Tsan-Hsing Shih

Ohio Aerospace Institute, Brook Park, Ohio 

Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio

Density Weighted FDF Equations for 

Simulations of Turbulent Reacting Flows

NASA/TM—2011-217012

May 2011
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Tsan-Hsing Shih

Ohio Aerospace Institute, Brook Park, Ohio 

Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio

Density Weighted FDF Equations for 

Simulations of Turbulent Reacting Flows

NASA/TM—2011-217012

May 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

This work is supported by the Subsonic Fixed Wing Project and the Supersonics Project of the NASA Fundamental 

Aeronautics Program. The authors highly appreciate the fruitful discussions with Dr. P. Tsao.

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Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by technical management. 

This report is a formal draft or working 

paper, intended to solicit comments and 

ideas from a technical peer group.

This report contains preliminary fi ndings, 

subject to revision as analysis proceeds.
NASA/TM—2011-217012 iii

Contents

Abstract.........................................................................................................................................................1

1.0 Introduction ..........................................................................................................................................1

2.0 Density Weighted Filtered Density Function (DW-FDF) and Filtered Turbulent Variables...............2

2.1 Fine Grained Probability Density Function (FG-PDF) f tf t U

′ ′ (Vx x ;,, ;, ) Φ (ψ ) ......................2

2.2 Filtered Turbulent Variables and DW-FDF F tf t U (Vx x ;,, ;, ) Φ (ψ ) ......................................4

2.2.1 Filtered Turbulent Variables......................................................................................... 4

2.2.2 Definition of DW-FDF.................................................................................................. 5

2.2.3 DW-FDF Mean 〈 〉L ....................................................................................................... 7

3.0 Traditional DW-FDF Equations for F tF t U ( ; , ), ( ; , ) Vx x Φ ψ .............................................................8

3.1 Traditional DW-FDF Equation for F t U ( ; ,) V x .........................................................................8

3.2 Traditional DW-FDF Equation for F t Φ( ; ,) ψ x ..........................................................................8

4.0 DW-FDF Equations Derived From Filtered Navier-Stokes Equations................................................9

4.1 Filtered Compressible Navier-Stokes Equations ........................................................................9

4.2 DW-FDF Equation for F xt U ( ; ,) V ............................................................................................11

4.3 DW-FDF Equation for F t Φ( ; ,) ψ x ............................................................................................13

4.4 Modeling of Unclosed Terms ...................................................................................................14

4.5 Summary...................................................................................................................................15

5.0 Concluding Remarks..........................................................................................................................16

References...................................................................................................................................................16
NASA/TM—2011-217012 iv
NASA/TM—2011-217012 1

Density Weighted FDF Equations for Simulations of 

Turbulent Reacting Flows

Tsan-Hsing Shih

Ohio Aerospace Institute

Brook Park, Ohio 44142

Nan-Suey Liu

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

In this report, we briefly revisit the formulation of density weighted filtered density function (DWFDF) for large eddy simulation (LES) of turbulent reacting flows, which was proposed by Jaberi et al.

(Ref. 1). At first, we proceed the traditional derivation of the DW-FDF equations by using the fine 

grained probability density function (FG-PDF), then we explore another way of constructing the DWFDF equations by starting directly from the compressible Navier-Stokes equations. We observe that the 

terms which are unclosed in the traditional DW-FDF equations are now closed in the newly constructed 

DW-FDF equations. This significant difference and its practical impact on the computational simulations

may deserve further studies. 

1.0 Introduction

The concept of filtered density function (FDF) method (Ref. 2) was extended to the compressible 

turbulent reacting flow by introducing a filtered mass density function (FMDF) (Ref. 1), which is a 

density weighted FDF and is referred as DW-FDF in this report. The work was considered as the 

extension of the probability density function (PDF) method (Pope (Ref. 3)) to the LES simulation with 

the advantage of its direct simulation of turbulence-chemistry interaction without modeling.

This report follows our previous work on the conservational PDF equations (Shih and Liu (Ref. 4)) to 

explore different ways of constructing the DW-FDF equations for simulations of filtered turbulent 

reacting flows. It will be shown that there are significant differences between the FMDF equations and 

the newly proposed DW-FDF equations, with the latter requiring much less empirical modeling. 

In Section 2.0, we review the definition of DW-FDF starting from a delta function of a turbulent 

random variable, which is called in the literature as the fine grained probability density function (FGPDF) f ′. The definition of a filtered turbulent variable used in LES is also briefly reviewed. Then we 

explore the relationship between the DW-FDF and the filtered turbulent variable. This relationship 

suggests the concept of “DW-FDF mean”, 〈 〉L which is the analogy to the statistical mean, 〈 〉 in the PDF 

theory. A few properties of the DW-FDF mean are explored, and they are found to be useful in 

constructing the transport equations for the DW-FDF. 

In Section 3.0, we first formulate a compressible transport equation for FG-PDF, and then apply a 

stationary or homogeneous filtering on it to construct the equations that we refer as the traditional DWFDF equations, which are consistent with the FMDF equations introduced by Jaberi et al. (Ref. 1).

In Section 4.0, we adopt a similar methodology described in Reference 4 to construct the DW-FDF 

equations by directly starting from the filtered compressible Navier-Stokes equations. The resulted

equations are significantly different from the traditional FMDF equations. For example, the unclosed 

terms appearing on the right hand side of FMDF equations are now closed in the newly constructed DWFDF equations.
NASA/TM—2011-217012 2

It is noted here that the quantity DW-FDF (or FMDF) is fundamentally different from the quantity 

PDF, namely, DW-FDF is a random quantity; but PDF is a deterministic quantity. The “DW-FDF mean”

defines a density-weighted filtered turbulent random variable; but the “PDF mean” defines a turbulent

mean variable. The DW-FDF, unlike PDF, does not represent a probability density of a random variable; 

it is only a density-weighted filtered FG-PDF and is still a random variable. These fundamental 

differences between DW-FDF and PDF may require a further investigation on their respective solution 

procedures based on the stochastic differential equation methods. 

2.0 Density Weighted Filtered Density Function (DW-FDF) and Filtered 

Turbulent Variables

In this section, we will first review the basic properties of FG-PDF and its density-weighted filtered 

quantity, DW-FDF, then explore the relationship between DW-FDF and filtered turbulent variable. This 

provides the basis for further establishing the transport equations of DW-FDF.

2.1 Fine Grained Probability Density Function (FG-PDF) f tf t U

′ ′ (Vx x ;,, ;, ) Φ (ψ )

The FG-PDF for turbulent velocity and scalars (e.g., species mass fraction, internal energy) are 

defined as follows (Pope (Ref. 3)),

( ) ( )

3

1

U ( ; ,) ( ,) ( ,) i i

i

f t t U tV

=

′ V x Ux V x ≡δ − ≡ δ − ∏ (1)

( ) ( )

1

1

( ; ,) ( ,) ( ,)

N

i i

i

ft t t

+

Φ

=

′ ψ Φψ xx x ≡δ − ≡ δ Φ −ψ ∏ (2)

where δ denotes the delta function, U(x,t) is the turbulent velocity vector (U1, U2, U3), Φ(x,t) is the 

turbulent scalar array (Φ1, Φ2,, ΦN, ΦN+1), for example, N species mass fractions and one internal 

energy ΦN+1 = e, the x,t denote the physical space variable (x1, x2, x3) and the time t, V ≡ (V1, V2, V3) and 

ψ ≡ (ψ1, ψ2,, ψN, ψN+1) are the sample space variables for U(x,t) and Φ(x,t), respectively. The FG-PDF 

has the following properties that will be used in this study:

i. The Unity Integral of FG-PDF

f td f td U ( ; , 1, ) ( ; , 1 ) +∞ +∞

Φ −∞ −∞ ′ ′ = = ∫ ∫ Vx V ψ ψ x (3)

Note that (– ∞, + ∞) represents the whole domain of sample space, for the species mass fractions it really 

means (0,1).

ii. The Differential Chain

U Ui Ui i

U

iii

f fU fU U f t Ut Vt V t∂ ∂∂ ∂∂ ′′ ′ ∂   ∂ = =− =− ′   ∂ ∂ ∂ ∂∂ ∂ ∂   (4)

U Ui Ui i

U

j ij ij i j

f fU fU U f x Ux Vx V x∂ ∂∂ ∂∂ ′′ ′ ∂   ∂ = =− =−   ′ ∂ ∂ ∂ ∂∂ ∂ ∂    

(5)
NASA/TM—2011-217012 3

Note that the summation convention on the same index, e.g., “i” is implied. Therefore, we have 

, (for incompressible flow) , 1,2,3 UU i i

j Uj

j i j

f f UU U f U i j t xV t x∂∂ ∂∂ ′ ′ ∂    

+ =− +   ′   = ∂ ∂∂ ∂ ∂      

(6)

and

, (for compressible flow) , 1,2,3 U i j U i j

U

ji j

f U U f U U

f i j tx V tx

∂ρ ′ ∂ρ ′ ∂     ∂ρ ∂ρ

+ =− +   ′   = ∂∂ ∂ ∂∂      

(7)

Note that the continuity equation of compressible flow has been applied here. Similarly,

ii i

iii

ff f f

tt t t

ΦΦ Φ

Φ

∂∂ ∂ ∂ ′′ ′ ∂Φ ∂Φ   ∂Φ = =− =− ′   ∂ ∂Φ ∂ ∂ψ ∂ ∂ψ ∂   (8)

ii i

j ij ij i j

ff f f

xx x x

ΦΦ Φ

Φ

∂∂ ∂ ∂ ′′ ′ ∂Φ ∂Φ   ∂Φ = =− =−   ′ ∂ ∂Φ ∂ ∂ψ ∂ ∂ψ ∂    

(9)

and

1,2, , 1 , (for incompressible flow) 1,2,3

i i

j j

j i j

f f i N

U fU

tx t x j

Φ Φ

Φ

∂∂∂ ′ ′     ∂Φ ∂Φ  = +

+ =− +   ′    ∂ ∂ ∂ψ ∂ ∂    =    

 (10)

1,2, , 1 , (for compressible flow) 1,2,3

j i j i

ji j

f U f U i N

f

t x t x j

Φ Φ

Φ

∂ρ ′ ∂ρ ′ ∂     ∂ρΦ ∂ρΦ  = +

+ =− +   ′    ∂ ∂ ∂ψ ∂ ∂    =    

 (11)

iii. The Sifting Property

U x, V x U x, U x, V V U x, V V V x ( tf t t t t f t ) U U ′ ′ ( ; ,) ≡ δ − =δ − = ( ) ( ( ) ) ( ( ) ) ( ; ,) (12)

( ,) ( ; ,) ( ; ,) tf t f t Φ Φ Φ xx x ′ ′ ψ ψψ = (13)

More generally,

( ( )) ( )

( ) ( )

( ; ,) ( ; ,)

( ,) ( ; ,) ( ; ,)

Q tf t Q f t U U

Q tf t Q f t Φ Φ

′ ′ =

′ ′ ψ ψψ =

U x, V x V V x

Φ xx x

(14)

where Q( ) denotes a general (but not the time or space differential) function form.

iv. The Statistical Mean of FG-PDF

Equations (1) and (2) indicate that FG-PDF is a random quantity, because it is the delta function of 

the random variable U(x,t) or Φ(x,t). The basic probability theory says that a random variable and its 

(non-differential) function share the same probability density. Therefore, the probability density function 

PDF of the random variable FG-PDF is fU (V; x,t) or fΦ(ψ; x,t), which is the PDF of turbulent velocity
NASA/TM—2011-217012 4

U(x,t) or scalars Φ(x,t), respectively. It is easy to verify that the statistical mean of FG-PDF is the 

corresponding PDF (Pope (Ref. 3)):

f t t U ( ; ,) ( ( ) ) ( ) f td f t U U ( ; ,) ( ; ,) +∞

−∞ ′ =δ − ≡ δ −′ ′′ = ∫ V x U x, V V V V x V V x (15)

Similarly,

f tf t Φ Φ ′ ( ; ,) ( ; ,) ψ ψ x x = (16)

The FG-PDF is a useful quantity for mathematical derivations. It is interesting to examine the following 

expressions:

U ( ; ,) ( ) ( ) FG f td t t ′ ≡ ∫V V x V U x, = U x, (17)

( ; ,) ( ,) ( ,) FG f td t t Φ

′ ≡ ∫ψψ ψ xxx Φ =Φ (18)

which indicate that the “FG-PDF mean” defines the original random variable itself. More generally,

( ) ( ( )) ( ( ))

( ) ( ) ( )

( ; ,)

( ; ,) ( ,) ( ,)

U FG

FG

Q f td Q t Q t

Q f td Q t Q t Φ

′ ≡ =

′ ≡ =

∫

∫ ψψ ψ

V V x V U x, U x,

xx x Φ Φ

(19)

2.2 Filtered Turbulent Variables and DW-FDF F tf t U (Vx x ;,, ;, ) Φ (ψ )

First we review the definition of filtered turbulent variables conventionally used in the turbulent

simulation and the definition of DW-FDF, and then we explore the relationship between the DW-FDF 

and the filtered turbulent variables.

2.2.1 Filtered Turbulent Variables

In the conventional simulation of compressible reacting flows, we often deal with two types of 

filtered turbulent variables: one with the density weighting, the other without the density weighting. The 

filtered turbulent variable without the density weighting is denoted by φ( ,) x t and is defined as

( ,) ( , ) ( ) t t G t t dt +∞

−∞ φ =φ − ′ ′′ ∫ x x (20)

where φ is the unfiltered turbulent variable, e.g., velocity components Ui, density ρ, pressure P, species

mass fraction Φi and internal energy 

1

N

m m

m

e e

=

= Φ ∑ . The integration is over the entire time domain

– ∞ < t′ < + ∞. G(t – t′) is the time filter with a constant filter width ∆T and satisfies the following 

condition and asymptotic property:

G t t dt () 1 +∞

−∞ − = ′ ′ ∫ (21)
NASA/TM—2011-217012 5

( , ) ( ) ( , ), as 0 T t G t t dt t +∞

−∞ φ − =φ ∆ → ′ ′′ ∫ x x (22)

The density-weighted filtered turbulent variable is denoted by φ( ,)t  x and defined as

( ,)t ρφ φ =

ρ

 x (23)

It is noted here that these filtered variables φ( ,) x t , φ( ,)t  x represent the large scale turbulent variables

(Shih and Liu (Refs. 5 and 6)), and they are still random but contain relatively low frequency part of the 

turbulent motion when comparing with the unfiltered turbulent variable φ(x,t).

2.2.2 Definition of DW-FDF

Jaberi et al. (Ref. 1) introduced the density-weighted space filter operation ρ(x′,t)G(x – x′) on the FGPDF f tf t U ( ; , ), ( ; , ) Φ ′ ′ Vx x ψ and referred to them as the filtered mass density function FMDF. However,

we prefer a more intuitive name related to the existing FDF methodology (Colluci et al. (Ref. 2)), and 

refer them as the density weighted FDF (DW-FDF). With the density-weighted time filter ρ(x,t′)G(t – t′)

we define the following marginal DW-FDF as

( )

( ) ( ( ) )

( )

( ) ( ( ) )

( ; ,) ( ; ,) ( )

 ( )

( ; ,) ( ; ,) ( )

 ( )

F t t' f t G t t dt U U

t' t' G t t dt

F t t' f t G t t dt

t' t' G t t dt

+∞

−∞

+∞

−∞

+∞

Φ Φ −∞

+∞

−∞

≡ ρ ′ ′ ′′ −

=ρ δ − − ′ ′

≡ ρ ′ ′ ′′ −

=ρ δ − − ′ ′

∫

∫

∫

∫

ψ ψ

ψ

V x x, V x

x, U x, V

x x, x

x, x, Φ

(24)

Obviously, DW-FDF, i.e., FU(V; x,t) or FΦ(ψ; x,t), is still a random quantity. It satisfies the following 

“normalization” property:

( ) ( )

( ) ( )

( ; ,) ( ; ,) ( ) ( )

( ; ,) ( ; ,) ( ) ( )

F t d U U t' f t G t t dt d t' G t t dt

F t d t' f t G t t dt d t' G t t dt

∞ ∞ +∞ +∞

−∞ −∞ −∞ −∞

∞ ∞ +∞ +∞

Φ Φ −∞ −∞ −∞ −∞

ρ ′ ′ ′′ − ρ − =ρ ′ ′

= ρ ′ ′ ′′ − = ρ − =ρ ′ ′

∫ ∫∫ ∫

∫ ∫∫ ∫ ψ ψψ

V x V = x, V x V = x,

x ψ x, x x,

(25)

With the definition of DW-FDF described in Equation (24), we can exactly deduce the filtered turbulent 

variables that are conventionally defined by Equations (20) and (23). For example,

( ) ( ( ) )

( ) ( ) ( ( ) )

( ) ( ) ( )

( ; ,) ( )

( )

( ) ( ,)

F t d U t' t' G t t dt d

t' t' t G t t dt d

t' t G t t dt t t

+∞ +∞ +∞

−∞ −∞ −∞

+∞ +∞

−∞ −∞

+∞

−∞

ρδ − − ′ ′

ρδ − − ′ ′′

ρ ′ ′′ − =ρ =ρ

∫ ∫∫

∫ ∫

∫ 

V V x V = V x, U x, V V

= U x, x, U x, V V

= x, U x, U x, U x

(26)

where the sifting property of FG-PDF has been used in the second line of (26). Similarly,
NASA/TM—2011-217012 6

F td ( ; , ) ( t t ) ( , ) +∞

Φ −∞ ρ =ρ ∫ ψψ ψ Φ x = x, x Φ (27)

If we denote the left hand side of Equations (26) and (27) as the operation 〈U〉L and 〈Φ〉L, we may write 

Equations (26) and (27) as 

U ( ; ,) ( ) ( , ) L F td t t +∞

−∞ ≡ = ρ ∫ U V V x V x, U x  (28)

( ; ,) ( ) ( , ) L F td t t +∞

Φ −∞ ≡ = ρ ∫ ψψ ψ Φ Φ x x, x (29)

and for the function Q(U(x,t)) or W(Φ(x,t)),

( ) ( ) ( )

( ) ( ) ( )

( ) ( ; ,) ( ,)

( ) ( ; ,) ( ,)

L U

L

Q Q F td tQ t

W W F td tW t

+∞

−∞

+∞

Φ −∞

= = ρ

= = ρ

∫

Φ ψψ ψ Φ ∫





U V V x V x, U x

x x, x

(30)

Furthermore, we may consider the derivatives ∇P, ∇U, ∇Φ as the new random quantities and legitimately 

write

 , ,   L LL ∇ =ρ∇ ∇ ∇ =ρ∇ ∇ =ρ∇ P PP U U Φ Φ (31)

However, the operation 〈 〉L does not have the differential commute property, i.e.

, , LL L L L L ∇ ≠∇ ∇ ≠∇ ∇ ≠∇ P P U U Φ Φ (32)

because   ( )   ( )   ρ∇ ≠∇ ρ ρ∇ ≠∇ ρ ρ∇ ≠∇ ρ P P , , U U Φ Φ ( ).

It can be verified that 

( ; ,) ( )( ,) i j i jU i j L

UU VV F t d t UU t +∞

−∞ = ρ ∫ V x V = x, x (33)

We can also write j i L

U Φ for the joint variables as

, ( , ; ,) ( )( ,) j i j iU j i L

U V F td d tU t +∞

Φ −∞ Φ= ψ =ρ Φ ∫ V x V x, x ψ ψ (34)

where, FU,Φ(V,ψ; x,t) is the joint DW-FDF defined as 

FU , ( , ; ,)t t t ( ) ( ( ) ) ( ( t G t t dt ) ) ( ) +∞

Φ −∞ = ρ δ −δ − − ′ ′ ′ ′′ ∫ V x x, U x, V x, ψ ψ Φ (35)

which also satisfies the normalization property:

FU , ( , ; ,)t d d ( t' G t t dt ) ( )

∞ ∞ +∞

Φ −∞ −∞ −∞

= ρ − =ρ ′ ′ ∫ ∫ ∫ V x V x, ψ ψ (36)
NASA/TM—2011-217012 7

2.2.3 DW-FDF Mean 〈 〉L

Analogous to ∫V fU (V; x,t)dV = 〈U (x,t)〉, which defines the statistical mean of the random velocity

by its PDF fU, we may refer the operation 〈 〉L as the “DW-FDF-mean”, which defines the density 

weighted filtered turbulent variable, e.g., L U U = ρ , ji ji L

U U Φ =ρ Φ (see Eqs. (28) to (34)). From 

Equation (35), we may follow Pope (Ref. 3) to define a “conditional” DW-FDF on the condition Φ = ψ as

, ( , ; ,) ( ; ,) ( ; ,)U

U

F t

F t

F t

Φ

Φ

Φ

≡ ψψ

ψ

V x

V x

x

(37)

and the “conditional DW-FDF-mean” as 

( ) ( ( ) ) ( ( ) )

( ) ( ) ( ( ) )

,

1 ( ,) ( ; ,) ( , ; ,) ( ; ,)

1 ( )

1 ( )

t L F td U F td U F t

t' t' t' G t t dt d

F

t' t' t G t t dt

F

+∞ +∞

Φ Φ −∞ −∞ Φ

+∞ +∞

−∞ −∞ Φ

+∞

−∞ Φ

≡

ρ δ −δ − − ′ ′

ρ δ −− ′ ′′

∫ ∫

∫ ∫

∫

ψ ψ ψ

ψ

ψ

ψ

U x V V x V= V V xV

x

= V x, U x, V x, V

= x, U x, x,

Φ

Φ

(38)

Then we have

( ) , ( , ) ( ; , ) ( ) U ( , ; ,) ( ,) L t' t f t G t t dt F t d F t +∞ +∞

Φ Φ Φ −∞ −∞ ρ ′′ ′ ′ ′ − = ⋅ ∫ ∫ x, U x x ψ ψψ V V x V= U x (39)

and

( )

( )

( )

, ( ,) ( , ; ,)

(,) ( )

( ,)

F t d L F td d U

t t G t t dt

t t

+∞ +∞ +∞

Φ Φ −∞ −∞ −∞

+∞

−∞

⋅ =

= ρ − ′ ′ ′′

= ρ

∫ ∫ ∫

∫

ψ ψ ψ ψ



U x V V x V

x, U x

x, U x

(40)

Equation (39) can be extended to any other turbulent quantities, for example, ∇P, ∇U, ∇Φ, Si(Φ):

( )

( )

( )

( ) ( ) ( ) ( )

(,) (; ,) ( )

(,) (; ,) ( )

(,) (; ,) ( )

(,) (; ,) ( )

U U L

U U L

L

i i i L

t' P t f t G t t dt F P

t' t f t G t t dt F

t' t f t G t t dt F

t' S t f t G t t dt F S F S

+∞

−∞

+∞

−∞

+∞

Φ Φ −∞

+∞

Φ ΦΦ −∞

ρ ∇ ′′ ′ ′ ′ − ⋅∇

ρ ∇ ′′ ′ ′ ′ − ⋅∇

ρ ∇ ′′ ′ ′ ′ − ⋅∇

ρ ′ ′ ′ ′′ − ⋅ =⋅

∫

∫

∫

∫

ψ ψ

ψ ψ ψ

x, x V x = V

x, U x V x = U V

x, x x =

x, x x =

Φ Φ

Φ Φ

(41)

where ∇P, ∇U, ∇Φ, are viewed as the random variables in addition to P, U, Φ.
NASA/TM—2011-217012 8

3.0 Traditional DW-FDF Equations for F tF t U ( ; , ), ( ; , ) Vx x Φ ψ

In this section, we briefly review the traditional way of deriving DW-FDF equations starting from the 

equation of FG-PDF. The “exact” but not closed DW-FDF equations are derived, which are consistent 

with the traditional FMDF equations introduced by Jaberi et al. (Ref. 1).

3.1 Traditional DW-FDF Equation for F t U ( ; ,) V x

The transport equation for the fine grained PDF with the variable density ρ(x,t) can be written as (see

Equation (7)):

2 2

3

U j U

U ij ij kk

j i ij

U f f P f SS t x V xx

∂ρ ′ ∂ρ ′ ∂ ∂∂         + = − − + μ − μδ   ′     ∂ ∂ ∂ ∂∂        

(42)

Where the right hand side of (42) is from the Navier-Stoke equation (47). 

Applying the time filtering operation G(t – t′) with a constant filter width on Equation (42) and using 

the sifting property of FG-PDF, we may obtain (noting that the differential commutation property of the

time filtering operation is valid with a constant time filter width)

1 1 2 ( ) 2 ( ) 3

U U

j U U ij ij kk

ji i j

F F P V f G t t dt f S S G t t dt

t xV x x

∞ ∞

−∞ −∞

∂ ∂ ∂ ∂         ∂   + = − − ρ   ′ − + ρ ′′ ′   μ − μδ − ′ ′     ∂ ∂ ∂     ρ ∂ ρ ∂       ∫ ∫

According to Equation (41), the DW-FDF equation for the velocity can be written as 

1 12 2

3

U U

j U U ij ij kk

ji i j L L

F F P V F F SS

t xV x x

  ∂ ∂ ∂∂ ∂     + = − − ⋅   + ⋅ μ − μδ   ∂ ∂ ∂ ρ∂   ρ ∂    

V V (43)

At this point, the velocity DW-FDF equation is general but unclosed because of the unknown terms of the 

conditional DW-FDF means. The last term in Equation (43) is referred as the molecular mixing term and 

was modeled in various empirical ways by different researchers. Later, we will see that, when using a 

different way to derive the DW-FDF equation, the molecular mixing term is automatically closed with no 

need of modeling. The same happens to the pressure gradient term.

3.2 Traditional DW-FDF Equation for F t Φ( ; ,) ψ x

Applying the time filtering G(t – t′) on the following FG-PDF equation for scalars, 

( ) ( ( ,)) j i

i i

j ij j

U f f f S t

tx x x

Φ Φ

Φ

∂ρ ′ ∂ρ ′ ∂ ∂         ∂Φ

+ = −   ′     ρΓ +ρ ∂ ∂ ∂ψ ∂ ∂          

Φ x (44)

we obtain the DW-FDF equation for the turbulent species:

( )

( ) ( ) j L 1 i

i i

j i jj L

F U F F F S

t x x x

Φ Φ

Φ Φ

∂   ∂ ∂ ∂   ∂Φ

+ = − ⋅ ρΓ   + ⋅

∂ ∂ ∂ψ ρ ∂ ∂    

ψ

ψ ψ (45)
NASA/TM—2011-217012 9

where SN+1(ψ) = 0. Equation (45) is also unclosed, because the conditional DW-FDF means need to be 

modeled. Later, we will also see that using a different derivation of DW-FDF equation, the molecular 

mixing term is closed. 

4.0 DW-FDF Equations Derived From Filtered Navier-Stokes Equations

4.1 Filtered Compressible Navier-Stokes Equations

The Navier-Stokes equations for a compressible reacting flow can be written as

0 j

j

U

t x

∂ρ ∂ρ

+ =

∂ ∂ (46)

1 2( ) 3

i j i

ij ij kk

j ij

U U U P S S

t x xx

∂ρ ∂ ρ ∂ ∂   + = − + ρν − δ   ∂ ∂ ∂∂   (47)

1 2

3

i i

kk ij ij ii kk

i i

e Ue q PS S S S S Q

tx x

∂ρ ∂ρ ∂   + = − + + ρν − +   ∂∂ ∂   (48)

( ) 1,2, , m im m m m

ii i

U Wm N

t xx x

∂ρΦ ∂ ρ Φ ∂   ∂Φ + = ρΓ + =   ∂ ∂∂ ∂  

 (49)

1

N n

n n

T P R

w =

Φ

= ρ ∑ (50)

( )

1

N m m i m

i i m

T qc h

x x υ

=

∂ ∂Φ = −ρ κ − ρΓ ∂ ∂ ∑ (51)

Applying the time filtering operation with a constant filter width on the above equations, we obtain the 

following filtered Navier-Stokes equations:

0 j

j

U

t x

∂ρ ∂ ρ

+ =

∂ ∂



(52)

 1 2( ) 3

i j i

ij ij kk

j ij

U U U P S S

t x xx

∂ρ ∂ ρ ∂ ∂  

+ = − + ρν − δ   ∂ ∂ ∂∂  

 (53)

 1 2

3

i i

kk ij ij ii kk

i i

e Ue q PS S S S S Q

tx x

∂ρ ∂ρ ∂   + = − + + ρν − +   ∂∂ ∂  

 (54)

 ( ) 1,2, , m im m m m

ii i

U Wm N

t xx x

∂ρΦ ∂ ρ Φ ∂   ∂Φ + = ρΓ + =   ∂ ∂∂ ∂  

  (55)

where
NASA/TM—2011-217012 10

 



1 1

1

1

or

N N n n

n n nv n

N

n N

v n n

T e R P R

wc w

R P

c w

= =

+

=

Φ Φ ρ =ρ =

ρ Φ Φ =

∑ ∑

∑

(56)

( )

1

N m m i m

i i m

T qc h

x x υ

=

∂ ∂Φ = −ρ κ − ρΓ ∂ ∂ ∑ (57)

In the above equations, κ, ν and Γ(m)

are the molecular heat conductivity, kinematic viscosity and the m-th 

species diffusivity, they have the same dimension (i.e., velocity ⋅ length). The hm, T are the enthalpy of 

species and temperature, Q is the radiation rate, Wm = ρSm is the chemical generation rate of the m-th 

species, ΦN+1 represents the internal energy e, R is the universal gas constant. These equations are general; 

however, unlike the constant density flows, further approximations for the terms on their right hand side

are required in order to complete the filtering process. One of such approximations leads to

1 2 2

3 3

j i k

ij ij kk ij

ji k

U U U

S S

xx x

    ∂ ∂ ∂

ρν − δ ≈μ + − δ       ∂∂ ∂  

  

(58)

In which, we have basically neglected the variations of μ and ρduring the filtering process, the value of μ

will be considered as the function of P T, , . Another type of approximation writes

1 2 2

3 3

j i k

ij ij kk ij

ji k

U U U

S S

xx x

    ∂ρ ∂ρ ∂ρ ρν − δ ≈ ν + − δ       ∂∂ ∂  

  

(59)

Similarly,

( ) ( ) 

1 1

N N m m m m m i m

i ii i m m

T T h qc h c

x xx x υ υ

= =

∂ ∂Φ ∂ρ ∂ ρ Φ = −ρ κ − ρΓ ≈ − κ − Γ ∂ ∂∂ ∂ ∑ ∑  (60)

() () m m m m

i i x x

∂Φ ∂ρΦ ρΓ ≈Γ ∂ ∂

 (61)

Furthermore, invoking the turbulent kinetic energy dissipation rate:

1 2

3 ij ij ii kk SS SS  

ρν − ≡ ρε      (62)

Where ν, κ, cυ and Γ(m)

are considered as the function of P T, , . 

Therefore, the filtered Navier-Stokes equations can approximately be written as 

0 j

j

U

t x

∂ρ ∂ ρ

+ =

∂ ∂



(63)
NASA/TM—2011-217012 11

 2

3

i j j i i k

ij

j ij j i k

U U U P U U U

t x xx x x x

∂ρ ∂ ρ ∂ ∂     ∂ρ ∂ρ ∂ρ + =− + ν + − δ     ∂ ∂ ∂∂ ∂ ∂ ∂    

   

(64)

i i

kk

i i

e Ue q PS Q

tx x

∂ρ ∂ρ ∂ + =− + +ρε+ ∂∂ ∂

  (65)

 ( ) 1,2, , m im m m m

ii i

U Sm N

t xx x

∂ρΦ ∂ρ Φ ∂   ∂ρΦ + = Γ +ρ =   ∂ ∂∂ ∂  

    (66)

 



1 1

1

1

or

N N n n

n n nv n

N

n N

v n n

T e R P R

wc w

R P

c w

= =

+

=

Φ Φ ρ =ρ =

ρ Φ Φ =

∑ ∑

∑

(67)

( ) 

1

N m m m i

i i m

T h

q c

x x υ

=

∂ρ ∂ρ Φ =−κ − Γ

∂ ∂ ∑  (68)

These equations are still considered as quite general, because i) they are exact if the flow becomes 

incompressible, ii) all the approximations made in Equations (64), (65) and (66) are related only to the 

molecular diffusion terms that are less important and even negligible comparing with the convection 

terms on the left hand side for turbulent flows at high Reynolds number (see Tennekes and Lumley

(Ref. 7) and Pope (Ref. 3)). For the LES simulation, Equation (63) to (66) are often used together with the 

further approximations for (67) and (68):

1 1

,

N N n n

n n n n

T RT RT P R M

w MM w = =

Φ Φ   ρ ρ =ρ = ≈ =     ∑ ∑   (69)

i

i

T

q c

x υ

∂ρ =−κ

∂

 (70)

The momentum flux ρU Ui j , the energy flux ρU e i and the species flux i m ρ Φ U are considered to be 

critically important in LES simulations and should be carefully modeled. Many models in the literature,

from the simplest Smagorinsky (Ref. 8) model to the complex two-equation models (Refs. 9 and 6)

including the dynamic procedure (Ref. 10) have been suggested. In the next Section 4.2 and 4.3, we will 

derive the DW-FDF equations directly from Equations (63) to (68).

4.2 DW-FDF Equation for F xt U ( ; ,) V

Using Equations (28) and (33), the left hand side of Equation (64) can be written as

i j i U U

i U i jU i j

j j j

U U U F F VF d VV F d V V d

t xt x t x

∞ ∞∞

−∞ −∞ −∞

∂ρ ∂ρ ∂ ∂   ∂ ∂

+= +   +

∂∂∂ ∂ ∂ ∂   ∫ ∫∫  V V = V (71)
NASA/TM—2011-217012 12

The pressure gradient term can be written as using (67)



2

1 ,

1

1

1 2

1

1

( , ; ,)

or

N

i nN U

k k vn n N

n N

i iv n n N

i nN U L k k vn n

R V F xtd d

V x cw

P R

x xc w

R V F d

V x cw

∞ ∞

+ Φ

−∞ = −∞

+

= ∞

+

−∞ =

     ∂      ψ ψ  ∂ ∂          ∂ ∂ρ Φ Φ  − =−   =  ∂ ∂   

∂      Φ Φ  ∂ ∂    

∫ ∫ ∑

∑

∫ ∑

ψ ψ V V

V V

(72)

During the above arrangement, no approximations have been made other than the integration by parts and 

a zero integration property like Eq.(74).

Note that the pressure gradient term can be closed only when the joint DW-FDF, FU,Φ, is considered,

otherwise, the “conditional DW-FDF mean” will be unavoidable. Now the molecular mixing term in 

Equation (64) can be written as

( ) ( )

2

3

2

3

2

3

or

j i m

ij

jji m

i U j U ij m U

j j i m

U

i i j U kj i m U

jj j k k m k

U U U

xx x x

VF d V F d VFd

xx x x

F V d V VF d V VF d

xx x x V x V

∞∞ ∞

−∞ −∞ −∞

∞ ∞∞

−∞ −∞ −∞

∂   ∂ρ ∂ρ ∂ρ ν + −δ   ∂∂ ∂ ∂  

  ∂∂ ∂ ∂

= ν  + − δ

∂∂ ∂ ∂  

    ∂ ∂∂ ∂ ∂ ∂ ∂   ν − ν  − δ

∂∂ ∂ ∂ ∂ ∂ ∂    

=

∂ −

∂

∫∫ ∫

∫ ∫∫

  

VV V

VVV

( ) ( ) ( )

2 2

2

3

2

3

or

i k U i j U kj i m U

jj k k k m k

U j U m U

i i i kj

j j j kk j km

i

j

V VF d V V F d V VF d

xx V x V x V

F V F V F V dV dV d

x x x Vx x Vx

V

x

∞∞ ∞

−∞ −∞ −∞

∞∞ ∞

−∞ −∞ −∞











    ∂∂ ∂∂ ∂ ∂

ν  + − δ  ∂∂ ∂∂ ∂ ∂   

  ∂∂ ∂ ∂ ∂   ∂     ν − ν +δ ν     ∂ ∂ ∂ ∂∂   ∂ ∂∂      

=

∂

− ν

∂

∫∫ ∫

∫∫ ∫

VV V

VV V

2 2 2 2

3

k U j U m U

i i kj

k j j kk j km

V F V F V F dV dV d

V x x Vx x Vx

∞∞ ∞

−∞ −∞ −∞











   ∂ ∂ ∂ ∂   ∂      − ν +δ ν      ∂ ∂ ∂ ∂∂   ∂ ∂∂        

∫∫∫ VV V

(73)

Note, on the third line of (73), we have applied the integration by parts and the following type of zero 

integration similar to the one used in PDF formulations (Ref. 3):

( ) 0 ik U

k

VV F d

V

∞

−∞

∂ = ∂ ∫ V (74)

Finally, collecting the terms that are in the integrands and factored by Vi, we obtain the transport equation 

for FU (V; x,t) as 
NASA/TM—2011-217012 13

2

1

1

2

1

1

2

3

or

N

U U U

j nN U L j k k vn n j j

U U

j kj m

k jk k jm

N

U U U

j nN U k L j k k vn n k j j

F F R F V F

t x V x cw x x

F F V V

V xx V xx

F F R F V F V

t x V x cw V x x

+

=

+

=

∂ ∂ ∂ ∂     ∂

+ =   Φ Φ + ν  ∂ ∂ ∂∂ ∂ ∂      

∂∂ ∂ ∂   ∂ ∂ 

− ν+ δ ν    ∂ ∂∂ ∂ ∂∂    

∂ ∂ ∂   ∂ ∂  ∂

+ =   Φ Φ − ν

∂ ∂ ∂∂ ∂ ∂ ∂  

∑

∑

V

V

2

3

U U

j kj m

k jk k jm

F F V V

V xx V xx

  

 

∂∂ ∂ ∂   ∂ ∂ 

− ν+ δ ν    ∂ ∂∂ ∂ ∂∂    

(75)

In Equation (75), all the terms are closed, except for the one involving the conditional DW-FDF mean 

originating from the pressure gradient. It should be noticed that one of the most important terms in the 

momentum equation, i.e., ρU Ui j , now is closed in the corresponding DW-FDF equation. And the less 

important molecular diffusion terms remain in the closed form. 

4.3 DW-FDF Equation for F t Φ( ; ,) ψ x

Similarly, we may obtain the DW-FDF equations for the scalars (i.e. species and internal energy) 

from Equations (65), (66) as follows: first, we write the terms on the left hand side of Equation (66) as

m

m m

F F d d

t t t

∞ ∞

Φ

Φ

−∞ −∞

∂ρΦ ∂ ∂

= ψ =ψ ∂ ∂ ∂ ∫ ∫ ψ ψ (76)



( )

,

, or

mi U

i

i m

i mU

i i

m i L

i

V F dd

x

U V F dd

x x

FU d

x

∞ ∞

Φ

∞ ∞ −∞ −∞

Φ

−∞ −∞ ∞

Φ

−∞

   ∂  ψ    ∂  

∂ρ Φ ∂ 

=ψ =  ∂ ∂ 

   ∂

ψ    ∂   

∫ ∫

∫ ∫

∫

ψ

ψ

ψ ψ

V

V (77)

Then, the terms on the right hand side of (66) can be written as 



( )

( ) ( )

2

( )

or

m m

i i

m m m

m

i ii i

m k m

i ki

F d

x x

F d

x xx x

F d

x x

∞

Φ

∞ −∞

Φ

−∞ ∞

Φ

−∞

 ∂ ∂    ψ Γ   ∂ ∂  

∂ ∂ρΦ ∂ ∂     

Γ =Γ ψ =      ∂ ∂∂ ∂     

 ∂   ∂ ψ −ψ Γ   ∂ ∂ψ ∂   

∫

∫

∫

ψ

ψ

ψ

(78)
NASA/TM—2011-217012 14

 ( ) ( ) k m m m

k

S F

S S Fd d

∞ ∞

Φ

Φ

−∞ −∞

∂

ρ = =− ψ ∂ψ ∫ ∫ ψ ψ ψ (79)

Where in Equations (78) and (79), we have applied the integration by parts and zero integrations similar 

to (74). Collecting all the integrand terms that factored by ψm, we obtain

( ) ( )

( ) ( )

( )

( )

, 1,2, , 1

or

, 1,2, , 1

i L m k

i i ik

i L m k k

i ki i

F F F U

FS m N

t x xx

F F F U

FS m N

tx x x

Φ Φ Φ

Φ

Φ Φ Φ

Φ

∂ ∂     ∂∂ ∂  

+ = Γ − ⋅  = +       ∂ ∂     ∂ ∂ ∂ψ  

∂ ∂ ∂∂ ∂      

+ =− Γ ψ + ⋅ = +     ∂ ∂ ∂ψ ∂  ∂    

ψ

ψ

ψ

ψ





(80)

This equation also represents the equation of internal energy m = N + 1, where SN+1(ψ) = 0 and other 

source terms in Equation (65) are neglected as suggested by Jaberi et al. (Ref. 1). Equation (80) is 

essentially closed if we consider the joint DW-FDF FU,Φ. For the marginal FΦ, the convection term is not 

closed because of the conditional DW-FDF mean i L U ψ . Then, this critically important term, 

corresponding to i m ρ Φ U in Equation (66), must be carefully modeled while the less important molecular 

diffusion term remains in the closed form. In addition, we noticed that the equally important chemistry 

source term ρS

m ( ) Φ in Equation (66) contains complex processes of turbulence-chemistry interaction, 

which is considered as very hard to be modeled. However, it is closed in the DW-FDF equation with no 

need of modeling. This direct calculation of turbulence-chemistry interaction represents one of the unique 

features of DW-FDF methodology. 

4.4 Modeling of Unclosed Terms

Consider the approximation described by Equation (69), i.e. P RT M ≈ ρ  , we may write from 

Equation (72)

1 1

1

1 ,

1

2

1

2

N

n N N

i iv n i v n

N U

i v

N U L

i v

i NU L kk v

i U

kk v

P R R

x x c w x cM

R F dd

x cM

R F d

x cM

R V F d

V x cM

R e V Fd

V x cM

+ +

=

∞ ∞

+ Φ

−∞ −∞

∞

+

−∞

∞

+

−∞

∞

−∞

∂ ∂ρ  Φ Φ ∂   ρ Φ − =−   ≈ −   ∂ ∂ ∂    

∂ = − ψ

∂

∂ =− Φ

∂

∂   =   Φ

∂ ∂  

∂   ≈   ∂ ∂  

∑

∫ ∫

∫

∫

∫

ψ





V

V V

V V

V

(81)

The last step in Equation (81) employs a rough approximation:
NASA/TM—2011-217012 15

N N 1 1 L

e Φ ≈Φ = + +  V  (82)

As for the convection term in the species Equation (80), i L F U Φ ψ , we may start from a more general 

model for the term ρ Φ U

i m (see Ref. 6):

   ( ) ( ) ( )

1 2 m m m m ij ij im im T T

i j

k U U cS c

x x

∂ρΦ ∂ρΦ ρ Φ =ρ Φ −Γ −Γ + Ω ∂ε ∂

    (83)

Where c1 = c2 = – 0.24. This will lead to the following model by directly applying the Equations (27) and

(39) described in Sections 2.2.2 and 2.2.3:

   ( )

   ( )

( ) ( )

1 2

( ) ( )

1 2

or

m m i ij ij i k L T T

i k j

m m i ij ij i k T T k L k ik j

Fk F F U UF cS c

x x

Fk F F U UF cS c

x x

Φ Φ

Φ Φ

Φ Φ

Φ Φ

  ∂ ∂   ∂ = −Γ + Γ + Ω ψ     ∂ ∂ψ ε ∂    

∂ ∂∂     ∂ = + Γψ + Γ +Ωψ     ∂ψ ∂ ∂ψ ε ∂    

ψ

ψ

(84)

4.5 Summary

With the models given by Equations (81) and (84), the marginal DW-FDF equations for FU(V; x,t)

and FΦ(ψ; x,t) can be written as

2

1

2

1

2

3

or

UU N U

j U

j kk v j j

U U

j kj m

k jk k jm

UU N U

j U k

j kk v k j j

U

j

k j

FF R F V F

t x V x cM x x

F F V V

V xx V xx

FF R F V F V

t x V x cM V x x

F V

V x

+

+

∂∂ Φ ∂ ∂ ∂    

+ = + ν    ∂ ∂ ∂∂ ∂ ∂      

∂∂ ∂ ∂   ∂ ∂ 

− ν+ δ ν    ∂ ∂∂ ∂ ∂∂    

∂∂ Φ ∂   ∂ ∂   ∂

+ = − ν     ∂ ∂ ∂∂ ∂ ∂ ∂    

∂ ∂ ∂

− ν

∂ ∂





2

3

U

kj m

k k jm

F V

x V xx

  ∂ ∂ ∂    +δ ν  ∂ ∂ ∂∂    

(85)
NASA/TM—2011-217012 16



( ) ( ) ( ( ))

  ( )



( ) ( ) ( )

( ) ( )

( )

1 2

( ) ( )

, 1,2, , 1

or

i m m

T k

i i i k

m k ij ij T

k i j

i m m k k T

i ki i

k

F F U F

F S

tx x x

k F cS c m N

x x

F F U F

F S

tx x x

Φ Φ Φ

Φ

Φ

Φ Φ Φ

Φ

∂ ∂ ∂∂ ∂       + = Γ +Γ − ⋅     ∂ ∂ ∂    ∂ ∂ψ  

∂ ∂       ∂ − ψ Γ +Ω     = +

∂ψ ∂ ε ∂      

∂ ∂ ∂∂ ∂       + =− ψ Γ +Γ + ⋅     ∂ ∂ ∂ψ ∂   ∂    

∂

− ψ

∂ψ

ψ

ψ



  ( ) ( )

1 2 , 1,2, , 1 m k ij ij T

i j

k F cS c m N

x x

Φ   ∂ ∂         Γ +Ω = +

∂ε ∂      



(86)

It can be verified that the DW-FDF Equations (85) and (86) can exactly deduce the filtered compressible 

Navier-Stoke equations (64) to (66) with the assumed approximations of (81) and (83). However, by no 

means, the models described by Equations (81) and (84) are unique. Furthermore, the variables (ΦN +1  , 

Ui

 , ij S

 and Ωij  ) are considered as available during the solution procedure of the DW-FDF equations.

5.0 Concluding Remarks

We have revisited the derivations of the traditional DW-FDF or FMDF equations by starting from the 

equation of FG-PDF that contains all the terms on the right hand side of the Navier-Stokes equations, see 

Equations (42) and (44). The resulting FMDF equations contain the “conditional DW-FDF mean”

quantities for all the terms on the right hand side (except for the reaction term), see Equations (43), (45),

which need empirical models to make the equations closed. 

From the relationship between the DW-FDF and the filtered turbulent variables, it is possible to 

construct the DW-FDF equations directly from the filtered compressible Navier-Stokes equations. Such 

DW-FDF equations have an outstanding feature that all the terms that were not closed on the right hand 

side of traditional FMDF equations are now in the closed form, see Equations (75) and (80), except for 

the pressure gradient term that involves joint DW-FDF.

In the marginal DW-FDF equations, the two unclosed terms, one related to the pressure gradient and 

the other related to the species convection, are expected to be more important than the molecular “mixing” 

or “diffusion” terms, especially in the case of high Reynolds number. Their physics-based models, such as 

Equations (81) and (84), should be further developed and evaluated. 

The fundamental differences between the DW-FDF (a random quantity) and the PDF (a deterministic 

quantity) warrant further investigation on their respective solution procedure using the stochastic 

differential equation methods.

References

1. 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,” J. Fluid Mech., vol. 401, pp. 85–121, 1999.

2. Colluci, P.J., Jaberi, F.A., Givi, P. and Pope, S., “Filtered density function for large eddy simulation 

of turbulent reacting flow,” Phys. Fluids 10, 499–515, 1998.

3. Pope, S.B, “Turbulent Flows,” Cambridge University Press, 2000.

4. Shih, T.-H. and Liu, N.-S. “Conservational PDF equations of turbulence,” NASA/TM—2010-216368.

5. Liu, N.-S. and Shih, T.-H., “Turbulence Modeling for Very Large-Eddy Simulation,” AIAA Journal, 

Vol. 44, No. 4, April 2006. 
NASA/TM—2011-217012 17

6. Shih, T.-H. and Liu, N.-S., “A non-linear Dynamic Subscale Model for Partially Resolved Numerical 

Simulation (PRNS)/Very Large Eddy Simulation (VLES) of Internal Non-Reacting Flows,” 

NASA/TM—2010-216323, May 2010.

7. Tennekes, H. and Lumley, J.L., A First Course in Turbulence, The MIT Press, 1972.

8. Smagorinsky, J., “General Circulation Experiments with the Primitive Equations,” Mon. Wea., 91, 

99, 1963.

9. Speziale, C.G., “Turbulence Modeling for Time-Dependent RANS and VLES: A Review,” AIAA J., 

Vol. 36, No. 2, 1998. 

10. Germano, M., Piomelli, U., Moin, P., and Cabot, W.H., “A Dynamic Subscale Eddy Viscosity 

Model,” Proc. Summer Workshop, Center for Turbulence Research, Stanford CA, 1990.
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Density Weighted FDF Equations for Simulations of Turbulent Reacting Flows 

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14. ABSTRACT

In this report, we briefly revisit the formulation of density weighted filtered density function (DW-FDF) for large eddy simulation (LES) of 

turbulent reacting flows, which was proposed by Jaberi et al. (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,” J. Fluid Mech., vol. 401, pp. 85-121, 1999). At first, we proceed the 

traditional derivation of the DW-FDF equations by using the fine grained probability density function (FG-PDF), then we explore another 

way of constructing the DW-FDF equations by starting directly from the compressible Navier-Stokes equations. We observe that the terms 

which are unclosed in the traditional DW-FDF equations are now closed in the newly constructed DW-FDF equations. This significant 

difference and its practical impact on the computational simulations may deserve further studies. 

15. SUBJECT TERMS

Turbulence 

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