density correction for turbulence model

8/10/2019 Density Correction for Turbulence Model

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Aerosp. Sci. Technol.4 (2000) 111 2000 ditions scientifiques et mdicales Elsevier SAS. All rights reservedS1270-9638(00)00112-7/FLA

Density corrections for turbulence models

Stphane Catris a,Bertrand Aupoix a,*

a ONERA, Department of Modelling in Aerodynamics and Energetics, Centre dtudes et de Recherches de Toulouse,B.P. 4025, 2 avenue douard Belin, 31055 Toulouse, cedex 4, France

Received 4 June 1999; revised 17 September 1999; accepted 8 November 1999

Abstract The present paper deals with the improvement of the predictions of high Mach number boundary layers.Two-equation models have been shown to poorly predict the logarithmic region of the boundary layer inpresence of large density gradients. This induces significant errors in the skin friction coefficient prediction.A correction of the diffusion terms in two-equation models is suggested to account for the density variationsand to retrieve the logarithmic law. The analysis yields the result that new quantities involved in transport arecombinations of the product kand the length scale. Standard forms of transport equations, with extra termslinked to density variations can be derived. Tests cases on flat plate boundary layers, over a wide range ofMach numbers, Reynolds numbers and wall to adiabatic temperature ratios as well as flat plate experimentshave confirmed the present analysis. 2000 ditions scientifiques et mdicales Elsevier SAS

turbulance models / compressibility / eddy viscosity / transport equations / logarithmic law

Rsum Amlioration des prvisions des modles de turbulence.Cet article traite de lamlioration des prvisions

des modles de turbulence pour les couches limites grands nombres de Mach. Il a t montr que les modles deux quations prvoient mal la rgion logarithmique dune couche limite en prsence de forts gradients demasse volumique. Les erreurs significatives commises sur la prvision du coefficient de frottement parital ensont une consquence. Une correction des termes de diffusion turbulente dans les modles deux quationsest propose afin de tenir compte des variations de masse volumique et de retrouver la loi logarithmique.Lanalyse mne considrer deux nouvelles quantits transporter qui sont des combinaisons du produitde lnergie cintique de turbulence par la masse volumique du fluide, k , et de lchelle de longueur .Une forme standard des quations de transport, avec des termes supplmentaires lis aux variations de massevolumique, peut tre obtenue. Des cas tests de couches limites de plaque plane, pour une large gamme denombres de Mach, de nombres de Reynolds ainsi que pour diffrents rapports de la temprature de paroi surla temprature de paroi adiabatique, et des expriences de plaque plane ont confirm cette analyse. 2000ditions scientifiques et mdicales Elsevier SAS

couche limite turbulente / compressibilit / modles de turbulence / viscosit tourbillonnaire / quations

de transport / loi logarithmique

Nomenclature

a1 0.15 Structure parameterC= (2 a1)2

* Correspondence and reprints

d1, d2, d3 Coefficients of density related terms

k Turbulent kinetic energy

Turbulent length scale


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2 S. Catris, B. Aupoix / Aerosp. Sci. Technol. 4 (2000) 111

Pk Turbulent kinetic energy productionq2 = 2 k Turbulent kinetic energyR Reynolds number based on the momen-

tum thickness(= eue /w )T Temperature

u Streamwise velocityu Friction velocity(

w/w)

v Normal velocityx Streamwise coordinatey Coordinate along the wall normal Turbulent kinetic energy dissipation rate Momentum thickness Von Krmn constantt Eddy viscosity Density Turbulent Prandtl number Shear stress=

k=

kSpecific dissipation

Ensemble average

Subscripts

e Boundary layer edge valuew Wall valueaw Adiabatic wall value

Superscripts

Fluctuation with respect tothe mass-weighted average

1. Introduction

This paper deals with predictions of the compressibleboundary layer using eddy viscosity models. The turbu-lence models currently used to predict compressible flowsare simple extensions of incompressible models with thehelp of mass-weighted averages. Coakley and Huang [4],Wilcox [24] and Aupoix et al. [3] have shown that mostof these models tend to underestimate the skin frictionand the wall heat flux as the Mach number increases.

Compressibility corrections have been proposed bySarkar et al. [15], Zeman [25] for compressible mixinglayers and by Wilcox [24] for compressible mixing lay-ers and boundary layers to account for the compressiblecharacter of the turbulent motion. Except Zemans [26]corrections for the compressible boundary layer, they in-troduce a dilatation dissipation term which damps the tur-bulence and yields lower skin friction and wall heat fluxvalues as pointed out by Wilcox [24] and Aupoix et al.[3]. Moreover, these compressibility corrections shouldact only when the turbulence motion is compressible, i.e.

for external Mach number greater than five for standardboundary layers while significant errors are observed atlower Mach number [3].

A key feature of compressible boundary layer, deducedfrom experimental data (Fernholz and Finley [8]), is that,using van Driest transformed velocity

u=

wdu,

a logarithmic velocity profile is retrieved as in incom-pressible flows. Wilcox [24] and Huang et al. [12] haveshown that standard turbulence models cannot reproducethe equilibrium in the logarithmic layer when densitygradients appear. Huang et al. have pointed out the in-fluence of the variable used to specify the length scale(, , , , . . .). Aupoix and Viala [2] used ad hoc tricksto force the models to reproduce the logarithmic law.

Thus, a good prediction of both the velocity profile andthe skin friction is achieved.Only zero pressure gradient boundary layer will be

considered in order to emphasize compressibility effects.

2. Huang, Bradshaw and Coakley (HB) analysis

The Huang, Bradshaw and Coakley [12] analysis ofthe prediction of the logarithmic layer of a compressibleboundary layer is considered. For the sake of simplicity,a generic two-equation model is considered. Its highReynolds number form is:

Dk

Dt= Pk + div

t

kgrad k

, (1)

D

Dt= C1Pk

k C2

k+ div

t

grad

, (2)

t= Ck2

, (3)

where the turbulent kinetic energy k provides the turbu-lence velocity scale while the turbulence length scale isgiven by= nkml .Pkrepresents the turbulent kineticenergy production. Its expression will be given below.The constants C

1and C

2are related to the correspond-

ing coefficients of the equation by:

C1= lC1+ m, C2= lC2+ m. (4)

In the logarithmic region of a zero pressure gradientboundary layer, by neglecting advection and viscouseffects, the Reynolds stress can be deduced from themomentum equation as follows:

uv= w= wu2. (5)


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S. Catris, B. Aupoix / Aerosp. Sci. Technol. 4 (2000) 111 3

The assumption of a logarithmic velocity profile gives:

u

y=

w

u

y, (6)

where is the von Krmn constant and characterizesthe slope of the logarithmic law. So, the Boussinesqhypothesis gives the eddy viscosity as:

t=

w uy . (7)

The dissipation rate could be linked to the kinetic tur-bulent energy with the help of equation (3), and its ex-pression could be used in the turbulent kinetic energytransport equation to determine a solution for k. Unfor-tunately, using this method, thekequation has no analyt-ical solution. Therefore, Huang et al. make use of Brad-shaws assumption, which relates the turbulent kinetic en-ergy and the Reynolds stress:

uv= 2 a1 k (8)so that the kinetic energy becomes, with C= (2a1)2:

k= w

u2C

. (9)

The dissipation rate, and hence , can be deduced fromthe Boussinesq hypothesis or equation (3) and is:

=

w

3/2u

y. (10)

With the above hypothesis (5)(9), the turbulent kineticenergy transport equation (1) is no longer balanced in thewall region where advection terms can be neglected. Toretrieve this balance, Huang et al. deduce the turbulentkinetic energy production,Pk , from equation (1), but thisis not consistent with Pk= uv(u/y). With thepreceding assumptions, equation (2) yields:

C

l22 (C2 C1)

=1

+1

l2d

1

y

y +d

2

y2

2

y 2+d

3y

y

2

. (11)Here the di coefficients are functions ofn, m, l , C1, kand :

d1=

n m 32

l

(1 2l) 1

2l+ C1

k, (12)

d2= n m 3

2l+ C1

k, (13)

d3=

n m 32

l

n m 3

2l+ 1

32

C1

k. (14)

If the models have to predict the same slope for thelogarithmic law in compressible as in incompressibleflows, i.e. the same value of , the right hand side of

the equation (11) must be equal to unity. Unfortunately,no combination ofn,m, l allows us to make d1, d2 andd3 all zero. At best, two coefficients can be set to zero.The terms involving C1 /k come from the diffusionterm ofk, which is not equal to zero (cf. equation (9)).The diffusion of provides the other terms of equations(12)(14). Therefore, the diffusion terms in the transportequations must be modified in order to ensure the samevalue of whatever the density variations.

The domain of validity of the generic two-equationmodel (1)(3) has to be pointed out. With the form ofequations (1) and (2), the left hand side of equation (11),which is equal to unity in incompressible flows, iswritten, using equations (4):

C

l 2 (C2 C1 ). (15)

Since C2 > C1 and must be positive, the onlychoice for l are positive values. As a matter of fact, thisapproach holds for k -or k -models, but not for a k -model, nor for the SpalartAllmaras model, which needa specific analysis similar to the one described below.

3. Compressibility corrections

3.1. Turbulent kinetic energy equation

In the turbulent kinetic energy equation, the only termto be modelled is the diffusion term, which requiresa divergence form, in order to be consistent with theexact form arising from the NavierStokes equations.The turbulent kinetic energy diffusion flux vector couldbe expressed as a combination of all gradients of theturbulent flow quantities, i.e.

fluxk tgrad k; tk

grad ; t

k

grad . . . .

(16)

A simple form can be proposed arguing that the diffusionacts upon fluid element the energy per unit volume ofwhich is k. Hence, the diffusion flux can be directlylinked to the gradient of k.

Thus, the proposed form of equation (1) is:

Dk

Dt= Pk + div

t

kgrad(k)

. (17)

In the logarithmic layer k is constant and the diffu-sion term is equal to zero. If advection is neglected, the


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Spalart and Allmaras [18,19] tmodel, as well as theirabove presented modified versions, have been comparedon flat plate boundary layer cases, using a compressibleboundary layer code [1]. The grid convergence was en-sured for all the tested models.

For the Chien model, the wall variable y+= wy/has been used in the damping functions as recommendedby Aupoix and Viala [2].

Concerning the SpalartAllmaras model (S.A. model),its results, being close to those of the Smith k -model,will not be presented here.

4.1. Tests cases

The velocity profile and the skin friction predictionshave been investigated for zero pressure gradient bound-

ary layers, with Mach numbers ranging from 3 to 8, wallto adiabatic temperature ratios between 0.2 and 1 andfor Reynolds number based on the momentum thicknessR=eue/wup to 105. Only typical results are pre-sented here for both k-and k-models.

In figure 1, the velocity profiles are plotted in semi-logarithmic form using van Driest transformed velocity

u+= 1u

wdu = f

wy u

w

for a Reynolds number R

equal to 105. The standard

logarithmic law

u+= 1

ln

wy u

w

+ C with= 0.41, C= 5.25

(54)is plotted as a reference (dotted line).

The prediction of the logarithmic layer is clearlyimproved. A constant slope is now predicted, with avalue close to 1/, whatever the Mach number and thewall temperature. It can be pointed out that the diffusionterm modifications also influence the prediction of theintercept C, by altering the buffer layer, whereas the wakeand the viscous sublayer are not changed. Although inthese figures the interceptCis well predicted, changes inthe Mach number or in the wall temperature can inducechanges of about 15% of the value of the intercept,especially for thek-model. Thek-and thek-modelsare slightly sensitive to the temperature ratio, while theprediction of the intercept by the k- model is almost

independent of Mach number and temperature ratio.The density corrections act upon the mean velocityprofile through the eddy viscosity level. So, they vanishnear the wall because of the damping function and at theedge of the boundary layer with the density gradient.

Figure 2 compares the evolution of the skin frictioncoefficient, plotted as the relative variation (Cf/Cf VDII1) 100 of the skin friction coefficient with respect to thevan Driest II [20] correlation (1951) versus the Reynoldsnumber R. In this figure, the symbolsCfHBCrefer to theHuang, Bradshaw and Coakley correlation [11] (1993).Viala [21] and Aupoix and Viala [2] have shown thatmost experiments are in good agreement (10%) withboth correlations.

(a) (b)

Figure 1.Tests cases Boundary layer profile. (a) Mach=5Tw/Taw0.4R=105 k-model. (b) Mach=5 Adiabaticwall R

=105 k-model.


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After a short transient, a nearly constant error isobserved, as well for the standard as for the modifiedversion of the models. The slope of the logarithmic lawusually decreases for the modified models and hence theskin friction coefficients increase. For all the models,the modification leads to a prediction closer to theHuang et al. [11] correlation. It can also be observedthat henceforth the skin friction coefficient evolves withrespect to the Mach number or to the wall temperaturelike the Huang et al. correlation.

The difference between a model and its modified ver-sion decreases when the wall to adiabatic temperature ra-

(a)

(b)

Figure 2.Tests cases Relative variation on Cf with respectto the van Driest correlation. (a) Adiabatic wall k -and k -models Mach=3 andMach=8. (b)Mach= 5 k-andk-models Tw/Taw=0.2 and 0.6.

tio decreases, since the density gradient effect is less im-portant.

The same conclusions hold for the other tested models,i.e. thek-,k-and S.A. models.

4.2. Comparisons with experiments

Four experiments of zero pressure gradient boundarylayer have been selected. The Mach number ranges from4 to 10, the wall to adiabatic temperature ratio from 0.5 to1.0 and the Reynolds number based upon the momentumthickness from 600 to 5000. Except for Winkler andChas experiment, the skin friction coefficient has beendirectly measured with a floating element balance.

The experimental boundary layer profiles, comple-mented by analytic laws in the wall region, are used asstarting profiles in the computations. The computed ve-locity profiles are compared to experimental ones at thelast station.

The first case is the Mabey et al. [10] experimentof a Mach 4 boundary layer over an adiabatic wall.The Reynolds number R varies between 3100 and5000. Figure 3 compares experimental and computedvelocity profiles at the last station. The skin friction

(a)

(b)

Figure 3.Mabey et al . [10] experiment Velocity profile. (a) Inwall variables. (b) Physical variables.


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Figure 4.Mabey et al. [10] experiment Skin friction coeffi-cient.

coefficient predictions are plotted in figure 4, an errorbar of10% being added to the experimental skinfriction coefficients. The model modifications improveboth the velocity profiles and the skin friction predictions,especially for the k-, k- and k- models. Thus, thelogarithmic layer and the wake region are correctlypredicted as shown infigure 3.The k- and k-models as

well as their modified versions fail to reproduce the nearwall region (cf.figure 3). The originalk-model alreadyreproduces this flow fairly well, so that the variation ishardly noticeable.

Winkler and Cha [9] investigated a Mach 5.3 boundarylayer over a cold wall (Tw/Taw 0.9). The ReynoldsnumberRranges from 600 to 900. Velocity profiles andskin friction predictions are plotted in figures 5 and 6.A featureof this experiment is its weak Reynolds number.Indeed, Purtell et al. [14] have studied the extent of thelogarithmic law versus the Reynolds number and haveshown that this extent, in wall variable, decreases withdecreasing Reynolds number. Even if the logarithmicregion is very small, the same improvements than those

for the Mabey et al. [10] experiment are observed.The last two cases deal with hypersonic boundary

layers. Watson [22] investigated a helium, Mach 10boundary layer over an adiabatic wall. The Reynoldsnumber Rranges from 800 to 1400. The skin frictionpredictions are plotted infigure 7(a). The Owen et al. [13]experiment deals with a Mach 7.2 boundary layer overa cold wall (Tw/Taw0.5). The Reynolds number Ris greater and ranges from 1400 to 3000. The evolutionof the skin friction coefficient is plotted in figure 7(b).The same trends as previously are observed: the model

(a)

(b)

Figure 5.Winkler et al. [9] experiment Velocity profile. (a) Inwall variables. (b) Physical variables.

Figure 6.Winkler et al. [9] experiment Skin friction coeffi-cient.


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(a)

(b)

Figure 7. Hypersonic cases Skin friction coefficient [13].(a) Watson experiment [22]. (b) Owen et al. experiment.

modifications increase the skin friction, more for the k-and k -models than for the k -model and very weaklyfor the k- model. For the Watsons experiment themodel modifications improve the k - model predictionsbut degrade the k-model. Otherwise, modified modelpredictions remain within the error bars.

5. Concluding remarks

A correction of the diffusion terms has been proposedin order to make turbulence models consistent withthe logarithmic law for compressible boundary layers.First, in the turbulent kinetic energy transport equation,the turbulent diffusion acts upon the energy of a unitvolume of fluid k . A more general form of the lengthscale equation has been investigated. It appears that theturbulent diffusion must act upon a combination of kand . From this approach, new forms of the k-, k-and k- models have been derived. A similar analysishas been used to extend the k- and SpalartAllmarasmodels.

Standard and modified model predictions have beencompared for zero pressure gradient boundary layers,over a wide range of Mach number, Reynolds numberand wall to adiabatic temperature ratio. The modificationsallow a better reproduction of the logarithmic layer and

therefore yield higher skin friction levels. The changeis larger for the k- and k- models than for the k- and SpalartAllmaras models and very slight for thek- model. Modified models generally improve thepredictions.

Acknowledgments

It is the authors pleasure to acknowledge SPA (Min-istry of Defense) who granted this research.

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