Tsan-Hsing Shih

Ohio Aerospace Institute, Brook Park, Ohio

Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio

Simulations of Spray Reacting Flows in a Single 

Element LDI Injector With and Without 

Invoking an Eulerian Scalar PDF Method

NASA/TM—2012-217676

August 2012
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Tsan-Hsing Shih

Ohio Aerospace Institute, Brook Park, Ohio

Nan-Suey Liu

Glenn Research Center, Cleveland, Ohio

Simulations of Spray Reacting Flows in a Single 

Element LDI Injector With and Without 

Invoking an Eulerian Scalar PDF Method

NASA/TM—2012-217676

August 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

This work is supported by the NASA Fundamental Aeronautics Program. The authors would like to thank Drs. Thomas Wey, 

Jeff Moder and M.S. Raju for their technical help on the Eulerian scalar PDF solver. The authors would also like to express our 

thanks to 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 lead 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

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

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—2012-217676 iii

Contents

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

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

2.0 Basic Equations for RANS and URANS Simulations........................................................................2

2.1 Averaged Turbulent Variables φ( ,) x t , φ ( ,) x t ..........................................................................2

2.2 Basic Equation............................................................................................................................3

2.3 Nonlinear Closure Models for Turbulent Fluxes........................................................................5

3.0 Basic Equation for Scalar Averaged Probability Density Function (APDF)......................................6

3.1 APDF for Scalars........................................................................................................................6

3.1.1 Fine Grained Probability Density Function (FG-PDF) for Scalars f t Φ

′ ( ; ,) ψ x ..............6

3.1.2 Scalar APDF F t Φ ( ; ,) ψ x ................................................................................................7

3.1.3 Relationship Between Scalar APDF and Averaged Turbulent Variables........................7

3.1.4 Joint APDF, , ( , ; ,) F t U Φ V x ψ and Its Conditional APDF ( ; ,) UF t Φ V x ψ ......................8

3.2 Transport Equation for Scalar APDF F t Φ ( ; ,) ψ x ......................................................................9

3.3 Nonlinear Model for Scalar Fluxes...........................................................................................10

3.4 Summary...................................................................................................................................11

4.0 Numerical Simulations of Single Element LDI Injector...................................................................11

4.1 Results of RANS Simulations With and Without Invoking an Eulerian Scalar APDF............13

4.1.1 Variation of Velocity and Temperature at Five Probes Along the Centerline...............13

4.1.2 Centerline Distributions of Mean Temperature and Axial Velocity..............................18

4.1.3 Mean Temperature and Velocity Profiles Along Z Axis at Downstream Locations.....18

4.1.4 Mean Flow Structure and Contour Plots of Variables in the Center x-z Plane .............23

4.2 Results of URANS Simulations With and Without Invoking an Eulerian Scalar APDF .........26

4.2.1 Convergence History .....................................................................................................27

4.2.2 Time Variation of Instantaneous Variables at Centerline Probes..................................28

4.2.3 Centerline Distributions of Mean Temperature and Axial Velocity..............................31

4.2.4 Mean Temperature and Velocity Profiles Along Z Axis at Downstream Locations.....31

4.2.5 Flow Structure and Contour Plots of Mean Variables in the Center x-z Plane .............35

5.0 Conclusions.......................................................................................................................................40

References...................................................................................................................................................41
NASA/TM—2012-217676 iv
NASA/TM—2012-217676 1

Simulations of Spray Reacting Flows in a Single Element

LDI Injector With and Without Invoking 

an Eulerian Scalar PDF Method

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

This paper presents the numerical simulations of the Jet-A spray reacting flow in a single element

lean direct injection (LDI) injector by using the National Combustion Code (NCC) with and without 

invoking the Eulerian scalar probability density function (PDF) method. The flow field is calculated by 

using the Reynolds averaged Navier-Stokes equations (RANS and URANS) with nonlinear turbulence 

models, and when the scalar PDF method is invoked, the energy and compositions or species mass 

fractions are calculated by solving the equation of an ensemble averaged density-weighted fine-grained 

probability density function that is referred to here as the averaged probability density function (APDF). 

A nonlinear model for closing the convection term of the scalar APDF equation is used in the presented 

simulations and will be briefly described. Detailed comparisons between the results and available 

experimental data are carried out. Some positive findings of invoking the Eulerian scalar PDF method in 

both improving the simulation quality and reducing the computing cost are observed.

1.0 Introduction

Many engineering applications of the computational fluid dynamics (CFD) for internal combustor

flows need to accurately account for the turbulence-chemistry interaction to facilitate a higher fidelity 

analysis of the design. In the conventional CFD simulation methods that involve an averaging procedure 

(e.g., RANS and URANS) or filtering process (e.g., large eddy simulation LES), the turbulence-chemistry 

interaction manifests itself as an unclosed term that must be modeled. Various such models have been 

proposed. The simplest one is the so called “laminar chemistry” which simply ignores the effects of the 

fluctuations of turbulence on the chemical reactions. However, in the PDF simulation methods (no matter 

Lagrangian or Eulerian approach, joint or scalar PDF), the chemical reaction source term in the PDF 

equations is always in a closed form, therefore it can be directly calculated without any modeling (Ref. 1). 

This unique feature will stay when the PDF method is extended to its large eddy simulation (LES) 

approach (Ref. 2).

In this paper, we present the preliminary numerical results of the complex spray reacting flow in a 

single element LDI injector. The simulation methods include both the conventional CFD methods 

(RANS, URANS) and an Eulerian scalar PDF method (Ref. 3), in which the velocity field is determined 

by the continuity and momentum equations of averaged Navier-Stokes equations. All simulations are 

done with the NCC code (Ref. 4) using the same numerical parameter setting and the same computational 

domain and grid resolution. In the RANS and URANS simulations, all the turbulent flux models, i.e., 

Reynolds stresses, heat and species fluxes, are nonlinear models (Ref. 5 and 6). The basic equations and 

models are described in Section 2.0. In the Eulerian scalar PDF method, we solve the transport equation 

of scalar APDF. It should be mentioned that the joint APDF equation for velocity and species is generally 
NASA/TM—2012-217676 2

in a closed form (Ref. 7). However, for the marginal or scalar APDF equation, the convection term 

contains a conditional mean that needs to be modeled. Here a nonlinear model is introduced, which

provides a diffusion process in the sample variable space. The brief description about the scalar APDF

equation and model is given in Section 3.0.

The liquid fuel is Jet-A, and C12H23 is adopted as its surrogate. At this stage of the simulations, we 

use a relatively simple spray injection model (i.e., prescribed droplet size and distribution) and a global 

five-species one-step kinetics for combustion chemistry (Ref. 8), so that we could concentrate more on 

the evaluation of different simulation approaches.

The main results of steady simulations, including both with and without invoking an Eulerian scalar 

PDF method, are described in Section 4.1. They are compared side by side to examine how much effects 

are produced by the invoked scalar PDF method. The results for unsteady simulations are described in a

similar manner in Section 4.2. 

One of the main objectives of this study is to search for a consistent and stable simulation tool for 

multiphase combustion flows. We require that this tool is not only to have the “physics” based models but 

is also able to produce the physically reasonable, numerically stable solution that can be sustained over a 

very long (infinite) time period. And also, this tool should be relatively economic. It is encouraging to 

observe from the present simulations that the adopted Eulerian scalar PDF method does show the 

potential for improving the simulation quality and also reducing the computing cost. 

2.0 Basic Equations for RANS and URANS Simulations

2.1 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 density weighting, the other without the density weighting. The 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 ∑ (1)

where φ(n)

is the n-th independent realization of turbulent variables, i.e., velocity components ( ) n Ui , 

density ρ(n)

, pressure P(n), m-th species mass fractions ( ) n Φm and internal energy () () ()

1

M n nn

m m

m

e e

=

= Φ ∑ . N is the 

total number of independent realizations. The density-weighted averaged turbulent variable is denoted by 



φ( ,) x t and is defined as

( ,)t ρφ φ =

ρ

x (2)

These ensemble averaged variables φ( ,) x t , φ( ,)t  x represent the turbulent mean variables that can be 

actually measured in experiments. They are deterministic in nature. They can vary with the time and

space but usually contain relatively low frequencies and low wave numbers of the turbulent mean flow, 

when compared with the ones contained in the non-averaged turbulent variables φ(x,t). 
NASA/TM—2012-217676 3

2.2 Basic Equation

Applying the ensemble averaging, Equations (1) and (2), to the compressible Navier-Stokes equations, 

we will obtain:

 0 j

j

U

t x

∂ρ ∂ ρ + =

∂ ∂ (3)

  1 2( ) 3

i i j

ij ij kk

j ij

U P U U

S S

t x xx

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

(4)

 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 − +   ∂∂ ∂  

 (5)

  ( ) 1,2, , m i m m m m

ii i

U Wm M

t xx x

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

 (6)

where

  1

11 1

 , or 

MM M

m m m M

mm m mv m v m

T e R R P R P

wc w c w

+

= = =

Φ Φ ρ ρ Φ Φ

=ρ = = ∑∑ ∑ (7)

( )

1

N m m i m

i i m

T qc h

x x υ

=

∂ ∂Φ = −ρ κ − ρΓ ∂ ∂ ∑ (8)

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 commonly assumed that 

Γ(m)

is the same for all species Φm, and wm is the molecular weight. The hm, T are the enthalpy of species 

and the temperature, Q is the radiation rate, Wm = ρSm is the chemical production rate of the 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 density weighted 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       ∂∂ ∂  

(9)

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

μ will be considered as a function of  P T, ,. Other types of approximations are also possible, for 

example,

   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       ∂∂ ∂  

(10)
NASA/TM—2012-217676 4

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 − Γ ρ ∂ ∂∂ ∂ ∑ ∑ (11)

 () () m m m m

i i x x

∂Φ ∂ρΦ ρΓ ≈Γ ∂ ∂ (12)

Where ,, , c ν κ υ and Γ(m)

are considered as functions of  P T, ,. Furthermore, we invoke the dissipation 

rate of turbulent kinetic energy,

1 2

3 ij ij ii kk SS SS  

ρν − ≡ ρε      (13)

Therefore, the ensemble averaged compressible Navier-Stokes equations can be written as 

 0 j

j

U

t x

∂ρ ∂ ρ + =

∂ ∂ (14)

     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     ∂ ∂ ∂∂ ∂ ∂ ∂    

(15)

i i

kk

i i

e Ue q PS Q

tx x

∂ρ ∂ρ ∂ + = − + +ρε + ∂∂ ∂

  (16)

   ( )  1,2, , m m i m m m

ii i

U Sm M

t xx x

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

 (17)

  1

11 1

 , or 

MM M

m m m M

mm m mv m v m

T e R R P R P

wc w c w

+

= = =

Φ Φ ρ ρ Φ Φ

=ρ = = ∑∑ ∑ (18)

 ( )

1

M m m i m

i i m

T qc h

x x υ

=

∂ρ ∂Φ = − κ − ρΓ ∂ ∂ ∑ (19)

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

incompressible, ii) all the approximations made in Equations (15), (16) and (17) are related only to the 

molecular diffusion terms which are less important and even become negligibly small, i.e., they are of 

O(1/Re) comparing with the convection terms on the left hand side for turbulent flows at high Reynolds 

numbers (see Ref. 9 and 1). For the steady and unsteady simulations (i.e., RANS and URANS), Equation 

(14) to (17) are often used together with the further approximations for Equations (18) and (19):



 



1 1 1

1 1 , ,

M M M

m m m

m m m m m m

T RT RT P R

= w M Mw w M M = =

Φ Φ   ρ ρ Φ

=ρ = ≈ = =     ∑ ∑∑ (20)
NASA/TM—2012-217676 5



i

i

T

q c

x υ

∂ρ =−κ

∂ (21)

The momentum flux ρU Ui j , the energy flux.. and the species flux U

i m ρ Φ are considered to be critically 

important and should be carefully modeled. Many models in the literature, from the simplest zeroequation model (Ref. 10) to the complex two-equation nonlinear models (Refs. 11and 12), have been 

suggested in terms of turbulent stresses and turbulent scalar fluxes: τ ≡ρ − ij i j i j ( ) UU UU    , 

Θ ≡ρ θ− θ i ii ( ) U U    , where θ represents the scalar quantities: and m e Φ .

2.3 Nonlinear Closure Models for Turbulent Fluxes

A general constitutive relationship between turbulent stresses ij τ and the strain rate of mean

turbulence ij S

 , Ωij  suggests (Ref. 13)

( )

( )

2

3

3 2

4

2 2

5 3

1 2 3

3

2 3 ( 3) ,

ij ij ij kk ij kk

ik kj ik kj

ik kj ik kj ik km mj kl lm mk ij s ij ij kk

k C SS

k AS S

k A S S S S II S S

μ τ =− ρ −δ + δ τ

ε

− ρ Ω −Ω

ε

+ ρ Ω − Ω +Ω Ω −Ω Ω δ + −δ     ε

 

   

         

(22)

where,   ( )   i j ji , , 2 , ( i j ji , , ) 2 , ( ) 2 ij ij s kk mm kl lk S U U = + Ω= − U U II S S S S = −      . The model 

coefficients Cμ, A3 and A5 are constrained by the realizability condition and the rapid distortion theory 

limit. They are formulated as (see Ref. 12):

2 2 22 *

3 5 2 4 ** * * * * *

2 3

1.0 1.6 1 , ,, 7 4.0 0.5 1.5

4

s

s

k k AC S C

CA A k k k SS A U S

μ μ

μ

  −   ρ   ε ε = = = +Ω Ω + +Ω ρ ε ε ε

(23)

in which, 

( ) ** *

* *

* 3

1 6 cos , arccos 6 , , 3 ( )

ij jk ki

s

SS S

A W W

S = φ φ= = (24)

* *2 *2 * * * * * 1 ( ) ( ), , , 3

U S ij ij ij ij ij ij ij kk = +Ω = Ω = ΩΩ = − δ S SS S S S     (25)

Note, here *

ij S is a zero trace strain rate tensor. Similarly, the nonlinear model for scalar fluxes is 

formulated as (Ref. 14)

i T T ij ij ( 1 2 )

i j

k cS c

x x

∂ρθ ∂ρθ Θ = −θ − θ + Ω

∂ε ∂

    (26)
NASA/TM—2012-217676 6

Where θT denotes the turbulent diffusivity for the corresponding scalar quantity θ. It is often 

approximated by Pr T T

θ θ =ν , and Prθ represents the turbulent Prandtl number or Schmidt number 

depending on whether the scalar quantity θ

 is the energy e or the species Φm  . The turbulent eddy 

viscosity is defined as 2

T C k ν= ⋅ ε μ . The coefficients, c1 and c1, are yet to be calibrated. In the current 

simulations they are set to be c1 = c2 = –0.24

The turbulent kinetic energy and its dissipation rate k, ε will be determined from the following model 

equations:

i ( T ) ij ij

ii i

k uk k s

tx x x

∂∂ ∂ ∂  

ρ + ρ = μ+μ − τ −ρε   ∂∂ ∂ ∂  

  (27)

( )

2

i T 1 2 ij ij

ii i

u Cs C

tx x x k k ε ε

∂ ∂ ∂ ∂   ε ρε ρε + ρ ε = μ+μ ε − τ −   ∂∂ ∂ ∂  

  (28)

where Cε1 and Cε2 are the model coefficients. We have adopted the commonly used values of Cε1 = 1.45 

and Cε2 = 1.92 in the present work while keeping in mind that they can be further constructed as functions 

of the local turbulence quantities (Ref. 15).

3.0 Basic Equation for Scalar Averaged Probability Density Function 

(APDF)

In this section, we will use the fine grained probability density function (FG-PDF) to define the 

averaged probability density function (APDF), then explore the relationship between APDF and the

ensemble averaged turbulent scalar variables. This will provide the basis for establishing the transport 

equation for scalar APDF.

3.1 APDF for Scalars

3.1.1 Fine Grained Probability Density Function (FG-PDF) for Scalars f t Φ

′ ( ; ,) ψ x

According to Pope’s definition (Ref. 1), the joint FG-PDF for turbulent velocity and scalars (i.e., 

compositions or species mass fractions, internal energy) can be written as

( ) ( ) ( ) ( )

3 1

1 1

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

M

ii m m

i m

f t t t U tV t

+

= =

′ V x Ux V x ψ ψ ≡δ − δ − ≡ δ − δ Φ −ψ Φ ∏ ∏ x x (29)

Its marginal FG-PDF for scalars is 

( ) ( )

1

1

( ; ,) ( ,) ( ,)

M

m m

m

ft t t

+

Φ

=

′ ψ ψ xx x ≡δ − ≡ δ Φ −ψ Φ ∏ (30)

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 (ΦΦ Φ Φ 1 2 , ,, ,  M M +1 ) , 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 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. 
NASA/TM—2012-217676 7

3.1.2 Scalar APDF F t Φ ( ; ,) ψ x

We define the following ensemble averaged density weighted fine-grained probability density 

function for scalars:

( ) ( ( ) ) ( ) ( )

1

1 ( ; , ) lim , ,

N n n

N n

F t t t

N Φ →∞ =

ψ ψ x xx ≡ ρδ − ∑ Φ (31)

Obviously, the scalar APDF, F t Φ ( ; ,) ψ x , is no longer a random quantity. And it satisfies the following 

“normalization” property:

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

1 1

1 1 ( ; , ) lim , , lim ,

N N n n n

N N n n

F td t t d t

N N

∞ ∞

Φ →∞ →∞ −∞ −∞ = =

∫ ∫ ψ ψ x = ρ δ − = ρ =ρ ∑ ∑ x x Φ ψ ψ x (32)

Note that here ∞

∫−∞ means that the integration is over the entire domain of the sample space ψ. 

3.1.3 Relationship Between Scalar APDF and Averaged Turbulent Variables

With the definition of scalar APDF described in Equation (31), we can exactly deduce the averaged

scalar turbulent variables that are defined in Equations (1) and (2). For example,

( ) ( ( ) )

( ) ( ) ( ( ) )

( ) ( )

( ) 

( )

1

( )

1

( )

1

1 ( ; , ) lim

1 lim

1 lim

( ,)

N n

N n

N n

N n

N n

N n

F td t t d

N

tt t d

N

t t

N

t t

+∞ +∞

Φ −∞ −∞ →∞ =

+∞

−∞ →∞ =

→∞ =

    ρδ −

 

    ρ δ−

 

ρ

ρ =ρ

∫ ∫ ∑

∫ ∑

∑

ψ ψ ψ ψ ψ ψ

ψ ψ

x = x, x,

= x, x, x,

= x, x,

= x, x

Φ

Φ Φ

Φ

Φ Φ

(33)

Equation (33) denotes that the left hand side is an operation Φ that defines the density weighed 

turbulent mean variable ρΦ : 

( )  F td t t ( ; ,) ( ,) +∞

Φ −∞ ≡ = ρ ∫ Φ Φ ψψ ψ x x, x (34)

For a function W t (Φ( ,) x ) , it is easy to verify that

( ) ( )  W W F td tW t ( ) ( ; ,) ( ( ,)) +∞

Φ −∞ = = ρ ∫ Φ Φ ψψ ψ x x, x (35)

Furthermore, we may consider the derivatives ∇Φ as a new random quantity and legitimately write

∇ =ρ∇ Φ Φ (36)
NASA/TM—2012-217676 8

However, because of the variable density, the “operation” does not have the differential commute 

property, i.e.,

∇ ≠∇ Φ Φ (37)

because   ρ∇ ≠∇ ρ Φ Φ ( ) .

We can also write the density weighted mean U j i Φ for the joint velocity and scalar 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 (38)

where, , ( , ; ,) F t U Φ V x ψ is the 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 (39)

which 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, (40)

3.1.4 Joint APDF, , ( , ; ,) F t U Φ V x ψ and Its Conditional APDF ( ; ,) UF t Φ V x ψ

From Equation (39), we may follow Reference 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

(41)

and the “conditional mean” 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

(42)

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 (43)

And the “complete” mean should be
NASA/TM—2012-217676 9

( )

( ) ( ( ) ) ( ( ) )

( ) ( ) ( ) 

( ) ( ) ( )

1

() ()

1

( ,)

1 lim

1 lim ( , )

N nn n

N n

N n n

N n

t Fd

t t t dd

N

t t tt

N

+∞

Φ −∞

+∞ +∞

−∞ −∞ →∞ =

→∞ =

⋅

= ρ δ −δ −

=ρ = ρ

∫

∫ ∫ ∑

∑

ψ ψ

ψ ψ

U x

V x, U x, V x, V

x, U x, x, U x

Φ (44)

Equation (43) can be extended to any other turbulent quantities, for example, ∇Φ, Si(Φ):

( ) ( ( ) )

( ( ) ) ( ) ( )

( ) ( ) ( )

1

() () ()

1

1 lim

1 lim

N n n n

N n

N nn n

i i i N n

t F

N

S t F S FS

N

Φ →∞ =

Φ Φ →∞ =

ρ ∇ δ − ⋅∇

ρ δ − ⋅ =⋅

∑

∑

ψ ψ

ψ ψψ

x, =

x, =

Φ Φ Φ

Φ Φ

(45)

where ∇Φ is viewed as a new random variable in addition to Φ.

3.2 Transport Equation for Scalar APDF F t Φ ( ; ,) ψ x

We may derive the transport equation for scalar APDF from Equations (16) and (17) as follows: first, 

we write the terms on the left hand side of Equation (17) as

 m

m m

F F d d

t t t

∞ ∞

Φ

Φ

−∞ −∞

∂ρΦ ∂ ∂

= ψ =ψ ∂ ∂ ∂ ∫ ∫ ψ ψ (46)



( )

,

, 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 (47)

Then, the terms on the right hand side of Equation (17) 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

(48)

 ( ) ( ) k m m m

k

S F

S S Fd d

∞ ∞

Φ

Φ

−∞ −∞

∂

ρ = =− ψ ∂ψ ∫ ∫ ψ ψ ψ (49)
NASA/TM—2012-217676 10

Where in Equations (48) and (49) we have applied the integration by parts and the following zero 

integration:

A Fd ( ) 0, if a finite mean exists. A

∞

Φ

−∞

∂  ⋅ =   ∂ ∫ ψ ψ

ψ

(50)

Collecting all the integrand terms, and let the sum be zero (we refer this as the conservation condition, 

which is in general a sufficient but not necessary condition, Ref. 16), 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





(51)

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

Equation (16) are neglected. Unlike the case of 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 (17), must be carefully modeled while the less important 

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

chemistry source term ( ) m ρS Φ in Equation (17), which involves complex processes of turbulencechemistry interaction, is closed in the scalar APDF equation, i.e., no need of modeling. This direct 

calculation of turbulence-chemistry interaction is one of the unique features of the PDF methodology. 

3.3 Nonlinear Model for Scalar Fluxes

The convection term in Equation (51) contains the term F U Φ i ⋅ ψ , which must be modeled. We 

may start from a nonlinear model, Equation (26), for the term ρ Φ U

i m :

      ( )  ( ) ( )

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 ∂ (52)

This will lead to the following model by directly applying Equations (34) and (43):

   ( )

   ( )

( ) ( )

1 2

( ) ( )

1 2

or

m m i ij ij iT T k

i k j

m m i ij ij i T k T k

k ik 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

(53)
NASA/TM—2012-217676 11

3.4 Summary

With the model given by Equation (53), the scalar APDF equation for F t Φ ( ; ,) ψ x can be written as



( ) ( ) ( ( ))

  ( )

() ()

( )

1 2 , 1,2, , 1

i m m

T k

i i i k

m ij ij k T

k i j

F F U F

F S

tx x x

k F cS c k M

x x

Φ Φ Φ

Φ

Φ

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

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

∂ψ ∂ ε ∂      

ψ



(54)

It can be verified that the scalar APDF Equation (54) can exactly deduce the ensemble averaged

Equation (17). However, the model described by this equation is by no means unique. In addition, the 

variables ( 

Ui , Sij and Ωij ) are considered to be available during the solution procedure of the scalar 

APDF equation. Furthermore, in order to apply the available stochastic solution procedure built in the 

NCC code, we further simplify the model term in Equation (54) as follows:

  ( ) ( )

1 2

m 1

ij ij k T k

k i j k

k F cS c F

x x

Φ

Φ

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

(55)

where

  ( ) ( )     1 2

1 1 m

T ij ij ij ij ij ij

i j

k F cS c S S

F x x

Φ

Φ

∂   ∂ ≡− Γ + Ω ≈ +Ω Ω   τ∂ ε ∂  

(56)

Equation (56) is a crude approximation based on the dimensional argument for a time scale, which is 

responsible for the diffusion of scalar APDF in the space ψk. We chose this time scale to be related to the 

rate of strain and rotation instead of its complex formulation. In order to prevent this time scale from 

being non-physically small during the simulation, we require that

τ≥ ν+ν ε ( T ) (57)

because the right hand side of Equation (57) represents the smallest time scale of the simulated flow field.

4.0 Numerical Simulations of Single Element LDI Injector

The lean direct injection (LDI) injector is a liquid fuel injector developed to reduce aircraft emissions. 

Stable combustion is essentially completed within a short distance through rapid fuel and air mixing. This 

design also allows for many small fuel injectors integrated into modules facilitating different fuel staging 

strategies, such as the one shown in Figure 1. So far, experimental observations have not fully clarified 

the dynamics of the mixing and combustion processes occurring in these injectors, and numerical studies 

need to be conducted to achieve a better understanding of the underlying physics of the LDI combustor. 

Figure 2 shows the air swirler and the convergent-divergent nozzle of the single element injector. Figure 3

depicts the single element LDI combustor geometry and its computational domain. Five probes are 

dispatched along the centerline (Figure 4) to monitor the evolution of turbulent variables during the 

simulations. The numerical grid is formed using hexahedral elements and the total number of elements is 

about 862,000, which is a relatively coarse grid used in a previous RANS simulation (Ref. 17).
NASA/TM—2012-217676 12

Figure 1.—Nine element LDI module. Figure 2.—Single element LDI injector.

Figure 3.—Computational domain and grid spacing.

Figure 4.—Probes located on centerline.

In this study, the liquid fuel is Jet-A, and C12H23 is adopted as its surrogate, the fuel is injected at the 

throat of the nozzle, mixing with the swirling air that comes from the air swirlers which consists of six 

helical, axial vanes with downstream vane angles of 60°. A prescribed droplet-size distribution spray model 

is used. In Section 4.1, we present the results of steady simulations. We first carry out a steady RANS 
NASA/TM—2012-217676 13

simulation with a “laminar chemistry” model, in which the generation rate of compositions is determined by 

the mean turbulent variables, which is commonly considered as a very crude assumption or model for the 

chemistry-turbulence interaction. This simulation has been carried out up to 900,000 iterations. Then we 

start a new simulation using the solution of the abovementioned simulation at the iteration of 460,000 as the 

initial condition and invoking an Eulerian scalar PDF method, in which the compositions and internal 

energy will be determined by the scalar APDF equation while the flow field is still determined by the 

continuity and momentum equations of the averaged Navier-Stokes equations. Since the production rate of 

composition is in a closed form in the scalar APDF equation, it can be calculated without modeling. This 

new simulation is carried out up to 760,000 iterations. In Section 4.2, we present the results of unsteady 

simulations URANS to consider the strong unsteadiness of LDI flow, using both with and without scalar 

PDF method. The stochastic numerical procedures for solving scalar APDF equation and fuel spray 

equations are described in Reference 3 and adopted in the present simulations.

4.1 Results of RANS Simulations With and Without Invoking an Eulerian Scalar APDF

In this section we present results of steady simulations with and without invoking the scalar APDF 

equation. Figure 5 shows the general pictures of simulated spray reacting flow in a single element LDI 

injector by both methods. Most of the results will be presented side by side for comparisons. The main

results are presented in terms of 1) the variations of velocity components and temperature with respect to 

the iteration at five probes along the centerline (see Figure 4), which indicate how much the simulated 

reacting flow is developed, 2) the centerline distributions of temperature and axial velocity, 3) the 

temperature and velocity profiles along Z axis at several downstream locations, 4) the center recirculation 

zone visualized by isosurface of zero axial velocity and the contour plots of various turbulent quantities in

the center x-z plane. These profiles, isosurface and contour plots will provide the information about flow 

and flame structures of the simulated spray reacting flow. Some available experimental data (Ref. 18) are 

also plotted for comparison with the calculated results from the simulations.

4.1.1 Variation of Velocity and Temperature at Five Probes Along the Centerline

Figure 6 and Figure 7 are the recorded iteration histories of velocity components and temperature at 

the five probes during the course of the steady simulations for the last 150,000 iterations. These figures 

indicate that the RANS simulation with the scalar APDF equation has reached its “steady” solution even 

before 600,000 iterations, the last 150,000 iterations are used for checking if the solution can be sustained 

or not. However, the conventional RANS without PDF method has the difficulty to reach its steady state 

and continues to vary around its “mean” value, which is clearly shown in the last 150,000 iterations. 

Figure 5.—Global pictures of spray reacting flow simulated with and without scalar PDF method.
NASA/TM—2012-217676 14

Figure 6.—Comparison of variations of simulated velocity components at 5 probes.
NASA/TM—2012-217676 15

Figure 6.—Concluded.
NASA/TM—2012-217676 16

Figure 7.—Comparison of variations of simulated temperature at 5 probes.
NASA/TM—2012-217676 17

Figure 7.—Concluded.
NASA/TM—2012-217676 18

4.1.2 Centerline Distributions of Mean Temperature and Axial Velocity

Figure 8 compares the centerline distribution of axial velocity with experimental data. The 

improvement in the nozzle region has been made by the PDF method. However, Figure 9 shows that the 

centerline distribution of temperature, comparing with the experimental data, is over predicted by the both 

methods in the region after the front face of dump combustor. 

4.1.3 Mean Temperature and Velocity Profiles Along Z Axis at Downstream Locations

Temperature profiles along Z axis at ten downstream locations have been plotted. Here we only show 

four locations at x = 5, 10, 20 and 200 mm downstream of the front face of dump combustor. Comparing 

with the experimental data, Figure 10 clearly reveals the improvement made by the RAN simulation with 

the APDF equation. Similar situations are observed in the profiles of velocity components V and U along 

Z axis, shown in Figure 11 and Figure 12. 

Figure 8.—Comparison of centerline distribution of axial velocity.

Figure 9.—Comparison of centerline distribution of temperature.
NASA/TM—2012-217676 19

Figure 10.—Comparison of temperature distribution along Z axis at downstream locations.
NASA/TM—2012-217676 20

Figure 10.—Concluded.

.

Figure 11.—Comparison of V distribution along Z axis at downstream locations.
NASA/TM—2012-217676 21

Figure 11.—Concluded.
NASA/TM—2012-217676 22

Figure 12.—Comparison of U distribution along Z axis at downstream locations.
NASA/TM—2012-217676 23

Figure 12.—Concluded.

Figure 13.—Comparison of center recirculation zone.

4.1.4 Mean Flow Structure and Contour Plots of Variables in the Center x-z Plane

To reveal the simulated flow and flame structures, we have plotted the center recirculation zone 

visualized by the isosurface of zero axial velocity (see Figure 13), the contour plots of velocity 

components, temperature, turbulent kinetic energy, species mass fraction C12H23, O2 and CO2 are 

shown in Figure 14, Figure 15, and Figure 16. From these figures we observed that all the structures 

simulated by RANS with APDF equation are more symmetric than the ones from the RANS without PDF 

method. This trend of symmetry seems more lined up with the experimental observation.
NASA/TM—2012-217676 24

Figure 14.—Comparison of U,V,W contours in center plane.
NASA/TM—2012-217676 25

.

Figure 15.—Comparison of temperature and k contours in center plane.
NASA/TM—2012-217676 26

Figure 16.—Comparison of C12H23, O2 and CO2 contours in center plane.

4.2 Results of URANS Simulations With and Without Invoking an Eulerian Scalar 

APDF

In this section, we present the results of unsteady simulations with and without scalar APDF equation. 

Most results will be presented in a similar way to what we have done for the steady simulations in 

Section 4.1. URANS without the scalar APDF equation is carried out up to 83,000 time steps, the size of 

time step dt = 2.×10–6

, which is also used in the solvers for spray and scalar APDF equation for 

synchronizing the simulation. After the simulated flow is fully developed, the mean turbulent quantities 

are calculated by averaging over the last 20,000 time steps, which is about 2.44 “flow through times” 

(FTT) which is defined here as the time period that is needed for the flow to go from the inlet to the outlet 

of the computational domain based on the speed at the inlet. Then the simulation is switched to URANS 

with the scalar APDF equation and carried out up to 153,000 time steps. Again, the mean values of 

turbulent quantities are obtained by averaging over the last 20,000 time steps. Figure 17 depicts the global 

mean structures from abovementioned two unsteady simulations of spray reacting flow.
NASA/TM—2012-217676 27

The main results to be presented are: 1) the convergence history of unsteady simulations, 2) the time 

histories of velocity components and temperature at four probes along the centerline, 3) the centerline 

distribution of temperature and axial velocity, 4) the profiles of temperature and velocity components

along Z axis at several downstream locations compared with available experimental data, 5) the center 

recirculation zone and the contour plots of various mean turbulent quantities in the center x-z plane.

4.2.1 Convergence History

Figure 18 shows the time history of the number of subiteration incurred in unsteady simulations with 

and without invoking the scalar APDF equation over the last 20,000 time steps. It is interesting to see that 

the simulation with the scalar APDF equation converges much faster than the simulation without the 

scalar APDF equation. For each time step, the former converges after 21 to 33 subiterations, but the later 

often needs much more subiterations to converge. Comparing the wall time of computing, the simulation 

with APDF is about 31 hr with 256 processors for the 20,000 time steps; however, the simulation without 

APDF needs about 42 hr to finish 20,000 time steps of calculations. 

Figure 17.—Global pictures of spray reacting flow simulated with and without scalar PDF method.

Figure 18.—Convergence histories of unsteady simulations.
NASA/TM—2012-217676 28

4.2.2 Time Variation of Instantaneous Variables at Centerline Probes

Time history of velocity components and temperature at four locations are recorded during the 

simulation. Here we list the time history at the probes 1, 2, 3, 5 (see Figure 19 and Figure 20), it can be 

seen that at the first three locations the flow appears to be strongly unsteady and fully developed. 

Figure 19.—Time history of velocity components u, v and w at 4 probes.
NASA/TM—2012-217676 29

Figure 19.—Concluded.
NASA/TM—2012-217676 30

Figure 20.—Time history of temperature at 4 probes.
NASA/TM—2012-217676 31

Figure 20.—Concluded.

Figure 21.—Centerline distribution of mean axial velocity U.

4.2.3 Centerline Distributions of Mean Temperature and Axial Velocity

Centerline distributions of mean axial velocity and temperature are shown in Figure 20 and Figure 21. 

The effect of scalar APDF equation on the unsteady simulation is quite similar to the case of steady 

simulation described in Section 4.1.2.

4.2.4 Mean Temperature and Velocity Profiles Along Z Axis at Downstream Locations

Mean temperature profiles at downstream locations x = 5, 10, 20 and 200mm are compared and 

shown in Figure 22. These comparisons have revealed a positive effect of the APDF equation on the 

prediction of temperature. However, the effect of APDF equation on the velocity field, shown in Figure 

23 and Figure 24, is not as obvious in present unsteady simulations. 
NASA/TM—2012-217676 32

Figure 22.—Centerline distribution of mean temperature T.

Figure 23.—Comparison of temperature distribution along Z axis at x = 5, 10, 20, 200 mm.
NASA/TM—2012-217676 33

Figure 23.—Concluded.
NASA/TM—2012-217676 34

Figure 24.—Comparison of velocity component V along Z axis at x = 5, 15, 29, 38, 92mm.
NASA/TM—2012-217676 35

Figure 24.—Concluded.

4.2.5 Flow Structure and Contour Plots of Mean Variables in the Center x-z Plane

To reveal the simulated mean flow and flame structures, we have plotted the center recirculation zone 

by the isosurface of zero axial velocity (see Figure 25), the contour plots of velocity components, 

temperature, turbulent kinetic energy and species mass fractions C12H23, O2 and CO2 are shown in 

Figure 26, Figure 27, and Figure 28. From these figures we observe that the structures of all scalar fields 

simulated by URANS with the APDF equation are quite different from the ones by URANS without the 

APDF equation. Also, we notice that the level of turbulent kinetic energy is significantly reduced in the 

unsteady simulation with the APDF equation (a phenomenon often found in the simulations with the 

conventional or standard PDF method); however, it does not severely affect the global structure of 

turbulent reacting flow in our simulations due to, we believe, the use of the APDF equation, in which the 

time scale or frequency for the APDF diffusion in the sample space is not directly related to the turbulent 

kinetic energy, instead, it is determined by the strain and rotation rate of the turbulent mean flow (see 

Eq. (56)). 
NASA/TM—2012-217676 36

Figure 25.—Comparison of velocity component U along Z axis at x = 5, 15, 29, 38, 92mm.
NASA/TM—2012-217676 37

Figure 25.—Concluded.
NASA/TM—2012-217676 38

Figure 26.—Comparison of center recirculation zone.

Figure 27.—Comparison of mean velocities U, V, W contours in center plane.
NASA/TM—2012-217676 39

Figure 28.—Comparison of mean temperature and k contours in center plane.
NASA/TM—2012-217676 40

Figure 29.—Comparison of C12H23, O2 and CO2 contours in center plane.

5.0 Conclusions

The numerical simulations of Jet-A spray reacting flow in a single element LDI injector have been 

carried out both with and without invoking an Eulerian scalar APDF method. The NCC code is used 

under the same numerical setting, same computational geometry and grid resolution. In addition, the same 

spray model and the same chemistry kinetics are used for all the steady and unsteady simulations. In this 

way, we hope to isolate and study only the effect of scalar PDF method (with the scalar APDF equation)

on simulations, paying particular attention to the simulation quality and numerical performance (e.g., 

computational stability and efficiency).

From both steady and unsteady simulations we have observed that there are noticeable improvements 

achieved on the temperature prediction in the region of strong turbulence by using the scalar PDF method. 

The distribution or structure of other scalar quantities appears to have significant differences between the 
NASA/TM—2012-217676 41

simulations using with and without the scalar APDF equation, and the scalar PDF method seems to 

produce more reasonable results. However, we need more reliable experimental data for a better

evaluation. For the flow field, we also notice a big difference and improvement in the steady simulation 

with the PDF method, but not much in the unsteady simulation. 

As to the numerical performance, the observation is that both steady and unsteady simulations 

invoking an Eulerian scalar PDF method appear to be more stable and faster than the conventional RANS 

and URAMS simulations.

Further evaluations of the scalar APDF equation and studies of stochastic numerical procedure for

Eulerian scalar PDF method will be continued and extended to the large eddy simulation (LES) approach.

References

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

2. 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.

3. M.S. Raju, EUPDF-II: “An Eulerian Joint Scalar Monte Carlo PDF Module Users’ Manual,” 

NASA/CR—2004-213073.

4. Liu, N.-S., “On the Comprehensive Modeling and Simulation of Combustion Systems,” AIAA-2001-

0805, January 2001.

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

Vol. 44, No. 4, 2006, pp. 687-697.

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

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

NASA/TM—2010-216323, May 2010.

7. Shih, T.-H. and Liu, N.-S., “Ensemble Averaged Probability Density Function (APDF) Equations for 

Compressible Turbulent Reacting flows,” NASA/TM (submitted).

8. Iannetti, A.C. and Liu, N.-S., “Effect of Spray Initial Conditions on Heat Release and Emissions in 

LDI CFD Calculations,” AIAA-2008-1150, January 2008.

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

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

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

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

Vol. 36, No.2, 1998. 

12. 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.

13. Shih, T.-H., “Constitutive Relations and Realizability of Single-Point Turbulence Closures” 

Turbulence and Transition Modelling, Chapter 4., Edited by Hallback, M., Henningson, D.S., 

Johansson, A.V. and Alfredsson, P.H., KLUWER ACADEMIC PUBLISHERS, 1996.

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

Research 20 (1997) 67-96.

15. Shih, T.-H., Liou, W.W., Shabbir, A., Yang, Z. and Zhu, J., “A New k − ε Eddy Viscosity Model for 

High Reynolds Number Flows,” Computational Fluids, Vol.24, No.3, pp227-238, 1995.

16. Shih, T-H. and Liu, N.-S., “Conservational PDF Equations of Turbulence,” NASA/TM—2010-

21638.

17. Davoudzadeh, F., Liu, N.-S. and Moder, J. P., “Investigation of Swiring Air Flows Generated by 

axial Swirlirs in a Flame Tube,” Proceedings of GT2006 ASME Turbo Expo 2006: Power for Land, 

Sea, and Air May 8-11, 2006, Barcelona, Spain.

18. Cai, J., Jeng, S.-M., and Tacina, R., “The Structure of A Swirl-Stablized Reacting Spray Issued from 

An Axial Swirler,” AIAA-2005-1424, 43rd AIAA Aerospace Science Meeting and Exhibit, 10-13 

January 2005, Reno, NV.
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Simulations of Spray Reacting Flows in a Single Element LDI Injector With and Without 

Invoking an Eulerian Scalar PDF Method 

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Shih, Tsan-Hsing; Liu, Nan-Suey 

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14. ABSTRACT

This paper presents the numerical simulations of the Jet-A spray reacting flow in a single element lean direct injection (LDI) injector by 

using the National Combustion Code (NCC) with and without invoking the Eulerian scalar probability density function (PDF) method. The 

flow field is calculated by using the Reynolds averaged Navier-Stokes equations (RANS and URANS) with nonlinear turbulence models, 

and when the scalar PDF method is invoked, the energy and compositions or species mass fractions are calculated by solving the equation of 

an ensemble averaged density-weighted fine-grained probability density function that is referred to here as the averaged probability density 

function (APDF). A nonlinear model for closing the convection term of the scalar APDF equation is used in the presented simulations and 

will be briefly described. Detailed comparisons between the results and available experimental data are carried out. Some positive findings 

of invoking the Eulerian scalar PDF method in both improving the simulation quality and reducing the computing cost are observed.

15. SUBJECT TERMS

Combustion CFD 

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