Tsan-Hsing Shih

Ohio Aerospace Institute, Brook Park, Ohio

Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio

Conservational PDF Equations of Turbulence

NASA/TM—2010-216368

July 2010
NASA STI Program . . . in Profi le

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA Scientifi c and Technical Information (STI) 

program plays a key part in helping NASA maintain 

this important role.

The NASA STI Program operates under the auspices 

of the Agency Chief Information Offi cer. It collects, 

organizes, provides for archiving, and disseminates 

NASA’s STI. The NASA STI program provides access 

to the NASA Aeronautics and Space Database and 

its public interface, the NASA Technical Reports 

Server, thus providing one of the largest collections 

of aeronautical and space science STI in the world. 

Results are published in both non-NASA channels 

and by NASA in the NASA STI Report Series, which 

includes the following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major signifi cant phase 

of research that present the results of NASA 

programs and include extensive data or theoretical 

analysis. Includes compilations of signifi cant 

scientifi c and technical data and information 

deemed to be of continuing reference value. 

NASA counterpart of peer-reviewed formal 

professional papers but has less stringent 

limitations on manuscript length and extent of 

graphic presentations.

• TECHNICAL MEMORANDUM. Scientifi c 

and technical fi ndings that are preliminary or 

of specialized interest, e.g., quick release 

reports, working papers, and bibliographies that 

contain minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientifi c and 

technical fi ndings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. Collected 

papers from scientifi c and technical 

conferences, symposia, seminars, or other 

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientifi c, 

technical, or historical information from 

NASA programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientifi c and 

technical material pertinent to NASA’s mission.

Specialized services also include creating custom 

thesauri, building customized databases, organizing 

and publishing research results.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page at 

http://www.sti.nasa.gov

• E-mail your question via the Internet to help@

sti.nasa.gov

• Fax your question to the NASA STI Help Desk 

at 443–757–5803

• Telephone the NASA STI Help Desk at

 443–757–5802

• Write to:

 NASA Center for AeroSpace Information (CASI)

 7115 Standard Drive

 Hanover, MD 21076–1320
Tsan-Hsing Shih

Ohio Aerospace Institute, Brook Park, Ohio

Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio

Conservational PDF Equations of Turbulence

NASA/TM—2010-216368

July 2010

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

This work was supported by the Subsonic Fixed Wing Project as part of the NASA Fundamental Aeronautics Program. 

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

Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://gltrs.grc.nasa.gov

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.
Contents 

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

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

2.0 PDF of Turbulent Variables ................................................................................................................. 2

2.1 PDF of Turbulent Velocity fU(V; x,t) .......................................................................................... 2

2.2 PDF of Turbulent Species fΦ(ψ; x,t) ........................................................................................... 3

2.3 Other Turbulent Quantities and Functions of Turbulent Variables ............................................. 4

2.3.1 Statistical Mean of Analytical Function of Turbulent Variable S(Φi) ........................... 4

2.3.2 Statistical Mean of Derivatives of Turbulent Variable ∂Ui /∂xj and ∂2

Ui /∂xj∂xj ........... 4

2.4 Joint PDF .................................................................................................................................... 5

2.4.1 Joint PDF and Marginal PDF ........................................................................................ 5

2.4.2 Conditional PDF and Conditional Mean ....................................................................... 6

2.4.3 Relationship Between Unconditional Mean and Conditional Mean ............................. 7

2.4.4 Summary ....................................................................................................................... 9

3.0 Transport Equation for Turbulent Velocity PDF fU(V; x,t) ................................................................ 10

3.1 Mean and Differentiation Commutation ................................................................................... 10

3.2 Taking Mean on Navier-Stokes Equations ............................................................................... 11

3.3 Traditional Velocity PDF Equation .......................................................................................... 12

3.4 Conservational Velocity PDF Equation .................................................................................... 13

4.0 Transport Equation for Turbulent Species PDF fΦ(ψ; x,t).................................................................. 15

4.1 Taking Mean on Species Equation ............................................................................................ 15

4.2 Traditional Species PDF Equation ............................................................................................ 17

4.3 Conservational Joint Species-Velocity PDF fU,Φ(V,ψ; x,t) Equation ........................................ 17

5.0 Concluding Remarks .......................................................................................................................... 19

Appendix A.—Fine Grained PDF, Velocity and Species PDF Equations .................................................. 21

A.1 Transport Equation for Fine Grained PDF f ′(V; x,t) ................................................................ 21

A.2 Transport Equation for Velocity PDF fU(V; x,t) ........................................................................ 21

A.3 Transport Equation for Joint Species-Velocity PDF fu,Φ(V,ψ; x,t) ........................................... 23

Appendix B.—Miscellaneous Formulations ............................................................................................... 25

References ................................................................................................................................................... 26

NASA/TM—2010-216368 iii
 
Conservational PDF Equations of Turbulence 

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 

Recently we have revisited the traditional probability density function (PDF) equations for the 

velocity and species in turbulent incompressible flows. They are all unclosed due to the appearance of 

various conditional means which are modeled empirically. However, we have observed that it is possible 

to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through 

conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the 

resulted PDF equations are neither general nor unique, they nevertheless lead to the exact transport 

equations for the first moment as well as all higher order moments. We refer these PDF equations as the 

conservational PDF equations. This observation is worth further exploration for its validity and CFD 

application. 

1.0 Introduction 

To explore the turbulence modeling employed in the PDF method, we revisited the constructions of 

the traditional PDF equations for turbulent velocity and species, which were described by several 

researchers in great details, for example, by Pope (Ref. 1). In those constructions, a Delta function (which 

is referred as the fine grained PDF) was extensively used, together with a few impressive mathematical 

techniques to invoke the Navier-Stoke equations into the identity relationship developed from the Delta 

function. Since the entry point for the Navier-Stokes equations was through the “conditional mean”, two 

or three conditional means appeared in the velocity PDF equation and additional two conditional means 

appeared in the species PDF equation. In these traditional PDF equations, all the conditional means were 

considered as unknowns and modeled empirically. 

In the present study, we started out by taking the statistical mean directly on the Navier-Stokes 

equations to form an integral equation, from which a PDF equation is then constructed. Depending on the 

way of taking mean on some particular terms, especially the pressure gradient ∇P and the molecular 

diffusions , , the resulted PDF equation can end up with different forms. Briefly speaking, 

if we treat, for example, the molecular diffusion term as a separate random variable from Ui, then 

its mean 

2 ν∇ Ui

2 Γ∇ Φi

2 ν∇ Ui

2 ν ∇ Ui will be expressed as an integration of the product of the velocity PDF and the 

conditional mean, i.e., 

2 ν ∇ Ui = ( ) 2 ; , ∫ ν ∇ U f td i V Vx V , 

where the conditional mean 2 ν∇ U Vi is a new unknown. However, if we take the mean based on the 

definition and the commutation rule between mean and differentiation, then we have 

( ) ( )

2

22 2 ; , ; , i ii i

f U U Vf t d V f t d f

⎛ ⎞ ∇ ν∇ = ν∇ = ν∇ = ν⎜ ⎟ ⎝ ⎠ ∫ ∫ Vx V Vx V , 

NASA/TM—2010-216368 1
which is in closed form. The former way will lead to the traditional PDF equations (see Sections 3.3 and 

4.2). The latter way will lead to the closed equations for the velocity PDF (see Section 3.4) and the joint 

species-velocity PDF (see Section 4.3). Since a sufficient but not necessary condition was involved in 

their construction, the resulted PDF equations are neither general nor unique. Nevertheless, we observed 

that they can lead to the exact transport equations for not only the first moments (i.e. mean velocity and 

species, etc.) but also all higher order moments deduced from the Navier-Stokes equations. This feature is 

referred as the conservation of the PDF equations. 

Later, we also found that, for the traditional way of utilizing the “fine grained PDF”, all the 

conditional means (e.g., 2 ν ∇ Ui V , 2 ν∇ Φi ψ , etc.) can also be systematically modeled through 

similar sufficient condition, hence the traditional PDF equations can be closed exactly as the PDF 

equations proposed in the present study. The details are described in the Appendix A. 

Section 2.0 introduces the basic definition of probability density function of turbulent velocity, 

species, other turbulent quantities and their joint PDF; the relationships between the marginal PDF, the 

joint PDF and the conditional PDF; and the relationship between the mean and the conditional mean. 

Section 3.0 demonstrates how to construct the velocity PDF equation starting directly from the NavierStokes equations. Section 4.0 demonstrates the construction of the transport equations for the species PDF 

and the joint species-velocity PDF. 

2.0 PDF of Turbulent Variables 

2.1 PDF of Turbulent Velocity fU(V; x,t) 

Turbulent velocity is a random vector variable, its probability density function PDF at a 

single point (in space and time) is denoted as 

( ,) U xt i i

fU U (V x;, , , ; , , , t f VV V x x x t ) ≡ ( 123 123 ), where 

 is the sample space coordinates of Ui ( , and 123 V ≡ VV V , , ) x ≡ ( x123 ,, , xx t ) are the location coordinates 

and the time. Formally, fU ( ; V x, )t

, 2, 3

 is a scalar function of V and x,t, but its arguments have different 

physical meaning and mathematical transformation. The V represents a point in the velocity sample space 

( ) at which the probability density fU of Ui is defined. The x,t indicate that fU is a 

field and a process. It is important to note that V is independent of x and t. The probability density 

function 

, 1 =

, )

−∞< <∞ V i i

fU ( ; V x t itself is not random and is fully defined at each V for every single point x,t, it is a 

differentiable (hence continuous) function of V, x, t. Furthermore, its integration over the whole sample 

space V at every single point x,t must be equal to one, because the total probability for all events must be 

100 percent, i.e., 

123 123 1 2 3 f V V V x x x t dV dV dV U ( , , ; , , ,) 1

∞∞∞

−∞ −∞ −∞

= ∫∫∫ . (1) 

Or written as 

f td U ( ; ,) 1 = ∫ Vx V . (2) 

Where is an abbreviation of . It is also important to note that the 

argument V in fU represents ‘ ’ so that 

( ) d ∫ V 123 ( ) dV dV dV

∞∞∞

−∞ −∞ −∞

∫∫∫

3 U ( ; ,) 1 2 VV V , , f V x t is a joint probability density function 

123 fU (, ) VV V, ;, x t of the turbulent velocity components . 1 2 U U, ,U3

NASA/TM—2010-216368 2
As Pope (Ref. 1) pointed out that, with the PDF fU ( ; ,) V x t

t)

, we can define various one-point

statistical properties of the random velocity ( , , for example, the mean velocity (or the first moment) Ui x

Ui : 

( ; ,) U Vf td i iU = ∫ Vx V (3) 

All other higher order moments at any single point of x,t (also referred as one point moments or one 

point correlations) can also be defined: 

( ; ,)

( ; ,)

i j i jU

i j k i j kU

UU VV f t d

UU U VV V f t d

=

=

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

∫

∫

Vx V

Vx V (4) 

In particular, the Reynolds stresses u ui j defined by the Reynolds decomposition uU U ii i = − are 

fully defined by fU ( ; ,) V x t : 

uu U U U U U U U U U U U U U U U U ij i i j j i j i j j i i j i j i j =− − = − − + = − ( )( ) (5) 

Equation (5) indicates that the turbulent kinetic energy k uu = i i / 2 is fully defined by the velocity PDF

fU ( ; ,) V x t . 

2.2 PDF of Turbulent Species fΦ(ψ; x,t) 

Turbulent scalar variables, for example, the species ( , ), 1,2, i Φ x t i = "n , are random variables. The 

PDF of species is denoted as fΦ ( ; ,) ψ x t , where ψ ≡ ψi is the n-dimensional sample space coordinates of 

the speciesΦi. Therefore, fΦ ( ; ,) ψ x t is a joint scalar PDF of multi species. Analogous to the turbulent 

velocity, the one-point statistical properties of the turbulent species Φi are fully defined by the PDF

fΦ ( ; ,) ψ x t . For example, the mean is written as 

( ) ; , , 1, 2, , i i Φ =ψ fΦ td i n = ∫ ψ ψ x " (6) 

And the higher order moment as: 

2 2

3 3

1 2 1 2

( ; ,)

( ; ,)

( ; ,)

i i

i i

n n

f td

f td

f t d

Φ

Φ

Φ

Φ =ψ

Φ =ψ

ΦΦ Φ = ψψ ψ

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

∫

∫

" " ∫

x

x

x

ψ ψ

ψ ψ

ψ ψ

 (7) 

In particular, the i

th species fluctuating intensity 2

i φ defined by the Reynolds decomposition 

φ =Φ − Φ ii i is fully defined by fΦ ( ; ,) ψ x t , 

( )2 2 2 φ = Φ−Φ =Φ −Φ Φ i i i i i i (8) 

NASA/TM—2010-216368 3
2.3 Other Turbulent Quantities and Functions of Turbulent Variables 

There are several other turbulent quantities, for example, the pressure P, the chemical reaction source 

S(Φi), the pressure gradient ∂ ∂ P xi , the velocity gradient ∇Ui , and the higher order derivatives such as

, . They may be viewed as other separate random (scalar or vector) variables, in addition to 

the turbulent velocity Ui and the species Φi. We will see later that these “random variables” often appear 

as a “conditional mean” in the traditional PDF equations.

2 ∇ Ui2 ∇ Φi

There are two types of quantities: one is the analytical function of the turbulent variables such as 

S(Φi) = 1 2 , and the other is the derivatives of the turbulent variables such as . The quantities 

S(Φi) and are both random because Φi is random. From the probability theory the statistical 

properties of S(Φi) and Φi can be determined by the same PDF

α β Φ Φ "

2 ∇ Φi

2 ∇ Φi

fΦ ( ; ,) ψ x t . However, the statistical 

properties of is not that straightforward. ∇ Φ2

i

2.3.1 Statistical Mean of Analytical Function of Turbulent Variable S(Φi) 

The statistical mean of S(Φi) is defined as 

( ) ( ) ( ; ,) S Sf t Φ= ψ i i ∫ Φ ψ x dψ (9) 

where S(ψi) is the functional form in the sample space ψ for S(Φi). The function S( ) is the assumed 

analytical function. 

2.3.2 Statistical Mean of Derivatives of Turbulent Variable ∂Ui /∂xj and ∂2Ui /∂xj∂xj

There are at least two ways to take means on the derivatives of velocity. Let us first use the 

commutation rule (Section 3.1) to express the following means: 

2

2

( ; ,) ( ; ,)

( ; ,)

U

i i iU i

j

U

i i

j j

f t U U Vf td V d

x

f t UV d

x x

∂ ∇ =∇ =∇

∂

∂ ∇ =

∂ ∂

∫ ∫

∫

V x V x V= V

V x V

 (10) 

Hence, the mean of derivatives can be determined by the derivatives of the PDF of turbulent variable. 

Equation (10) can be further expressed as 

2

2

( ; ,)

( ; ,)

i U

i U

U j

i U

i U

U jj

V f U f f x

V f U f f xx

⎛ ⎞ ∂ ∇ = ⎜ ⎟ ∂ ⎝ ⎠

⎛ ⎞ ∂ ∇ = ⎜ ⎟ ∂ ∂ ⎝ ⎠

∫

∫

Vx V

Vx V

t d

t d

or 

2

( ) ( ; ,)

( ) ( ; ,)

i jiU

i iU

U D V f td

U LV f t d

∇ =

∇ =

∫

∫

Vx V

Vx V

 (11) 

where Dj(Vi) and L(Vi) are defined as 

NASA/TM—2010-216368 4
2

( ; ,) ( ) ( ; ,)

( ; ,) ( ) ( ; ,)

i U

j i

U

i U

i

U j

V f D V

f tx

V f L V

f t xx

∂ = ∂

∂ = ∂ ∂

V x

V x

V x

V x

j

j

t

t

 (12) 

Analogy to Equation (9), we may view Dj(Vi) and L(Vi) as the functional forms of and in the 

sample space V. 

∇Ui 2 ∇ Ui

Another way to take mean on the derivatives of velocity is to view them as some different random 

variables from the velocity; hence it will involve the joint PDF or the condition means of the velocity 

derivatives. This will be described in the next Section, where the relationships between the conditional 

means and the functions Dj(Vi), L(Vi) will be revealed. 

2.4 Joint PDF 

2.4.1 Joint PDF and Marginal PDF 

The Navier-Stokes equations contain several random variables, for example, the velocity Ui, the 

pressure P (or the pressure gradient ∇P) and the molecular diffusion . The scalar transport 

equations contain the velocity Ui, the species Φi and its molecular diffusion

2 ν∇ Ui

2 Γ∇ Φi . Hence a joint PDF 

for the joint variables of Ui, Φi, P, , 2 Ui ∇ ≡ U L i

2 i i L ∇Φ ≡ Φ ," , i.e. U P ,,, , UL L f ( , , , , ; ,) p t V ll x ψ U Φ Φ

( ; ,)

Φ

U

, 

may be needed in the analysis together with the marginal PDF, such as f V x t , fΦ ( ; ,) ψ x t , and the 

joint PDF of , fU Φ ( , ; ,) V x ψ t . The arguments , , U p l lΦ are the sample space variables of P, and 

, respectively. 

2 ∇ Ui

2 ∇ Φi

By definition, PDF must satisfy the normalization condition: 

,,, , U ( , , , , ; ,) 1 U PL L f pt Φ d d dp d Φ d = ∫∫∫∫∫ V ll x V ll ψ ψ U Φ U Φ (13) 

The relationship between the joint PDF ,,, , U ( , , , , ; ,) U PL L f Φ p Φ V ll x ψ U Φ t and the marginal PDF

fU ( ; ,) V x t , fΦ ( ; ,) ψ x t , fP ( ; ,) p t x , U ( ; ,) Lf t Ul x ( ; ,) Lf Φ l x t Φ are defined as 

,,, ,

,,, ,

,,, ,

,,, ,

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

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

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

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

U

U

U

U U

U U PL L

U PL L

P U PL L

L U P L L

f t f p t d dpd d

f t f p t d dpd d

fp t f p td d d d

ft f p t

Φ

Φ

Φ

Φ

Φ

Φ Φ

Φ

Φ

=

=

=

=

∫∫∫∫

∫∫∫∫

∫∫∫∫

U U

U U

U

U U

V x V l l x l l

x V l l x V

x V l l x V

l x V ll x

U

l l

l l

Φ Φ

Φ Φ

Φ Φ

Φ

ψ ψ

ψ ψ

ψ ψ

ψ

,,, , ( ; ,) U ( , , , , ; ,) L U P L L

d d dpd

f Φ Φ t f p t d d dpd Φ =

∫∫∫∫

∫∫∫∫ U U

V l

l x V ll x V l

Φ

Φ Φ

ψ

ψ ψ

 (14) 

This is clearly required by the normalization condition for any type of PDF, including the marginal and 

joint PDF. With this basic concept, we can obtain various relationships between different joint PDF, for 

example, the joint PDF of Ui and Φi can be defined as 

, ,,, , U ( , ; ,) U ( , , , , ; ,) U PL L f Φ t f p t dpd d = Φ Φ ∫∫∫ V x ψ ψ V ll x ll U Φ U Φ (15) 

It should be noted from Equations (13) and (14) that the statistical mean of a random variable can be 

equally define by either the joint PDF or its marginal PDF, for example, 

NASA/TM—2010-216368 5
( )

,,, ,

,,, ,

( , , , , ; ,)

( , , , , ; ,)

( ; ,)

U

U

i i U PL L

i U PL L

i U

U V f p t d d dp d d

V f p t d dp d d

Vf td

Φ

Φ

Φ

Φ

=

=

=

∫∫∫∫∫

∫ ∫∫∫∫

∫

U U

U U

V ll x V ll

V ll x ll

Vx V

Φ Φ

Φ

ψ ψ

ψ ψ Φ dV (16) 

( )

,,, ,

,,, ,

( , , , , ; ,)

( , , , , ; ,)

( ; ,)

U

U

i i U PL L

i U PL L

i

f pt d d dp d d

f p t d dpd d

f td

Φ

Φ

Φ

Φ

Φ

Φ= ψ

= ψ

= ψ

∫∫∫∫∫

∫ ∫∫∫∫

∫

U U

U U

V ll x V ll

V ll x V ll

x

Φ Φ

Φ

ψ

d ψ Φ ψ

ψ ψ

ψ

 (17) 

( )

,,, ,

,,, ,

,

( , , , , ; ,)

( , , , , ; ,)

( , ; ,)

U

U

i j i j U PL L

i i U PL L

i iU

U Vf p t d d dp d

V f p t dpd d d d

V f td d

Φ

Φ

Φ

Φ

Φ

Φ= ψ

= ψ

= ψ

∫∫∫∫∫

∫∫ ∫∫∫

∫∫

U U

U U

V ll x V ll

V ll x ll V

VxV

Φ Φ

Φ Φ

ψ ψ d

ψ ψ

ψ ψ

 (18) 

Equations (16), (17) and (18) indicate that taking mean of a random variable only needs the relevant 

marginal PDF or the relevant joint PDF. 

2.4.2 Conditional PDF and Conditional Mean 

Let us examine an expression: 

 (19) , ( , ; ,) ∫Vf td i U Φ VxV ψ

which is taking mean on the random variable Ui at a fixed ψ, i.e. Φi = ψi . If we define the ratio of the 

joint PDF to the marginal PDF as 

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

U

f t

f t f t

Φ

Φ

Φ

= V xV x

x

ψ

ψ

ψ

 (20) 

Then the expression (19) can be written as 

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

( ; ,)

i U i U i

i

V f t d f t V f t dV

f tU

Φ Φ Φ

Φ

= ⋅

= ⋅

∫ ∫ VxV x Vx

x

ψψ ψ

ψ ψ

 (21) 

Where Ui ψ is an abbreviation of Ui Φ = ψ and is referred to as the conditional mean of Ui on the 

condition of : Φ =ψ i i

( ; ,) U Vf t i i ψ ψ ≡ ∫ U Φ Vx Vd (22) 

And f

U Φ ( ;, V x ψ t) is referred to as the conditional PDF of Ui on the condition of , which 

satisfies the normalization condition because of the definition 

Φ =ψ i i

(20): 

f td U Φ ( ; ,) =1 ∫ VxV ψ (23) 

NASA/TM—2010-216368 6
Similarly, the conditional mean of the random pressure P t ( ,) x on the condition of is denoted as U V i = i

P t ( ,) x V or P V and the corresponding conditional PDF is denoted as 

,, , U ( , , , ,; , ) PL L U f Φ p Φ

U ψ l lV x Φ t , then we may naturally write 

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

( ; ,)

PL L U U

P U

P t p f p t d dpd d

p f p t dp

= Φ Φ

=

∫∫∫∫

∫

U U x V l lV x l l

V x

ψ ψ Φ Φ

 (24) 

The above conditional PDF is defined as 

,,, ,

,, ,

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

U

U

U PL L

PL L U

U

f p t f pt f t

Φ

Φ

Φ

Φ =

U

U

V ll x

l lV x

V x

Φ

Φ ψ

ψ (25) 

or 

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

P U

U

f p t fp t f t = V x

V x

V x (26) 

In Equations (24) and (25), the sample space variables , , Ul l ψ Φ are irrelevant. 

2.4.3 Relationship Between Unconditional Mean and Conditional Mean 

It should be noted from Equation (24) that a conditional mean is a function of the sample space 

variable that has been excluded from the process of conditional mean (e.g., in this case). Obviously, Vi

P V is a function of sample space variable , and taking ‘mean’ on Vi P V over the sample space 

will lead to its unconditional mean (or just mean) pressure 

Vi

P , which is a function of , xi t .: 

( )

,

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

( , ; ,)

U UU P

U P

P f t d p f p t dp f t dV

p f p t d dp

P

=

=

=

∫ ∫ ∫

∫∫

V Vx V Vx Vx

V xV

i

Or vice versa, 

, P = = ∫∫ p f p t d dp f t P t d U P ( , ; ,) ( ; ,) ( ,) V x V Vx x V V ∫ U (27) 

Equation (27) is a general formula for two or more different random variables (P and Ui in this case) 

involved in the conditional mean of one random variable (P in this case). 

Now, following Equation (27), we may straightforwardly express the means of the derivatives of 

turbulent velocity , as ∇Ui 2 ∇ Ui

2 2

( ; ,)

( ; ,)

iU i

iU i

U f tU d

U f tUd

∇= ∇

∇= ∇

∫

∫

Vx V V

Vx V V

 (28) 

Comparing Equation (28) with Equation (11), a sufficient (but not necessary) condition for this identity is 

NASA/TM—2010-216368 7
2

2

( )

( )

i U

i ji

U j

i U

i i

U

V f U DV

f x

V f U LV

f

∂ ∇= =

∂

∇ ∇ ==

V

V

 (29) 

Equation (29) indicates the relationship between the conditional means and functions ( ) D Vj i , ( ) L Vi . It is 

noted here that these equations can also be derived by using the fine grained PDF through similar 

sufficient condition (see Appendix A: Equations (100) and (103)). 

For the mean of pressure gradient ( ,) ∂ ∂ P xt x i i , following Equation (27), we may write 

( ,) ( ; ,) i

U

i i

P P x t f t

x x

∂ ∂ = ∂ ∂ ∫ Vx V Vd (30) 

In the past, the conditional mean of the pressure gradient in Equation (30) has been considered as an 

unknown quantity even if P = P (U) was assumed. This conditional mean, like the other conditional mean

2 ∇ Ui V , is actually carried by the traditional velocity PDF equation and is modeled empirically. Now, 

if we use the rule of commutation and take the mean on the pressure gradient (assuming that

P t PU t P t PU t t ( , ) ( ( , )), or ( , ) ( ( , ), , ) x xx x = = x ), we may write 

( )

( ) ( ; ,)

( ) ( ; ,)

1 () ( ; ,)

( ) ( ; ,)

U

ii i

U

i

U

U

U i

g

i U

P P PV f t

xx x

PV f t d

x

PV f = f f x

P V f td

∂∂ ∂ = = ∂∂ ∂

∂ = ∂

⎛ ⎞ ∂ ⎜ ⎟ ⎝ ⎠ ∂

=

∫

∫

∫

∫

Vx V

Vx V

Vx V

Vx V

d

t d

 (31) 

Here, P V( ) is dependent on x,t in general and ( ) g Pi V is defined as 

1 [ ( ) ] ( ) g U

i

U i

P V f P V

f x

∂ = ∂ (32) 

Comparing Equations (30) and (31), and using a similar argument that leads to Equation (29), we obtain a 

relationship between the conditional mean and the function ( ) g Pi V : 

( ,) 1 [ ( ) ] ( ) i g U

i

i U

Px t PV f P V

x f

∂ ∂ = = ∂ ∂

V

xi

n

 (33) 

It is noted here that this equation can also be derived by using the fine grained PDF through the similar 

sufficient condition (see Appendix A: Eq. (104)) 

Now let us look at the mean of the product Ui ⋅Φ , i.e. Ui n Φ . By definition we may write 

NASA/TM—2010-216368 8
( , ( , ; ,) )

( ; ,)

i n n iU

n i

U V f t d

f tU d

Φ

Φ

Φ =ψ

= ψ

∫ ∫ d

∫

V xV

x

ψ ψ

ψ ψψ

 (34) 

or 

( , ( , ; ,) )

( ; ,)

i n i nU

i U n

U V f td

Vf t d

Φ= ψ Φ

= Φ

∫ ∫ d

∫

Vx V

Vx V V

ψ ψ

 (35) 

Equations (34) and (35) illustrate that, when taking joint mean, the joint PDF must be used to avoid the 

appearance of the conditional mean. The conditional mean Ui ψ is also carried by the current 

traditional species PDF equation. 

Finally, let us examine the mean and the conditional mean of a function of the variables Φi, say

, on the condition of 1 2 ( S ΦΦ Φ " n ) Φ ψ= . Let us first view 1 2 ( S n Φ Φ Φ " ) as a separate random 

variable with the sample space variable s which is independent of ψ, then, following Equation (27), we 

write the mean as 

1 2 1 2 ( ) ( ; ,) ( ) S ft ΦΦ Φ = n Φ S ΦΦ Φ " ∫ ψ x " n ψ ψd (36) 

The conditional mean in (36) is 

1 2 1 2

1 2

1 2

( ) ( ) ( ;,

( ) ( ;,

( )

n n S

n S

n

S Sf s

S fs

S

Φ

Φ

ΦΦ Φ = ψψ ψ

= ψψ ψ

= ψψ ψ

∫

∫

" "

"

"

x

x

ψ ψ

ψ

)

)

t ds

t ds (37) 

Then Equation (36) will end up an expression that is just the definition of its mean: 

1 2 1 2 ( () ( ; S Sf ΦΦ Φ = ψψ ψ " " n n ∫ Φ ψ x,t)dψ (38) 

This term is related to the chemical reaction rate in the traditional species PDF equation, which is in a 

closed form. Also note that the reason for the conditional mean of 1 2 ( S n Φ Φ Φ " ) in Equation (36)

becoming free from the constraint is that 1 2 ( S n Φ Φ Φ " ) is a known function of Φi without involving 

spatial differentiation. 

2.4.4 Summary 

2.4.4.1 Formulations With the Conditional Mean 

2 2 ( ; ,) ∇= ∇ U f t Ud iU i ∫ Vx V V (39) 

∇= ∇ P f tP U ( ; ,) ∫ Vx V Vd (40) 

2 2 ( ; ,) i i ∇Φ = fΦ t ∇ Φ ∫ ψ x ψ ψd (41) 

NASA/TM—2010-216368 9
1 2 ( ( ; , ) S ft in i ΦΦ Φ = Φ S " ∫ ψ x ψ ψd (42) 

( ; ,) U f tU in n Φ =ψ∫ Φ ψ x i ψ ψd (43) 

The conditional means (except the one in Eq. (42)) are all considered as unknowns in the traditional PDF 

equations. 

2.4.4.2 Formulations Without the Conditional Mean 

2 2

2

2

( ; ,)

or

( ) ( ; ,) ( )

i iU

i U

i iU i

U

U V f td

V f U LV f t d LV f

∇ =∇

∇ ∇ = =

∫

∫

Vx V

V x V,

 (44) 

[ ]

[ ]

( ) ( ; ,)

or

1 ( ) ( ) ( ; ,) ( )

U

i i

g g U

i i U

i U

P PV f t d

x x

P PV f P V f td P V x f

∂ ∂ = ∂ ∂

∂ ∂ = = ∂ ∂

∫

∫

Vx V

V x V,

xi

 (45) 

2 2

2

2

( ; ,)

or

( ) ( ; ,) ( )

i i

i

i i i

f td

f L f td L f

Φ

Φ

Φ

Φ

∇Φ = ψ∇

ψ ∇ ∇Φ = ψ ψ =

∫

∫

x

x ,

ψ ψ

ψ ψ

 (46) 

1 2 1 2 ( ( ; , ) ( S ft i n in ΦΦ Φ = Φ S ψψ ψ " ∫ ψ x " )dψ (47) 

, ( , ; ,) U V f td i n i nU Φ= ψ ∫∫ Φ V xV ψ dψ (48) 

These equations are exact and have no unknowns other than the PDFs fU, fΦ and U, f Φ . The functions

( ) L Vi , ( ) and L ψi ( ) g Pi V are all well defined by the corresponding PDF, and the function P(V) is 

assumed as known and will be defined later (Section 3.4, Equation (68)). 

3.0 Transport Equation for Turbulent Velocity PDF fU(V; x,t) 

3.1 Mean and Differentiation Commutation 

It is important to note the commutation property of taking mean and differentiation, that is 

2 2

, , i i ii i t t x x xx xx

∂∂ ∂ ∂ ∂ == = ∂ ∂ ∂ ∂ ∂∂ ∂∂ ,

i

∂

 (49) 

NASA/TM—2010-216368 10
These commutation properties have been shown in several text books, for example by Pope (Ref. 1), 

Tennekes and Lumley (Ref. 2). For example, 

0 0

( ,) ( ,) ( ,) ( ,) ( ,) ( , lim lim i i

i i

x x ii i

t + x t t +xt t t

xx x Δ → Δ →

∂ Δ − Δ− ∂ = = ∂Δ Δ

Ux Ux Ux Ux Ux Ux )

xi

= ∂ (50) 

where ( ) . 123 U = UU U , ,

3.2 Taking Mean on Navier-Stokes Equations 

Apply the mean operation on the Navier-Stokes equations for incompressible flow, i.e. 

2 1 . i j i

i

j i

U P U U

U

tx x

∂ ∂ ∂

+ = − + ν∇

∂ ∂ ρ∂ (51) 

Using the commutation rule, Equation (49), each term in Equation (51) can be expressed as follows, 

( ; ,)

( ; , ) Note, is independent of and 

i i

i U

i U i i

U U

Vf td

t tt

V f td V x

t

∂ ∂ ∂ = = ∂ ∂∂

⎡ ⎤ ∂ = ⎢ ⎥ ⎣ ⎦ ∂

∫

∫

Vx V

Vx V t

 (52) 

Similarly, 

( ; ,)

( ; ,)

i j

i jU

j j

ij U

j

U U

VVf td

x x

VV f td

x

∂ ∂ = ∂ ∂

⎡ ⎤ ∂ = ⎢ ⎥ ∂ ⎣ ⎦

∫

∫

Vx V

Vx V

 (53) 

Taking mean on the pressure gradient term can proceed in two different ways. One is with the conditional 

mean, i.e., via Equation (40): 

11 1 ( ; ,) ( ; ,)

1 1 ( ; ,) ( ; ,)

1 ( ; ,) Note, 

i

UU i

ii j j

i U i U

j jj

i U

j j

PP P V f t df t d

xx x V

P P Vf t d V f t d

V xVx

P V V ft d V x

∂∂ ∂ ⎛ ⎞ ∂ −= − = − ⎜ ⎟ ρ ∂ ρ ∂ ρ∂ ∂⎜ ⎟ ⎝ ⎠

⎛ ⎞⎡ ⎛ ⎞ ∂ ∂ ∂∂ = −− − ⎜ ⎟ ⎜ ⎟ ⎢ ⎥

∂ ρ ∂ ∂ρ ⎢ ⎥ ∂ ⎝ ⎠⎣ ⎝ ⎠

⎡ ⎤ ⎛ ⎞ ∂ ∂ ∂ = − ⎢ ⎥ ⎜ ⎟ − ⎢ ⎥ ∂ ρ ∂ ⎣ ⎦ ⎝ ⎠

∫ ∫

∫ ∫

∫

Vx V V Vx V V

Vx V V Vx V V

Vx V V

j

⎤

⎦

i

ij Vj

≡ δ

∂

 (54) 

Note that the following term in Equation (54) is zero (Pope (Ref. 1) or see Appendix B): 

1 ( ; ,) i U i 0

j j

P Vf t dV

V x

⎛ ⎞ ∂ ∂ ⎜ − ⎟ = ∂ ρ ⎜ ∂ ⎝ ⎠ ∫ Vx V ⎟ (55) 

NASA/TM—2010-216368 11
The other way without the appearance of the conditional mean is via Equation (45)

( ) ( )

1 1 ( ) ( ; ,)

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

( ) ( ; ,)

g i

j U

i j

g g

i U j j i U

j j

g

j

i U

j

P V PVf t d x V

VP V f t d V P V f t d V V

P V

V f td V

∂ ∂ − =−

ρ∂ ρ ∂

∂ ∂ = − +

ρ ∂ ρ ∂

⎡ ⎤ ⎛ ⎞ ∂ = ⎢ ⎥ ⎜ ⎟ ⎢ ⎜ ∂ ρ ⎟⎥ ⎣ ⎦ ⎝ ⎠

∫

∫ ∫

∫

Vx V

Vx V Vx V

Vx V

 (56) 

Note that (see Appendix B): ( ) 1 ( ) ( ; , ) 0. g

i U j

j

VP f t d V

∂ − =

ρ ∂ ∫ V Vx V

Finally, taking mean on the molecular diffusion term, one way with the conditional mean is via 

Equation (39), 

( ) ( )

( )

22 2

2 2

2

( ; ,) ( ; ,)

( ; ,) ( ; ,)

( ; ,)

i

i U i U j

j

i U j i U j

j j

iU j

j

V U f t Ud f t U d V

Vf t U d V f t U d V V

V f tU d V

∂ ν∇ = ν∇ = ν∇

∂

∂ ∂

= ν∇− ν∇

∂ ∂

⎡ ⎤ ∂ = − ⎢ ⎥ ν∇

∂⎣ ⎦

∫ ∫

∫ ∫

∫

Vx V V Vx V V

Vx V V Vx V V

Vx V V

 (57) 

Note that (see Appendix B): ( ) 2 i U ( ; ,) j 0

j

Vf t U d V

∂

ν∇ =

∂ ∫ Vx V V . 

The other way without the conditional mean is via Equation (44), 

( ) ( )

( )

2 ( ) ( ; ,)

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

( ) ( ; ,)

i

i jU

j

i jU i jU

j j

i j U

j

V U LV f t d V

V LV f t d V LV f t d V V

V LV f t d V

∂ ν∇ =ν

∂

∂ ∂

= ν − ν

∂ ∂

⎡ ⎤ ∂

=− ν ⎢ ⎥ ∂⎣ ⎦

∫

∫ ∫

∫

Vx V

Vx V Vx V

Vx V

 (58) 

Note that (see Appendix B): ( ) ( ) ( ; ,) i jU

j

V LV f t d V

∂

ν

∂ ∫ Vx V =0. 

3.3 Traditional Velocity PDF Equation 

Now substituting Equations (52), (53), (54), (57) into Equation (51) and collecting the terms factored 

by Vi , a sufficient but not necessary condition for satisfying the integral equation is that the factored, total 

integrand is zero. Thus, a transport equation for the turbulent velocity PDF fU ( ; ,) V x t is constructed: 

NASA/TM—2010-216368 12
( 2 U U 1

iU U

ii i i

ff P V f fU t xV x V∂ ∂∂ ∂ ∂ ⎛ ⎞

+ = ⎜ ⎟ − ν∇

∂ ∂ ∂ ρ∂ ∂ ⎝ ⎠

V i V ) (59) 

If the pressure term is decomposed into the mean and fluctuating parts, P P P' = +

( )' ' 1 1 11

i ii

P P P P

x xx

∂ ∂ + ∂

= =+

ρ∂ ρ ∂ ρ ∂ ρ∂

V V

i

P

x

∂ V (60) 

Then Equation (59) can be written as 

' 2 UU U 1

i U i

i ii i i

ff f P P V f U

t x xV V x

⎛ ⎞ ∂ ∂ ∂∂ ∂ ⎡ 1 ∂ ⎤

+ = − ν∇ − ⎜ ⎢ ⎥⎟

∂ ∂ ρ∂ ∂ ∂ ⎜ ρ ∂ ⎟ ⎝ ⎠ ⎢⎣ ⎥⎦

V V (61) 

Equation (61) is exactly the same as the traditional velocity PDF equation (Pope (Ref. 1)). The mean 

pressure in Equation (61) is considered as a known quantity through its Poisson equation. However, the 

last two terms with the conditional mean were considered as unknown and modeled empirically. Using 

the following identity relations (Pope (Ref. 1)): 

( ) ' ' ( ; ,) ( ; ,) ( ; ,) j

UUU

i i j i

P U

f t f tP f t P

x x V x

∂ ∂ ∂ ⎛ ⎞ ∂

= + ⎜ ⎟ ∂ ∂ ∂ ∂ ⎝ ⎠

Vx V Vx V Vx V ' (62) 

2

2 2 ( ; ,) ( ; ,) ( ; ,) j k

Uj U U

j j k i i

U U f t U f t f t V V V x x

∂ ∂∂ ⎛ ⎞ ∂ ⎡ ⎤ ∇ = ⎜ ⎟ − ∇

∂ ∂ ⎣ ⎦ ∂ ∂ ∂ ⎝ ⎠

Vx V V x V Vx (63) 

Equation (61) can be rewritten as 

2 '

2

2 '

1

1 2

2

U U U

i U U

i i i i i

j j i i

U U

i j i j k k

f f P f P V f f t x x V xV

PU U U U

f f V V x x x x

∂ ∂ ∂ ∂ ∂ ⎛ ⎞

+ =ν∇ + + ⎜ ⎟ ∂ ∂ ρ∂ ∂ ∂∂ ρ ⎝ ⎠

⎛ ⎞ ∂ ∂∂ ⎛ ⎞ ∂ ∂

+ + ⎜ ⎟ ⎜ ⎟ − ν

∂∂ ρ ∂ ∂ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠

V

V V (64) 

Now we have three unknown terms in Equation (64) that were modeled empirically (Pope (Ref. 1)). 

3.4 Conservational Velocity PDF Equation 

Alternatively, collecting the terms factored by Vi from (52), (53), (56) and (58), and following the 

same procedure that leads to Equation (61), we obtain a closed velocity PDF transport equation: 

( ) ( ) ( )

g

U U j

j j U

j j j

f f P V

V L V f t xV V

ν

ρ

∂ ∂ ∂ ∂ ⎛ ⎞

+ =− + ⎜

⎜ ∂ ∂∂ ∂ ⎝ ⎠ Uf ⎟

⎟ . (65) 

Here ( ) L Vj and ( ) g Pj V are known functions: 

NASA/TM—2010-216368 13
[ ] 2 1 ( ) ( ) , () j U g U

j j

U U

V f

j

P V f L V P V

f f

∇ ∂ = = ∂x

 (66) 

In order to determine P V( ), we first take mean on the Poisson equation 

2 2

2 2 1 1

, or i j i j

i j i j

U U U U

P P

x x x x

∂ ∂

∇ =− ∇ =−

ρ ∂∂ ρ ∂∂ (67) 

By the definition of the mean, we may write Equation (67) as 

2

2 1 ( ) ( ; ,) U i ( ; ,) j U

i j

P V f td VV f t d

x x

∂ ∇ = − ρ ∂ ∂ ∫ ∫ Vx V Vx V

or 

( )

2

2 1 ( ( ) ( ; ,) U

U i j

i j

f t ; , ) P V f t d VV d

x x

∂ ∇ = − ρ ∂ ∫ ∫ V x Vx V V

∂

By equating the integrands on both sides (again, this is a sufficient condition for satisfying the above 

integral equation), we obtain a model equation for P V( ):

[ ]

2

2 1 ( ) U

U ij

i j

f PV f VV

x x

∂ ∇ =−

ρ ∂ ∂ (68) 

Therefore we have constructed a velocity PDF Equation (65) that does not need extra turbulence 

modeling. 

Using Equation (65), it is easy to show that the exact equations for the first moment and higher order 

moments of the Navier-Stokes equations can be deduced by multiplying the Equation (65) with Vi and 

Vi Vj respectively and integrating over the velocity sample space V. For example, 

( ) ( ) ( )

g

U U k

ij k k U U

k k k

f f P

VV V LV f f d t xV V

⎧ ⎫ ⎛ ⎞ ⎪ ⎪ ∂ ∂∂ ∂ ⎨ ⎬ + =− ν + ⎜ ⎟

∂ ∂∂ ∂ρ ⎜ ⎟ ⎪ ⎪ ⎩ ⎭ ⎝ ⎠ ∫ V

V

The left two terms become 

i j i jk

k

UU UU U

t x

∂ ∂

+

∂ ∂ , 

and the first term on the right hand side is, 

( ) ( )

( )

2 2

( ) () ()

( ) ()

i j

ij k U ij k U k U

k k

j iU i jU

i jj i

V V

VV LV f d VV LV f d LV f d V V

LV V f LV V f d

U UU U

∂ ∂ ∂

− ν −ν ν

∂ ∂

ν +ν

ν ∇ +∇

∫ ∫∫

∫

V = V + V

= V

=

∂Vk

NASA/TM—2010-216368 14
Note that (see Appendix B): ( ) ( ) ij k U

k

VV LV f d V

∂

− ν

∂ ∫ V = 0. 

Similarly, the second term on the right hand side is 

( ) 1 g

k

i j U i j

k j

P P P V V f d U U V x

⎛ ⎞ ∂ ∂ ⎜ ⎟ =− +

∂ ρ ρ∂ ∂ ⎜ ⎟ ⎝ ⎠ ∫ V

V

xi

∂

Therefore we obtain the exact equation for the second moment U Ui j : 

2 2 i j i jk 1

i j i jj

k ji

UU UU U P P U U U UU U

t x xx

∂ ∂ ∂ ∂

+ = − ++ ν ∇ +∇

∂ ∂ ρ∂ ∂ i (69) 

4.0 Transport Equation for Turbulent Species PDF fΦ(ψ; x,t) 

4.1 Taking Mean on Species Equation 

Taking mean on the transport equation for species Φi 

2 1 2 ( i j i

ii n

j

U

S

t x

∂ Φ ∂Φ

+ = Γ∇ Φ + Φ Φ Φ

∂ ∂ " ) (70) 

The first term on the L.H.S. of (70) can be written as 

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

( ; ,)

i

i U i U

i

f t d d f td d

t t t

f td

t

Φ Φ

Φ

∂ Φ ∂ ∂

= ψ = ψ ∂ ∂ ∂

⎡ ⎤ ∂

= ψ ⎢ ⎥ ⎣ ⎦ ∂

∫∫ ∫ ∫

∫

Vx V V xV

x

ψ ψ ψ

ψ ψ

ψ

 (71) 

Equation (71) involves a marginal PDF fΦ . We can also write this term using the joint PDF U, f Φ : 

( )

( )

,

,

( , ; ,)

( , ; ,)

i

i U

i U

f t dd

t t

f t dd

t

Φ

Φ

∂ Φ ∂

= ψ ∂ ∂

⎡ ⎤ ∂

= ψ ⎢ ⎥ ⎣ ⎦ ∂

∫∫

∫∫

Vx V

Vx V

ψ ψ

ψ ψ

 (72) 

The second term on the L.H.S. of (70) can also be formed in two ways: 

, ( , ; ,)

( ; ,)

i j

i jU

j j

i j

j

U

Vf td d

x x

f tU d

x

Φ

Φ

∂ Φ ∂

= ψ ∂ ∂

⎡ ⎤ ∂

= ψ ⎢ ⎥ ∂⎣ ⎦

∫∫

∫

VxV

x

ψ ψ

ψ ψ ψ

 (73) 

or 

NASA/TM—2010-216368 15
,

,

( , ; ,)

( , ; ,)

i j

i jU

j j

ij U

j

U

Vf td d

x x

V f t dd

x

Φ

Φ

∂ Φ ∂

= ψ ∂ ∂

⎡ ⎤ ∂

= ψ ⎢ ⎥ ∂ ⎣ ⎦

∫∫

∫∫

VxV

Vx V

ψ ψ

ψ ψ

 (74) 

The next term of 2 ∇ Φi (70) can again be written in two different forms: Using Equation (41), we have 

( )

( )

22 2

2 2

2

( ; ,) ( ; ,)

( ; ,) ( ; ,)

( ; ,)

i

ii j

j

i ji j

j j

i j

j

ft dft d

f t d ft

ft d

Φ Φ

Φ Φ

Φ

∂ψ Γ ∇ Φ =Γ ∇ Φ =Γ ∇ Φ

∂ψ

∂ ∂ =Γ ψ ∇ Φ −Γ ψ ⎡ ⎤ ∇ Φ

∂ψ ⎣ ⎦ ∂ψ

⎡ ⎤ ∂

=− ψ ⎢ ⎥ Γ∇ Φ

∂ψ ⎣ ⎦

∫ ∫

∫ ∫

∫

x x

x x

x

ψ ψψ ψ ψ ψ

ψ ψψ ψ ψ

ψ ψψ

dψ (75) 

Note that (see Appendix B): 2 i j ( ; ,) 0

j

ft d Φ

∂ ⎡ ⎤ ψ ∇Φ ∫ ∂ψ ⎣ ⎦ ψ ψ x ψ = . 

Or we can straightforwardly write its mean as 

22 2 ,

2 ,

,

( , ; ,)

( , ; ,)

( ) ( , ; ,)

ii i U

i U

i U

f td d

f t dd

L f td d

Φ

Φ

Φ

Γ ∇ Φ =Γ∇ Φ =Γ∇ ψ

= ψ Γ∇ ⎡ ⎤ ⎣ ⎦

= Γψ

∫∫

∫∫

∫∫

VxV

Vx V

VxV

ψ ψ

ψ ψ

ψ ψ

 (76) 

Where ( ) is defined as L ψi

2 2 ,

,

( ) i U i

i

U

f f L

f f

Φ Φ

Φ Φ

ψ ∇ ψ ∇ ψ= = (77) 

Equation (76) can be further manipulated as 

( )

( )

( )

2 ,

,

,

,

( ) ( , ; ,)

( ) ( , ; ,)

( ) ( , ; , ))

( ) ( , ; ,)

i

i j U

j

i jU

j

i jU

j

i j U

j

L f t dd

L f t dd

L f t dd

L f t dd

Φ

Φ

Φ

Φ

∂ψ Γ∇Φ =Γ ψ ∂ψ

∂ =Γ ψ ψ ∂ψ

∂ −Γ ψ ψ ∂ψ

⎡ ⎤ ∂ = −Γ ψ ψ ⎢ ⎥ ∂ψ ⎣ ⎦

∫∫

∫∫

∫∫

∫∫

Vx V

Vx V

Vx V

Vx V

ψ ψ

ψ ψ

ψ ψ

ψ ψ

 (78) 

Note that (see Appendix B): ( ) , ( ) ( , ; ,) i jU

j

L f t dd Φ

∂ Γ ψψ ∂ψ ∫∫ Vx V ψ ψ = 0. 

NASA/TM—2010-216368 16
The reacting source term is readily available via Equation (42), 

( ) ( )

( )

1 2

1 2

( ) ( ; ,) ( ; ,)

( ; ,) ( ; ,)

( ; ,) ( )

i

in i j

j

iji j

j j

i jn

j

S ft S d f t S d

f tS d f tS d

f tS d

Φ Φ

Φ Φ

Φ

∂ψ ΦΦ Φ = = ∂ψ

∂ ∂

= ψ − ψ ∂ψ ∂ψ

⎡ ⎤ ∂

=− ψ ⎢ ⎥ ψψ ψ ∂ψ ⎣ ⎦

∫ ∫

∫ ∫

∫

"

"

x x

x x

x

ψ ψψ ψ ψ ψ

ψ ψ ψ ψ ψψ

ψ ψ

 (79) 

Note that (see Appendix B): ( ) i j ( ; ,) 0

j

f tS d Φ

∂

ψ = ∂ψ ∫ ψ ψψ x . 

Another expression for the reaction source term can be obtained as 

( )

1 2 1 ,

1 ,

( ) ( ) ( , ; ,)

( ) ( , ; ,)

i n i nU

i j nU

j

S S ft d d

S ft

Φ

Φ

ΦΦ Φ = ψψ ψ

⎡ ⎤ ∂

=− ψ ψψ ψ ⎢ ⎥ ∂ψ ⎣ ⎦

∫∫

∫∫

" "

"

VxV

Vx V

ψ ψ

ψ d dψ

 (80) 

Note that (see Appendix B): ( ) 1 , ( ) ( , ; ,) ij n U

j

S ft Φ

∂

ψ ψψ ψ ∂ψ ∫∫ " Vx V ψ d dψ = 0 

4.2 Traditional Species PDF Equation 

Collecting terms factored by from Equations ψi (71), (73), (75) and (79), and following the 

procedure similar to the construction of the velocity PDF equation, we obtain the species PDF equation: 

( ) ( ( )) 2 1 ( j j j

j j

f f U f S

t x

Φ

Φ Φ

∂ ∂ ∂

+ = − Γ∇ Φ + ψ ψ ∂ ∂ ∂ψ ψ ψ " ) ψn (81) 

By using Reynolds decomposition uU U ii i = − , Equation (81) becomes 

( ) ( ( )) 2 1 ( jj j j

j j j

f f U fu f S

t xx

Φ Φ

Φ Φ

∂ ∂∂ ∂ + + = − Γ∇ Φ + ψ ψ ψ ∂ ∂ ∂ ∂ψ ψ ψ " ) n (82) 

Equation (82) is also exactly the same as the traditional species PDF equation derived by Pope (Ref. 1), 

which contains two conditional means from the convection term and the molecular diffusion term. These 

terms were considered as unknowns and were modeled empirically. 

4.3 Conservational Joint Species-Velocity PDF fU,Φ(V,ψ; x,t) Equation 

Alternatively, by collecting terms factored by ψi from Equations (72), (74), (78), and (80) , we obtain 

the following equation: 

( ) ( , , , ,1 ( ) ( ) U U

j j U U j

j j j

f f V L f f S

t x

Φ Φ

Φ Φ

∂ ∂ ∂ ∂ + =− Γ ψ − ψψ ψ ∂ ∂ ∂ψ ∂ψ " n ) (83) 

NASA/TM—2010-216368 17
Equation (83) is in closed form. They can be reduced to the marginal species PDF fΦ( ; ,) ψ x t equation 

by integrating over the velocity sample space Vi. The result is 

( ) ( ) ( ) 1 ( ) ( ) j j jj

j j j j

f f U fu L f f S

t xx

Φ Φ

Φ Φ Φ

∂ ∂∂ ∂ ∂ + + =− Γ ψ − ψψ ∂ ∂ ∂ ∂ψ ∂ψ ψ "ψn (84) 

In Equation (84), there is an unavoidable conditional mean due to the convection term that involves both 

the velocity and the species. The conditional mean of fluctuating velocity conditioned on , Φ =ψ i i

u j ψ , is unknown and must be modeled. 

Now let us return to the subject of joint PDF. Equation (83) is derived from the species diffusion 

Equation (70), the two source terms on the right hand side represent the molecular mixing of “species” 

and the chemistry reaction, respectively. We also have to consider the contribution from the NavierStokes equations, which can be obtained in a way similar to that leads to Equation (65): 

( ) , , ,

( ) ( )

g

U U j

j j U

j j j

f f P

V L V f t xV V

Φ Φ

Φ

⎛ ⎞ ∂ ∂ ∂ ∂

+ =− ν + ⎜

∂ ∂ ∂ ∂ρ ⎜

⎝ ⎠

V

U , f Φ ⎟

⎟ (85) 

The terms on the right hand side of Equation (85) represent the molecular mixing of “momentum” and the 

pressure source. Therefore, the joint species-velocity PDF , ( , ; ,) U i ii i i f Φ V x ψ t equation should include all 

source terms on the right hand side of Equations (83) and (85), i.e. 

( )

( ) ( )

, , , ,

, ,1

( ) ( )

( ) ( )

g

U U j

j j U U

j j j

j U Uj

j j

f f P

V L V f f t xV V

L f fS

Φ Φ

Φ Φ

Φ Φ

⎛ ⎞ ∂ ∂ ∂ ∂

+ =− ν + ⎜ ⎟

∂ ∂ ∂ ∂ρ ⎜ ⎟ ⎝ ⎠

∂ ∂ − Γ ψ − ψψ ψ

∂ψ ∂ψ "

V

n

 (86) 

It is easy to verify that the joint PDF U, f Φ Equation (86) can be reduced to the equations for the marginal 

velocity PDF fU and the marginal species PDF fΦ by integrating over the corresponding sample space, 

and the results are the same as the Equations (65) and (84). It is also easy to verify that Equation (86) can 

deduce the exact equations for the first moments , Ui Φi and higher order moments , Φ Φi j Ui j Φ , 

etc. 

The functions involved in Equation (86), i.e., ( ) L Vj , , ( ) g Pj V ( ) L ψj are all known and given by 

Equations (66), (68) and (77): 

[ ]

2 2 ,

,

,

,

2 2 ,

2 2 ,

2 2 ,

,

( )

1 1 ( ) ( ) ( )

1 ( )

( )

jU jU

j

U U

g U U

j

U j Uj

i j U i j U

U ij U ij

i U i

i

U

Vf Vf L V

f f

PV f PV f P V

f x fx

V V f V V f P V

f xx f xx

f f L

f f

Φ

Φ

Φ

Φ

Φ

Φ

Φ Φ

Φ Φ

∇ ∇ = =

∂ ⎡ ⎤ ∂ ⎣ ⎦ = = ∂ ∂

∂ ∂ = − = − ρ ∇ ∂∂ ∇ ∂∂

ψ ∇ ψ ∇ ψ= =

 (87) 

NASA/TM—2010-216368 18
The Following two relations, which are similar to Equation (12), are also useful: 

,

,

,

,

( )

( )

i i U

j i

U jU

i i U

j i

U j

V V f D V U

j

j

f

f x fx

f f D

f x fx

Φ

Φ

Φ Φ

Φ Φ

∂ ∂ = = ∂ ∂

ψ ∂ ψ ∂ ψ= = ∂ ∂

(88) 

The solution of Equation (86) should be able to calculate all one-point statistics for the velocity and 

species, for example, 1 2 ,, , , U UU U i i ij ii Φ ΦΦ Φ " , etc. We can also calculate other turbulent 

quantities, such as j i

k k

U U

x x

∂ ∂

∂ ∂ , n n

xk kx

∂Φ ∂Φ

∂ ∂ , PUi and PSij , which are related to the dissipation rate

, , , the pressure transport and the pressure strain correlations, etc. For example, n ε εε ij Φ

j j i i

ij

kk k k

U U U U

xx x x

⎛ ⎞ ∂ ∂ ∂ ∂

ε =ν − ⎜ ⎟ ∂∂ ∂ ∂ ⎝ ⎠

n

nn n n

xx x x kk k k

Φ

⎛ ⎞ ∂Φ ∂Φ ∂Φ ∂Φ ε =Γ − ⎜ ⎟ ∂∂ ∂ ∂ ⎝ ⎠

, pu PU P U ps PS P S i i i ij ij =− =− ij

and 

,

,

() ( ) () ( )

()() ()()

j i

ki k jU ki k jU

k k

n n k n k nU knkn

k k

U U

DVDV f dd DVDV f d

x x

D D f dd D D f d

x x

Φ

Φ Φ

∂ ∂ = = ∂ ∂

∂Φ ∂Φ

= ψ ψ =ψ ψ ∂ ∂

∫∫ ∫

∫∫ ∫

V V

V

ψ

ψ ψ

 (89) 

( ) ( ) ( ) ( )

,

,

( ) ( )

1 1 () ( ) ( ) () ( ) ( ) 2 2

i i U i U

ij j i i j U j i i j U

PU P V V f d d P V V f d

PS PV D V D V f d d PV D V D V f

Φ

Φ

= =

= + =+

∫∫ ∫

∫∫ ∫

V V

V V

ψ

ψ d

(90) 

5.0 Concluding Remarks 

We have constructed a set of conservational PDF equations directly from the Navier-Stokes equations 

and the species diffusion equations through the use of some sufficient but not necessary conditions. 

Therefore, in theory, they are neither general nor unique. However, all these PDF equations are in closed 

form. They are able to deduce the exact transport equations for the first moment and all higher order 

moments. This feature has not been observed from any other existing modeled PDF equations. For 

example, the modeled traditional PDF equations can deduce the first moment equation correctly, but not 

for the higher order moments. 

In this study, we have also defined a few functions, for example, ( ) D Vj i , ( ) L Vi , ( ) Dj Φi , ( ) L Φi

and ( ) g Pi V , they may be viewed as the models for the conditional means of ∇Ui , ∇ , , 2Ui ∇Φi 2 ∇ Φi

and ∇P . These functions may be used to calculate other turbulent quantities of interest. 

NASA/TM—2010-216368 19
Future work includes further testing of the present conservational PDF equations for their application 

in the area of CFD and the development of conservational filtered density function (FDF) equations for 

compressible turbulent flows. 

NASA/TM—2010-216368 20
Appendix A.—Fine Grained PDF, Velocity and Species PDF Equations 

A.1 Transport Equation for Fine Grained PDF f ′(V; x,t) 

According to Pope (Ref. 1), the fine grained PDF f ′(V; x,t) is defined as 

( ( ) ) ' ft t ( ; ,) V x U x, V =δ − (91)

Taking the following operations, 

'' ' '

'' ' '

i i

iii

i i

i

i

j ij ij i j

f fU fU U f t Ut Vt V t

f fU fU U f

x Ux Vx V x

∂ ∂∂ ∂∂ ∂ ∂ ⎛ ⎞ = =− =− ⎜ ⎟ ∂ ∂ ∂ ∂∂ ∂ ∂ ⎝ ⎠

∂ ∂∂ ∂∂ ∂ ∂ ⎛ = =− =− ⎜ ⎟ ∂ ∂ ∂ ∂∂ ∂ ∂ ⎝ ⎠

⎞

 (92) 

We may form an identity equation for the fine grained PDF: 

' ' ' i

j

j i j

f f UU U fU t xV t x

i

j

⎧ ⎛ ⎞⎫ ∂ ∂ ∂∂ ∂ ⎪ ⎪ + =− + ⎨ ⎜ ⎟⎬ ⎜ ∂ ∂∂ ∂ ∂ ⎟ ⎩ ⎭ ⎪ ⎝ ⎠⎪

 (93) 

Inserting the Navier-Stokes equation into Equation (93), we obtain a transport equation for the fine 

grained PDF: 

' ' 2 ' 1 i

j

j i i jj

f f PU U f t x V x xx

⎧ ⎛ ⎞⎫ ∂ ∂ ∂ ∂∂ ⎪ ⎪ + =− − +ν ⎨ ⎜ ⎟⎬ ⎜ ∂ ∂ ∂ ρ∂ ∂ ∂ ⎟ ⎩ ⎭ ⎪ ⎝ ⎠⎪

 (94) 

A.2 Transport Equation for Velocity PDF fU(V; x,t) 

Now, taking mean on Equation (94) we will obtain a transport equation for the velocity PDF. First 

taking mean on the left hand side of (94), 

' ' ' '

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

( ; ,) ( ; ,)

j

j

j j

U j

j

U U

j

j

f f f fU U

tx t x

f t d Vf td U

t x

ftft V

t x

∂ ∂ ∂ ∂

+ =+

∂∂ ∂ ∂

∂ ∂ = δ− + δ−

∂ ∂

∂ ∂

= +

∂ ∂

∫ ∫ V V Vx V V V Vx V

Vx Vx

' '' ' ' ' (95) 

To explain the way of taking mean on the right hand side of (94), let us first take the mean on 

' i j f ∂ ∂ U x , i.e. 

NASA/TM—2010-216368 21
( ) ( )

( ) ( ) ( ) ( )

'

0

0

( ,) ( ,) lim

( ,) ( , lim

j

j

i ji i

x j j

i j i

x j

U U xt U t

f t

x x

t U xt t U t

x

Δ →

Δ →

∂ +Δ −

=δ −

∂ Δ

δ − +Δ −δ − = Δ

x x

U x, V

U x, V x U x, V x )

 (96) 

Now the first term on the R.H.S of (96) involves a two-point correlation between the fine grained PDF at 

the point x and the velocity at the point x + Δx j , which needs a two-point PDF to express the correlation. 

As an approximation (or modeling) for this two-point correlation at the limit Δx j →0, we express it with 

the one-point PDF as follows 

( ( ) ) ( ) ' ( ,) ( ; ,) i j i j δ − +Δ = δ − +Δ t U x t Vf x td ∫ U x, V x V V V x V ' ' ' (97) 

Then Equation (96) becomes 

( ) ( ) ' '

'

0

0

( ; ,) ( ; ,) lim

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

j

j

i i U j i U

x j j

iU j iU U

i x j j

U Vf x t Vf t f d

x x

Vf x t Vf t f t V

x x

Δ →

Δ →

∂ δ − +Δ −δ − = ∂ Δ

⎛ ⎞ +Δ − ∂ = = ⎜ ⎟ Δ ∂ ⎝ ⎠

∫ '' ''

' V V Vx V V Vx

V

V x V x V x

 (98) 

We note that Equation (98) itself is not exact correct due to the assumption made in Equation (97). On the 

other hand, the following integral equation is exactly correct: 

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

U i

j j j j

U U U f t f d ft d V

xx x x

∂ ∂ ∂ ∂ = = ∂∂ ∂ ∂ ∫∫ ∫ V x V V V x V= dV (99) 

This illustrates that the model (98) is just a sufficient condition for Equation (99) to be satisfied. 

Equation (98) also directly suggests 

i U 1

i

j U j

U f V

x f x

∂ ∂ = ∂ ∂

V (100) 

Using the same procedure to carry out the mean on the right hand side of (94), we may obtain 

2 2

' 1 ( ) i U

i i jj i i jj

P P U f V f V x xx V x xx

ν

ρ ρ

∂∂ ∂ ⎪ ⎪ ⎧ ⎫ ⎛ ⎞⎛ ∂ ∂

− − + =− − + ⎨ ⎬ ⎜ ⎟⎜ ∂ ∂ ∂∂ ∂ ∂ ∂∂ ⎪ ⎪ ⎩ ⎭ ⎝ ⎠⎝

Vi Uf ν

∂ ⎞

⎟

⎟

⎠

 (101) 

Therefore, we obtain a closed velocity PDF equation: 

2 U U ( ) U iU

j

j i ij j

f f P Vf V V

t x V x xx

∂∂∂ ∂ ∂ ⎛ ⎞

+ =− − +ν ⎜

⎜ ∂ ∂ ∂ ρ ∂ ∂∂ ⎝ ⎠

f ⎟

⎟ (102) 

Note that the conditional means appeared in the traditional PDF equations (Pope (Ref. 1)) are related to 

the terms on the right hand side of Equation (102). For example, 

NASA/TM—2010-216368 22
2 2 ' i i

U

2

i U

j j j j

U UV f f

j j

f

x x xx x

∂ ∂

ν = ⋅ ν =ν

∂∂ ∂∂ ∂∂

V

x

∂

This indicates that the conditional mean is 

2 2

i i

j j U jj

U V Uf

x x fx

ν ν

∂ ∂ = ∂ ∂ ∂ x

V

∂ (103) 

Similarly, 

1 1 [ ( ) U ]

i U

P P f

x i f x

∂ ∂ = ρ∂ ρ ∂

V

V (104) 

It is clear that these conditional means are actually the functional forms for and in the sample 

space . Let us denote them as 

2 ∇ Ui ∇P

V ( ) L Vi and ( ) g Pi V , then the velocity PDF equation can be simply written 

as 

[ ] 2

( ) ( )

1 ( ) ( ) , ( )

g

U U i

j U

j i

g U U

i i i

U i U

f f P V V f t xV

PV f

L Vi fU

f P V LV V

f x f

⎛ ⎞ ∂∂∂

+ =− − +ν ⎜ ⎟ ∂ ∂∂ ρ ⎝

∂ ∇ = = ∂

⎠ (105) 

Now, let us apply the fine grained PDF to the Poison equation of , which will lead to a functional 

form of : 

P t ( ,) x

P V( )

2

'2 ' 1 i j

i j

U U

fP f

x x

∂

∇ =−

ρ ∂ ∂ (106) 

The operation of the above mean, under a similar assumption made in Equation (97), gives the following 

equation for P V( ) 

[ ] 2 2 ( ) U U

i j

i j

PV f f V V

x x

∇ ∂ = − ρ ∂ ∂ (107) 

A.3 Transport Equation for Joint Species-Velocity PDF fu,Φ(V,ψ; x,t)

Similar to Equation (94), the transport equation for the fine grained species PDF 

( ) is ' ft t ( ; ,) ( ,) ψ Φ x x ≡δ −ψ

'' 2 ' ' 1 2 ( i

j

j i jj

f f U ff S

t x xx

∂ ∂ ∂ ∂Φ ⎛ ⎞

+ =− Γ − ΦΦ ⎜

⎜ ∂ ∂ ∂ψ ∂ ∂ ⎝ ⎠

") i ⎟

⎟

 (108) 

Analogy to the derivation of velocity PDF equation, we may construct the following species PDF 

equation: 

NASA/TM—2010-216368 23
( ) ( ) 1 2

2

() ( )

( )

j i i

j i

i i

f f U Lf S

t x

f L

f

Φ

Φ Φ

Φ

Φ

∂ ∂ ∂

+ = − Γ ψ − ψ ψ ∂ ∂ ∂ψ

∇

ψ =ψ

ψ " fΦ

 (109) 

It should also be recognized that the conditional mean 2 ∇ Φi ψ appearing in the traditional PDF 

equation is represented by ( ) here in Equation L ψi (109). Finally, Equations (105) and (109) will lead to a 

joint species-velocity PDF equation as the following: 

( ) , , , , ,1 2

( ) ( ) ( ) ( )

g U U i

j U i U i U i

j i i

f f P V V f L V f L f S

t xV

Φ Φ

U , Φ Φ Φ f

∂ ∂ ⎛ ⎞ ∂ ∂ + =− − +ν − Γ ψ − ψ ψ ⎜ ⎟ ∂ ∂∂ ρ ∂ψ ⎝ ⎠

" Φ (110) 

Where, 

2 2 , , ,

, ,

1 ( ) ( ) , ( ) , ( ) g U U U

i i i i i

U i U U

PV f f f P V LV V L

fx f f

Φ

,

Φ Φ

Φ Φ

∂ ⎣ ⎦ ⎡ ⎤ ∇ ∇ = = ψ ∂ Φ

= ψ

NASA/TM—2010-216368 24
Appendix B.—Miscellaneous Formulations 

In the derivation of PDF equations, the following relationships have been intensively used. The proof 

of Equation (111) has been described by Pope (Ref. 1). The same procedure can be followed to prove 

Equation (112). 

( ) U i ( ; ,) ( ) 0

j

f tA d V

∂ = ∂ ∫ Vx V V (111) 

Where ( ) is a vector function of sample space variable V, and it has a finite mean. Ai V

[ ] ( ; ,) ( ) 0 i

j

f tA d Φ

∂ = ∂ψ ∫ ψ ψψ x (112) 

Where ( ) is a vector function of sample space variable Ai ψ ψ , and it has a finite mean. 

Therefore, the following integrations are all zero: 

1

U i ( ; ,) 0

j j

P f tV d V x

⎛ ⎞ ∂ ∂ ⎜ ⎟ − = ∂ ρ∂ ⎝ ⎠ ∫ V Vx V

( ) [ ] 1

i U ( ) ( ; ,) 0

j

VP f t d V

∂ − =

ρ ∂ ∫ V Vx V

( ) 2 i U ( ; ,) i 0

j

Vf t U d V

∂

ν∇ =

∂ ∫ Vx V V

( ) 1 ( ) ( ; , ) 0. g

i U j

j

VP f t d V

∂ − =

ρ ∂ ∫ V Vx V

( ) ( ) ( ; ,) i jU

j

V LV f t d V

∂

ν

∂ ∫ Vx V =0. 

2 i j ( ; ,) 0

j

ft d Φ

∂ ⎡ ⎤ ψ ∇Φ = ∫ ∂ψ ⎣ ⎦ ψ ψ x ψ

( ) 1 , ( ) ( , ; ,) ij n U i

j

S f xt dd Φ

∂

ψ ψψ ψ = ∂ψ ∫∫ " V V ψ ψ 0

NASA/TM—2010-216368 25
NASA/TM—2010-216368 26

References 

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

2. Tennekes, H. and Lumley, J.L., “A First Course in Turbulence,” MIT Press, 1972. 
REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188 

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the 

data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this 

burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. 

Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB 

control number. 

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 

1. REPORT DATE (DD-MM-YYYY)

01-07-2010 

2. REPORT TYPE

Technical Memorandum 

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

Conservational PDF Equations of Turbulence 

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Shih, Tsan-Hsing; Liu, Nan-Suey 

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

WBS 561581.02.08.03.16.02 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

8. PERFORMING ORGANIZATION

 REPORT NUMBER

E-17304 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

10. SPONSORING/MONITOR'S

 ACRONYM(S)

NASA 

11. SPONSORING/MONITORING

 REPORT NUMBER

NASA/TM-2010-216368 

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited 

Subject Categories: 01 and 64 

Available electronically at http://gltrs.grc.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

14. ABSTRACT

Recently we have revisited the traditional probability density function (PDF) equations for the velocity and species in turbulent 

incompressible flows. They are all unclosed due to the appearance of various conditional means which are modeled empirically. However, 

we have observed that it is possible to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through 

conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the resulted PDF equations are neither 

general nor unique, they nevertheless lead to the exact transport equations for the first moment as well as all higher order moments. We refer 

these PDF equations as the conservational PDF equations. This observation is worth further exploration for its validity and CFD application.

15. SUBJECT TERMS

Turbulence 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

UU 

18. NUMBER

 OF 

 PAGES

33 

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov) 

a. REPORT

U 

b. ABSTRACT

U 

c. THIS 

PAGE

U 

19b. TELEPHONE NUMBER (include area code)

443-757-5802 

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18