Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, January 1996A9618387, N00014-93-1-0342, AIAA Paper 96-0425

High Reynolds number flow simulations using the localized dynamicsubgrid-scale model

S. MenonGeorgia Inst. of Technology, Atlanta

W.-W. KimGeorgia Inst. of Technology, Atlanta

AIAA 34th Aerospace Sciences Meeting and Exhibit, Reno, NV Jan 15-18, 1996

The localized dynamic subgrid-scale model introduced by Kim and Menon (1995) has been used in large-eddysimulations of decaying and forced isotropic turbulence and temporally evolving turbulent mixing layer for highReynolds numbers. In the simulations of isotropic turbulence, it is demonstrated that the low-resolution large-eddysimulation results accurately reproduce the characteristics of a realistic, high-Reynolds number turbulence such as thepower-law decay (decaying case), and the velocity statistics and the development of the non-Gaussian statistics (forcedcase). From the large-eddy simulations of temporally evolving turbulent mixing layer, the time-accurate results, whichagree very well with the existing high-resolution direct numerical simulation and experimental data, are obtained usinga new scaling which is capable of separating the distinct effects of initial development on the self-similar stage of themixing layer evolution. (Author)

Page 1

AIAA-96-0425

HIGH REYNOLDS NUMBER FLOW SIMULATIONS USING THELOCALIZED DYNAMIC SUBGRID-SCALE MODEL

Suresh Menon*and Won-WookSchool of Aerospace EngineeringGeorgia Institute of Technology

Atlanta, Georgia

ABSTRACTThe localized dynamic subgrid-scale model intro-duced by Kirn & Menon (1995), has been usedhi large-eddy simulations of decaying and forcedisotropic turbulence, and temporally evolving turbu-lent mixing layer for high Reynolds numbers. In thesimulations of isotropic turbulence, it is demonstratedthat the low-resolution large-eddy simulation resultsaccurately reproduce the characteristics of a realistic,high-Reynolds number turbulence such as the power-law decay (decaying case), and the velocity statis-tics and the development of the non-Gaussian statis-tics (forced case). From the large-eddy simulations oftemporally evolving turbulent mixing layer, the tune-accurate results, which agree very well with the ex-isting high-resolution direct numerical simulation andexperimental data, are obtained using a new scalingwhich is capable of separating the distinct effects ofinitial development on the self-similar stage of themixing layer evolution.

1 INTRODUCTIONThe dynamic subgrid-scale (SGS) model, introducedby Germano et al (1991), has been successfully ap-plied to various types of flow fields (Moin et al., 1991;Piomelli, 1993; Squires & Piomelli, 1994; Ghosal etal., 1995). Two desirable features make this model es-pecially attractive. First, the model coefficient is de-termined as a part of the solution, thus, removing themajor limitation of the conventional eddy-viscositytype SGS models which was the inability to parame-terize accurately the unresolved SGS stresses in differ-ent turbulent flows with a single universal constant.Second, as a result of the dynamic determination, the

•Associate Professor, Senior Member AIAA.t Graduate Research Assistant, Student Member AIAA.

Copyright © 1996 by S. Menon and W.-W. Kirn. Publishedby the American Institute of Aeronautics and Astronautics,Inc. with, permission.

model coefficient can become negative in certain re-gions of the flow field and, thus, appears to have thecapability to mimic backscatter of energy from thesubgrid-scales to the resolved scales. Although it hasbeen shown that Germano et aL's dynamic model issuperior to the conventional fixed-coefficient model,the dynamic procedure, as developed earlier, still hassome deficiencies. These deficiencies originate froma weakness of the Smagorinsky model used in Ger-mano et aL's dynamic model, as well as, from themathematically inconsistent derivation and the ill-conditioning of the dynamic formulation itself.

Recently, Kim and Menon (1995a,b) developeda new localized dynamic subgrid-scale (LDKSGS)model associated with the SGS kinetic energy equa-tion model. In this model, all deficiencies of Germanoet aL's dynamic subgrid-scale model were overcome.This model also provides a straightforward localizedevaluation of the model coefficients which do notcause any numerical instability. Moreover, the local-ized model coefficients obtained from this model areproved to be Galilean-invariant and very realizable.The localized dynamic model was applied to Taylor-Green vortex flows and, it was shown that this modelpredicts the turbulent flow field more accurately thanthe previously developed dynamic models. In this pa-per, the application of this model to high Reynoldsnumber decaying and forced isotropic turbulence, andturbulent mixing layers is discussed. The results arecompared with predictions using experiments, directnumerical simulation (DNS) and large-eddy simula-tions (LES) using Germano et aL's dynamic model.

The numerical simulations were carried out usinga finite-difference code that is second-order accuratehi tune and fifth-order (the convective terms) andsixth-order (the viscous terms) accurate in space us-ing upwind-biased differences. Time-accurate solu-tions of the incompressible Navier-Stokes equationsare obtained by the artificial compressibility approachwhich requires subiteration hi pseudotime to get the

divergence-free flow field. A significant acceleration ofthe convergence to a steady-state divergence-free so-lution in pseudotime is achieved by incorporating thefull approximation scheme (FAS) multigrid method.Earlier (Menon and Yeung, 1994), this code was vali-dated by carrying out DNS of decaying isotropic tur-bulence and comparing the resulting statistics withthe predictions of a well known psuedo spectral code.Further validation of this code is demonstrated in thisstudy by comparison with experiments.

In section 2, the LDKSGS model is described withthe basic equations indicating its advantages. In sec-tion 3, the LDKSGS model is applied to decayingand force isotropic turbulence, and temporally evolv-ing turbulent mixing layer. Conclusions are presentedin section 4.

2 LOCALIZED DYNAMIC AJ-EQUATI-ON SGS (LDKSGS) MODEL

In physical space, the incompressible Navier-Stokesequations for LES are obtained by filtering them us-ing low-pass filter of a computational mesh (hence,the characteristic length of this filter is the grid widthA) as follows,

= 0 (i)

where Ui(xi,t) is the resolved velocity field. Theapplication of the grid filter results in the follow-ing unknown SGS stress tensor, ry = TZiSJ — uftj,which needs to be modeled in terms of the resolvedvelocity field u. Eddy viscosity assumption impliesTij — —2i/r^ij + s&jTjub where VT is the eddy vis-cosity and Sij — \ (dui/dxj + cSZj/Szi). Simple di-mensional arguments suggest that the eddy viscos-ity VT should be given by the product of a velocityscale and a length scale. In LES, the length scale isusually related to the filter size (A), however, mod-els differ in their prescription for the velocity scalewhich can be estimated from the smallest resolvedscales. In the Smagorinsky model, an algebraicallydescribed velocity scale is obtained by assuming thatan equilibrium exists between energy production anddissipation in the small scales. One-equation SGSmodel solves the following transport equation (e.g.Menon et a/., 1995) for the subgrid-scale kinetic en-ergy) *»ff* = 2 (S5«i— «»«»), to provide the velocityscale:

- dk,g

Here, the three terms on the right-hand-side of (3)represent, respectively, the production, dissipationand diffusion of kig,. In the model of the diffusionterm, the direct effect of v has been dropped. In theoriginal model of this term, &*r/o*. is used in placeof VT. However, since a* = 1 is usually adopted(Yoshizawa, 1993), a* has been dropped from (3).The dynamic procedure can be used to determine a*as described in Kirn & Menon (1995b). The SGSstress tensor TV,- is parameterized according to theeddy viscosity assumption and the SGS eddy viscos-ity VT is modeled as: VT = Crklgt A where Cr is anadjustable coefficient that is determined dynamically,as shown below. Equation (3) is closed by provid-ing a model for the dissipation rate term, e. Usingsimple scaling arguments, e is usually modeled as:e = Cc A~ kf$ where, Ce is another coefficient thatis also obtained dynamically.

To obtain the coefficients Cr and Cc dynamically,a top-hat test filter characterized by A (typically,A = 2A) is used (the application of the test filter onany variable (j> will be denoted by <£). The dynamicmodeling procedure employed in the present study isbased on the existence of the similarity between theSGS stress ry = u&J — «<«,• and the following re-solved stress Ttj = ufuj — Sjttf which was observedin Liu et aL'a (1994) analysis of experimental datain the far field of a round jet at a reasonably highReynolds number (Re\ « 310). Using the existenceof the similarity, the model for Ty can be extended toT^ as follows,

^ (4)

Also, the dissipation rate at the test filter level canbe modeled in a similar manner:

E = (5)

'** **ST**- "i - UjtiiJ

Here, (y + VT) is used for E since E is described onlyby the resolved-range scales, while the definition of

the actual dissipation g includes the unresolved scaleinformation (dui/dxj)2. That is, the test filter level

/ —-~—. *** .x-fc \energy, K = ^ (u<«i — u^k \, is dissipated due to theSGS eddy viscosity as well as the molecular viscosity.Therefore, the effective viscosity for E is (y 4- VT).Now, Cr and Ce can be determined directly from (4)and (5). Equation (4) is a set of five independentequations for one unknown CT- To minimize the errorthat can occur solving this over-determined system,Lilly (1992) proposed a least-square method whichyields

where

^ _= -A uitli -

1/2

(6)

(7)

Equation (5) is a scalar equation for a single unknownand, hence, an exact value Ce can be obtained with-out applying any approximation:

Cf = A - luiu, -W)]-3/2

E (8)

As shown above, a mathematically consistent pro-cedure is employed in this dynamic formulation (Ger-mano et oi's dynamic model formulation contains amathematical inconsistency since the model coeffi-cient is taken out of the spatial-filtering operation inspite of its large spatial variation). Furthermore, thedenominators of the expressions for CT and Ce con-tain the energy information within the resolved scalerange which is well-defined (in Germano et aL's dy-namic model, the denominator in the final equationfor the model coefficient contains algebraically manip-ulated terms that can be very small causing numericalinstability and hence, the resulting expressions are ill-conditioned). Therefore, the ill-conditioning problemis not considered serious here. Also, the direct eval-uation of the SGS kinetic energy prevents the pro-longed occurrence of negative model coefficient whichhas been known to cause numerical instability in Ger-mano et aL's dynamic model. These properties of thecurrent model enable a stable localized evaluation ofthe model coefficients without employing any spatialor temporal averaging. Further, note that, the ex-pression for Ce does not have the unphysical propertyof vanishing at high Reynolds numbers since the effec-tive viscosity (y -f VT) is used instead of just v. Kimand Menon (1995b) showed that the LDKSGS modelis properly Galilean-invariant and, therefore, the re-sulting LES equations of motion for the large eddiesusing the LDKSGS model retain the same form in all

inertial frames of reference. They also proved thatthe model satisfies the realizability conditions givenby the inequalities T« > 0 and T?- < THTJJ (Vremanet al., 1994a). These ineqaulities provide the follow-ing realizable range for the dynamically determinedmodel coefficient CT,

L.V2l**tt* (9)

(10)

where Su and Snn denote, respectively, the largestand smallest eigenvalues of the strain rate tensor (i.e.,~Su > 0 and ~Snn < 0 in the incompressible case). Nu-merical experiments using decaying isofcropic turbu-lence (the detailed description of this flow field willbe presented in the section 3.1) show that more than99.9% (for the 483 grid resolution), 99.8% (for the 32s

grid resolution), and 99.6% (for the 243 grid resolu-tion) of the grid points satisfy both the realizabilityconditions, (9) and (10), at the same time during theentire simulation. Therefore, the LDKSGS model sat-isfies the realizability conditions hi a more strict sensewhen compared to Ghosal et al (1995) who reportedthat the DLM(£) model (which is the only other ex-isting localized dynamic model formulated withoutemploying the ad hoc averaging procedure) satisfiesthe realizability condition at about 95% of the gridpouits for the simulation of decaying isotropic tur-bulence using a 483 grid resolution. Furthermore, toexamine the realizability, they used only the condi-tion (9). According to our numerical experiments,the satisfaction of the condition (9) does not auto-matically guarantee the satisfaction of the condition(10). For definite realizability, both conditions, (9)and (10), should be used for accurate verification.

3 RESULTS AND DISCUSSIONThe LDKSGS model has been applied to decaying(section 3.1) and forced (section 3.2) isotropic turbu-lence, and turbulent mixing layer (section 3.3).

3.1 DECAYING ISOTROPIC TURBULENCEThe experiment of decaying isotropic turbulenceof Comte-Bellot and Corsin (1971) is simulated todemonstrate the capability of the LDKSGS model inpredicting the decay of the turbulent energy. First,the grid resolutions for possible DNS and LES are in-vestigated by computing the resolved energy at eachgrid resolution (i.e., by numerically integrating the

spectrum given by Comte-Bellot fe Corsin(1971) be-tween wavenumbers zero to the maximum wavenum-ber resolved by the grid resolution). The results showthat 3843, 1923, 963, 483, 323 and 243 grid resolu-tions resolve 99.5%, 96.6%, 87.3%, 70.3%, 59.3% and49.7% of the total turbulent kinetic energy, respec-tively. Therefore, to accurately resolve all excitedturbulence scales without employing any turbulencemodels, at least 3843 grid resolution is needed. Also,for LBS, the grid resolution used must be consistentwith the basic assumption of LES that the resolvedscales contains most of the energy. For this sameproblem, Ghosal et al. (1995) concluded that the 483

grid resolution is the smallest possible resolution forLF.S since, at the grid resolution coarser than 483, asignificant number of energy-containing eddies residesin the unresolved scales and, hence, the results aregreatly dependent on the quality of the SGS modelemployed. That is, the simulation of this experimentespecially using the grid resolution coarser than 483

(i.e., when the subgrid scales contain more than 30%of total turbulent kinetic energy) is a good test caseto measure the quality of the SGS model. For thispurpose, three grid resolutions (483, 32s, and 243)are used for the large-eddy simulations implementedhere.

In the experiment, measurements of the energyspectra were carried out at three locations down-stream of the mesh (which generated the turbulencein the wind tunnel). At the first measuring sta-tion, the Reynolds number based on the Taylor mi-croscale and based on the integral scale were, re-spectively, 71.6 and 187.9 (these values decreased to60.7 and 135.7, respectively, at the last measuringstation). Using the assumption of a constant meanvelocity across the cross section of the wind tunnel,the elapsed tune for the turbulent field traveling atthe mean velocity from the mesh (that is, propor-tional to the downstream distance) can be obtained.Therefore, this (spatially evolving) problem can bethought of as a decaying isotropic turbulence insidea cubical box which is moving with the mean flowvelocity. The size of the box is chosen to be greaterthan the integral scale of the measured real turbu-lence. The statistical properties of turbulence insidethe box are believed to be realistic even after apply-ing periodic boundary condition for numerical imple-mentation. All experimental data is nondimension-alized by the reference length scale 10Af/2ir (whereM = 5.08cm is the wind-tunnel mesh spacing) andthe reference time scale 0.1 sec for computational con-venience. (By this nondimensionalization, the threemeasuring locations correspond to the three dimen-sionless time levels, t' =2.13, 4.98 and 8.69, respec-

tively.)The initial velocity field (primarily the amplitudes

of the velocity Fourier modes) is chosen to match thethree-dimensional energy spectrum obtained at thefirst experimental measuring station. The phases ofFourier modes are chosen to be random so that theinitial velocity field satisfy Gaussian statistics. Theinitial pressure is assumed to be uniform throughoutthe flow field and the initial SGS kinetic energy is es-timated by assuming the similarity between the SGSkinetic energy and the resolved energy at the test fil-ter level: ksgl « %£• [UiUi —ikuij where a constantCK is determined by matching the magnitude of theSGS kinetic energy to the exact SGS kinetic energycalculated by integrating the experimental spectrumat the first measuring station. In situations where theinformation about the magnitude of the exact SGSkinetic energy is not known, a value of CK can bedetermined by adopting the similarity concept usedhi the dynamic procedure. This formulation is pre-sented in Kim & Menon (1995b).

Figure 1 shows the decay of the resolved turbulentkinetic energy computed using the LDKSGS model atthree grid resolutions, 483, 323, and 243. The resultsare compared with the predictions of the volume-averaged Germane et o/.'s model at the 483 grid res-olution and the experimental data of Comte-Bellotand Corsin (1971). The predictions of both models(at the 483 grid resolution) are in good agreementwith the experiment. As is well known, the turbu-lent kinetic energy undergoes a power law decay, i.e.,E ~ (<*)", in the asymptotic self-similar regime. Theexperimental data roughly confirms the existence ofthe power law by lying on a straight line on a log-log plot. The decay exponent n is obtained from aleast-square fit to each data: -1.17 (483), -1.13 (323),and -1.09(243) from the LDKSGS results and -1.20(483), -1.16 (S23), and -1.12(243) from the experi-mental data. These results confirm the agreementbetween the predictions of LES and the experiment.More importantly, the results of the LDKSGS modelat all three grid resolutions used (even for the 243 gridresolution where about a half of the turbulent kineticenergy is not resolved) show consistency in predict-ing the energy decay. This property of the model isa fascinating feature especially when the model is (tobe) applied to complex and high Reynolds numberflows where a significant amount of turbulent energypossibly lies in the unresolved scales. Without a self-consistent behavior, the SGS model can not simulatehigh Reynolds number flows in complex geometriesreliably. Therefore, the proposed LDKSGS modelseems to have a promising potential for application

to complex, high-Reynolds number flows.The computed and experimental three-dimensional

energy spectra resolved at the three different grid res-olutions, 483, 323, and 243, are shown in figures 2(a)at t* = 4.98 and (b) at £* =.8.69. Both of the LD-KSGS and volume-averaged Germane et aPs mod-els predict the spectra reasonably well. Especially,the LDKSGS model predicts the spectra consistentlywell for all three grid resolutions. Some discrepancybetween the experimental and LES-predicted energyspectra is observed around the cut-off wavenum-ber. This discrepancy is due to the fact that, forits discretized numerical implementation, the finite-difference code used in this study implicitly adoptsthe top-hat filter which yields a significant contri-bution to the subgrid-scale energy from the lowerwavenumbers than the cut-off wavenumber (when theFourier cut-off filter is employed, the subgrid-scale en-ergy is entirely due to the energy in the wave numberslarger than the cut-off wavenumber). Direct compar-ison between experimental and LES-predicted energyspectra is meaningful only when the full-resolutionexperimental flow field is filtered down to the LESgrid resolution to compare using the top-hat filter (tobe more consistent, the initial flow field for LES alsoshould be obtained from the experimental flow fieldusing the top-hat filter).

3.2 FORCED ISOTROPIC TURBULENCEA statistically stationary isotropic turbulence is simu-lated using a 32s grid resolution. The main purpose ofthis simulation is to determine whether a low resolu-tion LES using the LDKSGS model can reproduce thestatistics (of the large scale structures) in a realistic,high Reynolds number turbulent field. The resultsare compared with the existing high resolution DNSdata by Vincent and Meneguizzi (1991) and Jimenezet al. (1993) obtained at Re\ « 150 and Re* « 170,respectively.

A statistically stationary turbulent field is obtainedby forcing the large scales (i.e., the initial value of allFourier modes with wave number components equalto 1 is kept fixed). The initial condition is obtained bygenerating a random realization of the energy spec-trum (e.g., Briscolini and Santangelo, 1994),

E(k) =

where &o = 1 and C is a constant which normalizesthe initial total energy to be 0.5. In this study, LESare implemented under two different flow conditions.One is characterized by Taylor microscale Reynoldsnumber Re* « 260, the integral scale Reynolds num-ber Ret « 2400, and the large-eddy turnover time

T fa 3.7; the other is characterized by Re\ « 80,Ret « 220, and r « 4 (here, those values are es-timated in an approximate manner as described inKim & Menon, 1995b). The simulations have beenrun for 27 and 25 large-eddy turnover times, respec-tively. To ensure statistical independence, 20 fieldsare used in statistical analysis for both cases (i.e., thetime interval between successive fields is larger than(or at least same as) one large-eddy turnover tune).

The temporal evolution of the mean turbulent ki-netic energy is investigated (not shown here). Afteran initial decaying period, the mean turbulent kineticenergy remains at almost the same level, reflecting abalance between forcing at the large scales (the en-ergy injection rate) and dissipation at the small scales(the energy dissipation rate). Only this energy equi-librium period of time is used in statistical analysisbecause it is closer to a statistically steady state.

Figure 3(a) shows the probability distribution ofvelocity differences, 5tt(r) = u(x+r) — u(x), for vari-ous values of r (note that all values of r used here arecomparable with the inertial range scales). For gen-erality, Su is normalized so that a2 = (&*2} — 1- TheLES results (using the LDKSGS model at Re* « 260)clearly show that the distribution changes from a non-Gaussian (which has the wings) to a Gaussian, as rincreases. The same behavior of the distribution wasobserved in the high resolution DNS of Vincent andMeneguizzi (1991). In addition to the basic agree-ment regarding the development of the non-Gaussianstatistics, the LES accurately predicts the probabil-ity for each bin. Figure 3(b) shows an agreementbetween the distributions for r = 0.39 obtained fromthe LES and the DNS except for some deviation inthe wing region. However, as is well known, the wingsof the non-Gaussian distribution develop mainly dueto small-scale fluctuations. Therefore, the deviationbetween the LES and the DNS results in the wingsis somewhat natural; since in LES, most small scalesare not resolved and even the resolved portion of smallscales lies under strong influence of the top-hat filterimplicitly implemented in the finite-difference code.

The statistics of velocity and its derivatives are alsoinvestigated. While the statistics of velocity are theproperty of the large scales which are mostly resolvedin LES, the statistics of velocity derivative are theproperty of the dissipation range scales which are notresolved by LES. Therefore, the direct comparison ofLES and DNS using the statistics of velocity deriva-tive may be meaningless. A more useful comparisoncan be achieved by filtering the DNS field downtoihesame resolution as the LES. For the same grid reso-lution and flow conditions, the statistics of the LESand the filtered DNS should match well. However,

the velocity derivative statistics of the filtered DNSdata are not available, therefore, the DNS statisticsof velocity derivative obtained from the full resolu-tion simulation (shown in the table I) is being usedonly as a qualitative measure for the LES results. Wecomputed the nth-order moments of the velocity andits derivative distributions using

(12)

here, {•) denotes ensemble-averaging. The results ofthis calculation is summarized in table 1. The re-sults of the 5123 DNS (Rex « 170), the 2403 DNS(Rex w 150), and the 643 LES (Rex « 140) areobtained from Jimenez et aL (1993), Vincent andMeneguizzi (1991), and Briscolini and Santangelo(1994), respectively. (Note that different authors usea definition of Rex in different form, however, we usethe original value provided by the authors withoutany correction.) The 64s LES was implemented usingthe Kraichnan's eddy viscosity where the small scalesare parameterized reproducing a self-similar range ofenergy in spectral space. We simulated two differentReynolds number cases using the same grid resolu-tion to investigate the effect of the Reynolds numberon the statistics. As shown in the table, the veloc-ity statistics appear not to depend on either the gridresolution or the Reynolds number simulated. How-ever, the velocity derivative statistics were highly in-fluenced by the grid resolution employed (it can beobserved from the table that those values of the ve-locity derivative statistics are consistently decreasedas the grid resolution becomes coarse from 5123 to323). In the LES, the effect of the Reynolds num-ber on the velocity derivative statistics is not cap-tured (in the DNS of Jimenez et aL, 1993, consis-tent increase in the velocity derivative statistics withReynolds number increase was observed), since thevelocity derivative statistics are strongly determinedby the grid resolution employed.

Figures 4(a) and (b) show the temporal evolution ofthe model coefficient CT and the dissipation model co-efficient Ce (locally evaluated coefficients are volume-averaged for quantitative presentation). The LD-KSGS model predicts Cr « 0.056 and Ce « 0.33for higher Reynolds number case (Rex « 260) andCT « 0.05 and Ce « 0.44 for lower Reynolds num-ber case (Rex « 80). These values for Cr are wellmatched with that suggested by Yoshizawa & Horiuti(1985); they recommended Cr « 0.05 from the frame-work of the two-scale direct-interaction approxima-tion (TSDIA). (Note that, in the Reynolds-averagedturbulence models, a generally adopted value for CTis about 0.09; that is significantly larger than that for

LES.) However, there are some discrepancies betweenCe values dynamically determined and suggested byYoshizawa & Horiuti (Ce «1). Irom the observationof these figures, it can be roughly concluded that inLES of (forced) isotropic turbulence using the fixedgrid resolution, a larger value of the model coefficientCT and a smaller value of the dissipation model coeffi-cient Ce is required as higher Reynolds number flowsare simulated.

3.3 TURBULENT MIXING LAYERAlthough it is generally accepted that turbulent mix-ing layers achieve self-similarity after initial develop-ment, there is still a lack of agreement (between dif-ferent simulations or experiments) on the asymptoticgrowth rate and the turbulence properties in the self-similar period. The discrepancy becomes more severehi the numerical simulations (both DNS and LES) oftemporally evolving turbulent mixing layers since theevolution of the flows is very sensitive to the initialstate not only in the physical aspect but also in thenumerical aspect. That is, the evolution of the flowsvaries greatly in simulations using different numericalschemes, grid resolutions and turbulent models, eventhough extremely accurate schemes and models areemployed. Especially, since in LES, the SGS modelsare devised to predict the effect of the subgrid-scaleturbulence on the large-scale turbulence, they are notcapable of properly handling the artificial (somewhatunrealistic) turbulence prescribed in the initial field.Hence, the initial disturbance fields at different gridresolutions even obtained in a physically and a numer-ically consistent manner (such as by taking successivefiltering to a full-resolution field) are recognized bythe SGS model as being different initial states. Thisimplies that the LES would fail to provide resultswhich can be directly compared to the DNS data orthe LES results at different resolutions. The onlymeaningful comparison of time accurate results canbe achieved by a scaling using parameters which canremove the initial condition effects. This issue is ad-dressed below.

Generally, turbulent mixing layers are consideredself-similarly evolving if they grow linearly and theshapes of the mean velocity and turbulence pro-files are independent of time (or location in spatiallyevolving mixing layers) when scaled by the tune-dependent (or local in spatially evolving mixing lay-ers) momentum thickness Sm and the velocity dif-ference AC7. Itieniporally evolving mixing layers,the growth rate is generally scaled by the initial mo-mentum thickness 6^ and the velocity difference A£7.This scaling is typically adopted only for the pur-pose of nondimensionalization, hence, time-accurate

evolution can be obscured by the effects of the ini-tial conditions. To extract time-accurate information,the principal effects of initial development variabilityshould be removed. This can be achieved by choos-ing appropriate scaling parameters in the self-similarperiod and using a reasonable reference point at theinstant when self-similar evolution starts (i.e., whenthe mixing layer begins to grow linearly). However,both physically and numerically, it is difficult to pre-cisely estimate this starting point of self-similarity. Inthe present study, this point is estimated by numer-ical investigation. First, the points where the first,second, and third derivatives of momentum thicknessevolution curve have extreme (maximum and min-imum) values, are sought. And then, these pointsare used as reference points for the momentum thick-ness growth rate scaling. It is found that the pointwhere the second derivative of momentum thicknessevolution curve has a maximum value, provides themost reliable time-accurate results. The present scal-ing can be described as follows,

5-t5) (14)"m

where superscript * denotes scaled variables and su-perscript S indicates the scaling parameters at theself-similarity starting point. Subtractions by S^and fs are implemented to assign zero scaled tuneand scaled momentum thickness at the self-similaritystarting point. This scaling is applied to both DNSand LES data of temporally evolving mixing layersand its capability in generating tune-accurate resultsis demonstrated below.

Vreman et aL (1994b) simulated the temporal,weakly compressible (M=0.2) mixing layer in a cu-bic domain using 1923 grid resolution. In their study,the length of the domain is 29.55° (where 5° denotesthe initial vorticity thickness) which is correspond-ing to four times the wavelength of the most unsta-ble mode as predicted by linear stability theory atM=0.2. Periodic boundary conditions are imposed hithe streamwise (x) and spanwise (z) directions, whilein the cross-stream (y) direction the boundaries areassumed as slip walls. The initial velocity field isthe hyperbolic-tangent profile, u = At/tanh(2y/5^),on which is superimposed a three-dimensional large-amplitude eigenfunction disturbance obtained fromlinear stability analysis (Sandham & Reynolds, 1991).For the present study cf the iemporalrjrevolving in-compressible mixing layer, the initial LES field is ob-tained by filtering the initial DNS field (generated us-ing Vreman's code) for three different lengths of the

domain, 28.35® (which corresponds to four tunes thewavelength of the most unstable mode as predictedby linear stability theory at M=sO), 505°,, and 100 .̂Table 2 summarizes the test condition and the initialand self-similar starting point Reynolds number.

Figure 5(a) shows the evolution of the unsealed mo-mentum thickness as a function of the unsealed time.According to the plot, all four simulations seem toevolve differently, hence, the direct comparison of theflow properties at the same (unsealed) time resultspoor agreement (not shown here). The scaling de-scribed in (13) and (14) is applied to these data andthe results are plotted in figure 5(b). Now, curvesfor all three LES runs agree very well each other (theLDKSGS model seems to work consistently well forthe relatively coarser, same resolutions but larger do-main sizes, grids) but overpredict the growth rateover the DNS curve slightly. This difference betweenLES and DNS confirms the well-known feet that com-pressibility reduces mixing layer growth rate (Sand-ham & Reynolds, 1991). Figure 5(b) also includesthe asymptotic growth rate (slope) obtained from theexperiment of Bell and Mehta (1990) for a spatiallyevolving mixing layer begun from turbulent (tripped)splitter-plate boundary layers. The growth rates pre-dicted by LES runs agree very well with the experi-ment.

The mean velocity profiles of the LES Run 2 at fivetunes during the entire period after t* = 0 are plot-ted with self-similar scaling using the tune-dependentmomentum thickness 6m and the velocity differenceAJ7 hi figure 6. Also included are the experimentaldata of Bell and Mehta (1990) (the experimental pro-files have been shifted to center them at y = 0). Thecollapse of the data at the five times is excellent, andthe mean profile agrees very well with the data of Belland Mehta.

Figure 7 shows the Reynolds stress profile uu scaledusing self-similar parameters at t* — 110. The over-all agreement between the DNS and the LES runsis quite good. Interestingly, the LES Runs 2 and 3agree much better with the DNS than the LES Run1, although their computational domain lengths com-puted using initial parameters agree less with that ofDNS (see table 2). The poorer results of the LESRun 1 may stem from the fact that the computa-tional domain of the LES Run 1 is not large enoughto encompass the same structures resolved in theDNS at the chosen instant, even though both sim-ulations start with the initially identical computa-tional domain sizes. Therefore, from the above re-sults, it can be concluded that the simulation param-eters scaled by the self-similarity stating point param-eters are more useful than those scaled by the initial

parameters especially for comparing time-accurateDNS/LES results.

Figures 8 (a) and (b) show the evolution (with re-spect to the scaled time) of the model coefficient CTand the dissipation model coefficient Cc computedfrom the LES Run 2. Averaging over the two homo-geneous directions renders the coefficients as a func-tion of tune and the normal direction y. Five loca-tions are chosen between the center plane (y = 06^)and the lower boundary plane (y = —22.98^). Thevolume-averaged coefficients are also included. Att* = 0, both coefficients (especially, Ce) computed onthe center plane (y = 0) increase rapidly indicatingthat turbulence starts to decay by the viscous damp-ing and realistic turbulence is about to develop. Thisfact confirms that the self-similarity starting point isaccurately determined. The Ce values computed onthe planes other than the center plane (y = 0) are un-physically large during the initial period since the ini-tial disturbances are applied to very thin layer aroundthe center plane, however, these values rapidly de-crease as the mixing layer grows.

4 CONCLUSIONSIn this paper, the LDKSGS model has been success-fully applied to LES of high Reynolds number flowssuch as decaying and forced isotropic turbulence, andtemporally evolving turbulent mixing layer. The ca-pability of the LDKSGS model in predicting the en-ergy decay rate has been demonstrated by simulatingdecaying isotropic turbulence and comparing the re-sults to the experimental data (the LES results con-firmed the power law decay which was observed in theexperimental data). Furthermore, three different res-olutions LES (at the coarsest resolution, about a halfof the kinetic energy was not resolved) showed consis-tency in predicting the energy decay. This propertyof the LDKSGS model is very attractive, especiallywhen the model is (to be) applied to complex and highReynolds number flows where a significant amount ofturbulent energy possibly lies in the unresolved scales.The application of the present model to forced (sta-tistically stationary) isotropic turbulence also provesthe capability of the LDKSGS model in reproducingthe statistics (of the large scale structures) of a realis-tic, high Reynolds number turbulent field. The LESresults, when compared to the high resolution DNSdata, clearly show the accurate prediction of velocitystatistics and the development of the non-Gaussianstatistics (which was observed in the high resolutionDNS). The capability of the LDKSGS model in pre-dicting the time-accurate results has been further in-vestigated by simulating the temporally-evolving tur-

bulent mixing layer which is dominated by coherentstructures and evolves from the linear stability regimeup to a well developed turbulent flow field. Using thescaling which is capable of separating the distinct ef-fects of initial development on the self-similar stage ofthe mixing layer evolution, it was demonstrated thatthe coarse grid simulations employing the LDKSGSmodel can provide tune-accurate results which agreevery well with existing direct numerical simulationand experimental data.

ACKNOWLEDGMENTThe authors wish to thank Professor B. Vreman

for providing the initial disturbance field generator ofturbulent mixing layer and the data presented in fig-ures 5(b) and 7. This work is supported by the FluidDynamics Division of the Office of Naval Researchunder grant N00014-93-1-0342 (monitored by Dr. P.Purtell). This work was also supported in part by agrant of HPC time from the DoD HPC Centers: theNaval Oceanographic Office (NAVO) and the USAEWaterways Experiment Station (CEWES).

REFERENCESBell, J. H. & Mehta, R. D. 1990 Development of

a two-stream mixing layer from tripped and un-tripped boundary layers. AIAA J. 28, 2034.

Briscolini, M. & Santangelo, P. 1994 The non-Gaussian statistics of the velocity field in low-resolution large-eddy simulations of homogeneousturbulence. J. Fluid Mech. 270, 199.

Comte-Bellot, G. & Corsin, S. 1971 Simple Euleriantime correlation of full- and narrow-band velocitysignals in grid-generated, 'isotropic' turbulence. J.Fluid Mech. 48, 273.

Germano, M., Piomelli, U., Mom, P. & Cabot, W.H. 1991 A dynamic subgrid-scale eddy viscositymodel. Phys. Fluids A 3, 1760.

Ghosal, S., Lund, T. S., Moin, P. & Akselvoll, K.1995 A dynamic localization model for large-eddysimulation of turbulent flows. J, Fluid Mech. 286,229.

Jimenez, J., Wray, A. A., Saffman, P. S. & Rogallo,R. S. 1993 The structure of intense vorticity inisotropic turbulence. J. Fluid Mech. 255, 65.

Kim, W.-W. & Menon, S. 1995a A new dynamic one-equation subgrid-scale model for large eddy simula-tions. AIAA Paper 95-0356, AIAA 33rd AerospaceSciences Mtg, Reno, NV.

8

Kim, W.-W. & Menon, S. 1995b On the properties ofa localized dynamic subgrid-scale model for large-eddy simulations. Submitted to J. Fluid Mech..

Lilly, D. K. 1992 A proposed modification of the Ger-mano subgrid-scale closure method. Phys. Fluids 4,633.

Liu, S., Meneveau, C. & Katz, J. 1994 On the prop-erties of similarity subgrid-scale models as deducedfrom measurements in a turbulent jet. J. FluidMech. 275, 83.

Menon, S. & Yeung, P.-K. 1994 Analysis of sub-grid models using direct and large-eddy simula-tions of isotropic turbulence. Proc. AGARD 74thFluid Dynamics Symp. on Application of Directand Large Eddy Simulation to Transition and Tur-bulence, AGARD-CP-551, p. 10-1.

Menon, S., Yeung, P.-K. &: Kim, W.-W. 1995 Effectof subgrid models on the computed interscale en-ergy transfer in isotropic turbulence. To appear hiComputers Fluids.

Moin, P., Squires, K., Cabot, W. & Lee, S. 1991A dy-namic subgrid-scale model for compressible turbu-lence and scalar transport. Phys. Fluids A 3, 2746.

Piomelli, U. 1993 High Reynolds number calcula-tions using the dynamic subgrid-scale stress model.Phys. Fluids A 5, 1484.

Sandham, N. D. & Reynolds, W. C. 1991 Three-dimensional simulations of large eddies in the com-pressible mixing layer. J. Fluid Mech. 224, 133.

Squires, K. D. & Piomelli, U. 1994 Dynamic model-ing of rotating turbulence. In Turbulent shear flows9 (ed. F. Durst, N. Kasagi, B. E. Launder, F. W.Schmidt & J. H. Whitelaw). Springer-Verlag, Hei-delberg, p. 73.

Vincent, A. & Meneguzzi, M. 1991 The spatial struc-ture and statistical properties of homogeneous tur-bulence. J. Fluid Mech. 225, 1.

Vreman, B., Geurts, B. & Kuerten, H. 1994a Realiz-ability conditions for the turbulent stress tensor hilarge-eddy simulation. J. Fluid Mech. 278, 351.

Vreman, B., Geurts, B. & Kuerten, H. 1994b Onthe formulation of the dynamic mixed subgrid-scalemodel. Phys. Fluids 6, 4057.

Yoshizawa, A. 1993 Bridging between eddy-viscosity-type and second-order models using a two-scaleDIA. Proc. 9th Symp. on Turbulent Shear Flaws,p, 23-1-1.

5123 DNS(Rex«170)240PDNS

(Rex=150)643LES

(Rex«140)323LES

(Rex =260)323LES

(Rex -80)Gaussian

Us<

2.80

2.78

2.80

3.0

s.12.5

11.9

12.1

15.0

du/dx O«/dy)S3

-0.525

-0.5(-0.04)-0.35(0.06)-0.32

(-0.01)-0.30(0.03)0.0

s<6.1(9.4)5.9

(8.0)

(4.5)3.47

(4.87)3.59

(4.93)3.0

ss-12.0

-9

-3.48(-0-11)-3.57(0.23)0.0

S6125

(370)90

23.4(49.9)25.8

(51.3)15.0

Table 1. Higher-order moments for u-velodty and itsgradients. The 5123 DNS, 2403 DNS, and 643

LES results are obtained from Jimenez et al.(1993), Vincent & Meneguizzi (1991), andBriscolini & Santangelo (1994), respectively.

1923 DNS

LES Run 1

LES Run 2

LES Run 3

Domain size

55.98*(29.58°)39.58*

(28.38°)45.78*(508°)49.88*(1008°)

A£/8*V

106

143

219

402

A&8°V79

78

113

201

AP5°,V

200

200

200

200

Table 2. Temporally evolving mixing layer simulationparameters. The 1923 DNS results is obtainedfrom Vreman et al. (1994b).

0.10

0.08 -

0.06 -

0.04 -

0.02 -

0.00

—— LOKSOS (481.32'. 24*)••—•• Genuano aaL'* model (48*)

• Experiment (48*)• Experiment (32*)A Experiment (24*)

Time

Figure 1. Decay of turbulent kinetic energy resolved inLES; compared to the experimental data byComte-Bellot & Corsin (1971).

10"

OExpenmemO .

- LDKSGS(24')-• Gemano «al'« model (4S3)

10' itf

Figure 2(a). Energy spectra predicted by LES att' =4.98; compared to the experimental data byComte-Bellot & Corsin (1971).

E(k)

10- -

ID''

r.!&0s;

OExpcrimem——— LDKSOS(4S1)••••••» LWCSGSC521)

— - - Gemano a aL's model (48J)

10° 10' 102

Figure 2(b). Energy spectra predicted by LES att* =8.69; compared to the experimental data byComte-Bellot & Corsin (1971).

10°

Jio-3

f 10-

•10"W

w*

r»0.20.U>ICSGS(32>)

-1—i—1—•—I—i—l_-12 -10 -8 -6 -4 -1 0 2 4 6 8 10 12

Normalized velocity difference

Figure 3(a). Probability distribution of normalizedvelocity difference for five different scales (r)predicted by 323 LES at Re*. = 260.

10"

ID'1

1

•10"

10"

teJ.«260.LDKSGSC32*)

Rc1«l30,DNS(240^ ]

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12Normalized velocity difference

Figure 3(b). Probability distribution of normalizedvelocity difference for r = 0.39; compared to theDNS results by Vincent & Meneguizzi (1991).

0.07

0.06

0.05

0.04

0.03

Rex*80,LDKSOS(32>)

20 40 60Time

100

Figure 4(a). Time evolution of the model coefficientsdetermined from 323 LES at two differentReynolds numbers.

Time

Figure 4(b). Time evolution of the dissipation modelcoefficients determined from 325 LES at twodifferent Reynolds numbers.

10

25 -

O192'DNSofVrcman«iil32'LESRunl32'LESRun2

--- 32'LESRun3

100 200 300 400

Rgure 5(a). Time evolution of the momentumthickness in unsealed coordinates.

• 3

OI92'DNSofV[em»«<li.......... 32'LESRunl——— 32'LESRun2— - - • 32'LESRun3—— — Experime* of Bell md Meha (slope)

-40 40 80 120 160

Rgure 5(b). Time evolution of the momentumthickness in scaled coordinates using self-similarity starting parameters.

0.25

< 0.00

-0.25

-030

32'LES(T-0)32ILES(1»IO)

-32'LES(T-80>

• Experiment C»108.1 cm) .• Experiment (TK 1 28.4 cm)AExperiraenl (» 1 89.4 on)

-5JO 0.0 5.0

Hgure 6. Mean streamwise velocity profiles in self-similar scaled coordinates compared with theexperimental data of Bell and Mehta (1990).

aoo

-OJOl

-0.02

-0.03

0192' DNS of Vraran aaL— SZ'LESRunl— 32>LESRun2— - 32>LESRun3

-5.0 -IS 0.0 5.0

Rgure 7. Comparison of the time-accurate simulationresults in self-similar scaled coordinates for theReynolds stress component «v at f *=110.

0.08

aoo-40 80 120 160

Rgure 8(a). Time evolution of the plane-averagedmodel coefficients at five different locations alongthe normal direction.

1.0

g0-8

3 0.6

§ 0.4

1a 0.2

0.0

/ -OS*

-w 40 f 80t"

120 160

Rgure 8(b). Time evolution of the plane-averageddissipation model coefficients at five differentlocations along the normal direction.

11