Advanced Modeling of High Speed Turbulent Reacting...

Advanced Modeling of High Speed Turbulent Reacting Flows

Journal: 50th AIAA Aerospace Sciences Meeting Including the New Horizons

Forum and Aerospace Exposition

Manuscript ID: Draft

luMeetingID: 1964

Date Submitted by the Author:

n/a

Contact Author: Jaberi, Farhad; Michigan State University, Mechanical Engineering

http://mc.manuscriptcentral.com/aiaa-masm12

50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

American Institute of Aeronautics and Astronautics

1

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

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. 2 Graduate student, Department of Mechanical Engineering, Michigan State University. 3 Professor, Department of Mechanical Engineering, Michigan State University, and AIAA Associate Fellow.

T

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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 scalar-

FMDF 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 sub-

closures 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 two-

interacting 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:

( ; , ) ( , ) [ , ( , )] ( , )LP x t x t x t G x x dx

,

1

1[ , ( , )] ( ( , ))

Nsx t x t

(1)

where G denotes the filter function, is the scalar vector in the sample space, and is the “fine-

grained” density. The scalar vector )1,...,1(, sN 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:

)( cmpR

iii

i SSxxx

u

t

(2)

For the species mass fraction ),...,1( sN , the source/sink term RS in Equation (2) represents

the production or consumption of species α due to the chemical reaction. For the energy or

enthalpy )( 1 sN , the source term 1

( )sN

RS h W

represents the heat of combustion, and the term

1( )cmp i

i ij

i j

up pS u

t x x

is due to compressibility and viscous energy dissipation. The modeled FMDF

transport equation is obtained from the instantaneous unfiltered scalar equation (Equation (2)) as:

L

liii

LliL Pxxx

Pu

t

P|

1|

L

l

cmp

Ll

R PSPS

||

1

1

0 1,...,

1( ) ( )

s

R cmp

s

NR cmp i

i ij s

i j

S and S N

up pS h W and S u N

t x x

(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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4

LLm

i

lL

t

ii

LLiL Px

P

xx

Pu

t

P)([

/()(

L

cmp

L

R PSPS

~)(

1

1

0 )(1~

,...,10~

s

j

Li

Lij

i

l

Lil

l

cmpN

R

s

cmpR

Nx

u

x

pu

t

pSandWhS

NSandSs

(4)

In Equation (4), ttlt Pr/ is the turbulent diffusivity and Prt is the turbulent Prandtl number. The

SGS mixing frequency is calculated as 2

( )1

2 ( )

tm

l

C

. This equation can be solved by the Monte

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

tdWdtx

udX i

l

t

i

t

l

Lii

(5)

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:

dtSSdtd cmpR

Lm )~

()( (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.

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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.

-20 -10 0 10 200.01

0.02

0.03

0.04

0.05

0.06

DNS

FDNS

LES

LES+SGS

ek11

k x0

M1=2.0 LES with LJ Model

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.

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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.

-15 -10 -5 0 5 100

0.2

0.4

0.6

0.8

1

ks=8

ks=2

k x0

ek

t=0.0t=0.4t=0.8

t=0.0t=0.2t=0.4

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.

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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 LES-

FD 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 non-

equilibrium 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.

Page 10 of 11



http://hypersonicpropulsioncenter.us/


11

5 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).

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