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A realizable explicit algebraic Reynolds stress model for compressible turbulent flowwith significant mean dilatationI. A. Grigoriev, S. Wallin, G. Brethouwer, and A. V. Johansson Citation: Physics of Fluids (1994-present) 25, 105112 (2013); doi: 10.1063/1.4825282 View online: http://dx.doi.org/10.1063/1.4825282 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/25/10?ver=pdfcov Published by the AIP Publishing Advertisement:

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PHYSICS OF FLUIDS 25, 105112 (2013)

A realizable explicit algebraic Reynolds stress model forcompressible turbulent flow with significant mean dilatation

I. A. Grigoriev,1,a) S. Wallin,1,2 G. Brethouwer,1 and A. V. Johansson11Linné FLOW Centre, Department of Mechanics, KTH Royal Institute of Technology,SE-100 44 Stockholm, Sweden2Swedish Defence Research Agency (FOI), SE-164 90 Stockholm, Sweden

(Received 28 May 2013; accepted 2 October 2013; published online 23 October 2013)

The explicit algebraic Reynolds stress model of Wallin and Johansson [J. FluidMech. 403, 89 (2000)] is extended to compressible and variable-density turbulentflows. This is achieved by correctly taking into account the influence of the meandilatation on the rapid pressure-strain correlation. The resulting model is formallyidentical to the original model in the limit of constant density. For two-dimensionalmean flows the model is analyzed and the physical root of the resulting quarticequation is identified. Using a fixed-point analysis of homogeneously sheared andstrained compressible flows, we show that the new model is realizable, unlike theprevious model. Application of the model together with a K − ω model to quasi one-dimensional plane nozzle flow, transcending from subsonic to supersonic regime, alsodemonstrates realizability. Negative “dilatational” production of turbulence kineticenergy competes with positive “incompressible” production, eventually making thetotal production negative during the spatial evolution of the nozzle flow. Finally, anapproach to include the baroclinic effect into the dissipation equation is proposedand an algebraic model for density-velocity correlations is outlined to estimate thecorrections associated with density fluctuations. All in all, the new model can becomea significant tool for CFD (computational fluid dynamics) of compressible flows.C© 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4825282]

I. INTRODUCTION

Explicit algebraic Reynolds stress models (EARSM), proposed in Girimaji,1 Johansson andWallin,2 Ying and Canuto,3 and Wallin and Johansson,4 have been successfully applied to mod-eling incompressible and weakly compressible turbulent flows. The applications comprise two-dimensional and three-dimensional shear flows and boundary layers (Wallin and Johansson4), flowswith rotation and curved flows (Gatski and Wallin5), flows with passive scalar transport (Wikströmet al.6) and active scalars causing buoyancy forces (Lazeroms et al.7). The most apparent advantagesof EARSM over eddy-viscosity models are their ability to take into account flow rotation, changeof flow conditions and turbulence anisotropy in free shear and boundary layer flows. DifferentialReynolds stress models (DRSM), being computationally more complicated and expensive thanEARSM, effectively may be reduced to EARSM using the weak equilibrium assumption, which im-plies that the sum of advection and transport terms of anisotropy tensor is assumed to be negligible.There is a strong evidence, based on a posteriori examinations of EARSM (Wallin and Johansson4

and Marstorp et al.8), that the assumption holds well in many flows of interest, except in regionswhere the shear vanishes. Today various versions of incompressible EARSM are incorporated intocommercial CFD (computational fluid dynamics) codes.

In many situations the compressibility of a flow has to be properly accounted for. Thearea of compressible turbulence has been exposed to extensive and diverse studies. See, e.g.,

a)Author to whom correspondence should be addressed. Electronic mail: [email protected]

1070-6631/2013/25(10)/105112/26/$30.00 C©2013 AIP Publishing LLC25, 105112-1


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105112-2 Grigoriev et al. Phys. Fluids 25, 105112 (2013)

Friedrich9 for a review and references, in which one of the main issues is the modeling of thepressure-dilatation correlation, becoming relevant when the turbulent Mach number Mt =

√K/cs

(K - turbulence kinetic energy, cs - speed of sound) approaches unity and the turbulence level isextremely high. However, it may be argued using rapid-distortion theory (RDT) (Simone et al.10),that the primary influence on the turbulence is caused by the modification of the structure of theturbulence. Consequently, the examination of any particular turbulence test case has to be precededby a consideration of the behaviour of the rapid pressure-strain correlation.

For example, Aupoix11 modeled a compressible mixing layer in the framework of a K − �model, with EARSM by Wallin and Johansson4 incorporated, to examine the spreading rate evo-lution. Thacker et al.12 investigated the influence of compressibility on the rapid pressure-straincorrelation and the results were applied to characterize compressible mixing and boundary lay-ers depending on the values of the turbulent Mach number Mt and the gradient Mach numberMg = S l/cs (S - shear rate, l - characteristic length scale). The authors used the wave equationgoverning pressure fluctuations, arguing that RDT is not sufficient to derive the properties of sta-tistically stationary turbulence. Recently Girimaji et al. (Bertsch et al.,13 Gomez and Girimaji14)have identified three different pressure regimes in compressible turbulent flow, depending on themagnitude of Mg, and modeled the rapid pressure-strain correlation in the entire range of Mg forhomogeneous shear flow. Gomez and Girimaji15 also formulated an EARSM for compressibleflows based on the same approach but without considering strong mean flow dilatation. Kim andPark16 adopted a Girimaji-like compressibility factor function to obtain an explicit solution forcompressible ARSM, where dilatation-related terms were accounted for approximately with a re-sulting cubic algebraic equation. They investigated compressible mixing layer, supersonic flat-plateboundary flow, and planar supersonic wake flow demonstrating the performance of their model.However, the above studies considered flows without or with small mean compressibility, whereasin flows with shock-turbulence interaction and in high-speed nozzle flows the mean dilatation maybe significant. Moreover, high magnitudes of mean dilatation may also occur in low-speed flows,e.g., substantial mean density variations are found in combustion applications, when heat release issignificant.

EARSM versions, developed for compressible flows, e.g., Wallin and Johansson,4 Gomez andGirimaji,15 suffer from inconsistencies if applied to such cases. First, due to the influence of meandilatation, which manifests itself primarily through the rapid pressure-strain correlation. Althoughin the literature the general tensorial form, which would lead to the correct model for the rapidpressure-strain correlation, has been outlined several times (Speziale and Sarkar,17 Gomez andGirimaji14), no attention has been paid to the case of significant mean dilatation, when realizabilityissues become important. Second, the modification of the dissipation equation due to varying fluidviscosity in varying density flows, highlighted by Coleman and Mansour,18 may be important aswell. In isentropic compressible flow this effect is effectively related to the mean dilatation, but incases with heat release, the effect can become significant even in subsonic flows. The dissipationequation is also effected by the baroclinic term, but straightforward modeling of its influence turnedout not to be satisfactory (Aupoix11).

The baroclinic effect is closely connected with the influence of the density-velocity correlation,which ought to be correctly accounted for in compressible flow as well. The density-velocitycorrelation enters the turbulence equations in a different manner, depending on if Reynolds ordensity-weighted Favre averaging is used, and the consequences of this choice will be furtherdiscussed in the paper.

The main objective of the paper is to examine the influence of the mean dilatation in the rapidpressure-strain correlation on the turbulence behaviour, in particular the realizability of the Reynoldsstress tensor for large mean dilatation in the context of EARSM. Additional effects of thermodynamiccharacterization (Bertsch et al.,13 Gomez and Girimaji14) are not considered here.

The paper is organized as follows. In Sec. II we develop the model for the pressure-straincorrelation suitable for compressible flows and formulate an explicit algebraic Reynolds stressmodel. The modification of the dissipation equation due to the variable viscosity and the barocliniceffect along with a simple algebraic model for the density-velocity correlation are also presented. InSec. III we provide the general solution to the EARSM and consider the cases of two-dimensional


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mean flow, with the resulting fourth-order equation (for the production-to-dissipation ratio) solved,and irrotational flow. In Sec. IV we apply a fixed-point analysis to a homogeneously sheared andstrained compressible flows and consider quasi one-dimensional plane nozzle flow in the frameworkof an EARSM supplemented with equations for K and ω. The results indicate that the new modelis realizable in all turbulence regimes, unlike the earlier models. In Sec. V the conclusions aresummarized, and Appendices A–D contain the details of the model formulation.

II. GOVERNING EQUATIONS FOR COMPRESSIBLE FLOW

The governing equations necessary to describe compressible flow of a single-phase fluid are thecontinuity equation

∂tρ + ∂k (ρuk) = 0, (1)

where we denote time and space partial derivatives as ∂t ≡ ∂∂ t

and ∂k ≡ ∂∂ xk

, the Navier-Stokes

equation, which in momentum form reads

∂t (ρui ) + ∂k (ρ ui uk) + ∂i p = ∂k τik, τik =[μ(∂kui + ∂i uk) +

(μv − 2

3μ

)δik∂lul

], (2)

and the energy equation that will be discussed later. We assume the fluid to be Newtonian with thestandard expression for viscous stress τ ik; μ is dynamic viscosity, and μv is the bulk viscosity.

The common routine, preceding the investigation of turbulent flows, is to decompose the equa-tions into their mean part, obtained by applying Reynolds averaging to the equations, and fluctuatingpart, left after subtracting mean parts from the full equations. Denoting Reynolds averaging with abar, we split our flow variables in this manner: ρ = ρ̄ + ρ ′, p = p̄ + p′ and ui = ūi + u′i ; for furtherconvenience, we denote P = p̄ and Ui = ūi . We define the Reynolds stress tensor Rik = ρ u′i u′k anda characteristic velocity

ůi = ρ ′ u′i/ρ̄ (3)of the density-velocity correlation ρ ′u′i . We obtain from (1) and (2) the mean continuity equation

∂t ρ̄ + ∂k(ρ̄ Uk) + ∂k (ρ̄ ůk) = 0, (4)and mean momentum equation

∂t (ρ̄ Ui ) + ∂k (ρ̄ UiUk) + ∂k Rik + ∂i P = ∂kτik −(

∂t (ρ̄ ůi ) + ∂k (ρ̄ ůi Uk) + ∂k (ρ̄ ůk Ui ))

, (5)

where the average of the viscous stress tensor is

τik = μ̄(∂kUi + ∂iUk) +(

μ̄v − 23

μ̄

)δik∂lUl + μ′(∂ku′i + ∂i u′k) +

(μ′v −

2

3μ′

)δik∂lu′l . (6)

Terms containing the density-velocity correlation characteristic velocity ůi are present in the mo-mentum equation, adding complexity to the equations compared to the incompressible case. Notethat the presence of the terms with ůi in Eqs. (4) and (5) by itself does not tell us anything about theimportance of these terms, which can only be revealed by an analysis of the equation for ůi , as willbe illustrated later in the paper. To arrive at a more elegant form of Eqs. (4) and (5) it is common tointroduce density-weighted Favre averaging for the velocity ui = ũi + u′′i , where ũi = ρ ui/ρ̄. Themean equations then become

∂t ρ̄ + ∂k (ρ̄ ũk) = 0, ∂t (ρ̄ ũi ) + ∂k (ρ̄ ũi ũk) + ∂k R( f )i j + ∂i P = ∂k τik, (7)with the Reynolds stress tensor defined as R( f )ik = ρ u′′i u′′k = Rik − ρ̄ ůi ůk . Note that the averageof the viscous stress tensor τ ik is still expressed in terms of Ui and u′i according to (6), sinceby completely shifting to ũi = Ui + ůi and u′′i = u′i − ůi we would be forced to face the secondspatial derivatives of ůi even at this step. This is the main reason for expressing the average viscous


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stress tensor in terms of Reynolds-splitted velocity and velocity fluctuations, even when other termsare expressed in terms of Favre averaged quantities. Furthermore, this means that the necessity ofdetermining ůi is present both for the Reynolds and Favre averaging.

So, in both Reynolds and Favre formulations we need equations for mean pressure, Reynoldsstress tensor Rij or R

( f )i j , and density-velocity correlation ρ

′ u′i ≡ ρ ′ u′′i (or for ůi , equivalently). Asshown in Appendix A, equations for R( f )i j and ůi in Favre formulation are less complex than thecorresponding equations in Reynolds formulation. It may be argued, that “dynamical” experimentaltechniques, for example, hot-wire anemometry, measure the Favre averaged velocity, making ũiand u′′i physically more appropriate variables than Ui and u

′i . On the other hand, optical techniques

such as PIV and LDA measure Reynolds averaged quantities, making Ui and u′i more convenientmodeling variables instead. Note that dissipation expressed in terms of u′i is expected to be advectedby Ui not ũi , making the Favre formulation slightly inconsistent. Furthermore, aiming at explicitalgebraic Reynolds stress modeling, it is of primary importance to conveniently express mean strainand vorticity tensors and mean dilatation. Their definition through the kinematic variable Ui is morepractical, than through dynamic variable ũi , giving us the possibility to reveal easily the modificationsof the model due to non-zero density-velocity correlation, whereas in Favre formulation the changesare hidden in the definition of ũi . In the following we will use the Reynolds formulation.

Let us proceed with the following definitions

K = Rkk2 ρ̄

, ai j = Ri jρ̄ K

− 23

δi j , � = ερ̄

,

(8)

τ = K�

,D = τ ∂k Uk, Si j = τ2

(∂ j Ui + ∂i U j

)− D

3δi j ,i j = τ

2

(∂ j Ui − ∂i U j

),

where K - turbulent kinetic energy per unit mass, aij - anisotropy tensor, ε and � - dissipation rateper unit volume and mass, respectively (both to be introduced later on), τ - turbulence time scale,D - mean dilatation, Sij - traceless strain tensor, and ij - vorticity tensor. The reason why we chooseto work with K and not with turbulence kinetic energy per unit volume Rkk/2 is to exclude thestrong influence of the change of mean density, which is characteristic of many compressible flowsituations, for example, nozzle flow.

Most straightforward for the determination of pressure seems to be the investigation of theenergy equation, which may be written as

ρ T (∂t s + uk ∂k s) = τi j ∂i u j − ∂k qk (9)for the entropy function s(ρ, p). Here τ ij ∂ iuj is viscous dissipation and qk - heat flux due tothermal conductivity. In case of a calorically perfect gas with adiabatic constant γ the entropy is

s = R(γ − 1)M ln

p

ργ(R - universal gas constant, M - molar mass of the gas) and we rewrite (9)

as an equation for pressure

∂t p + uk ∂k p = −γ p ∂k uk + (γ − 1)(

τi j ∂i u j − ∂k qk)

, (10)

from which the equation for mean pressure is found as

∂t P + Uk ∂k P + ∂k p′ u′k = −γ P ∂k Uk − (γ − 1) p′ ∂k uk + (γ − 1) (εtot − ∂k qk), (11)where εtot = τi j ∂i u j is total dissipation rate. Here we encounter the pressure-dilatation correlationp′ ∂k uk .

Writing the equation for the fluctuating velocity in the form (A1) (see Appendix A), multiplyingit by ρ u′j , and adding the similar equation with i⇐⇒j, after using the continuity equation we averageto obtain the equation for the Reynolds stress tensor

∂t Ri j + ∂k (Ri j Uk) + ∂k Ti jk = Pi j − εi j + �i j + �i j . (12)The terms on the right-hand side represent production, dissipation, and pressure-strain rate whilethe meaning of the last term � ij, given by (A12), will be revealed later. The equation for turbulence


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kinetic energy is obtained by taking the trace of (12) and dividing by two:

∂t (ρ̄ K ) + ∂k (ρ̄ K Uk) + ∂k Tk = P − ε + � + �. (13)

One of the key issues in the paper is the proper derivation of the pressure-strain correlation forcompressible flows. To be able to express the corresponding terms in (12) and (13), we are in need ofsuitable modeling for the fluctuating pressure p′. Analysis of the pressure equation (10) to obtain anequation for p′ can, e.g., be found in Friedrich.9 Here we will follow another methodology, usuallyadopted in the study of incompressible turbulence, noting that the mean and fluctuating part of thepressure can be coupled to the other parameters by a global Poisson equation

∂i i p = −∂ik(ρ ui uk) + ∂t t ρ + ∂i i[(

μv + 43μ

)∂k uk

]+ 2 ∂i

[− ∂i μ∂k uk + ∂k μ∂i uk

]. (14)

Taking the fluctuating part, we obtain a global solution for the fluctuating pressure (A9). As shownin Appendix A, the use of this solution eliminates dilatational dissipation from the equation for theReynolds stress tensor, which means that dilatational dissipation energy goes directly into internalenergy passing over turbulent kinetic energy. The approach gives us the possibility to empiricallymodel the rapid part of the pressure-strain correlation �ij, which is crucial in the modeling of theReynolds stress tensor.

A. Pressure-strain correlation

As illustrated in Appendix A, considering the quantity p′ (∂i u′j + ∂ j u′i ) present in Eq. (A12) forthe Reynolds stress tensor, we split the fluctuating pressure into inertial, acoustic and viscous partsaccording to (A9). The influence of the acoustic part is not considered here (and is small as long asthe turbulent Mach number Mt is small). The viscous part is merged into the viscous stress tensor toget rid of the dilatational dissipation (note that all terms with bulk viscosity disappear), and we areleft with the inertial part, which we have to model. Hence, we define the pressure-strain correlationto be based on this inertial part:

�i j = p′in (∂i u′j + ∂ j u′i ). (15)

A number of studies (see Friedrich9 for a review) concern the investigation of the divergence part ofthe pressure-strain 23 �δi j , i.e., the pressure-dilatation correlation � = � j j/2 = p′in ∂k u′k . Here weassume that the turbulent Mach number Mt is small and consequently � is negligible. Primarily, weare going to highlight the effect of mean dilatation on the divergence-free part �i j − 23 �δi j .

We will model the divergence-free part of the pressure-strain correlation as a sum of slow andrapid parts. The slow part �(s)i j is taken to be the usual return-to-isotropy term, i.e., Rotta’s assumption(Rotta19)

�(s)i j = −c1 ε ai j . (16)

The rapid part �(r )i j is modeled in the spirit of the conventional scheme of Launder, Reece andRodi,20 which originally was applied to incompressible flow. The extension of the LRR model forcompressible flows that we present here is different from earlier approaches. Our approach is in linewith that outlined in Speziale and Sarkar,17 Gomez and Girimaji,14 but here we will take into accountsignificant magnitudes of mean dilatation. We base our consideration on the formalism developed inJohansson and Hallbäck,21 where, for the rapid pressure-strain, the most general form of nonlinearityin the anisotropy tensor model was proposed. Here, we aim for also incorporating the influence ofmean dilatation in the model, but we restrict ourselves to modifying the linear terms. We start withthe formal representation of the integral for �(r )i j , where the variation of the mean flow gradients is


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slow enough to be extracted from the integrals:

�(r )i j =

(Mimjl + M jmil

)∂mUl ≡ τ−1

(Mimjl + M jmil

) (Slm + lm

)+ D

3τ−1

(Mimjm + M jmim

).

(17)In homogeneous incompressible turbulence the expression is exact and Mimjl is

Mimjl = −(2 π )−1∫

∂2 ρ(x) um(x) ui (x + r)∂rl ∂r j

d3 rr

, (18)

where the symmetry conditions

Mimjl = Mmijl = Miml j ,Mimil = Mimjm = 0,Mimj j = 2 Rim ≡ ρ̄ K(

4

3δim + 2 aim

)(19)

are the result of the definition. Note that the second condition in (19) leads to the ruling out of theterm with dilatation from the expression for the rapid pressure-strain (17), and all the dependenceon dilatation is comprised into the definition of strain-rate tensor.

So, assuming that in compressible nonhomogeneous turbulent flow representation (17) is valid,we model Mimil as dependent only on the anisotropy tensor and its algebraic combinations, the mostgeneral expression for Mimjl is given by Johansson and Hallbäck,21 Sjögren and Johansson22

Mimjl = A1 δim δ jl + A2 (δi j δml + δil δmj ) + A3 δim a jl + A4 δ jl aim+A5 (δi j aml + δil amj + δmj ail + δml ai j ) + Mnon(aim a jl ...), (20)

where all tensor terms nonlinear in anisotropy are incorporated into Mnon(aim a jl ...) = A6 aim a jl +... (with finite number of terms due to Cayley-Hamilton theorem, see Johansson and Hallbäck21).The resulting expression for �(r )i j , with Mimjl linear in the anisotropy tensor and satisfying conditions(19), is

�(r )i j

ε= 4

5Si j + q3

(aik Sk j + Sik ak j − 2

3akm Smk δi j

)+ 7 q3 − 12

9

(aik k j − ik ak j

), (21)

which also may be written as

�(r )i j = −q3

(Pi j − 2

3P δi j

)+ ε 16 q3 − 12

9

(aik k j − ik ak j

)

−30 q3 − 1845

ρ̄ K

(∂ jUi + ∂iU j − 2

3∂l Ul δi j

)− ε 2

3q3 ∂lUl ai j . (22)

Here we prefer to express the freedom in �(r )i j through q3 (similar to Johansson and Hallbäck21), which

is a coefficient preceding the most complex tensor group in (21), rather than through c2 = A3/(ρ̄ K )(used in Wallin and Johansson4), representing a variable parameter in (20). The quantities are relatedas q3 = (9 c2 + 6)/11.

Note that (21) does not depend explicitly on the dilatation and formally is the same as inincompressible case. Further note that the last term in (22) serves to ensure that fact. In the originalmodel of Wallin and Johansson,4 the extension of LRR pressure-strain rate to compressible flowswas carried out just by utilizing the incompressible version, that is (22) with ∂ lUl ≡ 0, and thenmaking the third term traceless by subtracting (2/3 ∂ lUl δij) in the brackets, while the last (fourth)term was missed, following Vandromme.23 Both approaches are empirical, but the one chosen heremodels the physics in a more systematic way.

The real advantage of the new approach will be shown by test cases analyzed later in the paper.In particular, the new approach will remain realizable in flow cases where the original model fails.The model behaviour will be tested in an EARSM context, but advantages should be visible alsowhen used as a full differential Reynolds stress model.


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B. Explicit algebraic Reynolds stress model

The equation for the anisotropy tensor aij is derived from Eqs. (12) and (13) and reads

ρ̄ K Dt ai j + ∂k Ti jk −(

ai j + 23δi j

)∂k Tk =

Pi j − εi j + �i j −(

ai j + 23δi j

)(P − ε + � + �

)+ �i j . (23)

After modeling the transport terms for the Reynolds stress tensor Tijk and the turbulence kinetic energyTk, it would be possible to proceed with solving the differential equations for the anisotropy tensor,provided we have succeed in modeling �ij, εij, and � ij on the right-hand side. The production tensorPi j is the only a priori explicitly expressed quantity. However, the weak-equilibrium assumption,that is neglecting the left-hand side of (23), is generally considered as a good approximation inmany cases of interest. Conventionally it is implied that advection of aij is negligible, and assuming

that the relation ∂k Ti jk = Ri jρ̄ K

∂k Tk is valid, that is the transport of Reynolds stresses is modeled

by the transport of kinetic energy. The weak-equilibrium assumption breaks down in stronglynonhomogeneous or nonstationary cases, necessitating the employment of differential Reynoldsstress modeling in such regions.

Applying the weak-equilibrium assumption to (23) and dividing by ε we obtain, in matrixnotation,

−�(s)

ε− a = P

ε−

(a + 2

3I) P

ε+ �

(r )

ε+ ε−1

{� −

(a + 2

3I)

� −(

ε − 23

ε I)

− � a}.

(24)

It has to be made clear that the counterpart of the weak-equilibrium assumption for a( f )i j =R( f )i j /K

( f ) − 23 δi j , based on Favre averaged variables, effectively would lead to an equation, dif-ferent from (24). This means that one-to-one correspondence between Reynolds and Favre ap-proaches certainly is broken, and by choosing one of them, we obtain a solution different fromthe other. The question which of the approaches would give the best predictions requires a studyby itself, especially considering the difficulties with obtaining a fully consistent Favre formulationof the turbulence equations as well as the lack of experimental and DNS-data for compressibleturbulence.

As we will show in Subsection II D, ůi is directly connected to the mean density variation in theflow and is of the order of K/U unless low-strain or near uniform-density cases are considered. Thismeans that the influence of �-terms in (24) is, in general, comparable to the other terms, even underK/U2 → 0 condition, when the difference between Reynolds and Favre variables is erased and themean flow becomes unaffected by the turbulence. Since �-terms are expressed primarily throughthe gradients of mean density and mean pressure, we suggest that they represent the barocliniceffect. This implies that the dissipation equation also should be modified so as to include this effect,which is the subject of Subsection II C. Since the model for the density-velocity correlation (31)is still tentative and relies on rather strong assumptions, we consider that it would be somewhatpremature to try to include the combined effects at this moment. For this reason, we omit the�- and ůi - terms in Eqs. (4), (5), (13), (24), and (29) in the following investigation of the effect ofthe mean dilatation incorporated into the improved version of the rapid pressure-strain correlation.We assume the dissipation to be isotropic εi j = 23 ε δi j . The discussion of anisotropic modifications toaccount for flows near the wall may be found, e.g., in Hallbäck et al.24 (note, though, that correctionsto the dissipation tensor seem to have to be made in direct connection with corrections to the slowpressure-strain (Sjögren and Johansson22)). As we stated before, the pressure-dilatation correlation� is small for low values of turbulent Mach number.


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Making these simplifications, we note that the slow pressure-strain contributes with a term c1 ato the left-hand side of (24), while terms associated with the production tensor,

Pε

−(

a + 23

I) P

ε= tr (a S) a − 4

3S −

(a S + S a − 2

3tr (a S) I

)+ a � − � a, (25)

and rapid pressure-strain (21), represented by the last three tensor groups of (25) with coefficients4/5, q3, and (7 q3 − 12)/9, respectively, contribute to the right-hand side of (24). Now (24) may berewritten in the form identical to the incompressible case:

(c1 − 1 − tr (a S)

)a = − 8

15S + 7q3 − 3

9

(a � − � a

)+ (q3 − 1)

(a S + S a − 2

3tr (a S) I

),

(26)where everything is expressed in terms of S, �, and a, not containing dilatation explicitly. The earlierpublished compressible model by Wallin and Johansson4 is governed by the equation similar to (26),but with (c1 − 1) replaced by (c1 − 1 − 2/3D). The 23 D term appearing in the original model is dueto the inconsistent modeling of the rapid pressure-strain correlation there. As further considerationsshow, this linear dependence of the coefficient on D leads to the unrealizable values of anisotropytensor. The solution of (26) will be presented in Sec. III.

C. The dissipation equation

The anisotropy tensor aij, determined by solving (26), is a nonlinear function of ∂k Ui and τ =K/�. Recalling, that our primary goal is to find the Reynolds stress tensor Ri j = ρ̄ K

(ai j + 2

3δi j

)for solving (5), need for equations for turbulence kinetic energy and turbulence dissipation ratearises. The former is given by (13) and is exact except for the challenge to model the transport termTk. The exact equation for turbulence dissipation is extremely complex, serving mainly to indicatethe ways of modeling its terms empirically. The incompressible version of the equation, along withpossible modification of the standard �-equation for near-wall flows, may be found in Jakirlić andHanjalić.25 To arrive at a compressible modification of the dissipation equation, we are challengedprimarily to account for the mean dilatation, variable viscosity of the flow and the baroclinic effect.

The equation for the gradient of the fluctuating velocity, with only mean velocity terms retained,reads (denoting Dt ≡ ∂ t + Uk ∂k)

Dt ∂k u′i = −∂l u′i ∂k Ul − ∂k u′l ∂l Ui − u′l ∂lk Ui + . . . . (27)

Considering purely spherically symmetrical compression (or expansion) ∂k Ui = 1/3 ∂ l Ul δki, conse-quently gives Sij = 0 and P/ε = −2/3 τ ∂l Ul , and we see that the quantity ν ∂k u′i ∂m u′j is governedby

Dt ν ∂k u′i ∂m u′j = −

4

3∂l Ul ν ∂k u′i ∂m u

′j + ν ∂k u′i ∂m u′j

Dt ν

ν+ . . . , (28)

where the term (δik u′l ∂m u′j + δ jm u′l ∂k u′i ) ∂lp Up was neglected (because gradients of mean quan-

tities are much smaller than gradients of fluctuating quantities), and the viscosity fluctuations areassumed not to be important. Obviously, the second term in (28) is universal for any mean velocityfield. Observe, that whatever definition of dissipation is employed (i.e., solenoidal or dilatational,real- or pseudo-), it is made up of terms of the form ν ∂k ui ∂m u j . Further, if we subject the flowto one-dimensional compression (expansion) ∂k Ui = ∂ l Ul δi1 δk1 and assume that turbulent fluctu-ations are incompressible and isotropic (note, however, that in the frame of the developed EARSMthe anisotropy can be neglected only at low values of τ ∂k Ui), then the real solenoidal dissipation� = ν ∂k ui ∂k ui will be described by (28), where double summation k = m, i = j is performed.


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These arguments lead us to expect that in compressible flow the �-equation should be modified inthe following way:

∂t (ρ̄ �) + ∂k (ρ̄ � Uk) + ∂k T (ε)k = τ−1(

C�1

[Pε

+ 23D

]− 4

3D − C�2 + τ

Dt ν

ν+ C��

�

ε+ C�b

�

ε

)ε,

(29)where the additional terms are connected with D, ν̄, �, and �. The latter two terms are includedinto (29) in the usual spirit of formulating the �-equation in analogy with the K-equation, thatis introducing the coefficient C�b , responsible for the baroclinic effect, and the coefficient C�� ,presenting the effect of the pressure-dilatation correlation �.

For an ideal gas, μ is independent of density and μ∼ T n is considered to be a good approximationin different temperature regions (with n ≈ 2/3 for normal conditions). The following relations arevalid:

Dt ν

ν= n Dt T

T− Dt ρ

ρ,

Dt ρ

ρ= −∂k Uk, Dt T

T= (γ − 1) Dt ρ

ρ+ γ (γ − 1)�tot

c2s, (30)

where the first relation is the consequence of ν = μ/ρ and the second being the continuity equation.The last relation is a form of energy equation with negligible heat flux (cs – speed of sound), wherethe last dissipative term, leading to the deviation from isentropy, is negligible when the square ofthe turbulent Mach number M2t = K/c2s is much less than mean dilatation D. This is certainlytrue in the cases with significant mean dilatation that we are interested of in this paper, and thenτ Dt ν/ν = (1 − n (γ − 1))D.

The most substantial compressible modifications of the dissipation equation are connected withthe effect of mean dilatation and the baroclinic effect. The two terms directly related to D in (29)will reduce � in case of positive dilatation D while the viscosity term will enhance �. For standardvalues of the model coefficient C�1 the viscosity term will dominate and the net effect will be adamping of the turbulence level due to enhanced dissipation when D > 0. The relations (30) are notneeded in a CFD implementation since the mean advection of the viscosity easily can be derivedfrom the numerical solution without further assumptions. Though the influence of the barocliniceffect cannot be considered to be negligible, its influence cannot be quantified because of the lackof relevant reference data from DNS or experiments. Hence, we omit it later in the paper.

The considerations outlined above are similar to that in Coleman and Mansour,18 where theauthors were aiming to account for the effects of mean compressibility on the dissipation equationin the RDT limit.

D. Density-velocity correlation

The primary objective with this paper is to study the influence of mean density variationincluding mean dilatation. Density fluctuation correlations are usually considered to be importantonly in cases with significantly higher Mach number compared to when mean density effects are theonly significant compressibility effect.

However, the characteristic velocity of the density-velocity correlation ůi , entering in the � ijterms, is of significant importance as soon as there are mean density variations in, e.g., shear flows.Which is analogous with passive or active scalar flux ui θ for mean � variations in turbulent flows.Hence, we need to consider the influence of non-zero ůi , defined by (3), to be of the same magnitudeas that of mean density variations. Note that this is not necessarily connected to the mean dilatation.Also note that the need for modeling ůi also arises in the case of Favre averaged equations.

The most straightforward way to arrive at a plausible determination of the density-velocitycharacteristic quantity ůi is to look for an eddy-diffusivity model for this quantity. Here, we willutilize the improved anisotropy modeling for deriving an explicit algebraic model for ůi under the


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assumptions outlined in Appendix B:(G3 δik + cS Sik + c ik

)ůk = −

(aik + 2

3δik

)τ K

∂k ρ̄

ρ̄, G3 = 1

2

P − ε+� + �ε

− cρ + cS D3

.

(31)The equation is similar to that used in Wikström et al.6 for passive scalar transport modeling. In therapid-distortion limit with high τ ∂k Ui as well as the case of moderate values of the non-dimensionalstrain-rate, we estimate ůi ∼ (K/U ) (L/Lρ) (with U-characteristic mean velocity, L and Lρ - lengthsof the change of mean velocity and mean density). Thus, when Lρ ∼ L (for example, in nozzle flow),ůi may become of the order of U if the intensity of turbulence is high enough. From (A5) it maybe deduced that ρ ′2 ∼ ρ̄2 (K/U 2) (L/Lρ)2 for τ ∂ i Uk � 1. In the limit of weak turbulence withlow τ ∂k Ui the previously obtained ůi and ρ ′2 become damped by a factor τ ∂k Ui. It should bestressed, that the influence of the terms associated with ůi on the mean momentum and mean densityequations, the same as the effect of �-terms on the Reynolds stress tensor and dissipation equations,sits on an equal base with the other terms. Equation (A12) suggests that unless Dt Ui is close tozero, ů j ∂i P + ůi ∂ j P is the main contributor to the magnitude of � ij. We conclude that � ij maybe rendered negligible only by small values of the gradient of mean density or mean pressure. Themodel (31) gives ůi in terms of quantities which are themselves influenced by ůi . Therefore, to arriveat a fully explicit model, (31) must be solved jointly with (24) using an approach similar to that inLazeroms et al.,7 who consider flows with buoyancy forces. Moreover, a completely self-consistentanalysis must admit that the pressure-strain rate can have a constituent associated with �, that is�

(�)i j ∼ �i j , which will obviously modify the algebraic equation (24) for aij; an additional term

∼Dt Ui can also emerge in Eq. (31) for ůi . These issues are out of the scope of this paper.The conclusion is that accounting for density-velocity correlation within the framework of

algebraic model is in effect equivalent to accounting for large mean density variations in the flow.This modeling approach should so far be considered as tentative. Verification of the model form andcalibration of the coefficients must be based on experimental and/or numerical data of compressibleturbulence or turbulence with strong mean density variations.

III. SOLUTIONS TO THE EARSM

Apparently, the solution of (26) is identical to that given in Wallin and Johansson4 for generalthree-dimensional incompressible flows and is expressed through ten independent tensor groupswith coefficients depending on the five invariants

IIS = tr (S2), II� = tr (�2), IIIS = tr (S3), IV = tr (S �2), V = tr (S2 �2). (32)

Taking q3 = 1 (equivalently c2 = 5/9) we ensure that on the right-hand side of (26) the mostcumbersome term is eliminated, and the solution to the resulting equation

N a = −65

S +(

a � − � a)

, N = 94

[c1 − 1 − tr (a S)

](33)

is

a = β1 T (1) + β3 T (3) + β4 T (4) + β6 T (6) + β9 T (9), (34)with the coefficients depending on IIS, II�, IV, V , and N, which is determined by solving sixth-orderalgebraic equation which we write here as

N 4 − c′1 N 3 −(

2 II� + 2710

IIS

)N 2 + 2 c′1 II� N +

9

10II� D2 − 81

5

�

N 2 − II�/2 = 0, (35)

where c′1 =9

4(c1 − 1), � = V1 N 2 + D

3II� IV1 + IV 21 and IV1 = IV −

D6

II�, V1 = V

− II�2

(IIS − D

2

9

). The expressions for T(n) and βn are listed in Appendix A.


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The turbulence production-dissipation ratio is

Pε

= −tr (a S) − 23D,

where the trace is

tr (a S) = β1 IIS + β3 IV + 2 β6 V = −65

N

N 2 − 2 II�

(IIS − II�

3D2 N−2

)− 12 Q−1 N−1 �.

A. Two-dimensional mean flow

The evaluation of the anisotropy tensor (34) in general three-dimensional case is fairly com-plex, with five tensor groups (A15) and a sixth-order algebraic equation (35) for N involved.There is, however, a particular and nevertheless quite general type of flows, for which the analysisis significantly simplified, namely, two-dimensional mean flows with homogeneity in the span-wise direction. Restricting our scrutiny to such flows, we readily obtain IV1 = IV − D/6 II� = 0,V1 = V − II� (IIS − D2/9)/2 = 0, and the resulting equation for N (35) becomes of fourth-order

N 4 − c′1 N 3 −(

2 II� + 2710

IIS

)N 2 + 2 c′1 II� N +

9

10II� D2 = 0. (36)

To obtain the anisotropy tensor aij we can sum all tensor groups in (34), but we here choose anapproach, which fits the problem more naturally. Due to symmetry, ai3 = a3i ≡ 0 for i = 1, 2, buta33 may be non-zero. Furthermore, Si3 = S3i ≡ 0 for i = 1, 2, and S33 = −D/3 = 0, and we canintroduce the 2D traceless strain tensor:

S2D = S − D2

(I2 − 2

3I3

), I IS = I IS2D + D

2

6. (37)

The vorticity tensor has no diagonal terms and i3 = 3i ≡ 0 for i = 1, 2, 3. Now the generalrepresentation of the anisotropy tensor is given by

a = a2D + β0(

I2 − 23

I3

), a2D = β̂1S2D + β̂4(S2D� − �S2D), (38)

where non-zero a33 is accounted for by the tensor with factor β0, and the most general representationfor the other components involves only two tensor groups, due to Cayley-Hamilton theorem in twodimensions. Substituting this expression into (33) and utilizing linear independence of the threetensor groups, we obtain the coefficients:

β0 = −35

DN

, β̂4 = −65

1

N 2 − 2 I I� , β̂1 = N β4. (39)

The turbulence production can be expressed as

Pε

= −23D + 6

5

N

N 2 − 2 II�

(IIS − II�

3D2 N−2

)= −2

3D + 6

5

N

N 2 − 2 II� IIS2D + D

2

5N−1.

(40)The explicit expressions for the two-dimensional matrices and invariants are

S2D =(J σσ −J

), � =

(0 ω

−ω 0)

, S2D � − � S2D = 2 ω(−σ J

J σ

), II� = −2 ω2,

IIS2D = 2 (σ 2 + J 2),J = τ2 (∂x U − ∂y V ), σ =τ

2(∂y U + ∂x V ), ω = τ

2(∂y U − ∂x V ).

(41)

Note that ω as used above in (41) should not be confused with ω in the K − ω description.


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B. Solution of quartic equation

The physical root of the third-order counterpart of (36) with D ≡ 0 is well-known (Wallin andJohansson4). The physical root of the fourth-order equation can also be identified explicitly. Dividing(36) by c′41 , we obtain equation for Ñ = N/c′1:

Ñ 4 − Ñ 3 − a Ñ 2 − b Ñ − c = 0, b ≥ 0, a ≥ −b, b (a + b) ≥ c ≥ 0,(42)

a =(

2 II� + 2710

IIS

)c′−21 , b = −2 II� c′−21 , c = −

9

10II� D2 c′−41 .

Note that c = 0 in case of zero dilatation or zero rotation. The physical root is

Ñ1 = 14

+ 12

R1 + 12

√1

2− z1 + a + R1 + 2 b − z1

R1, R1 =

√1

4+ z1 + a ≡

∣∣∣∣ z12 − b∣∣∣∣√

z21 + 4 c, (43)

where z1 is given by

z1 = b2 − c − 4 a c

b + 4 c M̃−11 : M̃1 =

1

3

[1 +

(Q + √D

)1/3+

(Q − √D

)1/3]for D ≥ 0,

M̃1 = 13

+ 23

√P cos

(1

3arccos

Q√

P3

)for D < 0, (44)

D = Q2 − P3, P = 1 + 3 A, Q = 1 + 92

A + 272

B, A = a b2 − c − 4 a c(b + 4 c)2 , B =

(b2 − c − 4 a c)2(b + 4 c)3 .

Note that II� = 0 leads to b = 0, c = 0, and z1 = 0, which is obtained by taking the limit in (43). Theanalysis, which justifies the choice of the root, is given in Appendix C. Having the solution of thefourth-order equation, we can approximate the solution of (35) in general three-dimensional casesusing one or more iterations.

C. Irrotational mean flow

While the simplification of the EARSM for two-dimensional flows is considerable, there isanother type of mean flow allowing significant reduction of complexity even in three dimensions.This is the irrotational mean flow, which is an interesting case illustrating the concept of eddy-viscosity. Indeed, when ij = 0 only the first tensor group in (34) remains leading to

ai j = β1 Si j , −Ri j + 23

ρ̄ K δi j = μt(

∂ j Ui + ∂i U j − 23

∂k Uk δi j

)(45)

with

μt = −ρ̄ β1 K τ/2, β1 = −65

1

N, N 2 − c′1 N −

27

10IIS = 0, N = c

′1

2+

√(c′12

)2+ 27

10IIS.

(46)The sixth-order equation (35) has here been reduced to a quadratic equation (46) with unambiguouslydetermined physical root N (of course, (43) returns the same value of N), which specifies the turbulentviscosity μt . Now the turbulence production and N become

Pε

= −β1 IIS − 23D = 6 IIS

5 N− 2

3D, N = 9

4

(c1 − 1 + P

ε+ 2

3D

). (47)

Due to the condition of irrotationality, ∂ j Ui = ∂ i Uj, the divergence of the Reynolds stress tensor isgiven by

− ∂ j Ri j = −∂i(

2

3ρ̄ K

)+

(2 ∂i U j − 2

3∂k Uk δi j

)∂ j μt + 4

3μt ∂i ∂k Uk . (48)


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An analogous expression holds for the divergence of the viscous stress tensor τ ij (plus terms withμv).

Though the ordinary picture of turbulence is a tumultuous swirling stream, recall that one ofthe properties of turbulence is the tendency to smear out the transverse mean gradients far awayfrom boundaries, while vortical structures, generated by the action of viscosity near walls, do notpenetrate into the central region. In some geometries the mean velocity field of a turbulent flow awayfrom the wall is close to irrotational. In Sec. IV a close view will be given to an instance of such ageometry.

IV. MODEL BEHAVIOUR FOR GENERIC TEST CASES

The original EARSM of Wallin and Johansson4 has been found to result in physically unre-alizable solutions for large values of the dilatation. The main purpose with the generic test casesbelow is to show that the more physically correct modeling of the pressure-strain rate in Sec. II Aresults in physically reasonable and completely realizable results. Definitely, the most realistic wayfor a strict quantitative validation of the model would be to compare it with DNS-data. This canalso provide detailed information on the advantages and disadvantages of the Favre and Reynoldsaveraging approaches.

Generic and simple flow cases with significant mean dilatation suitable for testing EARSMapproaches are not easily defined. Shock waves are strong sources of negative mean dilatationand the interaction between shocks and homogeneous turbulence could be an interesting study.However, such cases are dominated by transient development of the Reynolds stress anisotropies,which cannot be captured by an EARSM. Shock-boundary layer interaction is another case wherethe mean dilatation can be of significant importance and where the EARSM approach can be justifiedbecause of high shear rates. However, such cases are far too complex for the generic testing that wewill perform. Heated pipe flow and combustion flow are other examples of flows with significantmean dilatation. Here, the dilatation is caused by the heat expansion and the Mach number does nothave to be in the compressible regime.

Accelerated flow through a Laval nozzle is another case with significant positive mean dilatationin the central part and where the EARSM approach can be justified with suitable choice of initialconditions. Such nozzle flow without considering the nozzle wall boundary layers will be consideredlater in this section.

A. Realizability in homogeneously sheared and strained flows: Fixed-point analysis

Here we will define two flow states of homogeneous straining and shearing including significantdilatation. The flow states are defined by constant normalized strain- and rotation-rate tensors, Sijand ij. Homogeneous flows with mean dilatation are somewhat artificial, but are inspired by thedifferent cases discussed above.

In the framework of EARSM, the time development of K and � can be computed for thosecases, leading to an asymptotic state with exponential growth or decay of K and � and where P/εapproaches a constant value. The flow is statistically homogeneous resulting in a set of ordinarydifferential equations that will approach an asymptotic state or fixed-point. Such an equilibrium P/εis then a result of the anisotropy and �-equation modeling, in particular the C�1 and C�2 coefficients,but a fixed-point analysis can also be performed with a pre-determined arbitrary P/ε.

The Reynolds stress tensor Rij must be positive definite. This means that the normal componentsρ u′αu′α ≥ 0 in every frame of reference. In two-dimensional mean flows necessary and sufficientconditions for the eigenvalues of the anisotropy tensor are

− 2/3 ≤ λ1,2,3, λ1,2 = −a332

∓√

a212 +(

a22 − a112

)2, λ3 = a33. (49)

One should note that unless the Cauchy-Schwarz inequality for the non-diagonal component |a12|≤ 1 is satisfied, λ1 or λ3 becomes unphysical; similarly, violation of −2/3 ≤ aαα for any of


130.237.233.116 On: Wed, 23 Oct 2013 15:25:48


diagonal components is unadmissible. Also note that these requirements on aαβ are only necessary,becoming also sufficient at a12 = 0. The Lumley invariant map in IIa ≡ aik aki = λ21 + λ22 + λ23 andIIIa ≡ ai j a jk aki = λ31 + λ32 + λ33 coordinates graphically illustrates realizability and the state of theturbulent flow. Points outside of the Lumley triangle, whose sides are IIa = 89 + IIIa and IIa =61/3 |IIIa|2/3, are unrealizable, the upper line indicates two-component turbulence and the two curves– axisymmetric turbulence.

The first case is developing in the x-direction with negligible values of derivatives in the otherdirections. The only non-zero component of the velocity gradient is D = τ ∂x Ux , while σ = ω ≈ 0.Such flow may be representative of the central part of the throat region of a nozzle flow or the centralregion of a heated pipe flow.

Assume that the flow is accelerated by a strong favourable pressure gradient or by heating, withresulting large positive D. The fixed-point analysis in the limit of large D results in the followingsolution for the anisotropy tensor and P/ε:

a12 = 0, N = 3√5D, a11 = − 4

3√

5, a22 = a33 = 2

3√

5,Pε

= −23

(1 − 2√

5

)D < 0, (50)

resulting in an axisymmetric turbulent state close to the two-component turbulence limit with a11≈ −0.6. Another observation in (50) is that P/ε becomes negative for positive dilatation andasymptotically approaches −0.07D at large D. Note that the values of the anisotropy tensor arephysical despite the fact that the production is negative. Clearly, the improved pressure-strain ratemodeling results in realizable turbulence while the original model by Wallin and Johansson4 wouldgive the following asymptotic fixed-point results

N = 34

(− 1 +

√21

5

)D, a11 = − 16

15(−1 +

√215

) , a22 = a33 = −a112

,Pε

= 49

N > 0, (51)

with positive P/ε → 0.35D and a11 ≈ −1 being unphysical.Near the wall in the cases discussed earlier a boundary layer is formed, and the magnitude of the

shear σ = ω becomes significant. Thus we proceed to the second test case which is an extension ofthe first case with a non-zero and fixed σ . In this region we expect that the turbulent production anddissipation may be comparable. σ 1 = −1.696 is the shear rate that gives P/ε = 1 for D = 0 and wechose σ = ω to be σ 1 and half of that value, 0.5 σ 1, as fixed values when studying the influence ofvarying dilatation D. The sign of σ only influences the sign of a12 and we prefer to keep a12 positivein this study, hence the negative σ . The first case above corresponds to σ = 0.

The behaviour of the improved model is illustrated by plotting the fixed point solution for thethree different σ for D varying from 0 to 5. Fig. 1(a) shows that the anisotropy lies within the

FIG. 1. Homogeneous shear flow. (a) Invariant map, arrows indicate the sides of the limiting curvilinear Lumley triangle,(b) λ1: σ = 0 – green, σ = 0.5 σ 1 – blue, σ = σ 1 – red, black line – λ1 = −2/3; diamonds – new model D = [0 : 0.25 : 5],dashed lines – original model D ∈ [0, 1.76].


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FIG. 2. Homogeneous shear flow. (a–d) a11, a22, a12, and P/ε, respectively: σ = 0 – green, σ = 0.5 σ 1 – blue, σ = σ 1 – red;solid lines – new model, dashed lines – original model (up to D = 1.76).

Lumley triangle for the present model while the original model becomes unrealizable. Moreover, theanisotropy approaches axisymmetric turbulence and eventually comes close to the two-componentlimit for increasing D. Fig. 1(b) shows the behaviour of the most critical eigenvalue, λ1, vs. thedilatation D where it is clear that the original model becomes unrealizable for D > 1.5. Fig. 2 showsthe corresponding anisotropies as well as P/ε. Observe that new model allows arbitrarily largenegative values of P/ε at sufficiently high D (depending on the σ chosen).

B. Quasi one-dimensional nozzle flow

Now we consider the plane quasi one-dimensional thin nozzle with turbulent flow, for which thetwo-dimensional expressions (41) are valid. The cross-sectional flow area as function of the stream-wise x-coordinate, which we assume in this section, is sketched in Fig. 3(a). We assume a uniformincoming mean streamwise velocity U and that the cross-stream variation of U remains negligiblethrough the nozzle. The y-component V is assumed to be negligibly small, while its y-derivative isfinite and is determined by the continuity equation, which itself becomes a consequence of the con-servation of mass in the entire cross section of the nozzle A(x), i.e., ∂t (ρ̄ A(x)) + ∂x (ρ̄ U A(x)) = 0.Moreover, the shear component of the strain-rate tensor, σ , as well as the shear stress Rxy and itsanisotropy axy are negligible but, similar to V , their y-derivatives are finite and must be accountedfor. These terms enter into the x-momentum equation through the divergence of the molecular andturbulence stresses. The analysis is given in the following.

Imposing the condition of irrotationality ∂y U = ∂x V , we arrive at ∂y ∂y U = ∂x ∂y V . Fur-thermore, σ = (τ/2) (∂y U + ∂x V ) = τ ∂y U = τ ∂x V and ∂y (τ−1 σ ) = ∂x (τ−1(D/2 − J )). Forthe anisotropy tensor axy and x-component of the divergence of the Reynolds stress tensor we


130.237.233.116 On: Wed, 23 Oct 2013 15:25:48


FIG. 3. Planar quasi one-dimensional nozzle flow. (a) Cross-sectional area A(x), arrow indicates the direction of the flow;(b) the corresponding dependence of Mach number M(x).

obtain

a = β1

⎛⎜⎝J + D/6 σ 0

σ −J + D/6 00 0 −D/3

⎞⎟⎠ with σ → 0,

(52)

−∂ j Rx j = −∂x(

2

3ρ̄ K

)+ τ−1

(2J + D

3

)∂x μt + 4

3μt ∂x

(τ−1 D

),

where the second equality may be extracted directly from (48).A statistically steady state of quasi one-dimensional irrotational mean nozzle flow with super-

imposed turbulence is considered here. The EARSM is complemented with equations for K and ωgiving the following closed system of equations for conservation of mass, momentum, and energy(recall, that we omit ůi -terms, � is negligible at small Mt, but p′ u′k may be of importance)

∂x ρ̄

ρ̄+ ∂x U

U+ ∂x A

A= 0,

(53)

ρ̄ U ∂x U = ∂x(

− P − 23

ρ̄ K

)+ τ−1

([2J + D

3

]∂x (μ + μt ) + D ∂x μv

)

+(

4

3(μ + μt ) + μv

)∂x

(τ−1 D

), U ∂x

P

ρ̄ γ+ ∂k p

′ u′kρ̄ γ

= (γ − 1) ερ̄ γ

(only the x-component of the momentum equation is essential in the problem) and for the turbulencequantities K and ω

ρ̄ U ∂x K = P − ε + ∂x T (K )x ,(54)

U ∂x ω = Cμ (C�1 − 1)Pε

ω2 − Cμ (C�2 − 1) ω2 + τ−1[

2

3(C�1 − 1) +

1

3− n (γ − 1)

]Dω + 1

ρ̄∂x T (ω)x ,

which should be complemented by relations for the dilatation D ≡ τ (∂x U + ∂y V ) = −τ U ∂x ρ̄/ρ̄and for the quantity J ≡ (τ/2) (∂x U − ∂y V ) = τ ∂x U − D/2. Variables D and J are obviouslythe only driving forces of the turbulence in the adopted approximation. Moreover, they represent theessential effects of compressibility. In incompressible flow D → 0 and J → τ ∂x U , which justifiesus to refer to the terms containing J as “incompressible,” and the terms with D as “compressible”(dilatational). Considering production-to-dissipation ratio in this spirit, we write

P/ε = (a11 − a22)J + (a11 + a22)D/6 − a33 D/3 − 2D/3


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and refer to the first term on the right-hand side as incompressible and to the remaining terms asdilatational. It results in the following split of P/ε, but note that N depends on both J and D:

Pε

=(P

ε

)in

+(P

ε

)dil

,

(Pε

)in

= 12J2

5 N,

(Pε

)dil

= D2

5 N− 2

3D. (55)

The equation for the “turbulence frequency” ω = �Cμ K

is the result of transforming the

�-equation discussed in Sec. II C with modeling coefficients C�1 and C�2 and the variable vis-cosity term. We do not give here the explicit expressions for the transport terms, but in Appendix Awe discuss their possible deviation from the commonly used form.

The system (53) and (54) can be easily integrated numerically, but only when the flow is ev-erywhere subsonic or supersonic. Otherwise, a time-marching technique must be applied (see, e.g.,Anderson26). For high magnitudes of turbulence kinetic energy, the interaction of mean flow withturbulence may become significant. Similarly, the behaviour of the transport terms gets importantunder high K or non-negligible viscosity. However, of primary importance for us is the response ofthe K − ω model to the EARSM proposed, before further effects are considered. In view of this wewill neglect the coupling with the turbulence in the mean-momentum and mean-pressure equationsand, hence, assume negligible values of K/U2, and neglect μ to arrive at the following simplifiedformulation after utilizing isentropic relations (Anderson26):

d U

U= M−2 d ρ̄

ρ̄= (M2 − 1)−1 d A

A= M−1 d M

1 + γ − 12

M2, (56)

where M = U/√γ R T is the Mach number. For a given A(x) or M(x), we can now directly calculateU and ρ̄ and solve the K- and ω-equations (54) in the presence of a predetermined mean flow.Now, also the ω-equation is fully decoupled from the K-equation and unless we are interested in theturbulence level itself, only the former equation for ω is needed to obtain J , D, N, P/ε, Sij, and aij.

In the following we will present the results based on the investigation of such a problem withisentropic mean flow. The profile of Mach number has been chosen as in Fig. 3(b), going fromM = 0.5 through M = 1 at the throat (x = 0.5) to M = 1.5, and is defined by M(x) = 0.25 + 1.5 x− 2 (x − 0.5)3. This corresponds to the cross sectional area distribution as in Fig. 3(a) with A(0)/A*= 1.34 and A(1)/A* = 1.176 being the inflow and outflow areas normalized by the throat crosssectional area A* = A(0.5).

Two cases with different inflow levels of the turbulence have been compared. In the first casethe nondimensional strain-rate is high, which corresponds to the RDT limit, and in the second casestrain-rate is low, representing turbulence almost in equilibrium with the imposed strain. A measureof the “total strain-rate” is given by S*:

S∗ =√

2 II ∗S , II∗S = S∗i j S∗i j , S∗i j =

τ

2

(∂i U j + ∂ j Ui

), II ∗S = IIS +

D23

. (57)

The variation in the magnitude of S*(x) is achieved by choosing different initial values of τ . Althoughinitial strain-rate is zero due to the definition of the geometry, we nevertheless characterize differentregimes of turbulence by a value S∗0 = τ |x=0

√(∂i U j + ∂ j Ui )2/2|x=0.5, where turbulence properties

are taken at the inlet of the nozzle, and kinematic properties at the throat. So, S∗0 = 11.3 andS∗0 = 1.22 have been chosen for the high and low strain-rate cases, respectively.

The two initial conditions have been used for computations with the EARSM based on theimproved pressure-strain model in Sec. II A as well as the modified �-equation presented inSec. II C with the variable viscosity term, but here transformed into the ω-equation (54). Com-putations without the variable viscosity term have also been done for comparison.

In Fig. 4(a) the evolution of the total strain-rate S* for both cases is plotted. The deviationsarising when we discard the variable viscosity correction are attributed to the suppressed growthof ω without the correction, as can be seen from Fig. 4(b). The correction makes the turbulencelife-time τ ∼ ω−1 shorter, especially in the supersonic region. In the high strain-rate case, we havedecreasing τ , thus causing the distinct drop in S* in the supersonic region, while in the low strain-rate


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FIG. 4. Spatial evolution: (a) “total strain-rate” S*, (b) “turbulence frequency” ω, (c) kinetic energy K, (d) J (thick lines)and D (thin lines). Red and blue lines – high and low strain-rate cases, respectively; solid and dashed lines – predictions withand without account for variable viscosity, respectively. Green lines in (a) the case of asymptotically low S∗0 .

case τ increases. Fig. 4(c) shows, that the behaviour of K is not very much affected by the variableviscosity correction, especially in the case of high strain-rate, implying that cancellation of somekind takes place. Fig. 4(d) describes the evolution of J and D, illustrating apparent exchange oftheir significance when going from subsonic to supersonic regime.

Fig. 5(a) presents a characteristic result, indicating that the dilatational production is essentiallynegative and overwhelms the incompressible production in the supersonic region, causing the total

FIG. 5. Spatial evolution: (a) incompressible (red) , dilatational (blue), and total (black) production; (b) a11 (red), a22 (blue),a33 (green). Solid and dashed lines – the cases of high and low strain-rate, respectively.


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production to change sign and become negative. Maximum and minimum peaks of the total produc-tion occur in the region of almost constant strain-rate, while the negative maximum of dilatationalproduction occurs just downstream of the throat and moves closer to the throat with increasing S∗0 .When we increase strain-rate, the point of zero total production moves towards the throat too, thoughremaining upstream the throat.

Fig. 5(b) shows that the diagonal components of the anisotropy tensor lie inside the realizableregion −2/3 ≤ aαα . axx is nearly constant, near its negative maximum, in the subsonic part of theplateau in S*. ayy and azz attain their nearly equal positive maxima in the subsonic and supersonicparts of the plateau, respectively. At the throat ayy = azz, which also follows from (52) with D =2J since ∂y V |x=0.5 = 0. When we asymptotically increase the strain-rate, the peak values of thecomponents approach axx → −0.6 , ayy → 0.49 , azz → 0.6 (at different positions). Hence, the valuesof J , D and P/ε grow in magnitude, but continue to ensure realizable values of the anisotropies.Moreover, by reducing the initial value of τ and consequently reducing S*(x), the resulting S*(x) curveasymptotically approaches the green curve in Fig. 4(a) with S∗max close to one. The correspondingcurves for J and D also tend to their asymptotic limits, causing P/ε and aαα to approach theirlimiting curves. In this case production-to-dissipation ratio is everywhere negative, and peak valuesof the components of the anisotropy tensor approach axx → −0.26 , ayy → 0.18 , azz → 0.2 (atdifferent positions). Indeed, a small value of S∗0 implies that the very large part of the turbulencescales are much smaller than that of the imposed strain, resulting in an initial period dominated byrapid decay of small scales until the peak of the spectrum becomes of comparable scale with theimposed strain. Such a condition would be associated with a low turbulence Reynolds number andasymptotically result in a value of S*(x) close to unity.

A few remarks concerning the original model for compressible turbulence of Wallin andJohansson4 follow. The proposed improvement of the pressure-strain rate term applied in the EARSMframework results in a difference only in the equation for N, which becomes N = (9/4) (c1 − 1 +P/ε + 2D/3) for the model presented in this study, unlike N = (9/4) (c1 − 1 + P/ε) in the previousmodel. Starting with asymptotically low strain-rate, we find that P/ε changes sign twice, eventuallybecoming negative at M ≈ 0.3. The negative peak of axx then occurs in the supersonic region, whilethe positive azz peak becomes significantly larger than the ayy peak. All the critical magnitudes arenearly twice those in the new model. Increasing the strain, we find that this correspondence holdsfurther, and near S∗0 = 2 the old model becomes unrealizable with axx < −2/3 near the throat, whichhappens after P/ε changes sign from positive to negative. Further increasing S∗0 , we eventuallyarrive at the situation when in the realizability region P/ε is positive.

Hence, the new explicit algebraic Reynolds stress model constitutes a robust approach formodeling compressible turbulent flow, ensuring realizability under arbitrary magnitude of dilatationand arbitrary turbulence state.

V. CONCLUSION

The systematic account for the effect of mean dilatation on the rapid part of the pressure-straincorrelation has allowed us to develop an explicit algebraic Reynolds stress model for compressibleturbulent flows. The EARSM, when expressed through vorticity- and traceless strain-tensors, isidentical to the corresponding incompressible model with a sixth-order algebraic equation deter-mining the model coefficients. The influence of variable viscosity on the dissipation equation isalso accounted for. Moreover, the baroclinic effect is included into the dissipation equation too,though being omitted from further analysis in this study, and an algebraic model for the density-velocity correlations is outlined, enabling us to estimate possible influence of corrections associ-ated with density fluctuations. A discussion of the Reynolds and Favre averaged equations revealsthe advantages and shortcomings of both approaches, leading us to adopt Reynolds averaging inthis paper.

In the case of two-dimensional mean flow we formulate the general framework for the EARSM.The sixth-order equation then reduces to a fourth-order equation, unlike a third-order one in theincompressible case, which is solved with the physical root univocally identified. The application ofa fixed-point analysis to a homogeneously sheared and one-dimensionally strained flow leading to


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significant mean dilatation, demonstrates the realizability of the model. Unlike the original modelby Wallin and Johansson,4 the production can attain arbitrarily negative values without loss ofrealizability.

For irrotational flow further simplifications can be made even in a general three-dimensionalcase. Only a quadratic equation is left and a pure eddy-viscosity expression for the Reynolds stresstensor is obtained. We proceed with the investigation of a quasi one-dimensional plane nozzle flow,incorporating the EARSM into a K − ω model and assuming that the mean flow is isentropic andtranscends from subsonic to supersonic regime at the throat. The model demonstrates realizabilityfor arbitrary strain-rate, which is controlled by the initial turbulence time scale τ x = 0. Asymptoticallydecreasing τ 0 at the inlet of the nozzle, we find that it rapidly increases with x so that strain-rateS*(x) tends to a limiting curve with S∗max ∼ 1. This indicates that an equilibrium between shear andturbulence time scale is established then. By splitting the turbulence production into incompressibleand dilatational parts, we find that the negative dilatational part changes from negligible to dominantwhen moving from subsonic to supersonic region. Thus, the total production attains negative valuesin any strain-rate regime, while in the original model at sufficiently high S* the realizability regionis restricted only to positive values of production .

There is an obvious need for accurate reference data for generic flow cases with significant meandilatation for calibration and validation of the models proposed herein. DNS-data would be idealfor a validation of the Reynolds and Favre formulations, since both Reynolds and Favre averagedquantities can readily be computed, while in experiments either of these quantities is measured.Note that it is not trivial to change from one formulation to the other. However, the incompressiblemodel has been extensively validated for a large number of cases and the proposed extension tocompressible flows is physically consistent and realizable, unlike earlier versions, which is of primaryimportance.

Although the new formulation for the rapid pressure-strain correlation has been incorporatedinto the EARSM framework here, its utilization in a DRSM context should retain the revealedadvantages, particularly leading to realizable values of the anisotropy tensor.

ACKNOWLEDGMENTS

Support from the Swedish Research Council VR through Grant Nos. 2010-3938, 2010-6965,and 2010-4147 is gratefully acknowledged.

APPENDIX A: DETAILS OF THE MODEL FORMULATION

Recall that the total velocity can be expressed with the Reynolds decomposition as Ui + u′ior with the Favre decomposition as ũi + u′′i . The equation for fluctuating velocity in the Reynoldsdecomposition form reads

∂t u′i + (Uk + u′k)∂ku′i + u′k∂kUi − u′k∂ku′i +

∂i p

ρ− 1

ρ∂i p = ∂k τik

ρ− 1

ρ∂k τik, (A1)

while the equation for fluctuating velocity in Favre form can be written

∂t u′′i + (ũk + u′′k ) ∂ku′′i + u′′k ∂k ũi −

∂k R( f )ik

ρ̄+ ∂i p

ρ− ∂i P

ρ̄= ∂k τik

ρ− ∂k τik

ρ̄. (A2)

From these we obtain the next equation for ůi = u′i − u′′i = ρ ′u′i/ρ̄:

∂t ůi + ũk ∂k ůi + ∂k ρ′ u′′i u

′′k

ρ̄= −ůk ∂k ũi − R( f )ik

∂k ρ̄

ρ̄2− u′′i ∂k u′′k −

ρ ′

ρ̄ ρ(∂i p − ∂k τik), (A3)

and this equation may be rewritten in terms of Reynolds decomposed variables as

∂t ůi + Uk ∂k ůi + ∂k ρ′u′i u

′k

ρ̄= −ůk ∂kUi − Rik ∂k ρ̄

ρ̄2− u′i ∂k u′k −

ρ ′

ρ̄ ρ(∂i p − ∂k τik) + ůi ∂k (ρ̄ ůk)/ρ̄,

(A4)


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where the only difference between (A3) and (A4) after the exchange ũi , u′′i → Ui , u′i formally is thepresence of ůi ∂k (ρ̄ ůk)/ρ̄ on the right-hand side of (A4).

For the density-density correlation we get the expression which looks the same in both Reynoldsand Favre variables:

∂t ρ ′2 + Uk ∂k ρ ′2 + ∂k ρ ′2 u′k = −2 ρ ′2 ∂k Uk − 2 ůk ρ̄ ∂k ρ̄ + (2 ρ̄ ρ ′ + ρ ′2) ∂k u′k . (A5)In Favre form the Reynolds stress tensor is governed by

∂t R( f )i j + ∂k (R( f )i j ũk) + ∂k ρ u′′i u′′j u′′k = −R( f )ik ∂k ũ j − R( f )jk ∂k ũi − u′′j (∂i p′ − ∂k τ ′ik)−u′′i (∂ j p′ − ∂k τ ′jk) + �( f )i j , �( f )i j = ů j (∂i P − ∂k τ ik) + ůi (∂ j P − ∂k τ jk), (A6)

and in Reynolds decomposition variables the corresponding equation reads

∂t Ri j + ∂k (Ri j Uk) + ∂k ρ u′i u′j u′k = −Rik ∂k U j − R jk ∂k Ui

−u′j (∂i p′ − ∂k τ ′ik) − u′i (∂ j p′ − ∂k τ ′jk) + �i j , (A7)where

�i j = −ρ̄ ů j Dt Ui − ρ̄ ůi Dt U j , Dt Ui = −u′k ∂k u′i −(

∂i p − ∂k τikρ

). (A8)

An alternative form of the momentum equation (5) is used here.To proceed further, we decompose fluctuating pressure into inertial, acoustic and viscous parts

p′ = p′in + p′ac + p′visc, where

p′in =1

4π

∫ {∂lm

(ρ u′lu

′m + 2ρ u′l Um + ρ ′ Ul Um − ρ u′lu′m + 2ρ u′l Um

)}y

d3y|x − y| ,

(A9)

p′ac =1

4π

∫ {−∂t t ρ ′}y d3y|x − y| , p′visc =(

μv + 43μ

)∂k uk −

(μv + 4

3μ

)∂k uk−

− 14π

∫ {2 ∂i

(∂k μ∂i uk − ∂i μ∂k uk − (∂k μ∂i uk − ∂i μ∂k uk)

)}y

d3y|x − y| ,

and the gradient of the viscous contribution is

∂i p′visc = ∂i

[(μv + 4

3μ

)∂k uk

]+ 2 (∂k μ∂i uk − ∂i μ∂k uk) − /.../, (A10)

whereby /.../ is denoted the average of the preceding terms. Now we observe that

∂i p′ − ∂k τ ′ik = ∂i p′in + ∂i p′ac − ∂k

(μ (∂k ui − ∂i uk)

)+ ∂k

(μ (∂k ui − ∂i uk)

). (A11)

We consider turbulence with low turbulent Mach number Mt, which means that p′ac is negligible.Now we rewrite Eq. (A7) as

∂t Ri j + ∂k (Ri j Uk) + ∂k Ti jk = Pi j + �i j − εi j + �i j ,Pi j = −R jk ∂kUi − Rik ∂kU j , �i j = p′in (∂i u′j + ∂ j u′i ), εi j = 2 μ ∂ku′i ∂ku′j − μ (∂i u′k ∂ku′j + ∂ j u′k ∂ku′i ),

Ti jk = ρ u′i u′j u′k + p′in (u′iδ jk + u′jδik ) + μ ∂k(u′i u′j ) − μ (u′j ∂i u′k + u′i ∂ j u′k), �i j = ψi j + ψ j i , (A12)

ψi j = −ρ̄ ů j Dt Ui − u′j ∂k[μ′ (∂k Ui − ∂i Uk)

], Dt Ui = −

(∂i p − ∂k τik

ρ

)− ∂k Rik

ρ̄+ ∂k ρ

′ u′i u′k

ρ̄+ u′i ∂k u′k ,

Tk = Tllk2

,P = Pi i2

, ε = εi i2

, � = �i i2

, � = �i i2

= ψi i .

Note that the correction to the dissipation by the variable part of dynamic viscocity is incorporatedinto the quantity � ij.


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Remarkable here is that under the above adopted split of pressure (A9) the bulk dissipationbecomes excluded from the equation for the Reynolds stress tensor. Moreover, the dissipationentering (A12) is the solenoidal dissipation ε = 12 μ (∂k u′i − ∂i u′k)2, independent of fluctuatingdilatation. It means that the cascade of dilatational dissipation goes directly to the internal energy, away round the turbulent kinetic energy.

In the expression for the transport of turbulent kinetic energy Tk = Tiik/2 the terms with molecularviscosity can be approximated in the following way. We assume that the contribution of ∂i u′i and ρ

′

is negligible, which leads to

μ∂k u′2j /2 − μ u′j ∂ j u′k = μ̄ ∂k K − μ̄ ∂ j(

2

3K δ jk + K a jk

)= μ̄

(∂k K/3 − ∂ j (K a jk)

).

(A13)Thus, we may expect that the transport terms in the K - and ω-equations are given by

T (K )k =(

μ̄ + μtσK

)(∂k K/3 − ∂ j (K a jk)

), T (ω)k =

(μ̄ + μt

σω

)(∂k ω/3 − ∂ j (ω a jk)

), (A14)

with σ K and σω tuned.For general three-dimensional mean flows the tensor groups T(i) and the corresponding coeffi-

cients β i, involved in (34), are given by

T (1) = S, T (3) = �2 − II�3

I3, T (4) = S � − � S, T (6) = S �2 + �2 S − 23

IV I3,

T (9) = � S �2 − �2 S �, Q = 56

(N 2 − 2 II�) (2 N 2 − II�),(A15)

β1 = −N (2 N 2 − 7 II�) Q−1, β3 = −12 N−1 IV Q−1, β4 = −2 (N 2 − 2 II�) Q−1,β6 = −6 N Q−1, β9 = 6 Q−1.

APPENDIX B: ALGEBRAIC MODEL FOR DENSITY-VELOCITY CORRELATION

The equation for the quantity ζ̊i = ůi ρ̄√ρ ′2

√K

in terms of �i = τ√

K√ρ ′2

∂i ρ̄ reads

∂t ζ̊i + Uk ∂k ζ̊i + ρ̄√ρ ′2

√K

∂kρ ′u′i u

′k

ρ̄− ζ̊i

2

(∂k Tkρ̄ K

+ ∂k ρ′2u′k

ρ ′2− ∂k (ρ̄ ůk)

ρ̄

)=

−ζ̊k ∂k Ui − τ−1 ζ̊i2

P − ε + � + �ε

+ τ−1 ζ̊i ζ̊k �k − τ−1(

aik + 23

δik

)�k−

− ρ̄√ρ ′2

√K

(u′i ∂ku

′k +

ρ ′

ρ ρ̄(∂i p − ∂k τik)

)− (ρ̄ ρ

′ + ρ ′2/2) ∂k u′kρ ′2

ζ̊i . (B1)

Trying to build an algebraic model for the density-velocity correlation, we equate the left-hand sideof (B1) to zero, making thus the assumption of weak-equilibrium. Upon that we face the need tomodel the last three terms on the right-hand side, none of them is obvious how to deal with. Toprogress, we suggest the following simplistic modeling:

u′i ∂ku′k = −cm ůk ∂k Ui − cn ůk ∂i Uk,

ρ ′

ρ ρ̄(∂i p − ∂k τik) = 2 cp ρ̄ ůi ůk

ρ ′2∂k ρ̄,

(B2)(ρ̄ ρ ′ + ρ ′2/2) ∂k u′k = −cρ ρ ′2/τ.

We, hence, assume that the first term is described by the interaction of ůi with the shear of meanvelocity. The effective response to the second combination is designed to be a non-linear interaction


130.237.233.116 On: Wed, 23 Oct 2013 15:25:48


of ζ̊i with mean density gradient. This is in line with modeling presented in Wikström et al.6 for thescalar heat flux. The third combination is directly connected to the pressure-dilatation correlation,and we expect that fluctuations are nearly isentropic, which leads to p′/ p̄ = γ ρ ′/ρ̄. So, we expectthe last combination to behave like ρ̄ �/c2s . There are analogies with the modeling of � (e.g.,Friedrich9), which lead us to infer that the value of cρ may be a function of flow parameters andhistory of the flow, easily changing sign during the simulation. Now the model is reduced to thealgebraic equation of the form

−(1 − cm) ζ̊k τ ∂k Ui + cn ζ̊k τ ∂i Uk − ζ̊i2

(P − ε + � + �ε

− 2 cρ)

+ (1 − 2 cp) ζ̊i ζ̊k �k

−(

aik + 23

δik

)�k = 0, (B3)

with the last term, proportional to the mean density gradient, presumably being an important term inthe equation. We expect that ζ̊i and �i are of order unity, so the penultimate term cannot be neglectedin general, unless cp = 1/2 or ζ̊k �k ∼ 0. We here make the further simplifying assumption, cp =1/2, thus rendering the equation linear, and omitting � and �. The equation can then be writtenas (31) in the main text with cS = 1 − cm − cn and c = 1 − cm + cn . Here the solution for thetwo-dimensional case is outlined as

ů = −A−1(

a + 23

I3

)τ K

∇ ρ̄ρ̄

, A−1 = G2 I2 − cS S2D − c �

det A2D+ I3 − I2

G0,

A = A2D + G0 (I3 − I2), A2D = G2 I2 + cS S2D + c �, (B4)

det A2D = G22 −1

2(c2S IIS2D + c2 II�), G2 =

1

2

P − εε

− cρ, G0 = 12

P − εε

− cρ + cS D2

.

If we still further imply zero rotation, then ůi will be expressed as

ůx = −(k1 + k2 J ) ∂x ρ̄ρ̄

− k2 σ ∂y ρ̄ρ̄

, ů y = −k2 σ ∂x ρ̄ρ̄

− (k1 − k2 J ) ∂y ρ̄ρ̄

,

(B5)

k1 = τ K[

2

3G2 + β̂1

(D6

G2 − cS IIS2D

2

)]/det A2D, k2 = τ K

[β̂1

(G2 +cS D

6

)+ 2

3cS

]/det A2D.

APPENDIX C: ANALYSIS OF THE QUARTIC EQUATION

The four roots of (36) are given by

Ñ1,2,3,4 = 14

+ 12

R1,2 ± 12

√1

2− z + a + R1,2 + 2 b − z

R1,2, R1,2 = ±

√1

4+ z + a = ±

∣∣∣∣ z2 − b∣∣∣∣

√z2 + 4 c ,

(C1)where z is any solution of the resolvent equation z3 + a z2 + (b + 4 c) z − (b2 − c − 4 a c) = 0,which we put as

z = b2 − c − 4 a c

b + 4 c M̃−1, M̃3 − M̃2 − A M̃ − B = 0, (C2)

with A and B from (44). Any root of (C2), when substituted into (C1), gives all the solutions, onlythe order

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