Velocity-scalar ﬁltered mass density function for large eddy … · 2012. 12. 21. · DOI:...

Velocity-scalar filtered mass density function for large eddy simulationof turbulent reacting flows

M. R. H. Sheikhia� and P. GiviDepartment of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh,Pennsylvania 15261, USA

S. B. PopeSibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853, USA

�Received 29 November 2006; accepted 11 July 2007; published online 26 September 2007�

A methodology termed the “velocity-scalar filtered mass density function” �VSFMDF� is developedand implemented for large eddy simulation �LES� of variable-density turbulent reacting flows. Thismethodology is based on the extension of the previously developed “velocity-scalar filtered densityfunction” method for constant-density flows. In the VSFMDF, the effects of the unresolved subgridscales �SGS� are taken into account by considering the joint probability density function of thevelocity and scalar fields. An exact transport equation is derived for the VSFMDF in which theeffects of SGS convection and chemical reaction are in closed forms. The unclosed terms in thisequation are modeled in a fashion similar to that in Reynolds-averaged simulation procedures. A setof stochastic differential equations �SDEs� are considered which yield statistically equivalent resultsto the modeled VSFMDF transport equation. The SDEs are solved numerically by a LagrangianMonte Carlo procedure in which the Itô-Gikhman character of the SDEs is preserved. Theconsistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed. Innonreacting flows, it is shown that the VSFMDF results agree well with those obtained by a“conventional” finite-difference LES procedure in which the transport equations corresponding tothe filtered quantities are solved directly. The VSFMDF results are also compared with thoseobtained by the Smagorinsky closure, and all the results are assessed via comparison with dataobtained by direct numerical simulation of a temporally developing mixing layer involving transportof a passive scalar. It is shown that all of the first two moments including the scalar fluxes arepredicted well by the VSFMDF. Moreover, the VSFMDF methodology is shown to be able torepresent the variable density effects very well. The predictive capabilities of the VSFMDF inreacting flows are further demonstrated by LES of a reacting shear flow. The predictions showfavorable agreement with laboratory data, and demonstrate several of the features as observedexperimentally. © 2007 American Institute of Physics. �DOI: 10.1063/1.2768953�

I. INTRODUCTION

The probability density function �PDF� approach hasproven to be useful for large eddy simulation �LES� of tur-bulent reacting flows.1,2 The formal means of conductingsuch LES is by considering the “filtered density function”�FDF�.3,4 The fundamental property of the FDF is to accountfor the effects of subgrid-scale �SGS� fluctuations in a proba-bilistic manner. Since its original conception,3,4 the FDF hasbecome very popular in the combustion research.5 Most con-tributions thus far are based on the marginal scalar FDF�SFDF�.6–10 This popularity is due to the capacity of thisformulation to provide a closed form for the chemical reac-tion effect. However, in the SFDF the effect of convectionneeds to be modeled similar to that in “conventional” LES.Gicquel et al.11 developed the marginal FDF of the velocityvector �VFDF� in which the effect of SGS convection is in aclosed form. However since the information about scalars isnot embedded in the VFDF, this method is only suitable for

constant-density, nonreacting flows. Following the develop-ments as cited above, the FDF methodology has experiencedwidespread usage. Examples are contributions in its basicimplementation,12–23 fine-tuning of its subclosures,24,25 andits validation via laboratory experiments.26–30 The FDF isfinding its way into commercial codes31,32 and has been thesubject of detailed discussions in several books.1,33–35 Givi2

provides a comprehensive review of the state of progress inLES/FDF.

The objective of the present work is to extend the FDFmethodology to account for the “joint” SGS velocity andscalar fields in variable-density flows. This is accomplishedby considering the joint “velocity-scalar filtered mass densityfunction” �VSFMDF�. This is the most comprehensive formof the FDF formulation to date. With the definition of theVSFMDF, the mathematical framework for its implementa-tion in LES is established. A transport equation is developedfor the VSFMDF in which the effects of SGS convection andchemical reaction �in a reacting flow� are in closed forms.The unclosed terms in this equation are modeled in a fashionsimilar to those in the Reynolds-averaged simulation �RAS�

a�Author to whom correspondence should be addressed. Telephone: �412�624-9755. Fax: �412� 624-4886. Electronic mail: [email protected]

PHYSICS OF FLUIDS 19, 095106 �2007�

1070-6631/2007/19�9�/095106/21/$23.00 © 2007 American Institute of Physics19, 095106-1

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http://dx.doi.org/10.1063/1.2768953http://dx.doi.org/10.1063/1.2768953http://dx.doi.org/10.1063/1.2768953

procedures. A Lagrangian Monte Carlo procedure is devel-oped and implemented for numerical solution of the modeledVSFMDF transport equation. The consistency of this proce-dure is assessed by comparing the moments of the VSFMDFwith those obtained by the Eulerian finite-difference of thesame moments’ transport equations. The results of theVSFMDF simulations are compared with those predicted bythe Smagorinsky36 closure. All the results are assessed viacomparisons with direct numerical simulation �DNS� data ofa three-dimensional �3D� temporally developing mixinglayer involving transport of a passive scalar variable. Thepredictive capability of the VSFMDF methodology in react-ing flows is assessed by comparison with experimental dataof Mungal and Dimotakis37 for a 3D reacting spatially devel-oping mixing layer.

There have been significant previous investigations onthe effect of density on turbulent flow38–42 characteristics.These studies show how density changes modify the turbu-lent structures in both low-speed and high-speed flows. Asignificant effect is reduction of the shear layer growth ratewhich is due to modification of Reynolds stresses by densityvariations. Now that LES is becoming a viable tool for thesimulation of turbulent reactive flows, it is useful to evaluatethe ability of SGS models in accounting for variable densityeffects. In this study, such an evaluation is performed on theVSFMDF methodology. The shear layers considered in thisstudy have low Mach number and hence, negligible com-pressibility effects. The spatial mixing layer simulations havemoderate amounts of heat release with the density ratios of1.3–1.5 for the range of concentration ratios considered. Inthe temporal mixing layer simulations, much higher free-stream density ratios �up to 8� are considered. These resultsare validated by comparing with DNS data of the same layer.

II. FORMULATION

In a turbulent flow undergoing chemical reactions in-volving Ns species, the primary transport variables are thedensity ��x , t�, the velocity vector ui�x , t� �i=1,2 ,3�, thepressure p�x , t�, the enthalpy h�x , t� and the species’ massfractions Y��x , t� ��=1,2 , . . . ,Ns�. The equations which gov-ern the transport of these variables in space �xi� �i=1,2 ,3�and time �t� are the continuity, momentum, enthalpy �en-ergy�, and species’ mass fraction equations, along with anequation of state

��

�t+

��uj�xj

= 0, �1a�

��ui�t

+��ujui

�xj= −

�p

�xi+

�� ji�xj

, �1b�

��t

+��uj��

�xj= −

�Jj�

�xj+ �S�, � = 1,2, . . . ,� = Ns + 1,

�1c�

p = �R0T��=1

Ns

Y�/M� = �RT , �1d�

where R0 and R are the universal and mixture gas constantsand M� denotes the molecular weight of species �. The

chemical reaction source terms S�� Ŝ��x , t�� are functionsof compositional scalars ��1 ,�2 , . . . ,�Ns+1��. Equation�1c� represents the transport of species’ mass fraction andenthalpy in a common form with

�� Y�, � = 1,2, . . . ,Ns, �� h = ��=1

Ns

h��, �2�

and

h� = h�0 + �

T0

T

cp��T��dT�. �3�

Here T and T0 denote the temperature field and the referencetemperature, respectively. In this equation, h�

0 and cp� denotethe enthalpy of formation at T0 and the specific heat at con-stant pressure for species �. For a Newtonian fluid, withFick’s law of diffusion, the viscous stress tensor �ij and thescalar flux Jj

� are represented by

�ij = �� ui�xj + �uj�xi − 23 �uk�xk �ij , �4a�Jj

� = − ��xj

, �4b�

where � is the fluid dynamic viscosity and �=� denotes thethermal and mass molecular diffusivity coefficients for allthe scalars. We assume �=�, i.e., unity Schmidt �Sc� andPrandtl �Pr� numbers. The viscosity and molecular diffusivitycoefficients can, in general, be temperature dependent but inthis study, they are assumed to be constants. In reactiveflows, molecular processes are much more complicated thanportrayed by Eq. �4�. Since the molecular diffusion is typi-cally less important than that of SGS, this simple model isadopted with justifications and caveats given in Refs. 43–45.

Large eddy simulation involves the spatial filteringoperation1,46–49

f�x,t�� = �−

+

f�x�,t�G�x�,x�dx�, �5�

where G�x� ,x� denotes a filter function, and f�x , t�� is thefiltered value of the transport variable f�x , t�. In variable-density flows it is convenient to use the Favre-filtered quan-tity f�x , t��L= �f�� / ��. We consider a filter function that isspatially and temporally invariant and localized, thusG�x� ,x��G�x�−x� with the properties G�x��0,�−

+G�x�dx=1. Applying the filtering operation to Eqs. �1�and using the conventional LES approximation for the diffu-sion terms, we obtain

095106-2 Sheikhi, Givi, and Pope Phys. Fluids 19, 095106 �2007�


��t

+��uj�L

�xj= 0, �6a�

��ui�L�t

+��uj�Lui�L

�xj

= −�p��xi

+�

�xj�� ui�L

�xj+

�uj�L�xi

−

2

3

�

�xi��uj�L

�xj − ��L�ui,uj�

�xj, �6b�

��L�t

+��uj�L��L

�xj

=�

�xj��L

�xj − ��L�uj,��

�xj+ ��S��L� ,

�6c�

where the second-order SGS correlations, defined by

�L�a,b� = ab�L − a�Lb�L, �7�

are governed by

��L�ui,uj��t

+��uk�L�L�ui,uj�

�xk= −

��L�uk,ui,uj��xk

− ��L�ui,uk��uj�L�xk

− ��L�uj,uk��ui�L�xk

+�

�xk��L�ui,uj�

�xk

+ �L�uj, ��xk��uk�xi + �L�ui, ��xk��uk�xj �−

2

3

�L�uj, ��xi��uk�xk + �L�ui, ��xj��uk�xk�

− ��uj �p�xi�� − uj�L�p��xi + ��ui �p�xj�� − ui�L�p��xj + 2��L� �ui�xk , �uj�xk� , �8a��L�ui,��

�t+

��uj�L�L�ui,��xj

= −��L�uj,ui,��

�xj− ��L�ui,uj�

��L�xj

− ��L�uj,��ui�L�xj

+�

�xj��L�ui,��

�xj + �L��, ��xj��uj�xi − 23�L��, ��xi��uj�xj�

− �� p�xi�� − ��L�p��xi + 2��L� �ui�xj , ��xj � + ��L�ui,S�� , �8b��L��,��

�t+

��ui�L�L��,��xi

= −��L�ui,��,��

�xi− ��L��,ui�

��L�xi

− ��L��,ui��L

�xi

+�

�xi��L��,��

�xi − 2��L� ��xi , ��xi �

+ ��L��,S�� + �L��,S�� . �8c�

In this equation, the third order correlations

�L�a,b,c� = abc�L − a�L�L�b,c� − b�L�L�a,c�

− c�L�L�a,b� − a�Lb�Lc�L �9�

along with the other terms within square brackets are un-closed. Equations �6� and �8� provide an “exact” form of thetransport equations.

III. VELOCITY-SCALAR FILTERED MASS DENSITYFUNCTION „VSFMDF…

A. Definitions

The “velocity-scalar filtered mass density function”�VSFMDF�, denoted by PL, is formally defined as

3

PL�v,�,x;t� = �−

+

��x�,t��v,�;u�x�,t�,��x�,t��

�G�x� − x�dx�, �10�

where

�v,�;u�x,t�,��x,t�� = �i=1

3

��vi − ui�x,t��

� ��=1

�

�� − ��x,t�� . �11�

In this equation, � denotes the Dirac delta function, and v ,�

095106-3 Velocity-scalar filtered mass density function Phys. Fluids 19, 095106 �2007�


are the velocity vector and the scalar array in the samplespace. The term is the “fine-grained” density.44,50 Equation�10� defines the VSFMDF as the spatially filtered value ofthe fine-grained density. With the condition of a positive fil-

ter kernel,51 PL has all of the properties of a mass densityfunction �mdf�.44 For further developments it is useful todefine the “conditional filtered value” of the variable Q�x , t�as

Q��x,t��u�x,t� = v,��x,t� = �� Q�v,�� =�−

+Q�x�,t��x�,t��v,�;u�x�,t�,��x�,t��G�x� − x�dx�PL�v,�,x;t�

. �12�

Equation �12� implies the following:

�i� for Q�x,t� = c, Q��x,t��v,�� = c; �13a�

�ii� for Q�x,t� � Q̂�u�x,t�,��x,t�� ,�13b�

�Q�x,t��v,�� = Q̂�v,��;

�iii� integral properties:

��x,t��Q�x,t��L = ��x,t�Q�x,t��

= �−

+ �−

+

�Q�x,t��v,��

�PL�v,�,x;t�dvd� . �13c�

From Eqs. �13� it follows that the filtered value of any func-tion of the velocity and/or scalar variables is obtained by itsintegration over the velocity and scalar sample spaces

��x,t��Q�x,t��L = �−

+ �−

+

Q̂�v,��PL�v,�,x;t�dvd� .

�14�

B. VSFMDF transport equations

To develop the VSFMDF transport equation, we con-sider the time derivative of the fine-grained density function�Eq. �11��

�

�t= − � �uk

�t

�

�vk+

��t

�

�� . �15�

Substituting Eqs. �1b� and �1c�, and Eqs. �4a� and �4b� intoEq. �15� we obtain

��

�t+

��uj

�xj= � �p

�xj−

��kj�xk

��v j

+ � �Jj��xj

− �S�� .�16�

Integration of this equation according to Eq. �10�, while em-ploying Eq. �12� results in

�PL�t

+�v jPL

�xj= −

�

��S��PL�

+�

�vi�� 1

�� p

�xi�v,��

�

PL−

�

�vi�� 1

�� ji

�xj�v,��

�

PL+

�

�� 1

�� Ji�

�xi�v,��

�

PL . �17�This is an exact transport equation and indicates that theeffects of convection, the second term on the left-hand side�LHS�, and chemical reaction, the first term on the right-handside �RHS�, appear in closed forms. The unclosed terms de-note convective effects in the velocity-scalar sample space.Alternatively, the VSFMDF equation can be expressed as

�PL�t

+�viPL

�xi=

�

�xi��PL/��

�xi − �

��S��PL� +

�

�vi

� 1

�� p

�xi�v,��

�

PL�−

�2

�vi � v j

� �

��ui�xk� �uj

�xk�v,��

�

PL� − ��vi�� 1�� xj��uj�xi �v,��PL�+

2

3

�

�vi

� 1

��

�xi��uj

�xj�v,��

�

PL� − 2 �2�vi � �� ui�xj� ��xj �v,��PL�−

�2

��

��

��xk

��xk

�v,��

PL� . �18�



This is also an exact equation. The unclosed terms are exhib-ited by the conditional filtered values as shown by the last sixterms on the RHS.

C. Modeled VSFMDF transport equation

For closure of the VSFMDF transport equation, we con-sider the general diffusion process,52 given by the system ofstochastic differential equations �SDEs�:

dXi+�t� = Di

X�X+,U+,�+;t�dt + BijX�X+,U+,�+;t�dWj

X�t�

+ FijXU�X+,U+,�+;t�dWj

U�t�

+ FijX��X+,U+,�+;t�dWj

��t� , �19a�

dUi+�t� = Di

U�X+,U+,�+;t�dt + BijU�X+,U+,�+;t�dWj

U�t�

+ FijUX�X+,U+,�+;t�dWj

X�t�

+ FijU��X+,U+,�+;t�dWj

��t� , �19b�

d��+�t� = D�

��X+,U+,�+;t�dt + B�j� �X+,U+,�+;t�dWj

��t�

+ F�j�X�X+,U+,�+;t�dWj

X�t�

+ F�j�U�X+,U+,�+;t�dWj

U�t� , �19c�

where Xi+, Ui

+, ��+ are probabilistic representations of posi-

tion, velocity vector, and scalar variables, respectively. TheD terms denote drift coefficient, the B terms denote diffu-sion, the F terms denote diffusion couplings, and the Wterms denote the Wiener-Lévy processes.53,54 To model thesecoefficients, following Refs. 9, 11, and 55–57 we utilize the

simplified Langevin model �SLM� and the linear meansquare estimation �LMSE� model50

dXi+ = Ui

+dt +� 2�

��

dWi, �20a�

dUi+ = − 1�� p��xi + 2�� xj��ui�L�xj

+1

��

�

�xj��uj�L

�xi − 2

3

1

��

�

�xi��uj�L

�xj�dt

+ Gij�Uj+ − uj�L�dt + �C0�dWi�

+� 2�

��

�ui�L�xj

dWj , �20b�

d��+ = − C��

+ − ��L�dt + S��+�dt , �20c�

where

Gij = − ��12 + 34C0�ij, � = �k ,�21�

� = C�k3/2

�L, k =

1

2�L�ui,ui� .

Here � is the SGS mixing frequency, � is the dissipation rate,k is the SGS kinetic energy, and �L is the LES filter size. Theparameters C0, C�, and C� are model constants and need tobe specified. The Fokker-Planck equation58 for FL�v ,� ,x ; t�,the joint PDF of X+ ,U+ ,�+, evolving by the diffusion pro-cess as given by Eq. �20� is

�FL�t

+�viFL�xi

=1

��

�p��xi

�FL�vi

−2

��

�

�xj��ui�L

�xj �FL

�vi−

1

��

�

�xj��uj�L

�xi �FL

�vi+

2

3

1

��

�

�xi��uj�L

�xj �FL

�vi

− Gij��v j − uj�L�FL�

�vi+

�

�xj��FL/��

�xj + �

�xj� 2�

��

�ui�L�xj

�FL�vi

+ �

��

�ui�L�xk

�uj�L�xk

�2FL�vi � v j

+1

2C0�

�2FL�vi � vi

+ C�� − ��L�FL�

��−

��S��FL��

. �22�

Equation �20� governs the evolution of Np notional particles,as explained in the subsequent section, which constitute a“stochastic particle system.” This system has inherent differ-ent characteristics from the “fluid system” governed by Eq.�1�.1 Providing a perfect model, the stochastic particle sys-tem can, at most, give a statistical description of turbulence.In view of this limitation, we aim to make a correspondencebetween the two systems at the level of one-point, one-time

statistics. This is done by equating the joint PDF of the par-ticle system with the VSFMDF, i.e.,

FL�v,�,x;t� = PL�v,�,x;t� . �23�

The stochastic processes X+�t�, U+�t�, and �+�t� are con-structed in such a way that Eq. �23� is satisfied. This resultsin similarity of the statistics obtained from the two systems.



Because of this similarity, no distinction is made betweenthese two statistics hereafter. The transport equations for thefiltered variables are obtained by integration of Eq. �22� ac-cording to Eq. �14�:

��t

+��uj�L

�xj= 0, �24a�

��ui�L�t

+��uj�Lui�L

�xj

= −�p��xi

+�

�xj�� ui�L

�xj+

�uj�L�xi

−

2

3

�

�xi��uj�L

�xj − ��L�ui,uj�

�xj, �24b�

��L�t

+��uj�L��L

�xj

=�

�xj��L

�xj − ��L��,uj�

�xj

+ ��S��L. �24c�

To have corresponding systems, the transport equations gov-erning the systems should also correspond. From Eqs. �24�, itis clear that the particle first order statistics satisfy the massconservation as well as momentum and scalar transportequations. The transport equations for the second order SGSmoments are

��L�ui,uj��t

+��uk�L�L�ui,uj�

�xk

= −��L�uk,ui,uj�

�xk+

�

�xk��L�ui,uj�

�xk

− ��L�ui,uk��uj�L�xk

− ��L�uj,uk��ui�L�xk

+ ��Gik�L�uj,uk� + ��Gjk�L�ui,uk� + ��C0��ij ,

�25a�

��L�ui,��t

+��uj�L�L�ui,��

�xj

= −��L�uj,ui,��

�xj+

�

�xj��L�ui,��

�xj

− ��L�ui,uj��L

�xj− ��L�uj,��

�ui�L�xj

+ ��Gij�L�uj,�� − ��C��L�ui,��

+ ��L�ui,S�� , �25b�

��L��,��t

+��ui�L�L��,��

�xi

= −��L�ui,��,��

�xi+

�

�xi��L��,��

�xi

− ��L��,ui��L

�xi− ��L��,ui�

��L�xi

+ 2��L

�xi

��L�xi

− 2��C��L��,��

+ ��L��,S�� + ��L��,S�� . �25c�

The implied closure for the SDEs �20� is obtained by com-paring the Fokker-Planck equation �Eq. �22�� to theVSFMDF transport equation �Eq. �18��

�

�vi

� 1

�� p

�xk�v,��

�

PL� − �2�vi � v j� �� ui�xk �uj�xk�v,��PL� − ��vi� 1�� xj��uj�xi �v,��PL�+

2

3

�

�vi

� 1

��

�xi��uj

�xj�v,��

�

PL� − 2 �2�vi � �� ui�xj� ��xj �v,��PL�−

�2

��

� �

��xi

� ��xi

�v,��

PL�=

1

��

�p��xi

�FL�vi

−2

��

�

�xj��ui�L

�xj �FL

�vi−

1

��

�

�xj��uj�L

�xi �FL

�vi+

2

3

1

��

�

�xi��uj�L

�xj �FL

�vi

− Gij��v j − uj�L�FL�

�vi+

�

�xj� 2�

��

�ui�L�xj

�FL�vi

+ �

��

�ui�L�xk

�uj�L�xk

�2FL�vi � v j

+1

2C0�

�2FL�vi � vi

+ C�� − ��L�FL�

��. �26�



The set of equations �24� and �25� may be compared withEqs. �6� and �8�. The closure at the second order level is

− 2��L� �ui�xk , �uj�xk + �L�uj, ��xk��uk�xi + �L�ui, ��xk��uk�xj − 23�L�uj, ��xi��uk�xk−

2

3�L�ui, ��xj��uk�xk − ��uj �p�xi�� − uj�L�p��xi

− ��ui �p�xj�� − ui�L�p��xj � ��Gik�L�uj,uk� + ��Gjk�L�ui,uk� + ��C0��ij ,

�27a�

− 2��L� �ui�xj , ��xj + �L��, ��xj��uj�xi −

2

3�L��, ��xi��uj�xj

− �� p�xi�� − ��L�p��xi � ��Gij�L�uj,�� − ��C��L�ui,�� , �27b�

��L� ��xi , ��xi � ��C��L��,��− �

��L�xi

��L�xi

. �27c�

It is clear that the transport equations implied by the modelare consistent with the original LES equations �Eqs. �6� and�8��. As indicated in Eq. �27c�, in the scalar covariance equa-tion, there is a spurious source term which is negligible athigh Reynolds number flows.

IV. NUMERICAL SOLUTION PROCEDURE

Numerical solution of the modeled VSFMDF transportequation is obtained by a hybrid finite-difference/MonteCarlo procedure. The basis is similar to those in RAS �Refs.59 and 60� and in previous FDF simulations,9–11,57 with somedifferences which are described here. For simulations, theFDF is represented by an ensemble of Np statistically iden-tical Monte Carlo �MC� particles. Each particle carries infor-mation pertaining to its position, X�n��t�, velocity, U�n��t�, andscalar value, ��n��t�, n=1, . . . ,Np. This information is up-dated via temporal integration of the SDEs. The simplestway of performing this integration is via Euler-Maruyammadiscretization.61 For example, for Eq. �19a�,

Xin�tk+1� = Xi

n�tk� + �DiX�tk��n�t + �Bij

X�tk��n��t�1/2� jX�tk��n

+ �FijXU�tk��n��t�1/2� j

U�tk��n

+ �FijX��tk��n��t�1/2� j

��tk��n, �28�

where Di�tk�=Di�X�n��tk� ,U�n��tk� ,��n��tk� ; tk�, . . ., and �tk�’sare independent standardized Gaussian random variables.

This scheme preserves the Itô character of the SDEs.62

The computational domain is discretized on equallyspaced finite-difference grid points. These points are used forthree purposes: �1� to compute the pressure field, �2� to iden-tify the regions where the statistical information from theMC simulations are obtained, and �3� to perform a set ofcomplementary LES primarily by the finite-difference meth-odology for assessing the consistency and convergence of theMC results. The LES procedure via the finite-difference dis-cretization is referred to as LES-FD and will be further dis-cussed below. Statistical information is obtained by consid-ering an ensemble of NE computational particles residingwithin an ensemble domain of characteristic length �E cen-tered around each of the finite-difference grid points. Forreliable statistics with minimal numerical dispersion, it isdesired to minimize the size of an ensemble domain andmaximize the number of the MC particles.44 In this way, theensemble statistics would tend to the desired filtered values,

a�E �1

NE�

n��E

a�n� →NE→

�E→0

a�L,

�29�

�E�a,b� �1

NE�

n��E

�a�n� − a�E��b�n� − b�E� →NE→

�E→0

�L�a,b� ,

where a�n� denotes the information carried by the nth MCparticle pertaining to transport variable a.

To reduce the computational cost, a procedure involvingthe use of nonuniform weights10 is also considered. This pro-cedure allows a smaller number of particles in regions wherea low degree of variability is expected. Conversely, in re-gions of high variability, a large number of particles is al-lowed. It has been shown10,44 that the sum of weights withinthe ensemble domain is related to filtered fluid density as

�� m

VE�

n��E

w�n�, �30�

where VE is the volume of ensemble domain and �m is themass of particle with unit weight. The Favre-filtered value ofa transport quantity Q�v ,�� is constructed from theweighted average as

Q�L ��n��Ew

�n�Q�v�n�,��n��

�n��Ew�n� . �31�

With uniform weights,44 the particle number density de-creases in regions of low density such as the reaction zone.The implementation of variable weights allows the increasein particle density without increasing the particle numberdensity in these regions. The LES-FD solver is based on thecompact parameter finite-difference scheme.63,64 This is avariant of the MacCormack scheme in which fourth-ordercompact differencing schemes are used to approximate thespatial derivatives, and second-order symmetric predictor-corrector sequence is employed for time discretization. All ofthe finite-difference operations are conducted on fixed gridpoints. The transfer of information from the grid points to theMC particles is accomplished via a linear interpolation. The



transfer of information from the particles to the grid points isaccomplished via ensemble averaging as described above.

The LES-FD procedure determines the pressure fieldwhich is further used in the MC solver. The transport equa-tions to be solved by LES-FD solver include unclosed sec-ond order moments which are obtained from the MC solver.The LES-FD also determines the filtered velocity and scalarfields. That is, there is a “redundancy” in the determinationof the first filtered moments as both the LES-FD and the MCprocedures provide the solution of this field. This redun-dancy is actually very useful in monitoring the accuracy ofthe simulated results as shown in previous works.10,11,57,59,60

V. RESULTS

A. Flows simulated

The following flow configurations are considered:

�i� A three-dimensional temporally developing mixinglayer involving transport of a passive scalar variable.

�ii� A three-dimensional spatially developing mixing layerinvolving chemical reaction with nonpremixedreactants.

Simulation �i� is used to assess the consistency and the over-all capabilities of the VSFMDF methodology. This predic-tion is compared with data obtained by direct numericalsimulation �DNS� of the same layer. Simulation �ii� is per-formed to demonstrate the predictive capabilities of theVSFMDF in reacting flows. The appraisal of these simula-tions is made by comparing to laboratory data.

In the representation below, x, y, and z denote thestreamwise, the cross-stream, and the spanwise directions,respectively. The velocity components along these directionsare denoted by u, v, and w in the x, y, and z directions,respectively. The temporal mixing layer consists of two par-allel streams traveling in opposite directions with the samespeed.65–67 The filtered streamwise velocity, scalar and tem-perature fields are initialized with hyperbolic tangent profileswith free-stream conditions as u�L=1, ��L=1 on the topand u�L=−1, ��L=0 on the bottom. These simulations areperformed with several density ratios defined as s=�1 /�2where �1 and �2 denote the �� on the top and bottom free-streams, respectively. The density ratios of s=1,2 ,4 ,8 areconsidered. With the uniform initial pressure field, the initial

T�L field is set equal to the inverse of �� field based onideal-gas equation of state. The length Lv is specified suchthat Lv=2

NP�u, where NP is the desired number of successivevortex pairings and �u is the wavelength of the most unstablemode corresponding to the mean streamwise velocity profileimposed at the initial time. The flow variables are normalizedwith respect to the half initial vorticity thickness, Lr= ��v�t=0�� /2 ��v=�U / ��u�L /�y�max, where u�L is the Reynolds-averaged value of the filtered streamwise velocity and �U isthe velocity difference across the layer�. The reference veloc-ity is Ur=�U /2.

Simulation �i� is conducted for a cubic box, 0�x�L,−L /2�y�L /2, 0�z�L where L=Lv /Lr. The 3D field isparameterized in a procedure somewhat similar to that by

Vreman et al.68 The formation of the large scale structuresare expedited through eigenfunction based initialperturbations.69,70 This includes two-dimensional66,68,71 andthree-dimensional66,72 perturbations with a random phaseshift between the 3D modes. This results in the formation oftwo successive vortex pairings and strong three-dimensionality. The flow configuration in simulation �ii� issimilar to the one considered in the laboratory experimentsof Mungal and Dimotakis.37 In these experiments, a heat-releasing reacting planar mixing layer consists of a low con-centration of hydrogen �H2� in one stream and a low concen-tration of fluorine �F2� in the other stream. Both reactants arediluted in nitrogen �N2� with the level of dilution determin-ing the extent of heat release. The computational domainextends 54.8 cm�36.6 cm�4.6 cm in x, y, and z directions,respectively, which covers the whole region considered ex-perimentally including x=45.7 cm where the measured dataare reported. This flow is dominated by large scale two-dimensional structures.38 Jaberi et al.10 demonstrated the suf-ficiency of their two-dimensional simulations to capture hy-drodynamic features of this flow and obtained goodagreement with laboratory data. Therefore, to reduce thecomputational costs, the domain size in the z direction isconsidered to be minimal. It is shown that this size is suffi-ciently large to let three-dimensional large-scale structuresdevelop. In order to simulate a “naturally” developing shearlayer, a modified variant of the forcing procedure suggestedin Ref. 73 is utilized. The cross-stream velocity componentat the inlet is forced at the most unstable mode as well asfour �sub- and super-� harmonics of this mode with a randomphase shift. In these simulations, the variables are normal-ized by the values in the high-speed stream. The referencelength Lr=45.7 cm which is the location in the experimentwhere the visual width of the layer is 7.4 cm.

Reaction mechanism

The chemical reaction considered in simulation �ii� in-volves the reaction of hydrogen �H2� and fluorine �F2� asrepresented by Dimotakis37

H2 + F2 → 2HF, �Q = − 130 kcal−1 mol−1, �32�

where �Q is the heat of reaction. This reaction is sufficientlyenergetic that 1% of F2 and 1% of H2 in nitrogen will pro-duce an adiabatic flame temperature of 93 K above ambient.Thus, dilute concentrations produce significant temperaturerise. The reaction actually consists of two second-order chainreactions with chemical times that are fast compared to thefluid mechanical time scales. Mungal and Dimotakis37 indi-cate that for the conditions of the experiment, the H2−F2mixture is in a stable region. Thus, for the chain reactions toproceed rapidly, it becomes crucial to provide some means toensure the presence of F atoms. The technique used in theexperiment consists of introducing a small amount of nitricoxide in the hydrogen reactant vessel. While it is necessaryto add nitric oxide to initiate the reaction, the addition ofexcessive amounts would deplete the available F atoms. Itwas determined experimentally that by keeping the productof nitric oxide and fluorine concentrations at 0.03% the re-



actions proceed rapidly. In this regard, it is important to notethat the addition of 50% more nitric oxide showed no sig-nificant changes in the mean temperature profile. Thus, thechemistry can be considered to be relatively fast. This is alsoshown in Ref. 10 where they considered both finite-rate andfast chemistry models and observed negligible differences.Therefore, the fast chemistry model is considered here.

B. Numerical specifications

Simulations are conducted on equally spaced grid points.Simulation �i� has grid spacings �x=�y=�z=� with thenumber of grid points 1933 and 333 for DNS and LES, re-spectively. In this simulation the Reynolds number is Re=UrLr� �=50. To filter the DNS data, a tophat function ofthe form below is used with �L=2 �,

G�x� − x� = �i=1

3

G̃�xi� − xi� ,

�33�

G̃�xi� − xi� = �1

�L, �xi� − xi� �

�L

2,

0, �xi� − xi� ��L

2.

No attempt is made to investigate the sensitivity of the re-sults to the filter function51 or the size of the filter.74

Simulation �ii� is conducted on 81�81�12 grid pointsin x, y, and z directions, respectively. The number of gridpoints in the z direction is sufficient and provides the samegrid resolution as in y direction. The LES filter size in thissimulation is �L=2� where �= ��x�y�z�1/3. Hyperbolictangent functions are utilized to assign the velocity, scalar,and temperature profiles for simulation �i�, and at the inletfor simulation �ii�. The same profiles are also used to initial-ize the particle values in both simulations and to assign theincoming particle values in simulation �ii�. In simulation �ii�,the characteristic boundary condition75 is used at the inletboundary. The pressure boundary condition76 is used at theoutflow boundary and a zero-derivative boundary conditionis implemented at cross-stream boundaries.

All simulations are performed with variable particleweights.10 In simulation �i�, the MC particles are initiallydistributed throughout the computational region uniformly ina random fashion. The particle weights are set according tofiltered fluid density at the initial time. In simulation �ii�, theMC particles are initially distributed randomly within region−0.15Lr�y�0.15Lr with a uniform distribution. In simula-tion �ii�, the composition and velocity components of theincoming particles are the same as those in the experimentand consistent with those on LES grid points. The initialnumber of particles per grid point is NPG=320 �NE=40� andthe ensemble domain size ��E� is set equal to half the gridspacing in each �x, y or z� direction. The effects of both ofthese parameters are assessed in the previous studies.9–11,57

All results are analyzed both “instantaneously” and “statisti-cally.” In the former, the instantaneous contours �snapshots�and scatter plots of the variables of interest are analyzed. Inthe latter, the “Reynolds-averaged” statistics constructedfrom the instantaneous data are considered. These are con-structed by spatial averaging over homogeneous directions

FIG. 1. Comparison of Reynolds-averaged values of different density esti-mates obtained from MC with LES-FD density �� in temporal mixinglayer simulations with s=2 at t=80. The thick solid line denote LES-FDprediction. The symbols denote: �triangle� particle weight density �Eq. �30��and �circle� MC density �Eq. �34��.

FIG. 2. Cross-stream variation of theReynolds-averaged values of �a� ��L,and �b� �L�� ,�� in temporal mixinglayer simulations at t=80 with s=2.The thick solid lines and circles denoteLES-FD and MC predictions,respectively.



�x and z� in simulation �i� and by time averaging in simula-tion �ii�. All Reynolds-averaged results are denoted by anoverbar.

No attempt is made to determine the appropriate valuesof the model constants; the values suggested in the literatureare adopted77 C0=2.1, C�=1, and C�=1. The influences ofthese parameters are assessed in Ref. 57. The value of Sc�=Pr� is 1 for all the simulations.

C. Consistency assessments

The objective of this section is to demonstrate the con-sistency of the VSFMDF formulation. Since the accuracy ofthe LES-FD procedure is well-established �at least for thefirst order filtered quantities�, such a comparative assessmentprovides a good means of assessing the performance of theMC solution.

The uniformity of the MC particles is checked by moni-toring their distributions at all times. The particle numberdensity inside the ensemble domain �NE� normalized by theinitial NE �here, initial NE=40� varies around unity �as itshould� while the particle weight density should be close tothe filtered fluid density. The Reynolds-averaged densityfields as obtained by both LES-FD and MC are shown in Fig.1. As depicted, the particle weight density �see Eq. �30�� andthe MC density, defined as

��E � ��n��Ew�n��RT�n�/p��n��Ew�n� −1

, �34�

are in very good agreement with the filtered density obtainedfrom LES-FD.

The consistency is checked for the first two moments. AsFig. 2�a� shows, the cross-stream variation of filtered scalaris consistently predicted by LES-FD and MC. The same con-sistency is also observed for all other first moments. These

FIG. 3. Scatter plots of several veloc-ity and scalar moments in temporalmixing layer simulations with s=2 att=80. �a� u�L, �b� v�L, �c� ��L, and�d� �L�� ,��. The solid and dashedlines denote the linear regression and45° lines, respectively. r denotes thecorrelation coefficient.

FIG. 4. �Color� Contour surfaces of the instantaneous ��L field in temporalmixing layer simulations with s=2 obtained from VSFMDF at t=80.



moments show very little dependence on the values of �Eand NE consistent with previous FDF simulations.

9–11,57 Theconsistency of the second order scalar correlation is alsoshown in Fig. 2�b�. The predictions via MC show closeagreement with LES-FD �the differences are due to statisticalerrors�. With NE and �E chosen, this demonstration is con-

sistent with previous assessment studies on the scalar,9,10 thevelocity,11 and the velocity-scalar FDFs.57 All other secondorder SGS moments behave similarly.

Complementary consistency assessment is obtained bypresenting the scatter plots of instantaneous results obtainedfrom LES-FD and MC. Figures 3�a� and 3�b� show the scat-

FIG. 5. �Color online� Contour plots of the ��L field on a spanwise plane at z=0.75L, t=80 in 3D temporal mixing layer simulations with s=2 as obtainedby: �a� DNS �filtered�, �b� VSFMDF, and �c� Smagorinsky.



ter plots of the velocity components in streamwise and cross-stream directions. For all the velocity components, there is ahigh level of correlation between LES-FD and MC results. InFig. 3�c�, the consistency of the filtered passive scalar field isdemonstrated. For all the first order moments the linear re-gression line almost coincides with the 45� line. The scatterplot of scalar correlation is shown in Fig. 3�d�. As shown, thescalar correlation shows increased statistical variations and

hence, decreased correlation coefficient. The high level ofcorrelations for all these quantities further establishes theconsistency of the VSFMDF methodology.

D. Validation via DNS data

The objective of this section is to analyze some of thecharacteristics of the VSFMDF via comparative assessments

FIG. 6. �Color online� Contour plots of the ��L field on a streamwise plane at x=0.25L, t=80 in 3D temporal mixing layer simulations with s=2 as obtainedby: �a� DNS �filtered�, �b� VSFMDF, and �c� Smagorinsky.



against DNS of a three-dimensional temporal mixing layer.In addition, comparisons are also made with LES via the“conventional” Smagorinsky36,78 model

�L�ui,uj� = − 2�t�Sij − 13Snn�ij� + 23k�ij ,

�L�ui,�� = − t��L

�xi,

�35�

Sij =1

2� �ui�L

�xj+

�uj�L�xi

,�t = C��L

2S, t =�tSct

,

C�=0.04, Sct=1, S=�SijSij and �L is the characteristiclength of the filter. The isotropic part of SGS stress is ex-pressed using Yoshizawa’s79 expression

k = CI�L2S2. �36�

Yoshizawa’s constant of CI=0.18 is adopted from the dy-namic simulations of Moin et al.80

For comparison, the DNS data are filtered from the origi-nal high resolution 1933 points to the coarse 333 points. Inthe comparisons, we also consider the “resolved” and the“total” components of the Reynolds-averaged moments. Theformer are denoted by R�a ,b� with R�a ,b�= �a�L− a�L��b�L− b�L�; and the latter is r�a ,b� with r�a ,b�= �a− ā��b− b̄�. In DNS, the “total” components are directly avail-able, while in LES they are approximated by r�a ,b��R�a ,b�+�L�a ,b�.

68

Figure 4 shows the instantaneous isosurface of the ��Lfield at t=80. By this time, the flow is going through pairingsand exhibits strong 3D effects. This is evident by the forma-tion of large scale spanwise rollers with the presence of sec-ondary structures in streamwise planes,69 as also illustratedin Figs. 5 and 6. These figures show the scalar fields obtainedfrom DNS, VSFMDF, and the Smagorinsky model on planesin the spanwise and streamwise directions. As Fig. 5 shows,the two neighboring rollers are being paired and in Fig. 6, theformation of secondary structures is evident. As illustrated inthese figures and consistent with the previous works,11 theresults obtained from the Smagorinsky closure are overlysmooth. This is due to the excessive amount of SGS diffu-

sion with the Smagorinsky model. As shown, there is moreresemblance in structures predicted by VSFMDF and DNS.

The effect of density variations on turbulence is studiedin simulation �i�. Free-stream density ratios of s=1,2 ,4 ,8are considered in both VSFMDF and DNS. Figure 7�a�shows the filtered fluid density field as predicted byVSFMDF and DNS. The level of agreement betweenVSFMDF and DNS is satisfactory. The streamwise velocityfields predicted by VSFMDF and DNS are shown in Fig.7�b� for selected density ratios. This figure exhibits the gen-erally good agreement between VSFMDF and DNS results.In addition, it is also indicative of the accurate prediction ofshear layer center location by VSFMDF. As the density ratioincreases, the shear layer center, defined as the dividingstreamline position �the position where u�L is equal to theaverage of the free stream velocities�, is shifted further to thelow-density side. As a result, the peak values of the Reynoldsstresses and scalar fluxes also show a shift to the low-densityside. This shift is known to be responsible for the decreasedcorrelation between density and velocity components39 andhence, reduction in turbulent production terms. The growthrate of a temporally developing mixing layer is proportionalto the integrated turbulent production terms.81 Therefore, de-crease in turbulent production results in reduction of shearlayer growth rate. This is evidenced in Fig. 8 which showsthe temporal evolution of the momentum thickness definedas81

��t� =1

�1��u�2�

−

+

��u1 − u�L��u�L − u2�dy , �37�

where �u=u1−u2, u1 and u2 are top and bottom free-streamstreamwise velocity components, respectively. As shown, the

FIG. 7. Cross-stream variation ofReynolds-averaged �a� density, and �b�streamwise velocity at t=80 in tempo-ral mixing layer simulations. The solidlines with white symbols denoteVSFMDF predictions. The black sym-bols denote DNS predictions. Thesymbols denote: �diamond� s=1;�circle� s=2; �triangle� s=4; and�square� s=8.

FIG. 8. Temporal variation of the momentum thickness in temporal mixinglayer simulations. The solid lines with white symbols denote the VSFMDFpredictions. The black symbols denote DNS predictions. The symbols de-note: �diamond� s=1; �circle� s=2; �triangle� s=4; and �square� s=8.



shear layer growth rate reduction with density ratio is repre-sented well by VSFMDF.

The following comparative assessments of VSFMDF areshown for the density ratio of s=2. Similar agreement isobserved for cases with other density ratios �but are notshown�. The Reynolds-averaged values of the filtered tem-perature field at t=80 are shown in Fig. 9�a�. The filtered andunfiltered DNS data yield virtually indistinguishable results.The Smagorinsky model underpredicts the spread of thelayer due to dissipative nature of this model. All VSFMDFpredictions compare well with DNS data in predicting thespread of the layer. This is also evident in Fig. 9�b� whichshows the temporal variation of the “scalar thickness,”

�s�t� = �y��L = 0.9�� + �y��L = 0.1�� . �38�

Several components of the Reynolds-averaged values of thesecond order SGS moments are compared with DNS data inFigs. 10 and 11. In general, the VSFMDF results are in better

agreement with DNS data than those predicted by the Sma-gorinsky model. In this configuration, there are no strongvelocity and scalar gradients in the streamwise and spanwisedirections and hence, a gradient-diffusion type model such asSmagorinsky is not capable of providing the correct predic-tion of scalar flux values in these directions. Consequently,the VSFMDF is expected to be more effective for LES ofreacting flows provided that the extent of SGS mixing isheavily influenced by these SGS moments.82,83

Several components of the resolved second order mo-ments are presented in Figs. 12 and 13. As expected, theperformance of the Smagorinsky model is not satisfactory asit does not predict the spread and peak values accurately. TheVSFMDF provides more reasonable predictions. The “total”components also yield very good agreement with DNS dataas shown in Figs. 14 and 15. The effects of model parametersare assessed in Refs. 11 and 57. It is important to note thatthe first and the “total” second order moments predicted by

FIG. 9. �a� Cross-stream variation ofthe Reynolds-averaged values of thefiltered temperature field at t=80, and�b� temporal variation of scalar thick-ness in temporal mixing layer simula-tions with s=2. The thick solid andthin dashed lines denote LES predic-tions using VSFMDF and Smagorin-sky closures, respectively. The whiteand black circles show the filtered andunfiltered DNS data, respectively.

FIG. 10. Cross-stream variation ofsome of the Reynolds-averaged com-ponents of �L at t=60 in temporal mix-ing layer simulations with s=2. Thethick solid and thin dashed lines de-note LES predictions using VSFMDFand Smagorinsky closures, respec-tively. The circles show the filteredDNS data.



VSFMDF are almost insensitive to these parameters. This ispleasing because these are the quantities we are primarilyinterested in when comparing with experimental data, etc.Obviously, the values cannot be set in such a way that thecontribution of the SGS components to the total components

becomes too large. With the constant values chosen forVSFMDF, while the SGS scalar flux in cross-stream direc-tion predicted by Smagorinsky is in closer agreement withDNS data, VSFMDF yields much more accurate predictionsof the resolved and consequently, the total fields.

FIG. 11. Cross-stream variation ofsome of the Reynolds-averaged com-ponents of �L at t=80 in temporal mix-ing layer simulations with s=2. Thethick solid and thin dashed lines de-note LES predictions using VSFMDFand Smagorinsky closures, respec-tively. The circles show the filteredDNS data.

FIG. 12. Cross-stream variation of

some of the components of R̄ at t=60in temporal mixing layer simulationswith s=2. The thick solid and thindashed lines denote LES predictionsusing VSFMDF and Smagorinsky clo-sures, respectively. The circles showthe filtered DNS data.



E. Validation via laboratory data

Simulation �ii� is consistent with the experimental stud-ies of Mungal and Dimotakis.37 These experiments are con-ducted with several concentration ratios, defined as �=c02/c01 where c01 and c02 denote the high- and low-speed

stream mole fractions, respectively. In the current simula-tions, the concentration ratios of �=1,2 ,4 are considered bykeeping F2 concentration at 1% and varying the H2 concen-tration from 1% to 2% and 4%. In addition, the flip experi-ments are also considered in which the low- and high-speed

FIG. 13. Cross-stream variation of

some of the components of R̄ at t=80in temporal mixing layer simulationswith s=2. The thick solid and thindashed lines denote LES predictionsusing VSFMDF and Smagorinsky clo-sures, respectively. The circles showthe filtered DNS data.

FIG. 14. Cross-stream variation of r̄ att=60 in temporal mixing layer simula-tions with s=2. The thick solid andthin dashed lines denote LES predic-tions using VSFMDF and Smagorin-sky closures, respectively. The whiteand black circles show the filtered andunfiltered DNS data, respectively.



compositions are simply switched to attain the inverse con-centration ratios ��=1,1 /2 ,1 /4�. These simulations are con-ducted only via VSFMDF, as implementation of DNS andLES-FD is not possible for this flow.

The three-dimensionality of the flow is evident by thepresence of primary and secondary structures, as shown inFig. 16�a�. This figure shows the contour surfaces of theinstantaneous filtered scalar field. Figures 16�b� and 16�c�show the instantaneous temperature field as obtained byLES-FD and VSFMDF. The resemblance of structures inthese figures, is an indication of the consistency of thesesimulations. The time series of the filtered temperature fieldrecorded by 15 probes across the layer, are shown in Fig.17�a�. These probes are located at x=45.7 cm downstreamand are symmetrically distributed in cross-stream directionabout the centerline with the vertical distance of 0.457 cmbetween each two. The high-speed stream located on top andcarries 1% H2 and the low-speed stream is in the bottomwith 1% F2 composition. In this figure, the horizontal axiscorresponds to the nondimensional time starting at one flow-through time. The vertical axis for each section representsthe temperature ranging from the ambient to the maximumtemperature recorded by each probe �denoted as Tmax�. Sev-eral features observed experimentally37 are also present inthese time series, namely, the presence of large, hot struc-tures; the cold regions extending deep into the layer and thenear-uniformity of temperature within the structure. The non-uniformity of temperatures in previous simulations10 was at-tributed to the lack of proper small-scale mixing due to two-dimensionality of their simulations. The present simulationssubstantiate this, as the more effective small-scale mixing inthree-dimensional simulations tends to make the temperature

more uniform inside the structures. Figure 17�b� shows acomparison of VSFMDF predictions with experimental data.The time-averaged filtered temperature profile correspondingto the case with concentration ratio �=1 is considered. Thepeak value of the temperature profile and the spread of layerare both predicted well.

FIG. 15. Cross-stream variation of r̄ att=80 in temporal mixing layer simula-tions with s=2. The thick solid andthin dashed lines denote LES predic-tions using VSFMDF and Smagorin-sky closures, respectively. The whiteand black circles show the filtered andunfiltered DNS data, respectively.

FIG. 16. �Color� Contour plots of the instantaneous filtered �a� passivescalar, and �b�, �c� temperature �K� fields on a spanwise plane as obtainedfrom LES-FD and VSFMDF, respectively, in 3D spatial mixing layersimulations.



The flip experiment predictions also demonstrate thesame features as the laboratory observations. The time-averaged filtered temperature profiles in these predictions areintegrated along the cross-stream direction to obtain theproduct thicknesses, as defined in the experiment,37

�P1 = �−

+ CpT�Lc01�Q

dy, �P2 = �−

+ CpT�Lc02�Q

dy , �39�

where Cp is the molar heat capacity of the carrier gas and �Qis the amount of heat release per mole of the reactant. Figure

18 shows the comparison of product thicknesses obtainedfrom VSFMDF with the experimental data. Consistent withthe experiment, the 1% thickness �1 is used to normalize theproduct thicknesses. The 1% thickness is defined as the dis-tance at which the mean temperature rise is equal to 1% ofthe maximum mean temperature. In the experiment, a meanvalue of �1 / �x−x0�=0.165 �where x−x0=45.7 cm� is used tonormalize all the product thicknesses. As shown in this fig-ure, at low concentration ratios, the product thicknesses varyalmost linearly with the concentration ratio, as the low-speedreactant reacts with an excessive amount of high-speed reac-tant. At high concentration ratios, the product thicknessesreach asymptotic limits. These limits correspond to the reac-tion of the high-speed reactant with an excessive amount oflow-speed reactant. As a result, the amount of product showslittle increase with the concentration ratio. As shown in thisfigure, the VSFMDF predictions compare reasonably wellwith the experimental data.

F. Computational times

To evaluate the computational requirements ofVSFMDF, the computational times are measured for simula-tion �i�. Table I lists the CPU times corresponding to LES viathe Smagorinsky36 SGS closure, VSFMDF, and DNS. Thesimulations are performed on a SGI Altix 3300 computerwith twelve 1.3 GHz Intel Itanium processors. In VSFMDF,320 particles per grid point �NE=40� are used. It is observedthat the computational time for VSFMDF is significantly lessthan that for DNS. Considering the close agreement betweenVSFMDF and DNS results, this suggests thatVSFMDF can be employed for simulations of reacting flowsfor which DNS is not feasible.

FIG. 17. �a� Time series of filtered temperature field at x=45.7 cm anddifferent cross-stream locations across the shear layer, and �b� cross-streamvariation of time-averaged filtered temperature field in spatial mixing layersimulations as predicted by VSFMDF. Tmax and Tflm denote the maximumrecorded by each probe and the adiabatic flame temperatures, respectively.The circles represent the experimental data.

FIG. 18. Product thickness based on�a� high-speed stream, and �b� low-speed stream concentrations as ob-tained by VSFMDF in spatial mixinglayer simulations.

TABLE I. Computational times for the three-dimensional temporal mixinglayer simulations.

SimulationGrid

resolutionNormalized CPU time

per unit simulation time

Smagorinsky 333 1

VSFMDF 333 15.6

DNS 1933 1655.2



VI. SUMMARY AND CONCLUDING REMARKS

The filtered density function �FDF� methodology hasproven to be very effective for large eddy simulation �LES�of turbulent reactive flows. In previous investigations, themarginal FDF of the scalar, that of the velocity, and that ofthe joint velocity-scalar with constant-density were consid-ered. The objective of the present work is to develop the jointvelocity-scalar filtered mass density function �VSFMDF�methodology for variable-density turbulent reacting flows.For this purpose, the exact transport equation governing theevolution of the FDF is derived. It is shown that effects ofSGS convection and chemical reaction appear in closedforms. The unclosed terms are modeled in a fashion similarto those typically followed in probability density function�PDF� methods in Reynolds-averaged simulations �RAS�.The modeled FDF transport equation is solved numericallyby a Lagrangian Monte Carlo �MC� scheme via consider-ation of a system of equivalent stochastic differential equa-tions �SDEs�. These SDEs are discretized via the Euler-Maruyamma discretization.

The consistency and accuracy of the VSFMDF are as-sessed in LES of a temporally developing mixing layer in-volving the transport of a passive scalar. This assessment ismade by comparing the moments obtained from the MCsolver with those obtained by solving the correspondingtransport equations directly by the finite-difference method�LES-FD�. The LES-FD equations are closed by includingthe moments from the MC solver. The consistency of the MCsolution are demonstrated by good agreement of the first twoSGS moments with those obtained by LES-FD. TheVSFMDF predictions are compared with those obtained us-ing the Smagorinsky36 SGS closure. All of the results arealso compared with direct numerical simulation �DNS� dataof the same flow. It is shown that the VSFMDF performswell in predicting some of the phenomena pertaining to theSGS transport. Most of the overall flow statistics, includingthe mean field, the resolved and total stresses are in goodagreement with DNS data. The temporal simulations are per-formed with several free-stream density ratios. The objectiveof these simulations is to evaluate the capability of theVSFMDF methodology to predict the variable density effectsin turbulence. It is shown that the features pertaining to vari-able density shear layers, such as shift in centerline positionand reduction in growth rate compare quite well with DNSdata. The VSFMDF methodology is also applied to a three-dimensional spatially developing shear layer. This flow in-volves a fast chemical reaction with nonpremixed reactants.The predictions are appraised by comparison with laboratorydata. The agreement is reasonably good and the VSFMDFpredictions capture many of the features of this flow as ob-served in the experiment.

ACKNOWLEDGMENTS

This work is sponsored by the U.S. Air Force Office ofScientific Research under Grant No. FA9550-06-1-0015�Program Manager: Dr. Julian M. Tishkoff�, the NationalScience Foundation under Grant No. CTS-0426857, and theOffice of the Secretary of Defense under Contract No.

FA9101-04-C-0014. The computations were performed onthe National Science Foundation Terascale Computing Sys-tem at the Pittsburgh Supercomputing Center and at the Na-tional Center for Supercomputing Applications at the Univer-sity of Illinois at Urbana.

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