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Modelling+streamline+curvature+effects+in+explicit+algebraic+Reynolds+stress+turbulence+models[1].pd

Modelling streamline curvature e?ects in explicit algebraic

Reynolds stress turbulence models

Stefan Wallin

a,*

,Arne V.Johansson

Aeronautics Division,FFA,Swedish Defence Research Agency (FOI),SE-17290Stockholm,Sweden

Department of Mechanics,KTH,SE-10044Stockholm,Sweden

Abstract

A curvature correction for explicit algebraic Reynolds stress models (EARSMs),based on a formal derivation of the weak-equilibrium assumption in a streamline oriented curvilinear co-ordinate system is presented.The curvature correction is given from the rotation rate of the curvilinear co-ordinate system following the mean ?ow.Two methods for de?ning that rotation rate are proposed,one is derived from the strain-rate tensor,and the other is derived from the local mean acceleration vector.Both methods are fully three-dimensional and Galilean invariant and the correction vanishes in cases without curvature or rotation e?ects.The EARSM proposed by Wallin and Johansson (J.Fluid Mech.403(2000)89)was extended with the proposed curvature corrections and recalibrated in such a way that the original model was retrieved in cases without curvature or rotation e?ects.Rotating ho-mogeneous turbulent shear ?ows with vanishing mean vorticity should be close to neutral stability according to linear stability theory,also observed from large eddy simulations.This was used for the recalibration.The importance of the curvature correction and the proposed recalibration is shown for rotating homogeneous shear and rotating channel ?ows.ó2002Elsevier Science Inc.All rights reserved.

1.Introduction

Turbulent ?ows over curved surfaces,near stagnation and separation points,in vortices and turbulent ?ows in rotating frames of reference are all a?ected by streamline curvature e?ects.Strong curvature and/or rotational e?ects form a major cornerstone problem also at the Reynolds stress transport modelling level,and pressure–strain rate models that are able to accurately capture rapidly rotating turbulence are rare.In more moderate situations the SSG model (Speziale et al.,1991),and derivations thereof,show rather good behaviour in ro-tating ?ows such as rotating homogeneous shear ?ows (see Gatski and Speziale,1993).Standard eddy-viscosity models without explicit corrections,completely fail in describing e?ects of local as well as global rotation.Algebraic Reynolds stress models (Rodi,1972,1976)are the results of applying the weak-equilibrium as-sumption on the full di?erential models.In the weak-equilibrium limit of turbulence,the Reynolds stress anisotropy tensor,a ij u i u j =K àe2=3Td ij ,is assumed to

be constant following a streamline.Neglecting also the di?usion of the anisotropy tensor results in an implicit purely algebraic relation for a ij .Algebraic modelling has had a renewal during the last decade after it was found that the resulting implicit algebraic relation for a ij may be formally solved resulting in an explicit relation (see e.g.Pope,1975;Taulbee,1992;Gatski and Speziale,1993;Girimaji,1996;Johansson and Wallin,1996).The material derivative that includes advection by the mean ?ow (in the following denoted D =D t )of a scalar ?eld is invariant of the choice of co-ordinate system.However,the material derivative of a tensor ?eld of higher rank than zero,e.g.vectors and second rank ten-sors,is not invariant of the choice of co-ordinate system for representing the tensor components.

It has been suggested by e.g.Rodi and Scheurer (1983)that the weak-equilibrium assumption is better evaluated for the anisotropy tensor expressed in a streamline-based co-ordinate system.In e.g.circular ?ows where the azimuthal direction is homogeneous,the weak-equilibrium assumption is then exactly ful?lled.This approach is,however,not Galilean invariant and should not be used in a general model.

Galilean invariant methods have been proposed by Girimaji (1997)and Gatski and Jongen (2000)which

are

Corresponding author.

E-mail address:stefan.wallin@foi.se (S.Wallin).

based on the rotation rate of the acceleration vector and the strain-rate tensor,respectively,following the mean ?ow.The latter is an extension of the ideas by Spalart and Shur(1997)for correcting eddy-viscosity models. These methods are,in principal,derived for general three-dimensional(3D)?ows,but so far,only complete expressions and methods for two-dimensional(2D) mean?ows have been presented.These ideas are ex-tended in the following section and fully3D closed form relations are presented.

In cases withmoderately curved streamlines th e choice of co-ordinate system has a rather minor e?ect, see e.g.Rumsey et al.(1999)for?ow over an airfoil.

H owever,in cases withstrong streamline curvature th is e?ect is dominating.In a study of a generic wing tip far-?eld vortex by Wallin and Girimaji(2000)it was found that the turbulent dissipation of the vortex was by far overpredicted using the standard algebraic Reynolds stress models while including the e?ect of the streamline curvature gave a qualitatively correct behaviour(see Fig.1).

2.Curvature corrected model

General quasi-linear Reynolds stress transport mod-els may be written in terms of a transport equation for the anisotropy tensor

D a ij

D t

àDeaTij

?A0A3

tA4

a ijtA1S ij

àa ik X kj

àX ik a kj

tA2a ik S kj

tS ik a kjà

a kl S lk d ij

e1T

(see Wallin and Johansson,2000).DeaTij is the di?usion of a ij and s?K=e is the turbulent time scale.The strain and rotation rate tensors,S ij and X ij,are normalized by s.This relation results from the general quasi-linear model for the pressure–strain rate and dissipation rate anisotropy,e ij,lumped together

722S.Wallin,A.V.Johansson/Int.J.Heat and Fluid Flow23(2002)721–730

P ij

àe ij?à1

tC1

a ijtC2S ij

tC3

a ik S kj

tS ik a kjà

a kl S lk d ij

àC4

a ik X kj

àX ik a kj

:e2T

The A coe?cients are related to the C coe?cients through

A0?C4

à1;A1?

3C2à4

3A0

;A2?

C3à2

2A0

;

A3?2àC0

2A0

;A4?

àC1

à2

2A0

e3T

2.1.The weak-equilibrium assumption

Usually,in deriving algebraic Reynolds stress models the l.h.s.of(1)is neglected in the computational (?Cartesian)co-ordinate system.The resulting algebraic relation may be formally solved leading to an explicit algebraic Reynolds stress model(EARSM),that is an explicit relation for a ij(see Wallin and Johansson,2000). The A0coe?cient in(1)in?uences the EARSM only if a contribution of the l.h.s.of(1)is included into the EARSM.

Girimaji(1997)and Sj€o gren(1997)realized that,by imposing the weak-equilibrium assumption in a general curvilinear co-ordinate system,e s e^t;^n;^sT,an addi-tional algebraic term appears which easily can be ac-counted for in the EARSM solution.

The Cartesian(or computational)co-ordinate system e e^x;^y;^zTis transformed to e s?Te using the or-thogonal transformation T(T t T?I).The advection of the anisotropy(or any second rank tensor)may be ex-panded to

D a D t ?T t

D TaT t

D t

Tàa XerT

àXerTa

;e4T

where the?rst term on the r.h.s.is interpreted as the advection of the transformed anisotropy,a s?TaT t, transformed back to the Cartesian system,T teáááTT.Th e second term contains the antisymmetric tensor

XerT?D T t

D t

T?àT t

D T

D t

:e5T

If the advection of the anisotropy is neglected in the e s system,the resulting term(a XerTàXerTa)may be fully accounted for and included into the EARSM formula-tion simply by replacing X in(1)with

X??Xà

XerT:e6T

The formal transformation of the material derivative has earlier been presented by Girimaji(1997)and Sj€o gren(1997),the later with an extension for non-orthogonal co-ordinates.They extended this analysis witha formal derivation of th e XerTtensor from the de?nition of the co-ordinate system.That analysis be-comes rather tedious especially in general3D?ows.The ?nal expression includes derivatives of the metrics of the curvilinear co-ordinate system which may cause nu-merical problems when utilized for real3D numerical computations(personal communications T.Rung, Technical University of Berlin).

In this paper we will not go further into the derivation of XerT,but rather try to understand the physical inter-pretation of XerT.The material derivative of e s is ex-pressed by using the transformation T

D e s

D t

D T

D t

T t e s?àXerT

e s;e7T

where XerT

?T XerTT t is XerTtransformed to the curvi-linear co-ordinate system.The XerTtensor is,thus,di-rectly related to the rotation rate of the co-ordinate system following the?uid particle

XerTij?à ijk xerT

;e8Twhere xerTis the co-ordinate system rotation rate vector, also noted by Gatski and Jongen(2000).

The problem is now reduced to?nding the xerTthat minimizes the advection of the anisotropy tensor in the e s system.In general3D?ow?elds it is not possible to know the optimal transformation a priori,and,thus, xerTmust be derived from the computed?ow?eld.

2.2.Strain-rate based co-ordinate system

If we assume that there is an algebraic relation for a in terms of S and X?,a?feS;X?T,the variation of a might be expressed in terms of the variation of S and X?.If there exist a curvilinear co-ordinate system where the variation of S and X?vanishes in the mean?ow direction,then also the variation of a must vanishin th at system.Let us try to?nd the curvilinear co-ordinate system,or equivalent the XerTtensor,for which the variation of S is minimal.The question of considering also the variation of X?will be discussed later.

The advection of the strain-rate tensor S may,simi-larly to the advection of the anisotropy tensor(4),be expressed as the advection in a curvilinear co-ordinate system plus an algebraic term arising from the trans-formation

D S

D t

?T t

D TST t

D t

TàS XerT

àXerTS

:e9TThe best approximation of the co-ordinate system where the advection of the S tensor is neglected may be obtained by?nding the solution for the XerTtensor from (9)where the?rst term on the r.h.s.is set to zero. However,that equation system is overdetermined since there are?ve(two in2D)independent equations for

S.Wallin,A.V.Johansson/Int.J.Heat and Fluid Flow23(2002)721–730723

D S =D t and three (one in 2D)independent com-ponents of X er T.

Let _S 0ij

be the advection of the transformed S ij (?rst term on the r.h.s.of (9)).By using that X er T à ijk x er T

the equation for _S 0ij becomes _S 0ij

?_S ij àeS il ljk tS jl lik Tx er Tk :e10T

_S 0ij

may be minimized in a least square sense by minimizing the norm _S 0ij _S 0ij ,which,for this case,is equivalent with S pl _S 0lq pqi ?0.That results in S pl _S

lq pqi ?A ij x er T

j ;e11T

where A ij ?2II S d ij à3S ik S kj (derived by using the rela-tion ijk rst ?d ir d js d kt td is d jt d kr td it d jr d ks àd ir d jt d ks àd js d it d kr àd kt d is d jr ).

The solution of (10)for the rotation vector x er T

i can now be determined by multiplying by the inverse of A ij x eS T

?A à1ij S pl _S lq pqj ;

e12T

where A ij is inverted by the aid of the Caley–Hamilton

theorem to

A à1ij

?II 2

S d ij t12III S S ij t6II S S ik S kj 2II S à12III S

:e13T

An interesting observation is that if the transforma-tion T is the eigenvectors of S ,then the transformed S is

the diagonal eigenvalue tensor K .The ?rst term on the

r.h.s.in (9)then becomes T t _K

T .Multiplying that term by S results in ST t _K

T ?T t K _K T ,which is symmetric since both K and _K

are real.That term,thus,vanishes when multiplied by ijk .In other words,the transfor-mation T that minimizes _S 0ij _S 0ij is given by the eigen-vectors of S .

The proposed method is,thus,identical to the Gatski and Jongen (2000)assumption of relating x er Tto the rotation rate of the principal directions of S following the mean ?ow.While Gatski and Jongen presented an explicit relation for x er Tonly for 2D mean ?ows,the present relation (12)is valid also for 3D ?ows.

The denominator in (13)may be investigated for the case where S is oriented in the principal direction.S is then diagonal with the components ?a ;b ;àa àb .Th e denominator then becomes

2II 3S à12III 2S ?4ea àb T2

e2a tb T2

ea t2b T

e14T

which is positive for all a and b except when two of the eigenvalues are equal or all eigenvalues are zero.The singularities at these points may be avoided by adding a small number to the denominator.

In 2D mean ?ows,x eS T

i reduces to

x eS T3

?S 11_S

12àS 12_S 112S 211t2S 2

e15T

which is identical to the Spalart and Shur (1997)and the

Gatski and Jongen (2000)corrections.

The variation of X ?may also be considered in de-termining the optimal x er T

i .By minimizing the norm

_S 0ij _S 0ij t_X 0?ij _X 0?ij ,where _X 0?ij is the advection of the

transformed X ij ,the transformation x eS –X Tthat mini-mizes bothth e variation of S and X is found.In 2D

mean ?ows,x eS –X T

reduces to x eS Tin (15),but in general x eS –X T?x eS T.However,due to the huge algebraic complexity,the complete x eS –X Tis of limited practical use.

2.3.Acceleration based system

It was proposed by Girimaji (1997)to use the accel-eration vector _U

D U =D t as the basis for constructing the curvilinear co-ordinate system.Since the accelera-tion vector is Galilean invariant the resulting streamline curvature corrected model would also be Galilean in-variant.Girimaji further suggested to let one of the

other unit vectors be in the direction of _U

?€U .In 2D mean ?ows that direction is known,but in 3D ?ows,one

needs to derive €U

D _U =D t from the mean ?ow ?eld.Since X er Talso depend on the rate of change of the unit vectors,one additional derivative of the velocity ?eld is needed in the resulting,quite complex,expression.

Is it possible to approximate the co-ordinate system rotation rate,x er T,directly from the acceleration vector and the rate of change of that?Let us investigate the following approximation proposed by Wallin (2000)x eapprox T

?_U

?€U _U

:e16T

This approximation obviously gives the correct x er Tin circular ?ows and was the motivation behind the ap-proximation.

The approximation may be related to the acceleration based system,x eacc T,proposed by Girimaji.In the cur-vilinear co-ordinate system,let ^n be in the direction of

the acceleration _U ,^s in the direction of _U

?€U and ^t ?^n ?^s .The acceleration vector may then be written as _U ?a ^n ,where a ?j _U

j .The rate of change of the ac-celeration is €U ?_a ^n ta _^n

and the approximation then becomes

x eapprox T?^n ?_^n ?x eacc Tàx eacc Tn

^n ;e17Twhere we have used relations (7)and (8)for expressing _^n

in terms of x eacc Tand e s .

The di?erence between the approximation (16)and (17)and the full transformation of the acceleration system is,thus,that the rotation rate in the direction of the acceleration is not accounted for in (16)and (17).2.4.Circular ?ows

Flows that can be described in a cylindrical (^r ;^h ;^z )co-ordinate system and that are homogeneous in the ^h and ^z directions are truly 3D if the ^z component of the

724S.Wallin,A.V.Johansson /Int.J.Heat and Fluid Flow 23(2002)721–730

mean velocity is not constant in the^r direction.Fully developed pipe?ow rotating around the symmetry axis and a free vortex witha velocity de?cit are examples of such?ows.The advection of the anisotropy represented in the(^r;^h;^z)system vanishes exactly and the corre-sponding co-ordinate system rotation rate vector is xerT?eVerT=rT^z,where VerTis the(absolute)angular velocity.The xerTvector is,thus,only constant if the ?ow is a rigid body rotation.In general xerTis a function of r.

This class of?ows may be used for testing if the fully 3D forms of the proposed corrections are consistent with the exact one.It is shown in Appendix A that both the proposed correction based on the strain-rate tensor in(12)as well as the one based on the acceleration in (16)are identical to the exact one.Spalart and Shur (1997)also proposed a3D correction which is di?erent from that proposed here and that fails in reproducing the exact xerT,see Appendix A.

3.Model calibration

Di?erent sets of A0–4coe?cients in Eq.(1)will be tested for some generic test cases where rotational e?ects are signi?cant.The EARSM derived from the linearized SSG(Speziale et al.,1991)denoted‘L-SSG’has the coe?cients shown in Table 1.That model,without curvature corrections,was proposed by Girimaji(1996). The Wallin and Johansson(2000)EARSM,Wallin and Johansson(WJ),was developed without any corrections related to the l.h.s.of Eq.(1)and,thus,the resulting model was independent of the A0coe?cient.However, the value for A0resulting from the original model is shown in the table.For some of the comparisons,the curvature correction for the WJ model will be switched o?to let the?ow be represented in an inertial system. That corresponds to A0!1and is denoted as‘iWJ’in the table.

Introducing the curvature correction for the original choice of the A0coe?cient in the WJ model leads to a model that predicts rotational e?ects poorly as will be seen later in this paper and also observed by Wallin and Girimaji(2000)for the vortex?ow.Wallin and Girimaji found that the WJ model behaviour was improved by increasing A0to a value closer to that of the L-SSG.Here,we will do a more thorough analysis of the e?ect

of the A0coe?cient.The resulting?nal calibration of the model is denoted CC-WJ in the table.

In calibrating the A0coe?cient,the long time as-ymptotic behaviour in rotating homogeneous shear?ow is considered.Fig.2shows the e?ective C l in the as-ymptotic limit for di?erent rotation numbers, Ro xerT

=ed U=d yT,and di?erent models.The e?ective C l is de?ned from

?CeeffT

o U

o y

:e18TDepending on the rotation number the turbulent kinetic energy grows exponentially withconstant P=e?eC e2à1T=eC e1à1T(for non-zero e?ective C l)or follows a power-law solution where e=eU y KT!0(Speziale and Mac Giolla Mhuiris,1989).

The bifurcation points between the two solution branches,Roàand Rot,correspond to the points where

CeeffT

becomes zero or where the?ow is close to neutral stability.There is,however,a weak algebraic growth associated with the power-law solution very close to the bifurcation points(see Durbin and Pettersson-Reif, 1999).

Neutral stability occurs near Ro?0:5and is also likely associated with the linear velocity pro?le in the core of a rotating channel(local Ro%0:5)according to Pettersson-Reif et al.(1999)(see also Oberlack,2001). Thus,one might calibrate the A0coe?cient such that the required bifurcation point Rotis obtained.Sucha re-lation reads

Table1

The values of the A coe?cients for di?erent quasi-linear pressure–strain models

A0A1A2A3A4

L-SSGà0.80 1.220.470.88 2.37

WJà0.44 1.200 1.80 2.25 iWJ1 1.200 1.80 2.25

CC-WJà0.72 1.200 1.80

2.25

S.Wallin,A.V.Johansson/Int.J.Heat and Fluid Flow23(2002)721–730725

2Ro t

A 0

?2Ro tà1à?????????????????????????????????????????????????????????????12A 1A 3C e 1à1C e 2à1t12A 1A 4t13A 22

s e19T

which gives A 0?à0:72for Ro t

?0:5withth e WJ val-ues in Table 1for the A 1–4coe?cients and C e 1?1:44and C e 2?1:83.That value is used for the CC-WJ model in Table 1and the resulting e?ective C l is given in Fig.2.The DNS data of the rotating channel by Alvelius and Johansson (1999)shows that there is an approxi-mate equilibrium (P %e )maintained by turbulent transport e?ects.One can also observe that the e?ective C l is small,but ?nite and non-constant through the core region.This is consistent with the CC-WJ curve in Fig.2for P ?e .At Ro ?1=2the model predicts a small,but ?nite e?ective C l and witha large negative derivative for increasing Ro .The slope is important,since this implies that if the velocity gradient in the core is given a small positive perturbation,then Ro is slightly decreased giv-ing an increased e?ective C l .This leads to an increased e?ective eddy viscosity that drives the velocity gradient back to the equilibrium state,because of the balance in

the momentum equation.Thus,balance is obtained

close to the Ro tpoint,but within the region of expo-nential asymptotic growth.

4.Generic test cases

4.1.Rotating homogeneous shear ?ow

Rotating homogeneous shear ?ow may be used as an illustration of the e?ect of including the streamline curvature correction.The ?ow is rotating with the rate x er Tz in the ^

z direction.In this speci?c case it is obvious to transform the anisotropy tensor to the rotating co-ordinate system.Exactly the same e?ect is obtained by applying the corrections proposed in this paper,and the weak-equilibrium assumption is then exactly ful?lled.Four di?erent cases were computed for rotation rates

Ro x er T

z =ed U =d y Tof 0,1=4,1=2and à1=2(see Fig.3).It is obvious that the eddy-viscosity model cannot dis-tinguishbetween th e di?erent rotation rates.

These cases were computed with the di?erent curva-ture corrected algebraic Reynolds stress models given in Table 1.The model based on the L-SSG gives

reason-

726S.Wallin,A.V.Johansson /Int.J.Heat and Fluid Flow 23(2002)721–730

able growthrates for most rotation numbers,th ough underestimating the most energetic case(Ro?1=4).The WJ model underestimates bothcases withpositive ro-tation.However,by increasing the A0coe?cient the CC-WJ model gives predictions close to the L-SSG model.

Switching o?the curvature correction,as in the iWJ model,degenerates the predicted growth rate for the Ro?1=4case,and for the Ro?1=2case the growth rate is severely overpredicted.From this,it is clear that the streamline curvature correction is important.Since the A0coe?cient is arbitrary without the curvature correction,the CC-WJ and WJ models are identical without correction or curvature e?ects.

4.2.Fully developed rotating channel

Fully developed rotating channel is considered.The channel co-ordinate system is^x,^y and^z which is ro-

tating withth e rate xerT

z in the^z direction.Also in this

case it is obvious to transform a ij to the rotational frame,and both the exact transformation and the pro-posed approximation exactly ful?ll the weak-equilib-rium assumption concerning the advection of the anisotropy tensor.

Direct numerical simulations of a fully developed rotating channel at di?erent Reynolds and rotational numbers were made by Alvelius and Johansson(1999). The two most rapidly rotating cases for Re s u s d=m?180are computed here.d is the half channel width and

the average wall friction velocity is de?ned as2u2

?eu s sT2teu u sT2where u s s and u u s are the stable and unstable side friction velocities,respectively.The rotation num-

ber Ro 2xerT

d=U m and the bulk Reynolds number Re m U m d=m are given in Table2,where xerTz is the ro-tation rate of the system and U m is the bulk velocity.

The Wallin and Johansson(2000)EARSM together withth e Wilcox(1988)K–x model is computed withth e proposed curvature correction.The curvature corrected CC-WJ EARSM agrees well withDNS data,th ougha somewhat overpredicted Re m(for the prescribed Re s)is seen in the Utplots in Fig.4and in https://www.wendangku.net/doc/b013083241.html,ing the curvature corrected original WJ EARSM the e?ect of rotation is slightly overestimated while the non-corrected iWJ EARSM underpredicts the rotation e?ects.Thus,the e?ect of curvature correction and the Table2

Rotating channel?ow.DNS speci?cation(Alvelius and Johansson, 1999)and computational results using the curvature corrected CC-WJ EARSM

DNS DNS CC-WJ CC-WJ Ro0.430.770.430.77

Re s

129.7133.0129.8138.4

Re u

218.3217.2218.8213.4

Re m309434463257

3804

choice of A0are important also for this case.In Table2 it is seen that the skin friction di?erences between the stable and unstable sides are well captured by the cur-vature corrected CC-WJ model.

The shear stress plots in Fig.4show that all models capture the laminarization on the stable side reasonably well.However,the DNS data show a small level of positive shear stress for the Ro?0:43case where all models give almost zero uv.

5.Concluding remarks

The local mean velocity gradient is not su?cient in determining the curvature or rotational e?ects and an independent additional measure is needed.Two basi-cally di?erent options for determining that measure are suggested by considering the rate of change of either the strain-rate tensor or the acceleration vector.In both methods,the second order derivative of the velocity?eld is needed.In situations where the strain rate or the ac-celeration vanishes the corrections might become almost singular.That situation is less severe for the strain-rate based method,since the turbulence production and the in?uence of the turbulence is less critical in vanishing mean?ow strain rate.Hellsten(submitted for publica-tion)has found that the acceleration based method leads to problems in some situations of mild curvature where the direction of the acceleration vector may vary rap-idly.This was demonstrated in a U-bend?ow,which showed an almost singular behaviour.For the same case,the strain-rate based method behaves much better.

The strain-rate tensor and the acceleration vector,as well as their material derivatives,ful?ll Galilean invari-ance,that is independence of solid-body motion of the frame of reference,and,thus,also the proposed cor-rections are invariant.However,any incompressible ?ow?eld should also be independent of a superimposed solid-body constant acceleration,according to Spalart and Speziale(1999),except for a modi?ed pressure?eld. The acceleration based modi?cation must thus be used withcaution in accelerated frames of reference.Exten-sions of EARSMs for including approximations of the usually neglected transport terms from the l.h.s.of Eq.

(1)could never be expected to be completely general,but could anyway be motivated by improved model per-formance in a reasonably wide class of?ows.

The EARSM proposed by Wallin and Johansson (2000)was originally without any corrections originat-ing from the l.h.s.of Eq.(1)and,thus,the choice of the A0coe?cient was arbitrary.Introducing the proposed curvature correction resulted in a rather poor behaviour in rotating?ows in contrast to the L-SSG.By calibrat-ing the A0coe?cient for consistency withneutral sta-bility in irrotational?ows,a value closer to that of the L-SSG was obtained.The behaviour of the resulting model,CC-WJ,became quite similar to the L-SSG without in?uencing the behaviour of the original WJ model in?ows without curvature e?ects.The rather limited complexity of the fully3D WJ EARSM,com-pared to the L-SSG,is,thus,retained also with the modi?ed A0coe?cient and withth e curvature correc-tion.In the WJ EARSM,A2?0(C3?2)which means that the last term in(1)vanishes leading to a reduction in the algebraic complexity for the resulting EARSM, especially in3D?ows(see Wallin and Johansson,2000).

It is,however,important to highlight that the corre-sponding pressure–strain rate model,given by Eq.(2), becomes completely di?erent by a modi?cation of the A0 coe?cient.The coe?cients are given in Table3.The CC-WJ model introduces e.g.a non-linear part through the non-zero C1

and also,the value of the C2coe?cient is not consistent with the rapid distortion theory,which gives C2?4=5.Also the L-SSG model has a departure from the theoretical C2?4=5even though the full non-linear SSG ful?lls the rapid distortion theory.It is,thus, reasonable that a lower value of the C2coe?cient is a better compromise over a wider parameter range than the more extreme state of rapid distortion and that a non-linear model is needed for covering also the rapid distortion limit.

The test cases considered in this paper were chosen primarily for validating the calibration of the model coe?cients.The cases also clearly demonstrate the im-portance of applying curvature correction even if the curvature is explicitly known for these particular cases. The novel feature in the proposed approximation methods of the local curvature in general?ows are mainly the explicit3D forms.Hopefully,this paper will encourage the use of the proposed corrections in more complex3D?ows.

Acknowledgements

The?rst author would like to acknowledge Karl Forsberg at SwedishDefence ResearchAgency(FOI) and Sharath Girimaji at Texas A&M University for helpful discussions.This work has partly been carried out within the HiAer Project(High Level Modelling of H ighLift Aerodynamics).Th e H iAer project is a col-laboration between DLR,ONERA,KTH,HUT,TUB, Alenia,EADS Airbus,QinetiQ and FOI.The project is Table3

The values of the C coe?cients for di?erent quasi-linear pressure–strain models

C2C3C4

L-SSG 3.4 1.80.36 1.250.40

WJ 3.600.82 1.11

CC-WJ 4.6 1.240.4720.56

728S.Wallin,A.V.Johansson/Int.J.Heat and Fluid Flow23(2002)721–730

managed by FOI and is partly funded by the European Union(project ref:G4RD-CT-2001-00448). AppendixA.Circu lar?ow with an ax ial component Circular?ows withan axial component is described in a cylindrical(^r;^h;^z)system.The velocity is given by U?VerT^htWerT^z:eA:1TThe exact transformation may be determined to

xerT?V

^z:eA:2T

The acceleration and rate of change of that are

_U?àV2

r ^r;€U?à

^h:eA:3T

The approximation of the system rotation rate based on the acceleration system,xeapproxTin(16)becomes

xeapproxT?V

^r?^h?

^zeA:4T

which is identical to the exact transformation.

The strain-rate tensor is then given by

S?1

0V0àV W0

V0àV00

W000

AeA:5T

and the rate of change of that is

_S?V

VàV000

0V0àV1W0

01W00

A:eA:6T

The approximation based on_S,xeSTin(12),can be determined by use of the S and_S relations

xeST?V

^zeA:7T

which also is identical to the exact transformation.

The Spalart–Shur correction e in3D can be written as

e?S pl_S lq pqi

2II S

2x i

eT;eA:8T

where x i is the vorticity.The quantity e is the inner product of the rotation rate of S following the stream-line and S rotated by the rate of the vorticity vector. That can be written as

e?xeS–ST

2x i

eTeA:9T

and by identi?cation,the approximation of the system rotation rate based on Spalart–Shur is given by

xeS–ST

i ?

S pl_S lq pqi

2II S

:eA:10T

By including the S and_S relations,one obtaines xeS–ST

?0;

xeS–ST

W0V

àV0

àá

VàV0

àá

tW02

;

xeS–ST

W02

VàV0

àá

tW02

eA:11T

Hence,the Spalart–Shur correction reduces to the exact transformation only in cases where the axial velocity is constant(W0?0).

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