Velocity-scalar filtered density function for large eddy simulation of ...

2324 Phys. Fluids, Vol. 15, No. 8, August 2003 Sheikhi et al.d tD X ,U , ;tdtB j X ,U , ;tdW j tF X j X ,U , ;tdW X j tF U j X ,U , ;tdW U j t, 17cwhere X i , U i , are probabilistic representations of position,velocity vector, and scalar variables, respectively. TheD terms denote drift in the composition space, the B termsdenote diffusion, the F terms denote diffusion couplings, andthe W terms denote the Wiener–Lévy processes. 29,30 FollowingHaworth and Pope, 31 Dreeben and Pope, 32 Colucciet al., 7 and Gicquel et al. 16 we consider the generalizedLangevin model GLM and the linear mean square estimationLMSE model 26dX i U i dt 1 dW X i ,dU i p 2 u i x 2 Gi x k x ij U j u j dtk18a 3u i x kdW k X C 0 dW i U , 18bd 2 S1 Cx k x S dtk S2 x kdW k X , 18cwhere the variables 1 , 2 ,... are all diffusion coefficientsto be specified, andG ij 1 2 3 4 C 0 ij , k , 19C k 3/2 L, k 1 2 u k ,u k .Here is the SGS mixing frequency, is the SGS dissipationrate, k is the SGS kinetic energy, and L is the LES filtersize. The parameters C 0 , C , and C are model constantsand need to be specified. The limit 1 3 S1 S2 0isthe standard high Reynolds number GLM–LMSE closure. 20The Fokker–Planck equation 33 for f (v,,x;t), the jointPDF of X , U , , evolving by the diffusion process asgiven by Eq. 18 is ft vx k fk p x 2 1 3 2 u i i x k x k f Gv i v ij v j u j f S1 1 S2 2 i x k x kC f 1 S2 x iS f 1 2 2 f 3 u i u j x i 2 x k x k 2 f u j x k x 1 3k x i 2 fx i v j 2 f 1 v i v j 2 C 2 fu i 0 v k v 3 S2k x k f x k 2 fv i S2 2 f.2 x k x k The transport equations for the filtered variables are obtained by integration of Eq. 20 according to Eq. 12:20u k x k0, 21au i t t u ku i x k px i 12 2 1 3 2u i u k ,u i , 21bx k x k x k u k x k S1 1 S2 12 2 Sx k x u k , . 21ck x kThe transport equations for the second-order SGS moments areu i ,u j t u ku i ,u j 1x k 2 2 u i ,u j x k x ku k ,u i u jux k ,u j u ik x 1 2 1 3 3 u i u j kx k x kG ik u k ,u j G jk u k ,u i C 0 ij u k ,u i ,u j x k, 22aDownloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

Phys. Fluids, Vol. 15, No. 8, August 2003Velocity-scalar filtered density function2325u i , t , t u ku i , 1x k 2 u k , 1x k 2 2 u i , x k x ku k ,u i ux k , u ikx k 1 1 3 1 S2 3 S2 u ix k Gx ik u k , C u i , ku i ,S u k ,u i , x k, 22b 2 , x k x k 1 2 1 S2 S2 x ku k , ux k , kx k 2Cx , ,S k ,S u k , , x k. 22cA term-by-term comparison of the exact moment transport equations Eqs. 4 and 6, with the modeled equations Eqs.21 and 22, suggests 1 2 3 S1 S2 2. However, this violates the realizability of the scalar field. A set ofcoefficients yielding a realizable stochastic model requires: 1 2 3 2 and S1 S2 0. That is,dX i U i dt2 dW i X ,23adU i p 2 2 u i Gx i x k x ij U j u j dt2 u idW Xk x k C 0 dW U i , 23bkd C S dt.23cThe Fokker–Planck equation for this system is ft vx k f pk x i f Gv i v ij v j u j f i2 u jx i 2 f u i u j x i v j x k x kC f S f 2 f x k x k 2 f 1 v i v j 2 C 2 f0v k v k24and the corresponding equations for the moments areu k x k0, 25au i t tu i ,u j tu i , t u ku i x k p 2 u i u k ,u i , 25bx i x k x k x k u k 2 Sx k x k x u k , , 25ck x k u ku i ,u j x k u ku i , x k 2 u i ,u j ux k x k ,u i u jkux k ,u j u iGk x ik u k ,u j G jk u k ,u i C 0 ij k u k ,u i ,u j x k, 26a 2 u i , x k x ku k ,u i x ku k , u iGx ik u k , C u i , ku i ,S u k ,u i , x k, 26bDownloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

Phys. Fluids, Vol. 15, No. 8, August 2003Velocity-scalar filtered density function2327TABLE I. Attributes of the computational methods.LES-FDvariablesVSFDFvariablesVSFDF quantitiesused by theLES-FD systemLES-FD quantitiesused by theVSFDF systemRedundantquantitiesVSFDF p,u i X i (u i ,u j ) u i , p/x i u i U i (u i , ) u i /x k , 2 u i /x k x k S () VSFDF-C p,u i X i (u i ,u j ) u i , p/x i u i , U i (u i , ) u i /x k , 2 u i /x k x k (u i ,u j )(u i ,u j ) (u i ,u j ,u k ) , k (u i , )(u i , ) (u i ,u j , ) ( , )( , ) (u i , , )information from the MC simulations are obtained; 2 toperform LES primarily by the finite difference methodologywhich is coupled to the MC solver. The LES procedure viathe finite difference discretization is referred to as LES–FDand will be further discussed below.Statistical information is obtained by considering an ensembleof N E computational particles residing within an ensembledomain of characteristic length E centered aroundeach of the finite-difference grid points. This is illustratedschematically in Fig. 1. For reliable statistics with minimalnumerical dispersion, it is desired to minimize the size ofensemble domain and maximize the number of the MCparticles. 20 In this way, the ensemble statistics would tend tothe desired filtered values,a E 1 a n ——→ a,N E nEN E → E →0 E a,b 1 a n aN E b n b E E nE——→N E → E →0a,b,30where a (n) denotes the information carried by nth MC particlepertaining to transport variable a.The LES–FD solver is based on the compact parameterfinite difference scheme. 39,40 This is a variant of the MacCormackscheme in which fourth-order compact differencingschemes are used to approximate the spatial derivatives, andsecond-order symmetric predictor–corrector sequence is employedfor time discretization. All of the finite differenceoperations are conducted on fixed grid points. The transfer ofinformation from the grid points to the MC particles is accomplishedvia a second-order interpolation. The transfer ofinformation from the particles to the grid points is accomplishedvia ensemble averaging as described above.The LES–FD procedure determines the pressure fieldwhich is used in the MC solver. The LES–FD also determinesthe filtered velocity and scalar fields. That is, there is a‘‘redundancy’’ in the determination of the first filtered momentsas both the LES–FD and the MC procedures providethe solution of this field. This redundancy is actually veryuseful in monitoring the accuracy of the simulated results asshown in previous work. 9,16,34–36 To establish consistencyand convergence of the MC solver, the modeled transportequations for the generalized second-order SGS momentsEq. 26 are also solved via LES–FD. In doing so, theunclosed third-order correlations are taken from the MCsolver. The comparison of the first and second-order momentsas obtained by LES–FD with those obtained by theMC solver is useful to establish the accuracy of the MCsolver. These simulations are referred to as VSFDF–C. Attributesof all the simulation procedures are summarized inTable I. In this table and hereinafter, VSFDF simulationsrefer to the hybrid MC/LES–FD procedure in which theLES–FD is used for only the first-order filtered variables. InVSFDF–C, the LES–FD procedure is used for both first- andsecond-order filtered values. Further discussions about thesimulation methods are available in Refs. 7, 16, 34–36.V. RESULTSA. Flows simulatedSimulations are conducted of a two-dimensional 2Dand a 3D incompressible, temporally developing mixing layersinvolving transport of a passive scalar variable. Since theperformance of the model in capturing the velocity-scalarcorrelations is of primary interest, only nonreacting flowsimulations are conducted. Inclusion of chemical reaction viathe joint FDF formulation is straightforward and is similar tothat in the marginal scalar FDF method. 7,9–12 The 2D simulationsare performed to establish and demonstrate the consistencyof the MC solver. The 3D simulations are used toassess the overall predictive capabilities of the VSFDF methodology.These predictions are compared with data obtainedby direct numerical simulation DNS of the same layer.The temporal mixing layer consists of two parallelstreams travelling in opposite directions with the samespeed. 41–43 In the representation below, x, y and z denotethe streamwise, the cross-stream and the spanwise directionsin 3D, respectively. The velocity components alongthese directions are denoted by u, v and w in the x, y andz directions, respectively. Both the filtered streamwise velocityand the scalar fields are initialized with a hyperbolictangent profiles with u1, 1 on the top stream andu1, 0 on the bottom stream. The length L isspecified such that L2 N P u , where N P is the desired num-Downloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

2328 Phys. Fluids, Vol. 15, No. 8, August 2003 Sheikhi et al.FIG. 2. Cross-stream variation of the Reynoldsaveragedvalues of at t34.3: a N E 40, b E/2.ber of successive vortex pairings and u is the wavelength ofthe most unstable mode corresponding to the mean streamwisevelocity profile imposed at the initial time. The flowvariables are normalized with respect to the half initial vorticitythickness, L r v (t0)/2 ( v U/u L /y max ,where u L is the Reynolds averaged value of the filteredstreamwise velocity and U is the velocity difference acrossthe layer. The reference velocity is U r U/2.All 2D simulations are conducted for 0xL, and2L/3y2L/3. The formation of large scale structures isfacilitated by introducing small harmonic, phase-shifted, disturbancescontaining subharmonics of the most unstablemode into the streamwise and cross-stream velocity profiles.For N p 1, this results in formation of two large vortices andone subsequent pairing of these vortices. The 3D simulationsare conducted for a cubic box, 0xL, L/2yL/2 (0zL). The 3D field is parametrized in a procedure somewhatsimilar to that by Vreman et al. 44 The formation of thelarge scale structures are expedited through eigenfunctionbased initial perturbations. 45,46 This includestwo-dimensional 42,44,47 and three-dimensional 42,48 perturbationswith a random phase shift between the 3D modes. Thisresults in the formation of two successive vortex pairings andstrong three dimensionality.B. Numerical specificationsSimulations are conducted on equally spaced grid pointswith grid spacings xyz for 3D. All 2D simulationsare performed on 3241 grid points. The 3D simulationsare conducted on 193 3 and 33 3 points for DNS andLES, respectively. The Reynolds number is ReU r L r /50. To filter the DNS data, a top-hat function of the formbelow is used3Gxx G˜ x i x i ,i11, LG˜ x i x ix i x i L2 ,0, x i x i L2 .31No attempt is made to investigate the sensitivity of the resultsto the filter function 27 or the size of the filter. 49The MC particles are initially distributed throughout thecomputational region. All simulations are performed with auniform ‘‘weight’’ 20 of the particles. Due to flow periodicityin the streamwise and spanwise in 3D directions, iftheparticle leaves the domain at one of these boundaries newparticles are introduced at the other boundary with the samevelocity and compositional values. In the cross-stream directions,the free-slip boundary condition is satisfied by themirror-reflection of the particles leaving through theseboundaries. The density of the MC particles is determined bythe average number of particles N E within the ensemble domainof size E E ( E ). The effects of both of theseparameters are assessed to ensure the consistency and thestatistical accuracy of the VSFDF simulations. All results areanalyzed both ‘‘instantaneously’’ and ‘‘statistically.’’ In theformer, the instantaneous contours snap-shots and scatterplots of the variables of interest are analyzed. In the latter,Downloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

Phys. Fluids, Vol. 15, No. 8, August 2003Velocity-scalar filtered density function2329FIG. 3. Color Temporal evolution of the scalar with superimposed vorticity iso-lines top and the vorticity bottom fields for LES–FD, with E /2 andN E 40 at several times.the ‘‘Reynolds-averaged’’ statistics constructed from the instantaneousdata are considered. These are constructed byspatial averaging over x and z in 3D. All Reynoldsaveragedresults are denoted by an overbar.C. Consistency and convergence assessmentsThe objective of this section is to demonstrate the consistencyof the VSFDF formulation and the convergence ofits MC simulation procedure. For this purpose, the results viaMC and LES–FD are compared against each other inVSFDF–C simulations. Since the accuracy of the FD procedureis well-established at least for the first-order filteredquantities, such a comparative assessment provides a goodmeans of assessing the performance of the MC solution. Noattempt is made to determine the appropriate values of themodel constants; the values suggested in the literature areadopted 50 C 0 2.1, C 1, and C 1. The influence ofthese parameters is assessed in Sec. V D.The uniformity of the MC particles is checked by monitoringtheir distributions at all times, as the particle numberdensity must be proportional to fluid density. The Reynoldsaveraged density fields as obtained by both LES–FD and byMC are shown in Fig. 2. Close to unity values for the densityat all times is the first measure of the accuracy of simulations.Figures 3 and 4 show the instantaneous contour plotsof the filtered scalar and vorticity fields at several times.These figures provide a visual demonstration of the consistencyof the VSFDF. This consistency is observed for all firstDownloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

2330 Phys. Fluids, Vol. 15, No. 8, August 2003 Sheikhi et al.FIG. 4. Color Temporal evolution of the scalar with superimposed vorticity iso-lines top and the vorticity bottom fields for VSFDF with E /2 andN E 40 at several times.order moments without any statistical variability. Also, all ofthese moments show very little dependence on the values of E and N E consistent with previous FDF simulations. 7,9,16 Inthe presentation below we only focus on second-order moments.Specifically, the scalar-velocity correlations areshown since all other second-order SGS moments behavesimilarly.Figures 5 and 6 show the statistical variability of theresults for simulations with N E 40. It is observed that thesemoments exhibit spreads with variances decreasing as thesize of the ensemble domain is reduced. Figures 7–10 showthe sensitivity to N E and E . All these results clearly displayconvergence suggested by Eq. 30. As the ensemble domainsize decreases, the VSFDF results converge to those of LES–FD. Ideally, the LES–FD results should become independentof the MC results, as the latter become more reliable, i.e.,when (N E →, E →0). It is observed that best match isachieved with E /2 and N E 40. This conclusion is consistentwith previous assessment studies on the scalar FDF, 7,9and the velocity FDF. 16 All the subsequent simulations areconducted with E /2 and N E 40.D. Comparative assessments of the VSFDFThe objective of this section is to analyze some of thecharacteristics of the VSFDF via comparative assessmentsagainst DNS data. In addition, comparisons are also madewith LES via the ‘‘conventional’’ Smagorinsky 18,51 model:Downloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

Phys. Fluids, Vol. 15, No. 8, August 2003Velocity-scalar filtered density function2331FIG. 5. Statistical variability of LES–FD and VSFDF–C simulations withN E 40 for Reynolds-averaged values of (u,) at t34.4. Solid lines,LES–FD; dashed lines, VSFDF–C.FIG. 6. Statistical variability of LES–FD and VSFDF–C simulations withN E 40 for Reynolds-averaged values of (v,) at t34.4. Solid lines,LES–FD; dashed lines, VSFDF–C. L u i ,u j 2 3 k ij 2 t S ij , L u i , t Lx i,S ij 1 2 u i Lx j t C L 2 S, u j Lx i t tSc t.,32C 0.04, Sc t 1, SS ij S ij and L is the characteristiclength of the filter. This model considers the anisotropic partof the SGS stress tensor a ij L (u i ,u j )2/3k ij . The isotropiccomponents are absorbed in the pressure field.For comparison, the DNS data are transposed from theoriginal high resolution 193 3 points to the coarse 33 3 points.In the comparisons, we also consider the ‘‘resolved’’ and the‘‘total’’ components of the Reynolds-averaged moments. TheFIG. 7. Cross-stream variations of theReynolds-averaged values of (u,)a E /2, b E , c E2.Downloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

2332 Phys. Fluids, Vol. 15, No. 8, August 2003 Sheikhi et al.FIG. 8. Cross-stream variations of theReynolds-averaged values of (v,)a E /2, b E , c E2.former are denoted by R(a,b) with R(a,b)(aa)(bb); and the latter is r(a,b) with r(a,b)(aā)(bb¯). In DNS, the ‘‘total’’ SGS components are directlyavailable, while in LES they are approximated byr(a,b)R(a,b)(a,b). 44 Unless indicated otherwise, thevalues of the model constants are C 0 2.1, C 1, C 1;but the effects of these parameters on the predicted resultsare assessed.Figure 11 shows the instantaneous iso-surface of the field t80. By this time, the flow has gone through pairingsand exhibits strong 3D effects. This is evident by the formationof large scale spanwise rollers with presence of mushroomlike structures in streamwise planes. 45 Similar to previousresults, 16 the amount of SGS diffusion with theSmagorinsky model is significant. Thus, the predicted resultsare overly smooth. The Reynolds-averaged values of the filteredscalar field at t80 are shown in Fig. 12, and thetemporal variation of the ‘‘scalar thickness,’’ s ty0.9y0.133is shown in Fig. 13. The filtered and unfiltered DNS datayield virtually indistinguishable results. The dissipative natureof the Smagorinsky model at initial times resulting in aslow growth of the layer is shown. All VSFDF predictionscompare well with DNS data in predicting the spread of thelayer.Several components of the planar averaged values of thesecond-order SGS moments are compared with DNS data inFIG. 9. Cross-stream variations of theReynolds-averaged values of (u,)a N E 20, b N E 40, c N E 80.Downloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

Phys. Fluids, Vol. 15, No. 8, August 2003Velocity-scalar filtered density function2333FIG. 10. Cross-stream variations of the Reynolds-averaged values of (v,) a N E 20, b N E 40, c N E 80.Figs. 14 and 15 for several values of the model constants. Ingeneral, the VSFDF results are in better agreement with DNSdata than those predicted by the Smagorinsky model. In thisregard, therefore, the VSFDF is expected to be more effectivethan the Smagorinsky type closures for LES of reactingflows since the extent of SGS mixing is heavily influencedby these SGS moments. 52,53 However, it is not possible tosuggest ‘‘optimum’’ values for the model constants, exceptthat at small C and C values, the SGS energy is very large.Several components of the resolved second-order momentsare presented in Figs. 16 and 17. As expected, theperformance of the Smagorinsky model is not very good as itdoes not predict the spread and the peak value accurately.The VSFDF yields reasonable predictions except for smallC values. However, the total values of these moments arefairly independent of the model constants and yield verygood agreement with DNS data as shown in Figs. 18 and 19.It is also noted that while the SGS moments and/or the resolvedmoments may be overestimated and/or underestimateddepending on the values of the model coefficients, thetotal values of the moments are fairly independent of thesecoefficients, at least in the range of values as considered. ButFIG. 11. Color Contours surface of the field in the 3D mixing layer at t80 as obtained by a DNS, b Smagorinsky, c VSFDF.Downloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

2334 Phys. Fluids, Vol. 15, No. 8, August 2003 Sheikhi et al.FIG. 14. Cross-stream variations of some of the components of at t60.FIG. 12. Cross-stream variations of the Reynolds-averaged values of thefiltered scalar field at t80.low values of C , C are not recommended as they wouldresult in too much SGS energy in comparison to the resolvedenergy.The computational cost of VSFDF simulations relativeto those required by DNS and by the Smagorinsky model isthe same as that reported previously. 16 The typical ratios ofthe normalized Smagorinsky–VSFDF–DNS run times are1O30O200.VI. SUMMARY AND CONCLUDING REMARKSThe filtered density function FDF methodology hasproven effective for LES of turbulent reactive flows. In previousinvestigations, either the marginal FDF of the scalar, orthat of the velocity were considered. The objective of presentwork is to develop the joint velocity-scalar FDF methodology.For this purpose, the exact transport equation governingthe evolution of VSFDF is derived. It is shown that effects ofthe SGS convection and chemical reaction appear in a closedform. The unclosed terms are modeled in a fashion similar tothose typically followed in PDF methods. The modeledVSFDF transport equation is solved numerically via a LagrangianMonte Carlo MC scheme via consideration of asystem of equivalent stochastic differential equationsFIG. 13. Temporal variations of the scalar thickness.FIG. 15. Cross-stream variations of some of the components of at t80.Downloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

Phys. Fluids, Vol. 15, No. 8, August 2003Velocity-scalar filtered density function2335FIG. 16. Cross-stream variations of some of the components of R¯ at t60.FIG. 18. Cross-stream variations of r¯ at t60.SDEs. These SDEs are discretized via the Euler–Maruyamma approximation. The consistency of the VSFDFmethod and the convergence of its MC solutions are assessedin LES of a two-dimensional 2D temporally developingmixing layer. This assessment is done by comparing the resultsobtained by the MC procedure with those of the finitedifferencescheme LES–FD for the solution of the transportequations of the first two moments of VSFDF. Byincluding the third moments from the VSFDF into the LES–FD, the consistency and convergence of the MC solution aredemonstrated by good agreements of the first two SGS momentswith those obtained by LES–FD.The VSFDF predictions are compared with LES resultswith the Smagorinsky 18 SGS model. All of these results arealso compared with direct numerical simulation DNS dataof a three-dimensional, temporally developing mixing layer.It is shown that the VSFDF performs well in predicting someof the phenomena pertaining to the SGS transport. Most ofthe overall flow features, including the mean field, the resolvedand total stresses as predicted by VSFDF are in goodagreement with DNS data. However, the model does requirethe input of three empirical constants. Also, the numericalimplementation of VSFDF is more expensive than the traditionalmodels. It may be possible to improve the predictivecapabilities of the VSFDF by two ways: 1 development ofa dynamic procedure to determine the model coefficients,and/or 2 implementation of higher order closures for thegeneralized Langevin model parameter G ij . 50 Future work isrecommended for development and application of the jointfiltered velocity-scalar mass density function VSFMDF toallow for LES of variable density flows with/or without thepresence of chemical reaction.FIG. 17. Cross-stream variations of some of the components of R¯ at t80.FIG. 19. Cross-stream variations of r¯ at t80.Downloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

2336 Phys. Fluids, Vol. 15, No. 8, August 2003 Sheikhi et al.ACKNOWLEDGMENTSThe authors are indebted to Dr. P. J. Colucci, Dr. T. D.Dreeben, Dr. M. Germano, Dr. L. Y. M. Gicquel, Dr. S.Heinz, Dr. M. Lesieur, Dr. H. Steiner, and Dr. C. Tong fortheir excellent and very valuable comments on the first draftof this manuscript. Part of this work was conducted when thefirst three authors were at the State University of New Yorkat Buffalo. The work is sponsored by the U.S. Air ForceOffice of Scientific Research under Grant No. F49620-03-1-0022 to University of Pittsburgh and Grant No. F49620-03-1-0015 to Cornell University. Dr. Julian M. Tishkoff is theProgram Manager for both these grants. Additional supportfor the work at University of Pittsburgh is provided by theNASA Langley Research Center under Grant No. NAG-1-03010 with Dr. J. Philip Drummond as the Technical Monitor.Computational resources are provided by the NCSA atthe University of Illinois at Urbana and by the PittsburghSupercomputing Center PSC.1 S. B. Pope, Turbulent Flows Cambridge University Press, Cambridge,UK, 2000.2 S. B. Pope, ‘‘Computations of turbulent combustion: Progress and challenges,’’Proc. Combust. Inst. 23, 591 1990.3 C. K. Madnia and P. Givi, ‘‘Direct numerical simulation and large eddysimulation of reacting homogeneous turbulence,’’ in Large Eddy Simulationsof Complex Engineering and Geophysical Flows, edited by B. Galperinand S. A. Orszag Cambridge University Press, Cambridge, UK,1993, Chap. 15, pp. 315–346.4 F. Gao and E. E. O’Brien, ‘‘A large-eddy simulation scheme for turbulentreacting flows,’’ Phys. Fluids A 5, 12821993.5 S. H. Frankel, V. Adumitroaie, C. K. Madnia, and P. Givi, ‘‘Large eddysimulations of turbulent reacting flows by assumed PDF methods,’’ inEngineering Applications of Large Eddy Simulations, edited by S. A.Ragab and U. Piomelli ASME, New York, 1993, FED-Vol. 162, pp.81–101.6 A. W. Cook and J. J. Riley, ‘‘A subgrid model for equilibrium chemistry inturbulent flows,’’ Phys. Fluids 6, 2868 1994.7 P. J. Colucci, F. A. Jaberi, P. Givi, and S. B. Pope, ‘‘Filtered densityfunction for large eddy simulation of turbulent reacting flows,’’ Phys. Fluids10, 499 1998.8 J. Réveillon and L. Vervisch, ‘‘Subgrid-scale turbulent micromixing: Dynamicapproach,’’ AIAA J. 36, 3361998.9 F. A. Jaberi, P. J. Colucci, S. James, P. Givi, and S. B. Pope, ‘‘Filteredmass density function for large eddy simulation of turbulent reactingflows,’’ J. Fluid Mech. 401, 851999.10 S. C. Garrick, F. A. Jaberi, and P. Givi, ‘‘Large eddy simulation of scalartransport in a turbulent jet flow,’’ in Recent Advances in DNS and LES,Fluid Mechanics and its Applications, edited by D. Knight and L. SakellKluwer Academic, Dordrecht, 1999, Vol. 54, pp. 155–166.11 S. James and F. A. Jaberi, ‘‘Large scale simulations of two-dimensionalnonpremixed methane jet flames,’’ Combust. Flame 123, 465 2000.12 X. Y. Zhou and J. C. F. Pereira, ‘‘Large eddy simulation 2D of a reactingplan mixing layer using filtered density function,’’ Flow, Turbul. Combust.64, 2792000.13 K. H. Luo, ‘‘DNS and LES of turbulence-combustion interactions,’’ seeGeurts Ref. 24, Chap. 14, pp. 263–293.14 T. Poinsot and D. Veynante, Theoretical and Numerical Combustion R. T.Edwards, Philadelphia, PA, 2001.15 C. Tong, ‘‘Measurements of conserved scalar filtered density function in aturbulent jet,’’ Phys. Fluids 13, 2923 2001.16 L. Y. M. Gicquel, P. Givi, F. A. Jaberi, and S. B. Pope, ‘‘Velocity filtereddensity function for large eddy simulation of turbulent flows,’’ Phys. Fluids14, 11962002.17 P. Givi, ‘‘A review of modern developments in large eddy simulation ofturbulent reacting flows,’’ in DNS/LES-Progress and Challenges, edited byC. Liu, L. Sakell, and R. 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O’Brien, ‘‘The probability density function PDF approach to reactingturbulent flows,’’ see Libby and Williams Ref. 19, Chap.5,pp.185–218.27 B. Vreman, B. Geurts, and H. Kuerten, ‘‘Realizability conditions for theturbulent stress tensor in large-eddy simulation,’’ J. Fluid Mech. 278, 3511994.28 S. Karlin and H. M. Taylor, A Second Course in Stochastic ProcessesAcademic, New York, 1981.29 N. Wax, Selected Papers on Noise and Stochastic Processes Dover, NewYork, 1954.30 C. W. Gardiner, Handbook of Stochastic Methods Springer-Verlag, NewYork, 1990.31 D. C. Haworth and S. B. Pope, ‘‘A generalized Langevin model for turbulentflows,’’ Phys. Fluids 29, 387 1986.32 T. D. Dreeben and S. B. Pope, ‘‘Probability density function andReynolds-stress modeling of near-wall turbulent flows,’’ Phys. Fluids 9,154 1997.33 H. Risken, The Fokker–Planck Equation, Methods of Solution and ApplicationsSpringer-Verlag, New York, 1989.34 S. B. 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