el liptic r - stanford university

Center for Turbulence Research

Proceedings of the Summer Program ��

��

Second moment closure analysisof the backstep �ow database

By S� Parneix� D� Laurence� AND P� Durbin�

A second moment closure computation �SMC� is compared in detail with the di�rect numerical simulation �DNS� data of Le and Moin for the backstep �ow atRe � �� in an attempt to understand why the intensity of the back�ow and�consequently� the friction coe�cient in the recirculation bubble are severely under�estimated The data show that this recirculation bubble is far from being laminarexcept in the very near wall layer A novel di�erential a priori� procedure was used�in which the full transport equation for one isolated component of the Reynoldsstress tensor was solved using DNS data as input Conclusions are then di�erentfrom what would have been deduced by comparing a full simulation to a DNS Onecause of discrepancy was traced back to insu�cient transfer of energy to the nor�mal stress by pressure strain� but was not cured A signi cant nding� con rmedby the DNS data in the core region of a channel �ow� is that the coe�cient thatcontrols destruction of dissipation� C�� should be decreased by a factor of � whenproduction is vanishing This is also the case in the recirculation bubble� and a newformulation has cured �� of the back�ow discrepancy

�� Introduction

The �ow over a backward�facing step has been probably the most popular sepa�rated �ow test case of the past �� years� for which numerous experiments �by Kim�Johnston� Eaton� Vogel� Durst� Driver� etc� provide data on the e�ects of geome�try� inlet conditions� and Reynolds number With the improvement of turbulencemodels and numerical methods� it is now generally possible to recover the reattach�ment length� but the intensity of the back�ow and� as a consequence� the negativepeak in skin friction are always underestimated by nearly a factor of � when secondmoment closures �SMC� are usedThe recent DNS database of Le and Moin �� at Re � �� well corroborated

by the experiments of Jovic � Driver �� and of Kasagi et al� �� is analyzedhere to understand this severe defect common to all SMC

�� Full simulation of the backward�facing step

The �ow was computed using INS�D� a nite di�erence code in generalized co�ordinates written at NASA Ames Research Center A ne� non�uniform grid of

� Electricit�e de France� DER� Lab� Nat� d�Hydraulique� � quai Watier� �� Chatou� France

Stanford University

�� S� Parneix� D� Laurence � P� Durbin

�� cells was used to cover the region x�h � �� to �� x � � being thelocation of the sudden expansion and h being the step height The inlet values forthe mean velocities� Reynolds stresses� and dissipation were taken from the DNSdatabase The elliptic relaxation procedure of Durbin �� has been combinedwith the Speziale� Sarkar� Gatski �SSG� pressure strain model in the neutral� for�mulation as in Laurence et al� �� see Appendix�

�a� �b�

Figure �� Streamlines �a� Second moment closure� �b� DNS

Figure � shows the predicted streamlines compared to the DNS data The reat�tachment is correct and a secondary bubble is found However� the size of this cornerbubble is much smaller in the simulation than in the data In fact� if one looks at thepredicted friction coe�cient compared to the DNS and experimental data �Fig ��the intensity of the main recirculation is underpredicted by a factor �� the slightimprovement shown by a dashed curved is discussed further in section �� Thestagnating �ow between the two recirculations at x�h � � is also missed Since it isbelieved that the underestimation of the secondary bubble is a consequence of theunderestimation of the primary recirculation� we will concentrate in the followingon curing the latter discrepancy Note that the recovery after reattachment is alsotoo slow�this is a problem in virtually all turbulence transport modelsThe above observations are believed to re�ect what can be expected from any

state of the art SMC In Fig �� the mean streamwise velocity U is shown The centerof the recirculation is well predicted� but its intensity is severely underestimated�see station x�h � �� Since Cf is too weak� this is not due to an overprediction

Elliptic relaxation for second moment closures ��

Cf

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y�h

U�U�

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�b�

Figure �� a� Friction coe�cient� � DNS �Le � Moin�� N experiment �Jovic �Driver�� SMC� modi ed SMC �cf section �� b� DNS U�pro les atlocations x�h � �� M�� C� and �� O�� model

of turbulent mixing in the near wall region� but rather to an underestimation ofthe entrainment from the shear layer Since the �ow in the upper layer splits atthe stagnation point into the recirculation and the downstream �ow� this samevelocity defect is transported into the recovery region Le � Moin noted� in goodagreement with the Jovic � Driver experiment� that the log pro le of the law ofthe wall was still not recovered at x�h � �� this as a consequence of the lowReynolds number In the Reynolds�averaged Navier�Stokes �RANS� computation�the recovery at x�h � �� is� however� still slightly underestimated

Figure � focuses on the recirculation and shows the budgets of the U momentumequation The underestimation of the back�ow is most severe at station x�h � �As expected� the adverse pressure gradient driving the back�ow is fairly constantacross the whole height and balances the viscous shear stress at the wall� hence�the pressure gradient determines the value of Cfmin

Jovic and Driver �� found

that the minimum of Cf follows a laminar like� law� Cfmin� ��Reh

�� forReynolds numbers between �� and �� It is very clear� however� that therecirculation is not laminar like� except for the very near wall region below themaximum of the reverse �ow

A possible explanation for the Reynolds number dependence� consistent with thefact that the turbulent Reynolds number remains high in the recirculation bubble�might be as follows The pressure eld is a consequence of the general form of theseparated layer �which causes �ow expansion and pressure rise� The form of theseparated layer is determined by the turbulence The Reynolds number in�uencenoted by Jovic and Driver �� might come from a wider �compared to h� shearlayer detaching from the step� causing a stronger adverse pressure gradient at lowervalues of Re This view is supported by our observation that RANS computationswere more sensitive to model changes in the shear layer than in the recirculating�ow The near�wall� viscous layer of reversed �ow results from a balance of thepressure force and the viscous friction� and covers only a few percent of the bubble

S� Parneix� D� Laurence � P� Durbin

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�b� �c� �d�

Figure �� a� Zoom of U in recirculation at locations x�h � � �� M�� C� and � �O�� model �bcd� Budget of U at locations �b� x�h � ��c� x�h � � �d� x�h � �� convection �� DNS� SMC� turbulent force �� DNS�

SMC�� viscous force � DNS� SMC�� pressure force �� DNS� SMC�

Elliptic relaxation for second moment closures �

height�so this cannot lead to laminar like� behavior The turbulent Reynoldsk�� ie� � ��t�� in the bubble is in the range �� Fig �� similarto its value in the shear layer Incidently� this makes any low Reynolds dampingfunction� ine�ective in the main portion of the bubbleThe momentum budget station at x�h � � shows that entrainment by the tur�

bulent shear stress is underestimated right below the shear layer detaching fromthe corner At x�h � � it was checked that uv and V dU�dy are by far the majorcontributors to the turbulent and convection terms The shear is weakly opposingthe recirculation �a pair of arrows in Figs � and � indicates the position of U � ��while advection is driving the recirculation in its upper part Overall� the modelseems only slightly to underestimate this turbulent shear force in the full simula�tion� even if at this stage the defect is traced to insu�cient entrainment at the toppart of the recirculation However� we will see that uv itself is in errorThe V component in the shear layer shown in Fig � is severely underestimated

The turbulence force was found to be due almost entirely to v� Again� right nearthe corner� at x�h � �� the turbulent force is underestimated by about a factor of ��although not far upstream at x�h � �� the RANS results were in accordance with theDNS data At x�h � � and � �near reattachment�� it is still v� which is driving the�ow downward Advection �inertia or streamline curvature� e�ects are negligible�showing that a reattaching �ow is di�erent from an impinging �ow Fig � showsthat the streamlines become smoothly tangent to the wall� some RANS simulationshave produced a kink in the streamlines at this reattachment point �Hanjalic� ��

�� Di�erential a priori tests

�� Reynolds stresses

The Reynolds stresses will now be analyzed from two di�erent sets of computa�tions The rst corresponds to the full simulation and explains the mean velocitybudgets shown previously The second is from a di�erential a priori test and per�mits an analysis of the true e�ects of the pressure�strain and transport models Thelatter results are obtained by solving the full di�erential equations of each individ�ual stress uiuj one by one� while the other stresses and the mean �ow are takendirectly from the DNS databaseAn overall glance at Figs �� explains why a comparison using only the full

simulation may entail erroneous conclusions� the full simulation could lead one tobelieve that the model has problems in the recirculation bubble� whereas the apriori test shows that discrepancies are mainly located in the shear layerThe u� streamwise �uctuation in the a priori test is overestimated in the shear

layer� but improved in the recirculation when the correct mean velocity �used inthe a priori test� enters its production In both the full and a priori simulations�the v� component is seen to be underestimated in the shear layer� and elsewhereat station x�h � � On the other hand� the shear stress seems to be� on average�correct in the full simulation and overestimated in the a priori test This is becausein the former an erroneously small value of v� is entering its production term Theorigin of the problem lies in insu�cient return to isotropy in the SSG pressure�strain

� S� Parneix� D� Laurence � P� Durbin

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Figure �� a� Zoom of V in recirculation� legend cf Fig�a �bcd� Budget of Vat �b� x�h � � �c� x�h � � �d� x�h � �� legend cf Fig�bcd


y�h

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Figure �� u� pro les �a� full computation� �b� a priori test� at locations x�h � �� M�� C� and � �O�� model

model� which should increase v� and decrease both u� and uv� but only in the shearlayer

�� Budgets

For the analysis of budgets� the DNS data has been processed in the same formas the elliptic relaxation model �see appendix�� ie� some anisotropy e�ects in thedissipation are lumped with the so�called pressure�strain term

For the budgets of u� in Fig �� the production terms coincide perfectly of course�since the mean velocities and Reynolds stresses other than u�� are taken from theDNS That is the method of this di�erential� a priori test The model for tur�bulent transport �Daly�Harlow� performs well� but the pressure gradient�velocitycorrelation �kf�� is underestimated in the free shear layer

The budgets of v� �Fig �� show here again that in the shear layer the pressurecorrelation term is underestimated�at x�h � � by a factor of � The pressure termincludes pressure�transport e�ects �in the present case countergradient transporte�ects� which partially balance the turbulent di�usion terms� and which� as a con�sequence� are also underestimated by the model Note that the production term ismaking a signi cant contribution� and since it is mainly composed of �v�DV�dy�underestimations of both v� and V �which is a�ected by v�� self�amplify throughthis term Near the wall� turbulent di�usion is generating the wall normal �uctua�tions� while the wall blocking e�ect is impeding them� the latter is represented by


y�h

v��U��

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v��U��

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Figure �� v� pro les �a� full computation� �b� a priori test� legend cf Fig�

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Figure �� uv pro les �a� full computation� �b� a priori test� legend cf Fig�

Elliptic relaxation for second moment closures

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�a� �b� �c�

Figure �� A priori test� budget of u� at locations �a� x�h � �� b� x�h � ��c� x�h � �� production �� DNS� SMC�� Rij�k� �� DNS� SMC�� con�vection � DNS� SMC�� transport �M triple correlations DNS� SMC�� vis�cous di�usion �� DNS� SMC�� pressure�deformation � dissipation anisotropye�ects �O��vpg

ij � �ij � �Rij�k� DNS� �kfij � SMC�


y�h

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�a� �b� �c�

Figure � A priori test� budget of v� at locations �a� x�h � �� b� x�h � ��c� x�h � �� legend cf Fig�

the elliptic relaxation e�ect �the homogeneous solution to Eq � of the appendix isactually positive in this area�


y�h

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�a� �b� �c�

Figure �� A priori test� budget of uv at locations �a� x�h � �� b� x�h � ��c� x�h � �� legend cf Fig�

The budget of uv� on the other hand� shows an overestimation of the pressure�correlation� though again this compensates for an underestimation of the turbulent


transport At x�h � �� near the wall� the production term is seen to change sign� butstill uv remains approximately zero because of the strong transport term Hence�in the narrow region between the maximum of the back�ow and the wall� the mean�ow is largely viscous� as seen previously With increasing Reynolds number� onecan expect the ratio of production to transport terms to become larger� and theturbulent shear stress would then decelerate this back�ow� leading to a smallerpeak in Cf Because uv is countergradient with respect to the velocity gradient�

the production of u� and k becomes negative in this area

�� Model parameters

Fine tuning of models is often based on functions of the following parameters�anisotropy of the Reynolds stresses� A� turbulent Reynolds number�Ret� productionover dissipation� P�� sometimes the non�dimensional rate of strain� Sk�� is used��and turbulent lengthscale� k�� In seeking improvements here� one should lookfor parameters that exhibit di�erent values from those in simpler shear �ows� forwhich the model should not be changed The above parameters have been computedfrom DNS data to see if they are pertinent

The range of variation of A �Fig �� from �� to �� in the recirculation bubbleshows no particularity� the Ret values in the range of �� to �� Fig �� is toohigh to invoke low Reynolds e�ects� the ratio P�� Fig �� decreases from � to�� in the shear layer� but is seen to be particularly weak in the lower half ofthe recirculation bubble This last is a feature that is signi cantly di�erent fromnear wall regions of boundary layers and should be considered further Indeed�even negative production occurs along the wall from the reattachment to x�h � �The production of dissipation is usually modeled as proportional to that of k� andnegative values might lead here to unphysical e�ects

y�h

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Figure �� DNS pro les �a� Production over dissipation P�� legend cf Fig��a�b� Turbulent length scale L � ��k�� at locations x�h � � �� DNS�

SMC�� DNS� SMC�� DNS� SMC�� DNS� SMC�and �� DNS� SMC�� y


y�h

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A0.0 0.2 0.4 0.6 0.8

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Figure �� DNS pro les �a� Turbulent Reynolds number Ret � k�� atlocations x�h � � �� and �� b� Anisotropy A �� A� �A�� A� and A� are the second and third invariants of aij � legend cfFig��a

� Modeling dissipation

Several attempts were made to increase the pressure�strain in the shear layer�but all resulted in a �sometimes dramatic� shortening of the reattachment length�without amplifying the strength of the recirculation Though the previous analysisindicates that this is a route to pursue� the following only reports some success inimproving the dissipation equation

An a priori test of the k � � equations was carried out by solving the coupledk � � system with DNS data for uiuj and U Dissipation is� of course� the exactsource term for k� but k also has a strong relation to dissipation through the inversetimescale ��k in front of its source term� this is why the equations were solved as acoupled system

Figure �� seems to indicate that k is overestimated and dissipation is correct in thecoupled� di�erential test In the recirculation bubble the e�ect of the source term inthe � equation is destruction of �� since production is low The modi cation� citedin the gure caption that is detailed below was intended to increase dissipation� butactually it leaves � unchanged and decreases k� bringing it closer to the DNS Thisis because when the source term coe�cient in � equation is decreased� the balanceof the dissipation budget is re�established by a decrease of the time�scale� ie adecrease of k�� that occurs by k decreasing with little change of �


y�h

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Figure �� A priori test� �a� k� �b� �� at locations x�h � �� M�� C� and � �O�� model� � modi ed model �� equation�

Tuning of the dissipation equation has been a popular game for the past twodecades� so it needs to be shown that the present modi cation should not deterioratepredictions in other �ows� and what the rationale is behind it

Various procedures have been developed to enhance dissipation In near wall�ows below y� � �� an extra viscous production term is usually included in lowRe models However� it is ine�ective here because of the relatively high valueof Ret Another dissipation enhancement is the Yap correction� �Launder ��which consists in a positive source term in the dissipation equation that is activatedwhenever the turbulent lengthscale L is larger than the mixing length �y Thoughrather ad hoc� this Yap correction� has been particularly e�ective for backstepor sudden expansion �ows �Hanjalic �� and shows that something peculiar ishappening to the dissipation that is still not understood Indeed Fig �� shows thatL is overestimated in the recirculation bubble near the wall Although the Yapcorrection goes in the right direction� there is no justi cation for forcing L to besmaller than �y since the DNS data shows it to be considerably larger than this

Another way to increase the production of � in the near�wall region is to usethe non�dimensionalized parameter P�� Durbin �� This proved e�ective forchannel and boundary layer �ow� but showed unfortunately high levels of numericalinstability in more complex �ows Moreover� this does not su�ce in the back�owregion since P�� is fairly small Durbin and Laurence �� recently proposedto replace this unstable term by a ratio of k and v� in the k � � � v� model�


C�� a�

qk�v�� with a� � �� In this study� we have generalized this idea

in the full SMC by introducing the following� C�� a�pPk��jPSMC j� with

a� � �� Pk�� kTSijSji and PSMC � uiujSij are respectively the k � �formula for production and the exact Reynolds�stress production This correctionhas been found to have similar e�ects to P�� in the near�wall region of channel�ow� without any numerical instabilities in more complex situations It does notcure the underestimation of the back�ow in the present case� of course� that wasnot its intent

Very little data is available concerning the dissipation equation budget asidefrom the channel �ow DNS at CTR From that data� the following adjustment todestruction of dissipation �which we called modi ed model�� was devised� C�� f�I�� with I� � �P��D��D� This measures the weight of transport in thebudget of � by using the imbalance of production minus destruction The function fvaries from � �for shear �ows� to �� when production is zero In order to preservenumerical stability� we combined this modi cation with the same adjustment forC�� multiplication by f�I�� The following function f was chosen to avoid non�realistic coe�cients� f�x� � max�min�x� �� The result for the backstep asconcerns the Cf pro le was found to be modest �see Fig �� yet it is larger thanany results obtained through modi cations of the pressure strain model

The idea of re�adjusting C�� actually came from the analysis of channel �ow atRe� � �� for which the budgets of dissipation are available �Mansour � Kim�private communication� The near wall region has been analyzed in detail by Rodi� Mansour� but what interests us here is the central part of the channel It is wellknown to modelers that dissipation is underestimated in the core region� but withlittle consequence except that the modeled L is continuously increasing instead ofleveling o� just outside of the log�layer Since � decreases as y�� and the terms inits budget are as y�� all terms in this budget were multiplied by y� to produceFig ��

The dissipation budget� as discussed by Mansour� Kim � Moin �� or by Man�sour � Moin �� comprises a viscous transport term� negligible in the core region�a turbulent transport� very well modeled here by gradient di�usion with �� and ve source terms It is known on fundamental grounds that these ve termscannot be clearly grouped into production and destruction terms For the present�the terms involving gradients of velocity are grouped as P� � P� � P� and com�pared to the rapid� part of the model� ��kPk� while the remaining slow� termsare compared to ��k The DNS values of k and � are used in the model terms�hence the jagged appearance of the model transport� due to double di�erentiationof this DNS data It would seem from Fig �� that both constants C�� and C�� areseverely overestimated� but again� the present split is arguable It is� however� veryclear that near the center of the channel� where the rapid terms go to zero� a valueof C�� is too large by a factor of � On the other hand� the transport isaccurately modeled with the standard value ��


��budgetterms��y�

��

y�0 100 200 300 400

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Figure �� Budget of � for the Re� � �� channel �ow� all the terms havebeen multiplied by �y�� rapid part� �� P� � P� � P�� modelisation term��Pk��k�� slow part� �� P� � �� modelisation term� ��k�� trans�port � DNS� modelisation term�� viscous di�usion �� sum of the modeli�sation terms � �

� Conclusion

A detailed comparison of a SMC computation with the DNS data for the backstep�ow at Re � �� leads to the following conclusions�

�� The intensity of the back�ow and the friction coe�cient in the recirculation bub�ble are severely underestimated

�� The recirculation bubble is far from being pseudo�laminar� an understanding ofthe problems encountered by SMC should� thus� be of general interest

�� The SMC underestimates entrainment out of the recirculating bubble by thedetaching shear layer The mechanism is the following� pressure�strain �� gen�erates normal �uctuations v�� which create the transverse mean velocity V � thisin turn provides a momentum impulse to the bubble In the shear layer� �� v��V are underestimated


�� A new di�erential a priori procedure was used� in which the full transport equa�tions are solved one by one

�� A modi cation was proposed to enhance dissipation in� and only in� the recircu�lation bubble The new formulation cured �� of the back�ow discrepancy

�� The model is supported by a signi cant nding from the DNS data in the coreregion of a channel �ow� the constant related to destruction of dissipation� C��should be decreased by a factor of � as production vanishes

REFERENCES

Durbin� P� �� A Reynolds�stress model for near�wall turbulence J� Fluid Mech��

Durbin� P� A� � Laurence� D� �� Nonlocal e�ects in single point closureTurbulence Research Associates� meeting� Seoul Korea�

Hanjalic� K� �� Some resolved and unresolved issues in modeling non�equilibriumand unsteady turbulent �ows Engineering Turbulence Modeling and Measurements� W Rodi and G Bergeles Eds� Elsevier Pub� ��

Jovic� S� � Driver� D� �� Reynolds number e�ect on the skin friction inseparated �ows behind a backward�facing step Experiments in Fluids� ��

Kanniche� M�� Laurence� D�� Wizman� V� �� Combining a second mo�ment closure with elliptic relaxation in rotating turbulent �ows ��th� Symp�on Turbulent Shear Flows� Penn State Pennsylvania� ��

Kasagi� N� � Matsunaga� A� �� Three�dimensional particle�tracking ve�locimetry measurements of turbulence statistics and energy budget in a backward�facing step �ow Int� J� Heat and Fluid Flow� � � ��

Ko� S� � Durbin� P� �� Separated turbulent �ows computed with a near�wallReynolds stress model ASMEFED Summer Meeting� ��

Launder� B� E� �� Second�moment closure� present and future!Int� J� Heatand Fluid Flow� �� N � � ��

Laurence� D�� Durbin� P� A� � Demuren� A� O� �� Modeling near wall ef�fects in second moment closures by elliptic relaxation ��th� Symp� on TurbulentShear Flows� Penn State Pennsylvania� ��

Le� H�� Moin� P� � Kim� J� �� Direct numerical simulation of turbulent �owover a backward�facing step Proc� th Symp� on Turbulent Shear Flows� Kyoto�Japan� ��

Mansour� N� N�� Kim� J� � Moin� P� �� Reynolds stress and dissipation�ratebudgets in a turbulent channel �ow J� Fluid Mech� ��

Rodi W� � Mansour� N� N� �� Low Reynolds number k � � modeling withthe aid of direct simulation data J� Fluid Mech� ��

Speziale� C� G�� Sarkar� S� � Gatski� T� B� �� Modeling the pressure�strain correlation of turbulence� an invariant dynamical systems approachJ� Fluid Mech� ��

Elliptic relaxation for second moment closures

APPENDIX � Elliptic relaxation

The Reynolds stress transport equation is written as�

Dtuiuj � Pij � �ij � uiuj�

k� Tij � �r�uiuj ��

withPij � �uiukkUj � ujukkUi

�ij � �uijp� ujip� ��ij � uiuj�

k� �

�

�ukkpij

Tij � �k�ukuiuj ��

�ukpij �

��

The term �ij di�ers from the usual pressure�strain �ij since it includes a deviatoricdissipation tensor in the form

�ij � �ij ��ij � uiuj

�

k

��

The following neutral formulation for the elliptic relaxation is now obtained �Durbinand Laurence ��

�ijk

� Lr�L�ijk�

�hijk

��

For homogeneous turbulence �ij �� kfij� in Eq � reduces to �hij� for which any

standard redistribution model �hij can be used The SSG rapid model is

�hijrapid � �C�dev �Pij �� C�dev �Dij �� CskSij ��

The coe�cients are�

C� �g� � g��

� C� �g� � g��

�

Cs ��

�g� � g� �

g��

pA�

The slow term is of the form

�hijslow � � ��C� � ��aij � C �

�dev�aikakj��k

T��

C� � � ��

�

�g� � g��

P

�

�� C �

� � �g��

The dissipation equation is

Dt� �C �

��P � C��

T� k

��

C�ukulT

��

�l�

��

The time scale� T � is de ned as�

T �

rk�

��

�

��


The length scale L appearing in Eq � also is prevented from going to zero at thewall by using the Kolmogorov scale as a lower bound�

L � CL

rk�

��C�

�

��

��

Lastly� the Daly�Harlow expression for the turbulent di�usion is used�

Tij � l�C� ulum Tmuiuj

��

The constants used in this report are�

C� � �� CL � �� C� � ��

C�� C��

Alsoaij � dev�uiuj��k� A� � aijaij �

A� � aijajkaki� A � �� A� �A��

was used in the text

el liptic r - stanford university

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