subgrid mixing and molecular transport modeling in a reacting shear layer

59

Twenty-Sixth Symposium (International) on Combustion/The Combustion Institute, 1996/pp. 59–66

SUBGRID MIXING AND MOLECULAR TRANSPORT MODELING IN AREACTING SHEAR LAYER

SURESH MENON and WILLIAM H. CALHOON JR.School of Aerospace EngineeringGeorgia Institute of Technology

Atlanta, Georgia 30332-0150, USA

The importance of molecular transport in high–Reynolds number reacting flows is now well acceptedbecause of incontrovertible experimental evidence. However, most mixing models neglect this small-scaleprocess and, as a result, fail to predict the observed variation of the mean mixed composition and theReynolds number, Schmidt number, and Damkohler number dependence observed in the experiments. Amixing model that explicitly accounts for the small-scale mixing, molecular transport, and chemical kineticsprocesses was developed earlier as a stand-alone method by Kerstein. Here, this linear eddy model (LEM)is implemented as a subgrid model within a configuration-independent simulation methodology based onlarge eddy simulation (LES) and used to study scalar mixing in a low–heat release mixing layer. The abilityof the present method to capture correctly the small-scale mixing and molecular transport effects is dem-onstrated by carrying out both qualitative and quantitative comparisons with experimental data obtainedin high–Reynolds number flows. The present method provides a unique modeling capability to handle thesmall-scale processes within the context of a general LES approach without any ad hoc closure of thescalar equations.

Introduction

Although the importance of large-scale structuresin scalar mixing and chemical reactions has beenknown for some time, it is only recently that exper-iments [1–5] have proven that molecular transporteffects cannot be neglected even for high–Reynoldsnumber flows. This observation explains why modelsthat neglect molecular transport fail to predict theobserved features in turbulent shear layers such asthe variation of mean mixed composition and theReynolds number (Re), the Schmidt number (Sc),and the Damkohler number (Da) dependence.

A model that incorporates both large-scale andmolecular diffusion effects into scalar mixing was de-veloped by Kerstein (called the linear eddy model)and was applied in a stand-alone mode to scalar mix-ing and chemical reactions in shear layers and jets[6–9]. In this implementation, the formulation in-corporates the entrainment of air parcels into themixing region, their breakdown by turbulent eddies,molecular mixing of fuel and air, and chemical ki-netics using a stochastic simulation in a one-dimen-sional domain that may represent the jet centerline[8,10,11] or an axially advancing radial line in theshear layer [6,7]. Studies have shown that this modeldeals explicitly with the Re, Sc, and Da effects andcan predict correctly the experimental trends in bothgaseous and liquid shear layers.

A limitation of the stand-alone approach is that theflow-specific processes such as air entrainment,

shear layer spreading, and so forth must be incor-porated using inputs based on experimental or em-pirical data. This precludes application of this ap-proach to more complex flows where the large-scaleprocesses are configuration dependent. An approachthat combines the advantage of this linear eddymodel (LEM) with a deterministic simulation of thelarge-scale processes was developed recently [12,13]for large eddy simulations (LES). In LES, all scalesof motion larger than the grid resolution are com-puted explicitly and the unresolved small scales aremodeled using an eddy viscosity model. Recent ad-vances include the development of dynamic subgridmodels [14,15] (in which the constants of the modelare determined as a part of the simulation). Eddyviscosity models are reasonable for momentum/en-ergy closure since the small scales are known to pro-vide a mechanism for energy dissipation and trans-fer. However, because of the subtle interactionbetween turbulent stirring, molecular diffusion, andchemical reactions at the smallest scales, a similar(i.e., eddy diffusivity) model for the scalar fieldwould be seriously flawed. Here, a subgrid combus-tion model (based on LEM) is implemented withinthe LES cell to simulate locally the small-scale mix-ing and molecular transport processes. Preliminaryevaluation of this method was done earlier for bothdiffusion and premixed flames [12,13]. In this study,both qualitative and quantitative comparison withdata obtained in a low heat-release turbulent shear

60 NON-PREMIXED TURBULENT COMBUSTION

layer is carried out to demonstrate the viability ofthis approach.

Model Formulation

The formulation of the simulation model is sum-marized briefly here. More details are available else-where [12,14,16] and, therefore, are omitted herefor brevity.

The Subgrid Closure for Large-Scale Transport

The LES equations are obtained by applying aspatial filter (top-hat) to the compressible Navier–Stokes equations. These equations contain unre-solved terms such as the subgrid stresses in the mo-mentum equation 4 1 uiuj], the subgridsgs ;s q[u uij i jenergy flux and the viscous work terms in the energyequation 4 1 Eui] 1 1 pui] and; ;sgsH q[Eu [pui i i

4 1 ui ij], and the scalar correlation sgssgsr [u s s wi i ij k

4 1 TYk in the state and the energy equations.;

TYkHere, tilde indicates Favre filtering and bar indicatesconventional filtering. Also, ui, and E are, re-q, s ,ijspectively, the resolved velocity, density, viscousstresses, and total energy per unit volume. To close

and , a modeled equation for the sub-sgs sgs sgss , H rij i i

grid kinetic energy ksgs 4 1 uiui is solved along;u ui iwith the LES equations [16]. The subgrid eddy vis-cosity is then determined as mt 4 Cv Dg, where1/2sgskDg is the characteristic grid size. The coefficients ap-pearing in this closure are determined using a dy-namic procedure, whereby they are computed lo-cally (in space and time) during the simulation. Thedetails of the dynamic procedure have been reportedelsewhere [14] and therefore are omitted here.

With ksgs known, the unknown subgrid terms areclosed using the following relations: 4 (Sij

sgss 12qmij t1 Skkdij /3), 4 /rk)(]H/]xi) and 4sgs sgsH 1(qm ri t ijuj . Here, Prt 4 1 is the turbulent Prandtl num-sgssijber, Sij 4 (]ui /]xj ` ]uj /]xi)/2 is the resolved rate-of-strain tensor, and H 4 CpT ` 0.5uiui ` ksgs isthe total enthalpy. It should be noted that all third-order correlations are neglected in the precedingformulation.

The Subgrid Closure for Scalar Transport

To model reacting flows conventionally, the fil-tered species conservation equations of the form

˜]qY ]k sgs ¯˜ ˜` [qY (u ` V ) ` S ] 4 x (1)k i ik ik k]t ]xi

must be solved. Here, Yk, Vik, and are the kthxkspecies mass fraction, the ith component of the dif-fusion velocity, and the production term, respec-tively. Assuming Fickian diffusion, the diffusion ve-locity can be approximated as Vik 4 1(Dk/Xk)]Xk/]xi, where Xk and Dk are, respectively, the kth species

mole fraction and diffusion coefficient. The closurefor 4 1 uiYk), Vik, and is prob-;sgs sgs¯S q(u Y x , wik i k k klematic because to estimate these terms, the small-scale turbulent stirring and molecular transport mustbe modeled accurately. The new subgrid model wasdeveloped to address primarily the closure of theseterms.

In the subgrid combustion model, the speciesequations, Eq. (1), are not solved with the LESequations. Rather, between each LES time step,Dtles, four processes occur within each grid cell:small-scale turbulent stirring, molecular diffusion,chemical kinetics, and thermal expansion. Theseprocesses are governed by their respective timescales and are simulated on a one-dimensionasl do-main within each grid cell. This one-dimensional do-main is an instantaneous slice through the local(three-dimensional) scalar field and is a space curvealigned with the local scalar gradients [6–8]. Here,the length of this domain is approximated as Dg, al-though it can be larger. The resolution within thisdomain is chosen to resolve the smallest scale (forgases, Sc 4 0.7 and so the Batchelor scale is largerthan the Kolmogorov scale, gk). A minimum reso-lution of six subgrid cells is used to resolve gk, de-termined using the scaling

3/4 sgs 3/4D /g 4 Re 4 ( k D /m)!g k D g

which was found sufficient. Molecular transport (andchemical production) is incorporated by the finite-difference solution (using zero-gradient boundaryconditions) of the reaction-diffusion equationswithin the one-dimensional domain: ]Yk/]t 4 1[xk](qYkVk)/]s]/q, where s is the subgrid spatial coor-dinate. For finite-rate kinetics, a one-dimensionalthermal diffusion equation is also solved in the sub-grid [11,16]. The diffusion and chemical time scales,Dtdiff and Dtchem for this integration are chosen tosatisfy the local stability criteria.

Turbulent stirring by eddies ranging from thesmallest (gk) to the largest (Dg) is implemented asrandomly occurring (at a local time interval Dtstir)events that interrupt the ongoing reaction-diffusionprocesses. These events represent the effect of tur-bulent eddies on the scalar field and are imple-mented using the “triplet” mapping [8–11]. Threeparameters are needed for this mapping: the eddysize l (determined randomly from a length scalePDF, f(l), gk # l # Dg), the mapping frequency km('1/DtstirDg) and the mapping location (also deter-mined randomly on the one-dimensional domain).The details of how these parameters are determinedhave been reported in many cited references and aretherefore omitted. However, note that, earlier, allconstants in the scaling relations used to determinethese parameters were set to unity. Earlier, to com-pare with DNS results, the stirring time Dtstir was

SUBGRID MIXING IN A REACTING SHEAR LAYER 61

rescaled by relating the LEM diffusivity to the largeeddy turn over time in the DNS [17]. In the presentcase, the time scales Dtdiff and Dtstir are rescaled us-ing a single constant to reproduce the experimentallyobserved [1,2] Reynolds number-variation of theproduct thickness (see below).

Within the subgrid, volumetric expansion is alsoincluded [12] to account for heat release. This is im-plemented as an expansion of the LEM elements toaccount for temperature rise and results in a de-crease in the local scalar gradients near the flamezone. Although this feature is implemented for thepresent study of shear layers with negligible heat re-lease, no thermal expansion is required.

From the temporal evolution of the local subgridfield, the joint scalar PDF P(T,Yk) can be computedand used to calculate the local and . The sub-sgs¯x wk kgrid approach also obviates the need for a closure ofVik and , since the former is exactly included insgsSikthe diffusion equation and the latter is implicit in thelocal stirring process. However, to couple the sub-grid processes and the large-scale evolution (com-puted by the LES equations), two additional pro-cesses must be incorporated. These processes arethe large-scale convection of the scalar fields (due tothe resolved velocity field) and the effect of subgridevolution on the large-scale resolved fields. Thelarge-scale convection of the scalar fields from oneLES cell to another is accomplished by the splicingalgorithm [12,13], which is implemented at a con-vective time scale Dtconv (based on the local resolvedvelocity). The splicing algorithm essentially modelsthe convective terms in the scalar conservation equa-tions and involves (1) calculating the volume transferacross the LES cell faces caused by both the resolvedvelocity (ui) and the fluctuating velocity field ( 'u8i

), (2) choosing the subgrid elements to besgsk!spliced, and (3) splicing the chosen elements acrossLES cell boundaries into the neighboring subgriddomains.

Although details are given elsewhere [12,16],some comments are worth noting. In step 1, the vol-ume transfer is computed so that no mass conser-vation errors and no nonphysical upstream propa-gation of the species occurs. In step 2, the subgridelements are chosen to ensure that the splicing al-gorithm recovers laminar convection when ksgs iszero. This is because the convective transport of sca-lars across LES cell surfaces is only modified by tur-bulent fluctuations. Finally, in step 3, to avoid spu-rious diffusion, the “age” of the spliced elements istracked using an inflow-outflow process, wherebyonly the older elements (in the LES cell) are eitherspliced out or truncated (since the splicing processcan increase locally the elements in the LES cell).Detailed discussion regarding spurious diffusion andmethods used to avoid this phenomenon is givenelsewhere [12,13,16] and is therefore omitted herefor brevity.

The splicing algorithm is key to the LEM imple-mentation as a subgrid model in LES since it ensuresproper convective transport of the scalar fields. Itdepends only on the local velocity field and not onthe magnitude or gradient of species that are beingtransported. Thus, this allows scalar transport with-out false numerical diffusion typically associatedwith conventional finite differencing of the convec-tive terms in Eq. (2). In addition, since gradient dif-fusion approximation is completely avoided, the ef-fects of small-scale molecular transport andturbulent stirring effects are included implicitly inthe convected scalar fields.

The splicing algorithm incorporates the large-scaleeffects on the subgrid fields. The effects of the sub-grid evolution on the large-scale (LES) motion areincluded by using Yk and computed by filteringsgswkthe subgrid field in the LES equations. Note thatthe resolved temperature T can be obtained fromthe LES energy equation and, also, by filtering thesubgrid temperature field. However, since all theterms in the energy equation (such as viscous work)are not included in the subgrid simulation, T is de-termined from the energy equation and comparedto the subgrid prediction. Significant disagreementbetween these two predictions would indicate thatthe assumptions used in the subgrid thermal equa-tion (e.g., no viscous work and constant pressure) arebeing violated and serves as a measure of the appli-cability of the current implementation.

Results and Discussion

The LES equations are solved using a high-orderfinite volume scheme. The inviscid fluxes are ob-tained using a fifth-order scheme [16]. The viscousstresses, heat flux, and all other derivatives are ap-proximated using a fourth-order scheme, and thetime integration is carried out using a second-orderRunge–Kutta scheme. The reaction-diffusion equa-tions in the subgrid field are solved using a second-order scheme on an equally spaced grid. The finite-volume scheme was validated by carrying out severalDNS simulations, and the LES implementation(with the dynamic subgrid model) was validated bycomparing with DNS results of decaying isotropicturbulence. The subgrid implementation was veri-fied by comparing with an exact solution of the dif-fusion equation, and the subgrid accuracy waschecked by comparing with results obtained using amore expensive sixth-order compact scheme. Thesplicing algorithm was used to reproduce accuratelya pure convection problem thereby validating thisimplementation. Because space limitations, these re-sults are not shown here.

The experimental results [1–5], obtained in a low-heat release shear layer, are used for both qualitativeand quantitative evaluation. In the near field of the


Fig. 1. Mean streamwise velocity profile plotted in asimilarity coordinate, y/dm, where dm is the momentumthickness. Experimental data are from Ref. 19 for a trippedspatial shear layer measured at three different locations.

shear layers studied in these experiments, the three-dimensional effects were primarily confined to thesmall scales, and the large scales were primarily twodimensional. This near-field evolution is studiedhere using LES of a temporally evolving two-dimen-sional mixing layer with the new subgrid model.Since the scalings used in the LEM are known tocapture three-dimensional mixing effects [6–9], thisapproach should properly characterize (if the modelis correct) the near field of a spatially developingshear layer. In contrast, two-dimensional DNS hasbeen shown unable to properly capture the three-dimensional small-scale effects [18]. To simulate thebreakdown of the shear layer in the far field, a fullthree-dimensional LES–LEM approach is needed;however, the far field is not the focus of this study.

Since the simulated mixing layer contains only di-lute species, several simplifying assumptions are pos-sible. Assuming equal diffusivity of all species allowsthe mixing process to be described completely interms of the mixture fraction n, which is the nor-malized mass fraction of the atomic species in thefuel stream and is, therefore, unity in the fuel streamand zero in the oxidizer stream. The LEM subgriddiffusion equation for n becomes ]n/]t 4 []{qD]n/]s}/]s]/q, where the diffusion coefficient D 4 m/Sc.The unity Lewis number assumption is also invokedto simplify the closure of the energy equation.

An irreversible reaction F ` O → P is studied [1]with negligible heat release. This eliminates the tem-perature equation from the LEM formulation, andthe subgrid temperature is specified as the LES pre-dicted temperature T. As a result, the subgrid cor-relation is zero. The capability of the presentsgswkapproach is demonstrated here using infinite-ratechemistry. For infinite-rate chemistry, the productformation can be calculated from n by assumingcomplete conversion of all reactant species. Finite-

rate kinetics (Da effects) has also been studied [16]but is not reported here because of space limitations.

The computational domain is 2p 2 2p ` ys withperiodic boundary conditions in the streamwise di-rection and slip walls in the normal direction. Uni-form grid spacing is employed in both directions inthe (2p)2 domain and then the grid is stretched tothe walls (located at 5(2p ` ys)/2) to minimize thewall effects. The resolution in the uniform grid re-gion is 128 2 128 with an additional 20 points usedin each stretched region. A subgrid resolution of300–600 cells, depending on the problem, was foundto provide adequate resolution of gk.

The resolved mean velocity field is initialized by ahyperbolic tangent profile [û& 4 0.5U0 tanh(y/a),^m& 4 0] and an isotropic velocity field perturbationthat decays as a k12 energy spectrum in wave num-ber (k) space. Here, ^ & indicates the mean value anda 4 0.5dx,0, where dx,0 4 U0[]û&/ is the initial11]y]maxvorticity thickness. A value of a 4 0.1496 is chosenthat causes the third mode (from linear stability) tobe the most unstable. For a characteristic velocity ofU0 4 71 m/s and an initial temperature of 300 K,Rex,0 4 U0dx,0 /m 4 7000, which is sufficiently highto be beyond the mixing transition (characterized bythe appearance of small-scale three-dimensionalmotion that significantly increases mixing and prod-uct formation) [5]. Thus, the capability of the LEMsubgrid model to capture the essence of three-di-mensional small scales can be evaluated.

The subgrid kinetic energy is initialized as ksgs ùCkkres, where kres is the test filter (42Dg) energy andCk 4 0.45 (obtained from scale similarity analysis instationary isotropic turbulence [14]). Since heat re-lease is neglected, the subgrid density is set to theLES filtered value and the subgrid n is initializedconstant in each LES cell and equal to a mean value^n& 4 0.5(1 ` tanh(y/a)).

Large-Scale Velocity Field Evolution

As noted in Ref. 1, for negligible heat release anddilute reactants, the fluid dynamic becomes decou-pled from the scalar evolution. The velocity fieldevolves as expected from the initial state with themost unstable mode initially growing linearly andthen finally saturating at around s 4 tU0/dx,0 ' 20.Subsequently, the primary vortices begin to pair asthe second mode becomes more energetic. Thestreamwise velocity field evolves self-similarly as ex-pected for unconfined mixing layer (note that theslip walls are located a considerable distance fromthe shear layer). Figure 1 shows the plot of û& acrossthe layer in terms of the similarity coordinate y/dm.Here, dm is the momentum thickness. The predictedmean velocity is shown averaged for s $ 5 and com-pares very well with the experimental data [19] ob-tained in a spatial mixing layer.


Fig. 2. The normalized mean product mass fraction ^Yp&/Yp,st as a function of y/dPm at s 4 tUo/dx,0 4 19. The mixedprobability thickness, dPm is defined as the width of theregion ^Pm& $ 0.99, where ^Pm& is the mean mixed fluidprobability defined [5] as ^Pm& 4 (n)&dn. Here, e11e sgs* ^pe

4 0.033 and ^psgs(n)& is the mean subgrid mixture fractionPDF calculated from all the subgrid fields at a particulartransverse station. The pure mixed fluid outside the rangee # n # 1 1 e is excluded from this integration.

(a)

(b)

Fig. 3. Trace plots of filtered product thickness (nor-malized by its maximum) at eight transverse locations inthe mixing layer. Arrows indicate the ramplike structures.(a) LEM-predicted product thickness for f 4 1/8; (b) ex-perimental data for normalized temperature for f 4 1/8[1].

Small-Scale Scalar Field Evolution

Simulations were performed for different concen-trations of the reactants in the fuel and oxidizerstreams specified here in terms of the equivalenceratio f 4 YO,0 /YF,0 [1]. Figure 2 shows the normal-ized mean product mass fraction ^Yp&/Yp,st across thelayer at s 4 19. Here, st indicates stoichiometricconditions. The computed profiles exhibit severalobserved trends: the asymmetry for f ? 1, the skew-ness toward the lean reactant side, and the tendencyto become symmetric as f → 1. The asymmetry re-sults from the vertical shift of the flame toward thelean reactant side and was also observed in the con-tour plots of ^Yp& (not shown). Experimental datashowed a higher peak for f 4 8 than for f 4 1/8because of an entrainment ratio of 1.3 [1]. This fea-ture is not captured since the entrainment ratio isunity in the temporal simulation and is not due toany limitation of the model. The mean product massfraction is maximum for f 4 1 and drops off for f? 1 consistent with data [1]. The peak values are inthe range 0.57 # [^Yp&/Yp,st]max # 0.65, in goodagreement with the measured range 0.54 # [^Yp&/Yp,st]max # 0.67.

Quantitative comparison of the mean productmass fraction at f 4 1 is also shown in Fig. 2. Theexperimental data have been calculated from themeasured mean temperature (as noted in Ref. 1).The agreement is quite good for both the peak valueand the shape of the profile even though the Reyn-olds numbers (based on mean mixed probability [1])are slightly different: Rep 4 6.5 2 104 [1] and

ù 4.1 2 104 (present). However, the product for-mation is known [1] to be a slow function of Re (6%decrease for a factor of 2 increase in Re).

The importance of molecular transport can beseen in the filtered mean product field shown in Fig.3a (for f 4 1/8) and Fig. 4a (for f 4 8). For com-parison, the experimental mean temperature dataare shown in Figs. 3b and 4b, respectively. Thesefigures show that at nearly all transverse locations, asignificant amount of product is formed in the largestructures that are separated by entrained freestream fluid (regions with negligible product). Theexperimentally observed ramplike structures in thestreamwise direction (due to the formation ofstreamwise gradients in the mixed fluid composition)have been captured in the simulation. Furthermore,the change in the sign of the gradient (i.e., the prod-uct shifts from the trailing edge to the leading edgeof the primary vortices) when f 4 1/8 is changedto 8 is also captured. These results suggest that thesubgrid LEM accurately reproduces the role of small


(a)

(b)

Fig. 4. Trace plots of filtered product thickness (nor-malized by its maximum) at eight transverse locations inthe mixing layer. Arrows indicate the ramplike structures.(a) LEM-predicted product thickness for f 4 8; (b) ex-perimental data for normalized temperature for f 4 8 [1].

Fig. 5. Averaged mixed mean mixture fraction ^nm&across the layer. Predicted results at s 4 19 and experi-mental data for gases from Refs. 5 and 20. The ^nm& datafrom Ref. 5 were calculated from the mean temperaturedata using an approximate technique [1]. The reliability ofthis technique at the extremities of the layer, where nm isintermittent, is not known.

Fig. 6. Normalized product thickness dp/d2 dependenceon Reynolds number, Re2 4 U0d2/m at s 4 19. Here, d2 isthe 1% thickness of ^Yp&/Yp,st. The experimental data ofMungal et al. [2], which has a slope of 0.05, was used todetermine the calibration coefficient used to rescale thesubgrid stirring and diffusion timescales.

scales, which is to mix the free stream fluid entrainedby the large-scale vortices. This three-dimensionalmixing capability is absent in two-dimensional DNS[18] (and in three-dimensional LES with gradientdiffusion closure), resulting in wrong transverse gra-dients in the mean mixed composition ^nm&.

The variation of ^nm& across the shear layer isshown in Fig. 5. The predicted magnitude and vari-ation of ^nm& are in good agreement with experi-mental data in the central portion of the shear layer.The experimental data at the extremities are not con-sidered reliable because of intermittency effects [5].This figure gives the clear evidence of the impor-tance of correctly modeling the molecular transporteffects since conventional models (that ignore mo-lecular effects) fail to predict the experimentaltrends [4].

Figure 6 shows the variation of product thicknesswith Reynolds number. As noted previously, the sub-grid timescales were rescaled using a coefficient(constant for all simulations) to reproduce this Redependence of product decay. It is worth noting that

a linear variation (in the log plot) of product thick-ness with Re (as seen in the experiments) was ob-tained even when this coefficient was set to unity.The observed Re dependence has yet to be properlyexplained and is still under study. Regardless, thepresent computations appear to capture this trendquite well in spite of the slight overprediction of theproduct thickness that is due to the dependence ofthe mean product profile on the entrainment ratio(noted earlier to be different from the experimentalvalue).


Conclusions

A formulation that employs the LEM as a subgridmodel in a LES has been used to study turbulentnon-premixed combustion in a shear layer. It hasbeen shown that it is possible to properly couple thelarge-scale (LES-computed) processes and thesmall-scale (LEM-computed) processes. The abilityof the present approach to capture correctly thesmall-scale three-dimensional effects has been dem-onstrated using a mixing layer in which the largescales are primarily two-dimensional and the three-dimensional effects are confined to the small scales.Finite-rate kinetics and heat release can be incor-porated into the present formulation without anymajor modifications. Extension to full three-dimen-sional LES can also be accomplished easily but at anadded computational cost. However, this model isoptimal for parallel processing and therefore, fullthree-dimensional LES–LEM may become practi-cally feasible in the near future (an issue currentlybeing addressed). Both qualitative and quantitativeexperimental trends in high Reynolds number shearlayers (that are known to be due partly to moleculartransport effects) have been reproduced. This ap-proach provides a new modeling capability to handlesmall-scale turbulent mixing, molecular transport,and chemical kinetics within the context of a generalLES approach.

REFERENCES

1. Mungal, M. G. and Dimotakis, P. E., J. Fluid Mech.148:349 (1984).

2. Mungal, M. G., Hermanson, J. C., and Dimotakis,P. E., AIAA J. 23:1418 (1985).

3. Mungal, M. G. and Frieler, C. E., Combust. Flame71:23 (1988).

4. Broadwell, J. E. and Mungal, M. G., Phys. Fluids A3:1193 (1991).

5. Koochesfahani, M. M. and Dimotakis, P. E., J. FluidMech. 170:83 (1986).

6. Kerstein, A. R., Combust. Sci. Technol. 60:391(1988).

7. Kerstein, A. R., Combust. Flame 75:397 (1989).8. Kerstein, A. R., J. Fluid Mech. 216:411 (1990).9. Kerstein, A. R., Combust. Sci. Technol. 81:75

(1992).10. Menon, S., McMurtry, P. A., Kerstein, A. R., and

Chen, J.-Y., J. Prop. Power 10:161 (1994).11. Calhoon, W., Menon, S., and Goldin, G., Combust. Sci.

Technol. 104:115 (1994).12. Menon, S., McMurtry, P. A., and Kerstein, A. R., Large

Eddy Simulations of Complex Engineering and Geo-physical Flows, Cambridge University Press, 1993, p.287.

13. McMurtry, P. A., Menon, S., and Kerstein, A. R.,Twenty-Fifth Symposium (International) on Combus-tion, The Combustion Institute, Pittsburgh, 1992, pp.271–278.

14. Menon, S. and Kim, W.-W., AIAA 96-0425, 1996.15. Moin, P., Squires, W., Cabot, W., and Lee, S., Phys.

Fluids A 3:2746 (1991).16. Calhoon, W. and Menon, S., AIAA 96-0516, 1996.17. McMurtry, P. A., Gansauge, T. C., Kerstein, A. R., and

Krueger, S. K., Phys. Fluids A 5:1023 (1993).18. Park, K.-H., Metcalfe, R. W., and Hussain, F., Phys.

Fluids A 6:885 (1994).19. Bell, J. H. and Mehta, R. D., AIAA J. 28:2034

(1990).20. Konrad, J. H., Ph. D. Thesis, CalTech, 1976.

COMMENTS

S. B. Pope, Cornell University, USA. A useful distinctionis between LES in which all of the energy-containing rangeis resolved, and VLES (Very Large Eddy Simulation) inwhich only the largest eddies are resolved. In the VLESreported, the dynamic subgrid scale model is used whichrelies on scale similarity. Is it appropriate to assume scalesimilarity within the energy-containing range?

Author’s Reply. Similarity between the scales at the gridand test filter levels is essential for dynamic modeling andis also required in our approach. However, with increasein Reynolds number the dynamic range of scales wherescale similarity holds also increases. Our approach to dy-namic subgrid modeling is applied to high Re flows andscale similarity is invoked at the upper end of the dynamicrange (i.e., at lower wave numbers) relative to conventionalLES where the scale similarity is used at the high wavenumber end of the inertial range. In our approach, it is not

necessary to resolve all scales, whereas, all scales (abovethe viscous range) must be resolved in the classical LESapproach. A consequence of our approach is that equilib-rium models, such as the algebraic eddy viscosity model,cannot be used as a subgrid model. To allow for local non-equilibrium between production and dissipation of the sub-grid kinetic energy we employ a one-equation dynamicmodel for the subgrid kinetic energy.

●

R. W. Bilger, The University of Sydney, Australia. At theNon-premixed Workshop preceding this symposium, sev-eral jet diffusion flames with well-established experimentallaser-based data for major species, temperature, NO, OH,and the velocity field. Is it feasible for your method to com-pute this flames and what computer resources are needed?


Author’s Reply. We have already begun using the avail-able data sets for hydrogen flames from Sandia (R. Barlow),and methane flames from the University of Sydney (A.Masri) and are currently studying these flames using the

linear-eddy modeling approach. Results of these studieswill be reported in the near future. Since this approach isuniquely suited for parallel simulation, future LES will becarried out on massively parallel systems.

subgrid mixing and molecular transport modeling in a reacting shear layer

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