nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows

*Corresponding author. Tel.: 001-757-864-5552; fax: 001-757-864-8816.E-mail address: [email protected] (T.B. Gatski).

Progress in Aerospace Sciences 36 (2000) 655}682

Nonlinear eddy viscosity and algebraic stress modelsfor solving complex turbulent #ows

T.B. Gatski!,*, T. Jongen"

!NASA Langley Research Center, Hampton, VA 23681-2199, USA"Unilever Research, PO Box 114, 3130 AC Vlaardingen, Netherlands

Abstract

Nonlinear eddy viscosity and algebraic stress models are currently providing an invaluable link between the morecommon linear eddy viscosity turbulence models and the full di!erential Reynolds stress forms. With the increasedpopularity has come an abundance of di!erent formulations. The purpose of this review is to provide a cohesiveframework for the variety of models proposed and to highlight the various similarities and di!erences among the models.Their link with di!erential Reynolds stress models and their improved predictive capability over linear eddy-viscositymodels is also highlighted. ( 2000 Elsevier Science Ltd. All rights reserved.

Contents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6562. Reynolds-averaged equations and single-point closures . . . . . . . . . . . . . . . . . . . . . . . . 658

2.1. Reynolds stress models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6582.2. Linear eddy viscosity models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660

3. General Reynolds stress tensor representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6623.1. Representation of the Reynolds stress anisotropy tensor as a linear combination of basis

tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6623.2. Representation of the Reynold stress anisotropy tensor as a projection on a tensor basis . . . . 6643.3. Applications to "ve- and three-term representations . . . . . . . . . . . . . . . . . . . . . . . 6653.4. Representation of higher-order terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667

4. Nonlinear eddy-viscosity models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6694.1. Examples of nonlinear eddy-viscosity models . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

5. Algebraic stress models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6715.1. Implicit algebraic stress model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6725.2. Explicit representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680Appendix. Fundamental relation for 3]3 tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

0376-0421/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 3 7 6 - 0 4 2 1 ( 0 0 ) 0 0 0 1 2 - 9

Nomenclature

bij, b Reynolds stress anisotropy tensor

Ce1 closure coe$cient for production-of-dissipa-tion rate term

Ce2 closure coe$cient for destruction-of-dissipa-tion rate term

D/Dt material derivative("L/Lt#u6

jL/Lx

j)

Dij

represents the combined e!ect of turbulenttransport and viscous di!usion

e6(i)

eigenvector of strain rate tensoreijk

permutation symbolG(n) Gram matrixI identity matrixL arbitrary tensor containing additional anisot-

ropy e!ectsK turbulent kinetic energy (,q

ii/2)

p@ #uctuating pressurePM , p6 mean pressureQ orthogonal transformation matrixR2 #ow parameter ("!g

2/g

1)

Sij, S mean strain rate tensor (,(Lu6

i/Lx

j#Lu6

j/

Lxi)/2)

T(n) tensor basis elementu@i

#uctuating velocity componentu6i

mean velocity component=

ij, W mean rotation rate tensor in noninertial

frame=H

ij, WH mean absolute rotation rate tensor

WM mean rotation rate tensor intransformed frame

Ws modi"ed mean rotation rate tensor in inertialframe

xi

coordinate direction in inertial(Cartesian) frame (x, y, z)

x6i

coordinate direction in transformedcoordinate frame

X, XM orthogonal transformation matricesan,b

ntensorial expansion coe$cients

ci

model coe$cientsdij

Kronecker deltae isotropic turbulent energy dissipation rategi

scalar invariants of kinematic tensorsji

eigenvalue of strain rate tensorl kinematic viscositylt

turbulent eddy viscosityq turbulent time scale ("K/e)qij, s Reynolds stress tensor (,u@

iu@j)

/ angle between principal axes andCartesian frame

uij

mean rotation rate tensor(,(Lu6

i/Lx

j!Lu6

j/Lx

i)/2)

Xi, X arbitrary time-independent

rotation rate of noninertialframe

SymbolsII

bsecond invariant of Reynoldsstress anisotropy

IIIb

third invariant of Reynolds stress anisotropy

SuperscriptsT transpose of matrix

representation base

SubscriptsR equilibrium value

1. Introduction

The development of turbulent closure models withinthe framework of a Reynolds averaged Navier}Stokes(RANS) approach has been the subject of intense studyand numerous reviews (e.g. [1,2]) over the last threedecades. In the RANS formulation for an incompressible#uid #ow, the mean momentum conservation equationsare closed by a speci"cation of the single-point, secondmoments of the #uctuating velocity "eld, that is, theturbulent Reynolds stress correlation. Over this period ofdevelopment, a signi"cant number of closure schemeshave been proposed which range from simple algebraicspeci"cation of turbulent velocity and length scales to thesolution of full di!erential transport models. The formerbeing classi"ed as algebraic or zero-equation modelsand the latter as Reynolds stress models (RSMs) orsecond-moment closures (SMCs), for example. During

this period, utilization of these single-point correlationmodels has increased signi"cantly and the RANS ap-proach has surpassed the integral method approachas the common engineering tool in numericallypredicting features of turbulent #uid #ows. Unfortu-nately, the rapid proliferation of both models and usershas not necessarily had a totally positive impact on the"eld.

With the appearance of a wide variety of single-pointclosure models, each with the tag of `new and bettera, ithas become less apparent to many how the individualproposals di!er, if at all, from one another. From theusers standpoint, the wide choice has led to confusionabout which is the best model for the particular #ow tobe studied. Such is the consequence of the fact that thereis no single universal model available. Fortunately, somerecent reviews (e.g. [3,4]) on the subject have attemptedto organize and provide a cohesive framework for

656 T.B. Gatski, T. Jongen / Progress in Aerospace Sciences 36 (2000) 655}682

Fig. 1. Hierarchy of single-point turbulent closure models.

the wide variety of turbulence models that have beenproposed.

Nevertheless, the introduction of new lower-orderone- and two-equation models still occurs despite thewell-known de"ciencies associated with such linear eddyviscosity models (LEVMs). The reason for this is theoverall computational robustness of such models and thefact that many turbulent #ows of practical interestare simply turbulent shear stress dominated and as suchonly require a reasonable closure for this one componentof the Reynolds stress tensor. For such shear stress clos-ures, a Boussinesq-type approximation relating the shearstress to the mean velocity gradient is usually su$cient.

Higher-order models, where closures for the pres-sure}strain rate correlation and triple-velocity correla-tion are needed and which require the solution of partialdi!erential equations for the individual components ofthe Reynolds stress tensor, have not been received asenthusiastically as the lower-order models by the usercommunity. Such models for the second moments shouldprovide a better representation of the physics since clos-ure is at the higher moments. In addition, development ofsuch models is more demanding and requires a moredetailed knowledge of the turbulence physics in order todevelop closures which are both mathematically andphysically consistent. For this reason, as well as theweaker demand for such models, fewer alternative RSMformulations currently exist.

Fortunately, an intermediate level of closure is avail-able which retains some aspects of the LEVMs and the

RSMs. This level of closure is founded within the two-equation level of turbulence modeling, but replaces theusual Boussinesq-type eddy viscosity relationship be-tween the Reynolds stress tensor and the mean strain ratewith a higher-order expansion in terms of powers of themean strain rate and rotation rate tensors. In the process,the turbulent eddy viscosity is additionally sensitized toinvariants associated with the mean strain and rotationrate. This alternative representation for the Reynoldsstress couples into the two-equation formulation throughthe production term appearing in the turbulent transportterm. As outlined, such models retain the di!erentialfeatures of a two-equation formulation, but also to somedegree the Reynolds stress anisotropy e!ects associatedwith a full di!erential closure which contains evolutionequations for all the components of the Reynolds stresstensor. In general, this level of closure is referred to asnonlinear eddy viscosity models (NLEVMs). Fig. 1 illus-trates the position the NLEVMs occupy in a hierarchy ofclosure schemes when balanced between cost/complexity(e.g. the number of di!erential transport equations tosolve) and realism/dynamic range (e.g. the number of#ow classes which can be correctly simulated withouthaving to recalibrate the model coe$cients). Within theclass of NLEVMs lies the group of models known asalgebraic stress models (ASM). This group is more close-ly linked to the full di!erential Reynolds stress closure(see Fig. 1) since it formally retains the e!ect of thepressure}strain correlation model contained within thesecond-moment transport equation, and inherits some of

T.B. Gatski, T. Jongen / Progress in Aerospace Sciences 36 (2000) 655}682 657

the most essential features of the behavior of theReynolds stresses in a rigorous way.

The purpose here is to provide the reader with a cohe-sive outline of the development of such models and thevarious advantages and shortcomings of the approachwithin a fairly rigorous mathematical framework. This isbecoming more necessary since even this level of closureis yielding to a variety of newly proposed models whichsuper"cially may appear di!erent but actually retain anunderlying commonality with other proposals. The samegeneral framework will be used to show how turbulencemodels can be developed and assessed in a systematicway. With this motivation, the governing di!erentialequations for the mean and turbulent "elds are presentedin Section 2, along with the development, at the second-moment level, of the transport equations for theReynolds stresses. Although not a central focus of thisarticle, the concept of linear eddy viscosity models ispresented. This level of closure is shown to be a consis-tent subset of the full Reynolds stress closure level and assuch provides the motivation for the extension to morephysically realistic and mathematically complete consti-tutive equations for the turbulent second moments.

Tensor representation theory and its relevance to thedevelopment of algebraic stress models is discussed inSection 3. This discussion details the methodology usedin forming tensor polynomial expansions from a set ofbasis tensors as well as the determination of the corre-sponding expansion coe$cients. Optimal representa-tions are constructed from a subset of the full integritybasis needed to characterize a tensor function such as theReynolds stress anisotropy.

In Section 4 the class of nonlinear eddy viscosity mod-els (NLEVMs) are discussed in the context of tensorrepresentations. The methodology used in the determina-tion of the expansion coe$cients is identi"ed as theprimary characteristic distinguishing the class of nonlin-ear eddy viscosity models from the sub-class of explicitalgebraic stress models (EASMs). In Section 5 the devel-opment of EASMs using the methodology discussed inSection 3 is presented along with a detailed analysis ofthe various assumptions required. Included in theanalysis is an extension to nonequilibrium #ows and theframe-invariance properties of the models.

The material presented herein should provide thereader with a detailed knowledge of the concepts andmathematical tools currently being used and developedin connection with nonlinear eddy viscosity and alge-braic stress turbulence closure models.

2. Reynolds-averaged equations and single-pointclosures

In order to develop the Reynolds-averaged form of theNavier}Stokes equations, a decomposition of the #ow

variables (Reynolds decomposition) into mean and #uc-tuating components is assumed. Decomposing the velo-city (u

i) and pressure (p) "elds into their mean (u6

i, PM ) and

#uctuating parts (u@i, p@), the resulting Reynolds-averaged

Navier}Stokes (RANS) equations for an incompressible#ow in a noninertial frame [5] can be written as

Lu6j

Lxj

"0, (1)

Du6i

Dt"

Lu6i

Lt#u6

j

Lu6i

Lxj

"!

1

oLPMLx

i

!

Lqij

Lxj

#

LLx

jAl

Lu6i

LxjB

!eijk

eklm

XjX

lxm!2e

ijkX

ju6k, (2)

where Xi

is a time-independent rotation rate of thenoninertial frame. The "rst term involving X

ion the

right-hand side of (2) is the centrifugal acceleration andthe second term involving X

iis the Coriolis acceleration.

As Eq. (2) shows, for closure the RANS formulationrequires a model for the single-point correlation (or

Reynolds stress) qij

("u@iu@j).

As discussed in the Introduction, both the nonlineareddy viscosity and algebraic stress models are seen asintermediate approaches between the linear eddy-viscos-ity models and the Reynolds stress models. It may there-fore be useful to review the formulations used for thesetwo levels of modeling before focusing on nonlineareddy-viscosity and algebraic stress models.

2.1. Reynolds stress models

The transport equation in the noninertial frame whichgoverns the behavior of the second-moment or Reynoldsstress tensor q

ijis readily formed from the #uctuating

momentum equation [2,5] and is given by

Dqij

Dt"!q

ik

Lu6j

Lxk

!qjk

Lu6i

Lxk

#Pij!e

ij#D

ij

!2(eimk

Xmqkj#e

jmkX

mqki), (3)

where the "rst two terms on the right-hand side are theturbulent production, P

ijis the pressure}strain rate cor-

relation, eij

is the turbulent dissipation rate tensor, Dij

isthe combined e!ect of turbulent transport and viscousdi!usion, and the last two terms on the right-hand siderepresent the e!ect of the Coriolis acceleration. Note thatthe evolution of the turbulent second moments is notdependent on the centrifugal acceleration, but is onlydependent on the reference frame through the Coriolisacceleration. The modeling of the higher-order correla-tions appearing in this equation has been the subject ofmany studies; however, within the development of thealgebraic stress models the choice of particular closuremodel for these correlations is not of issue.

The turbulent kinetic energy equation K"qii/2 is

easily derivable from the contraction of the Reynolds


stress transport equation (3), using the fact that the pres-sure}strain redistribution term is traceless (P

ii"0),

DK

Dt"P!e#D, (4)

where the right-hand side represents the transport ofK by the turbulent production P"!q

ikLu6

i/Lx

k, the

isotropic turbulent dissipation rate, e"eii/2, and the

combined e!ects of turbulent transport and viscous di!u-sion D"D

ii/2. A comparison of the transport equations

for the Reynolds stresses (3) and the turbulent kineticenergy (4) clearly shows the fundamentally di!erent re-sponse of these turbulence quantities in noninertialframes. The turbulent kinetic energy K is described bya transport equation which is materially frame-indi!er-ent (MFI); whereas, the Reynolds stress tensor is de-scribed by a transport equation which is not MFI. Thesepoints are clearly discussed in detail in [6]. The issue ofMFI will arise later in an assessment of the underlyingassumptions used in the development of algebraic stressmodels.

It is becoming common practice in the development ofturbulence models to work with a tensor quantity de-rived from the Reynolds stress tensor, the Reynolds stressanisotropy tensor b

ijde"ned as

bij"

qij

2K!

dij3

, (5)

which is a traceless, symmetric tensor. Similarly, a turbu-lence dissipation rate anisotropy tensor can be de"ned as

dij"

eij

2e!

dij3

. (6)

When Eqs. (3) and (4) are used, the (exact) transportequation for the Reynolds stress anisotropy tensor b

ijis

given by

Dbij

Dt"

1

2KADq

ijDt

!

qijK

DK

Dt B"!b

ijAP

K!eB!

2

3Sij!(b

ikSkj#S

ikbkj

!

2

3bmn

Smn

dijB

# [bik(u

kj!2e

mkjX

m)!(u

ik!2e

mikX

m)b

kj]

#

Pij

2K#

1

2KADij!

qij

KDB!

1

qdij, (7)

where q"K/e, and the velocity gradient tensor has beendecomposed into the sum of the symmetric strain ratetensor S

ijand the antisymmetric rotation rate tensor u

ij

(Sij#u

ij"Lu6

i/Lx

j). Solving Eq. (7) for the turbulence

anisotropy tensor, together with Eq. (4) for the turbulentkinetic energy is equivalent to solving the originalReynolds stress tensor transport equation (3).

While it is outside the scope here to discuss themodeling of the pressure}strain rate correlation P

ij, it

will be necessary for the development of an algebraicstress model to specify a form for the pressure}strain ratecorrelation model. A rather general form which is nonlin-ear in the anisotropy tensor b

ij, and valid in noninertial

frames is given by

Pij"!AC0

1#C1

1

P

e Bebij#C2KS

ij

#C3K(b

ikSjk#b

jkSik!2

3bmn

Smn

dij)

!C4K(b

ik=H

kj!=H

ikbkj

)

#C5e(b

ikbkj!1

3bmn

bnm

dij), (8)

where

=Hij"u

ij!e

mijX

m(9)

is the absolute rotation rate tensor or the rotation ratetensor relative to the inertial frame. It is necessary tointroduce the absolute (or intrinsic) rotation rate tensorsince it has been shown that the pressure}strain ratecorrelation must be frame invariant [7]. For consistency,it is then necessary to develop models which retain thissame property. It is well known that the strain rate tensorSij

is frame invariant (or objective), while the rotationrate tensor u

ijis not. However, the absolute tensor=H

ijis

frame invariant, and as such can be used in the develop-ment of a pressure}strain rate model. The closure coe$-cients required in (8) can, in general, be functions of theinvariants of the stress anisotropy tensor, and the meanstrain rate and absolute mean rotation rate tensors. Sub-stituting Eq. (8) into Eq. (7) and rewriting yields

Dbij

Dt!

1

2KADij!

qij

KDB

"!Cbij

a4

#a3AbikSkj

#Sikbkj!

2

3bmn

Smn

dijB

!a2(b

ik=

kj!=

ikbkj

)#a1Sij!

1

qa5(b

ikbkj

!

1

3bmn

bnm

dijB#

1

qdijD, (10)

where now the rotational e!ects enter through the fol-lowing tensor:

=ij"u

ij!A1#

1

a2Bemij

Xm"u

ij!c

wemij

Xm

(11)

and the coe$cients aiand c

ware directly related to the

closure coe$cients of the pressure}strain correlation


model by

cw"

4!C4

2!C4

, a1"

1

2A4

3!C

2B, a2"

1

2(2!C

4),

a3"

1

2(2!C

3), (12)

a4"qC

C01

2!1#A

C11

2#1B

P

e D~1

"qCc1#c0

P

e D~1

,

a5"

1

2C

5.

For example, for the Launder, Reece and Rodi (LRR)model [8] the pressure}strain rate closure coe$cients areC0

1"3.0, C1

1"0, C

2"0.8, C

3"1.75, C

4"1.31 and

C5"0, and for the Speziale, Sarkar and Gatski (SSG)

model [9] the coe$cients are C01"3.4, C1

1"1.8,

C2"0.36, C

3"1.25, C

4"0.4 and C

5"4.2. Note that

the rotation rate tensor=ij

is not frame-indi!erent dueto the coe$cient c

wwhich, in general is not unity. This is

a further re#ection of the fact that such turbulent sec-ond-moment models are not frame-invariant.

2.2. Linear eddy viscosity models

For linear eddy viscosity models (LEVMs), themean momentum equation, Eq. (2), is closed by usinga Boussinesq-type approximation between the turbulentReynolds stress and the mean strain rate tensors

qij"2

3Kd

ij!2l

tSij, (13)

where ltis the turbulent eddy viscosity. In terms of the

Reynolds stress anisotropy tensor, the closure is given by

bij"!

lt

KSij, (14)

which clearly shows at this level of closure that an anisot-ropy of the Reynolds stress tensor can only be sustainedby a local mean strain due to the insensitivity of theclosure to the mean rotation rate tensor. The sole de-pendence on the strain rate tensor coupled with the formof the turbulent kinetic energy equation (4) shows thatthis level of closure is frame indi!erent. Unfortunately,such frame indi!erent behavior is contrary to direct nu-merical simulations for even the simplest of turbulent#ows, that is, decaying isotropic turbulence. The numer-ical simulations have shown that the turbulence decaydepends on the frame rotation which is necessarily pre-cluded with the closure model in Eq. (14). When Eq. (13)is substituted into Eq. (2), the mean velocity "eld isdescribed by

Du6i

Dt#

1

oLp6Lx

i

"

LLx

jC(l#l

t)Lu6

iLx

jD, (15)

where the isotropic part of the closure model 2K/3 isassimilated into the pressure term so that p6 "PM #2K/3.In the EVM formulation, the turbulence "eld is coupledto the mean "eld only through the turbulent eddy viscos-ity, which appears as part of an e!ective viscosity (l#l

t)

in the di!usion term of the Reynolds-averaged Navier}Stokes equation. The turbulent eddy viscosity l

tcan be

thought of as a di!usivity determined by the macroscopicvelocity and length scales of the large energetic eddies ofthe turbulence, in contrast to the molecular viscosityl which is determined by the scales of the Brownianmotion on the molecular level. Since in general, l

t'l,

this formulation of the problem can be numerically ro-bust, especially when compared to the alternative form ofretaining the stress gradient Lq

ij/Lx

jexplicitly in Eq. (2).

Dimensional analysis considerations dictate that theeddy viscosity l

tbe given by the product of a turbulent

velocity scale and a turbulent length scale. The naturalvelocity scale, used in practically all existing eddy-viscos-ity-based models, is the square root of the kinetic energy,K1@2. The eddy viscosity can then be expressed as

lt"C@kK1@2l"CAkKnZm, (16)

where l is a characteristic measure of the length scale ofthe energetic eddies. Alternatively, some other relatedquantity Z can be used as a measure of a characteristic`scalea of the turbulence. The exponents m and n are thenchosen so as to guarantee correct dimensionality of theeddy viscosity.

Common choices of Z are a scalar dissipation ratee (the isotropic turbulent dissipation rate), a speci"c dissi-pation rate u (&e/K), and a turbulent time scale q (e.g&K/e). (The book by Wilcox [10] provides a morecomplete discussion.) Any nonlocality in the descriptionwithin the eddy-viscosity framework enters through theuse of transport equations for the velocity and lengthscales (or any other combination). Depending on thetotal number of scalar transport equations included inthe model, the closures are referred to as zero, half, one,or two-equation models. In principle, the class of non-linear eddy viscosity and algebraic stress models to bediscussed later can be utilized within the framework ofany linear EVM which requires a knowledge of theReynolds stress components q

ij. However, inclusion of

such closures has been con"ned to the two-equationplatform up to this point. Many variations on this ap-proach exist, but for the most part, they also use thetransport equation for the turbulent kinetic energy K asthe turbulent velocity scale equation. In contrast, thelength scale equation has generally been the most contro-versial piece of the two-equation formulation. The par-ticular choice of scale equation is not of muchsigni"cance, since in the nonlinear eddy viscosity andalgebraic stress formulations, it is the combination


with the turbulent velocity scale which forms eithera characteristic time or length scale that is of interest.

For the present purposes, attention will be focused ontwo-equation formulations using e to obtain the lengthscale. In this case, the eddy viscosity l

tis given by the

relation

lt"Ck

K2

e"CkKq, q"

K

e(17)

and the turbulent length scale readily follows as l"K3@2/e. The modeling coe$cient Ck usually assumesa value of 0.09. In the two-equation K}e context, theturbulent kinetic energy transport equation is derivedfrom the contraction of the Reynolds stress transportequation (3), and is given in Eq. (4). The turbulent pro-duction term in the kinetic energy equation can be re-written in terms of the eddy viscosity as

P"2lt(S

ikSki)"2l

tg1, (18)

where g1"S

ikSki. In order to complete the closure of

Eq. (4) a model is needed for the turbulent transportterm D. Consistent with the simpli"cation thatresults from the two-equation formulation, a simpleform for the turbulent transport is usually used. Theclosure of D is then based on a gradient transport modelsuch that

D"

LLx

jAl#

lt

pKB

LK

Lxj

, (19)

where the "rst term on the right is the viscous contribu-tion, and the second is the model for the turbulent trans-port. The coe$cient p

Kis an e!ective Prandtl number for

di!usion, which is taken as a constant in incompressible#ows, but is dependent on the particular scale variableused. The resulting simple form of the modeled turbulentkinetic energy equation is an obvious appeal of theformulation.

There are several variations to the modeled form of thetransport equation for the isotropic dissipation rate e.A rather general expression from which many of theforms can be derived and which can be integrated to thewall is given by

DeDt

"

1

q(Ce1P!Ce2e)#

LLx

kCAl#

lt

peBLeLx

kD, (20)

where Ce1+1.45 is usually "xed from calibrations withhomogeneous shear #ows, and Ce2 is usually determinedfrom the decay rate of homogeneous, isotropic turbu-lence (+1.90). The closure coe$cient pe acts like ane!ective Prandtl number for dissipation di!usion and is

speci"ed to ensure the correct log-law slope of i~1,

pe"i2

JCk (Ce2!Ce1 ). (21)

In the early development work on lower-order zero-and half-equation models, attention was clearly focusedon the construction of models suitable for direct integra-tion to the wall through a two-layer structure for theeddy viscosity. In addition, while less explicit about itssuitability for direct integration to the wall, the one-equation formulation was also developed with the capa-bility of being used unaltered in wall-bounded #ows.However, in the two-equation K}e formulation, Eq. (17)and (20) in their high-Reynolds number form, do notprovide the correct asymptotic behavior in the near-wallregion. Moreover, the destruction-of-dissipation rateterm Ce2e2/K, is singular at the wall since e is "nite, andthe turbulent kinetic energy vanishes at the wall. Therehas been an extensive list of near-wall modi"cations overthe last two decades for both the eddy viscosity,

lt"fkCk

K2

e, (22)

where fk is a damping function, and the transport equa-tion for the dissipation rate

De(Dt

"f1Ce1

e(KP!f

2Ce2

e( 2K

#

LLx

kCAl#

lt

peBLe(Lx

kD#E,

(23)

e"e(#D, (24)

where f1

is a damping function, f2

is used to ensure thatthe destruction term is "nite at the wall, and D and E areadditional terms included to better represent the near-wall behavior. A partial list of the various forms for thesefunctions can be found in [11}13]. While the list is not allinclusive, it does provide the functional forms which areused today for these near-wall functions.

The linear eddy-viscosity models have proven to bea valuable tool in turbulent #ow-"eld predictions. How-ever, inherent in the formulation are several de"ciencieswhich do not exist within the broader Reynolds stresstransport equation formulation. Two of the most notablede"ciencies are the isotropy of the eddy viscosity and thematerial-frame indi!erence of the models. The isotropiceddy viscosity is a consequence of the Boussinesqapproximation which assumes a direct proportionalitybetween the turbulent Reynolds stress and the meanstrain rate "eld. The material frame indi!erence is a con-sequence of the sole dependence on the (frame-indi!er-ent) strain rate tensor. These de"ciencies preclude, forexample, the prediction of turbulent secondary motionsin ducts (isotropic eddy viscosity) and the insensitivity ofthe turbulence to noninertial e!ects such as imposedrotations (material frame-indi!erence). Remedies for


these de"ciencies can be made on a case-by-case or adhoc basis; however, within the framework of a lineareddy-viscosity formulation such defects cannot be "xedin a rigorous manner.

The nonlinear eddy-viscosity models and their subsetof algebraic stress models extend the range of applicabil-ity of the linear eddy-viscosity models by replacing theBoussinesq approximation (13) where q

ij"q

ij(S

ij, q) with

the more general relation qij"q

ij(S

kl,=

kl,q) (q "K/e for

the K}e formulation). As mentioned earlier, there isa close linkage between the LEVM formulation and theNLEVM and ASM formulations. In all these cases, thecommon thread lies in the need for transport equationsfor the characteristic turbulent scales (e.g. K}e). Thedistinguishing feature among these approaches is thechoice of constitutive equation for the turbulentReynolds stresses or second moments, but all the strat-egies used to obtain the turbulence scales required forclosure in the Boussinesq formulation are directly trans-ferable to these more sophisticated models. In addition,while virtually all the NLEVM and ASM formulationsare based on a two-equation closure, any other (lower)closure (zero-, half-, one-, or two-equation) level, requir-ing a knowledge of a Reynolds stress component, couldin principle be connected to a NLEVM or ASM.

3. General Reynolds stress tensor representations

As just discussed, nonlinear eddy-viscosity and explicitalgebraic stress models attempt to represent theReynolds stress tensor as a suitably general tensor func-tion of the kinematic tensors. The continuum mechanicscommunity has dealt with such questions on tensor rep-resentations for several decades [14,15]. Within this con-text, the objective of this section is to brie#y review thetheoretical basis of tensor representations from whicha general framework for a rigorous and systematicdevelopment and analysis of turbulence models can bepresented. It will "rst be shown that the previous devel-opments on tensor representations for general turbulencemodels can be interpreted in terms of projections ontotensor bases. Optimal three- and "ve-terms representa-tions of the anisotropy tensor for three-dimensional #owswill then be developed. These representation results willthen be used in the following sections to formulate non-linear eddy-viscosity models, and to derive models in theimportant sub-class of explicit algebraic stressmodels.

The class of nonlinear eddy-viscosity models simplyextends the one-term tensor representation in terms ofthe strain rate, used in the linear EVMs (see Eq. (13)), tothe generalized form

qij"

2

3Kd

ij#

N+n/1

a@n¹(n)

ij, (25)

or, in terms of the stress anisotropy bij,

bij"

N+n/1

an¹(n)

ij, (26)

where ¹(n)ij

(n"1,2, N) is a given tensor basis, withN "nite, and the a

nare expansion coe$cients which need

to be determined.As was shown in Section 2.2, the linear EVMs, through

the Boussinesq approximation, couple to the RANSequations through a simple additive modi"cation to thedi!usion term (see Eq. (15)). In the case of generalReynolds stress representations, this coupling can bemore complex. The coupling can be either through thedirect use of Eq. (25) in Eq. (2) or through a modi"edform of Eq. (15) given by

Du6i

Dt#

1

oLp6Lx

i

"

LLx

jC(l#l

t)Lu6

iLx

jD#S, (27)

where S represents nonlinear (source) terms from thetensor representation (25). The degree of complexity as-sociated with the nonlinear source terms is dependent onboth the number and form of the terms chosen for thetensor representation.

The choice of the proper tensor basis is, of course,dependent on the functional dependencies associatedwith the Reynolds stress q

ijor the corresponding aniso-

tropy tensor bij. As seen from the transport equation for

the Reynolds stress anisotropy, Eq. (10), the only depend-ency on the mean #ow is through the mean velocitygradient (the mean strain rate tensor S

ijand a mean

rotation rate tensor =ij). Thus, it has been generally

assumed [2] in developing turbulent closure models forthe Reynolds stresses, that in addition to the functionaldependency on the characteristic turbulent scales, thedependence on the mean velocity gradient should beincluded as well. Within this context, the stress aniso-tropy tensor considered here is assumed to have thefunctional dependency

bij"b

ij(S

kl,=

kl, q), (28)

where q"K/e is a scalar representing the characteristicturbulent time scale. As noted previously, this turbulentscale could be obtained by any strategy (e.g. zero-, half-,one- and two-equation models).

3.1. Representation of the Reynolds stress anisotropytensor as a linear combination of basis tensors

Equation (28) expresses the fact that the elements ofthe symmetric, traceless, 3]3 turbulence anisotropy ten-sor b

ijin any rectangular Cartesian coordinate system

are only functions of the elements of the two independentkinematic tensors S

ijand =

ijin the same coordinate

system. Since it must also be required that the formsof these functional relationships be independent of the


1 In dealing with tensor representations, it is sometimes better,for notational convenience, to use matrix notation to eliminatethe cumbersome task of accounting for several tensor indices.For this reason, both tensor and matrix notation will be used indescribing the models in this and subsequent sections.

2By extension, B(0) is de"ned as the 5]5 matrix whosecolumn j is the vector (¹(j)

11,2,¹(j)

23), the coe$cient matrix of the

linear system (31). Similarly, B(1) (resp. B(5)) has the "rst (resp.last) column formed by the elements of b

ij.

particular coordinate system in which they are expressed[16], the relation between b

ij, S

ijand=

ijis isotropic, and

has therefore to satisfy the following property:1

b(QSQT,QWQT, q)"Qb(S,W, q)QT, (29)

which must hold for all orthogonal transformation ma-trices Q such that QQT"QTQ"I and detQ"$1,where QT is the transpose of Q, and det denotes thedeterminant.

Rivlin and Ericksen [17] have shown that a linearrelation of the form

a0b"

N+n/1

anT(n) (30)

can be obtained between the tensor b (the dependenttensor) and a "nite number N of other tensorsT(1), T(2),2,T(N) (the basis tensors) formed from the ele-ments of the tensors S and W (the independent tensors).The scalar coe$cients in this linear relation are scalarinvariants of the independent tensors and of b. Moreover,an in"nite number of such linear relations can beobtained by making di!erent choices for the tensorsT(n) formed from the elements of the independent tensors.For each such choice, there exist certain conditions onthe independent tensors for which the coe$cient of thedependent tensor in the representation is zero and forwhich the linear relation does not provide an expressionfor the dependent tensor. As will be shown, this repres-entation result can be interpreted as projections onto thesubspace generated by the basis tensors, and allows forthe determination of general tensor representations aswell as optimal approximations of b in terms of a givenset of basis tensors.

Since the interest here is in the development of modelsfor the Reynolds stress tensor, the above existence resultsare not su$cient. It is necessary to have a method toconstruct such a tensor representation. Following Rivlinand Ericksen [17], let T(1),T(2),2,T(N) be any symmetric3]3 traceless matrices formed from the elements of thematrices S and W. If a linear relation of the form (30)can be established between b and the matricesT(n) (n"1, 2,2, N), which are not necessarily linearlyindependent, such that the coe$cient of b is nonzero, thenthis expression is referred to as a representation of b interms of T(n) (n"1, 2,2,N). Eq. (30) is a system ofequations for the N#1 unknown scalar quantities a

n.

Since b is a symmetric, traceless matrix, only 5 elementsare independent, and Eq. (30) can be rewritten as 5 linearequations

a0bij"

N+n/1

an¹(n)

ij, (31)

where (i, j) takes, for example, the "ve values (1, 1), (2, 2),(1, 2), (1, 3) and (2, 3).

If N"5, Eq. (31) forms a linear system of "ve equa-tions in the six unknown coe$cients a

n(n"0, 1,2, 5)

which can be determined in terms of the elements of thematrices, apart from an arbitrary nonzero multiplier K,if the equations are linearly independent. Linear indepen-dence means that the tensors T(1),T(2),2,T(5) are lin-early independent, or that the only solution of the linearsystem in the unknowns a

n

5+n/1

¹(n)ij

an"0 (n"1, 2,2, 5) (32)

is the trivial solution an,0 (n"1,2, 5).

If Cramer's rule is applied to the linear system (31),

an"K(!1)ndetB(n) (n"0,2, 5), (33)

where the 5]5 matrices B(n) are de"ned as2

B(n)"C¹(1)

11 2 ¹(n~1)11

b11

¹(n`1)11 2 ¹(5)

11¹(1)

22 2 ¹(n~1)22

b22

¹(n`1)22 2 ¹(5)

22. . . . .

. . . . .

¹(1)23 2 ¹(n~1)

23b23

¹(n`1)23 2 ¹(5)

23D.(34)

It is clear that if det B(0)"0, then a0"0, which would

contradict the assumption that the matricesT(1),T(2),2, T(5) are linearly independent. If det B(0)O0,representation (30) can be rewritten as

b"5+n/1Aan

a0BT(n) (35a)

with

an

a0

"(!1)ndet B(n)

detB(0). (35b)

However, since b is required to be an isotropic function ofS and W, the form of Eq. (30) must be unaltered by


3The trace of a tensor is denoted in matrix notation asMaN"a

ii.

4 It is easy to show that this bilinear form indeed satis"es therequirements of a scalar product.

simultaneous orthogonal transformations of the matricesb, S and W. If the T(n) (n"1, 2,2,N) are isotropicfunctions of S and W, then the a's are expressible in termsof the elements of S, W and b in a form which is unalteredby an orthogonal transformation. It must, therefore, bepossible to rewrite Eq. (33) in a form that is independentof the coordinate system used (see [17] for details), suchthat the coe$cients are algebraic functions of the invari-ants formed with b and the basis tensors. A simple way toobtain this is to form the N matrix equations

N+n/1

anT(n)T(m)"a

0bT(m) (m"1, 2,2,N) (36)

by multiplying Eq. (30) by T(1),T(2),2,T(N). Forming thetrace3 of each of the equations in (36) yields

N+n/1

anMT(n)T(m)N"a

0MbT(m)N (m"1, 2,2,N). (37)

These N scalar equations in the N#1 coe$cients anmay

be solved, apart from an arbitrary nonzero multiplier K,to obtain the desired expressions for the a

n's as functions

of the invariants

an"K(!1)ndetD(n) (n"0,2,N) (38)

where the N]N matrices D(n), (n"0,2,N) are

D(n)kl"G

MT(k)T(l)N, lOn,

MbT(k)N, l"n.(39)

For the cases where the matrix D(0)kl"MT(k)T(l)N is nonsin-

gular, the coe$cient a0O0, and representation (30) can

be written as

b"N+n/1Aan

a0BT(n), (40a)

with

an

a0

"(!1)ndet D(n)

detD(0). (40b)

There is a fundamental di!erence between the formula-tions given in (35b) and (40b) for the coe$cients a

n. The

expression for the coe$cients an

given by Eq. (33) canonly be obtained for N"5, and with T(1),T(2),2, T(5)

linearly independent. On the other hand, the expressionfor a

nobtained by Eq. (38) is valid for any number N of

basis tensors T(n), not necessarily linearly independent.When N"5 and the tensors are linearly independent,both expressions are equivalent. When NO5 or whenthe tensors are not linearly independent, only Eq. (37) isa solution of problem (40a). According to the results of

Rivlin and Ericksen [17], there can only be "ve indepen-dent symmetric 3]3 traceless matrices. This is consistentwith the present results, since there must be at least "veterms in the tensor basis to provide a valid representationfor any 3]3 tensor b. For N(5, Eq. (40a) has to beviewed as an approximation in a least-squares sense.

3.2. Representation of the Reynolds stress anisotropytensor as a projection on a tensor basis

It is possible to further quantify these results by recast-ing the previous manipulations in terms of projectionswithin vector spaces. Let V be the set of all the 3]3symmetric, traceless tensors such that V is a real, linear,vector space, in which a scalar product (.,.) on V can bede"ned,4

(.,.) :V]VPR : (X,Y),MXYN,Xmn>

mn. (41)

V is, therefore, a real Hilbert space. The natural normassociated with the scalar product is

DD ) DD :VPR` : DDXDD2,MX2N,(Xmn

)2. (42)

The linear vector subspace generated by the tensorsT(1),T(2),2, T(N) can now be de"ned as

V(N),MX3V: a0X"

N+n/1

anT(n),

∀(a0, a

1,2, a

N)3RN`1,a

0O0N. (43)

With these de"nitions, Eq. (37) can now be interpretedas a projection onto the subspace generated by the ten-sors T(1),T(2),2, T(N). When N"5, Eq. (37) is an exactrepresentation of b in terms of the basis tensorsT(1),T(2),2, T(5); however, under certain conditions onthe independent tensors, this representation may fail toexist (a

0"0). When N(5, Eq. (37) yields an approxi-

mation of b in terms of the basis tensors, and for N'5,redundancies would, in principle, exist in the tensor basis.The conditions, if any, under which the representationwould fail to exist will, in general, be more severe, and thecoe$cients in the expansion may take simpler forms.

In addition, the representation problem (30) can nowbe identi"ed as a problem of "nding the representationb(N) in V(N) such that DDb!b(N)DD is minimal,

DDb!b(N)DD4DDb!b@DD, ∀b@3V(N). (44)

Following the classical theory of approximation,the positive-de"nite quadratic form which has to


be minimized,

Q(a0,2, a

N)"a2

0DDb!b(N)DD2

"a20Mb2N#

N+n/1

N+

m/1

amanMT(m)T(n)N

!2a0

N+n/1

anMbT(n)N (45)

can be de"ned. For this, it is required that

LQ

Lam

"0"2N+n/1

anMT(n)T(m)N!2a

0MbT(m)N

(m"1,2,N) (46)

and

LQ

La0

"0"2a0Mb2N!2

N+n/1

anMbT(n)N. (47)

Note that Eq. (47) implies that Mbb(N)N"Mb2N whena0O0. The N equations (46) provide for the determina-

tion of the N#1 representation coe$cients an* apart

from an arbitrary nonzero multiplier K. De"ning theN]N Gram matrix G(N) as

G(N)kl

,MT(k)T(l)N (k, l"1, 2,2,N), (48)

the coe$cients an

are the solution of the linear system

N+n/1

G(N)nm

an"a

0Mb,T(m)N, (49)

given by (40b).It is important to realize that, although this interpreta-

tion of tensor representation results is a powerful way tosystematically and rigorously derive the best representa-tion coe$cients for a given representation basis, it doesnot guarantee the choice of the best representation basis.As long as the semi positive-de"nite, real symmetricGram matrix G(N) is nonsingular, the choice of the basistensors T(1),T(2),2,T(N) is arbitrary. Under certain con-ditions on the independent tensors S and W, system (49)may become singular, and no valid representation can beobtained (a

0"0). In some cases, it is possible to remove

the degeneracy, and a valid representation can be ob-tained in fewer basis tensors [17]. The selection of basistensors should ideally be among the set of tensors form-ing the integrity basis corresponding to S and W.

In such a representation, the scalar coe$cients givenin Eq. (38) are scalar invariants of the tensors b,T(1),T(2),2, T(N). Since the basis tensors are isotropicfunctions of the kinematic tensors S and W, it followsthat the coe$cients of the expansion must ultimately beexpressed as polynomial functions of the scalar invari-ants of b, S and W which are, at most, linear in b.

Eq. (30), with Eq. (38) for the coe$cients, is a simplemathematical identity, and does not constitute, in itself,an explicit formulation for b. Indeed, the coe$cients a

nof

the expansion are dependent on b through the invariantsformed with b and T(n), (n"1,2, N). Genuine explicitrepresentations for b would, of course, require additionalinformation on the properties of b that must be providedin some external way. As will be seen in Section 5,algebraic stress relations are suitable for this, since therequirement that b satis"es the algebraic stress relationwill then provide the necessary constraints to express thescalar invariants containing b in terms of known invari-ants formed on the kinematic tensors. Nonlinear eddy-viscosity models, on the other hand, only take intoaccount the formal tensor polynomial representation,and provide expressions for the scalar representationcoe$cients in an empirical way.

For any choice of basis tensors, it is possible to use theCayley}Hamilton theorem, which states that tensorpolynomial terms beyond a certain order are redundant(see Appendix), to express the entries of the matrix G(N) interms of lower-order polynomials [14,15]. Due to theisotropy condition (29), the coe$cients of the expansionmust be scalar invariants of the kinematic tensors S andW, and of the anisotropy tensor b which are at mostlinear in b. An integrity basis is de"ned as a set ofinvariants that can be used to express any invariant asa single-valued function of the given set. The integritybasis is minimal if it contains the smallest possible numberof members. It has been shown [16,18] that the minimalintegrity basis for the tensors S and W is given by

g1,MS2N, g

2,MW2N, g

3,MS3N, g

4,MW2SN,

g5,MW2S2N, g

6,MSWS2W2N. (50)

It then follows that the coe$cients of the expansionwill be single-valued functions of the invariants g

i(i"1,2, 6) together with some of the irreducible invari-ants formed with b, S and W that are linear in b. It shouldbe pointed out here that the use of the Cayley}Hamiltontheorem to reformulate the entries of G(N) in terms of theinvariants g

i(i"1,2, 6) is not essential in the present

context, and only constitutes a convenient simpli"cation.That is why the inclusion of a sixth invariant g

6into the

integrity basis is not further discussed. Nevertheless, it ismore convenient to include it in the present derivations,but as pointed out in Lund and Novikov [19], g

6may

also be expressed in terms of g1,2, g

5. In the limit,

a direct computation of the entries of G(N) from Eq. (48),without further simpli"cation, would lead to exactly thesame representation, although with a more complex formfor the expansion coe$cients.

3.3. Applications to xve- and three-term representations

Consider the following "ve-term tensor base:

T(1)"S, T(2)"SW!WS, T(3)"S2!13MS2NI,

T(4)"W2!13MW2NI, T(5)"WS2!S2W. (51)


G(5)"Cg1

0 g3

g4

0

0 g1g2!6g

50 0 g

1g4#g

2g3

g3

0 16g21

!13(g

1g2!3g

5) 0

g4

0 !13(g

1g2!3g

5) 1

6g22

0

0 g1g4#g

2g3

0 0 !12g1(g

1g2!2g

5)!2g

3g4D. (52)

These tensors correspond to 5 of the 10 tensor terms[14,15] that form a tensor basis which is always linearlyindependent. Any other group of "ve tensors could havebeen chosen (see [19]). The present choice is arbitrary,and is motivated by the fact that T(1), T(2) and T(3) forman integrity basis for two-dimensional #ows, and hasbeen chosen by Pope [20] and Gatski and Speziale [21]for an explicit algebraic stress model (EASM). It will beshown that these "ve tensors are always su$cient fora valid representation of a symmetric, traceless tensor b,except in some degenerate cases.

3.3.1. Five-term representations for three-dimensionalyows

It is possible to show, after successive applications ofthe Cayley}Hamilton theorem, that the matrix G(5), ex-pressed in terms of the irreducible invariants g

i, is given

by [22]

The inversion of this matrix leads to the following repres-entation result which is valid for any symmetric, tracelesstensor that is an isotropic function of the tensors Sand W,

b(5)"a1

a0

S#a2

a0

(SW!WS)#a3

a0AS2!

1

3MS2NIB

#

a4

a0AW2!

1

3MW2NIB#

a5

a0

(WS2!S2W), (53)

where b(5) is the representation of b in V(5), and thecoe$cients, found using Eq. (49), are functions of theinvariants g

i(i"1,2, 6), as well as the invariants for-

med with b, S and W that are linear in b

a0"K[2q2

1#g2

1g2q5#4g

3g4q4#12g

1g25],

a1"K[(g2

1g22q5#12g2

5)MbSN#2(g2

2g3#2g

4q3)MbS2N

#2(2g3q3#g2

1g4)MbW2N,

a2"!2K[(4g

3g4#g

1q2)MbWSN!2q

1MbWS2N],

a3"2K[(g2

2g3#g

4q3)MbSN!(g

1g22!6g2

4)MbS2N

!4(3g3g4#g

1q3)MbW2N],

a4"2K[(2g

3q3#g2

1g4)MbSN!2(3g

3g4#g

1q3)MbS2N

#(6g23!g3

1)MbW2N]

a5"!4K[q

1MbWSN#q

4MbWS2N], (54)

where q1"g

1g4#g

2g3, q

2"g

1g2!2g

5, q

3"g

1g2!

3g5, q

4"g

1g2!6g

5, q

5"g

1g2!8g

5, and KO0 is

an arbitrary scalar.(Eq. (53) is an identity. The rather lengthy and tedious

steps leading to the form Eq. (52) for the matrix G(5) havebeen veri"ed with a symbolic mathematical package byexpanding each entry of the right-hand side of Eq. (53),without any assumption on the form of S, W or b, andchecking that it could indeed be simpli"ed to the corre-sponding entry on the left-hand side of Eq. (53).)

When the coe$cient a0"0, Eq. (53) no longer gives

a representation of b in terms of the basis T(1),2,T(5). Inthe principal coordinate system of S, S is a diagonal

matrix of eigenvalues j1, j

2and j

3, and the expression

for the coe$cient a0

reduces to

a0"!8(=2

12=2

13#=2

12=2

23#=2

13=2

23)

](j1!j

2)2(j

1!j

3)2(j

2!j

3)2, (55)

where=12

,=13

and=23

are the coe$cients of W ex-pressed in the principal coordinate system of S. It is seenthat a

0is nonpositive, but can vanish in two cases (see

[19]): (1) two eigenvalues of the mean strain rate matrixS are equal (the case of three equal eigenvalues is notpossible, since j

1#j

2#j

3"0 by incompressibility),

and (2) when expressed in the principal coordinatesystem of S, two components of the vorticity tensorW vanish (the case S"0 is discarded since it wouldsimply reduce everything to zero). Case (1) correspondsto a purely axisymmetric strain case, while case (2) is fora #ow having the rotation vector aligned with one of theprincipal directions of the strain rate S. It is easy to seethat if the dependent tensor b is restricted to also havetwo equal eigenvalues (for case (1)), or one eigenvectoraligned with one of the principal directions of the strainrate tensor (for case (2)), then the degeneracy in therepresentation can be removed, for instance by using lessterms in the expansion.


In Gatski and Speziale [21], the 10-term integrity basisproposed in [17] was utilized, which is the minimal basissuch that the coe$cient a

0never vanishes. Thus, the

10-term integrity basis is the minimal one that alwaysprovides a representation for b. In the cases where repres-entation (53) is valid, both representations exist concur-rently and o!er two di!erent valid ways to express b asa function of S and W.

3.3.2. Three-term representations for two- andthree-dimensional yows

In the case of mean #ows that are two-dimensional(S has one vanishing eigenvalue (say j

1) and in the

principal coordinate of S, the vorticity vector is alignedwith the eigenvector of S corresponding to j

1) only the

"rst two invariants g1

and g2

are independent since

g3"0, g

4"0, and g

5"1

2g1g2. (56)

In addition, the only dependent tensors b that can berepresented must have one eigenvector aligned with theeigenvector of S corresponding to j

1, and only the "rst

three basis tensors are needed to represent b. The matrixG(3) is given by the "rst three rows and columns of G(5)

G(3)"Cg1

0 g3

0 g1g2!6g

50

g3

0 16g21D, (57)

which can be simpli"ed to

G(3)"Cg1

0 0

0 !2g1g2

0

0 0 16g21D (58)

in the two-dimensional case using relations Eq. (56).Consistent with the discussion in Section 3.2, the

matrix G(3) can be used as an approximate representa-tion for the three-dimensional case. In this case,

b(3)"a1

a0

S#a2

a0

(SW!WS)#a3

a0AS2!

1

3MS2NIB (59)

with

a0"

K

6(g

1g2!6g

5)(g3

1!6g2

3),

a1"

K

6(g

1g2!6g

5)(g2

1MbSN!6g

3MbS2N),

a2"!

K

3(g3

1!6g2

3)MbWSN,

a3"K[g

1(g

1g2!6g

5)MbS2N!6g

3MbSN], (60)

where KO0 is arbitrary. Once again, for two-dimensional mean #ows the case N"3 provides a

representation of b that is always valid

b(2d)"MbSNg1

S#MbWSNg1g2

(SW!WS)

#6MbS2Ng21AS2!

1

3MS2NIB. (61)

In the two-dimensional case, a0("!g4

1g2/3) never van-

ishes unless W vanishes. In this case, the correspondingtensors in the basis containing W vanish, and the degen-eracy is automatically removed.

In the general three-dimensional case, the expressionfor b(3) in Eq. (59) provides an approximation of b. Thisapproximation is optimal in the least-squares sense, sincethe di!erence between b and its projection onto thethree-term basis is minimal from Eq. (46),

DDb!b(3)DD4DDb!b@DD, ∀b@3V(3) (62)

The N"3 representation can degenerate though, sincethe coe$cient a

0vanishes when two of the eigenvalues of

S are equal, that is when W"0. Here again, if b has thesame properties, the degeneracy can be removed since thecorresponding tensors in the basis containing W vanish.

The tensor basis for N"3 can be (and usually is) usedas a simpli"cation to more complex and costly basesusing N55. It constitutes a suitable approximationsince it is exact for two-dimensional #ows. When used inthe general case, it is Eq. (60) for the scalar coe$cientsthat has to be used instead of Eq. (61), since relations (56)are no longer valid. The optimal representation Eq. (60)is, in fact, also exact for a larger class of mean #ows thanplanar #ows. It is a valid representation of mean #ows forwhich the rotation rate tensor has one eigenvector alig-ned with one eigenvector of S (this eigenvector does notnecessarily have a corresponding eigenvalue equal tozero).

3.4. Representation of higher-order terms

The present tensor representation technique can befurther exploited to characterize the Reynolds stresses.For instance, any integer power of the anisotropy tensorb can also be represented in terms of the same tensorrepresentation base. First, consider the traceless part ofbp, with p integer. As it is a traceless, symmetric tensor, itcan be represented as

bp!1

3MbpNI"

N+n/1

bn

b0

T(n) (63)

with bn

the representation coe$cients to be determined.Note that since the same representation base is used asfor b, the same conditions of existence of representation(63) apply, and b

0"0 if and only if a

0"0. Since in the


present formalism b is represented as

b"N+n/1

an

a0

T(n), (64)

the following expression can be obtained:

b0Abp!

1

3MbpNIB"b

0A+n

an

a0

T(n)Bp

!

1

3b0GA+

n

an

a0

T(n)Bp

HI

"

b0

ap0

+n1

2+np

an1 2

anp

T(n1 )2T(np )

!

b0

3ap0

+n1

2+np

an12

anp

MT(n1 )2T(np )NI. (65)

In order to transform this set of tensor relations intoan equivalent system of invariant scalar relations whichcan be used to determine the representation coe$cientsbn, it is su$cient to project relation (65) onto the repres-

entation basis

N+n/1

bn

b0

(T(n),T(m))"1

ap0

+n1

2+np

an12

anp

(T(n1 )2T(np ), T(m))

(m"1,2,N), (66)

where the property (I,T(m))"0 for all m has been used.The system described by (66) is a linear system that canbe solved to "nd the representation coe$cients b

n. As

was noted previously, since T(n) (n"1, 2,2, N) formsa representation basis, the N]N Gram matrix G(N) isnonsingular. The system (66) can easily be solved sincethe inverse of G(N) is already known from the representa-tion of b itself. In addition, since the right-hand side ofsystem (66) contains pth-order products of the coe$cientsan, the representation coe$cients b

nof the traceless part

of bp will "nally be expressed as polynomials in pth-orderproducts of the representation coe$cients a

n.

As an illustration, consider the three-term representa-tion of the quadratic tensor b2!Mb2NI/3 for two-dimen-sional #ows. The expansion coe$cients are the solutionof

3+n/1

G(3)mn

bn

b0

"

1

a20

3+i/1

3+j/1

aiajMT(i)T(j)T(m)N

(m"1, 2, 3), (67)

where G(3) is given by (58). The coe$cients aiare found

from (61) and the 27 invariants MT(i)T(j)T(m)N,(i, j, m"1, 2, 3) must be evaluated. As a result of thesymmetry properties (( j,m, i)"(i, j,m)"(m, i, j)), only 11invariants need to be computed: (i, j,m)"(1, 1, 1), (1, 1, 2),(1, 1, 3), (1, 2, 2), (1, 2, 3), (1, 3, 2), (1, 3, 3), (2, 2, 2), (2, 2, 3),(2, 3, 3) and (3, 3, 3). With the generalized version of the

Cayley}Hamilton theorem (see Appendix and the refer-ence by Spencer [14]), the only resulting nonzero invari-ants are

MT(1)2T(3)N"16g21, MT(2)2T(3)N"!1

3g21g2,

MT(3)3N"! 136

g31, (68)

together with the invariants that result from the cyclicpermutations of the indices. Therefore, the relations

b1

b0

"

1

3g1

a1a3

a20

,

b2

b0

"

1

3g1

a2a3

a20

,

b3

b0

"

a21

a20

!2g2

a22

a20

!

1

6g1

a23

a20

(69)

lead to the desired representation for the quadratic term

b2!1

3Mb2NI"

g1a1a3

3a20

S#g1a2a3

3a20

(SW!WS)

#Aa21

a20

!

2g2a22

a20

!

g1a23

6a20BAS2!

1

3g1IB.(70)

This result will be used later when algebraic stressmodels are developed. Another quantity which is oftenused to characterize the turbulence is the trace of integerpowers of the anisotropy tensor, MbpN, and more parti-cularly the second and third invariants of b, de"ned as

IIb"!1

2Mb2N and III

b"1

3Mb3N, (71)

which are used to assess realizability of the turbulence[23]. Using the representation for b and following theprocedure just outlined,

MbpN"1

ap0

N+

n1/12

N+

n2/1

an1 2

anp

MT(n1 )2T(np )N. (72)

The invariants of products of the T(n) tensors are easilyevaluated, and may be simpli"ed using theCayley}Hamilton theorem (see Appendix). In particular,relation (72) leads to the following expressions for theinvariants

IIb"!

1

2

1

a20

N+i/1

N+j/1

aiajMT(i)T(j)N and

IIIb"

1

3

1

a30

N+i/1

N+j/1

N+k/1

aiajakMT(i)T(j)T(k)N. (73)

This result shows that as soon as the representationcoe$cients a

nare known (in a manner to be discussed in

subsequent sections), it is straightforward to obtain theexpression of the second and third invariants of the


anisotropy as a function of these parameters. Froma modeling standpoint, this is a very important featuresince many of the turbulence models to be presented inlater sections have their representation coe$cients deter-mined based on the ful"llment of a number of con-straints, among which the realizability of turbulence.With the closed-form and explicit expression of the sec-ond and third invariants in (73), it is possible to accurate-ly determine the constraints to be put on the expressionof the model coe$cients in order to guarantee the realiza-bility of the model.

Referring again to the three-term representation fortwo-dimensional #ows, the expression of the second andthird invariants becomes in this case

IIb"!

1

2g1C

a21

a20

!2g2

a22

a20

#

1

6g1

a23

a20D and

IIIb"

1

6g21

a3

a0Ca21

a20

!2g2

a22

a20

!

1

18g1

a23

a20D. (74)

These can be linked together by

IIIb"!

1

3g1

a3

a0CIIb#

1

9g21

a23

a20D. (75)

These relations are a direct consequence of representa-tion (61) of the Reynolds stress anisotropy tensor, andare therefore always valid, independent of the way theReynolds stress tensor evolution is modeled.

4. Nonlinear eddy-viscosity models

As pointed out in Section 3.3.1, the tensor representa-tion consisting of the integrity basis is the minimal basissuch that the coe$cient a

0of the expansion Eq. (30)

never vanishes in the case of fully three-dimensionalmean #ow. In the three-dimensional case for the tensorb(S,W, q), this basis consists of the linearly independentelements

T(1)"S,

T(2)"SW!WS,

T(3)"S2!13MS2NI,

T(4)"W2!13MW2NI,

T(5)"WS2!S2W,

T(6)"W2S#SW2!23MSW2NI,

T(7)"WSW2!W2SW,

T(8)"SWS2!S2WS

T(9)"W2S2#S2W2!23MS2W2NI,

T(10)"WS2W2!W2S2W. (76)

The expansion coe$cients anassociated with this repres-

entation will, in general, be functions of the scalar invari-ants (the integrity basis) of the representation and othercharacteristic scales of the turbulence:

an"a

n(g

i, q,2). (77)

The linear term T(1) is the strain rate S, as in the linearEVM case, and its coe$cient a

1/a

0is now proportional

to the turbulent eddy viscosity lt, which is used in

Eq. (27) (see also Eq. (14)). The nonlinear source termsare the remaining terms T(n) (n52) in the polynomialexpansion.

What is usually termed nonlinear eddy-viscosity mod-els are models in which a polynomial expansion is as-sumed that is a subset of Eq. (76), with the distinguishingfeature that the expansion coe$cients are determinedbased on calibrations with experimental or numericaldata, and on some physical consistency constraints. Onthe other hand, what is usually termed explicit algebraicstress models, or explicit algebraic Reynolds stress mod-els, are models in which a polynomial expansion is, onceagain, assumed from Eq. (76), but the expansion coe$-cients are derived consistent with the results ofSection 3 from the full di!erential Reynolds stress equa-tion. Thus, in both cases, an explicit tensor representa-tion for b is obtained in terms of S and W; however,despite this formal functional equivalence between thesenonlinear eddy viscosity and algebraic stress models,fundamental di!erences exist which "nd their origin pre-cisely in the way the expansion coe$cients are obtained.

4.1. Examples of nonlinear eddy-viscosity models

As might be expected, there have been a signi"cantnumber of nonlinear eddy-viscosity models proposedover the last decade. Prior to that, attempts at usingalgebraic relations for the Reynolds stress s (or aniso-tropy b) mainly focused on the solution of implicit alge-braic stress models which will be discussed in Section 5.While it is not possible to list or discuss all the NLEVMswhich have been proposed, it is possible to highlighta few models which represent the types of derivationsused. The following examples of nonlinear eddy-viscositymodels represented by the tensor expansions in Eqs. (25)or (26) include forms where both terms quadratic andcubic in the mean strain rate and rotation rate tensors areretained. Each of the examples (while not all inclusive)provide insight into the variety of assumptions requiredin identifying the expansion coe$cients a

nneeded in the

algebraic representation of the Reynolds stresses.Consider the quadratic NLEVM proposed by Shih

et al. [24]

b"a1S#a

2(SW!WS). (78)


Fig. 2. Evolution of the stress anisotropy in the invariant map(III

b,!II

b) in the case of a fully developed plane channel #ow:

(L), DNS [30]; (2), LEVM; (- - - ), NLEVM [24]; ( ) - ) -),EASM; (**), DSM [31].

This expansion uses the basis tensors T(1) and T(2). Thequadratic basis element T(4) was removed from the ex-pansion when the RDT result for rapidly rotating turbu-lence (no shear) was used. This result showed that therewas no e!ect on the (isotropic) turbulence due to therapid mean rotation. In order for the algebraic representa-tion for b to be consistent with this result, the dependencyon T(4) needs to be removed. The a

icoe$cients in (78)

were determined by applying the realizability constraints

qbb*0, no sum (79)

and

q2bc)qbbqcc , Schwarz inequality (80)

to the limiting cases of axisymmetric expansion and con-traction. Application of these constraints yielded a

3"0

which removed the remaining quadratic basis elementT(3) from the expansion for b. The coe$cients were opti-mized by further comparison with experiment and nu-merical simulation of homogeneous shear #ow and theinertial sublayer. Initial validation studies were run onrotating homogeneous shear #ow, backward-facing step#ows, and con"ned jets with overall improved predic-tions over the linear eddy-viscosity models. The valuesused for the coe$cients and other details of the calib-ration process are given in Shih et al. [24]. A similarfunctional form was also used by Lien and Leschziner[25] to study high-lift airfoil #ows; although, the coe$c-ient values for the corresponding K and e equations usedwere di!erent. Irrespective of the calibration process, aninherent de"ciency associated with the functional formarises already in the case of simple unidirectional shear#ow (say Lu

i/Lx

j"Sd

12). In this case, b

11"!b

22, and

b33

"0 always which is inconsistent with observedbehavior of such #ows.

Earlier attempts at developing quadratic models of theform given in (78) include the models of Yoshizawa[26,27], Myong and Kasagi [28], and Rubinstein andBarton [29]. The functional forms of the representationswere expressed in terms of the mean velocity gradientsand were not formulated based on tensor representationideas. Nevertheless, the forms presented could easily berewritten in terms of the (quadratic) tensor bases in (76).It is obvious from relation (75) that quadratic representa-tions which omit the T(3) term yield the condition thatIII

b,0 in any two-dimensional yow* a condition which

is always obtained with linear EVMs (see Eq. (14)). AsFig. 2 shows, such models yield predictions for planechannel #ow which are inconsistent with numerical simu-lation data [30] (DNS) as well as predictions froma Reynolds stress model [31] (RSM). (Results from theformulation of the EASM model will be discussed later inSection 5.2.3.)

In each of these studies the expansion coe$cients weredetermined in a di!erent manner. In [27], the a

i's

were determined from a plane channel #ow calibration.In [28], the a

i's were determined by optimization from

#ow predictions, and in [29], the expansion coe$cientswere determined from the renormalization group ap-proach used to construct the model.

While quadratic models have been widely used, morerecently some have argued (e.g., [32]) that the range ofapplicability of such models is limited and that higher-order terms are needed to be able to predict #ows withcomplex strain "elds. Craft et al. [32] considered a modelof the form

b"a1S#a

2(SW!WS)

#a3(S2!1

3MS2NI)#a

4(W2!1

3MW2NI)

#a6(W2S#SW2!2

3MSW2NI). (81)

Calibration of the closure coe$cients was based on anoptimization over a wide range of #ows including; planechannel #ow, circular pipe #ow, axially rotating pipe#ow, fully developed curved channel #ow, and impingingjet #ows. In order to improve the predictive capability ofthe NLEVM in the near-wall region, Craft et al. [33] (seealso [34]) introduced the transport equation for the sec-ond invariant of the Reynolds stress anisotropy II

b(or

A2

in the Craft et al. notation). These ideas were extendedby Suga [35] who introduced a transport equation forthe Lumley #atness factor [23] (replacing the transportequation for II

b). The motivation was to improve the

predictions of #ows where the turbulence evolves to itstwo-component limit.

The examples just presented showed models developedbased on the functional dependency for the Reynoldsstress anisotropy b given in Eq. (28). As a last example,


the quadratic model proposed by Speziale [36] is con-sidered. Speziale's approach, while also motivated by theneed to include Reynolds stress anisotropy e!ects intoa linear eddy-viscosity type of formulation, di!ered inits development of the tensor representation for theReynolds stress anisotropy. Speziale assumed that theanisotropy tensor b

ijwas of the form

bij"b

ijAK, e, Skl,dS

kldt B, (82)

where

dSkl

dt"

DSkl

Dt!

Lu6k

Lxm

Sml!

Lu6l

Lxm

Smk

(83)

is the Oldroyd convective derivative (e.g., [37, p. 185]).The dependency on the convective derivative was used toensure that the nonlinear polynomial approximationwould be frame-indi!erent, in keeping with the frame-indi!erent properties of the anisotropy tensor itself.These proposals also extended the earlier ideas of Rivlin[38] who likened the behavior of a turbulent Newtonian#uid to the behavior of the laminar #ow of a non-Newtonian #uid. While Rivlin's early ideas were basedon the observed in#uence of the normal stresses on the#ow in tubes, and a functional form for the stress tensorwas proposed [38], it appears these ideas were never fullypursued.

The resulting tensor representation for the Speziale[36] model was

b"a1S#a

2(SW!WS)#a

3(S2!1

3MS2NI)#a

D

DS

Dt,

(84)

where aD(K, e) was a closure coe$cient determined from

the calibration. Written in this form, it can be seen thatthe introduction of the frame-indi!erent convective de-rivative simply modi"es two of the tensor bases given inEq. (76). Calibrations were based on fully developedchannel #ow predictions using both K}l and K}e two-equation models. Preliminary validation studies weredone for a rectangular duct #ow and for a backstep #owto highlight the improved predictive capabilities of thenonlinear model over the corresponding linear eddyviscosity K}l and K}e forms.

The previous examples of NLEVMs focused on theform of the representation used for the Reynolds stresses.Associated with each of these forms, suitable scale equa-tions (K}e, K}u, 2) need to be used. As might beexpected, a wide variety of scale equations have beenused in the application of these models to a variety of#ows; although, in the development of the models, theK}e formulation was usually used. These applicationsincluded both the incompressible and compressibleregimes, and wall-bounded and free shear #ows. This

variability and range of applications can easily exceed thebounds within which the models were developed. As thepreceding discussion showed, the tensor representationsthemselves are simply kinematic relations dictated by thefunctional dependency of the Reynolds stresses and therelated anisotropies. The in#uence of any #ow dynamicsis (indirectly) con"ned to the calibration process asso-ciated with the expansion coe$cients and the choice ofdi!erential scale equations to be solved in conjunctionwith the algebraic models. Both the calibration processand choice of scale equations can have a signi"cant (if notprimary) in#uence on the model predictions. This is par-ticularly true if the key dynamics is dictated by factorsother than the anisotropy of the turbulence.

5. Algebraic stress models

The identifying feature of algebraic stress models(ASMs) is the technique used to obtain the expansioncoe$cients a

n. In the development of these models, the

coe$cients have a direct relation to a correspondingReynolds stress model, or more speci"cally, to the pres-sure}strain rate correlation model used in the Reynoldsstress model. The development of such models dates backalmost 30 years. The implicit algebraic form for the mod-els is extracted from the second-moment closure for theReynolds stress (or Reynolds stress anisotropy) usingassumptions on the advection, turbulent transport, andviscous di!usion terms. From this implicit form, anexplicit representation can be extracted which has thefunctional form given in Eq. (10), and using the basis(or a subset) given in (76).

For the most part, the implicit form of the algebraicstress model has been derived from Eq. (10) according toa procedure "rst proposed by Rodi [39,40]. The implicitform was obtained by assuming a weak equilibrium con-dition on the turbulent stress anisotropy

Dbij

Dt"0 (85)

and by assuming the following anisotropic behavior forthe turbulent transport and viscous di!usion term

Dij"

qij

KD. (86)

With these conditions, an implicit algebraic model for thestress anisotropy b

ijcould be extracted from the form

given in Eq. (10). Unfortunately, the implicit algebraicforms were not computationally very robust. Thus, it wasnecessary to provide corresponding explicit algebraicforms (or solutions) to the implicit equations.

Variations on both Eqs. (85) and (86) have been pro-posed. In Section 5.2.2, the implications on the frame-invariance properties of the algebraic stress model due to


(85) will be discussed as well as an alternative condition.Another variation to Eq. (85) has been proposed byTaulbee [41] who invoked the equilibrium condition

D

DtAbij

qJg1B"0. (87)

This had the e!ect of including the closure coe$cientsfrom the dissipation rate equation into the a

4coe$cient

appearing (see Eq. (12)) in the implicit algebraic stressmodel.

An alternative to Eq. (86) can be readily extracted byrewriting the condition as

Dij!

qij

KD"D

ij!

2

3Dd

ij!2Db

ij. (88)

The right-hand side of Eq. (88) is now the sum of thedeviatoric part of D

ijand a term proportional to the

anisotropy tensor bij

with scalar coe$cient D. With thisform, the term proportional to the anisotropy b

ijcan be

combined with similar terms appearing in (10), and theterm proportional to the deviatoric part of D

ijis then

assumed to vanish

Dij!2

3Dd

ij"0. (89)

This has a similar e!ect on the form of the algebraicmodel as the Taulbee [41] alternative just discussed.Carlson et al. [42] have used this assumption on theturbulent transport and viscous di!usion term to im-prove their EASM predictions in turbulent wake #ows.Both assumptions, Eqs. (86) and (89), can be combined inthe expression

Dij!

qij

KD"!2c

2Db

ij. (90)

The value c2"0 leads to the original equilibrium

assumption for the turbulent transport and viscousdi!usion term (86), whereas c

2"1 corresponds to the

alternative assumption (89).Pope [20] appears to be the "rst to systematically use

a tensor polynomial expansion to obtain an explicit alge-braic representation for the turbulent stresses. The pro-cedure outlined was for two-dimensional mean #ows andconsisted of a tensor basis from (76) with N"3. Taulbee[41] also used the Pope procedure to obtain an explicitrepresentation. Gatski and Speziale [21] used a similarprocedure as Pope [20] to extend the explicit algebraicmodel to three-dimensional mean #ows. This requiredthe use of the full 10 term basis given in (76). Other morerecent approaches include the work of Apsley and Les-chziner [43] and Wallin and Johansson [44]. These arecubic models (although Wallin and Johansson also pro-pose a quartic form which includes T(7) from (76)) whichhave been re-calibrated to account for near-wall or low-Reynolds number e!ects. Nevertheless, even with these

higher-order models, the starting point was an implicitalgebraic form derivable from the di!erential stressmodel.

In the remainder of this section, these ideas will berecast into both more general implicit and explicit forms.In particular, the methodology introduced in Section 3for obtaining tensor representations will be applied to thedevelopment of an explicit algebraic stress model.

5.1. Implicit algebraic stress model

An implicit stress relation can be obtained from themodeled transport equation for the Reynolds stress an-isotropy (10) using the assumption on the turbulenttransport and viscous di!usion term (90). Since Eq. (90) isexactly veri"ed in homogeneous turbulent #ows (even forc2"1), the model development to be detailed will be

valid within the context of homogeneous turbulence. Ofcourse, such models are developed for practical #owsituations which are inhomogeneous, so that the conse-quences of using the algebraic model in #ows whereEq. (90) is not satis"ed will also be discussed.

With Eq. (90) satis"ed, Eq. (10) becomes

Dbij

Dt"!

bij

a4

!a3AbikSkj

#Sikbkj!

2

3bmn

Smn

dijB

#a2(b

ik=

kj!=

ikbkj

)

#

1

qa5Abikbkj!

1

3bmn

bmn

dijB!a

1Sij!

1

qdij, (91)

or rewritten using matrix notation

Db

Dt"!

1

a4

b!a3AbS#Sb!

2

3MbSNIB

#a2(bW!Wb)#

1

qa5Ab2!

1

3Mb2NIB!L. (92)

The generalization implied in (92) by using L is intendedto indicate that the right-hand side of Eq. (92) can con-tain any known symmetric, traceless tensor function ofthe tensors S and W [22]. In the context of Eq. (91),L"a

1S#d/q contains the e!ects of dissipation rate

anisotropy. The coe$cients ai

have been given pre-viously in Eq. (12); however, the coe$cient a

4will di!er

from its form given by Eq. (12) when Eq. (90) is used,

a4"qC(c0!c

2)AP

e B#c1#c

2A1#1

eDK

Dt BD~1

, (93)

where Eq. (4) for the turbulent kinetic energy has beenused to express the turbulent transport and viscous di!u-sion term D as

D"

DK

Dt#e!P. (94)


For c2"0, the expression of a

4corresponding to

Eq. (12) is recovered.Clearly, the expression given in (93) when coupled with

Eq. (92) could be a rather di$cult system to solve numer-ically. In the discussion of the explicit representationsin the next section, additional assumptions will lead toa simpli"ed approximation to (93).

Eq. (92) has to be solved for b and is an implicit,time-dependent di!erential equation. Most of the pre-vious modelling e!orts have focused on its asymptoticsolution, corresponding to the case of equilibrium turbu-lence. Although the present methodology will be appliedto the derivation of algebraic stress models accountingfor the full evolution of the Reynolds stresses, the asymp-totic form of such a di!erential equation could be ap-proximated by solving numerically in an iterative fashionthe implicit equation obtained by imposing Db/Dt"0 in(92). Unfortunately, such procedures can be numericallysti!, depending on the complexity of the #ow to besolved. As shown earlier in Eq. (85), the imposition ofthe weak equilibrium condition in Eq. (92) has beenthe usual starting point for the explicit representa-tions [20,21]. A general methodology will nowbe presented that allows for the systematic identi-"cation of the time-dependent coe$cients a

nfrom an

implicit evolution equation such as that given in Eq. (92).All the previous algebraic stress models mentioned abovecan be included within this framework as particularcases.

5.2. Explicit representation

The tensor equation (92) governing the evolution of thestress anisotropy cannot be manipulated further becauseit involves matrix products and their transpose. Evenwithout the quadratic term (a

5"0), the terms that factor

b cannot all be grouped to allow the simple integration ofthe ordinary di!erential tensor equation. Moreover, thedependence of the coe$cient a

4upon b through the

production-to-dissipation ratio renders Eq. (92) genu-inely nonlinear. The following technique, however, trans-forms the tensor relation into an equivalent system ofscalar ordinary di!erential equations which is bettersuited for further analysis and which in turn can besolved using the standard tools of the theory of systemsof scalar di!erential equations. Following the formalismintroduced in Section 3, Eq. (92) can be solved a% laGalerkin by projecting this relation onto the tensor basis(T(1), T(2),2,T(N)) itself. For this solution, the scalarproduct of Eq. (92) is formed with each of the tensors ofthe representation basis. Using representations (30)and (63),

b"N+n/1

an

a0

T(n) and b2!1

3Mb2NI"

N+n/1

bn

b0

T(n), (95)

this procedure leads to the following system of equations:

N+n/1A

D

DtCan

a0

T(n)D,T(m)B"N+n/1

an

a0C!

1

a4

(T(n),T(m))

!2a3(T(n)S,T(m))#2a

2(T(n)W,T(m))D

#

1

qa5

N+n/1

bn

b0

(T(n),T(m))!(L,T(m)) (96)

with m"1,2,N. As shown in Section 3, the representa-tion coe$cients b

nof the quadratic term are functions of

the products aiajonly. Using Eq. (66), the terms in b

nin

(96) can be directly replaced by the quadratic terms in an,

N+n/1

bn

b0

(T(n),T(m))"N+n/1

N+k/1

anak

a20

(T(n)T(k),T(m)). (97)

Assuming that the kinematic tensors S and W of thehomogeneous mean #ow are constant in time, the deriva-tive on the left-hand side of (96) can be further simpli"ed,and the equation can be rewritten in a more compactform as

N+n/1

G(N)nm

d

dtAan

a0B"

N+n/1CAnm

#

N+k/1

QnkmA

ak

a0BD A

an

a0B

!¸m

(m"1,2,N), (98)

where the N]N Gram matrix G(N)nm

is de"ned as

G(N)nm

,(T(n),T(m)), (99)

the coe$cient matrices Anm

and Qnkm

are de"ned as

Anm

,!

1

a4

(T(n),T(m))!2a3(T(n)S,T(m))

#2a2(T(n)W,T(m)), (100)

and

Qnkm

,

1

qa5(T(n)T(k),T(m)), (101)

and the independent term ¸m

is given by

¸m,(L,T(m)). (102)

The coe$cient matrices G(N)nm

, Anm

, Qnkm

and ¸m

are alleasily evaluated once the representation basis(T(1)T(2),2,T(N)) has been chosen. As required from theisotropy condition (29), the coe$cients are all scalarinvariants, and will ultimately be single-valued functionsof the invariants g

i(i"1,2, 6) forming an integrity

basis, as de"ned in (50). It should be pointed out here thatthe use of the Cayley}Hamilton theorem to reformulatethe entries of the coe$cient matrices G(N)

nm, A

nm, Q

nkmand


¸m

in terms of the invariants gi(i"1,2, 6) is not essen-

tial, and only constitutes a convenient simpli"cation. Inthe limit, a direct computation of the entries of G(N)

nm, A

nm,

Qnkm

and ¸m

from their de"nitions, without further sim-pli"cation, would lead to exactly the same representa-tion, although with a more complex formal expressionfor the coe$cients.

Eq. (98) is a system of N algebraic ordinary di!erentialequations in the N unknown representation coe$cients(a

n/a

0)(t). This dynamic system is dependent on the initial

conditions an,0

"an(0) which can be obtained from the

initial value of the anisotropy tensor b0"b(0) using

representation (30). System (98) of scalar ordinary di!er-ential equations with representation (30) is equivalent tothe original tensor evolution equation for the Reynoldsstress anisotropy (92). However, it has a signi"cant ad-vantage in that it is much more tractable and bettersuited for analysis than the original tensor equation. Anyexpression of the extra anisotropy tensor L can be pro-vided which involves the stress anisotropy itself to anydegree of complexity. It then su$ces to study the result-ing dynamical system (98) to have a complete descriptionof the evolution of the Reynolds stress anisotropy tensoras governed by (92).

As mentioned before, the equation governing theevolution of the stress anisotropy (92), or equivalently(98), has to be solved together with equations for turbu-lence scale-determining quantities. These quantities onlyenter in the anisotropy evolution equation through theturbulent time-scale q"K/e, and through the nondimen-sional ratio e~1DK/Dt when c

2"1. This is true even if

no quadratic terms are included in the pressure-straincorrelation (a

5"0), and no additional anisotropy

(L"a1S) or relaxation (c

2"0) e!ects are taken into

account. When Eq. (93) for a4

is used, it can be expressedas

a4"qC!2(c

0!c

2)q

N+n/1Aan

a0BMT(n)SN#c

1#c

2

]A1#1

eDK

Dt BD~1

, (103)

where the production-to-dissipation rate ratio has beenrewritten as

P

e"!2qMbSN"!2q

N+n/1Aan

a0BMT(n)SN. (104)

Note also that although the present formulationuses constant values for the closure coe$cientsC0

1,C1

1,C

2,2, C

4, they could, in principle, be general

functions of the #ow-"eld invariants g1,2, g

6and the

turbulence time scale q (or related quantity such as inEq. (93)). This case is also covered by the present solutionframework. In the case of a K}e formulation, theequation governing the evolution of the turbulence time

scale is obtained from the evolution equations for theindividual scales K and e,

DqDt

"

1

eDK

Dt!

K

e2DeDt

. (105)

Eq. (105), coupled with Eq. (98) for the representationcoe$cients for the Reynolds stress anisotropy tensor,fully determines the evolution of the turbulent quantitiesin the framework of second-moment closures in a homo-geneous mean #ow. In the context of the derivation ofexplicit algebraic Reynolds stress models, the additionalweak equilibrium condition (85) is imposed on system(92),

Db

Dt"0, (106)

or equivalently on system (98) since the homogeneousmean #ow is assumed to be time-independent,

d

dtAan

a0B"0. (107)

In this case, the asymptotic value (an/a

0)=

of the repres-entation coe$cients is obtained by solving the nonlinearalgebraic system of N scalar equations

N+n/1CAnm

#

N+k/1

QnkmA

ak

a0B=DA

an

a0B=

!¸m"0

(m"1,2,N). (108)

As mentioned before, the nonlinearity in system (108)enters through the coe$cient a

4(see (103)), through the

quadratic term in the pressure}strain correlation(a

5O0), and possibly through the expression of the extra

anisotropy tensor L. The resulting explicit algebraicstress model is then simply obtained by using the repres-entation with the equilibrium values of the coe$cients a

n,

b="

N+n/1Aan

a0B=

T(n). (109)

Strict equilibrium [45] in homogeneous turbulence isachieved when both Dq/Dt"0 and Db/Dt"0. In solv-ing system (108) in the context of homogeneous turbu-lence at equilibrium, the value of the turbulence timescale is consistently taken as its equilibrium value ob-tained by imposing Dq/Dt"0 in (105). In the case ofa K}e closure using (4) and (20) in homogeneous #ows,the equilibrium condition Dq/Dt"0 leads to the follow-ing relations for the relative turbulence production rateand the production-to-dissipation rate ratio,

1

eDK

Dt"

K

e2DeDt

"Ce1AP

e B=

!Ce2 (110)


and

1#1

eDK

Dt"A

P

e B=

"

Ce2!1

Ce1!1. (111)

Note that the second equality in Eq. (111) is only validwhen the standard evolution equation for the dissipationrate (20) is used. The equilibrium value of the turbulenttime scale q

=is then indirectly obtained by inverting

relation (104) which links the equilibrium values of theproduction-to-dissipation rate ratio, of the turbulencetime scale, and of the stress anisotropy.

These results extracted from the time scale equilibriumcondition can also be used to develop an alternate formfor the expression for a

4. Inserting the equilibrium rela-

tion (111) into Eq. (93) leads to

a4"qC(c0!c

2)AP

e B#c1#c

2AP

e B=D

~1. (112)

The values for Ce1 and Ce2 used to determine (P/e)=

in(111) should be the same as those used in the isotropic,dissipation rate equation (20) in conjunction with thecalibration of the particular pressure}strain rate model.This modi"cation of the expression of the coe$cienta4

meets the requirement that the results in homogene-ous #ows are unaltered and only slightly a!ects theresults for the log-layer [42].

Virtually, all the previously developed explicit alge-braic stress models (EASM) were based on equilibriumconsiderations, and they can be obtained within the pres-ent framework as particular cases of system (108). Theearly EASM of Pope [20] and the EASM of Gatski andSpeziale [21] were obtained for two-dimensional (N"3)and three-dimensional #ows (N"10), respectively, in thecase of a linear pressure}strain correlation model(a

5"0) and without extra anisotropy e!ects (L"a

1S).

Gatski and Speziale [21] linearized system (108) byassigning a constant value to the coe$cient g,a4/q ("0.233) with c

2"0. This value was based on the

equilibrium value of the ratio P/e"1.88 in homogene-ous #ows [21] obtained from (111) using calibratedvalues for the closure coe$cients Ce1 and Ce2 . Due to thelinearization, #ow "elds could exist where singularities inthe expansion coe$cients would arise. This problem wasremedied through a regularization [21] of the analyticform for the expansion coe$cients. The three-term rep-resentation used by Gatski and Speziale [21] was ob-tained by retaining only the "rst three coe$cients of thesolution of (108) with N"10. It is important to realizethat this truncation to the "rst three terms from the10-terms solution leads to an expression of the repres-entation coe$cients which is diwerent from the expres-sion obtained by solving (108) with N"3 in the case ofthree-dimensional #ows. Due to its construction as a

projection, the latter solution leads to the optimal repres-entation of b with a three-term basis for three-dimen-sional #ows.

The singularity which arose due to the linearization ofa4

(with c2"0) has been removed in later models by

taking into account the nonlinearity introduced intoa4

through P/e. This improvement was applied by Yingand Canuto [46] and Girimaji [47] to a three-termsrepresentation (N"3), and by Wallin and Johansson[44] to a "ve-terms representation (N"5) using a sim-pli"ed pressure strain correlation model. In all thesecases, the solution of system (108) was obtained by solv-ing a cubic (resp. "fth order) scalar polynomial. In thecase of the cubic scalar polynomial equation, Jongen andGatski [48] further analyzed the equilibrium statesof a more general time-dependent formulation andrigorously determined the proper choice among the rootsof the polynomial.

Analysis of algebraic stress models taking into accounta quadratic term in the pressure}strain correlation modelwas performed by Jongen and Gatski [49] and an ex-plicit algebraic stress model accounting for extra aniso-tropy e!ects generated by the dissipation rate tensor wasderived in the case of a three-term representation byJongen et al. [50].

5.2.1. Extension to inhomogeneous yowsExplicit algebraic stress models are developed for ap-

plications in nonhomogeneous and nonequilibrium situ-ations. In this case, the same formal expression for thecoe$cients (a

n/a

0)=

obtained by solving (108) is used.The turbulence time scale q entering in the expression ofthe representation coe$cients is, in general, now a func-tion of both space and time. As such, q is obtained bysolving the transport Eqs. (4) and (20) for K and e in thesemore complex #ow "elds. Additional consequences arediscussed in Section 5.2.3 in the case of two-dimensionalmean #ows.

5.2.2. Frame-invariance propertiesOne of the prime motivating factors in developing

ASMs is the hope that their predictive behavior willclosely replicate that of the full di!erential stress model.Another aspect to this coupling which has not beenaddressed up to this point is the comparison between theframe-invariance properties of the algebraic stress modeland the di!erential stress model. Speziale [6,7,51] hasaddressed this issue in the context of the Reynolds stressor second-moment closure transport equations. Heshowed that while the Reynolds stresses themselves areframe-invariant, the transport equations which governtheir evolution in the #ow "eld will only be frame-invari-ant under the extended Galilean group. This is due to thefact that the frame-dependence enters into the #uctuationdynamic equations through the appearance of the abso-lute (or intrinsic) mean rotation rate tensor (see Eq. (9)).


The previous development of algebraic stress models wassu$ciently general to be applicable to #ows in noniner-tial frames (with rigid-body rotation relative to the iner-tial frame given by X

i).

In the present analysis, an alternative formulation tothe development of the algebraic model will be followedwhich will exploit more general group transformationproperties of the full di!erential stress anisotropy equa-tions. This will allow for a broader applicability of theresultant algebraic stress model.

Consider "rst a general Euclidean transformation ofDb/Dt from an initial inertial frame

D

Dtb(SM ,WH,q)"QC

D

Dtb(S,x, q)DQT

#

DQ

Dtb(S,x, q)QT#Qb(S,x, q)

DQT

Dt, (113)

or rewritten using bM "b(SM ,WH, q)

Db1Dt

"QDb

DtQT#bM XM !XM b1 , (114)

where

SM "QSQT and WH"Qx QT#X, (115)

and

XM "QDQT

Dt(116)

is the rate of rotation of the proper orthogonal tensorQ(t), or the relative angular velocity of the two frames. Asis well known, the rotation rate tensor is not frame-invariant (not objective) and as such transforms accord-ing to Eq. (115) (e.g., see [7]). With these relations, thetransformation of Eq. (92) under the more generalEuclidean group is given by

Db1Dt

"!

1

a4

bM !a3Ab1 S1 #SM b1 !

2

3Mb1 S1 NIB

#a2(b1 WM !WM b1 )#

1

qa5Ab1 2!

1

3Mb1 2NIB!L1 , (117)

where

WM "WH#1

a2

XM "x6 #A1#1

a2BXM (118)

and the overbar denotes the tensorial form in the trans-formed system.

A simple example will serve to link the general trans-formation group framework described here with the for-mulation used earlier which accounted for noninertial

frames. Consider the case of rotating homogeneous shear#ow relative to a "xed Cartesian frame with rigid-bodyrotation X about the axis (z-axis) perpendicular to theplane of shear (x}y plane). This #ow has been usedpreviously, for example, in the study of equilibrium statesof homogeneous turbulence [52], and in the calibrationof a pressure}strain rate correlation model [9]. At equi-librium Db1 /Dt"0, and the #ow "eld is described in thenoninertial (barred) frame by an implicit equation forb1 given by

1

a4

b1 #a3Ab1 S1 #S1 b1 !

2

3Mb1 S1 NIB

!a2(b1 WM !WM b1 )#a

1S1 "0, (119)

where only the linear form is used (a5"0) and L1 "a

1S1 .

In this rotating shear case, the relative rotation ratebetween the barred and "xed frames is given byX

r"(0, 0,X), and is simply related to XM (,XM

ij) through

!eijk

Xk

so that

=Mij"u6

ij!A1#

1

a2BeijrXr

. (120)

Eq. (120) is the intrinsic rotation rate tensor and isconsistent with the form used previously in Eq. (11).Thus, the expression given in (119) is consistent with theearlier formulations for rotating shear #ows [9,52] andthe formulation used here applicable to noninertialframes.

Clearly, the situation which arises in the simpleexample of rotating homogeneous shear can be generaliz-ed to more general #ows where the weak equilibriumcondition Db/Dt"0 is not satis"ed but where the condi-tion Db1 /Dt"0 is. In the rotating case just discussed, the#ow "eld in the noninertial frame (the barred frame) wasa simple homogeneous shear; however, in more complexcases where the #ow "eld is simply solved for numer-ically, the computations are based in some "xed frame.Nevertheless, if an equilibrium condition can be estab-lished in some other frame (Db1 /Dt"0) then the math-ematical developments used in this section can still beapplied.

At the outset, assume that Db1 /Dt"0 so that Eq. (114)can be written as

Db

Dt"bX!X b, (121)

where

X"QTDQ

Dt. (122)

If Eq. (121) is now used as the condition on Db/Dt ratherthan Db/Dt"0, then the resulting implicit algebraicequation for the stress anisotropy in an inertial frame is


given by

1

a4

b#a3Ab S#S b!

2

3MbSNIB!a

2(b Ws!Ws b)

!

1

qa5Ab2!

1

3Mb2NIB#L"0, (123)

where

Ws"x#

1

a2

X. (124)

The remaining question now is the choice of frame wherethe condition Db1 /Dt"0 should be applied. It should beemphasized that the only e!ect this new condition onDb/Dt has on the implicit formulation is through therotation rate tensor Ws.

Astarita [53] (see also [54,55]) addressed a relatedissue of "nding a generally applicable criteria for #owclassi"cation, and determined that any relevant criteriashould be made based on a rate of rotation which couldbe objectively measured. This could be done if X wasbased on the principal axes of the strain rate tensor S.(Den Toonder et al. [56] later used these ideas in a studyidentifying strong #ow regions in turbulence.) Therefore,the transformation matrix Q is simply identi"ed with theeigenvectors of the strain rate matrix S, that is

Q"(e(1)

, e(2)

, e(3)

)T. (125)

At each point in the #ow, the local eigenvalues andeigenvectors of the strain rate tensor as well as theirmaterial derivative need to be evaluated. While thisprocedure has not been implemented in three-dimen-sional #ows, it has been applied in a two-dimensionalcurved #ow case where the condition Db/Dt"0 wasviolated in the reference frame used for the numericalcomputations. The formulation and results are discussedin the next section.

5.2.3. Explicit solution for two-dimensional mean yowsWhile it is possible to implement the methodology

described earlier in this section using any number ofterms in the tensor representation T(N), it is di$cult toobtain closed-form analytic expressions beyond N"3.Thus, this section will be limited to N"3 where thethree-term basis representation,

a0b"a

1S#a

2(SW!WS)#a

3(S2!1

3MS2NI) (126)

is used. In Eq. (126), the aiare scalar coe$cient functions

of the invariants of S and W given in (50). For a two-dimensional mean #ow, the three-term representation isalways exact and for three-dimensional mean #ow, thethree-term representation is an approximation, as shownin Section 3.3.2.

As discussed earlier, the coe$cient a4

in (100) intro-duces a nonlinearity into the system due to its depend-ence on the production-to-dissipation ratio P/e. For thethree-term representation basis (126) used here, this ratiocan be rewritten as (see Eq. (104))

P

e"!2qMbSN"!2qg

1Aa1

a0B (127)

and the coe$cient a4

then expressed as

a4"qC!2qg

1(c

0!c

2)A

a1

a0B#c

1

#c2A1#

1

eDK

Dt BD~1

. (128)

The alternative form for a4

given in Eq. (112) and extrac-ted when an additional equilibrium assumption on theturbulent time scale is used, can be written in the two-dimensional case as

a4"qC!2qg

1(c

0!c

2)A

a1

a0B#c

1#c

2AP

e B=D

~1.

(129)

The e!ect of using the expression for a4

given in (129)with c

2"1 rather than c

2"0 is readily seen in this

two-dimensional case (three-term basis). Fig. 3a showsthe theoretical values of CHk ("a

1/qa

0) as a function of

the #ow parameter [49] R2 ("!g2/g

1) for various

levels of P/e when c2"0 is used in the expression of a

4.

Note that for regions in an equilibrium log-layer, whereR"1 and P/e"1, the value of CHk is approximately0.096, which is the `equilibriuma level employed in themodel. However, in other regions of the #ow "eld, CHk canassume unreasonably high levels. For example, near thecenterline of a wake, P/e can be small and R tendstoward zero. In this case, the a

4given by Eq. (12), or

Eq. (129) with c2"0, yields unrealistically large levels of

CHk near 0.68! Carlson et al. [42] found that as a result,the turbulent eddy viscosity produced near the center ofa wake is very high, and the velocity pro"les tend to besomewhat `#attened.a Results obtained using the expres-sion for a

4given in (129) with c

2"1 are shown in

Fig. 3b. The "gure shows that CHk+0.0885 in the log-layer (again corresponding to the `equilibriuma levelemployed in the model), and also more reasonable levelswhen P/e is small. For example, the maximum level ofCHk is less than 0.19, as opposed to 0.68. With c

2"1,

Carlson et al. [42] found that the resulting wake pro"lesmore closely replicated the experimental form. It shouldbe noted that Fu et al. [57] may have been the "rst torecognize the adverse e!ects on free shear #ow predic-tions of the large values of the eddy viscosity which thealgebraic stress models could produce.


Fig. 3. Values of CHk as a function of R2"!g2/g

1for di!erent

levels of P/e using a4

given by Eq. (129): (a) with c2"0; (b) with

c2"1.

For a two-dimensional mean #ow "eld, the coe$cientmatrix A

nmis easily evaluated as

Anm

"C!

1

a4

g1

!2a2g1g2

!13a3g21

2a2g1g2

2

a4

g1g2

0

!13a3g21

0 !

1

6a4

g21D. (130)

The entries of the coe$cient matrices G(3)nm

andQ

nkmhave already been obtained in Sections 3.3.2 and

3.4, respectively. In summary, system (98) of three alge-braic ordinary di!erential equations in the three un-known representation coe$cients (a

n/a

0), n"1, 2, 3, is

given by

d

dtAa1

a0B"!

1

a4Aa1

a0B#2a

2g2A

a2

a0B!

1

3a3g1A

a3

a0B

#

1

6qa5g1A

a1

a0BA

a3

a0B!

1

g1

¸1,

d

dtAa2

a0B"a

2Aa1

a0B!

1

a4Aa2

a0B#

1

6qa5g1A

a2

a0BA

a3

a0B

#

1

2g1g2

¸2,

d

dtAa3

a0B"!2a

3Aa1

a0B!

1

a4Aa3

a0B#

1

qa5CA

a1

a0B

2

!2g2A

a2

a0B

2!

1

6g1A

a3

a0B

2

D!6

g21

¸3, (131)

where the tensor L appears through the invariants

¸1"MLSN, ¸

2"!2MLWSN, ¸

3"MLS2N. (132)

When solved together with the evolution equation forthe turbulence time scale q (105), the system gives theevolution of the Reynolds stress tensor in planar homo-geneous #ows. In the case of linear pressure}strain mod-els (a

5"0), without extra anisotropic e!ects (L"a

1S),

and without relaxation of the turbulent transport andviscous dissipation term (c

2"0), the exact, closed-form

temporal solution of (131) has been derived in a recentanalysis of second-moment closures [48].

At equilibrium (d/dt(an/a

0)"0), the resulting set of

three nonlinear algebraic equations is

Aa1

a0B=

"!a4,=

(¸1!a

2a4,=

¸2!2a

3a4,=

¸3)

g1(1!2

3a23a24,=

g1!2a2

2a24,=

g2),

Aa2

a0B=

"a4,=Ca2A

a1

a0B=

#

1

2g1g2

¸2D,

Aa3

a0B=

"!2a4,=Ca3A

a1

a0B=

#

3

g21

¸3D. (133)

The subscript a4,=

is introduced in order to emphasizethe nonlinearity introduced by a

4through Eq. (128),

which can be rewritten as

a4,=

"qC!2qg1cH0A

a1

a0B=

#cH1D

~1(134)

with

cH0,c

0!c

2,

cH1,c

1#c

2A1#1

eDK

Dt B. (135)


5The invariant-map analysis is a stringent one, since errors inthe anisotropy b

ijare squared (cubed) for II

b(III

b), respectively.

See [58] for a full comparison between the model results in thecase of the fully developed plane channel #ow.

If the form for a4

given by Eq. (129) is used, then usingrelation (111), the coe$cients cH

0and cH

1are de"ned as

cH0,c

0!c

2,

cH1,c

1#c

2ACe2!1

Ce1!1B. (136)

Inserting the expression for a4,=

into the "rst equation ofsystem (133) leads to a cubic equation for (a

1/a

0)=

givenby

cH20 A

a1

a0B

3

=

!

cH0cH1

g1q A

a1

a0B

2

=

#

1

4g21q2CcH21 !2q2cH

0¸1

!2g1q2A

1

3a23#

g2

g1

a22BDA

a1

a0B=

#

1

4g31q[cH

1¸1!q(a

2¸2#2a

3¸3)]"0. (137)

The proper solution root to this equation has been ex-tracted in [48] from an asymptotic analysis of the equi-librium state solutions to Eqs. (131). Using Eq. (127), thecubic relation (137) is easily transformed into a cubicequation for the determination of the equilibrium pro-duction-to-dissipation rate ratio (P/e)

=as a function of

the normalised strain rate g1q2, the #ow-type parameter

g2/g

1, and the normalised extra anisotropy e!ects q2¸

1,

q3¸2

and q3¸3,

cH20 A

P

e B3

=

#2cH0cH1A

P

e B2

=

#CcH21!2cH

0(q2¸

1)

!2g1q2A

1

3a23#

g2

g1

a22BDA

P

e B=

!2[cH1(q2¸

1)!a

2(q3¸

2)!2a

3(q3¸

3)]"0. (138)

Since the tensor L is assumed to be a known, symmetric,traceless tensor function of the kinematic tensors S andW, the invariants ¸

mare ultimately expressible as single-

valued functions of g1

and g2, the integrity basis corre-

sponding to planar #ows. For instance, when no extraanisotropy e!ects are included in the model formulation,L"a

1S, and ¸

m"(a

1g1, 0, 0).

Eq. (138) must be satis"ed for any equilibrium state ina planar homogeneous #ow, and is therefore a simpleconstraint imposed by the Reynolds stress equations onthe possible states in the (P/e, g

1q2, g

2/g

1) space. Since

(138) contains three unknowns (P/e, g1q2, g

2/g

1), any one

of these variables can be "xed, and the solution depend-encies of the remaining two variables studied [49]. Addi-tional information such as the knowledge of q, forexample (through the K}e equations), will simply serve asan external constraint imposed on the system in thedetermination of its equilibrium.

In planar homogeneous #ows, g2/g

1is "xed by the

particular #ow "eld under investigation (homogeneous

shear, homogeneous strain, etc.), and the equilibriumvalue of the production-to-dissipation rate ratio P/e isgiven by (111) when the standard equation for the dissi-pation rate (20) is used. In this case, the cubic relation(138) "xes the value of the normalised strain rate g

1q2.

In the case of planar inhomogenous #ows, the values ofthe kinematic invariants g

iare determined by the mo-

mentum equations (2), whereas the value of the turbu-lence time scale q is obtained through the K}e equations.Since the ASM model is directly derived from the stressanisotropy equation (91) for homogeneous #ows, it willstill satisfy the state equation (138) even in in-homogeneous #ows. In other words, every point in the(P/e, g

1q2, g

2/g

1) space attained by the system must sat-

isfy (138), even in inhomogeneous conditions. The modelis sensitized to inhomogeneities indirectly through thespatial distributions of the turbulence time scale q"K/eand of the mean #ow invariants g

i. Referring to Fig. 2

introduced in Section 4.1 for the case of a fully developedplane channel #ow, the EASM (with c

2"0, a

5"0 and

L"a1S) is shown to lead to improved predictions5 over

the NLEVM. In the log-layer region, where the equilib-rium assumptions (85) and (86) are valid [59], the EASMcomes very close to the RSM results, as could be ex-pected.

As a "nal application to two-dimensional #ows of thegeneral results obtained previously, consider the frame-invariant analysis in the previous section. An example ofwhere these results can be applied is the case of a stronglycurved #ow. Rumsey et al. [60] have shown that theequilibrium condition Db/Dt"0 does not hold and thatthe explicit algebraic stress model is not able to replicatethe results of a full di!erential stress model. In order toapply the frame-invariant results, consider once again thethree term basis representation given in Eq. (126). Forthis example, the rotation rate tensor W is simply re-placed with Ws, and all that is needed is to identifythe quantity X measuring the (local) relative velocitybetween the strain rate principal axes and the baseCartesian frame.

In order to "rst identify the transformation tensor Q,consider the transformation between the unit vectors ofthe "xed and principal axes system

e6 k(i)"X

1jiek(j)

(139)

with

X1ji"X

1"

Lxj

Lx6 i, (140)


Fig. 4. Sketch of eigenvector orientation in principal axes frame relative to Cartesian base frame.

where subscript (i) ("1, 2) is the particular unit vector,and superscript k ("1, 2) is a component of the unitvector (i). Since in this analysis, the "xed (unbarred)system is a Cartesian system where ek

(j)"dk

j, the unit

vectors in the principal axes frame are simply given by

e6 k(i)"X

1ki. (141)

Comparing Eq. (141) with Eq. (125) shows that XM"Q,

where XMis a proper orthogonal tensor with X

MXM "I, and

that from Eq. (122)

X"X1DX

1Dt. (142)

In a two-dimensional mean #ow, the eigenvalues of the

strain rate tensor are j1"!j

2"Jg

1/2, and the

transformation matrices are

XMji"C

e6 1(1)

e6 2(1)

e6 1(2)

e6 2(2)D"

1

J(S11

!j1)2#S2

12C

S12

!(S11

!j1)

(S11

!j1) S

12D,(143)

where the elements of the matrix are simply the cos andsin of an angle / with the Cartesian base (see Fig. 4). Therelative velocity X between the coordinate frame consist-ing of the principal axes and the base Cartesian frame canthen be written as

X"C0 !

D/

DtD/

Dt0 D , (144)

where

D/

Dt"

1

g1CS11

DS12

Dt!S

12

DS11

Dt D. (145)

As noted previously, Rumsey et al. [60] used this proced-ure to correct for de"ciencies in the prediction of aU-bend #ow. This strongly curved #ow was poorly

predicted using an EASM which did not account for thecurvature through the methodology outlined here. Whenthe modi"cations just discussed were introduced, thepredictions were consistent with the predictions of thefull di!erential model. Rumsey and Gatski [61] alsoapplied this technique as well as other recent proposalsfor curvature corrections to multi-element airfoil #ow"elds.

6. Summary

The development of the class of single-point closuremodels consisting of nonlinear eddy-viscosity and alge-braic (Reynolds) stress models has been presented. Thisclass of models is shown to be intermediary between thetwo-equation linear eddy-viscosity models and the fulldi!erential Reynolds stress models. As such, these modelsincorporate the robustness associated with a two-equa-tion eddy-viscosity formulation as well as the ability toaccount for turbulence anisotropy e!ects.

An outline of tensor representation theory relevant tothe construction of the tensor polynomial expansionsassociated with the algebraic representation of theReynolds stress anisotropy has been presented. This the-ory is shown to provide the framework for the develop-ment of a wide variety of nonlinear eddy-viscosity modelscurrently in use. In addition, it is also shown how tensorrepresentation theory can be used to develop a newmethodology for the systematic derivation of explicitalgebraic stress models. Utilizing this methodology, it ispossible to construct, within a consistent mathematicalframework, explicit algebraic stress models which caninclude e!ects from nonlinear pressure}strain rate cor-relation models and anisotropic dissipation rate models.In addition, nonequilibrium e!ects are also formally in-cluded in the development as well as a means for retain-ing some of the same frame-invariant properties of thefull di!erential stress closure. Further, the power of thisnew analysis tool for single-point closure models hasbeen illustrated by providing perspectives on the charac-terization of the performance of various closure levels.


This material should provide the reader with a newperspective on this class of models both from the stand-point of more e!ective utilization of the models in practi-cal computations and also from the standpoint ofdeveloping improved physical models within a single-point closure framework.

Appendix. Fundamental relation for 3]]3 tensors

The following fundamental relation, written here fortraceless tensors, is satis"ed by any 3]3 tensors a, b, c,whether or not they are symmetric [14,16]:

abc#bca#cab#bac#acb#cba

"aMbcN#bMacN#cMabN#(MabcN#McbaN)I.

If we take a"b"c, the Cayley}Hamilton theorem isobtained:

a3!13Ma3NI!1

2Ma2Na"0.

Any polynomial term in a, b and c given by

P"aa1bb1cc1aa2bb2cc22aapbbqccr ,

where ai, b

iand c

imay take any positive integer value,

can be reduced to a sum of polynomials of lower degreesthan P by repeated applications of the fundamentalrelation [14].

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