the counterpart of the probability density function (PDF) method in RANS and is now ... speed considered (the so called flame D), the flame was burning near ...
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Advanced Modeling of High Speed Turbulent Reacting Flows Z. Li1 , A. Banaeizadeh1, S. Rezaeiravesh2 and F.A. Jaberi3 Michigan State University, East Lansing, MI, 48910
This paper provides a brief overview of the compressible scalar filtered mass density function (FMDF) model and its application to high speed turbulent combustion. The FMDF is a subgrid-scale probability density function model for large eddy simulation (LES) of turbulent combustion and is obtained by the solution of a set of stochastic differential equations with a Lagrangian Monte Carlo method. The applicability and the validity of the LES/FMDF are established by simulating various high speed reacting and non-reacting flows. The LES/FMDF results are found to be consistent and comparable to experimental and numerical (DNS) data in different flows.
I. Introduction
T
he modeling and simulation team in the National Center for Hypersonic Combined Cycle Propulsion1 is developing and using three different types or generations of computational models for high speed flows. Generation I models are based on Reynolds-averaged Navier-Stokes (RANS) closures and are currently being used by center researchers for the development and testing of new concepts and design of high speed propulsion systems. Our generation II and III models are primarily based on the large-eddy simulation (LES) method and filtered mass density function (FMDF) 4 methods. The FMDF is the counterpart of the probability density function (PDF) method in RANS and is now widely recognized as one of the best models for turbulent combustion2. Earlier applications of the FMDF/FDF model (FDF or filtered density function is the constant-density version of the FMDF) were for relatively simple problems and were focused on the development and testing of the model for low-speed single-phase flows.3-8 However, with the advancements in computational power and with the development of more efficient parallel numerical algorithms for the hybrid Eulerian-Lagrangian equations, the FMDF model has been used for the simulations of increasingly more sophisticated flows over the past several years. These simulations have been conducted in conjunction with non-equilibrium and equilibrium reaction models and reduced and detailed chemical kinetics mechanisms for various non-premixed, partially-premixed and premixed turbulent flames. For example, Yaldizli et al. 9 employed the scalar FMDF for LES of Sandia’s partially-premixed methane jet flames10 with complex chemical kinetics mechanisms, using the flamelet assumption or direct finite-rate chemistry solver. Sandia experiments were conducted for several turbulent jet speeds. For the lowest jet speed considered (the so called flame D), the flame was burning near equilibrium with limited local extinction. For this condition, the scalar FMDF results as obtained with the flamelet model and detailed mechanisms were found to be close to the experimental data. However, for the higher jet speeds (flames E and F), with significant local extinction, the flamelet model fails to reproduce the experimental data. In contrast the LES/FMDF with finite-rate multi-step reaction mechanisms was shown to be able to predict “high speed” flames E and F. This clearly indicates that the SGS turbulence-combustion interactions and finite-rate chemistry effects are important and should be considered at high speed flames. 1
Research Associate, Department of Mechanical Engineering, Michigan State University and AIAA Member. Graduate student, Department of Mechanical Engineering, Michigan State University. 3 Professor, Department of Mechanical Engineering, Michigan State University, and AIAA Associate Fellow. 1 American Institute of Aeronautics and Astronautics 2
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In the previous applications of the LES/FMDF, the effect of pressure on the FMDF was not considered. This effect could be ignored at low Mach number flows or constant pressure combustion. However, it is important and should be included in the FMDF for compressible (subsonic or supersonic) flows. Compressibility effect can be implemented in the scalar and velocity-scalar formulations of the FMDF both. In the later formulation, the pressure and energy are coupled with the velocity, temperature, density and species mass fractions; therefore are included in the definition of the joint FMDF and its transport equation. The joint energy-pressure-velocity-scalar (EPVS) FMDF is the most complete and complex formulation of the FMDF that has been ever considered 11. However, it is still under developed and cannot be used for simulations of practical combustion systems. The EPVS-FMDF is considered to be the Generation III model. The level of sophistication in FMDF can be reduced with consideration of fewer variables. This will obviously have the drawback of the need for more modeling. The most popular version of the FMDF is the scalar FMDF 4,12. This is also the most practical and efficient form of FMDF. The scalar-FMDF has been successfully utilized for prediction of a variety of low speed turbulent flames in the past but only recently was extended and used for high speed flows 13. The compressible scalarFMDF model is being used in Generation II models for combustion simulations. This paper describes our recent efforts on the development and application of compressible scalar-FMDF model to high speed reacting flows. It presents some of the basic components of the LES/FMDF and discussed issues related to its validation and efficiency. The compressible LES/FMDF and its subclosures are relatively new and have not been fully tested for high speed reacting flows. Therefore, they must be carefully appraised before they can be applied to actual systems. In doing so, we are making extensive use of DNS data for high speed flows (with and without reactions) to examine the extent of validity of our principal sub-closures. This assessment is being done via both a priori and a posteriori analysis of the DNS data. For the former, the performance of sub-closures is tested against DNS data assembled for the related physics. For the latter, the final predictions are assessed by direct comparisons with both (filtered) DNS data (and also experimental data). In all validations, the flow/chemistry parameters are virtually identical in LES and DNS, but the grid resolution in DNS is significantly higher. Most of our a priori and a posteriori assessments are done in the context of turbulent, compressible, homogeneous-isotropic and shear flows which provide an excellent setting for model validations, particularly SGS closures. We are also employing such simulations for capturing various physical phenomena such as scalar mixing, chemical reactions and various effects of compressibility and exothermicity in high speed flows. II. LES/FMDF Model for High Speed Turbulent Combustion The LES/FMDF model is implemented via a hybrid Eulerian-Lagrangian numerical scheme. The twointeracting fields modeled by the hybrid scheme are: (i) the Eulerian grid-based finite difference field, describing the gas dynamic variables, and (ii) the grid-free Lagrangian Monte Carlo field, describing gaseous species and temperature through FMDF. The Eulerian gas-phase flow solution is based on the generalized high-order multiblock finite difference methods applicable to compressible turbulent flows in complex geometries. The SGS combustion is modeled with the compressible scalar FMDF and its stochastic Lagrangian Monte Carlo solver. The LES/FMDF calculations may be conducted in conjunction with non-equilibrium and equilibrium reaction mechanisms, and reduced and flamelet-based detailed chemical kinetics. Figure 1 illustrates basic components of LES/FMDF. Details of the model are presented below. Also see Ref. 13.
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Figure 1: Basic components of LES/FMDF and its hybrid Eulerian-Lagrangian numerical solution method.
As mentioned before, in the hybrid LES/FMDF methodology, two sets of Eulerian and Lagrangian equations are solved together for the velocity, pressure, and scalar (temperature and mass fraction) fields. The first set of equations includes the standard filtered continuity, momentum and energy equations13. The second set of equations governs the evolution of the scalar FMDF, which represents the joint PDF of the scalar vector at the subgrid-level, defined as:
PL (; x , t )
( x, t )[, ( x, t )]G ( x, x )dx ,
[ , ( x , t )] 1 ( ( x , t )) (1) Ns 1
where G denotes the filter function, is the scalar vector in the sample space, and is the “finegrained” density. The scalar vector , ( 1,..., N s 1) includes the species mass fractions and the specific enthalpy. The scalar FMDF transport equation is obtained from the transport equation for the unfiltered scalar equation:
u i ( SR Scmp ) (2) t xi xi xi For the species mass fraction ( 1,..., N s ) , the source/sink term SR in Equation (2) represents the production or consumption of species α due to the chemical reaction. For the energy or Ns
enthalpy ( N s 1 ) , the source term SR (hW ) represents the heat of combustion, and the term 1
cmp
S
u 1 p p ( ui ij i ) is due to compressibility and viscous energy dissipation. The modeled FMDF t xi x j
transport equation is obtained from the instantaneous unfiltered scalar equation (Equation (2)) as: PL u i | l PL 1 SR | PL Scmp | PL | PL l l t xi xi xi l SR and Scmp 0 1,..., N s Ns ui 1 p p R cmp S (hW ) and S ( t ui x ij x ) N s 1 1 i j (3) The FMDF equation cannot be solved directly because of three unclosed terms. Following the suggested models for these terms13, the closed form of FMDF transport equation for a compressible reacting system is obtained,
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( PL / l ~ cmp PL ui L PL SR ( ) PL S PL [ m ( L ) PL ( t ) t xi xi xi ~ SR and Scmp 0 1,..., N s Ns p p u ~ 1 i L S R h 0W l l and Scmp ( ui L ij ) N s 1 L t x x 1 i j l (4) In Equation (4), t l t / Prt is the turbulent diffusivity and Prt is the turbulent Prandtl number. The
SGS mixing frequency is calculated as m 1 C ( t ) . This equation can be solved by the Monte 2 ( 2 l ) Carlo (MC) procedure. In this procedure, each MC particle undergoes motion in physical space due to filtered velocity and molecular and subgrid diffusivities. The particle motion represents the spatial transport of the FMDF and is modeled by the following stochastic differential equation (SDE): 2( ) 1 ( t ) t (5) dX i u i L dWi (t ) dt x i l l where Wi denotes the Wiener process. The scalar value of each particle is changed due to mixing, reaction, viscous dissipation, and pressure variations in time and space. The change in scalar space is described by the following SDEs:
d m (
L
~ )dt (SR Scmp )dt
(6)
When combined, the diffusion processes described by Equations (5) and (6) have a corresponding Fokker–Planck equation that is identical to the FMDF transport equation (Equation (4)).
III. Results and Discussions Successful implementation of the scalar FMDF model for high speed turbulent combustion required a systematic and step-by step examination and improvement of: (1) numerical methods for compressible turbulence/shock simulations, (2) subgrid-scale models for supersonic flows with shock wave, (3) Lagrangian Monte Carlo methods for supersonic combustion, (4) SGS mixing and scalar flux models for compressible FMDF, (5) efficient parallel algorithms for the implementation of LES/FMDF in complex geometries, (6) efficient and reliable multi-step reaction models for FMDF. For the past few years, center researchers have worked on all of these elements of the model. The ability of LES to capture the turbulence and compressibility/shock is dependent on the accuracy of the numerical method and also the SGS turbulence models. DNS and LES of various flows have been conducted to look at these issues. Figure 2 shows the interactions of an isotropic turbulence with a normal shock, obtained by DNS with a new high-order Monotonicity-preserving (MP) numerical method. As the turbulence passes through the shock, its characteristic size decreases but its strength, as measured by the vorticity magnitude is increased. For moderate flow Mach numbers, the size of turbulent structures decreases more in the streamwise direction than in the transverse direction. However, for relatively high Mach numbers (~ 5) the size of these structures decreases in all directions. The flow in Figure 2 is at Mach number of 5 before the shock which is high, yet our MP numerical method can accurately capture the shock and turbulence, particularly the small-scale turbulence generated by the shock. Our LES results for the turbulent kinetic energy (not shown) confirm that the numerical dissipation is negligible and nearly all of energy dissipation is due SGS model dissipation when the 7th order MP method is employed. Additionally, Figure 3 shows that the streamwise component of turbulent kinetic energy at resolved scales as predicted by LES and Li and Jaberi’s (LJ) SGS model14 is in good agreement with that of DNS. The total (resolved plus SGS) energy is also very well predicted by the LJ model. Similar results are observed for other turbulent kinetic energy components. 4 American Institute of Aeronautics and Astronautics http://mc.manuscriptcentral.com/aiaa-masm12
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Figure 2: Vortex structures, identified by the iso-surface of second structure function, colored with the vorticity in a Mach 5 isotropic turbulent flow interacting with a normal shock wave. 0.06 LES with LJ Model
M1=2.0 0.05
0.04
ek11 0.03 DNS FDNS 0.02 LES LES+SGS 0.01 -20
-10
0
k 0x
10
20
Figure 3: Streamwise variations of the streamwise turbulent kinetic in the shock-isotropic turbulent flow.
The generated/modified turbulence has a significant effect on the scalar mixing. This is demonstrated in Figure 4, where it is shown that the scalar mixing is increased and the characteristic size of scalar field is decreased as the scalar passes through the shock wave. Figure 4 also indicates that the decrease in scalar length scale is more pronounced for scalars with smaller pre-shock scalar length scales, indicating that the small scale scalar fluctuations are affected more by the shock wave. These effects are more pronounced in reacting flows. This is observed in Figure 5, where the DNS predicted contours of instantaneous temperature, scalar (hydrogen mass fraction) and vorticity in a reacting Mach 2 isotropic turbulent flow interacting with a normal shock wave are shown. Figures on the left show the results for the scalar field, initially made of larger scales and the ones on the right are for initially smaller scalar length scales. The combustion between air and hydrogen is simulated with a simple global hydrogen-air mechanism. Evidently, the effect of shock on the temperature and species fields is very significant and very much dependent on the scalar scales. The large-scale scalar field has a more significant effect on the shock. The vorticity field seems to be also affected more by the shock when the scalars are initially larger. This is consistent with the results in Figure 6 which show that the turbulent kinetics energy grows much more in time with the shock and combustion when the fuel-air length scales are initially larger. 5 American Institute of Aeronautics and Astronautics http://mc.manuscriptcentral.com/aiaa-masm12
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Figure 4: Contours of scalars with different initial length scales in a Mach 2 isotropic turbulent flow interacting with a normal shock wave. (a) large initial scalar scales; (b) moderate scalar scales; (c) small scalar scales.
Figure 5: Contours of instantaneous temperature, scalar (hydrogen mass fraction) and vorticity in a reacting isotropic turbulent flow interacting with a normal shock wave. Figures on the left show the results for the scalar field initially made of larger scales and the ones in the right are for initially smaller scalar length scales. 1
t=0.8
t=0.0
t=0.4
0.8
ks=8 ks=2 0.6
ek 0.4
t=0.4
t=0.2
t=0.0
0.2
0 -15
-10
-5
k 0x
0
5
10
Figure 6: Turbulent kinetic energy in a reacting shock-isotropic flow for different initial scalar length scales at different times. ks is the pick of initial scalar variance spectra.
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As mentioned before, the FMDF model has been applied to a variety of high speed flows. One of these flows is the shock “tube” flow. The initial condition for the thermodynamic variables is based on Sod’s shock tube solution. However, unlike Sod’s problem, the initial velocity is not zero rather it is an isotropic turbulent velocity with an intensity of 6 per cent of the laminar shock upstream velocity. Also, the flow is homogeneous and periodic in directions perpendicular to the shock/flow. Figure 7 shows the iso-levels of instantaneous filtered density as obtained by the finite difference (FD) and Monte Carlo (MC) parts of the hybrid LES/FMDF model for the shock-tube problem. Evidently, the shock wave has a significant effect on the turbulence and mixing. Nevertheless, the LES-FD and FMDF-FD predictions are shown to be consistent, even in the vicinity of the shock, indicating the ability of the compressible scalar FMDF model to capture the shock effects on the turbulence. The MC particle number density is also shown to compare well with the filtered density computed from LES-FD and FMDF-MC data as predicted by the FMDF theory. The computed mean and rms of the resolved temperature by the FMDF-MC and LES-FD (not shown) are also found to be in good agreement with each other; further indicating the consistency and the reliability of the LES/FMDF. It is to be noted here that the LES-FD and FMDF-MC predictions deviate noticeably when the pressure term is removed from the FMDF formulation.
(a)
(b)
Figure 7: Filtered density obtained by LES-FD and FMDF-MC in a shock tube flow. (a) Instantaneous contours obtained from LES-FD data. (b) Instantaneous contours obtained from FMDF-MC data. (c,d) Density at the center of shock tube obtained from LES-FD, FMDF-MC data also the MC particle number density. 7 American Institute of Aeronautics and Astronautics http://mc.manuscriptcentral.com/aiaa-masm12
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The reaction effect, even though it is highly nonlinear, appears in a closed form in the FMDF formulation. This allows simulations of various types of reactions (slow, fast, premixed, non-premixed, etc.) with different kinetics mechanisms, as long as the mechanism is known and is computationally affordable. Very recently, the LES/FMDF model is used for the simulations of hydrogen-air flames with various reaction models including a 37-step detailed15 mechanism. In these simulations, the method/data/formulas developed by NASA16 are used to compute the molecular viscosity, the thermal conductivity and other molecular properties of the species.
Figure 8: Schematic of supersonic mixing layer along with initial and boundary conditions.
To test the LES/FMDF model, two- and three-dimensional simulations of the subsonic and supersonic planar hydrogen-air reacting mixing layer were conducted. The flow condition for the supersonic case is shown in Figure 8. Evidently, the fuel stream has a lower speed and temperature than the oxidizer one. Also for this flow, the flame is ignited and stabilized by preheating of the air stream and by using appropriate fuel-air equivalence ratios. The 3D contours of the temperature obtained from LES/FD and FMDF-MC data for the 3D mixing layer are shown in Figure 9. Figure 10 shows a sample of the predicted temperatures in the computational domain. Even with the detailed reaction and molecular transport models, the LES-FD and FMDF-MC solvers predict similar results for the temperature and species mass fractions. The filtered values of temperature predicted by the LES-FD and FMDF-MC are clearly in close agreement (Figure 9). The instantaneous values of the fuel mass fraction and temperature in Figure 10 also demonstrate that the LES-FD and FMDF-MC predictions are highly correlated and fully consistent. The results obtained for other species are similar to those shown in Figure 10. It should be noted here that the LES-FD results are computed by using the reaction source/sink terms obtained from the FMDF and MC particles. This is only possible in our hybrid LES/FMDF solver since the reaction is closed in the FMDF formulation. In other LES models the highly nonlinear and complex SGS reaction terms have to be modeled!
Figure 9: Contours of instantaneous filtered temperature in a hydrogen-air reacting mixing layer obtained by LESFD and FMDF-MC with a detailed 37-step hydrogen-air mechanism.
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Figure 10: Scatter plots (a) H2 mass fraction, and (b) Temperature, predicted by LES-FD and FMDF-MC in a reacting hydrogen-air mixing layer.
An issue intimately related to (any) LES is its actual feasibility. For a SGS closure to be practical it must be implemented in a computationally efficient manner, especially if employed for prediction of complex flows. This important issue can be the overriding constraint. Monte Carlo simulations typically require order of millions to billions of particles. The computational requirements can become very significant in simulations of practical flows, especially those involving complex kinetics. Therefore, scalable parallelization at the MC particle level is required. The major challenge in scalability is the extreme load imbalance associated with stiff chemistry. At any time during the simulation, different regions of the flow experience different stages of chemical reactions. Even though the particle number density is statistically uniform, the computational load per particle varies significantly. A popular parallelization strategy in CFD is via temporally invariant block decomposition where the mesh is partitioned into equally sized boxes, and each box is assigned to a processor. This uniformity is relatively easy to implement and yields a minimal communication overhead. But for unsteady and inhomogeneous flows, it usually leads to a poor load distribution. Processors with lighter loads must wait (and remain idle) until the synchronization at the end of each time step. It has been shown that the load imbalance problem can be fully resolved by portioning the domain irregularly and adaptively17. In doing so, the Eulerian mesh is represented as an undirected graph where particle cells are the vertices of the graph and are weighted by the computational load. Each vertex is assigned a computational weight, i.e. a computation-load metric, which is a function of heterogeneous and homogenous computational loads. This weighted graph is then fed into a graph partitioning algorithm which subdivides the domain into clusters of particle cells on which the computational load is evenly distributed. The resulting scheme is termed “irregularly portioned Lagrangian Monte Carlo” (ILPMC)17 and allows efficient FMDF simulations on massively parallel platforms. With the efficient parallel algorithms and chemistry solvers, the LES/scalar-FMDF model is being applied to more practical combustion systems. Figure 11 for example shows a schematic view of one of injection/flame holding systems simulated by LES/FMDF and the contours of normalized temperature we have obtained from some of our preliminary simulations. We are in the processes of establishing the consistency of MC and FD parts of the LES/FMDF for this flow and then simulating the hydrogen-air combustion with the scalar FMDF model already tested for simpler (e.g. mixing layer) flows.
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Figure 11: Schematic view and contours of normalized temperature predicted by LES for a cavity flame holder.
IV. Conclusions The compressible scalar filtered mass density function (FMDF) model is used for the large eddy simulation (LES) of high speed turbulent combustion. LES/FMDF provides a convenient means of capturing some of the complicated processes in turbulent combustion, regardless of the type of reaction. The most challenging modeling aspect, namely modeling of turbulence-chemistry interaction, appears in a closed form in FMDF. Furthermore, the model is readily adaptable to systematically including closures for increasing detail of subgrid effects. In our previous work, we have been able to successfully implement LES/FMDF for a variety of low speed turbulent flow simulations with equilibrium and nonequilibrium reaction mechanisms. The compressible version of the model recently developed and extended and tested for various non-reacting and reacting flows involving simple and complex kinetics mechanisms. Well characterized and relevant DNS data have also been/are being developed for systematic validation of FMDF models. Primary barriers to utilizing LES/FMDF in production codes are related to computational implementation and algorithmic implementations with plenty of room for improvements.
Acknowledgement This research was sponsored by the National Center for Hypersonic Combined Cycle Propulsion grant FA 9550-09-1-0611. The technical monitors on the grant are Chiping Li (AFOSR), and Aaron Auslender and Rick Gaffney (NASA).
References 1
National Center for Hypersonic Combined Cycle Propulsion at University of Virginia, http://hypersonicpropulsioncenter.us/. 2 Givi, P., "Filtered Density Function for Subgrid Scale Modeling of Turbulent Combustion," AIAA Journal, Vol. 44, No. 1, 2006, pp. 16-23. 3 Colucci, P. J., Jaberi, F. A., Givi, P., and Pope, S. B., "Filtered Density Function for Large Eddy Simulation of Turbulent Reacting Flows," Physics of Fluids , Vol. 10, No. 2, 1998, pp. 499-515. 4 Jaberi, F. A., Colucci, P. J., James, S., Givi, P., and Pope, S. B., “Filtered Mass Density Function for Large Eddy Simulation of Turbulent Reacting Flows,” Journal of Fluid Mechanics., Vol. 401, 1999 pp. 85-121. 10 American Institute of Aeronautics and Astronautics http://mc.manuscriptcentral.com/aiaa-masm12
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Jaberi, F. A., “Large Eddy Simulation of Turbulent Premixed Flames via Filtered Mass Density Function, AIAA Paper 99-0199, AIAA, 1999. 6 James, S. and Jaberi, F. A., “Large Scale Simulations of Two-Dimensional Nonpremixed Methane Jet Flames,” Combustion and Flame, Vol. 123, 2000, pp. 465-487. 7 Gicquel, L. Y. M., Givi, P., Jaberi, F. A, and Pope, S. B., “Velocity filtered density function for large Eddy simulation of turbulent flows,” Physics of Fluids, Vol. 14, 2002, pp. 1196–1213. 8 Sheikhi, M. R. H., Drozda, T. G., Givi, P., and Pope, S. B., “Velocity–scalar filtered density function for large Eddy simulation of turbulent flows,” Physics of Fluids, Vol. 15, No. 8, 2003, pp. 2321–2337. 9 Yaldizli, M., Mehravaran, K., Jaberi, F. A., “Large-Eddy Simulations of Turbulent Methane Jet Flames with Filtered Mass Density Function,” International Journal of Heat and Mass Transfer, Vol. 53, 2010, pp. 2551-2562. 10 Barlow, R., Sandia/TUD Piloted CH4/Air Jet Flames, Available at: http://www.ca.sandia.gov/TNF/DataArch. 11 Nik, M. B., Mohebbi, M., Sheikhi, M. R. H. and Givi, P., "Progress in Large Eddy Simulation of High Speed Turbulent Mixing and Reaction" AIAA Paper: AIAA-2009-0133;2009. 12 Afshari, A., Jaberi, F. A., and Shih, T. I.-P., “Large-Eddy Simulation of Turbulent Flows in an Axisymmetric Dump Combustor,” AIAA Journal, Vol. 46, No.7, 2008, pp. 1576-1592. 13 Banaeizadeh, A., Li, Z., and Jaberi, F. A., “Compressible Scalar FMDF Model for Large-Eddy Simulations of High speed Turbulent Flows,” AIAA Journal, 49(10):2130–2143 (2011). 14 Li, Z. and Jaberi, F.A. Large-Eddy simulation of shock-isotropic turbulence, Physics of Fluid, to be submitted, 2011. 15 G. Stahl and J. Warnatz, Numerical Investigation of Time-Dependent Properties and Extinction of Strained Methane- and Propane-Air Flamelets, Combustion and Flame, Vol. 85, pp. 285-299 (1991). 16 B.J. McBride, S. Gordon, M.A. Reno, Coefficients for Calculating Thermodynamic and Transport Properties of Individual Species, NASA Technical Memorandum 4513 (1993). 17 S.L. Yilmaz, M.B. Nik, M.R.H. Sheikhi, P.A. Strakey and P. Givi, An Irregularly Portioned Lagrangian Monte Carlo Method for Turbulent Flow Simulation, Journal of Scientific Computing, Vol. 47(1), pp. 109–125 (2011). 5
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