Advanced Modeling of High Speed Turbulent Reacting Flows

Transcription

1 50th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace Exposition Advanced Modeing of High Speed Turbuent Reacting Fows Journa: 50th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace Exposition Manuscript ID: Draft umeetingid: 1964 Date Submitted by the Author: n/a Contact Author: Jaberi, Farhad; Michigan State University, Mechanica Engineering

2 Page 1 of 1150th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace Exposition Advanced Modeing of High Speed Turbuent Reacting Fows Z. i 1, A. Banaeizadeh 1, S. Rezaeiravesh 2 and F.A. Jaberi 3 Michigan State University, East ansing, MI, This paper provides a brief overview of the compressibe scaar fitered mass density function (FMDF) mode and its appication to high speed turbuent combustion. The FMDF is a subgrid-scae probabiity density function mode for arge eddy simuation (ES) of turbuent combustion and is obtained by the soution of a set of stochastic differentia equations with a agrangian Monte Caro method. The appicabiity and the vaidity of the ES/FMDF are estabished by simuating various high speed reacting and non-reacting fows. The ES/FMDF resuts are found to be consistent and comparabe to experimenta and numerica (DNS) data in different fows. T I. Introduction he modeing and simuation team in the Nationa Center for Hypersonic Combined Cyce Propusion 1 is deveoping and using three different types or generations of computationa modes for high speed fows. Generation I modes are based on Reynods-averaged Navier-Stokes (RANS) cosures and are currenty being used by center researchers for the deveopment and testing of new concepts and design of high speed propusion systems. Our generation II and III modes are primariy based on the arge-eddy simuation (ES) method and fitered mass density function (FMDF) 4 methods. The FMDF is the counterpart of the probabiity density function (PDF) method in RANS and is now widey recognized as one of the best modes for turbuent combustion 2. Earier appications of the FMDF/FDF mode (FDF or fitered density function is the constant-density version of the FMDF) were for reativey simpe probems and were focused on the deveopment and testing of the mode for ow-speed singe-phase fows. 3-8 However, with the advancements in computationa power and with the deveopment of more efficient parae numerica agorithms for the hybrid Euerian-agrangian equations, the FMDF mode has been used for the simuations of increasingy more sophisticated fows over the past severa years. These simuations have been conducted in conjunction with non-equiibrium and equiibrium reaction modes and reduced and detaied chemica kinetics mechanisms for various non-premixed, partiay-premixed and premixed turbuent fames. For exampe, Yadizi et a. 9 empoyed the scaar FMDF for ES of Sandia s partiay-premixed methane jet fames 10 with compex chemica kinetics mechanisms, using the fameet assumption or direct finite-rate chemistry sover. Sandia experiments were conducted for severa turbuent jet speeds. For the owest jet speed considered (the so caed fame D), the fame was burning near equiibrium with imited oca extinction. For this condition, the scaar FMDF resuts as obtained with the fameet mode and detaied mechanisms were found to be cose to the experimenta data. However, for the higher jet speeds (fames E and F), with significant oca extinction, the fameet mode fais to reproduce the experimenta data. In contrast the ES/FMDF with finite-rate muti-step reaction mechanisms was shown to be abe to predict high speed fames E and F. This ceary indicates that the SGS turbuence-combustion interactions and finite-rate chemistry effects are important and shoud be considered at high speed fames. 1 Research Associate, Department of Mechanica Engineering, Michigan State University and AIAA Member. 2 Graduate student, Department of Mechanica Engineering, Michigan State University. 3 Professor, Department of Mechanica Engineering, Michigan State University, and AIAA Associate Feow. 1

3 50th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace ExpositionPage 2 of 11 In the previous appications of the ES/FMDF, the effect of pressure on the FMDF was not considered. This effect coud be ignored at ow Mach number fows or constant pressure combustion. However, it is important and shoud be incuded in the FMDF for compressibe (subsonic or supersonic) fows. Compressibiity effect can be impemented in the scaar and veocity-scaar formuations of the FMDF both. In the ater formuation, the pressure and energy are couped with the veocity, temperature, density and species mass fractions; therefore are incuded in the definition of the joint FMDF and its transport equation. The joint energy-pressure-veocity-scaar (EPVS) FMDF is the most compete and compex formuation of the FMDF that has been ever considered 11. However, it is sti under deveoped and cannot be used for simuations of practica combustion systems. The EPVS-FMDF is considered to be the Generation III mode. The eve of sophistication in FMDF can be reduced with consideration of fewer variabes. This wi obviousy have the drawback of the need for more modeing. The most popuar version of the FMDF is the scaar FMDF 4,12. This is aso the most practica and efficient form of FMDF. The scaar-fmdf has been successfuy utiized for prediction of a variety of ow speed turbuent fames in the past but ony recenty was extended and used for high speed fows 13. The compressibe scaar- FMDF mode is being used in Generation II modes for combustion simuations. This paper describes our recent efforts on the deveopment and appication of compressibe scaar-fmdf mode to high speed reacting fows. It presents some of the basic components of the ES/FMDF and discussed issues reated to its vaidation and efficiency. The compressibe ES/FMDF and its subcosures are reativey new and have not been fuy tested for high speed reacting fows. Therefore, they must be carefuy appraised before they can be appied to actua systems. In doing so, we are making extensive use of DNS data for high speed fows (with and without reactions) to examine the extent of vaidity of our principa sub-cosures. This assessment is being done via both a priori and a posteriori anaysis of the DNS data. For the former, the performance of sub-cosures is tested against DNS data assembed for the reated physics. For the atter, the fina predictions are assessed by direct comparisons with both (fitered) DNS data (and aso experimenta data). In a vaidations, the fow/chemistry parameters are virtuay identica in ES and DNS, but the grid resoution in DNS is significanty higher. Most of our a priori and a posteriori assessments are done in the context of turbuent, compressibe, homogeneous-isotropic and shear fows which provide an exceent setting for mode vaidations, particuary SGS cosures. We are aso empoying such simuations for capturing various physica phenomena such as scaar mixing, chemica reactions and various effects of compressibiity and exothermicity in high speed fows. II. ES/FMDF Mode for High Speed Turbuent Combustion The ES/FMDF mode is impemented via a hybrid Euerian-agrangian numerica scheme. The twointeracting fieds modeed by the hybrid scheme are: (i) the Euerian grid-based finite difference fied, describing the gas dynamic variabes, and (ii) the grid-free agrangian Monte Caro fied, describing gaseous species and temperature through FMDF. The Euerian gas-phase fow soution is based on the generaized high-order mutibock finite difference methods appicabe to compressibe turbuent fows in compex geometries. The SGS combustion is modeed with the compressibe scaar FMDF and its stochastic agrangian Monte Caro sover. The ES/FMDF cacuations may be conducted in conjunction with non-equiibrium and equiibrium reaction mechanisms, and reduced and fameet-based detaied chemica kinetics. Figure 1 iustrates basic components of ES/FMDF. Detais of the mode are presented beow. Aso see Ref

4 Page 3 of 1150th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace Exposition Figure 1: Basic components of ES/FMDF and its hybrid Euerian-agrangian numerica soution method. As mentioned before, in the hybrid ES/FMDF methodoogy, two sets of Euerian and agrangian equations are soved together for the veocity, pressure, and scaar (temperature and mass fraction) fieds. The first set of equations incudes the standard fitered continuity, momentum and energy equations 13. The second set of equations governs the evoution of the scaar FMDF, which represents the joint PDF of the scaar vector at the subgrid-eve, defined as: ( ;, ) P (, ) [, (, )] (, ) x t x t x t G x x dx, Ns 1 [, ( x, t)] ( ( x, t)) 1 (1) where G denotes the fiter function, is the scaar vector in the sampe space, and is the finegrained density. The scaar vector,( 1,..., Ns 1) incudes the species mass fractions and the specific enthapy. The scaar FMDF transport equation is obtained from the transport equation for the unfitered scaar equation: ui R cmp ( S S ) t xi x i x (2) i R For the species mass fraction ( 1,..., N s ), the source/sink term S in Equation (2) represents the production or consumption of species α due to the chemica reaction. For the energy or enthapy N ), the source term S S cmp ( s 1 i j N s R hw 1 ( ) represents the heat of combustion, and the term 1 p p u ( i ui ij ) is due to compressibiity and viscous energy dissipation. The modeed FMDF t x x transport equation is obtained from the instantaneous unfitered scaar equation (Equation (2)) as: P u i P 1 R cmp P S P S P t xi xi xi R cmp S and S 0 1,..., Ns Ns R cmp 1 p p ui S ( hw ) and S ( ui ij ) Ns 1 1 t xi x j (3) The FMDF equation cannot be soved directy because of three uncosed terms. Foowing the suggested modes for these terms 13, the cosed form of FMDF transport equation for a compressibe reacting system is obtained, 3

5 50th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace ExpositionPage 4 of 11 P ui P ( P / ( t ) [ m ( ) P R ~ cmp S ( ) P S P t xi xi xi R ~ cmp S and S 0 1,..., N s Ns 0 ~ 1 p p u R cmp i S h W and S ( ui ij ) N s1 1 t x i x j (4) In Equation (4), / Pr is the turbuent diffusivity and Pr t is the turbuent Prandt number. The t SGS mixing frequency is cacuated as t t m 1 ( ) t C 2 2 ( ). This equation can be soved by the Monte Caro (MC) procedure. In this procedure, each MC partice undergoes motion in physica space due to fitered veocity and moecuar and subgrid diffusivities. The partice motion represents the spatia transport of the FMDF and is modeed by the foowing stochastic differentia equation (SDE): 1 ( ) 2( ) t t dx i ui dt dwi ( t) (5) x i where W i denotes the Wiener process. The scaar vaue of each partice is changed due to mixing, reaction, viscous dissipation, and pressure variations in time and space. The change in scaar space is described by the foowing SDEs: R ~ cmp d ) dt ( S S dt (6) m ( ) When combined, the diffusion processes described by Equations (5) and (6) have a corresponding Fokker Panck equation that is identica to the FMDF transport equation (Equation (4)). III. Resuts and Discussions Successfu impementation of the scaar FMDF mode for high speed turbuent combustion required a systematic and step-by step examination and improvement of: (1) numerica methods for compressibe turbuence/shock simuations, (2) subgrid-scae modes for supersonic fows with shock wave, (3) agrangian Monte Caro methods for supersonic combustion, (4) SGS mixing and scaar fux modes for compressibe FMDF, (5) efficient parae agorithms for the impementation of ES/FMDF in compex geometries, (6) efficient and reiabe muti-step reaction modes for FMDF. For the past few years, center researchers have worked on a of these eements of the mode. The abiity of ES to capture the turbuence and compressibiity/shock is dependent on the accuracy of the numerica method and aso the SGS turbuence modes. DNS and ES of various fows have been conducted to ook at these issues. Figure 2 shows the interactions of an isotropic turbuence with a norma shock, obtained by DNS with a new high-order Monotonicity-preserving (MP) numerica method. As the turbuence passes through the shock, its characteristic size decreases but its strength, as measured by the vorticity magnitude is increased. For moderate fow Mach numbers, the size of turbuent structures decreases more in the streamwise direction than in the transverse direction. However, for reativey high Mach numbers (~ 5) the size of these structures decreases in a directions. The fow in Figure 2 is at Mach number of 5 before the shock which is high, yet our MP numerica method can accuratey capture the shock and turbuence, particuary the sma-scae turbuence generated by the shock. Our ES resuts for the turbuent kinetic energy (not shown) confirm that the numerica dissipation is negigibe and neary a of energy dissipation is due SGS mode dissipation when the 7 th order MP method is empoyed. Additionay, Figure 3 shows that the streamwise component of turbuent kinetic energy at resoved scaes as predicted by ES and i and Jaberi s (J) SGS mode 14 is in good agreement with that of DNS. The tota (resoved pus SGS) energy is aso very we predicted by the J mode. Simiar resuts are observed for other turbuent kinetic energy components. 4

6 Page 5 of 1150th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace Exposition Figure 2: Vortex structures, identified by the iso-surface of second structure function, coored with the vorticity in a Mach 5 isotropic turbuent fow interacting with a norma shock wave M1=2.0 ES with J Mode ek DNS 0.02 FDNS ES ES+SGS k 0 x Figure 3: Streamwise variations of the streamwise turbuent kinetic in the shock-isotropic turbuent fow. The generated/modified turbuence has a significant effect on the scaar mixing. This is demonstrated in Figure 4, where it is shown that the scaar mixing is increased and the characteristic size of scaar fied is decreased as the scaar passes through the shock wave. Figure 4 aso indicates that the decrease in scaar ength scae is more pronounced for scaars with smaer pre-shock scaar ength scaes, indicating that the sma scae scaar fuctuations are affected more by the shock wave. These effects are more pronounced in reacting fows. This is observed in Figure 5, where the DNS predicted contours of instantaneous temperature, scaar (hydrogen mass fraction) and vorticity in a reacting Mach 2 isotropic turbuent fow interacting with a norma shock wave are shown. Figures on the eft show the resuts for the scaar fied, initiay made of arger scaes and the ones on the right are for initiay smaer scaar ength scaes. The combustion between air and hydrogen is simuated with a simpe goba hydrogen-air mechanism. Evidenty, the effect of shock on the temperature and species fieds is very significant and very much dependent on the scaar scaes. The arge-scae scaar fied has a more significant effect on the shock. The vorticity fied seems to be aso affected more by the shock when the scaars are initiay arger. This is consistent with the resuts in Figure 6 which show that the turbuent kinetics energy grows much more in time with the shock and combustion when the fue-air ength scaes are initiay arger. 5

7 50th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace ExpositionPage 6 of 11 Figure 4: Contours of scaars with different initia ength scaes in a Mach 2 isotropic turbuent fow interacting with a norma shock wave. (a) arge initia scaar scaes; (b) moderate scaar scaes; (c) sma scaar scaes. Figure 5: Contours of instantaneous temperature, scaar (hydrogen mass fraction) and vorticity in a reacting isotropic turbuent fow interacting with a norma shock wave. Figures on the eft show the resuts for the scaar fied initiay made of arger scaes and the ones in the right are for initiay smaer scaar ength scaes. 1 t=0.8 t=0.4 t= ks=8 ks=2 ek t=0.4 t=0.2 t= k 0 x Figure 6: Turbuent kinetic energy in a reacting shock-isotropic fow for different initia scaar ength scaes at different times. ks is the pick of initia scaar variance spectra. 6

8 Page 7 of 1150th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace Exposition As mentioned before, the FMDF mode has been appied to a variety of high speed fows. One of these fows is the shock tube fow. The initia condition for the thermodynamic variabes is based on Sod s shock tube soution. However, unike Sod s probem, the initia veocity is not zero rather it is an isotropic turbuent veocity with an intensity of 6 per cent of the aminar shock upstream veocity. Aso, the fow is homogeneous and periodic in directions perpendicuar to the shock/fow. Figure 7 shows the iso-eves of instantaneous fitered density as obtained by the finite difference (FD) and Monte Caro (MC) parts of the hybrid ES/FMDF mode for the shock-tube probem. Evidenty, the shock wave has a significant effect on the turbuence and mixing. Nevertheess, the ES-FD and FMDF-FD predictions are shown to be consistent, even in the vicinity of the shock, indicating the abiity of the compressibe scaar FMDF mode to capture the shock effects on the turbuence. The MC partice number density is aso shown to compare we with the fitered density computed from ES-FD and FMDF-MC data as predicted by the FMDF theory. The computed mean and rms of the resoved temperature by the FMDF-MC and ES-FD (not shown) are aso found to be in good agreement with each other; further indicating the consistency and the reiabiity of the ES/FMDF. It is to be noted here that the ES-FD and FMDF-MC predictions deviate noticeaby when the pressure term is removed from the FMDF formuation. (a) (b) Figure 7: Fitered density obtained by ES-FD and FMDF-MC in a shock tube fow. (a) Instantaneous contours obtained from ES-FD data. (b) Instantaneous contours obtained from FMDF-MC data. (c,d) Density at the center of shock tube obtained from ES-FD, FMDF-MC data aso the MC partice number density. 7

9 50th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace ExpositionPage 8 of 11 The reaction effect, even though it is highy noninear, appears in a cosed form in the FMDF formuation. This aows simuations of various types of reactions (sow, fast, premixed, non-premixed, etc.) with different kinetics mechanisms, as ong as the mechanism is known and is computationay affordabe. Very recenty, the ES/FMDF mode is used for the simuations of hydrogen-air fames with various reaction modes incuding a 37-step detaied 15 mechanism. In these simuations, the method/data/formuas deveoped by NASA 16 are used to compute the moecuar viscosity, the therma conductivity and other moecuar properties of the species. Figure 8: Schematic of supersonic mixing ayer aong with initia and boundary conditions. To test the ES/FMDF mode, two- and three-dimensiona simuations of the subsonic and supersonic panar hydrogen-air reacting mixing ayer were conducted. The fow condition for the supersonic case is shown in Figure 8. Evidenty, the fue stream has a ower speed and temperature than the oxidizer one. Aso for this fow, the fame is ignited and stabiized by preheating of the air stream and by using appropriate fue-air equivaence ratios. The 3D contours of the temperature obtained from ES/FD and FMDF-MC data for the 3D mixing ayer are shown in Figure 9. Figure 10 shows a sampe of the predicted temperatures in the computationa domain. Even with the detaied reaction and moecuar transport modes, the ES-FD and FMDF-MC sovers predict simiar resuts for the temperature and species mass fractions. The fitered vaues of temperature predicted by the ES-FD and FMDF-MC are ceary in cose agreement (Figure 9). The instantaneous vaues of the fue mass fraction and temperature in Figure 10 aso demonstrate that the ES-FD and FMDF-MC predictions are highy correated and fuy consistent. The resuts obtained for other species are simiar to those shown in Figure 10. It shoud be noted here that the ES-FD resuts are computed by using the reaction source/sink terms obtained from the FMDF and MC partices. This is ony possibe in our hybrid ES/FMDF sover since the reaction is cosed in the FMDF formuation. In other ES modes the highy noninear and compex SGS reaction terms have to be modeed! Figure 9: Contours of instantaneous fitered temperature in a hydrogen-air reacting mixing ayer obtained by ES- FD and FMDF-MC with a detaied 37-step hydrogen-air mechanism. 8

10 Page 9 of 1150th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace Exposition Figure 10: Scatter pots (a) H 2 mass fraction, and (b) Temperature, predicted by ES-FD and FMDF-MC in a reacting hydrogen-air mixing ayer. An issue intimatey reated to (any) ES is its actua feasibiity. For a SGS cosure to be practica it must be impemented in a computationay efficient manner, especiay if empoyed for prediction of compex fows. This important issue can be the overriding constraint. Monte Caro simuations typicay require order of miions to biions of partices. The computationa requirements can become very significant in simuations of practica fows, especiay those invoving compex kinetics. Therefore, scaabe paraeization at the MC partice eve is required. The major chaenge in scaabiity is the extreme oad imbaance associated with stiff chemistry. At any time during the simuation, different regions of the fow experience different stages of chemica reactions. Even though the partice number density is statisticay uniform, the computationa oad per partice varies significanty. A popuar paraeization strategy in CFD is via temporay invariant bock decomposition where the mesh is partitioned into equay sized boxes, and each box is assigned to a processor. This uniformity is reativey easy to impement and yieds a minima communication overhead. But for unsteady and inhomogeneous fows, it usuay eads to a poor oad distribution. Processors with ighter oads must wait (and remain ide) unti the synchronization at the end of each time step. It has been shown that the oad imbaance probem can be fuy resoved by portioning the domain irreguary and adaptivey 17. In doing so, the Euerian mesh is represented as an undirected graph where partice ces are the vertices of the graph and are weighted by the computationa oad. Each vertex is assigned a computationa weight, i.e. a computation-oad metric, which is a function of heterogeneous and homogenous computationa oads. This weighted graph is then fed into a graph partitioning agorithm which subdivides the domain into custers of partice ces on which the computationa oad is eveny distributed. The resuting scheme is termed irreguary portioned agrangian Monte Caro (IPMC) 17 and aows efficient FMDF simuations on massivey parae patforms. With the efficient parae agorithms and chemistry sovers, the ES/scaar-FMDF mode is being appied to more practica combustion systems. Figure 11 for exampe shows a schematic view of one of injection/fame hoding systems simuated by ES/FMDF and the contours of normaized temperature we have obtained from some of our preiminary simuations. We are in the processes of estabishing the consistency of MC and FD parts of the ES/FMDF for this fow and then simuating the hydrogen-air combustion with the scaar FMDF mode aready tested for simper (e.g. mixing ayer) fows. 9

11 50th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace Exposition Page 10 of 11 Figure 11: Schematic view and contours of normaized temperature predicted by ES for a cavity fame hoder. IV. Concusions The compressibe scaar fitered mass density function (FMDF) mode is used for the arge eddy simuation (ES) of high speed turbuent combustion. ES/FMDF provides a convenient means of capturing some of the compicated processes in turbuent combustion, regardess of the type of reaction. The most chaenging modeing aspect, namey modeing of turbuence-chemistry interaction, appears in a cosed form in FMDF. Furthermore, the mode is readiy adaptabe to systematicay incuding cosures for increasing detai of subgrid effects. In our previous work, we have been abe to successfuy impement ES/FMDF for a variety of ow speed turbuent fow simuations with equiibrium and nonequiibrium reaction mechanisms. The compressibe version of the mode recenty deveoped and extended and tested for various non-reacting and reacting fows invoving simpe and compex kinetics mechanisms. We characterized and reevant DNS data have aso been/are being deveoped for systematic vaidation of FMDF modes. Primary barriers to utiizing ES/FMDF in production codes are reated to computationa impementation and agorithmic impementations with penty of room for improvements. Acknowedgement This research was sponsored by the Nationa Center for Hypersonic Combined Cyce Propusion grant FA The technica monitors on the grant are Chiping i (AFOSR), and Aaron Ausender and Rick Gaffney (NASA). 1 References Nationa Center for Hypersonic Combined Cyce Propusion at University of Virginia, 2 Givi, P., "Fitered Density Function for Subgrid Scae Modeing of Turbuent Combustion," AIAA Journa, Vo. 44, No. 1, 2006, pp Coucci, P. J., Jaberi, F. A., Givi, P., and Pope, S. B., "Fitered Density Function for arge Eddy Simuation of Turbuent Reacting Fows," Physics of Fuids, Vo. 10, No. 2, 1998, pp Jaberi, F. A., Coucci, P. J., James, S., Givi, P., and Pope, S. B., Fitered Mass Density Function for arge Eddy Simuation of Turbuent Reacting Fows, Journa of Fuid Mechanics., Vo. 401, 1999 pp

12 Page 11 of 11 50th AIAA Aerospace Sciences Meeting Incuding the New Horizons Forum and Aerospace Exposition 5 Jaberi, F. A., arge Eddy Simuation of Turbuent Premixed Fames via Fitered Mass Density Function, AIAA Paper , AIAA, James, S. and Jaberi, F. A., arge Scae Simuations of Two-Dimensiona Nonpremixed Methane Jet Fames, Combustion and Fame, Vo. 123, 2000, pp Gicque,. Y. M., Givi, P., Jaberi, F. A, and Pope, S. B., Veocity fitered density function for arge Eddy simuation of turbuent fows, Physics of Fuids, Vo. 14, 2002, pp Sheikhi, M. R. H., Drozda, T. G., Givi, P., and Pope, S. B., Veocity scaar fitered density function for arge Eddy simuation of turbuent fows, Physics of Fuids, Vo. 15, No. 8, 2003, pp Yadizi, M., Mehravaran, K., Jaberi, F. A., arge-eddy Simuations of Turbuent Methane Jet Fames with Fitered Mass Density Function, Internationa Journa of Heat and Mass Transfer, Vo. 53, 2010, pp Barow, R., Sandia/TUD Pioted CH4/Air Jet Fames, Avaiabe at: 11 Nik, M. B., Mohebbi, M., Sheikhi, M. R. H. and Givi, P., "Progress in arge Eddy Simuation of High Speed Turbuent Mixing and Reaction" AIAA Paper: AIAA ; Afshari, A., Jaberi, F. A., and Shih, T. I.-P., arge-eddy Simuation of Turbuent Fows in an Axisymmetric Dump Combustor, AIAA Journa, Vo. 46, No.7, 2008, pp Banaeizadeh, A., i, Z., and Jaberi, F. A., Compressibe Scaar FMDF Mode for arge-eddy Simuations of High speed Turbuent Fows, AIAA Journa, 49(10): (2011). 14 i, Z. and Jaberi, F.A. arge-eddy simuation of shock-isotropic turbuence, Physics of Fuid, to be submitted, G. Stah and J. Warnatz, Numerica Investigation of Time-Dependent Properties and Extinction of Strained Methane- and Propane-Air Fameets, Combustion and Fame, Vo. 85, pp (1991). 16 B.J. McBride, S. Gordon, M.A. Reno, Coefficients for Cacuating Thermodynamic and Transport Properties of Individua Species, NASA Technica Memorandum 4513 (1993). 17 S.. Yimaz, M.B. Nik, M.R.H. Sheikhi, P.A. Strakey and P. Givi, An Irreguary Portioned agrangian Monte Caro Method for Turbuent Fow Simuation, Journa of Scientific Computing, Vo. 47(1), pp (2011). 11