Tsan-Hsing Shih

Ohio Aerospace Institute, Brook Park, Ohio

Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio

Ensemble Averaged Probability Density Function 

(APDF) for Compressible Turbulent Reacting Flows

NASA/TM—2012-217677

August 2012
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Tsan-Hsing Shih

Ohio Aerospace Institute, Brook Park, Ohio

Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio

Ensemble Averaged Probability Density Function 

(APDF) for Compressible Turbulent Reacting Flows

NASA/TM—2012-217677

August 2012

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 would also like to thank Professor John L. Lumley, Professor Zellman Warhaft and Professor J. Y. Chen 

for their critical comments, suggestions and encouragements to our previous related work on “Conservation PDF Equations of 

Turbulence” that leads to our current work. Finally, we would like to thank Professor Stephen B. Pope for his excellent text book 

“Turbulent Flows” that guides us in our PDF studies and this work.

Available from

NASA Center for Aerospace Information

7115 Standard Drive

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National Technical Information Service

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Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identification 

only. Their usage does not constitute an official 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 findings, 

subject to revision as analysis proceeds.
NASA/TM—2012-217677 iii

Contents

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

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

2.0 Ensemble APDF..................................................................................................................................2

2.1 Fine Grained Probability Density Function ( ; , ), ( ; , ) Uf tf t Φ ′ ′ Vx x ψ .........................................2

2.2 Ensemble Averaged Turbulent Variables and APDF ( ; , ), ( ; , ) F tF t U Vx x Φ ψ ..........................2

2.2.1 Definition of Ensemble Averaged Turbulent Variables φ( ,) x t , φ ( ,) x t ........................3

2.2.2 Definition of ensemble APDF .........................................................................................3

2.2.3 Relationship Between APDF and Ensemble Averaged Turbulent Variables..................3

2.2.4 Conditional APDF and Its Conditional Mean .................................................................5

3.0 Traditional Form of APDF Equations.................................................................................................6

3.1 Traditional Form of ( ; ,) F t U V x Equation .................................................................................6

3.2 Traditional Form of F t Φ ( ; ,) ψ x Equation .................................................................................7

3.3 Traditional Form of Joint APDF , ( , ; ,) F t U Φ V x ψ Equation.......................................................7

4.0 Conservational Form of APDF Equations ..........................................................................................8

4.1 Ensemble Averaged Compressible Navier-Stokes Equations ....................................................8

4.2 Conservational form of ( ; ,) F xt U V equation ..........................................................................11

4.3 Conservational Form of F t ( ; ,) Φ ψ x Equation ........................................................................13

4.3.1 Modeling of Unclosed Terms........................................................................................14

4.3.2 Summary........................................................................................................................15

4.4 Conservational Form of Joint APDF , ( , ; ,) F t U Φ V x ψ Equation ...............................................16

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

References...................................................................................................................................................17

NASA/TM—2012-217677 1

Ensemble Averaged Probability Density Function (APDF) 

for Compressible 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 paper, we present a concept of the averaged probability density function (APDF) for studying

compressible turbulent reacting flows. The APDF is defined as an ensemble average of the fine grained 

probability density function (FG-PDF) with a mass density weighting. It can be used to exactly deduce

the mass density weighted, ensemble averaged turbulent mean variables. The transport equation for

APDF can be derived in two ways. One is the traditional way that starts from the transport equation of 

FG-PDF, in which the compressible Navier-Stokes equations are embedded. The resulting transport 

equation of APDF is then in a traditional form that contains conditional means of all terms from the right 

hand side of the Navier-Stokes equations except for the chemical reaction term. These conditional means 

are new unknown quantities that need to be modeled. Another way of deriving the transport equation of 

APDF is to start directly from the ensemble averaged Navier-Stokes equations. The resulting transport 

equation of APDF derived from this approach appears in a closed form without any need for additional

modeling. The methodology of ensemble averaging presented in this paper can be extended to other 

averaging procedures: for example, the Reynolds time averaging for statistically steady flow and the 

Reynolds spatial averaging for statistically homogeneous flow. It can also be extended to a time or spatial 

filtering procedure to construct the filtered density function (FDF) for the large eddy simulation (LES) of 

compressible turbulent reacting flows. 

1.0 Introduction

The advantage of PDF method in the turbulent reacting flow simulations has been studied and 

examined (Refs. 1, 2, 3, and 4). Here we present the concept of an ensemble averaged probability density 

function (APDF) that may be considered as an extension of the conventional probability density function 

(PDF) method for dealing with compressible turbulent reacting flows. We do not assume that the mass 

density is a function of temperature and composition only, because this is only valid for the Mach number 

being near to zero (see Ref. 4). The APDF is defined by an ensemble average of a mass density weighted 

fine grained probability density function. It can be used to exactly define the mass density weighted

ensemble averaged turbulent mean variables, very similar to the way in which the conventional PDF 

defines the turbulent mean variables.

Two ways of deriving APDF transport equation will be described in detail. One is to start from the 

transport equation of the fine grained probability density function (FG-PDF), in which the compressible 

Navier-Stokes equations are embedded. By taking ensemble average over the equation of FG-PDF, the 

resulting transport equation of APDF in its traditional form contains conditional means of all terms from

the right hand side of the Navier-Stokes equations, except for the chemical reaction term. These 

conditional means are new unknown quantities that need to be modeled. Another way of deriving the 

transport equation of APDF is to start directly from the ensemble averaged compressible Navier-Stokes 
NASA/TM—2012-217677 2

equations. Using the relationship between APDF and the mass density weighted, ensemble averaged

turbulent mean quantities, and under a “conservational” condition, the derived transport equation of

APDF results in a closed form without any need for additional modeling. No terms of conditional means 

will appear in this conservational form of APDF equation, except when a marginal APDF equation such 

as the scalar APDF is considered. 

The methodology of ensemble averaging procedure presented in this paper can be extended to other 

averaging procedures: for example, the Reynolds time average for statistically steady flows and the 

Reynolds spatial average for statistically homogeneous flows. In these cases, the Reynolds time (or spatial) 

averaged probability density function will be defined, which then defines the mass density weighted 

Reynolds time (or spatial) averaged turbulent mean variables.

It should be noted that for the constant density turbulent flow, the APDF will reduce exactly to the

conventional PDF if the expectation of FG-PDF can be viewed as the ensemble average of FG-PDF. The

conservational PDF transport equation has been reported in Reference 5.

The methodology described in this paper can also be extended to a time or spatial filtering to 

construct the transport equation for a mass density weighted filtered FG-PDF for large eddy simulations

(LES). In this case, the mass density weighted filtered FG-PDF is named as “FMDF” or “DW-FDF” (see 

Refs. 3 and 6), which is a random quantity and can be used to define the mass density weighted filtered 

turbulent variables to describe large scales of compressible turbulent reacting flow.

Finally, it is noted that the equations for various (PDF, APDF, DW-FDF) methods are sought to be 

similar, so that the solution procedures would be similar regardless of the solution variables.

2.0 Ensemble APDF

In this section, we will use the FG-PDF to define the ensemble APDF, and then explore the 

relationship between APDF and ensemble averaged turbulent mean variables. This will provide the basis 

for establishing the transport equation of APDF.

2.1 Fine Grained Probability Density Function ( ; , ), ( ; , ) Uf tf t Φ ′ ′ Vx x ψ

The FG-PDF for turbulent velocity and scalars (e.g., species mass fraction or composition, internal 

energy) are defined as follows (Pope (Ref. 1)),

( ) ( )

3

1

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

i

f t t U tV

=

′ Vx Ux V x ≡δ − ≡ δ − ∏ (1)

( ) ( )

1

1

( ; ,) ( ,) ( ,)

M

m m

m

ft t t

+

Φ

=

′ ψ ψ xx x ≡δ − ≡ δ Φ −ψ Φ ∏ (2)

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

the turbulent (random) scalar array ( ) 1 2 1 , ,, , ΦΦ Φ Φ  M M + , for example, M species mass fractions and 

one internal energy M 1 e Φ = + , the x,t denote the physical space variable (x1, x2, x3) and the time t, V ≡

(V1, V2, V3) and 1 2 1 ψ ≡ψ ψ ψ ψ (, ,, , )  M M + are the sample space variables for U(x,t) and Φ(x,t), 

respectively. 

2.2 Ensemble Averaged Turbulent Variables and APDF ( ; , ), ( ; , ) F tF t U Vx x Φ ψ

We start with the definition of ensemble averaged turbulent variables and the definition of ensemble 

averaged probability density function APDF. And then explore the relationship between the two.
NASA/TM—2012-217677 3

2.2.1 Definition of Ensemble Averaged Turbulent Variables φ( ,) x t , φ ( ,) x t

In the case of compressible turbulent reacting flow, we often deal with two types of ensemble 

averaging: one with the mass density weighting, the other without the mass density weighting. The 

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

as

( )

1

1 ( , ) lim ( , )

N n

N n

t t →∞ N =

φ= φ x x ∑ (3)

where φ(n)

is the n-th independent realization of turbulent variables, for example, the velocity 

components ( ) n Ui , the densityρ(n)

, pressure P(n), the species mass fraction ( ) n Φm and the internal energy 

() () ()

1

M n nn

m m

m

e e

=

= Φ ∑ . N is the total number of independent realizations. The mass density weighted 

ensemble averaged turbulent variable is denoted by φ ( ,) x t and is defined as

( ,)t ρφ φ =

ρ

x (4)

These ensemble averaged variables φ( ,) x t , φ ( ,) x t represent the turbulent mean variables and are 

measurable in experiments. They only contain relatively low frequency and low wave number 

parts of the turbulent motion comparing with the non-averaged turbulent variables φ(x,t).

2.2.2 Definition of Ensemble APDF

We define the following ensemble averaged probability density functions APDF:

( ) ( ( ) )

( ) ( ( ) )

( ) ( )

1

( ) ( )

1

1 ( ; , ) lim

1 ( ; , ) lim

N n n

U N n

N n n

N n

F t t t

N

F t t t

N

→∞ =

Φ →∞ =

≡ ρδ −

≡ ρδ −

∑

ψ ψ ∑

V x x, U x, V

x x, x, Φ

(5)

Obviously, thus defined ( ; ,) F t U V x or F t Φ ( ; ,) ψ x is no longer a random quantity. And they satisfy the 

following “normalization” property:

( ) ( ( ) ) ( )

( ) ( ( ) ) ( )

( ) ( ) ( )

1 1

( ) ( ) ( )

1 1

1 1 ( ; , ) lim lim

1 1 ( ; , ) lim lim

N N n n n

U N N n n

N N n n n

N N n n

F td t t d t

N N

F td t t d t

N N

∞ ∞

→∞ →∞ −∞ −∞ = =

∞ ∞

Φ →∞ →∞ −∞ −∞ = =

ρ δ − ρ =ρ

= ρ δ − = ρ =ρ

∫ ∫ ∑ ∑

∫ ∫ ψ ψ ∑ ∑

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

x ψ ψ x, x, Φ x,

(6)

Where the integration proceeds in the entire sample space of ( ,) Vi −∞ ∞ and (0,1) ψi .

2.2.3 Relationship Between APDF and Ensemble Averaged Turbulent Variables

With the definition of APDF described in Equation(5), we can exactly deduce the ensemble averaged

turbulent variables that are defined in Equations (3) and (4). For example,
NASA/TM—2012-217677 4

( ) ( ( ) )

( ) ( ) ( ( ) )

( ) ( )

( ) 

( ) ( )

1

() () ( )

1

() ()

1

1 ( ; , ) lim

1 lim

1 lim

( ,)

N n n

U N n

N nn n

N n

N n n

N n

F td t t d

N

t t td

N

t t

N

t t

+∞ +∞

−∞ −∞ →∞ =

+∞

−∞ →∞ =

→∞ =

    ρδ −

 

    ρ δ−

 

ρ

ρ =ρ

∫ ∫ ∑

∫ ∑

∑

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

= x, U x, U x, V V

= x, U x,

= U x, U x

(7)

where the sifting property of FG-PDF has been used in the second line of Equation (7). 

Similarly, we have

( )  F td ( ; ,) t t ( ,) +∞

Φ −∞ ρ =ρ ∫ ψψ ψ x = x, x Φ Φ (8)

If we denote the left hand side of Equations (7) and (8) as a “mean” operation using APDF: U and Φ , 

we may write Equations (7) and (8) as 

( )  ( ; ,) ( ,) F td U t t +∞

−∞ ≡ =ρ =ρ ∫ U V V x V U x, U x (9)

( )  F td ( ; ,) t t ( ,) +∞

Φ −∞ ≡ =ρ =ρ ∫ Φ ψ ψ x ψ Φ Φ x, x (10)

For any function Q t (U x( ,)) or W t (Φ( ,) x ) , it is easy to verify that

( ) ( )  ( )

( ) ( ) ( )

( ) ( ; ,) ( ,)

( ) ( ; ,) ( ,)

Q Q F td tQ t U

W W F td tW t

+∞

−∞

+∞

Φ −∞

= = ρ

= = ρ

∫

∫ ψψ ψ

U V V x V x, U x

Φ Φ x x, x

(11)

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

write

∇ =ρ∇ ∇ =ρ∇ ∇ =ρ∇ P P, , U U Φ Φ (12)

However, because of the variable density, the mean does not have the differential commute property, 

i.e.,

∇ ≠∇ ∇ ≠∇ ∇ ≠∇ P P , , U U Φ Φ (13)

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

It can be verified that the following “second moment” is true

( ; ,) ( )( ,) UU VV F t d UU t UU t i j i jU i j i j

+∞

−∞ = ρ =ρ ∫ V x V = x, x (14)
NASA/TM—2012-217677 5

We can also write U j i Φ for the joint variables as

, ( , ; ,) ( )( ,) U V F td d tU t j i j iU j i

+∞

Φ −∞ Φ= ψ =ρ Φ ∫ V x V x, x ψ ψ (15)

where, , ( , ; ,) F t U Φ V x ψ is the velocity-scalar joint APDF defined as 

( ) ( ( ) ) ( ( ) ) ( ) ( ) ( )

,

1

1 ( , ; , ) lim

N nn n

U N n

F t tt t

N Φ →∞ =

V x ψ ψ = ρ δ −δ − ∑ x, U x, V x, Φ (16)

Here, the velocity-scalar joint fine grained probability density function U , f Φ′ is introduced:

f xt t U

′ ,Φ ( , ; ,) V U x, V x, ψ ψ =δ − δ − ( ( ) ) (Φ( t) ) (17)

The joint APDF , ( , ; ,) F t U Φ V x ψ also satisfies the normalization property:

( ) ( )

,

1

1 ( , ; , ) lim

N n

U N n

F td d t

N

∞ ∞

Φ →∞ −∞ −∞ =

∫ ∫ V x V ψ ψ = ρ =ρ ∑ x, (18)

2.2.4 Conditional APDF and Its Conditional Mean

From Equation(16), we may follow Pope (Ref. 1) to define a “conditional” APDF on the condition

Φ = ψ as

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

U

F t

F t

F t

Φ

Φ

Φ

≡ ψψ

ψ

V x

V x

x

(19)

and it has the following property,

( ; ,) 1 UF td

∞

Φ

−∞

= ∫ VxV ψ (20)

The “conditional mean” will be defined as

( ) ( ( ) ) ( ( ) )

( ) ( ( ) )

,

( ) ( ) ( )

1

() () ()

1

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

1 1 lim

1 1 lim

U U

N nn n

N n

N n nn

N n

t F td F td

F t

t t td

F N

t t

F N

+∞ +∞

Φ Φ −∞ −∞ Φ

+∞

−∞ →∞ Φ =

→∞ Φ =

≡

    ρ δ −δ −

 

ρ δ−

∫ ∫

∫ ∑

∑

ψ ψ ψ

ψ

ψ

ψ

U x V V x V= V V x V

x

= V x, U x, V x, V

= x, U x,

Φ

Φ

(21)

Then we have

( ) ( ( ) ) () () ()

,

1

1 lim ( , ; ,) ( ,)

N n nn

U N n

t t F td F t

N

+∞

Φ Φ →∞ −∞ =

∑ρ δ −= ⋅ ∫ x, U x, Φ ψψ ψ V V x V = U x (22)
NASA/TM—2012-217677 6

And the total mean should be defined as

( )

( ) ( ( ) ) ( ( ) )

( ) ( ) ( ) 

( ) ( ) ( )

1

() ()

1

( ,)

1 lim

1 lim ( , )

N nn n

N n

N n n

N n

F td

t t t dd

N

t t tt

N

+∞

Φ −∞

+∞ +∞

−∞ −∞ →∞ =

→∞ =

⋅

= ρ δ −δ −

= ρ =ρ =ρ

∫

∫ ∫ ∑

∑

ψ ψ

ψ ψ

U x

V x, U x, V x, V

x, U x, U x, U x

Φ (23)

Equation (22) can be extended to any other turbulent quantities, for example, ∇P ,∇U ,∇Φ , Si (Φ):

( ) ( ( ) )

( ) ( ( ) )

( ) ( ( ) )

( ( ) ) ( ) ( )

( ) ( ) ( )

1

( ) ( ) ( )

1

( ) ( ) ( )

1

() () ()

1

1 lim

1 lim

1 lim

1 lim

N n n n

U N n

N n n n

U N n

N n n n

N n

N nn n

i i i N n

P t FP

N

t F

N

t F

N

S t F S FS

N

→∞ =

→∞ =

Φ →∞ =

Φ Φ →∞ =

ρ ∇ δ − ⋅∇

ρ ∇ δ − ⋅∇

ρ ∇ δ − ⋅∇

ρ δ − ⋅ =⋅

∑

∑

∑

∑

ψ ψ

ψ ψψ

U x, V = V

U U x, V = U V

x, =

x, =

Φ Φ Φ

Φ Φ

(24)

where ∇∇ ∇ P, , U Φ are viewed as new random variables in addition to P , U, and Φ .

3.0 Traditional Form of APDF Equations

In this section, we perform the traditional way of deriving the transport equation for APDF, i.e., we 

start from the equation of FG-PDF, and the resulting equation for APDF contains conditional means that 

require closure modeling.

3.1 Traditional Form of ( ; ,) F t U V x Equation

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

Ref. 6):

2 2

3

U j U

U ij ij kk

j i ij

f U f P f SS t x V xx

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

(25)

where the compressible continuity and momentum equations (31), (32) have been applied. Applying the 

ensemble average operation, Equation (5), on Equation (25) and using the sifting property of FG-PDF, we 

may obtain
NASA/TM—2012-217677 7

( )

( )

( )

( ) ( )

1

( )

( ) ( )

1

1 1 lim

11 2 lim 2

3

n N

U U n n

j U N j i n i

n N n n

ij ij kk U N i j n

F F P V f t xV N x

S Sf VN x

→∞ =

→∞ =

  ∂ ∂ ∂ ∂    

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

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

∑

∑

According to Equation (24), the APDF equation for the velocity can be written as 

1 12 2

3

U U

j U U ij ij kk

ji i j

F F P V F F SS

t xV x x

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

V V (26)

At this point, the velocity APDF equation is general but unclosed because of the unknown terms of the 

conditional means. The last term in Equation (26) is referred as the molecular mixing term in which

significant modeling efforts have been devoted over several decades, nevertheless, major issues still 

remain. Later, we will show that when using a different way to derive the APDF equation, the molecular 

mixing term is in a closed form, and the pressure gradient term is also essentially closed. In addition, we 

noticed that the molecular mixing term is of the order of O(1/Re), and may become vanishingly small 

when comparing with other terms in the equation in the case of large Reynolds number (Re).

3.2 Traditional Form of F t Φ ( ; ,) ψ x Equation

Applying the same procedure (in Section 3.1) 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 (27)

we obtain the APDF equation for the turbulent scalars:

( ) ( ) 1 ( ) 1,2, 1 j i i

i

j i jj

F U F F FS i M

t x x x

Φ Φ

Φ Φ

∂   ∂ ∂ ∂   ∂Φ

+ = − ⋅ ρΓ + ⋅ = +   ∂ ∂ ∂ψ ρ ∂ ∂    

ψ

ψ ψ  (28)

where SM 1 ( ) 0 + ψ = . Equation (28) is also unclosed because the conditional means are unknown. Later, 

we will also see that the molecular mixing term is closed when using a different way of derivation. Again, 

it is of the order of O(1/Re) and could become vanishingly small at large Reynolds numbers. 

3.3 Traditional Form of Joint APDF , ( , ; ,) F t U Φ V x ψ Equation

We may obtain the velocity-scalar joint APDF equation from the joint fine grained probability density 

function U , f Φ′ defined in Equation (17). Similar to the relationship Equation (25), we may write
NASA/TM—2012-217677 8

, , '

, ,

,

( )

,

2 2

3

U j U i k i j j k

U U

j i j k j

U ij ij kk

i ij

k k

U

kjj

f U f u u u u

f f t x V tx t x

P f SS V xx

f

x x

Φ Φ

Φ Φ

Φ

Φ

∂ρ ′ ∂ρ ′     ∂ρ   ∂ρ Φ  ∂ ∂  ∂ρ   ∂ρΦ  + =− + − +  ′       ∂ ∂ ∂ ∂ ∂ ∂ψ ∂ ∂         

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

∂ ∂  ∂Φ − ′ ρΓ ∂ψ ∂ ∂ 

S t k ( ( ,))

            + ρ         

Φ x

(29)

Applying the ensemble average operation (5) on Equation (29) we obtain 

( )

, ,

, ,

( )

, ,

1 12 2

3

1

U U

j U U ij ij kk

ji i j

k k

U U k

k jj

F F P V F F SS

t xV x x

F F S

x x

Φ Φ

Φ Φ

Φ Φ

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

  ∂ ∂   ∂Φ − ⋅ ρΓ   + ⋅

∂ψ ρ ∂ ∂    

ψ ψ

ψ ψ

V, V,

V,

(30)

It is easy to verify that Equation (30) will reduce to Equations (26) and (28) by an integration over the 

sample space ψ and V, respectively. The equation (30) is general and is not restricted to low Mach 

numbers. 

4.0 Conservational Form of APDF Equations

In this section, we first review the ensemble averaged compressible Navier-Stokes equations, which 

are unclosed due to the unknown second order moments. Then we use the relationship between APDF 

and the first and second order moments described in Section 2.2 to express the ensemble averaged 

compressible Navier-Stokes equations in terms of an integral equation of APDF. Under a sufficient 

“conservation” condition, the APDF transport equations are established, which are in a closed form 

without additional modeling, unless a marginal APDF equation is considered. 

4.1 Ensemble Averaged Compressible Navier-Stokes Equations

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

0 j

j

U

t x

∂ρ ∂ρ

+ =

∂ ∂ (31)

1 2( ) 3

i j i

ij ij kk

j ij

U U U P S S

t x xx

∂ρ ∂ ρ ∂ ∂   + = − + ρν − δ   ∂ ∂ ∂∂   (32)

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 − +   ∂∂ ∂   (33)

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

ii i

U Wm M

t xx x

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

 (34)
NASA/TM—2012-217677 9

1

M

m

m m

T P R

= w

Φ

= ρ ∑ (35)

( )

1

M m m i m

i i m

T qc h

x x υ

=

∂ ∂Φ = −ρ κ − ρΓ ∂ ∂ ∑ (36)

Applying the ensemble averaging on the above equations, we obtain the following Reynolds averaged

Navier-Stokes equations:

 0 j

j

U

t x

∂ρ ∂ ρ + =

∂ ∂ (37)

  1 2( ) 3

i i j

ij ij kk

j ij

U P U U

S S

t x xx

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

(38)

 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 − +   ∂∂ ∂  

 (39)

  ( ) 1,2, , m i m m m m

ii i

U Wm M

t xx x

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

 (40)

where

 



1 1

1

1

or

M M

m m

m m mv m

M

m M

v m m

T e R P R

wc w

R P

c w

= =

+

=

Φ Φ ρ =ρ =

ρ Φ Φ =

∑ ∑

∑

(41)

( )

1

N m m i m

i i m

T qc h

x x υ

=

∂ ∂Φ = −ρ κ − ρΓ ∂ ∂ ∑ (42)

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). It is also commonly assumed

that Γ(m)

is the same for all species Φm. The hm, T are the enthalpy of m-th species and the temperature, Q

is the radiation rate, Wm = ρSm is the chemical generation rate of m-th species, ΦM+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 ensemble averaging process. One of such approximations leads to

   1 2 2

3 3

ij k

ij ij kk ij

ji k

UU U S S

xx x

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

(43)
NASA/TM—2012-217677 10

In which, we have basically neglected the variations of μ and ρ during the averaging process, the value of 

μ will be considered as the function of  P T, ,. Alternatively,

   1 2 2

3 3

ij k

ij ij kk ij

ji k

UU U S S

xx x

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

(44)

Similarly,

( )  ( )

1 1

M M m m m m im m

i ii i m m

T T qc h c h

x xx x υ υ

= =

∂ ∂Φ ∂ρ ∂Φ = −ρ κ − ρΓ ≈ − κ − ρΓ ∂ ∂∂ ∂ ∑ ∑ (45)

 () () m m m m

i i x x

∂Φ ∂ρΦ ρΓ ≈Γ ∂ ∂ (46)

Furthermore, invoking the turbulent kinetic energy dissipation rate:

1 2

3 ij ij ii kk SS SS  

ρν − ≡ ρε      (47)

Where ,, , c ν κ υ and Γ(m)

are now considered as functions of  P T, ,. 

Then, the ensemble averaged Navier-Stokes equations can approximately be written as 

 0 j

j

U

t x

∂ρ ∂ ρ + =

∂ ∂ (48)

     2

3

i i j ij k

ij

j ij j i k

U P UU U U U

t x xx x x x

∂ρ ∂ ρ ∂ ∂ ∂ρ ∂ρ ∂ρ    

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

(49)

i i

kk

i i

e Ue q PS Q

tx x

∂ρ ∂ρ ∂ + = − + +ρε + ∂∂ ∂

  (50)

   ( )  1,2, , m m i m m m

ii i

U Sm M

t xx x

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

 (51)

 



1 1

1

1

or

M M

m m

m m mv m

M

m M

v m m

T e R P R

wc w

R P

c w

= =

+

=

Φ Φ ρ =ρ =

ρ Φ Φ =

∑ ∑

∑

(52)

 ( )

1

M m m i m

i i m

T qc h

x x υ

=

∂ρ ∂Φ = − κ − ρΓ ∂ ∂ ∑ (53)
NASA/TM—2012-217677 11

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 (49), (50) and (51) are related only to the 

molecular diffusion terms that are less important and even become negligibly small (1/Re) comparing 

with the convection terms on the left hand side for turbulent flows at large Reynolds numbers (see 

Tennekes & Lumley (Ref. 7) and Pope (Ref. 1)). For the conventional unsteady Reynolds averaged 

Navier-Stoke simulation (URANS), Equation (48) to (51) are often used together with the further 

approximations for Equations (52) and (53):

 

1 1 1

1 ,

M M M

m m m

m m m m m m

T RT RT P R

= w M MM w w = =

Φ Φ   ρ ρ Φ

=ρ = ≈ = ≈     ∑ ∑ ∑ (54)



i

i

T

q c

x υ

∂ρ =−κ

∂ (55)

The momentum flux ρ U Ui j , the energy flux ρ U ei and the species flux U

i m ρ Φ are considered to be 

critically important in URANS simulations and should be carefully modeled. Many models in the 

literature, from the simplest zero-equation model (Ref. 8) to the complex two-equation models (Refs. 9

and 10) have been suggested. In the following Sections 4.2 and 4.3, we will derive the APDF equations 

directly from Equations (48) to (53). It is important to note that in the joint PDF method, the terms 

corresponding to these important turbulent fluxes are always in closed forms with no need of modeling. 

4.2 Conservational form of ( ; ,) F xt U V equation

Using Equations (9) and (14), the left hand side of Equation (49) can be written as

i i j 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 (56)

The pressure gradient term can be written as (via Eq. (52))



2

1 ,

1

1

1 2

1

1

( , ; ,)

or

M

i mM U

k k vm m M

m M

i iv m m M

i mM U

k k vm m

R V F xtd d

V x cw

P R

x xc w

R V F d

V x cw

∞ ∞

+ Φ

−∞ = −∞

+

= ∞

+

−∞ =

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

∫ ∫ ∑

∑

∫ ∑

ψ ψ V V

V V

(57)

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

a zero integration property like Equation (59). Note that the pressure gradient term can be in a closed 

form only when the joint APDF, FU ,Φ , is considered, otherwise, a conditional mean will be unavoidable. 

Now the molecular mixing term in Equation (49) can be written as
NASA/TM—2012-217677 12

  

( ) ( )

2

3

2

3

2

3

or

ij m

ij

jj i 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

j

UU 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

x

∞∞ ∞

−∞ −∞ −∞

∞ ∞∞

−∞ −∞ −∞

∂ ∂ρ ∂ρ ∂ρ  

ν + −δ   ∂∂ ∂ ∂  

  ∂∂ ∂ ∂

= ν  + − δ

∂∂ ∂ ∂  

    ∂ ∂∂ ∂ ∂ ∂ ∂   ν − ν  − δ

∂∂ ∂ ∂ ∂ ∂ ∂    

=

∂

− ν

∂

∫∫ ∫

∫ ∫∫

VV V

V VV

( ) ( ) ( ) 2

3 i k U i j U kj i m U

jk kk m k

V VF d V V F d V VF d

xV xV x V

∞∞ ∞

−∞ −∞ −∞











    ∂∂ ∂∂ ∂ ∂   + − δ  ∂∂ ∂∂ ∂ ∂    ∫∫ ∫ VV V

and further rearranged as 

2 2

2 2 2

2

3

or

2

3

U j U m U

i i i kj

j j j kk j km

k U j U m U

i i i kj

j k j j kk j km

F V F V F V dV dV d

x x x Vx x Vx

V F V F V F V dV dV

x Vx x Vx x Vx

∞∞ ∞

−∞ −∞ −∞

∞ ∞

−∞ −∞

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

=

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

∫∫ ∫

∫ ∫

VV V

V V d

∞

−∞











     

  ∫ V

(58)

Note, on the third line of Equation (58), we have applied the integration by parts and the following type of 

zero integration similar to the one used in PDF formulations (Pope (Ref. 1)):

( ) 0 ik U

k

VV F d

V

∞

−∞

∂ = ∂ ∫ V (59)

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

transport equation for ( ; ,) F t U V x by setting the sum of the integrands to zero, which will sufficiently 

satisfy the Navier-Stoke equations (48), (49) and (52). We refer to this condition as the “conservation 

condition”.

2

1

1

2

1

1

2

3

or

M

U U U

j mM U

j k k vm m j j

U U

j kj m

k jk k jm

M

U U U

j mM U k

j k k vm m 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

+

=

+

=

∂ ∂ ∂ ∂     ∂

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

∂∂ ∂ ∂   ∂ ∂ 

− ν+ δ ν    ∂ ∂∂ ∂ ∂∂    

∂ ∂ ∂   ∂ ∂   ∂

+ =   ΦΦ ⋅ − 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    ∂ ∂∂ ∂ ∂∂    

(60)
NASA/TM—2012-217677 13

In Equation (60), all the terms are closed, except for the one involving the conditional mean originating 

from the pressure gradient. It should be noted that the unclosed term in the momentum equation,

i.e., ρU Ui j , now is closed in the corresponding APDF equation while the less important molecular 

diffusion terms remain in closed forms. 

4.3 Conservational Form of F t ( ; ,) Φ ψ x Equation

Similarly, we may obtain the APDF equation for the scalars (i.e., species and internal energy) from 

Equations (50), (51) as follows: first, we write the terms on the left hand side of Equation (51) as

 m

m m

F F d d

t t t

∞ ∞

Φ

Φ

−∞ −∞

∂ρΦ ∂ ∂

= ψ =ψ ∂ ∂ ∂ ∫ ∫ ψ ψ (61)



( )

,

, 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 (62)

Then, the terms on the right hand side of (51) 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

(63)

 ( ) ( ) k m m m

k

S F

S S Fd d

∞ ∞

Φ

Φ

−∞ −∞

∂

ρ = =− ψ ∂ψ ∫ ∫ ψ ψ ψ (64)

Where in Equations (63) and (64), we have applied the integration by parts and zero integrations similar 

to Equation (59). Collecting all the integrand terms that factored by ψm , and invoking the sufficient, 

conservation condition, we obtain

( ) ( )

( ) ( )

( )

( )

, 1,2, , 1

or

, 1,2, , 1

i m

k

i i ik

i m

k k

i ki i

F F F U

FS k M

t x xx

F F F U

FS k M

tx x x

Φ Φ Φ

Φ

Φ Φ Φ

Φ

∂ ∂ ∂ ⋅     ∂ ∂  

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

∂ ∂ ∂ ⋅ ∂ ∂      

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

ψ

ψ

ψ

ψ





(65)
NASA/TM—2012-217677 14

This equation also represents the equation of internal energy when SM 1 ( ) 0 + ψ = and other source terms 

in Equation (50) are neglected. Equation (65) is essentially closed if we consider the joint APDF FU ,Φ . 

For the marginal FΦ , the convection term is not closed because of the conditional mean Ui ψ . Then, 

this critically important term, corresponding to U

i m ρ Φ in Equation (51), must be carefully modeled 

while the less important molecular diffusion term remains in a closed form. In addition, we noticed that 

the equally important chemistry source term ( ) m ρS Φ in Equation (51), which involves complex 

processes of turbulence-chemistry interaction and is very difficult to be modeled accurately, now is closed 

in the APDF equation with no need of modeling. This direct calculation of turbulence-chemistry

interaction is one of the unique features of PDF methodology. 

4.3.1 Modeling of Unclosed Terms

Consider the approximation described by Equation (54), i.e.,  P RT M ≈ ρ , we may write from 

Equation (57)

 1

1 ,

2

1 ,

2

1

M

i iv

M U

i v

i M U

kk v

i M U

kk v

P R

x x cM

R F dd

x cM

R V Fdd

V x cM

R V F d

V x cM

+

∞ ∞

+ Φ

−∞ −∞

∞ ∞

+ Φ

−∞ −∞

∞

+

−∞

∂ ∂ ρΦ   − ≈−   ∂ ∂  

∂ = − ψ

∂

  ∂ =   ψ ∂ ∂  

∂   = Φ ⋅   ∂ ∂  

∫ ∫

∫ ∫

∫

ψ

ψ

V

V

V V

(66)

For the marginal APDF equation of FU , we must model the unknown “conditional” mean ΦM +1 V . 

While seeking its physical based model the roughest one could be suggested as follows: 

 1 M 1 M + e Φ ≈Φ = + V  (67)

As for the term containing conditional mean, F U Φ i ⋅ ψ , in Equation (65), we may start from a more 

general model for the term ρ U

i m Φ (see Refs. 11 and 12):

      ( )  ( ) ( )

1 2

m m m m i m ij ij i m T T

i j

k U U cS c

x x

∂ρΦ ∂ρΦ ρ Φ = ρ Φ −Γ −Γ + Ω ∂ε ∂ (68)

Where 1 2 c c, are model coefficients and were temporally set to 1 c = 2 c = - 0.24, ( ) m ΓT is the turbulent 

diffusivity for species Φm and is assumed to be the same for all species. This will lead to the following 

model by directly applying Equations (8) and (22) described in Sections 2.2.2 and 2.2.4:
NASA/TM—2012-217677 15

   ( )

   ( )

( ) ( )

1 2

( ) ( )

1 2

or

m m i ij ij iT T m

i m j

m m i ij ij i T m T m

m im j

F F k F U UF cS c

x x

F F k 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

(69)

4.3.2 Summary

With the models given by Equations (66) and (69), the marginal APDF equations for ( ; ,) F t U V x and 

F t Φ ( ; ,) ψ x can be written as





2

1

2

1

2

3

or

U U M U

j U

j kk v j j

U U

j kj m

k jk k jm

U U M U

j U k

j kk v k j j

U

j

k j

F F R F V F

t x V x cM x x

F F V V

V xx V xx

F F R F V F V

t x V x cM V x x

F V

V xx

+

+

∂ ∂ ∂Φ ∂     ∂

+ =   + ν  ∂ ∂ ∂∂ ∂ ∂      

∂∂ ∂ ∂   ∂ ∂ 

− ν+ δ ν    ∂ ∂∂ ∂ ∂∂    

∂ ∂ ∂ Φ ∂∂     ∂

+ =   − ν   ∂ ∂ ∂∂ ∂ ∂ ∂    

∂ ∂ ∂

− ν

∂ ∂∂

2

3

U

kj m

k k jm

F V

V xx

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

(70)

and



( ) ( ) ( ( ))

  ( )



( ) ( ) ( )

() ()

( )

1 2

() ()

, 1,2, , 1

or

i m m

T k

i i i k

m ij ij k T

k i j

i m m

kT k

i ki i

k

F F U F

F S

tx x x

k F cS c k M

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 ij ij k T

i j

k F cS c k M

x x

Φ

    ∂   ∂     Γ +Ω = +

∂ε ∂      



(71)

It can be verified that the APDF Equations (70) and (71) can exactly deduce the ensemble averaged

compressible Navier-Stoke equations (48) to (51) with the assumed approximations of (66) and (68).

However, the models described by Equations (66) and (69) are by no means unique. Furthermore, the 

variables ( 

ΦM +1 , 

Ui , Sij and Ωij ) are considered as available quantities during the solution procedure of

APDF equations.
NASA/TM—2012-217677 16

4.4 Conservational Form of Joint APDF , ( , ; ,) F t U Φ V x ψ Equation

The transport equation for the velocity-scalar joint APDF , ( , ; ,) F t U Φ V x ψ can be constructed from 

Equations (60) and (65) as 

( )

2 , , ,

1 ,

, ,

( ) ,

,

2

3

, 1,2, , 1

U U U

j M U

j kk v j j

U U

j k j

k jk k j

m U

U m

i im

F F R F

V F d

t x V x cM x x

F F

V V

V xx V xx

F

F S m M

x x

∞

Φ Φ Φ

+ Φ

−∞

Φ Φ

Φ

Φ

∂ ∂ ∂ ∂     ∂

+ = ψ +ν     ∂ ∂ ∂∂ ∂ ∂    

∂∂ ∂ ∂   ∂ ∂ 

− ν +δ ν    ∂ ∂∂ ∂ ∂∂    

∂ ∂   ∂

+Γ − ⋅     = +   ∂ ∂ ∂ψ  

∫ ψ

ψ







(72)

or,

( )

2 , , ,

1 ,

, ,

( ) ,

,

2

3

, 1,2, ,

U U U

j M U k

j kk v k j j

U U

j k j

k jk k j

m U

m U m

mi i

F F R F

V Fd V

t x V x cM V x x

F F

V V

V xx V xx

F

FS m M

x x

∞

Φ Φ Φ

+ Φ

−∞

Φ Φ

Φ

Φ

∂ ∂ ∂   ∂ ∂   ∂

+= ψ − ν     ∂ ∂ ∂∂ ∂ ∂ ∂    

∂∂ ∂ ∂   ∂ ∂ 

− ν +δ ν    ∂ ∂∂ ∂ ∂∂    

∂ ∂       ∂

− ψ Γ +⋅     = ∂ψ ∂ ∂      

∫ ψ

ψ





 +1

(73)

It is easy to verify that Equation (72) or (73) will reduce to Equations (70) and (71) by an integration over 

the entire space ψ and V, respectively. It is noted that the transport equation for joint APDF 

, ( , ; ,) F t U Φ V x Φ is in a closed form; however it contains an integration of M U 1 , F d ∞

+ Φ −∞ ψ ∫ ψ from the 

pressure gradient term.

5.0 Concluding Remarks

We have derived the traditional form of APDF equations by starting from the equation of FG-PDF.

The resulting equations contain the “conditional means” for all the terms on the right hand side of the 

embedded Navier-Stokes equations except for the reaction term (see Equations (26) and (28)), which need 

empirical models to make the equations closed. 

We have also derived the conservational form of APDF equations directly from the ensemble 

averaged compressible Navier-Stokes equations using the relationship between APDF and the ensemble 

averaged turbulent variables. In this way, the resulting joint APDF equation is in a closed form, see 

Equation (72) or (73). 

APDF equations both in the traditional and in the conservational forms derived in this study do not 

subject to the low Mach number limitation. 

In the marginal APDF equations, (60) and (65), there are two unclosed terms, one is related to the 

pressure gradient and the other is related to the scalar convection. They are generally much more 

important than the molecular “mixing” terms, especially at large Reynolds numbers. The models

proposed in Equations (66) and (69) should be further developed and evaluated. 
NASA/TM—2012-217677 17

It should be pointed out that the methodology used in the present study can be extended to other 

averaging operations: for example, the Reynolds time (or spatial) averaging. Furthermore, it can also be 

extended to a filtering operation to derive the filtered density function (FDF) for large eddy simulation 

(LES) such as the mass density weighted filtered density function DW-FDF for LES of compressible 

turbulent reacting flows (see Ref. 6). 

References

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

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. 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.

4. Haworth, D.C. “Progress in probability density function methods for turbulent reacting flows.” 

Progress in Energy and Combustion Science, 36 (2010), 168-259.

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

6. Shih, T.-H. and Liu, N.-S., “ Density weighted FDF equations for simulations of turbulent reacting 

flow,” NASA/TM–2011-217012.

7. Tennekes, H. and Lumley, J. L., A FIRST COURSE IN TURBULENCE, The MIT Press, 1972.

8. Baldwin, B. S. and Lomax, H., “Thin Layer Approximation and Algebraic Model for Separated 

Turbulent Flows,” 1978. AIAA paper 78-257.

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

Vol. 36, No. 2, 1998. 

10. Shih, T.-H., Zhu, J., Liou, W., Chen, K.-H., Liu, N.-S. and Lumley, J. L., “ Modeling of Turbulent 

Swirling Flows,” NASA/TM—1997-113112, 1997.

11. Shih, T.-H., “Some developments in computational modeling of turbulent flows,” Fluid Dynamic 

Research 20 (1997) 67-96.

12. 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.
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Ensemble Averaged Probability Density Function (APDF) for Compressible Turbulent 

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14. ABSTRACT

In this paper, we present a concept of the averaged probability density function (APDF) for studying compressible turbulent reacting flows. 

The APDF is defined as an ensemble average of the fine grained probability density function (FG-PDF) with a mass density weighting. It 

can be used to exactly deduce the mass density weighted, ensemble averaged turbulent mean variables. The transport equation for APDF can 

be derived in two ways. One is the traditional way that starts from the transport equation of FG-PDF, in which the compressible NavierStokes equations are embedded. The resulting transport equation of APDF is then in a traditional form that contains conditional means of all 

terms from the right hand side of the Navier-Stokes equations except for the chemical reaction term. These conditional means are new 

unknown quantities that need to be modeled. Another way of deriving the transport equation of APDF is to start directly from the ensemble 

averaged Navier-Stokes equations. The resulting transport equation of APDF derived from this approach appears in a closed form without 

any need for additional modeling. The methodology of ensemble averaging presented in this paper can be extended to other averaging 

procedures: for example, the Reynolds time averaging for statistically steady flow and the Reynolds spatial averaging for statistically 

homogeneous flow. It can also be extended to a time or spatial filtering procedure to construct the filtered density function (FDF) for the 

large eddy simulation (LES) of compressible turbulent reacting flows. 

15. SUBJECT TERMS

Combustion CFD 

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