating the intial transients and should beevaluated based on their ability to capture the long-time be-havior. The ARSM~curved!, after a short initial phase~St'2!, reproduces the behavior of RSCM very accurately. Theother models do not reproduce the long-time RSCM resultswell at all. The performances of ARSM~G! and ARSM~GS!are somewhat better than that of theK-« model. TheK-«model is, of course, insensitive to curvature and indicates nosuppression of turbulence due to stabilizing curvature. TheP/« ratio predicted by the various models is shown in Fig. 3.The RSCM and ARSM~curved! are again indistinguishably

close after the initial phase. TheP/« at the end of the simu-lation is about 0.312~as compared to 1.88 for rectilinearhomogeneous shear turbulence!, clearly implying a suppres-sion of turbulence. The other models are not close to RSCMand generally indicate a lesser degree of turbulence suppres-sion.

In Fig. 4, the comparison in the destabilizing curvaturecase~Cf520.15 and initialS56.0! is presented. The RSCMcalculations show a clear augmentation of turbulence re-flected by higher values ofP/« and ub12u. Again, the presentARSM ~curved! captures RSCM behavior extremely well.The other models do not duplicate the RSCM behavior asaccurately.

From the above figures it is quite clear that the presentARSM ~curved! is an excellent alternative to RSCM forcurved homogeneous shear flows. The other algebraic mod-els, while not too bad for destabilizing curvature~energeticturbulence case!, perform poorly for stabilizing curvature.

Validation against experimental data. The ARSM~curved! is now validated against the curved homogeneousshear flow data of Holloway and Tavoularis.10 The experi-mental equilibrium values ofb12 andP/« ratio are comparedagainst ARSM~curved! calculations for various curvaturefactors. In order to isolate the present model evaluation fromthe well-known shortcomings of the dissipation equation~seeRef. 17!, the experimental value of dissipation is used tonormalize the shear rate. The normalized shear rate values

FIG. 3. Comparison between Reynolds stress closure model and other mod-els in the curved homogeneous shear flow caseCf50.15: — RSCM; --curved ARSM; –-– ARSM~G!; –– ARSM~GS!; and ——K-« model.

FIG. 4. Comparison between Reynolds stress closure model and other mod-els in the curved homogeneous shear flow caseCf50.15. ~Legend same asFig. 3.!

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taken from the experiment for various curvature factors aregiven in Table I.

Comparison is performed only for those cases which at-tain structural equilibrium in the experiment. Calculationsfrom ARSM~GS! and the standardK-« model are also pre-sented. Figure 5 shows the comparison of shear stress anisot-ropy, b12. As is to be expected, the experimental data showincreased shear stress magnitudes for negative curvature fac-tors ~destabilizing curvature! and diminished values for posi-tive curvature factors~stabilizing curvature!. On the whole,the ARSM ~curved! captures this variation as a function ofcurvature reasonably well: it appears to slightly overpredictthe shear stress levels for destabilizing curvatures and sup-press the shear stress a little more than indicated by data forstabilizing curvatures. The standardK-« model, on the otherhand, predicts much higher levels of shear stress for all cur-vatures. The prediction of ARSM~GS! is the least sensitiveto curvature. It predicts anisotropy reasonably well in theconcave curvature case in which the actual production todissipation ratio is close to the universal equilibrium valueassumed by the model. However, ARSM~GS! does not pre-dict the reduction~in magnitude! of the shear stress anisot-ropy in the convex curvature cases. Recall that in the convexcurvature case, the production is much diminished and hencethe actualP/« ratio is much smaller than the assumed uni-versal equilibrium value.

One of the key issues that needs to be predicted withaccuracy in curved turbulent flows is the relaminarizationeffect induced by stabilizing curvature. In a plane homoge-neous shear flow, the turbulence production is greater thandissipation~P'1.8«! resulting in an exponential growth ofturbulent kinetic energy. When the shear flow is subjected toconvex curvature, the production decreases due to reducedlevels of turbulent viscosity and effective strain rate, result-ing in slower growth of the turbulent kinetic energy. Whenthe convex curvature is strong enough, the turbulence pro-duction becomes smaller than dissipation resulting in the de-cay of kinetic energy. The value of the curvature factor atwhich this kinetic energy decay~interpreted as onset ofrelaminarization! begins is of great interest. The equilibriumvalues ofP/« calculated from the various models are com-pared in Fig. 6 against experimental data. Whereas all themodels capture the correct trend of decreasing productionwith increasing curvature factor, the ARSM~curved! valuesare closer to the experimental data over the entire range ofcurvatures. Again the ARSM~GS! values are quite close todata for the concave curvature cases, but poor for the convexcurvature flows.

For a fully developed homogeneous curved shear flowwith circular streamlines, the weak-equilibrium assumptioninvoked in this paper is an exact statement. Since the flow ishomogeneous, the neglect of the turbulent transport term in

FIG. 5. Comparison between experimental data, ARSM, and standardK-«model: anisotropy of shear stressb12.

FIG. 6. Comparison between experimental data, ARSM, and standardK-«model: production-to-dissipation ratio.

TABLE I. Table of parameters.

Cases

1 2 3 4 5 6 7 8 9 10 11 12

S 4.7 4.6 4.26 5.85 6.54 5.66 4.58 3.89 3.14 4.17 3.63 3.10Cf 20.033 20.039 20.066 20.079 20.093 20.150 0.032 0.040 0.076 0.078 0.10 0.18

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the ARSM methodology is also completely valid. This begsthe question, why is there still a distinct discrepancy betweenthe model and experimental data? If the experimental resultsare reliable, the lack of complete agreement between the ex-periment and ARSM~curved! must be due to the pressure–strain correlation model. The pressure–strain correlationmodels have generally been optimized for non curved homo-geneous flows. If one were to use a nonlinear pressure–strainmodel, a fully explicit, self-consistent ARSM may not bepossible. This calls for the development of quasilinearpressure–strain correlation models@of the form given in Eq.~27!# that are optimized for curved homogeneous flows.~Afully nonlinear model derived specifically for rotating andcurved flows is presented in Ref. 18.!

V. CONCLUSION

A new weak-equilibrium hypothesis for flows withstreamline curvature has been proposed and developed. Thishypothesis ensures Galilean invariance and is also consistentwith known physics of circular flows. In the absence ofstreamline curvature, the hypothesis reduces to the Rodiweak-equilibrium assumption. Employing the new assump-tion, an explicit and self-consistent algebraic Reynolds stressmodel is derived from the Reynolds stress transport equation.This model expression is the analytical fixed point solutionof the anisotropy evolution equation in two-dimensionalflows for a variety of quasilinear pressure–strain correlationmodels. The algebraic model computations are validatedagainst Reynolds stress closure model~RSCM! computationsand experimental data~of Holloway and Tavoularis10! incurved homogeneous shear flows. The agreement is excellentwith RSCM and adequate with experimental data.

ACKNOWLEDGMENT

This research was supported by the National Aeronauticsand Space Administration under NASA Contract No. NAS1-19480 while the author was in residence at the Institute for

Computer Applications in Science and Engineering~ICASE!.

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