AD-A282 340 VIIIIIIhIII

A NEW APPROACH TO VALIDATE SUBGRJID MODELSIN COMPLEX HIGH REYNOLDS NUMBER FLOWS

Semi-Annual Report for the Period

October, 1993 - May, 1994

Grant Number: N00014-93-1-0342DTIC

ELECTE

JUL 1.5 1994D

Suresh MenonSchool of Aerospace EngineeringGeorgia Institute of Technology

Atlanta, Georgia 30332-0150

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Office of Naval Research Availability CodeSStaemet eleonAvail andlIor

Statement Aper telecon Lawrence Purtell Dist SpecialOffice of the Chief of Naval Research800 north Quincy StreetArlington, VA 22217-5000 .1NWW 7/8/94

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A NEW APPROACH TO VALIDATE SUBGRID MODELS IN COMPLEX HIGHREYNOLDS NUMBER FLOWS

Semi-Annual Report for the Period: October, 1993 - May, 1994Grant Number: N00014-93-1.0342

Sure& MeonSchool of AerospaceEniergGeorgia Insltute of TechnologyAtanta, Georgia 30332.0150

1. INTRODUCTION

This second semi-annual report summarizes the progress made in the last six months. The overallobjectives of this research are to develop new methods to evaluate subgrid models and then to uti-lize these methods to improve the chosen subgnd models. The subgrid models investigated in thisresearch are chosen primarily for application in high Reynolds number complex flows. Prelimi-nary studies of these models have been completed. A priori analysis wng data from direct numer-ical simulations (DNS) homogeneous isotropic flows was carried out, and then the models wereimplemented in large-eddy simulation (LES) codes and further evaluated. Two types of analysismethods have been developed so far. The first method uses information in Fourier (spectral) spaceand evaluates the interscale energy transfer as a function of the wavenumbers resolved in the LES."The second, uses information in the physical space and uses cross correlation analysis to investi-gate the behavior of subgrid models. The physical space analysis method will be the primary anal-ysis tool for the next year's study, since the next phase of research will focus on complex flowssuch as flows past rearward facing steps and swirling flows.

In the following, we discuss recent developments of the last six months. Some of the results havebeen described in more detail in the papers (Menon and Yeung, 1994a, 1994b; Kim et al., 1994)and, therefore, will not be repeated here. These papers (or the extended abstracts) are included asappendices to this report.

2. ANALYSIS OF SUBGRID MODELS USING DNS AND LES OF ISOTROPIC TURBU-LENCE

In this phase of research, homogeneous isotropic turbulence was used as a test flow field todevelop analysis methods. This was motivated by the fact that isotropic turbulence has been stud-ied extensively in the past and detailed DNS data is available. In addition, the computationaldomain is very simple and therefore, methods using spectral techniques can be used to identifyfeares of the flow which are not possible using physical space methods and vice versa. Bothinc mnpressible and compressible isotropic fields have been studied using two different finite dif-ference codes. These codes have been validated by comparing the DNS predictions by these codeswith the results obtained using the well know spectral code of Rogallo.

2.1 Analysis of Incompressible Isotropic Turbulence

Various subgrid models were evaluated in both spectral and physical space using high resolutionDNS dama Subsequently, these models were implemented in coarse grid LES. The spectral spacemethod was used primarily for a priori analysis while the physical space method was used forboth a priori and posteriori analysis. For a priori analysis, very high resolution data (using 1283grid) was used. All the posteriori analysis were carried out using LES data and without using anyDNS information. This approach is essential for future application to high Reynolds number flowssince DNS data for such flows will not be available.

For the physical space analysis of the subgrid models, LES using different grid resolutions is firstcarried out. Then, using top hat filtering, the modeled subgrid stresses and the energy transfer arecorrelated between the two LBS data fields. For example, LS was carried out using 323 gridand 163 grid using identical initial flow field and using the same subgrid model. Then, at a choseninstant, the resolved field in the 323 grid is filtered to compute the subgrid stresses and energytransfer in the 163 grid. This field is considered 'exact', as far as the coarser grid is concerned,since the length scales that are unresolved (and hence modeled) in the coarser grid are supposed tobe resolved in the finer grid LES. Therefore, if the model is working properly this 'exact' fieldmust be reproduced by the subgrid model in the 16x16xl6 grid. Cross correlation analysisbetween the modeled and 'exact subgrid stresses and the energy transfer was carried out. If thecorrelation is high, it would suggest that the subgrid model behaves consistently for different gridresolutions and that the subgrid energy transfer is modeled correctly in the 16x16x16 LES. Withthis approach, model validation does not require DNS data and, more importantly, since the sameflow field is being investigated, the model can be investigated directly in the flow field and geom-etry of interest. Furthermore, this approach allows an immediate assessment of the capability ofthe subgrid models in the coarsest grid.

Note that, the results of the above analysis methods do not provide any information on the accu-racy of the results. Comparison with experimental data (or DNS data, where ever possible) isessential to demonstrate the accuracy of the LES. So far, for isotropic flows, DNS data have beenused to evaluate the accuracy of the LES results, but future studies will be directed to more com-plex flows for which no DNS data is available. Comparison with experimental data will have tobe carried out in such cases. It is expected that the analysis methods for model validation will alsohave to be further developed for complex flows, for example, to handle the near wall effects.

Various subgrid models have been implemented and evaluated using the techniques describedabove. More details of the analysis methods are given in Menon and Yeung (1994a, 1994b). Theaubgrid models studied so far are:

(a) the classical Smagorinsky's eddy viscosity model(b) the dynamic (Germano's) eddy viscosity model(c) a spectral eddy viscosity model(d) a new scale similarity model(e) a one-equation model for the subgrid kinetic energy with and without stochastic backacatter(f) a dynamic one-equation model for the subgrid kinetic energy(g) a two-equation model for the subgrid kinetic energy and subgrid helicity (k-h model)

Smagorinaky's model is very popular in literature; however, it has been shown to require signifi-cant modifications (primarily adjustment of the 'constant') for good agreement with experimentaldata. The major 'breakthrough' in subgrid model development is the application of the algebraicidentity of Germano to evaluate dynamically the constant in Smagorinsky's model. In Menon andYeung (1994a), we studied the classical model with fixed constant while in Menon and Yeung(1994b) and Kim et al. (1994), we studied the dynamic eddy viscosity model.

Uhe scale similarity model is a modified version of the original Bardina's model and was pro-posed by Meneveau (Liu et al., 1993) based on analysis of high Reynolds number experimentaldata on turbulent jets. It is based on the idea that the energy transfer at the resolved grid resolutionis self similar to the energy transfer occuring at a resolution twice as coarse. This method, there-fore, uses a test filter (an approach very similar to Germano's) to compute the scale-similar sub-grid stresses in terms of the resolved field. Computationally, this model is very simple and easy toimplement. However, as discussed in Menon and Yeung (1994a), there are some inherrant limita-tions to this model when used in LES. This model can predict backscaner but the amount of back-scatter may exceed the real backscatter. This can result in numerical instability and, therefore,some sort of backscatter control is necessary. More details of the analysis of this model are givenbelow and in Menon and Yeung, (1994a).

The one equation model for the subgrid kinetic energy (k-equation model) was chosen keeping inmind the requirements for practical high Reynolds number LES. It is expected that for high Rey-nolds number LES of complex flows, the grid resolution practically possible (due to resource con-straints) will be limited. Therefore, simple dissipative models (even with dynamic evaluations)may not be sufficient for practical LES. In addition, the assumption of local equilibrium betweenthe production and dissipation of the kinetic energy (an assumption implicit in all algebraic eddyviscosity models) may be violated. The k-equation model with fixed coefficients was investigatedin Menon and Yeung (1994a), while the dynamic k-equation model is investigated in Menon andYoung (1994b) and Kim et al. (1994).

We are also investigating more advanced models for high Reynolds number flows. One suchmodel is a two-equation model for the subgrid kinetic energy and subgrid helicity (the k-h model).Helicity is non-zero only if the flow is locally 3D. Thus, if the small scales are anisotropic or non-homogeneous (which can occur if the grid is coarse, the geometry is complex, and the Reynoldsnumber is very high) then simple eddy viscosity models or even the one-equation models, maynot be able to take into account this small-scale, local 3D effects. Some preliminary studies of thismodel have been completed and results are discussed below.

2.1.1 Summary of the Results

Mbe results of the analysis of these subgrid models have been reported in the papers attached inthe Appendices. Here, we briefly summarize those results and then discuss some new resultsrecently obtained.

The analysis of the eddy viscosity models (models (a) and (c)), the scale similarity model (model(d)) and the one-equation model (model (e)) were reported in Menon and Yeung (1994a). The apriori anlysis showed that for fine grid resolution, the scale similarity model had the highest cor-

IJ

relation with the exact subgrid stresses and energy transfer. However, for a coarse grid LES, thiscotTelation dropped significantly indicating that this model is not appropriate for coarse grid LES.This result was understandable since the scale similarity concept implies that the energy transerat two grid levels are self similar. As the grid is coarsened, this similarity begins to breakdown.For the low Reynolds number flows studied in Menon and Yeung (1994a), there was no clear iner-dal range resolved in the DNS. This made it difficult to fully evaluate this model. This model wasproposed for high Reynolds number flows (based on experimental data at Re=3 10) where a dis-dnct inerital range existed. Scale-similarity assumptions hold very well in the inertial range and,thus, this model would be applicable. However, this implies that to use this model an inertialrange must be resolved in the LES. This may be an unacceptable requirement in high Reynoldsnumber flows (since, it implies a very high grid resolution) and, thus, at present, this model(although very elegant) appears to have limited use for LES of complex, high Reynolds numberflows.

Smagorinsky's model (with fixed constant) was quite poor when compared to the other models.The k-equation model (with fixed coefficients) was better than the eddy viscosity model but had acorrelation lower than the scale similarity model. Interestingly, when the guid was coarsened, thescale similarity model became poorly correlated; however, the k-equation model did not showsuch a behavior. This suggests that the k-equation model has the potential for modeling the sub-grid stresses and energy transfer in coarse grids.

Subsequent to this a priori analysis, these models were implemented in a LES code and simula-dions were carried out using different grd resolutions. To compare the results, all simulationswere begun with nearly identical initial conditions. Thus, at t=O, all the flow fields were highlycorrelated. The analysis was carried out at an instant when the flow had evolved to realistic turbu-lence. The correlation analysis using just the LES data showed a completely different picture. Allmodels showed very poor correlation (compared to a priori tests). Both the eddy viscosity modeland the scale similarity model appeared to model the subgrid stresses quite poorly compared tothe one-equation model.

These results clearly highlight the fact that subgrid models that appear to be quite good in a priorianalysis may not be as good when implemented in actual LES. Furthermore, it appeared that thek-equation model had the best capability to model the subgrid stresses in coarse grid LES.

iTh study in Menon and Yeung (1994a), employed the models with fixed coefficients. Since it hadbeen shown by other researchers that the dynamic eddy viscosity model is quite superior to Sma-gorinsky's classical model, it was decided to revisit these models but by allowing for dynamicevalutation of the coefficients. The dynamic eddy viscosity model and a dynamic version of the k-equation model was then analyzed using the methods developed above. Some of these results willbe reported in details in Menon and Yeung (1994b) and Kim et al (1994). It has been shown thatthe dynamic procedure significanly improves the correlations, An interesting observation wasthat, compared to the dynamic Germano's model, the dynamic k-equation model showed a muchbetter improvement and was clearly superior for coarse grid LES. This has given confidence thatfor coarse grid LES for high Reynolds number flows, the use of such higher order models may bebeneficial. This is an issue that will be revisited using more complex flows in the second year ofthis research.

We are now investigating a new two-equation model. The governing equations are shown in theAppendix and the subgrid model is essentially the model proposed by Yoshizawa (1993). Weexpect that in the course of our study, this model (if useful) will undergo some changes. Forexample, so far, results have been obtained using fixed coefficients. We expect the dynamic pro-cedure will improve this model and, therefore, we are now in the process of including a dynamicprocedure to solve this model.

We simulated a simple periodic flow field for the Taylor-Green Vortex. This field is primarily 2Dand will not contain any helicity. This was confirmed by carrying out LES using the k-h modeland showing that the subguid helicity was negligible. Next, the spanwise velocity field waschanged by adding a cosx term. This allows the large-scales to become 3D while still satisfyingcontinuity. We were interested in determining if this flow field would generate small-scale 3DStructures and, if so, would the new k-h model be able to predict the helicity in the unresolvedscales.

We looked at three quantities: (I) the correlation between subgrid helicity and large-scale vortic-ity, (2) the accuracy with which subgrid helicity is being modeled, and (3) the relevance of helic-ity to the subgrid stresses (that is, the effect of non-eddy viscosity terms in the stress model thatappears explicitly due to helicity).

In Fig. la and I b, we show 3D visualization of the vortex tubes (constant vorticity isosurfaces)along with contours of subgrid helicity as predicted by the model. The subgrid helicity is onlyshown on a certain plane. Figure lI shows the field as seen in a 32x32x32 LBS, while in Fig. lb,the results are shown for aIt 16x16x16 LES. It can be seen that regions of intense vorticity are sur-rounded by regions of high-magnitude subgrid helicity. This indicates significant production ofmUl-scale helicity by the breakdown of the large scale Structures. This result also implies that theunresolved scales may be locally anisotropic.

Correlation between the large-scale helicity (due to the resolved fields) and the subgrid helicitywas also computed. This correlation should be very low, in fact, it should ideally be zero, sincethe model is supposed to compute the subgrid helicity only due to the anisotropy or non-homoge-neity in the small-scales and, therefore, should not correlate with the large-scale helicity. Thecomputed correlations were also very small with the 32x32x32 LES predicting a value of 1.03E-4and the 16x16x16 LES predicting a value of 5.6E-4. These results showed that the k-h model hasbeen implemented correctly and appears to be predicting the correct physics.

Coreation between the subgrid helicity modeled in the 16x16x16 LES and the subgrid helicityprdicted by filtering the 32x32x32 LES data into a 16x16x16 grid was also carried out. If themodel behaves accurately in both grids then this correlation should be high. Our preliminary datashowed a correlation of 0.736 which is reasonably high. Figure 2a shows the subgrid helicitycomputed using the 32x32x32 LES data filtered to 16x16x16 grid, and Fig. 2b shows the modelprediaion in the 16x16x16 LES. Clearly, the model is behaving reasonable well in both grids.

This study is not yet completed and there are still some unresolved issues. For example, the inclu-sion of helicity model did not improve the subgrid energy transfer correlation. However, our pre-vious study using fixed coefficients for the k-equation model also showed poor correlation(Menon

and Yeung, 1994a), while with dynamic evaluation, the correlation improved significantly (Kim etal., 1994). Therefore, we are now beginning to evaluate dynamically the constants in this k-hmodel

Note that, adding one more equation will increase the computational cost. Therefore, before suchmodels are proposed for LES application, it must be clearly demonstrated that it is superior to theconventional eddy viscosity model. The tests using Taylor-Green vortex or isotropic turbulencemay not be appropriate to evaluate this model. Therefore, we are now starting to implement thismodel into the code developed to simulate more complex flows such as flows past rearward facingsteps. If this model is superior for such flows, then the additional cost of computation may be bal-anced by the ability of the new model to handle complex flows using relatively coarser grids(thereby, decreasing computational cost). This is the primary goal of this research.

2.2 Analysis of Compressible Isotropic Thrbulence

Some studies were also carried out to extend the analysis methods to study compressible flows.As noted before, the analysis methods are supposed to be independant of the type of flow fieldstudied and, therefore, with some minor modifications should be applicable in compressibleflows. To compare with the incompressible flow results, we began by simulating low Mach num-ber (essentially incompressible) isotropic turbulent flow fields. So far, only the compressible ver-sions of the eddy viscosity model (Edebacher et al., 1987), the dynamic eddy viscosity model,and the scale similarity model have been implemented and evaluated. Here, we will summarizesome of the more recent results of this study. More details of this work will be included in thefinal version of the paper Menon and Yeung (1994b).

Figures 3a and 3b show, respectively, the correlation of the exact subgrid stress - (obtainedfrom 643 DNS data) with the eddy viscosity model and scale-similarity model predictions as afunction of filter width. Again, as before, box filters have been employed. The results for the ear-lier incompressible data are also shown. These figures show the characteristic decrease in correla-tion when the grid is coarsened with the scale similarity model showing most rapid decrease. Thecompressible data is quite close to the incompressible data since very low Mach number flow hasbeen simlated. However, note that, two completely different numerical solvers and subgrid mod-els were employed for this comparison.

Figures 4a and 4b show, respectively, the correlation between exact energy transfertjij and themodeled energy transfer for the two models. Again, both models show that with decrease in gridresolution, the correlation decreases. The scale similarity model again shows a strong dependanceon the grid resolution. However, for relatively fine mesh, the scale similarity model is quite supe-rior to the eddy viscosity model. This is in agreement with the incompressible flow results dis-cussed in Menon and Yeung (1994a).

Since the incompressible study showed that the dynamic subgrid models are superior to the mod-els with fixed coefficients, we are now in the process of evaluating dynamic subgrid models forompreible flows. Preliminary results show good agreement with the results of Moin et al.

(1993). More results of this study will be be reported in the near future.

3. LES OF FLOWS PAST REARWARD FACING STEPS

We are now getting ready to simulate more complex flows such as flows past rearward facingsteps. We have completed preliminary validation studies using the simple eddy viscosity model(with no dynamic evaluation) and have demonstrated the ability of our numerical solver to repro-fce results consistent with earlier studies. For example, Fig.5 shows the variation of reattache-

meat distance (normalized by step height) as a function of Reynolds number. Also shown areresults obtained by other researchers. Clearly, our LES solver is in good agreement with earlierstaudies. Figures 6a and 6b show, respectively, the vortical structures downstream of the step forthe two Reynolds numbers. As Reynolds number increases, more complex flow patterns areformed as expected.

The above results were obtained using the classical eddy viscosity model. These earlier calcula-tions were carried out to determine the accuracy of the code and to resolve all the programmingissues. Therefore, detailed analysis of the data have not been carried ouL Since the analysis ofsubgrid models in isotropic turbulence clearly suggest that the dynamic models are superior, weare now in the process of including the dynamic models into this code. It is expected that allfuture calculations in this configuration will be carried out using dynamic models (such as thedynamic k-equation and dynamic k-h models) and for relatively high Reynolds numbers. Theexact test conditions have not yet been finalized since we want to first make sure that there is suf-ficient experimental data for model validation in such flows.

4. PLANS FOR THE NEXT YEAR

The research carried out in the first year focussed on simple flows such as isotropic turbulence.The methods developed for analysis of subgrid models will now be used for more complex flows.Therefore, from now on, all studies will focus of complex, high Reynolds number flows. Two testflows have been chosen for subgrid model validation studies. The first configuration is the rear-ward facins step described above. The second configuration is a co-axial jet shear layer with andwithout swirl.. This type of highly 3D swirling flow occurs in many flow situations and has someinteresting features associated with swirl induced mixing. This flow is also sufficiently complexand there is some experimental data for comparison. Both these configurations will be studiedusing the same numerical solver and subgrid models. We are interested in determining if the sametype of subgrid model is capable of simulating accurately these two different types of flows.

Rdbmuew

Gennano, M., Piomelli, U., Moin, P., and Cabot, W. H. (1991) "A Dynamic Subgrid-Scale EddyViscosity Model", Physics of Fluids A. Vol. 3.

Kim, W.-W., Menon, S., and Chakravarty, V. K. (1994) "A New Dynamic One-Equation SubgridModel for Large-Eddy Simulations", submitted for presentation at the 33rd AIAA Aerospace Sci-ences Meeting, January 9-12, Reno, NV.

Menon, S., and Yeung, P-K. (1994a) "Analysis of Subgrid Models using Direct and Large-EddySimulations of Isotropic Turbulence" Paper No. 10, 74th AGARD Fluid Dynamics Panel Sympo-sium on "Application of Direct and Large Eddy Simulation to Trnsition and Turbulence".

lenon, S., and Yeung, P-K. (1994b) "Effect of Subgrid Models on the Computed InterscaleEnergy Transfer in Compressible and Incompressible Isotropic Turbulence" ALAA Paper No. 94-2387, to be presented at the AIAA 25th Fluid Dynamics Conference, June 20-23, 1994, ColoradoSprings, CO.

Moin, P., Squires, K., Cabot, W., and Lee, S. (1991) "A Dynamic Subgrid-Scale Model for Com-pressible Turbulence and Scalar Transport", Physics of Fluids, Vol. 3., pp. 2746-2752.

Yoshizawa, A. (1993) "Bridging between eddy-viscosity type and second-order models using atwo-scale DIA" presented at the 9th Symposium on Turbulent Shear Flows, Kyoto, Japan, August16-18, 1993.

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APPENDIX 1I

ANALYSIS OF SUBGR1D MODELS USING

DIRECT AND LARGE EDDY SIMULATIONS

OF ISOTROPIC TURBULENCE

S. Menon and P.-K. Yeung

School of Aerospace Engineering

Georgia Institute of Technology

Atlanta, Georgia 30332-0150

Paper No. 10

74th AGARD Fluid Dynamics Panel Symposium

an

"Application of Direct and Large Eddy Siuon

to Tramtion and Turbulence"

April 18-21 1994

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09" aub hima Shhe) emse ad Sem mww abd Pl5m A sU man somp Sm umawV Ama toa ft ---onl SoIs ft maft" Wk 7Ws ma P" *f * ma moesis efe 0 mm v at ai. Nor Pift Ae osw onamon" on Ame. ma on &a ma meiii. weu asm amb- nu mimmmamftt"p M"amma "ilmd b

~v ma.ma ~pa ma.. b *pb~ mlyn ala "i elmsu it 5A.rIs do liem whisk £ as umaownONdm mma~t do avyo X~sWOlib) sambnemsp.I 0maebpedmaism~blebuwgasaft

a do sftmmils. m 0V db otdd aw MupLm mim Ae4 sum . 12. 5)mm dm maDCW O dohmd-- 1, 1 lbdu Ukmobis ~vim. As £iqW whambe vt ma oarpst miles smauh-

amma is ma g1imal by ma #W on mimiulb W. ma ma ftk mism.OWNs V4 ON ad doad isy I m IImii weibi

Wima mod gu a%= ,. bN ad ~ go bobmm sue 81010160mgm --MA -l tma aw*@p9Vmac(A04 6 Tim neam. bueve hiB6 sma -w bar %wa. dbd be of ma Older of so iipwhns on a mempoleftSol IS 116by (ha sm) di of 3D lIi hIs slw Im SNO ma h as maime" memi hum lb semiw

00 don) bib..t ha am&s pohoi uds. bus Nipli Mid le M uld 135.S an is ihipi.mk va i as mmpda oau ARKi um.

am s - (Wm 490b wasts -Sm mu i *"I. ht lb Iu ha wemu by ammyl so is ineadow-d ý- ima -ru so so obv by Lis a d. unly-o. Tooly adyig L3O maift. Ow OW3 dow Aaba143 710 uwhG at M q Sob bphAb . a is doobsL 4@G U' modolui ad do L3S haf edushmi doPwaminple k wodwm So) do ism Sm at sau 52' p14 um i~ Ns 14' pM. i S. to NW36m asf mupeumbi by sumbdmes hownS do %.4 pmibmi 6m 6m We dudms atS6 161 pU mmhim.

spbohr umbbusoI!s i4 ftua Onsy is am. op do S.mad psfUoa be do ma 140=.N a~ at ams he*a pm u13. lm msof amuid has on d08be pwimmiam ~ ~ ~ i AL. so mm1wo w my Pe mwm (b am in) m~ as 16, pm i L

% un a ý to. a6m. lb Mo fm pmin S. e IU b sopl bwanE Go h me LS. p1 d d

vmý A ad 2AAf ft~ ism mami =puma S 6m vyumumu is ' S. hrwakdids f mai svory-W 00d tos Onsh 8 ibs ha AmP. A Nok be. (send 0&12) uldleSo &qp uI peulma # ubvdy6wwvm a..mmr we ýi is wmqumm hrlo 3mwbw idoson M 1 so* losi ha wbeTAM. lbWsuds (spbe. as dons far Nowy) busing a m p55 VW mall,dI be LU do kw maid "mu oe-,wweb hiulw~s(-m b s 10,mils(2&) ur iWM m mo bowu. At h6 p~ssom boom a 6 isubi-w -mlv SO mswum do bo* IM ýy Man w.. udhiuy 33dm mo be is WIde ha U mub am.

buwud amed.t A mi. high umods ad bobsww i s V S* poWuS dnonisseA "a fe =nw VWso mftbby Lb a O. ioftka pewfindyub16 fs mumi dsiluisywium my be 5sbm bun dm I i-~L"f -~ -)s a IDuy sWid 9 mudb aft LU at ma

bigh ha% siba omsemb inbemIs st 6e buu imp .Th ses nd , e& 6k 5. 15 usimus by Lksa aL (9 iM& hu I" bim so"" ohmist qsmimif. pubY aighug ha S. =MWm mWOM d1 dhpulmý sm be inindam nd mummiusiw mnusum.pedissi by do amid. ibm,shar -C 2,us aimib MUA6 oqis. A v&4ate u1md mad d U d5 1 sMilS mm sud so ubeg 6s 32'0A15 us uhammmi isis ighRqymdb mmbir sm. md 14' P14 mibeh la ids U tf aI W~u. IM6 be a vbum! hau P93. S. d mubium bads ohma be Fis. NO Reo h aw55 a. d be pebbl beysm S. opihf

wamI a IL, c~*M @u mopisysi. Hoswmm in do.rhl asemn e sWy hd. suipuss SmOy a uipusk usiiam bs Go unuudalo *I a ot d ft im ISIy be qwMhWI7y ambuu by smrftabsod a a t~wisi hat S.6 doow TAM5 mi saft. 5 40 I. nk sks. .1 S.f beheebs OfS. sulOWd uL.ibWM sks hA aSm " rn inusioaf ofg 860 'h USWW amdt ha be hig B 3.mob w ftw& S. pmi

semw aui.. P~g..9d isusi eiw u oca, s rr, inedufs dot ke usamumd p.mof! be bel a be wowe of o1w vim ad It%. Us maimt swmI ha 6bs s =IW is hud lme.uaishshamissiblmuuusbL-lbsOwbmifadinmm

beesapis af. momm As=A bpl'&Jf I - mba. fm to am awiuiumy ww" or ai W upS. hin W ea di. lb W vd ula~ dbym Lki yatks p)k M Widd taO f I au 13.7 Itd dli asd imb b*s bi

be a Uas of is go"u ump of mud "=houd ho is adi ainia for ham isuiuimi is"@ UUis (wibnuss DoEs adi a S. pabIm wib ft dmfws mo"e Amu) is .) wo upiloommod by a iSum~mby' a*

obimsd 8""v) Whim do pr be"NW I&mg lb ha hoay modaL be siha. mwftg okhMa amid S od va-096 -1mhad do ob f So aimed inui popsedV Irnie. lb spaw

dou ha 60S isd mailed diimis amid mud ow AN,

M si Obe. ha hab mud hr nprbnpk si LU pose. one ID beelbvdg of at wmim is - kbom adk kedons (au dosumd be ft. 4ý amubm at is Now dir 6d Go sowsd at5 0 S 5615m atat-wof "ud as lowad. aswoh IrubunII ,-A@. doawed 37Umxys somktad osh u3a1Wh Ildspaw my daft mbba *A mi ad*gMO R% (00aiem. be U ~id is Wi hgh emUI5 s wmid...6W M ast Mwsa lbl admA ky amid sumowlydsw sash hbar madum is haS kesum ad as lb Ono ad imUp In sodsl basn to 52' p~i

33 g ha n nim do toi &W iso mi wW is ad do to, pM LUS ur ursl aso hr obum. Pig..buguhas maid a ftud ppe wy low .og'a. 118. ad ib ds. mipseiwdy ob ow. . adamp haleSmom, udo lis i.sumd siO 6um be is o515hU (dsbeii bei 614' pOW U. ai toof do bel.

gmw"Ift U Re&. AsuudbW IuI, bmS aruod -u. be -ist

SO. wplim on" woSmbamplememolb LBulS2 sft 322.b 0. smbi s ii adat lot pON nkmsk. hr3 is& doam 5.14 us Med. "UN 60006 a maim Iul. sIb S. dpubM

46' OWL NOW. at I*d Oyde Ww I". wmu Maid (imw 15) hdw a wo*l bw osh~ 6&S eMell mmmid (disugh be is huhe qm ha us h vbS. ~ibmkw maidf(miuugveMwya ~ uimi)

Wboil ha I -a ui.M ftmd. isf Dd3 .4 LU au is -ine bIn is P mai *. amua* hr al

highl an. ibm4 do DmI G d ad by m i iný* ao ha w 614 muds... f tb ~au ha dihowbeg s Ss U h m be is DM .4 L 5b7 ib~ st is I aids ubb p v m .ft somumin M N ap a bees..m

pha mis haw SIM O bse ha oniu a d yb L s A" p.M6O &V mb- Iae isW hl m bet hight Iui~ arS

Ift iSo pegs ld4m ft evduft ft i~sw atl i S

IS C4~W0 ZmW CJL. Smbsec u Sehsw Im a kbdl~idsaOb" wl lip a*y .ibdm of hee wd *cf . Uuiui: ftu 11W Miuk LeW". ftyS. Pidd A. 2,i.awk awdou ho bum pI ImI ubig a peamlupoml 297-29. 1990.ad -t" bumhws "ub abpll mob *A bends a 7. Pbm*IL~ V., MO~ Reynods Wokber Cimkuims aft

IU4~~S ubid used. GummodumbI mldci tobW Dpwakl Sobguld.eSOO lNs Mader. Phys. FkAds A.S,mebumis binimm &mkg an mod.a am mos iflaly H 1414.19 1993.GON m bes L-das NJ Is bv ft %wis sposs ad pby."ii aft 60 Msivolmlm MdoJ 1mg we" upJ . DommunkhL A.. Us, W.. ad Debet*@ M. IL OAm

I ~d Ibmldt som wg - I~ amos do owl- Aid5 Ot kmMi Sodsi hinmlmudm b Nmiuegy Sim-.mmub. IA mimmd by In II iwa Cmvko bla I~b1om.m" Iby. Paids A. 5. pp. 1747.-0* 4f 66 mud"e ml Sm smmomi - mod dosm

OW m Iin Obml "p on u my Ams 9* Lk I. uaw. c.. ad Kam L. vom fk Ppmwu of

in - in aebTsbuhwW sI. uli.mb.1w dintpub~lndom.Omf ousmm1m 1.6mm keSU vd #adk mdw nodded

"mr Tmi I I eig so fo bw f2. somis dw so 11 011 lesdi. L L no od Bquwkb he hNogmsomm*@ usad smV moSmf podbbi by do boob do mpdW ThWbSS NASA 7M 11315.1961.miub mmli. wemy VweE1 wiab te O d w wh kw mae It. bwam, V. ad P"p, U.. -Am ftnednof P1 cbig

wu~ 0b1 on" a 'mvy WO somins m rwwomos 6~e 06 im in detdoume of ftb"Wi, Compol.-p mdeis. Hosmeva wft scummeus in V meoem, bo so Phd, K p. 217-ZM Ml.wmd aliliky mold hsum oweai mum-ls uu-idt to H6 sums modd 1. ItYmig P. L. ml J.nw .0., nUh Ruposs of ho.-

mlek Thwumesw ID bupic ad AmbobashupcFmig a uVft . tu mbgW moleb urns seed far IS, coJjggo wo Laps Imi., ftl. Phib A. A pp.3=44K6 191.doa O ý smal vus my love. 'Eb Wes in $PI of kepepsn 13. IfWM R.M, OVudesy. lmdur mad buufe Spins in68 MbW low GAS h 80 Onb Ambdome ighl monabld. Wumminui Taodenw 3. Pht Mochk, 211, pp. 3W9332.

Mud 0 Fe" d. blibavio of S.nbr mopdd hsie actedoUN3. 'lbs amlyis of5 So US damn cabimed as very on 14. PWO&SII U. main, IP.. ad Fuziger mlL ib cow-pUds sbowad Sdo do ml. dmmiuiy modul k 16 - vory downcy b Logs Idly Mosioaulomf Thusiw Camuigoody whos -ip a ODd w=&)* wvj& usosimly Plowsm. Phbl. 1Mb. A1.W .114.91.19IM.aswe smaboh w ouehdes. Usemins - uagast *at IS. Mann 5. -A Nusuisa Stud of Suemlusy Fullajec-be.l usA Soiuty model a be used -ml for inkdwsy Ins doo CumW of Cambutli amubft in a Swnjor. AlAMp1U mumbe.whm .,tebr~ ug ode eue Pkper No. 92477.19M2.w"eb an uU Mab Was furnb.dwmma by GY-he as U t a3 Ms y blab aft flo wU-ch the sos lef ituiis yo h A. "suftin betums.dyvmukeinbwI bed moody =au uuelwm whltub *8 &P mod emm aid Seiinmd-sinde Mdals ofug a Two-lush VIA. Huhlbed Whkb) dds vuWas. Symposium sm "'I~boime SehFow sbW Itym. lapin.

Ac-m-0011 31-n RM 17. lnmms. U., "kipi Sads Modld hr Pbi. Diffe-am Xulmin. Of Twbelsot Fwsin Pku tine Cmwlsa

k wok in maisnd by do 0(1. of Naval Resuchb mider Amor. ). Compm t.hys. A pp. 376404, 1975.Ono No. H00014-93-44342. Coaupan Orne wu po*Wid

by S. Husuba Muodywmb Sinnmlesd (WAS) at NASA I$. NHINiL L., ad YOMhhW16 A., "Lug Sddy libabduomAmo ieummh Curad is IA pbiby sebtawldgul Swp of Tubeolme Chmmd Paow by Iqumlis. bodl-b he Fm&aport pueldd by Palau. smlomb W. soi we K Chlbs- Appo~bdoss be Phil Mosthud.5 ad. LIL Hikueb, pp.Inyl asioldd 119-134,1965.

1t. PImelA U. -W"Aums of Lap Widly Slodadm ft3.~mlg:AN OmviW. to -Lopaftdl Sbhun, of

1.14 mu l. J. a.33..m Mukopib, U. "Tesoof C onm ~noSaf ad osphyii Plows. D. 06*A~aamo bbgi.ukal Molub is Itsabo Uhndm AIAA ad 3. Can&ag Wd. Cambaide Udvudy Paw I. p99.99ft H Wo3953.1Inc. 319-137.

8. Man 1P. ml Eli. J., nbmWisd hvuumiftaf of Tw. 30. 3in5, J.. Fuea, llo. mIn sayeihs, W.C.,6" ome Fd P .. Lflail Mocb. hIf pp.341.77. ¶EPWI lfthil Sds Mdsk for Lop. Bdly Sisal.-

&_ mum uIA Pap. 904W, U"3960.^ &*

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Georgia eadmueo(TechnologAimif, Georgi 3M32-0150

(404)44-34

EXTENDED ABSTRACTMhe -ono saut-cf4bmhaw mIn ,Il Imodels bs numudy vam.of the siml edfy vscosity

model doi iwpoed by Smagwkuky In 1963. ThS s bin qua. of the mdow IMidoms of "i modelsdothhowe basn bioght Into stiveadan In nowt ausmdW Far exumle, bebcitmm of amesy km the umtovedto e reslved mssd has bern dised lo ocu In dirct smeulul dIobms (DNS) d (Pam %ulli atAL, 1991) dmly daimamsumatg doa dhe voesolvud sdbgdd , ,oeP. inomm be =&Wle by a pornl dbsal--av anmim. Ins adio., the 'emamatw (the Somiorlky cousins qeus w vary Wsm Bow to Blowa4*dws tewhpldmode requited fardher fise-tusig for a nudcfow. Raw st m"i to himprvs do y af iscosity model involve the alupsive valuation of the mbgdd comiinat (Umurn et AL. 1991)mola in qctmodeling of the ssabgl bacekcuttw Ioe- (aamov, 1991). Ins qulu of thee deveop-mosti, It ks m yet dew Kf suc simple models we sufficlent for hempeddy smulatdo O12) of hig Rey-mole nmuber Sow

The ojecslveof his pmprbstoevadme Oeueffectof *e lbamof hdo uosessupd*Mmodon shecomupued immesle amg transfer piuoem baemr the nesoled mod monuolved scale in LES. The even-ad! pdat dfths reeId Isb to davelop a su*W smdd m del h witj peffu adopawl In IMS of high Rey-moib nuber flows 7b lovesiatie the ffeta ofth 69 9 grl od m ondel ba.x Col 1.m ! mip i pro-amrn we begb the study by ayfng oas both DMS ad LES for Idem"o Saow conditionsmadthu inslyl-bg theewgy Iase poce . waft a method deeloped remedy byDaomdd at L (199). They hame

p inomsv d sht sihgldb arnie(SOS)- F Igysase qantiddes am be reafil imcied Dam DNIS dataIs INcqrubl Sow, thsgy quedna ELMk of doe resolv miss at wavanumber k evalves by

a ELk- -2vk 2EL(k)+T~kik,)+T(ktIk,), tk (1)

whore v b the kkaeuasic vlscoul, k, bsa cot-off waveoumbe qupwalog fth resolved anml V sac-rdmulsT(k 1k,) nusaus ierg wuander ham. Iaractioms wit meolved mule only. and 760k 1k) zepmesenimesactions wit 30S modsIft theo form of the quectu SOS eddy viscosty is cose as

* -2-

C(t , kC16 (2)

6% ft btsvl cim fd ath viscouity as a timala.s ithdo agaft wavig.mer csbe deatiloed, iKZ,hi halWo s0401 (1) S owu In R - -7owesm pobesam with ths q ýr who. IS is

L.hd bn LNM doe 10 WIN bi NPa (1) is modeled by 60. siiul modd. Thus the aovean&pnw u.mu do s mff w.~mb wil be afiecoed by trhefas (or choke) ofth Segdd model

ued o~a inet cthS faledu.6u.

ow eddyvbsualymoddels h u is. Ailed ulleby ~inov(1991).6 Oem ataL (1991). baii Warn a moospilwam modd fo Oae .bid unbule Mastic iner (lMo.. 199) aid a two-equaiomodi o So. metul d dim katcinde a mid mtgId hilcity (Yodiowa 199) w~l be evalmd.Now aer .6w pM -od uan id1 - ulmo6i for dhouhohg Nog Repualds amucr Am&e damvado- in -al phM sublalo do 0 ns doe ci.~. imlwved ca cornahi a sigmklm snwiof Ihokc Merv. As WNWa don. uwulved oaks be tPedr~lmlsutre Models thathiuhide Winspi hinilcy Oawn dft belely dalmd ale, is exastly wo for two,4uumlml meuctwes)may be ebb ao acot for te asyeuy k d mbe Ingd scales. ThMen models wil be sWile oft bothquoal wi It, ph spinSqe(mwvbn low solves. At v fte medhod of mmy*l I simiilar to#Wt 1 1 em1 by DomwW at al. (1993); however, we hope, to develop au equivalewm idahd of odmysisdo -a be med dimedy fa dke phy"sical aud vEwil me m re ui smualmf th. Saw lAdd hiwowas~ apoe. ThU Is partdumly kqpam damu hio Reynods nube lmow f pci ietu Irem

emI hi GIIPls dmhie (04g9- movwed S saps) " d,^ tuIt will Me be possbleI toevdalue theasog msbh af Fowls -mbm

As Shi pqin, we andder do iergy Iumtera yp lipe of deamyi hicompeumsbl mid compesibb 11 c-scic ---6dim c A the kiklolug we show ind comyue iuullmlwoy amolb obtained by four dif-bow uithadeos DS afmgtheo m 4dek~y Iuedoaeusal abgddim ci RogalI(1961). DN4S using a

W~~ Mw-volum code (Nemo. 19M) at low Mu*h mber, IMS usin a caqausmhe eddyv tymodd his So lWo-voloe code. mid LES aftg a -i~frM e hI 'Jcoqirulse code The

DIES ad US dftan me uhimu wfth 64' mud 32 gSW polmm respectively. Excpt for diflocie.. In20.~ lc " rem6o mud mthds, the Pollmn me has tom statistically tksame Bow bi &I

Cmess. bodase d.6b li cih m Im methaside 6th s IphImodel willbe give hi te hi payper.Degilogt km a Isesqic Oulu andoo lAdd with a qmuelled iniild inerg qec~um. the

%*yk dymc SUMl is Slowed tobevolvei and a Nuulmstlc" melt-sailer sm is developed (a bi Yaewsn sodEmurnr 1991). Reets ki. a 64' aicobar sho a powerohiw decay of amer, with mxpome 1.31comiow with giddgmued mNA -Ime-d&Wmud mll-scae nhuivlu ilutraed by tha collapse ofWO -wouviner spech at IMr -don sotsmd Kalmogarov cihig a smhom hi Figies Uad It is

homuhi tomo" both the DME amodisa gre gesukdiby wall v4, thwecomo that the, LES data alsovawsny pWe @rum" *ith the DME due.

lb ow ideat ulos Ses ue uiyi.here. in amibmiud at adua a-9.0 (whchmaq~Im woomd 203 MWud edgy-Surover daes) was seletd for to 0s inmyuls. By this time, Oth

tbIm fa. Is eL delop ed, ad ftthe aylr-ecm Reayrnldsaw~r (based ou the rime velocty mud Thsy-

aimbsed mergy mud diulpath speiam at the choe Swee for the Im ifferm ame. lb DIGS dama*am USt dilmono me well resoved mud Also aiw wih Acote suba minkiy weD Tho pets ofaEly mad dhbupO. qiecyat me qimued by a hmc of two. lb LMBda also apme a w 1 1 withft OME moults, howevr doe pet value of both th e aigy aud dissipala. Is slihtl highe.

Aqugu 3,4a.5 show the efd viscosity Olqualon (Z)), stubod eteraf, 7*(klIRJ .6 resolvedam& I*k Ike). reqecdvaly for M -cuooffwavoomuabw A6. Although ftheg cp oscale. availilefa ow 64' ciculatlos is Ihlled uveahelms same hI I -I S coikhulaoe my be *aaia, to beowiftad by 123' laf dia wEl be obtaied by thetime of the Symposum. Risults for do lowest Cut-off chue (k6 -5) me aumewIa- arnie, becaus they waumspoud to IM wth& a voy caum AIMer mdbecme of abdassle vuhbilky uasuled with the resoveid-scale OMeg spectum (ELOk) at low

* ~ ~ e -i ou Lb For hNOW cu~lft Wmpcal eddy viscosty snimimply as thet CW~ wavminwibe sq--ouied (Filme 3@4). NtIes thai for do LBS cow OVg 34)6 to cuoff wawumiber ke-S is SMUfyto mdms a wa -m bis btdo resolved Aed.L At eniaf k/k,. SOe SOCuul edfy visosty is Ieptive,

sowt bbmso usluam of ls eddy vimeask wmcwt mufef. Thee fam~es me cominss with the- of cDewsM at IL (1993)6 who studied a TayW-OCkm vane ad jauese~d *9 vebdom Of

', Ak1 ). &AMi do 1 1 (7*k1k 4) sd7T(k 1k,)) asaIim ci ok/k,.FPgue 4 iownm ft ~s efec Solb-UI mules Is minlY 10 UNI~ migy Bm So lerg mules but

dmb isdo a moaqwglgble b~acooe (7(6t 1k. ý 0) dftct maing Cms So agust scmlif We PlmI so tudyAnd Sis oduct by suplicly mocdelint *a beckmuue paus as doe -Iur by Cbuv (1991). FigueS elo is do doe -nr ci fI --F -m smmg do resold mscls Oimmuelws Is jwinwlly forwed Casco&-6* wfthinwwgyle(T(k lk,) <0) at owk /k6sadmmoWg pin(T(&lJkj ) as k/1k, Apomhes, wiy.

Ibas nooks wIN be toer P valloed for the And pqer by cmayin out 128e DNS asolailomi Asmaood asko vemigi models, such a dw eofy viscosity amod with adapdv evabolodm i ofSo comsm(0-so at AL. 1991) and the mmeoeqadow said twooequadmla models, wil be Imhoaptemd in both theSube-vohne ccdw sad So effct of Os subgdd mode wm ft energ wmfe pomss wil be bnvestiptedin dsua usft a aides of LES at venou gri reoludo. 7be effect i of masin dk Reynolds woauaevwI also be knvudgatud.t rh esuit of thee studies hoUl died lgtk an l. deicleocies soid sumgih of*e &Agid models and provid a dkwutim for inquviag the models so tho high Reynolds umber Sows- be mmaltdw

Chumov, J. IL (1991) "Shimulsom of the Kclogxorv Inetia sauges using on inapoved uabgildmodelPhyL Fuidi A. VoL3. 3pp. 188.

Dmanwduki J. A.. Lha W.. and Dtmcbu M~ .R(199) -An Analysb iso " m sca b mule inmetda inindicaly uimulatud Isotrpic toibuleace", PkyL Fluidi A. VolS.5 Wp 1747.Gemuxk bl, R'omeUI U., Main P., sod CiOM W. IL (1991) *A dylck smbgvdel edfy viscsitmoodeLO PhJL ?keidA. Vol. 3. pp. 1760.Memo. S. (1992) OSecmduy fladI lecdom camedof aicmbustiom, ksfiksaly im a unmjWet, o appm Incom&"=i Scienc amd Techaokvg.Pleuil, U., Cubs. W. H., Main, P., sod L^e S. (1991) -SW ltIcak bacitscote In nubiulantsd mel-domd lows," Phyz Flids A, Vl. 3. pp. 1766.Roapfe L. S. (1981) -Nunerica expeiments in Hammogemeou 1Tubabeucs NASA MWI 81315.

Yamg P. L soid Bromseur, J.0G., (1991) "The reqian of sotopic: Surbuleace D som ioapc sod solsolropiforcin at dos bWgs mmis, Ph. Fluis A, Vol. 3, pp. 884.Yadihiwa A. (199) "Brddgig betwen Eddy-Visosty Type sod Swcmd-(Oder Models =sinp Two-Scale NA" Symposium on "TIubulent Shim lows", Iqmpoo Auput 16-18.

10-1. .

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.0016 .8322-

.0014 ."I2s

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.0004 .33062

.0002 .9004

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.00004.-. 00001 .00002

-. 000002T(k ik) 0 --

-.0000 -.00002-".00004

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-. 00007 -. 00008(a) m8, 64, 4ecrd mehod-. 00008 '1 _ " . ... . ..... -. 0010" ° '. ... . . .

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-. fuel .00844

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Joa"y 9-1Z 1995, Ru NV

A MW DYNAMIC ONE-EQUATION SUDGRID MODELFOR LARGE EMY SIMULATIONS

Wrn-Week Uas, Swab Meme md Vermu K. ChakravarthySchol Aern~me Eqieerag

Alma, G"rgla 3033241S

1. INThODUCTION

The dynamic sugM scale (SOS) modling tecnue, lmuikacd by Germma e al. (1991),has been smocessibly SqWped to vulous types of flow fields (see Main e! al.. 1994 for a reentravlew). Thm are dm •tmphotanm feat-res oa, this modei approach. Fit, tm el cgmsant,die so-aled Smaglariu's Cotantm. Is not prelbed a prior, rather, It Is deurmined a a partof te soluion. This qVrosch renoved one of dhe major probems associated with te

al eddy viscosity modd which was the det-Pmination (and fi&nunng) of Wecausta far different flows. Second, he dynamic model d tws t coarect asymptotic behaviorwar walls ad trd, as a resit of th dynamic evaluation, th wmu can become negatve Incrtain reglons of de flow field and daws, oppem to have Ute caspdsh to mimic becaner ofenergy Jkon t subgrld sale to the reolved scale Tlhs last fatize Is ptioularly attuactvesice direct mnmercal simulation (DNS) dat has howl that becaca effect can dominateover a dpgnfican pardion of the grid polats In th flow OWel (e.g., Piomelli et al., 1991).Eowever, lmpg-eddy simulations 02S) using this dynamic model has dmon that th backscatedlect of Oemano's model can be cumssive and can caue mnmerical instability. Some rmomtefforts (Moi el at., 1994; Pkmel and Llu. 1994) were directed paricuarly to develop bettermethods to evaluate Mewcmml. However, these studies Mill assume dth the form of the eddyvibsosty, as poposedP by Smagaulnsky. Is sil aqn opuate and vald for the enire rnge of flows.Reemnt DNS sudies of decaying boroc tubulence sggest tht assumption of the localequillbium between Uth SOS ewy production ad dissipation re (which was used to derivethe Smagarinky model) may not be satisfied ovr a large porion of the gid points. T7is lack a(local equilibrium can be significant wen tde flow ilWds at dIlf n llgrid resolutions we relatedto each other, a is Ut cae In the dynamic model.

In this Ipqlr, these Isue•s addMessed by coomldma two new subgrid modes that do notIun A local equMbdum between Ut SOS enery poucon ad dissipaion me. These models

have bee develped using do dynamic modeig mp- mad Uefam therewe nic no wnwus110 have to be prescibed a plo,. The first mod tht Is studied Is a dynamic version of theKmoulm xovs scaling eP., usl or aftw edyisosity and the seodmdlIs a one-equationmodel for SOS kineic eneirgy.

Lling LES of decaying isotropic urbuleace at variou Reynolds mumber, Mae behaevior ofdme nw dynamic models have been compared to Ute pndcnm of ft dynamic model based.

8 the Ss edd viscty concptl An evenual goal of this ireA: is to determie

do farm of a dynamic suwbgld model that will provide realistic results over a wide range ofleymods number and grid resolution.

The mumerical simulations were carried out using a i f c code that Is second-ordermacy in time and Wb-order (the convective terms) and sixth-order (the viscous terms)accuracy in space using uwin•ased diffrences (Rai and Mon, 1991). Thie-accurateselutioms Of the Inpres sible Navier-Stokes equations we obained by the artificialcog Wssbity approach (Roge r at1.. 1991) which fauires smubiteaton in psadotime to getft dKv lec-ftn flow field. Earlier (Menon and Yom& 1994), this code was validated byCarrying out DNS of decaying IsoUpi nrbulence and comparing the resulting statistics with thepredictions of a well kmnwn psedo qsetal code (Rogallo, 1981).

2. DYNAMIC SUBGRID MODEIUNG

In the fbilowilng, the firms of the dynamic subgrld models studied here me briefly described.Morn details of the modeling approach will be given In the fnal paper.

2.1 Dynamic Smagormnsky Model

The primary requirement In LES Is the modeling of the SOS resses dat result from thespatial Altering of the Instantaneous Incompressible Navier-Stokes equations. If an appropriatefilter is employed (here, a top hat filur with a filter width A is used, where, AIs the gridspacing), then the tue SOS s tenor Is: ' u u, j-911. ITis euss tensor must be modeledto close the LES eqations of motion. "be Smagodnsky eddy viscosity closure for the SGS stresstnsor Is of the form:

19 8YT V oVT=C~p1 (I)

where C is the model coefficient, is the resolved scale strain rate tensor

'WD O (2)

and 1-( )".Her, the owrba on the flow variables indicates the result of filtering at

en grid scle 1. In the dynamic od Mproach, a mathematical Identity beween theMum resolved at the Vid scale filter A and a tt filler A (ticany dcoen to be twice of thegrid filter A) Is used to detrmiue the model coefficient C a a part of the simulation. Thus, If the

oilcatio onfthe mt filer on any variable* Is denoted by j or (t). then It ca be shown that

| f - t (3)

MMre, 7(w r"aa)-At,) Is defined by using the teat fiter. Using the assumption of the self-ulmilarity of tie subgrid sess, Tw can be modeed in the same way as T, resulting in the

*Mowing expression:

f f *=-2V VTrWCAI& (4)

Soboatuti•g (1) and (4) Into (3), one can obtain an equation for C

LW -1 La = 2CMf (5)

N -k11Y -P 1 Y) (6)Equation (5) Is a set of Mi WvmWed equations for one unknown C. To minimize the error*om solving this overew••nd system, Ully (1992) proposed a least square method whichyields

=I (7)2M, M,

Past studies (e.g., Zang et at., 1992; Yang and ferzlpgr, 1993) using this approach have shownta the value of C obtained ftom Equadon (7) can vary widely In the flow field. Ths can cousenumedcal instability. To relieve this problem, spatial averaging is typically performed for boththe numerat and the denomnaho on the RHS of (7) (Plomelli, 1993). Usually, this averaging isdone only in the directions of flow homogeneity (e.g., Mon et aL, 1991). In the present study ofhomogneou Isotroprc ulence, aveaging is implemented over the entire computationaldomain, hence C is a function of time only.

2.2 Dynamic Kolmogorov Scaling Model

Recently, Wong and Lilly (1994) revaluated the Iotmogarov•s scaling expression for theeddy viscosity. The rpqPremadve vatdabes M - -e- __ SOS fluctuations are the SGS enerjydissipaion rate a ad the SOS lenth scale which can be apprmg m&W by the giid interval A.Using dimensional argumen de subgMd eddy viscosity can be writen as,

nlm, ft model coeficient C. as the dimension rad to C (precistly, C. -= ). This simpleujeuto lfor V.r can be robust became It does not eqloy the usumption of local epWbrumbewee SOS engy producton rate and dissipation me which was adopted to derive theS I model (and which is implicit in the dynamic model described in Section 2.1).A i advata of this model is that, due to the simplicity of the model, the total

computationAl Ime can be signifcantly reduced. However, this model can be valid only byamlyoYing a dynamic modeling method to determine the model ceffclienL Otherwise, there maybe no way to pIescrlbe this no1..nmensionless coefficient by a fixed value. The resultingaOatim for C. Is expesed as follows,

It - i ,8# =- 2 C9

2.3 Dyamic k-equation Model

A higher order model Is also cornired in this study. This is a one equation model for the

SOS kinetic eergy k ( j - ?) /2). This model was recently evaluated by Menon andYMng (1994) through a priori tests using DNS of decaying and forced Isotropic turbulence. Inthis model, an evolution equation for k can be written in Uthe following form

8k + 1 k _T# s C+ a..8k(10

where

T -2vTY, + 80k V --C,kV2& (11)

MAlo, one can model Uth dissipation rate c (a v [~/~) - (aria)2]) using k, as

c k V2 (12)

Menon md Yeung (1994) chose the model coefficients, based on an earlier study (Yoshizawa,1993), to be C,=0.0854 and C,=0.916. However, these values were found to cause excessivedissipation of large-scale energy for Isorouc turbulence. In the present study, we apply thedynamic modeling technique to obtain apropriate values of the coefficients. To implement thedynamic approach, the bgrdd kinetic eergy K at th test fiter level Is required. This Is obtainedby unt the trace of(3), K = L, / 2+k. Thususing the pocduredesibed earlier, equationsfor both C, and C can be derved:

L,- -2 C, (a K41 A,()

W\• cVI-• a,,} c , -u

Her, equation (14) is a scalar equation for a scalar unknown and thus, we can obtain Uth exactvalue for C. wIthoult applying Me least square mefthd.

Orne May Othn that Uit merit of the LES approach les In the simplicity of the model used forfte sImulatdons sad that this merit may be lost by using ft hihe-order models such as the one'equation model. However, the Increased cost of this type of model is compensated by Uth

I~ovemu0 I6eU accuracy resulting by not assuming local balance between Uth SOS energyprouctonand diaipalo rafte Accordin to our a prior test using reltvely high reisolution

DS, (using 128x128xI28 uvJ 64z64z64 grid riesolutIons), over a large portion of th grid pointsIna rIa r;Iic turbulent flow, local equilibrium assumption is violated, This Imple that modelsdoa do not require local equilibium such as Ut one-equation model has Uth potential to producebette reslt than Uth Smagorinsky-type model. lTb ealier sxtude using the: one-equation modelwith fixed values of Uth cod~effcets (Meoon and YeMng 1994) Improved Uth results whencoMpaed to Uthe amrak' model with fixed value of coefficient. However, Ute results werepoom than the results Obtained usin Uth dynamic Smagorluhkys model. Analysis of Uthsimulation data showed tha this was due to a poor prediction of both Uth production anddissipation tens in (10) using fixed coefficents. Especially, Uth prediction of Uth dissipationterms was very poor. In Uth present study, the use of Uth dynamic p irocedurm, to evaluate Uth

coeficin results In a much btrprediction of both Uth production and dissipation terms in Uthk-equation model. This in turn Improves significantly Uth results of Uth LES using Uth dynamic k-eqation sbrdmodel

3. PRELIMINARY RESULTS AND DISCUSSION

To evaluate UtthreeO different dynamic SOS models (described In Sec. 2). LES of decayingWom-g--ou Isocropic turbulence were conducted. Starting from an initially divergent free

Adctyfed with a prescribed energy spectumur, Uth flow Is allowed to develop Into realisticdecaying turbulence. The flow under consideration is modeled in a culc box with periodicboundarvy conditions and two grid resolutions, 32x32x32 and 16x16x1 6, were employed for Uthsimulations. The simulations were prformed for Wree differen Initial Taylor Reynolds numbersRex. =30, 100 and 1000. In ft following. preliminary results we discussed to highlght Uthbehavior of the new dynamic subgrld models. The results discussed here we obtained by usingUth flow fied at an instant in time where Uth flow field has relaxed to a realistic decayingturbulence and Uth Reynolds number of the flow fied Is decreasing very slowly.

To evaluate Uth self consistency of Utb dynamic models, Uth LES results obtained on the32x32x32 grid resolution were compared to Uth LES results obtained on Uth 16x16x16 grid. Toensre tha both grid resolution simulations wre being performed using newrly identical initialconditions, Uth Initial flow field for Uth 16x16xl6 grid simulation is obtained by filtering Ut32xWx2 grid resolved flow fied, ltb resulting flow fieds from thes two differet resolutionsimulatons can be areatd using die mathmatical Identity, (3). By this approach, Uth modeled

quantiy (uv,u can be obtained fro Ute two data acts which should be Identical or at least

highy correlate if Uth subgrid model performed correctly at Uth two grid levels. To quantify ftbehaio of Uth subpl model, correlation coefficients (define In the u=9a manne) are

comptedusing the anisotropc parts of expressions for Mua,. Tevraino taeae

cor-eltion coefficients of thesethree anisotropic components wre dmon in Fig. I as a function ofMhe Taylo Reynolds mune at Me Instant of the comarwison. The correation coefficient is veryhog for all Ut models ow er Utntre: rang of Taylor Reynolds numbers. However, Ut

cwieatlon coefficient for fth dynamic Smagorinsky model decreases with increase: in theReynolds mmnbers. Although the correlation coefficient for the dynamic KolnMoorov scalingmocdel Is always blogw thanf th othe it was fOWW using spectra space anailysis that this modelhas som problems In predicting the dissipation rate of the SOS turbulent energy. This model iswore dissipative in the earlie ransitlonal stage and less dissipative in the fully developedbutulent stage WTis Vroblem may be cauased by the lack of direct modeling of the dissipationrate when the Kolmogorovik scaling for the eddy viscosity Is used. In any event, the resultsShown In Fig. I clearly suggest that the models diet do not mae- fth MUMption Of localeipllbrium between, the production ad! dissipation fate produmebte results than the dynamic

~agodzmka model. ibis is important since Uth eventual goal of LES methodology is todevielop subgrld models tha will allow simulation of high Reynolds nume flows usingseatvely coarse grids (grid resolution testriction we typically Imposed due to computerreouc limitations).

Fig. 2a shows the variation In the dynamically evaluated constants with time: during thesimulation for Re. =lOO and Fig. 21b shows the variation of the model constant with the TaylorReynolds numbers. Obviously, the model coefficients go though changes In the earliertransitional stage. However, after some time, all coefficients reach an asymptotic state with theexception of Cc In Kolmogarov's scaling expression which keeps decreasing as the kineticenergy decreatses. MAlo, Uth values of Uth coefficients at this asymptotic stat me almostIdeedn on Reynolds number except for C, in k-equation model because E, and hence Ccwhich is generated ftom direct modeling of C. is very sensitive to the grid resolution andReynolds amnber. Mhe resulting constants wre similar to the values obtained in earlier studies.For e~xample, In fth presen study, the dynamic Smagrnsky model predicts that theSmagoulnsiky model coefficient C. (which Is the square-root of dynamic mdlcoefficien 0)should be about 0.165. ihis Is very close to the value of 0.17 suggested by Lilly (1966) forhomoeeu Isotropic turbulence with cutoff in fth Inerial subrange. It Is noteworthy thatdenterming temodel coefficient usin the dynamic Smagorlnsiy model may have somelimitations. Tbis model predicts Its model coefflcient to be negative for a long period of time inthe earlier transitional stage. Even this period of time Incre=e with Increasing Initial Reynoldsnumber. Obviously It leads to numerical instability. To prevent this problem from happening, weImpose the constraints that C should be always larger than 0.01. Tbis kind of numericalInstability was not brought about by the other models.

Fig 3h show contours of the SOS kinettic energy on 32x32x32 grid directly obtained by LESusing the dynamic k-equation model. Tbis result Is compared to the prediction by the LES usingthe same model but dlffernt resolution, 16xl6x16 grid, (Fig. 6b) at an hatbtary (but same) sliceof the 3D flelcL Mbe comparison sOws tha fthr is significant: similarity between two LESresults of " fIp 1t grid widfth and thus confirme the self consistency of the dynamic k-equationmodel. The peak values of the coarser grid (16x16x 16) results me approximately twice of thoseftom the finer gri (32x32x32). This is reasonable becwse the coma grid which has the lowercatoff wmavenmber should contain the larger SOS knetic energy inside of Its subgrld regons forthe samne Reynolds number.

4. FUTUVRE WORK TO BE INCLUDED IN THE FINAL PAPER

The remas obtained so fur show that the dynamic moes that do not assume local* ~~~quilibrium between fth SOS energy prowduco and dissipation rate perform significantly bte

than Uth dnamic model based on the classical Smagoulnsky's eddy viscosy model. Furthersimulations =e underway to confirm this result using fime grid resolution and for highuerReynolds numbers. Comparison of the LES results with Ute DNS results for representative(relatively low Reynolds numbers) caes will also be carried out. Finally, Utme models will beevalumatd for Mfeant flow cuas. For example, ft Taylor-Came vorte flow, which has beenused by Dmemradzk es al. (1993), is considered a good test problem. Ibis Is. due to flowsymmetry, a flow at a relatively high Reynolds number can be simulated using relatively comsemahatimn The results; ofthewsetuadies will be described In amo details In die Anal paper.

M RRNCES

J. A. Domaradzkl, W. Uu, and M. E. Drachet (1993), "An analysis of subgrld-scale Interactionslanummerically simulated isotropic turbulenece" Phys. Fluids A S.,1747.

M. Gemnano, U. Piomeflli, P. Moin, and W.I. Cabot (1991), "A dynamic subplid-scale eddyviscosity model,- Flays. Fluids A 3,1760.

D. K. Lilly (1966), "On th application of Uth eddy viscousity concept in the inrgtial sub-rage ofturbulence," NCAR Manuscript 123, National Cetrfor Atmospheric Research, Boulder, CO.

D. IL Lilly (1992), "A proposed modification of Uth Germano mubgrid-scale closure medxtho,plays. Fluids 4,633.

S& Memow and P. X. YeMg (1994), "Analysis of subgrl models using direct and large eddysimulations of Isotropic turbulence," presented at the 74th AGAJIDOFDP Symposium onApplication of Direct and Large Eddy Simulation to Transition and Turbulence, Chanla, Greece,April 18.21, 1994.

P. Moim, X. Squires W. Cabot, and S. Lee (1991), "A dynamic subgrld-scalc model forcoempressible turbulence and scalar transport," Flays. Fluids A 3.2746.

P. Moin, D. Carati, T. Lund, S. Ghosal, and L Akselvoll (1994), -Developments andapplications of dynamic models for large eddy simulation of complex flows," presented at the74th AGARD/FDP Symposium on Applicatio, of Direct and Large Eddy Simulation toThmldon and Turbulence, Chanla, Greece, Api 18-21, 1994.

U. Plomeilli, W. KL Cabt, P. Moin, and S. Lee (1991), -Subgrid-scale backscte in turbuentand tranitional flows," Flays. Fluids A 3.1766.

U. Plomelli (1993), 111gb Reynolds number calculations using die dynamic aabgrid-scale stresmnode, Flys. Fluids A 5, 1484.

U. PloMel and J. Llu (1994). "Lug-oddy simulation Of rotating channe flows using a localizeddynumi model." presented at Uth 74th AGARDIFDP symposium on Application of Direct a&dLarg Eddy Simulation to Thnsiton and 'Dubulenme Chania, Greem. April 18-21,.1994.

iM.I. Rai and P. Main (1991), "Direct simulations of turbulent flow using finie-diffeenceadumes," J. CompaL Phys. 96 1s.

L. S. Rogallo (1981), "Numaical expaiments in homogeneous untulrence," NASA Tech.Memo 81315.

S. P. Rotors, D. Kwak, and C. Kids (1991), "Steady and unsteady solutions of themcoaqsuible Navier-Siokes equaions," AlAA J. 29, 603.

V. C. Wong and D. K. illy (1994), "A comparson of two dynamic subPrd closure methods forbfamlnt mtermad covectlion" Pbys. Fluids 6,1016.

K-S. Yang and J. H. Ferwiger (1993), nxp-eddy simulation of tabulent obstacle flow usinga dymmic oubsd-scale model," AIAA J. 31, 1406.

A. Yslizawa (1993), "BDrdging between eddy-viscouslty-type and second-order models using atwo-scale DIA." prisented at the 9th symposium on Turbulen Shear Flows, Kyot, Japan,August 16-18, 1993.

Y. Zang, RJ.L.Sueet, and J.R. Kosef (1992), "Applcation of a dynamic subgrid-scale model totubulent recirculating flows," In CTR Annual Research Briefs 1992 (Center for TurbulenceResearch, Stanford University/NASA Ames, 1992), p. 85.

MMOW 2

FIG. 1. Camidado betee fth LES result co 32x2x2 gil sad on 16x16x16 gild as a Ifzzwoa

oftht Taylor Reyk m~Isfmber at ftheA ofa ftth cwmpulSoM

j 4--Coeff. inq EQ.

@-ff ine hEQ. (13)

- Cow. i bEQ (14)

() 0 6 12 i6 264 30 3

'I--*- COW. Im EQ. cmSCOWf. In EQ. (9)

I _ _ _ _ _ _ _ _ C w .Im (1

(b) IM41IO4

FI. 2. Vmdiuiof fit dymuuicaiy do P , Sd nowl mfclew (A) AS a famctS Of (1UmiCRexom100, (b) u A fiucOa of t Taylo Rcyudt mmaue.

COuMMu lnW "M2-4 TO 6.UZI-4 JOY OMC5-4

MI. I. omquadm of ft miu of ft SOS Wc umu foma to AM I , L. a"n fit#jgc kgq.Mlam MOM: (&) LMS as 3W32z3 VW (b) LBSS an l~zl~xl6 gld.