Nan-Suey Liu and Thomas Wey

Glenn Research Center, Cleveland, Ohio

Tsan-Hsing Shih

Ohio Aerospace Institute, Brook Park, Ohio

Time-Filtered Navier-Stokes Approach and 

Emulation of Turbulence-Chemistry Interaction

NASA/TM—2013-217873

May 2013

AIAA–2013–0707
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Nan-Suey Liu and Thomas Wey

Glenn Research Center, Cleveland, Ohio

Tsan-Hsing Shih

Ohio Aerospace Institute, Brook Park, Ohio

Time-Filtered Navier-Stokes Approach and 

Emulation of Turbulence-Chemistry Interaction

NASA/TM—2013-217873

May 2013

AIAA–2013–0707

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

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Grapevine, Texas, January 7–10, 2013
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NASA/TM—2013-217873 1

Time-Filtered Navier-Stokes Approach and Emulation of 

Turbulence-Chemistry Interaction

Nan-Suey Liu and Thomas Wey

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Tsan-Hsing Shih

Ohio Aerospace Institute

Brook Park, Ohio 44142

Abstract

This paper describes the time-filtered Navier-Stokes approach capable of capturing unsteady flow 

structures important for turbulent mixing and an accompanying subgrid model directly accounting for the 

major processes in turbulence-chemistry interaction. They have been applied to the computation of twophase turbulent combustion occurring in a single-element lean-direct-injection combustor. Some of the 

preliminary results from this computational effort are presented in this paper.

Nomenclature

CFD computational fluid dynamics

DNS direct numerical simulation

LES large-eddy simulation

RANS Reynolds-averaged Navier-Stokes approach

URANS unsteady RANS

TFNS time-filtered Navier-Stokes approach

SGS subgrid scale

SFC subfilter component

FCP filtering-control parameter

LEM linear-eddy mixing model

LDI lean-direct injection

VBB vortex-breakdown bubble

PVC precessing-vortex core

1.0 Introduction

High-fidelity simulation of liquid combustion in practical engineering devices remains an elusive 

target in spite of significant advances in physical models and numerical methods over the past decade.

Within these devices, there are a broad range of intricate physical and chemical phenomena, with liquid 

fuel atomization and spray, as well as turbulence-chemistry interaction, being two of the most important 

processes. The most accurate and straightforward numerical approach to fluid flow problems is to solve 

the Navier-Stokes equations without filtering and approximation other than numerical discretizations 

whose errors are to be controlled by highly accurate numerical schemes. This approach is known as the 

direct numerical simulation (DNS). Although the governing equations are solved directly in DNS, the use 

of models to accommodate the multiphase formulation and interaction is still unavoidable. Much more 

research in this area is needed. Modeling and simulations of fuel injection and spray combustion is a very 

difficult task, in fact, many existing large-eddy simulation (LES) of spray flow and combustion invoke 

the same liquid-phase models as those used in the traditional Reynolds-averaged Navier-Stokes (RANS) 
NASA/TM—2013-217873 2

approach, with a subgrid-scale (SGS) model used for the gas-phase turbulence. LES uses spatial filtering 

to explicitly account for flow structures larger than the filter width, the filtering in LES leads to unknown 

terms in the filtered equations, and these so-called SGS terms need to be modeled to form a closed set of 

the fluid flow governing equations. For chemically reacting flow, modeling of the filtered term arising 

from the production and destruction of chemical species is quite challenging, whether the approach is the 

traditional RANS or LES, as the turbulence-chemistry interaction is present throughout the turbulent 

spectrum. In any event, for liquid fueled combustion processes, unsteadiness is a dominant feature of the 

fluid dynamics, furthermore, the flame structure is very complex and locally (both in space and time) can 

range from nonpremixed to premixed burning. Thus, the prerequisites for improved simulation of 

multiphase turbulent combustion include the ability of the turbulence model to capture the unsteady 

turbulent structures responsible for the fuel-oxidizer mixing, and that the modeled process of turbulencechemistry interaction can account for the multiregime-flame structure in these combustors. This paper 

describes some of the recent progress made in these two areas in the framework of the time-filtered 

Navier-Stokes (TFNS) approach.

Before we focus on the TFNS approach, a few words on the fundamental characteristics of the 

conventional LES is in order. In the framework of conventional LES, the filtered equations are 

established by applying a spatial filter to the exact form of the governing equations. The filter width is 

typically the local grid size; in addition, the eddy viscosity uses the local grid size as a model parameter. 

Therefore, the grid resolution and the model fidelity are formally linked, and, in principle, a grid 

independent solution cannot be reached. An explicitly filtered LES approach can be used to mitigate this 

issue, and this is an area which needs further development. Last but not the least, the need to use fine 

grids when performing LES is of paramount importance, because, in LES, the subfilter field coincides 

with the subgrid field.

In the framework of TFNS, the filtered equations are established by applying a temporal filter to the 

exact form of the governing equations. The filter width does not relate to the time step of the numerical 

solution, and the eddy viscosity contains a so called filtering-control parameter, which is defined as the 

ratio of a (conceptual) temporal filter width to a characteristic integral time scale of the turbulent flow. 

Since the grid resolution and the model fidelity are not formally linked, in principle, a grid independent 

solution can be attained. It should be pointed out that TFNS is not LES, nor hybrid RANS/LES, nor, in 

general, unsteady RANS (URANS). When performing TFNS, grids must numerically support the spatial 

gradients of the filtered variables under investigation. Since the smoothness of these spatial gradients 

correlates to the value of the filtering-control parameter, requirement of grid sizes and distribution varies 

according to the specified value of the filtering-control parameter. This is a reflection of the characteristic 

of TFNS, i.e., the subfilter field is not the subgrid field.

This paper is organized as follows. In the next section, a description of the TFNS formulation for twophase flow and the subfilter closure models are presented. In Section 3.0, we describe an approximate

evaluation of the filtered, chemical-reaction, source terms via a subgrid model of turbulence-chemistry 

interaction in the framework of TFNS. In Section 4.0, we summarize the experimental setup and the 

computation set up employed to assess the current, overall TFNS approach. This is followed by 

presenting some of the preliminary results in Section 5.0 and concluding remarks in Section 6.0.

2.0 Mathematical Formulation

The conservation equations for compressible reacting flow are solved using the TFNS approach. To 

simulate spray combustion, Lagrangian droplet model is concurrently solved with the Eulerian gas phase.

In the following, we will briefly describe the definitions of time-filtered quantity, the gas-phase equations, 

and the coupling between the gaseous field and the spray field. We will then make a very general 

comment on the spray modeling without presenting the liquid-phase equations.
NASA/TM—2013-217873 3

2.1 Time-Filtered Quantities φ( ,) x t and φ ( ,) x t

In the case of compressible turbulent flow, we often use two distinct yet closely related time-filtered 

quantities. One is denoted by φ( ,) x t , which is defined as 

( , ) ( , ') ( ') ' t t G t t dt +∞

−∞ φ =φ − ∫ x x (1)

The integration is over the entire time domain –∞ < t′ < + ∞. Where φ is an unfiltered turbulent variable

and G(t – t′) is a temporal filter with a constant filter width ∆T. Furthermore, this temporal filter satisfies 

the following conditions:

G t t dt ( ') ' 1 +∞

−∞ − = ∫ (2)

( , ') ( ') ' ( , ), as 0 T t G t t dt t +∞

−∞ ∫ φ − =φ ∆ → x x (3)

The other filtered quantity is denoted by φ ( ,) x t , which is defined as

( ,)t ρφ φ =

ρ

x (4)

That is, φ ( ,) x t is determined by the familiar Favre filtering, which is density-weighted. 

2.2 Gas-Phase TFNS Equations

Applying a temporal filter with a constant filter width to the exact equations of conservation of mass, 

momentum, energy, and chemical species, as well as the equation of state, the following filtered 

governing equations in the TFNS framework are obtained,

 j

liq

j

U

t x

∂ρ ∂ ρ + =ρ ∂ ∂

 (5)



,

i sfc

i j ij ij ij i liq

j

U UU P F

t x

∂ρ ∂ + ρ + δ −τ +τ =     ∂ ∂

    (6)

  ( )   ,

sfc sfc sfc i j i ij i i i k i liq

i

E E P U q U E II S Q

t x

∂ρ ∂ + ρ + + −τ + + +σ =− +  

∂ ∂  

 (7)

   , m sfc

m mi mi m m liq i

i

Y YU g F S

t x

∂ρ ∂

+ ρ − + =ρ +ρ     ∂ ∂

  (8)

1 1

N N s s

m sfc

u u m

m m m

Y T PR RT

= MW =

=ρ +ρ ∑ ∑   (9)
NASA/TM—2013-217873 4

Where ρ is filtered mass density, Ui

 is filtered velocity vector, P is filtered pressure determined from 

filtered equation of state, E is filtered total energy per unit mass, ij τ is filtered viscous stress, i q is 

filtered heat flux vector, and Sk is the source term from the subfilter kinetic energy. In addition, Ym

 is 

filtered mass fraction of the m-th species, mi g is filtered mass flux vector of m-th species. mS

 is the 

filtered reaction-source term of the m-th species, MWm is the molecular weight of the m-th species, Ru is 

the universal gas constant, and T is filtered temperature, Ns is the total number of species. Subscripts ‘liq’ 

denote source terms from the liquid phase. Superscripts ‘sfc’ denote subfilter component (SFC) that 

require closure.

The filtered viscous stress tensor, filtered heat flux vector, and filtered mass flux vector of the m-th 

species are approximated as

2 2

3 ij ij kk τ=μ − μ S S   (10)



,

1 1

N N s s

m sfc

i m m i m

i i m m

T Y

q Dh q

x x = =

∂ ∂

=−κ − ρ + ∂ ∂ ∑ ∑   (11)

m mi m

i

Y g D

x

∂

= ρ ∂

 (12)

Where ij S

 is the filtered strain-rate tensor, and mh

 is the specific enthalpy of species m. Here, the 

molecular viscosity (μ ) is approximated by the Sutherland’s law based on filtered temperature (T ), the 

thermal conductivity ( κ ) and the diffusion coefficient of the m-th species are approximated as

Pr p κ=μc

Sc m

T

D μ ≈ ρ

Where p c is the specific heat at constant pressure for the gaseous mixture, Pr is the Prandtl number, and 

ScT is the turbulent Schmidt number.

The SFC terms that require closure are

τ ≡ρ − ij i j i j

sfc ( ) UU UU   

sfc

,

m m im m m m m

i i

Y Y q hD hD

x x

∂ ∂ ≡ − ∂ ∂

 

( ) sfc i E EU EU i i ≡ρ −  

sfc

i ii II PU PU ≡ − 

σ ≡ τ− τ i j ji j ji

sfc U U  
NASA/TM—2013-217873 5

( ) sfc F UY UY mi i m i m ≡ρ −  

T YT YT mmm

sfc ≡ −   

In the present study, we neglect terms such as ,

sfc

i m q , sfc

i II , and sfc Tm . The closure of the rest of the SFC 

terms is described below.

2.2.1 Momentum Transport Closure

A general constitutive relationship between subfilter turbulent stresses and filtered strain rate (Ref. 1) 

leads to a nonlinear model for sfc

ij τ , i.e.,

( )

( )

3

3 2

4

2 2

5 3

2 2 3

3

2 ( 3) ,

sfc

ij T ij ij kk ij

ik kj ik kj

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

SS k

k Af S S

k A f S S S S II S S

τ = − ρν −δ + ρ δ

− ρ Ω −Ω

ε

+ ρ Ω − Ω +Ω Ω −Ω Ω δ + −δ     ε

 

   

         

(13)

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 (Ref. 2):

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

in which, 

( ) ** *

* *

* 3

1 6 cos , arccos 6 , , 3 ( )

ij jk ki

s

SS S

A W W

S = φ φ= =

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

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

The coefficient f is a function of the filtering-control parameter (FCP) which is defined as the ratio of a

(conceptual) time filter width ∆T to an integral time scale T, i.e., FCP = ∆T/T. Furthermore (Ref. 3),

2

2 T TT f T TT

     ∆ ∆∆ ≈ −           (14)

By definition, the value of the parameter FCP and the value of the coefficient f are always between 0 and 1.

This model uses the concept of subfilter eddy viscosity which is defined as
NASA/TM—2013-217873 6

2 / T fC k ν= ⋅ ⋅ ε μ

μ =ρν T T

Here, the subfilter turbulent kinetic energy and its dissipation rate (k, ε) are determined from the 

following model equations:

T

i k

i k i

k k uk S

t x x

∂ ∂     μ ∂

ρ + ρ − μ+ =     ∂ ∂ σ ∂    

 (15)

T

i

i i

u S

t x x ε

ε

∂ ∂     μ ∂ ε

ρε + ρ ε− μ+ =     ∂ ∂ σ ∂    

 (16)

where σk and σε are model constants, the source terms are given by

sfc

k ij ij S s = − τ −ρε  (17)

2

1 2

sfc

ij ij S Cs C

k k εε ε

ε ρε =− τ −  (18)

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.

2.2.2 Energy Transport Closure

The subfilter total energy flux, sfc Ei , is also modeled using the subfilter eddy viscosity and gradient 

assumption, a nonlinear model is formulated as (Ref. 2)

( 1 2 ) 1

Pr

sfc T

i T ij ij

i T j

T k c T E cS c

x x

υ ∂ μ ∂ρ = −κ − + Ω

∂ρ ε ∂

    (19)

Here, PrT is the turbulent Prandtl number, cυ is the specific heat at constant volume for the gaseous 

mixture. The coefficients c1 and c2 are yet to be calibrated. In the current simulations they are set to be 

c1 = c1 = –0.24. The subfilter thermal conductivity (κT) is evaluated by

/ Pr T Tp T κ =μ c

It is noted here that the total energy ( E ) is evaluated as

1

2

E e UU = + i i   

where e is the specific internal energy of the mixture. The subfilter viscous work is modeled as

sfc sfc σ= τ i j ji U (20)
NASA/TM—2013-217873 7

2.2.3 Species Transport Closure

Similar to the subfilter total energy flux, the subfilter mass flux of m-th species is modeled as

( 1 2 ) sfc m m mi T T ij ij

i j

Y Y k F Dm Dm c S c

x x

∂ ∂ρ = −ρ − + Ω

∂ε ∂

    (21)

with

T

T

T

Dm

Sc

μ = ρ

where ScT is turbulent Schmidt number.

In the present study, the filtered reaction rate of m-th species ( mS

 ) is approximately evaluated via a 

subgrid model of turbulent mixing and combustion, and this model is described in Section 3.0.

2.3 Coupling between Gaseous Field and Spray Field

This coupling is through the interphase exchange terms. The effects of the spray field on the gaseous

field are accounted for by the source terms from the liquid phase in the gas-phase equations (Eqs. (5), (6),

(7), and (8)), and they are modeled as (Ref. 4)

1

liq k k

c k

n m

V ρ =  ∑  (22)

3

,

1 4

3

ki

i liq k k ki k k

c k

d u F n mu r

V dt

  π = − ρ     ∑   (23)

( , ) 1

liq k k s k eff

c k

Q nm h V= ∑ −    (24)

,

1

m liq m k k

c k

n m

V ρ= ε 

 ∑  (25)

where Vc is the volume of the computational cell, nk is the number of droplets in k-th drop group, mk  is 

the vaporization rate of droplets in k-th group, uki is the i-th velocity component of k-th group, ρk is the 

density of droplets in k-th group, rk is the droplet radius in k-th group, hs is the specific enthalpy at the 

droplet surface, k,eff is the effective latent heat of vaporization of droplets in k-th group, and εm is the 

fractional vaporization rate for species m.

2.4 Liquid-Phase Modeling

The governing equations of the liquid phase are based on a Lagrangian formulation where the spray 

particle position and velocity are described by a set of ordinary differential equations. Various submodels, 

such as the droplet drag model and the drop vaporization model, are needed to simulate the transport of a 

vaporizing spray particle. The specification of the fuel injection condition plays a major role in the 

fidelity of the simulation. Common practice is to specify the starting droplet condition using correlations 

of droplet sizes calibrated by relevant experimental data. In addition to the use of correlation, various 
NASA/TM—2013-217873 8

models for primary atomization and secondary droplet breakup also have been employed. A more detailed 

description of the liquid phase modeling and the two-way coupling between the liquid-phase and gasphase transport can be found in Reference 4.

3.0 Subgrid Model of Mixing and Combustion

Simulation of turbulent combustion requires modeling the turbulence-chemistry interaction processes. 

To this end, various approaches have been invoked in many previous studies. Examples of these 

approaches include the so-called unmixed model, eddy-break-up model, thickened flame model, flameletbased method, conditional moment closure method, filtered mass density function/probability density 

function method, and linear-eddy mixing (LEM) model. In the present study, we focus on the LEM 

model, because it does not treat molecular mixing and small scale turbulence effects as a single process 

(micro mixing), and it can be applied to both nonpremixed and premixed burning. The major drawback of 

this model is its relatively high computational cost which can be very high when detailed chemical 

kinetics is employed. A fairly detailed description of the LEM model when used in the LES framework 

can be found in Reference 5.

Closely following the approach described in Reference 5, we have adapted the LEM model into the 

TFNS framework, and we call it LEM-like to emphasize that (i) instead of using LEM model to directly 

establish the entire species field without actually solving filtered equations of species, the LEM model is 

used locally (both in space and time) to only provide approximations of filtered reaction source terms in 

the filtered equations of the species to be solved, (ii) instead of solving unfiltered quantities using the 

LEM equations, the same forms of the LEM equations are employed as a subgrid model of mixing and 

combustion of time-filtered scalars, with the molecular transport coefficients substituted by the ‘effective’ 

transport coefficients.

The LEM-like model is implemented in terms of a fractional splitting technique; it is divided into a

supergrid process and a subgrid process. The supergrid process emulates the convection of the scalar field 

by grid-resolved velocity field across the surfaces of the computational cell. The subgrid process, which 

occurs within each computational cell, consists of four operators: (a) (effective) molecular diffusion, (b) 

finite-rate chemical reaction, (c) volumetric expansion caused by the heat release, and (d) stochastic 

stirring due to the subgrid eddies.

3.1 Supergrid Process

The macro mixing of the scalars is implemented by a Lagrangian transfer of LEM elements across the 

surfaces of the CFD computational cells. This Lagrangian transport is also known as splicing. Referring 

to Figure 1, for example, the outward mass through the right side of a cell computed from filtered velocity 

and density is equivalent to 1.5 LEM elements and colored in red. The outward mass through the bottom 

side of the mesh is equivalent to 2.5 LEM elements and colored in magenta. Similarly, the inward mass 

through the top and left sides of the mesh are equivalent to 6 LEM elements. Splicing will result in 

14 LEM elements in this computational cell. In general, splicing will cause different computational cell to 

have different number of LEM elements. To avoid programming complexities in a parallel environment, 

the LEM domain is re-gridded to have the same fixed number of elements, and each element is of the 

same volume. Conservation of mass is maintained during the re-gridding procedure.
NASA/TM—2013-217873 9

Figure 1.—Schematic illustrating the splicing algorithm used for scalar convection. (Assuming the 

number of LEM elements is 12. Each element carries its own volume, filtered temperature, and 

filtered species density).

3.2 Subgrid Process

Within each CFD computational cell, a one-dimensional domain consisting of a fixed number of 

LEM elements is employed, the governing equations have the following form:

, , 1 2 (, , , ) m s

m m

Y stirring m eff m N

Y Y F D YY Y

t ss

∂ ∂ ∂  

ρ = + ρ +ρω   ∂ ∂∂   

      (26)

, ,

1 1

N N s s

m p T stirring pm m eff eff m m

m m

T Y T T c F c D h

t ss s s = =

∂   ∂ ∂∂ ∂  

ρ = +ρ  +κ − ω   ∂ ∂∂ ∂ ∂      ∑ ∑       (27)

where Y stirring m , F and T stirring , F represent the subgrid turbulent mixing, and they are accounted for by 

employing a stochastic rearrangement of the LEM elements, known as the triplet mapping, more details 

about this mapping procedure can be found in Reference 5. Here, Dm,eff is the effective mass diffusion 

coefficient, κeff is the effective heat conductivity, and 1 2 (, , , ) m Ns ω YY Y     indicates that the reaction 

kinetics of unfiltered species mass fractions is formally used for the reaction kinetics of filtered species 

mass fractions.

The solution of the above one-dimensional LEM-like equations over NLEN elements gives a subgrid 

field of Ym

 , namely, 1 2

, ,, l l l NLEN YY Y mm m

  = = =  . Their average, ( ) LEM Y tt 

m + ∆ , provides a prediction of 

( ) Yt t 

m + ∆ at the center of the computational cell. This predicted value is then used to construct an 

approximation of the filtered source term ( mS

 ) needed to solve Equation (8). In the current study, the 

approximation is

( ) () LEM TFNS

m m m

Y t tY t S

t

+∆ − = ∆

   (27)

where ( ) TFNS Y t m

 is the TFNS solution of the (filtered) species mass fraction at the computational cell 

center.

Arrows -- Convection direction

Initial number of lumps – 12

Light Green – inward mass (1.5)

Dark Green – inward mass (4.5)

Red – outward mass (1.5)

Magenta – outward mass (2.5)

Current number of lumps -- 14 

Final number of lumps will became 12 after re-gridding 

via equalizing the volume.
NASA/TM—2013-217873 10

Figure 2.—Swirler geometry and computational domain of a single-element 

LDI combustor.

4.0 Experimental and Computational Setup

Target application is low-emissions, fuel-flexible combustors, including the lean-direct-injection 

(LDI) combustors. A typical single-element LDI combustor is illustrated in Figure 2. More detailed 

description of the combustor geometry and the test rig can be found in Reference 6. Each element consists 

of an air passage with an upstream air swirler and a converging-diverging venturi section. The fuel is 

injected through the center of swirler and the fuel tip is at the throat of the venture. The air swirlers have 

six helical, axial vanes with downstream vane angles of 60°. The air then dumps into a combustion 

chamber with a square cross-section. Velocity measurements were taken with a two-component Laser 

Doppler Velociometry (LDV) system, temperature measurements were taken with thermocouples, and 

emissions data was gathered via an isokinetic probe and gas analyzer. Quartz makes up the combustion 

section. In regions close to the dump plane, measured gas-phase velocities may suffer from some 

contamination by the injected spray. The combustor experiments also may have significant convective 

and radiation heat losses. The temperature measurements reported are not corrected to adiabatic 

conditions. Experimental droplet measurements are collected with a Phase Doppler Particle Analyzer 

(PDPA).

The computational domain consists of approximately 862,000 hexahedral elements (see Figure 2). At 

the air inflow boundary, the air flow speed is 20.14 m/s, the density is 1.19 kg per cubic meter, and the 

static temperature is 294.28 °K. The operating pressure of the combustor is approximately 1 atm, and the 

measured pressure drop (as a percentage of the air inlet pressure) during the experiments was measured at 

4 %. At the combustor chamber exit, an outlet boundary condition facilitating the convection of pressure 

disturbances out of the computational domain is applied.

The fuel is injected at 0.415 g/s, which gives a global equivalence ratio of 0.75. The specification of 

the starting condition for the fuel spray is particularly critical for accurate predictions. In this study, the 

following droplet size distribution is used (Ref. 4):

0.4

32

3.5 16.98

6

32 32

4.21 10

d

dn d d dd

e

nd d

  −      

= ×    

where n is the total number of the droplets and dn is the number of droplets in the size range between d

and d + dd. This correlation also requires the specification of a Sauter mean diameter, d32 and the number 

of droplet classes. These specified inflow droplets will undergo evaporation without secondary breakup. 

Experimental data suggests a Sauter mean diameter around 32 μm, and the spray cone angle is 90°.
NASA/TM—2013-217873 11

In this study, the liquid fuel C12H23 is used as the surrogate for the experimental Jet-A fuel, and a 

single-step, five-species global reduced mechanism (see e.g., Ref. 7) is employed for the chemical 

reactions (Table I).

TABLE I.—SINGLE STEP (GLOBAL) CHEMISTRY MODEL

Reaction A, 

mole – cm – sec– K

n E,

cal/mole

1 4 C12H23 + 71 O2 => 48 CO2 + 46 H2O

GLO / C12H23 0.10 /

GLO / O2 1.65 /

8.60×1011 0.00 3.00×104

The current TFNS (with FCP = 0.255) solutions are obtained by an iteratively implicit, finite-volume

scheme that is second-order accurate in space and time. The LEM-like equations are solved using a 

standard fractional splitting, finite-difference scheme, and 24 LEM elements per CFD computational cell

are employed.

5.0 Results

While recognizing that the inherent uncertainties in current models of liquid fuel atomization, fuel 

evaporation, and fuel chemistry often can be overwhelming, this LDI burner and its measured data has 

been used to assess as well as guide the development of modeling and simulation of two-phase turbulent 

combustion in LDI combustors. Results from RANS approach are reported in Reference 7. Results from

LEM-LES approach are presented in Reference 5, and results from flamelet-LES approach are described 

in Reference 8. In the following, some of the results obtained from the TFNS approach are presented.

The results of non-reacting flow and the comparison with experimental data can be found in 

Reference 9. They reveal the major flow structures in the LDI combustor. Figure 3 is a snapshot of the 

unsteady flow field. Embedded in this figure are the instantaneous iso-surface of the zero axial velocity 

component colored by the effective eddy viscosity and six instantaneous stream lines emanating from the 

upstream of the swirler, going through the converging-diverging nozzle, then passing through the 

combustion chamber. Major flow structures in the LDI combustor are visualized via the iso-surface of the 

zero axial velocity and the iso-surface of a relatively low pressure. The iso-surface of the zero axial 

velocity is also known as the vortex-breakdown bubble (VBB). The iso-surface of a sufficiently low 

pressure captures the precessing-vortex core (PVC). Figure 4 is a snapshot of the PVC and VBB. The 

dark blue region is a vortex core, which is formed near the venturi throat and extends into the combustor 

chamber. This spiraling vortex core rotates and breaks, it changes randomly in space and time. The light 

green surfaces are the iso-surfaces of the zero axial velocity. In addition to the VBB, there are some small 

structures near the dump plane and in the corner region. Figure 5 shows the instantaneous streamlines 

around the PVC, they start from the upstream of the swirler, and yield a complex, seemingly random 

pattern. Some of them spiral around the dark blue surface indicating that the dark blue region is indeed a 

vortex core. As demonstrated in Reference 5, the dynamics of the PVC and the VBB, as well as their 

interactions, are critical to the fuel-air mixing and the flame stability in the LDI combustors.
NASA/TM—2013-217873 12

Figure 3.—A snapshot of the unsteady flow field.

Figure 4.—A snapshot of precessing-vortex core (PVC) and 

vortex-breakdown bubble (VBB).

Figure 5.—Instantaneous streamlines around the precessingvortex core (PVC).
NASA/TM—2013-217873 13

The same grid is used for the two-phase reacting flow calculations. Figure 6 is a snapshot of the spray 

droplet field. The performance of the LEM-like model is highlighted in terms of the calculated 

temperature field. The comparison between the temperature field from the LEM-like model and the 

temperature field from the so-called unmixed (umx) model is shown in Figure 7. The unmixed model 

neglects the direct effects of turbulent fluctuations when evaluating the filtered reaction source terms in 

the species equations. The temperature field shown in Figure 7 is averaged temperature in the center plane 

(z=0). In the unmixed case, the filtered temperature is averaged over 10,000 time-steps (i.e., 0.01 sec). In 

the LEM-like case, the filtered temperature is averaged over 5,000 time-steps (i.e., 0.005 sec). The feature 

of the temperature field calculated by using the LEM-like model is closer to what is typically observed in 

the experiments than the one from the unmixed model.

Figure 8 is the (gage) pressure trace recorded at a centerline point which is 15 mm downstream of the 

dump plane having a cross-section of 25.4- by 25.4-mm. In the following, time-averaged quantities are 

presented. The time period used to construct these averages is from time-step 380,001 to time-step 

500,000, and the corresponding duration is 0.12 sec. All contour plots illustrate the distributions in the 

center plane (z=0). Time-average of filtered temperature is shown in Figure 9, also included are two 

snapshot solutions. Similarly, time-average of filtered fuel (C12H23) vapor mass fraction, along with two 

snapshot solutions, are given in Figure 10. Time-averaged field gives the impression of an orderly, 

symmetric burning, the snapshots suggest that, in parts of the flame region, the supply of fuel vapor is

intermittent, and pockets of significant temperature variation exist.

Figure 6.—A snapshot of the spray-droplet distribution in the center plane 

and a cross-section.

Figure 7.—Temperature field in the center plane (z=0).
NASA/TM—2013-217873 14

Figure 8.—Pressure trace at a center point which is 15 mm downstream of the dump plane.

Figure 9.—Temperature distribution in the center plane: time-averaged field and two snapshots.
NASA/TM—2013-217873 15

Figure 10.—Fuel vapor distribution in the center plane: time-averaged field and two snapshots.

In the following, comparisons between time averages of filtered quantities and experimental mean 

values are shown. The discrepancies may be due to a variety of reasons. In addition to the obvious 

computational reasons such as grid resolution, there are model-related causes, and we will, based on 

information available to us from other closely related studies, suggest some of these possible causes 

without discussion. 

Centerline averaged temperature downstream of the dump plane (located at x=0.0072 m) along the 

length of the combustor is shown in Figure 11. Near the dump plane, the major cause of the discrepancy 

is the spray injection and vaporization models, the discrepancy in the downstream region is mainly due to 

the employed subgrid model of turbulent mixing. Figure 12 presents the centerline averaged axial velocity 

along the combustor length. In the immediate neighborhood of the dump plane, there was difficulty in 

sorting seeder particles from high-momentum spray particles in the experimental study; nevertheless, 

heat-release tied to the spray models is also a cause of the discrepancy. These possible causes of 

discrepancies between computational averages and experimental mean values also are the possible 

reasons for discrepancies shown in Figure 13 and Figure 14. Figure 13 is the radial profile of the averaged 

temperature at 5 mm downstream of the dump plane, and Figure 14 is the radial profile of the averaged 

axial velocity at 5 mm downstream of the dump plane. Note that similar discrepancies relative to 

experiments have been reported in Reference 8. The radial profile of the averaged azimuthal velocity at 

this location is presented in Figure 15. It suggests the need of improved grid resolution in the crosssection.
NASA/TM—2013-217873 16

Figure 11.—Averaged temperature along the center line.

Figure 12.—Averaged axial velocity along the center line.
NASA/TM—2013-217873 17

Figure 13.—Radial profile of averaged temperature (5 mm downstream of 

the dump plane).

Figure 14.—Radial profile of averaged axial velocity (5 mm downstream of 

the dump plane).
NASA/TM—2013-217873 18

Figure 15.—Radial profile of averaged azimuthal velocity (5 mm downstream 

of the dump plane).

Figure 16.—Radial profile of averaged temperature (20 mm downstream of 

the dump plane).

Radial profile of the averaged temperature at 20 mm downstream of the dump plane is shown in 

Figure 16. Radial profiles of the averaged axial velocity and the averaged azimuthal velocity at 29 mm 

downstream of the dump plane are shown in Figure 17 and Figure 18, respectively. There are still 

discrepancies, it is apparent that grid resolution in the near wall region needs to be improved. Radial 

profiles of averaged temperature, averaged axial velocity, and averaged azimuthal velocity at further 

downstream locations are presented in Figure 19, Figure 20, and Figure 21, respectively. Figure 19

indicates that some experimental uncertainty in the temperature measurement may exist, as suggested by 

its cross-stream gradient.
NASA/TM—2013-217873 19

Figure 17.—Radial profile of averaged axial velocity (29 mm downstream of 

the dump plane).

Figure 18.—Radial profile of averaged azimuthal velocity (29 mm downstream 

of the dump plane).
NASA/TM—2013-217873 20

Figure 19.—Radial profile of averaged temperature (40 mm downstream of 

the dump plane).

Figure 20.—Radial profile of averaged axial velocity (46 mm downstream of 

the dump plane).
NASA/TM—2013-217873 21

Figure 21.—Radial profile of averaged azimuthal velocity (46 mm downstream 

of the dump plane).

6.0 Concluding Remarks

Two-phase turbulent combustion in a single-element LDI combustor is calculated by employing the 

TFNS/LEM-like approach. Together with an assumed injection condition of fuel droplets and a singlestep, five-species global reduced kinetics for the combustion, preliminary results are obtained by using a 

computational grid consisting of approximately 862,000 hexahedral cells. These results are useful in 

gaining deepened insight into the unsteady physical processes in the LDI combustor, but their quantitative 

accuracy suffers from the inherent uncertainties in current spray-related models. Future plan includes

replacing the present combustion chemistry with, say, a three-step, seven-species global reduced 

mechanism, achieving a grid-independent solution, and employing atomization as well as breakup models 

to characterize the fuel injection processes.

References

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

Turbulence and Transition Modeling, Chapter 4, edited by Hallback, M., Henningson, D.S., 

Johansson, A.V. and Alfredsson, P.H., Kluwer Academic Publishers, 1996.

2. Shih, T.-H., “Some Developments in Computational Modeling of Turbulent Flows,” Fluid Dynamic 

Research, Vol. 20, 1997, pp. 67-96.

3. Shih, T.-H. and Liu, N.-S., “A Nonlinear Dynamic Subscale Model for PRNS/VLES of Internal 

Combustor Flows,” AIAA-2009-0467, 47th AIAA Aerospace Science Meeting and Exhibit, 4-8

January 2009, Orlando, FL.

4. Raju, M.S., “LSPRAY-III: A Lagrangian Spray Module,” NASA/CR—2008-215290, 2008.

5. Patel, N., and Suresh, M., “Simulation of Spray-Turbulence-Flame Interactions in a Lean Direct 

Injection Combustor,” Combustion and Flame Vol. 153, 2008, pp. 228-257.

6. Cai, J., Jeng, S.-M., and Tacina, R., “The Structure of a Swirl-Stabilized Reacting Spray Issued from 

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

January 2005, Reno, NV.
NASA/TM—2013-217873 22

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

in LDI CFD Calculations,” AIAA-2008-1150, 46th AIAA Aerospace Science Meeting and Exhibit, 7-

10 January 2008, Reno, NV.

8. Knudsen, E. and Pitsch, H., “Large Eddy Simulation of a Spray Combustor Using a Multi-regime 

Flamelet Approach,” Annual Research Briefs 2010, Center for Turbulence Research, Stanford 

University, pp. 337-350.

9. Liu, N.-S. and Shih, T.-H., “A Very Large Eddy Simulation of the Non-reacting Flow in a SingleElement Lean Direct Injection Combustor Using PRNS with a Nonlinear Subscale Model,” 

ISROMAC-13-36, Proceedings of the 13th International Symposium on Transport Phenomena and 

Dynamics of Rotating Machinery, 4-9 April 2010, Honolulu, HI.

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Time-Filtered Navier-Stokes Approach and Emulation of Turbulence-Chemistry Interaction 

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14. ABSTRACT

This paper describes the time-filtered Navier-Stokes approach capable of capturing unsteady flow structures important for turbulent mixing 

and an accompanying subgrid model directly accounting for the major processes in turbulence-chemistry interaction. They have been 

applied to the computation of two-phase turbulent combustion occurring in a single-element lean-direct-injection combustor. Some of the 

preliminary results from this computational effort are presented in this paper.

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Combustion; CFD 

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