This document provides a review of turbulent combustion modeling closures for large eddy simulations (LES), with a focus on models applicable to propulsion applications. It identifies three classes of models that can provide broad-based modeling: flamelet-library/presumed PDF models, linear eddy based (LEM) models, and transported PDF or filtered density function (FDF) based models. The document discusses the fundamental physical assumptions of these models, particularly regarding the presumed size of the turbulent scalar manifold. It also provides novel results from direct numerical simulations testing some assumptions using flames with detailed and reduced chemistry models.

1. Survey of Turbulent Combustion
Models for Large Eddy Simulations of Propulsive
Flowfields
Justin W. Foster⇤
Corvid Technologies, Mooresville, NC 28117
Richard S. Miller †
Department of Mechanical Engineering, Clemson University, Clemson, SC 29634
AIAA Science and Technology Forum and Exposition
January 5-9, 2015, Kissimmee, FL
(Session: Turbulent Combustion Models, Their Foundations and Major Assumptions)
A general review of turbulent combustion modeling closures applicable to large eddy
simulations (LES) is provided. The focus is on regime-independent models able to provide
turbulent combustion closures ranging from purely premixed to purely non-premixed and
all regimes between these two limits. Special emphasis is placed on primary propulsion
applications, including liquid rocket engines, diesel engines, gas turbines, and scramjets.
These applications span a large range of physical phenomena including both ideal- and
real-gas behavior, single-phase and multi-phase combustion, relatively low Mach number
to supersonic and hypersonic combustion, and relatively simple geometries to highly com-
plex geometries. Three classes of models are identified as possibly providing such broad
based modeling closures: flamelet-library/presumed probability density function (PDF)
models, linear eddy based models (LEM), and transported PDF or filtered density func-
tion (FDF) based models. This review focuses both on fundamental physical assumptions
that apply across all of the models and assumptions that apply to each of the models
individually. Namely, assumptions regarding the presumed size of the large dimensional
turbulent scalar manifold apply to all of the models; however, flamelet models almost al-
ways presume only a few dimensions are necessary to yield adequate representation of the
larger, turbulent manifold. In contrast, LEM and FDF models are not, in theory, bound
by any manifold size assumptions (i.e. direct calculation of the turbulent scalar manifold
is possible); however, due to current computational limitations, these models often employ
manifold reduction techniques which are usually not as restrictive as those used by flamelet
models. Individual assumptions associated with the specific formulation of each model are
also analyzed. From these discussions, additional novel results testing some of the fun-
damental physical assumptions associated with each model are provided from a unique
database of DNS of high pressure turbulent reacting temporally developing shear flames.
The DNS database includes simulations of H2/O2, H2/Air, and Kerosene/Air flames with
both detailed and reduced chemistry. The DNS include real property models, a real-gas
equation of state, and generalized heat and mass di↵usion derived from non-equilibrium
thermodynamics. The simulations span a large range of Reynolds numbers and pressures
(up to 125 atm), with resolutions approaching 1 billion grid points. Finally, some general
comments towards the future challenges related to LES combustion modeling are o↵ered.
I. Introduction
This review of multi-regime large eddy simulation (LES) approaches is motivated by propulsive applications.
Such applications cover a broad range of combustion regimes with primary applications being liquid rockets,
⇤Computational Analyst, 145 Overhill Drive.
†Associate Professor, Department of Mechanical Engineering, Senior Member AIAA
1 of 20
American Institute of Aeronautics and Astronautics

2. diesels, gas turbines, and scramjets. These applications cover a broad range of thermodynamic and chemical
physics, including both liquid and gaseous fuels, ideal and real thermodynamic states, and low speed to
supersonic speed combustion. Although many LES approaches are available in the literature, only relatively
few are capable of capturing the multi-regime combustion phenomena associated with such applications.1
Such a general purpose model must be able to capture all combustion regimes from purely non-premixed,
to purely premixed, to partially premixed, from low speed to high speed, as well as from ideal to real
thermodynamic behavior. Three classes of models are identified as possibly providing such broad based
modeling closures: flamelet-library/presumed probability density function (PDF) models, linear eddy based
models (LEM), and transported PDF or filtered density function (FDF) based models. The primary objective
of the current paper is to re-examine the fundamental physical assumptions associated with each of these
modeling approaches and to compare their limitations to use in general propulsive applications. This is by
far not the first such work to review these LES models and many recent reviews exist including Refs.1–8
Nevertheless, some fresh perspective and novel results are presented in what follows. In addition, other
potentially multi-regime “geometric” models such as the flame surface density, flame wrinkling, and thickened
flame models are not addressed in this review. A recent review of these models can be found in Ref.1
Despite the primary emphasis on reviewing the fundamental physical assumptions of LES combustion tech-
niques, several novel results are also presented to test the validity of these assumptions under conditions
relevant to today’s common propulsion applications. The authors have at their disposal a database of direct
numerical simulations (DNS) of turbulent temporally developing shear layer flames that include detailed and
reduced mechanisms for H2/O2, H2/Air, and Kerosene/Air combustion (additional details below).9,10
The
formulation is based on the general compressible Navier Stokes equations, a cubic Peng-Robinson equation of
state, generalized molecular di↵usion derived from non-equilibrium thermodynamics and fluctuation theory,
and realistic property models. The simulations are conducted with eighth order central finite di↵erencing
for all spatial derivatives and fourth order Runge Kutta time integration. Initial Reynolds numbers based
on the initial vorticity thickness, free stream velocity di↵erence, and mean free stream densities and vis-
cosities range from 850 to 4, 500. Pressure range from 1 atm to 125 atm. Resolutions up to 3/4 billion
finite di↵erence grid points are employed. These flames represent very di↵erent thermochemical conditions
with the H2/O2 and Kerosene/Air exhibiting fast chemistry with no local extinction, whereas the H2/Air
flame is characterized by large amounts of scatter due to strong local extinction. Figure 1 portrays these
local extinction events by showing instantaneous center plane temperature contours and temperature scatter
plots on the stoichiometric surface as a function of normalized scalar dissipation for the H2/Air flame at
Re0 = 4500. This simulation utilized 575 ⇥ 106
numerical grid points and achieved a final time Reynolds
number based on instantaneous vorticity thickness of 26, 000 and a centerplane Taylor Reynolds number of
240 based on the streamwise rms velocity and Taylor length scale.9
(a) (b)
Figure 1. Temporally developing H2/Air shear flame at Re0 = 4500 and P = 35atm showing long time (a) instan-
taneous center plane temperature contours and (b) scatter plots of temperature as a function of normalized
scalar dissipation on the stoichiometric surface
2 of 20
American Institute of Aeronautics and Astronautics

3. The outline for this document includes an overview of the governing equations for LES in Section II followed
by a discussion of the assumptions associated with the reduction of turbulent scalar manifolds in Section III.
Sections IV, V, and VI discuss the particular physical assumptions associated with LES-Flamelet, LES-LEM,
and LES-FDF closure models, respectively. Finally, Section VII o↵ers conclusions and comments about the
remaining challenges associated with LES of turbulent combustion modeling going forward.
II. LES Governing Equations
We begin this section by o↵ering the exact (unclosed) LES equations that describe the applications of interest
for this study. Consistent with traditional LES formulations, each instantaneous variable is decomposed into
a “resolved” scale plus a subgrid scale fluctuating component, mathematically represented by, = ¯ + 0
.
The filtered variable, ¯, is mathematically defined by the convolution integral:
¯(x) =
Z
⌦
(x
0
j)G(xj x
0
j)dx
0
j, (1)
where G(xj) represents the filter kernel defined over the domain, ⌦. Similar to a PDF, G(xj) must integrate
to unity to maintain the conservation of constants. Compressible flows use the concept of Favre filtering,
which is a density-weighted filter related to the standard filter represented by the expression, ˜ = ⇢ /¯⇢.
Filtered variables can be decomposed by Favre filter as = ˜ + 00
. In a rigorous LES approach, the set of
governing equations is filtered term by term to derive the exact (albeit unclosed) LES equations:
@¯⇢
@t
+
@¯⇢ euj
@xj
= 0, (2)
@¯⇢ eui
@t
+
@¯⇢ eui euj
@xj
=
@P(¯)
@xi
+
@⌧ij(¯)
@xj
@
@xi
[P( ) P(¯)] +
@
@xj
[⌧ij ⌧ij(¯)]
@Tij
@xj
, (3)
@¯⇢eet
@t
+
@¯⇢eet euj
@xj
=
@P(¯) euj
@xj
+
@ eui⌧ij(¯)
@xj
@Qj(¯)
@xj
+
@
@xj
[
NsX
↵=1
[H,↵(¯)Jj,↵(¯)] + Se (4)
@
@xj
[Qj( ) Qj(¯)] +
@
@xj
[ui⌧ij eui⌧ij(¯)] (5)
@
@xj
[¯⇢ getuj ¯⇢eet euj]
@
@xj
[P( ) P(¯)] euj (6)
+
@
@xj
[
NsX
↵=1
H,↵( )Jj,↵( )
NsX
↵=1
H,↵(¯)Jj,↵(¯)], (7)
@¯⇢fY↵
@t
+
@¯⇢fY↵ euj
@xj
=
@Jj,↵(¯)
@xj
+ SY↵
@
@xj
[¯⇢ gY↵uj ¯⇢fY↵ euj]
@
@xj
[Jj,↵( ) Jj,↵(¯)], (8)
where gravity and radiation have been neglected and a single phase gaseous or supercritical fluid has been
assumed for the sake of simplicity. In the above, we define to represent the set of primitive variables
(eg. density ⇢, velocity vector ui, energy et, and mass fraction Y↵). In contrast, ¯ represents the set of
appropriately filtered primitive variables (eg. ¯⇢, eui, eet, fY↵), while (¯) represents a variable calculated from
the filtered primitive variables (eg. P(¯), T(¯)). As such, P( ) is the filtered pressure. The “resolved”
pressure P(¯) is the pressure obtained from the chosen equation of state as a function of only the available
resolved field LES variables (i.e. that can be calculated in an actual LES).
The above set of equations additionally require a chemical kinetics mechanism for the source terms, ¯Se and
SY↵ , an equation of state along with constituitive relations for the viscous shear stress tensor, ⌧ij, the heat
flux vector, Qj, and the mass flux vector of species ↵, Jj,↵. The pressure dependent kinetics schemes available
in Clemson’s code (and the results herein) pertain to hydrogen-oxygen (8 species, 19 steps) and hydrogen-air
(12 species, 24 steps),11
as well as a surrogate kerosene mechanism (10 species, 34 steps) commonly used in
3 of 20
American Institute of Aeronautics and Astronautics

4. the rocket community at large pressure.12
Clemson’s DNS code utilizes the cubic Peng-Robinson equation
of state whose resolved form would be:
P(¯) =
RT(¯)
v(¯) Bm(¯)
Am(¯)
v(¯)2 + 2v(¯)Bm(¯) Bm(¯)2
, (9)
where Am(¯) and Bm(¯) are the “resolved” mixture parameters calculated from a set of mixing rules
obtained from the literature.13
The partial molar enthalpy of species ↵, H,↵( ) is derived from the above
equation of state. The shear stress tensor is assumed to take the standard Newtonian form whereas the heat
and mass flux vectors are derived from nonequilibrium thermodynamics and fluctuation theory required
for high pressure dense fluids. Their forms contain terms proportional to temperature, pressure, and all
species mole fractions (ie. Dufour and Soret e↵ects as well as di↵erential and multicomponent di↵usion).
Additional details of the formulation can be found in Refs.9,14,15
However, the LES formulation above, and
the discussion to follow, apply equally to general choices of kinetics, equations of state, as well as constituitive
relations.
In the above, many LES terms are unclosed and may require modeling. These include the subgrid turbulent
stress Tij = ¯⇢( guiuj eui euj), the subgrid viscous term [⌧ij ⌧ij(¯)], the filtered chemical reaction source terms
fSe and gSY↵ , the subgrid pressure [P( ) P(¯)], the subgrid pressure dilatation [P( )uj P(¯) euj], the
subgrid heat flux [Qj( ) Qj(¯)], the subgrid convective energy [¯⇢ getuj ¯⇢eet euj], the subgrid viscous energy
[ gui⌧ij eui⌧ij(¯)], the subgrid molecular enthalpy flux [
PNs
↵=1 H,↵( )Jj,↵( )
PNs
↵=1 H,↵(¯)Jj,↵(¯)], the
subgrid scalar flux [¯⇢ gY↵uj ¯⇢fY↵ euj], and the subgrid mass flux vectors [Jj,↵( ) Jj,↵(¯)]. Of these unclosed
terms, the subgrid turbulent stresses and the filtered chemical source terms have received the most attention
in the literature. The subgrid stresses are most commonly modeled with either constant or dynamic versions
of either the Smagorinsky or the mixed similarity models.16–23
For transported FDF methods, the chemical
source term is in closed form and does not require any direct modeling.
The subgrid pressure, the subgrid molecular enthalpy flux, the subgrid molecular shear stress tensor, the
subgrid heat flux vector, and the subgrid mass flux vectors are typically neglected in standard LES. However,
the Clemson group,9
among others, have shown that the subgrid pressure, heat flux, and mass fluxes are
not necessarily negligible for reacting flows due to their non-linear forms and the fact that reacting flows
have larger scalar gradients than non-reacting flows in general. Nevertheless, these terms are not discussed
in what follows.
III. Turbulent Scalar Manifold Reduction Assumptions
The most fundamental assumptions in general LES approaches to turbulent combustion modeling are cen-
tered around the size of the (typically) large dimensional scalar manifold (of size Ns, usually on the order
of tens to hundreds) that contains the species, energy, and pressure dependent variables and whether or
not a smaller attracting manifold exists. Ideally, a turbulent combustion simulation would transport the
physical processes of convection, di↵usion, and reaction for all of the scalars (contained in the vector Y↵,
where 1  ↵  Ns) that characterize a given flow. In reality, this task is quite often untenable due to the
large computational requirements associated with such a level of description. It is then often necessary to
select a reduced set of “represented” scalars, or linear combination of scalars, (denoted by the vector Y r
I ,
where 1  I  M and M ⌧ Ns) to be transported instead while the remaining “unrepresented” scalars
(denoted by Y u
i , where 1  i  Ns M) are not transported, but rather calculated from a defined mapping
function that maps the unrepresented scalars in terms of the represented ones [i.e. Y u
i = Y m
i (Y r
I ) where Y m
i
is the mapping function]. These reduction arguments suggest that the large Ns-dimensional manifold lies
on, or at least “close” to, a much smaller M-dimensional manifold.
The reduced manifolds associated with each of the closure models discussed in this review vary quite substan-
tially in their size, complexity, and implementation. In general, LES-Flamelet models assume that the large
Ns-dimensional manifold can be described by a very low dimensional reduced manifold (typically 1  M  3)
derived from one dimensional non-premixed and/or premixed laminar flames at similar thermodynamic con-
ditions to the turbulent flame of interest and stored in a pre-constructed library that can be accessed during
the turbulent simulation to obtain the necessary chemical information. Examples of these very Low Dimen-
sional Manifolds (LDMs) generated from one dimensional flames include Flamelet Generated Manifolds24–27
4 of 20
American Institute of Aeronautics and Astronautics

5. (FGM), Flame Prolongation of ILDM28–30
(FPI), and Flamelet Progress Variable31–33
(FPV) techniques,
among others. A detailed review of each of these methods is not provided here as the major assumption
associated with all of these techniques is related to the assumed very small size of the reduced manifold, not
the generation of the reduced manifold itself.
Although LES-LEM and LES-FDF models commonly integrate manifold reduction techniques into their
formulations, they do not usually assume such strict limitations as those associated with LES-Flamelet
models. In fact, most LEM and FDF modeling approaches do not assume that the Ns-dimensional turbulent
scalar manifolds can be accessed from laminar flame libraries parameterized by only a few parameters during
an LES and therefore typically rely on either directly solving or tabulating the (sti↵) chemical kinetics for
most of the scalars during the simulation. In this context, LEM and FDF methods are therefore able to o↵er
a reduced description of the overall Ns-dimensional manifold that is computationally tractable but not so
restrictive as to limit the reduced manifold to sizes of 1 3. A recent detailed review of the majority of the
various reduced manifolds associated with turbulent combustion is contained in Ref.7
therefore very little
elaboration is provided further here.
It is, however, important to discuss the manifold reduction argument itself as it begs several questions
including the overall validity of reduced manifolds in general turbulent flames, the physical mechanisms
responsible of increasing or decreasing the size of reduced manifolds, and whether or not commonly used
parameters o↵er suitable projections onto reduced manifolds. Recently, Ref.7
provided an analytical look
at reduced manifolds by examining the conservation equation for the “departure vector” which represents
the di↵erence between the true value for the unrepresented quantities and the mapping function [i.e. yi =
Y u
i Y m
i (Y r
I )]. In order for the large Ns-dimensional scalar manifold to lie on a LDM, the departure vector
must be null for some definition of Y m
i (Y r
I ). The analysis in Ref.7
concluded that, due to the curvature
properties of Y m
i (Y r
I ) and the invariance properties of the reaction manifold, yi cannot be zero for the LDMs
associated with turbulent combustion. At best, the true Ns-dimensional scalar manifold lies close to a LDM.
At this point; however, nothing has been said regarding how close, or not, turbulent combustion lies to a
LDM. Furthermore, the reduced manifolds commonly associated with Flamelet, LEM, and FDF models are
usually associated with specific assumptions that may or may not be true in the turbulent flow field of
interest. Therefore, these a priori generated manifolds do not o↵er a direct measure of the departure vector
to a real turbulent flame therefore a posteriori methods are more appropriate.
To this end, several researchers have analyzed so-called Empirical Low Dimensional Manifolds (ELDM)
where high fidelity experimental or numerical datasets are post-processed using dimension reduction tech-
niques (most common are Principal Component Analysis [PCA] and Multivariate Adaptive Spline Regression
[MARS]).34–37
These ELDM’s are not limited by any physical assumptions (since they are only based on
regression techniques) and the resulting M parameters from these dimension reduction techniques make up
a strong attracting M-dimensional reduced manifold. It should also be noted that these PCA and MARS
generated manifolds have been shown to be stronger attracting than other common LDM’s used in turbulent
reacting flows, including flamelet-based manifolds.7
Figure 1 shows results from DNS of a high pressure
H2/Air shear flame10
that further elucidates the idea that common flamelet parameters are not always
strong attractors for LDM’s of turbulent combustion. The turbulent nature of this flame is depicted in Fig.
1a as pockets of extinction/re-ignition are evident. The corresponding scatter plot in Fig. 1b of tempera-
ture as a function of normalized scalar dissipation on the stoichiometric surface shows the low correlation
between commonly used flamelet parameters (mixture fraction and scalar dissipation) and the scalar field
(temperature in this case) within turbulent manifolds.
Because PCA and MARS are known to generate strong attracting M-dimensional reduced manifolds, these
ELDM analyses provide a well-founded approach for quantifying the departure vector for each of the scalars
lying on the larger Ns-dimensional manifold. Yang et al.35
conducted both PCA and MARS analyses on
DNS datasets of a non-premixed temporally evolving 11-species CO/H2 jet flame and a 22-species lifted
Ethylene jet flame. The results showed that, for the CO/H2 flame, in order for the scalars to maintain an
overall departure less than 5% from the original 12-dimensional manifold, 5 and 2 dimensions were required
for PCA and MARS, respectively. For the 23-dimensional Ethylene manifold, 9 and 7 dimensions were
required for PCA and MARS, respectively. It should also be noted that the chemical source terms (which
are unclosed in reduced manifold space and also require modeling) required even higher dimensions to obtain
 5% departure for both PCA and MARS analyses.
5 of 20
American Institute of Aeronautics and Astronautics

6. To put these results into perspective, DNS capable flames, even when projected onto LDM’s much stronger
attracting than those predicted by flamelet parameters, are not well described by 2 or 3 dimensions except in
“simple” configurations. Given the complex nature (highly turbulent, many species, extinction, re-ignition,
pollutant formation, etc.) of the vast majority of flames prevalent in propulsion applications, it is highly
unlikely that LDM’s (on the order of two or three dimensions) parameterized by flamelet variables will
be able to accurately reconstruct all of the pertinent physics within these flames. Therefore, the primary
fundamental physical assumption associated with all flamelet models is largely inconsistent with the physics
describing today’s propulsion applications. Further physical assumptions associated with flamelet models,
as well as LEM and FDF models are reviewed in detail below.
IV. LES-Flamelet
Turbulent combustion models based on flamelet concepts remain some of the most widely researched topics
in the field. The concept of the flamelet is that locally a one-dimensional laminar flame subject to appro-
priate initial and boundary conditions can describe a turbulent flame front. These fully-resolved solutions
to one-dimensional flames are stored in pre-constructed “flamelet libraries” usually generated from simple
canonical geometries (typically purely non-premixed counterflow opposed laminar jet flames and simplified
one-dimensional laminar premixed flames) and are parameterized by a reduced set of M-number of scalars.
The turbulent LES then only needs to transport the filtered M-number of parameters (and usually the
variance as well). Chemical information is then accessed from the library as needed, using a statistical
relationship (usually a beta PDF) between the stored instantaneous and transported filtered values. Peters
originated the flamelet concept under the argument of non-premixed flames being much thinner than the
Kolmogorov or other flow length scales.38
It has since been extended well beyond this flame regime and
in many modern applications it is assumed that nearly all turbulent regimes (non-premixed, premixed, and
partially premixed) can be modeled from flamelet libraries during the LES. Put in terms of the manifold
perspective, the flamelet library solutions make up Y m
i (Y r
I ); the manifold of unsolved scalars parameterized
by the M-number of solved scalars (usually mixture fraction, progress variable[s], and dissipation[s]; typically
two to three at most).
The flamelet concept has been extensively studied throughout the literature for purely premixed24–26,39–44
and purely non-premixed27,32,33,45–47
conditions; and recently it has been extended to regime-independent
flames.30,48–50
Regime-independent flamelet-based models, which are required to simulate modern propul-
sive applications, are similar to purely premixed and non-premixed flamelet models in that they typically
parameterize the flamelet library by the same set of solved scalars (mixture fraction, progress variable, etc.).
However, the generation of the regime-independent flamelet manifold is usually done by either solving the
multi-dimensional flamelet equations in reduced parameter space such that a range of laminar flames from
the purely non-premixed limit to the purely premixed limit are stored,49
or solving the purely non-premixed
and purely premixed one-dimensional flame equations separately and then blending the solutions based on
a “regime identifier” that measures the degree to which a local state is non-premixed, premixed, or partially
premixed.48,50
When considering regime-independent flamelet models, the first task carrying physical implications is the
specific choice of the M parameters that make up the reduced manifold as these parameters will a) makeup
the basis that spans the Ns-dimensional scalar space and b) be transported (in filtered form) during the LES
with appropriate transport equations that must be derived (and closed). The basis of M-parameters must
constitute a well-defined generalized coordinate system for which the Ns scalar equations can be transformed
into from physical space.48
The vast majority of regime-independent LES-Flamelet models employ mixture
fraction and progress variable(s) as the selected M-parameters. For purely non-premixed combustion, the
mixture fraction, , can be viewed as a passive scalar (ranging in value from 0 to 1 in pure oxidizer and
pure fuel, respectively) making up a “flame-attached” coordinate system with one direction normal to iso-
surfaces and two directions tangential to the iso- surface.38
This three dimensional mixture fraction
space constitutes a well-defined coordinate system; however, the vast majority of LES-Flamelet models only
consider the single flame normal coordinate in the transformed Ns scalar equations. This is the result of
the “thin-flame” argument suggesting gradients normal to iso- surfaces are much greater than those in
the tangential directions. Gradients in the tangential directions have been shown to be non-negligible in
recirculation regions and flames with substantial curvature and di↵erential di↵usion e↵ects.51
Su ce it
6 of 20
American Institute of Aeronautics and Astronautics

7. to say, many of the flames associated with propulsive applications are not thin and are known to exhibit
substantial amounts of recirculation, curvature, and di↵erential di↵usion e↵ects.
For purely premixed flamelet combustion, progress variables, usually made up of a single or linear combination
of the Ns-scalar(s), are selected as the dimension reduction parameters. Unlike the mixture fraction, progress
variables are not passive scalars and they do not o↵er a description of the fuel/oxidizer mixed state but rather
the overall burnt state. This description leaves the exact definition of progress variables somewhat open-
ended and this is reflected by the many di↵erent progress variable definitions that have been used throughout
the literature. Generally speaking, progress variables should vary monotonically from fresh to burnt gases
such that species concentrations remain single-valued functions of the progress variable. It is also desirable
to minimize the non-linear nature of the progress variable from fresh to burnt gases as this can cause large
interpolation errors within the flamelet library. A recent study from Niu et al.52
showed that commonly used
progress variables do not always satisfy one or both of these constraints, especially when complex chemical
kinetics mechanisms are used (e.g., hydrocarbon-based mechanisms).
The shortcomings associated with the mixture fraction and progress variables within complex turbulent
flames are central to the idea that variables based on laminar flame arguments do not always o↵er strong
attractors for reduced descriptions of turbulent manifolds. Furthermore, for regime-independent flamelet
models, Knudsen et al.48
showed that the statistical dependence between mixture fraction and progress
variables is such that they do not form a well-defined coordinate system for all combustion regimes thereby
calling into question the validity of some of the solutions generated from flamelet equations solved in mixture
fraction and progress variable spaces. The multi-regime approach of Knudsen et al.50
alleviates the statistical
dependence issue by formulating a mixture fraction and statistically independent “progress parameter” based
coordinate system. However, their partially premixed solutions obtained from linear combinations of non-
premixed and premixed flamelet models via a “regime identifier” are somewhat ad hoc as there are no
physically based arguments that suggest partially premixed behavior is simply a linear combination of non-
premixed and premixed behavior.
It should be noted that while the majority of flamelet models tend to build laminar flamelet libraries through
solving the flamelet equations in reduced variable space, e↵orts have been made to incorporate flamelet
libraries from physical space solutions to certain laminar flames. One recent study further explored this idea
in H2/O2 counter-flow di↵usion flames by solving (in physical space) a transport system including detailed
chemistry, a real gas state equation, real property models, and a conserved passive scalar.53
Using this
technique, the physics associated with finite rate kinetics, real gas e↵ects, fluid strain rate, and di↵erential
di↵usion are able to be captured naturally. Furthermore, since the reduced manifold variable of interest
evolves with this system, a direct mapping function that implicitly includes all of the aforementioned physics
also emerges. Although the particular study cited here was applied only to non-premixed combustion, the
idea could theoretically be extended to multi-regime environments where mixture fractions and progress
variables are transported in physical space amongst various non-premixed, premixed, and partially premixed
laminar flames.
In summary, a wide array of techniques are available for generating the M-dimensional reduced manifold
and corresponding mapping function Y m
i (Y r
I ) from flamelet arguments. However, regardless of the imple-
mentation method, the very restrictive assumption that the large Ns-dimensional manifold can be mapped
using only two or three parameters remains. Furthermore, in an actual LES, the instantaneous quantities
stored in the flamelet library need be combined with a PDF in order to obtain their filtered values. Several
PDF’s are used in practice including Delta functions,32
Beta functions,42,54
and Statistically Most-Likely
Distribution (SMLD) functions;55
each of which carry further physical assumptions and restrictions regard-
ing the description of the unrepresented quantities in the actual turbulent flow field. However, the fact that
the majority of propulsive flow fields do not likey lie only on very low dimensional manifolds remains, in
these authors’ views, the dominant physical assumption that restricts the LES-Flamelet approach’s use for
general multi-regime predictive propulsion applications.
V. LES-LEM
LES-LEM56
is based on the assumption that all length and time scales must be fully resolved. In its most
rigorous definition this assumption would require that a full DNS be performed, including the resolution of
7 of 20
American Institute of Aeronautics and Astronautics

8. all chemical processes. Because DNS of the applications considered here are computationally intractable,
the LEM approach assumes that the small scale scalar features below the LES filter width can be resolved
using a one-dimensional (1D) domain at each finite di↵erence grid point, or within each finite volume cell,
aligned with the mean scalar gradient. The fourth-dimension to the general LES solver has a 1D length at
each grid point that ranges from the Kolmogorov (or Batchelor) scale to the length scale of the filter width.
Molecular di↵usion and chemical source terms are fully resolved on the 1D domain, whereas turbulent stirring
is accounted for via Kerstein’s stochastic triplet mapping.57–59
The LES-LEM approach first departs from the rigorous LES approach by returning to the instantaneous
equations for the scalar mass fractions and energy rather than the filtered forms.56,60–64
In the LES-LEM
approach the instantaneous velocity is first decomposed in the traditional LES sense as ui = eui + u00
i where,
as usual, the contributions are from the resolved (LES mesh) velocity and the subgrid scales, respectively.
At this point in the LES-LEM approach, the transport equations for the instantaneous (not filtered) mass
fractions are then decomposed as:
⇢
@Y↵
@t
+ ⇢ euj
@Y↵
@xj
= 0, (10)
⇢
@Y↵
@t
+ ⇢u00
j
@Y↵
@xj
=
@Jj,↵
@xj
+ SY↵
, (11)
where the first equation is meant to represent the large (LES) scale advection and the second equation
represents the instantaneous subgrid scale processes. A similar approach is employed for the energy equa-
tion. So, LES-LEM involves the solution of a set of modeled transport equations for the filtered continuity,
momentum and energy equations coupled with a set of modeled transport equations for the instantaneous
scalar mass fractions and temperature. Modeling the scalar transport begins with with the 1D assumption
which replaces Eq. (11) with:
⇢
@Y↵
@t
+ Fstir =
@J↵
@s
+ SY↵ , (12)
where s is the 1D subgrid refined mesh coordinate and Fstir represents Kerstein’s triplet mapping used to
model the e↵ects of subgrid turbulent stirring.61
Several assumptions are made at this point. First, the resolved scalar fields only “see” advection, therefore
all molecular di↵usion occurs at the small subgrid scales (note that this is counter to most LES in which
the subgrid molecular di↵usion is commonly neglected). Second, the resolved field advection is only via
the filtered velocity even though the scalar is an instantaneous quantity. It is to be noted that in some
approaches62
the e↵ects of the velocity fluctuations are modeled by decomposing the velocity alternatively
as ui = eui + (u00
i )R
+ (u00
i )S
, where the fluctuation has been decomposed into an “LES resolved subgrid
fluctuation” (superscript R) and an “unresolved subgrid fluctuation” (superscript S). In this case the resolved
subgrid fluctuation can be modeled using a transported subgrid kinetic energy: (u00
i )R
=
q
2
3 ksgs. However,
this is somewhat questionable since the subgrid kinetic energy is positive definite whereas velocity fluctuations
are by definition both positive and negative. The next assumption in the LES-LEM approach is that the
3 dimensional “subgrid” scalar transport, di↵usion, and reaction described by Eq. (11) can be accurately
resolved in a purely one dimensional (1D) domain located at each finite di↵erence grid point or within each
finite volume cell. The assumption is justified by assuming that this 1D domain resides along a notional
direction aligned “in the flame normal or the maximum scalar gradient direction and thus, does not represent
any physical Cartesian direction.”56
Implicit in this argument is that there is only a single scalar gradient
direction; ie. all scalars are aligned in the same direction. This will be returned to below.
Other assumptions implicit to the LES-LEM approach reside within the triplet mapping procedure. The
basic argument is that the subgrid turbulent stirring can be modeled by re-mapping the 1D scalar field
according to a set of mapping rules which conform to a k 5/3
spectrum. Many assumptions exist and details
can be found in Refs.56,57,61
among others. However, for present purposes the primary question is whether
or not the inertial k 5/3
scaling is relevant to the di↵usive sub-inertial scales resolved by the LEM. These
scales make up a large part of the kinetic energy typically residing within the subgrid in “academic” LES;
however, this may not be the case for industrial or other propulsion applications and their associated much
higher Reynolds numbers.
The next assumption of the model involves the treatment of the resolved field scalar advection on the
LES mesh. Rather than treating this in the traditional Eulerian approach, in LES-LEM the advection is
8 of 20
American Institute of Aeronautics and Astronautics

9. treated by a Lagrangian “splicing” procedure.56
In practice, portions of each 1D subgrid domain are moved
“downwind” into adjacent cells based on a set of mass conserving rules. Again, the details of the approach
are not considered here. However, it is noted that the approach does have a directional bias; ie. di↵erent
results can be obtained depending upon the order in which the three directions are treated. In practice, the
largest mass flux direction is treated first, followed by the second, then the third.
Other issues implicit to the LES-LEM approach include the following. Boundary conditions for the 1D
subgrid domains are not clear for the small scales. Typically, Neumann boundary conditions are employed
without physical justification. The pressure within each computational cell is also be assumed to be uni-
form (thereby neglecting any possible subgrid pressure e↵ects). Small scale isotropy is assumed which is
questionable particularly for high speed flows (although this assumption is common to many LES models).
Another issue is that two sets of energy equations are required. One is the typical filtered energy equation
necessary for the resolved field equations (continuity, momentum, energy, plus constitutive equations and an
equation of state). The second is the instantaneous energy equation decomposed similarly to Eqs. (10) and
(12) needed for the chemical source terms. These two sets of equations are not mathematically consistent
and the e↵ects of this on the transported energy are not clear.
Finally, two LES-LEM physical assumptions are tested using the Clemson database. Figure 2 presents
example results showing that assuming that molecular di↵usion resides only within the subgrid 1D scales is
highly questionable, at least at academic scale LES. The figure shows PDFs of the ratio of the resolved field
mass flux vector magnitudes to the exact filtered mass flux vector magnitudes for the H2/O2 shear flame
at P0 = 100atm and Re0 = 4, 500 with a filter width of 4.7 times that of the initial vorticity thickness and
conditioned on the stoichiometric surface. Although not a strict interpretation of a subgrid variable, we can
decompose the filtered mass flux vector as: Jj,↵( ) = Jj,↵(¯) + Jsgs
j,↵ . That is, the exact filtered mass flux
vector calculated by the set of scalars is equal to the resolved field mass flux vector (i.e. that calculated
from the set of filtered scalars ¯ available within an actual LES), plus the di↵erence defined as the subgrid
mass flux vector. The data in the figure was originally obtained for the purpose of determining whether
or not subgrid molecular di↵usion is significant and may need modeling.10
By substitution the plots are
PDFs of |Jj,↵( ) Jsgs
j,↵ |/|Jj,↵( )|. Therefore, if the subgrid portion of the mass flux vector is completely
negligible the PDF would be a delta function located at unity; which is essentially the case in the figure.
Therefore, Fig. 2 shows that resolved field molecular di↵usion is dominant in this flame at least along the
stoichiometric surface, which is inconsistent with the physical assumptions of LES-LEM. Additional PDFs
of the same ratio but for di↵erent filter widths, Reynolds numbers, and conditioning regions may be found
in Refs.9,10
The results show that resolved field di↵usion is substantial is all cases. Again, this is at least
true for academic Reynolds number flames. However, the assumption of negligible resolved field di↵usion
may well be much better for higher industrial scale Reynolds numbers (except near walls, laminar-turbulent
boundaries, and other potential low Reynolds number regions).
The second assumption tested is that the entire subgrid scale scalar field evolutions represented by Eq.
(12) can be resolved on a 1D domain residing along the direction of the maximum scalar gradient. Again,
representative example results shown in Fig. 3 illustrate that there is no single direction along which all
scalar gradients reside even in a nonpremixed Kerosene/Air parallel shear flame from Clemson’s database at
P0 = 35atm and Re0 = 2, 500. There are many scalars present including all of the species mass fractions and
the temperature. So, there is no single particular scalar gradient to assume the 1D domain resides along.
However, if correct, the assumption would have all scalars along the same gradient, therefore, any one could
be used as the representative test direction. In the case of Fig. 3 the mixture fraction gradient is used as the
reference direction. The figure contains scatter plots of the cosine of the angle between the temperature and
the O2 mass fraction gradients with respect to the mixture fraction gradient. If the LES-LEM directional
assumption holds true then only values of -1 and +1 would be expected. However, as the figure shows there
is no one single direction along which all of the scalar lie even for this nonpremixed flame.
Finally, despite the numerous (questionable) assumptions implicit to LES-LEM, the model has produced
good results for a variety of conditions, including nonpremixed and premixed flames, sooting flames, two-
phase flames, as well as flames involving extinction and reignition.56
Despite its successes, there are numerous
fundamental physical assumptions implicit to the approach, many of which are not consistent with combus-
tion regimes associated with common propulsion applications. These make the approach di cult to justify
rigorously.
9 of 20
American Institute of Aeronautics and Astronautics

10. Figure 2. The PDF of the ratio of the mass flux vector magnitudes as a function of the resolved (filtered)
scalar variables that can be calculated within an actual LES to the exact filtered mass flux vector magnitudes
obtained from Clemson’s DNS of the H2/O2 shear flame at P0 = 100atm and Re0 = 4, 500. The ratio of the filter
width to the initial vorticity thickness is 4.7 (spherical top hat with diameter equal to 22 times the streamwise
DNS grid spacing) and the data are conditioned on the stoichiometric surface.
(a) (b)
Figure 3. Scatter of the instantaneous angles between the gradient of the mixture fraction and the gradient of
the (a) temperature and (b) O2 mass fraction from Clemson’s Kerosene/Air DNS at P0 = 35atm and Re0 = 2, 500.
10 of 20
American Institute of Aeronautics and Astronautics

11. VI. LES-FDF
LES-FDF methods share the same lack of implicit assumption as LES-LEM that turbulent combustion lies
on a LDM. In contrast to the LES-Flamelet and the LES-LEM approach, the LES-FDF methods solve a
modeled transport equation for the joint FDF [or filtered mass density function (FMDF) for variable density
flows] typically of at least the (represented) scalars needed to close the chemical source terms. These chemical
source terms are highly nonlinear and computationally sti↵. Various approaches have been implemented to
speed the computations including reduced manifold techniques, flamelet libraries, to the In Situ Adaptive
Tabulation (ISAT) method,65,66
which dynamically generates a similar manifold library which can be used
for interpolation purposes later in the simulation. This method has been used in several FDF simulations and
dramatically decreases the computation time.67–69
Other methods of manifold reductions were referenced in
Section III. However, these types of approaches to dealing with the chemical source terms are not specific to
LES-FDF approaches and will not be discussed further in what follows.
Directly related to LES-FDF approaches though is in choosing the proper set of variables to be transported.
This FDF or FMDF may include the scalars only (S-FMDF)70
to close the chemistry, the velocity only
(V-FMDF) that can be used to close terms in the LES momentum equation,71
the scalars plus the velocity
components (VS-FMDF) that can additionally be used to close more terms including the subgrid convection,
subgrid pressure-work, and subgrid pressure in the energy equation,72,73
the velocity plus the scalars, plus
a subgrid turbulence frequency (FVS-FMDF),74
or the most comprehensive form which solves a transport
equation for the joint statistics of the energy, pressure, velocity, and scalars (EPVS-FMDF)75
needed for
high Mach number and pressure dependent chemistry.
The FDF or FMDF o↵ers a unique form of combustion closure in that knowing the joint FMDF of the
scalars the chemical source terms appear in closed form and no direct modeling assumptions are required for
these terms. The modeling of unclosed terms is thus shifted from the chemical kinetics to other terms; the
majority of which are chosen to reduce to commonly used models in traditional LES as described below. As
such, the modeling of these terms is essentially the same as in any form of LES and therefore modeling errors
associated with these terms are not specific to the FMDF approach. However, there remain three primary
issues relating specifically to the FMDF approach. The first involves accurately modeling the conditional
filtered molecular di↵usion (or the conditional filtered dissipation although the majority of available models
are for the former). The second involves the high dimensionality of the FMDF transport equations and the
corresponding choice in solution procedures. Both of these are discussed in detail below following a review of
the basic LES-FMDF approach. An additional issue associated with the model is whether or not to solve both
the FMDF and the Eulerian LES transport equations, or only the FMDF equations. In the former approach
the FMDF is only used to close specific terms in the traditional LES equations. In the latter approach the
FMDF provides the resolved field evolution directly. The approach discussed hereinafter is to solve both sets
of equations. This both minimizes errors in the FMDF modeling and provides a self-consistency check (ie.
by ensuring that the two sets of equations produce the same resolved field evolution).
Now we present the basic FMDF by examining the (relatively simple) S-FMDF transport equation and
its modeling. The statistical scalar field ↵ is characterized by a filtered mass density function (FMDF),
denoted by FL,
FL( ; x, t) =
Z +1
1
⇢(x0
, t)⇣[ , (x0
, t)]G(x0
x)dx0
, (13)
⇣[ , (x0
, t)] = [ (x0
, t)] =
Y
↵=1
[ ↵ ↵(x, t)], (14)
where is the delta function, the composition domain of the scalar array (which are enthalpy and all mass
fractions here). The term ⇣[ , (x0
, t)] is the “fine-grained” density, and Eq. (13) implies that the FMDF is
the mass-weighted spatially filtered value of the fine grained density. The integral property of the FMDF is
such that: Z +1
1
FL( ; x, t)d =
Z +1
1
⇢(x0
, t)G(x0
x)dx0
= ¯⇢(x, t). (15)
11 of 20
American Institute of Aeronautics and Astronautics

12. For further development, the mass-weighted conditional filtered mean of the variable Q(x, t) is defined as:
gQ(x, t)| =
R +1
1
⇢(x0
, t)Q(x0
, t)⇣[ , (x0
, t)]G(x0
x)dx0
FL( ; x, t)
. (16)
From its properties, it follows that the filtered value of any function of the scalar variables is obtained by
integration over the composition space,
gQ(x, t) =
Z +1
1
⇢(x0
, t)Q(x0
, t)FL( , x, t)]d . (17)
As such, any LES term that is only a function of the variables within the FMDF can be directly calculated;
including the chemical source terms. Finally, following the procedures in76
and,70
the exact transport
equation for the S-FMDF is:
@FL( ; x, t)
@t
+
@[ gui| ]FL( ; x, t)
@xi
=
@
@ ↵
[<
1
¯⇢
@Ji,↵
@xi
| > FL( , x, t)]
@[S↵( )FL( , x, t)/⇢]
@ ↵
, (18)
where the brackets within the first term on the right hand side indicate the standard filter operator for
convenience. It is important to note that the above equation is exact, albeit unclosed. Also, due to the high
dimensions of the FMDF transport equation a direct solution procedure is not computationally feasible.
Even if it were, correct boundary conditions are unclear. Therefore, an alternative solution procedure is
necessary. By far the vast majority of LES-FDF is solved using a set of stochastically equivalent “notional”
particles in a Monte Carlo sense. However, an alternative Eulerian approach exists and has recently seen
use in solving the LES-FDF equations based on the Direct Quadrature Method of Moments (DQMOM)
which may have advantages over the particle approach for high speed flows involving shocks.77–81
However,
as the vast majority of LES-FDF is based on the particle approach only it will be discussed hereinafter.
The notional particle approach is based on an ensemble of Monte Carlo particles to represent the FMDF.
It is noted that the particles are not Lagrangian fluid particles and therefore only represent the statistics
of certain quantities. Note that once a set of models is chosen and incorporated into the original FMDF,
the particles form a set of equivalent stochastic equations and transport the identical FMDF as the modeled
Eulerian set of equations in the limit of large numbers of particles.5,77,82,83
The last term on the right hand side of the S-FMDF [Eq. (18)] is due to chemical reaction and is in a closed
form. This is the primary advantage of the LES-FDF approach. The unclosed nature of SGS convection and
mixing is indicated by the conditional filtered values. The convection term is typically modeled in a manner
consistent with conventional LES by decomposing the term as:
gui| FL( ; x, t) = euiFL + [ gui| eui]FL, (19)
where the second term on the right hand side denotes the influence of SGS convective flux. This term is
typically modeled as:
[ gui| eui]FL = µt
@(FL/¯⇢)
@xi
. (20)
The first Favre moments corresponding to Eqs. (19) and (20) are:
gui ↵ = eui
f↵ + ( gui ↵ eui
f↵), (21)
¯⇢[ gui ↵ eui
f↵] = Dt
@f↵
@xi
, (22)
where Dt = µt
Sct
is the subgrid molecular di↵usivity, Sct is typically assumed constant and equal to 0.7, and
unity Lewis number and simple Fickian di↵usion have been assumed. Therefore, when integrated to recoup
the standard corresponding Eulerian LES transport equation, the above model is the standard gradient
di↵usion hypothesis. Other models can be used; however, it is important that they be consistent with the
choice of models used for the actual Eulerian LES transport equations for self-consistency.
The only term left unclosed, and unique to the FDF approach, is the conditional filtered di↵usion term,
@
@ ↵
[< 1
¯⇢
@J↵
@xi
)| > FL]. This is the typically considered to be the most challenging term to model is
12 of 20
American Institute of Aeronautics and Astronautics

13. a physically consistent manner within the LES-FDF approach. The primary reason for this is that the
conditional filtered di↵usion is not only local in physical space but it is also local in composition space due
to the gradient in ↵ and the conditional filtering.
Physically, the conditional di↵usion defines how the stochastic particles map the molecular mixing process
into composition space and physical space. Statistically, a successful mixing model should 1) preserve
scalar means, 2) correctly describe the scalar variance decay rates, 3) preserve realizibility, and 4) mix with
“nearby” quantities (i.e. interactions local in both physical space and in composition space). There are also
other criteria that can be prescribed, but these four are the main requirements.
To illustrate the LES-FDF approach, we begin with the simplest and most widely used conditional filtered
di↵usion model, the Interaction and Exchange with the Mean model (IEM). The IEM, also known as the
Least Mean Square Error (LMSE) model, was first proposed by Dopazo and O. Brien.84
The IEM divides
the conditional di↵usion into two processes; one in physical space as a random walk (modeled as a gradient
di↵usion) and a “mean drift” towards their local filtered value:
@
@ ↵
[<
1
¯⇢
@
@xi
(D
@ ↵
@xi
)| > FL] =
@
@xi
(
D
¯⇢
@FL
@xi
) +
@
@ ↵
[⌦m( ↵
f↵)FL], (23)
where ⌦m is the “subgrid mixing frequency” which is not known a priori. This frequency is typically modeled
as ⌦m = C⌦(D + Dt)/(¯⇢ 2
G) where C⌦ is a modeling constant. It is noted that in its standard form simple
Fickian and Fourier mass and thermal di↵usion are assumed with all species having the same di↵usivities:
i.e. Jj,↵ = ⇢D@ ↵/@xj. In the above, Dt is the turbulence scalar di↵usivity, typically modeled as being
proportional to the momentum turbulence viscosity.
While relatively accurate for some flows, the IEM has significant deficiencies. First, it presumes a Gaussian
distribution which violates the physics of bounded scalars,85
second, it can generate an unphysical spurious
scalar variance for pure scalar mixing in which an exponential decay of the scalar variance is expected,76
and,
in addition, its treatment of the composition space interaction is not entirely correct since the interactions
are only with particles physically local to one another. Furthermore, most of its development is based on
scalar mixing despite the fact that the form of the conditional filtered di↵usion can be quite di↵erent for
reacting scalars (more on this below).86
Several improvements to the original IEM model have been suggested in the literature in order to over-
come several of shortcomings. The IEM, while relatively simple, fairs poorly even for some relatively simple
dispersion problems because it does not adhere to the dispersion-consistency condition.87
An alternative
Interaction by Exchange with the Conditional Mean (IECM) model was introduced to overcome this defi-
ciency.88,89
However, the model remains local only in physical space and not in composition space. The IEM
plus mean drift model was developed to address the spurious variance behavior introduced by the random
walk modeling:76,90
@
@ ↵
[<
1
¯⇢
@Ji,↵
@xi
| > FL] =
@
@
[FL
1
¯⇢
@
@xi
( gDi,↵
@f↵
@xi
)]
@
@ ↵
[⌦m( ↵
f↵)FL] (24)
It also allows for scalars to have di↵erent di↵usivities. Other variations include the fractal IEM developed
by Shetty.85
More recently, Meyer and Jenny91
proposed the parametrized scalar profile (PSP) model,
which is quite similar to the IEM formulation but uses one dimensional scalar profiles to construct the
three-dimensional scalar field.91
To illustrate the LES-FDF approach for the S-FMDF with the IEM plus mean drift model, the corresponding
stochastic di↵erential equation (SDE) governing the notional particles are:
dX⇤
i (t) = Di(X(t), t)dt + E(X(t), t)dWi(t), (25)
d ⇤
↵(t)
dt
= ⇥↵ + ˆS↵( ⇤
), (26)
d ⇤
h(t)
dt
= ⇥h + ✏ + ˆSh, (27)
where X⇤
i is the Lagrangian particle location, ⇤
↵ = ↵(X(t), t) is the scalar value of the particle at that
location, f↵ is evaluated by interpolation, the “drift” coe cient is Di = ( eui + 1
¯⇢
@ t
@xi
)⇤
, ⇥↵ is the conditional
13 of 20
American Institute of Aeronautics and Astronautics

14. di↵usion model (e.g., in standard IEM ⇥↵ = ⌦⇤
m ( ⇤
↵
f↵
⇤
)), E = (
p
2 t/⇢)⇤
, and the Wiener process is
dWi = ( t)1/2
⇠i, where ⇠i is modeled by a standard Gaussian PDF. The enthalpy equation can be modeled
to match the form of the mass fraction equations, or it can include additional terms, ✏. As mentioned above,
the chemical source terms, ˆS↵( ⇤
), are directly integrated based on local particle values, ⇤
. These particles
traverse the flow field, each transporting their own values of species mass fractions, and enthalpy. They
individually undergo chemical reactions and interact within computational volumes. In its simplest form,
the mean value of all of the particles’ scalars being transported within a cell corresponds to the Eulerian
filtered value at that cell. Therefore, the filtered chemical source terms needed for the Eulerian LES closure
are simply local averages over particle reaction rates. Details on convergence rates with numbers of particles,
required numbers of particles, and other attributes of particle approaches are not covered in this review but
can be found in Refs.5,82,83
among others.
As noted above, the IEM, while popular, retains significant fundamental assumptions not entirely consistent
with either mixing or reacting scalar di↵usion. Other models do exist that are also fairly widely used, some
of which alleviate these fundamental physical assumptions. Historically, Curl’s model92
is the first pairwise
mixing model. Particles are stochastically selected in pairs and exchange values. The model was improved by
Pope93
by allowing the scalar PDF to approach a Gaussian form in homogeneous turbulent flow. Nooren94
extended Curl’s model to include non-uniform particle weights. However, as with the IEM, the model is only
local in physical space and not in composition space.
The Mapping Closure (MC) model was developed by Pope95
which maps the composition space with a
Gaussian reference field. Both the modified Curl’s and the MC models satisfy the preserved mean, localness,
and scalar variance decay. The original MC method requires that particles be sorted in increasing order,
which is intuitive if composition space is only 1 D. However, if the composition space has higher dimensions,
which is the case in reacting flow, creating a unique mapping function is questionable. Therefore, the Multiple
Mapping Conditioning (MMC) model was developed by Klimenko and Pope96
which is a multidimensional
extension of MC. Nevertheless, each of these models thus far restrict particle interactions to be local in only
physical space, but not in composition space as required by the physics of the conditional filtered di↵usion.
In order to overcome this localness issue, Subramaniam and Pope97
developed the Euclidean Minimum
Spanning Tree (EMST) model in which particles mix with connected particles. The model is local in
both physical and composition space. The connections among particles are made based on minimizing the
Euclidean distance in each ensemble domain. An “interaction matrix,” Mij, is iterated in time along with
the particles such that a physical variance decay rate is preserved. To prevent the “stranding” problem,
each particle has one additional property, age, which determines whether the particle is undergoing mixing
or not. Besides the four criteria mentioned before, the EMST also ensures for mixing scalars: 1) linearity in
which the particle composition evolution equation should transform unchanged when the scalars are subject
to an arbitrary linear transformation, 2) independence in which any scalar is not influenced by other scalars,
and 3) the scalar FDF carried by particles relaxes towards a Gaussian distribution. Richardson and Chen98
introduced a modified version of the EMST to adopt di↵erent decay coe cients for di↵erential di↵usion.
While the EMST addresses the issue of localness in physical and composition space it still retains problematic
issues, including; its convergence with the number of particles (see below) is not clear, it violates linearity and
independence for conserved passive scalars, it does not guarantee Gaussian statistics in all limits where they
are expected, and it can strand particles in composition space.87
In addition, despite its more physically
justifiable modeling, the EMST does not always out perform the IEM and other models. For example,
Krisman et al.99
found in a comparison of mixing models for LES-FDF of several large DNS flames that
only the EMST correctly predicted reignition at high Damkohler numbers. However, at low Damkohler
numbers, the EMST over predicts reignition while the IEM and MC under predict it. In addition, if the IEM
and MC modeling constants are re-defined based on the DNS mixing frequency, both models were found to
be able to predict the reignition. While achieving reignition in these models is desirable, having to “tweak”
modeling constants to obtain this objective is not indicative of any truly predictive turbulent combustion
closure.
The latest model for the conditional filtered di↵usion is Pope’s Shadow-Position Mixing Model (SPMM).87
In
this model Pope addresses the primary deficiencies of previous mixing models. Rather than having particles
relax to their local spatial filtered value as in the IEM model, in the SPMM method a new variable is
introduced called the “shadow position.” In this model the particles relax to their shadow position rather
14 of 20
American Institute of Aeronautics and Astronautics

15. than to their spatial filtered value. The model was only recently introduced and therefore only limited testing
has been done. However, preliminary results are highly promising and the model appears to satisfy all of
the mathematical constraints placed on models for the conditional filtered di↵usion.87
In summary of the LES-FDF approach, hydrodynamic closure in incompressible, non-reacting flows has
been successfully achieved via the velocity-FDF (V-FDF).71
Peyman Givi and collaborators conducted the
first LES of a hydrocarbon flame, the Sandia-Darmstadt piloted di↵usion flame,100
in 1998 via both S-
FMDF101
and VS-FMDF.102
They have also been successful in predicting the more complex field of the
blu↵-body Sandia-Sydney flame103,104
and a premixed Bunsen burner105
as reported in Ref.106
and Ref.,107
respectively. Some of the important contributions in FDF by others are in its basic implementation,76,108–120
fine-tuning of its sub-closures121–123
and its validation via laboratory experiments.111,124–127
For real world
applications, the LES-FDF method has been extended for use on unstructured meshes.128,129
A hybrid and
highly scalable version of the model has also been introduced which is scalable to thousands of processors.130
Finally, LES-FDF is finding its way into industry, into commercial and government codes (eg. ANSYS-Fluent
and NASA’s VULCAN), and has received broad coverage in several text- and hand-books.77,109,131–134
See
Refs.5,135
for recent reviews of the state of progress in LES via FDF.
VII. Future Challenges of LES Combustion Modeling
Despite substantial progress in each of the LES-Flamelet, LES-LEM, and LES-FDF approaches, a truly
predictive, universal, multi-regime, multi-application turbulent combustion model remains elusive. Future
challenges remain for all three modeling approaches. Flamelet based models remain constrained by the
limitations of the manifold reduction assumption. In addition, by definition such a universal combustion
closure must be applicable outside of the so-called “flamelet regime” in which the model implementation
was originally justified. For the LES-LEM model, although molecular di↵usion is treated directly along
some direction within the flow field, the strong multidimensional nature of most propulsion application
relevant flames and ad hoc description of subgrid scale turbulent convection via the triplet mapping procedure
correspond to physical limitations of the LES-LEM model that are not always easily justified. Finally, for
the LES-FDF, while convection can be treated directly, the di culties in modeling the conditional filtered
di↵usion, or the conditional filtered dissipation, in reacting flows remain challenging. Most models for the
conditional filtered di↵usion or dissipation are based on purely mixing flows, which is seen by some as
decoupling the di↵usion and the chemistry (as the behavior of the conditional filtered di↵usion can be quite
di↵erent in reacting flows in comparison to pure mixing86
). In reacting flow, criteria such as the preservation
of scalar means and bounds may or may not directly apply (although they certainly need to be recovered
in the limit of no chemistry). In the majority view, the conditional di↵usion and chemical source terms
are indirectly coupled as molecular di↵usion “smooths” towards the scalar means, etc. while reactions alter
means, gradients, as well as implicitly the form of the conditional filtered di↵usion. However, some attempts
have been made to include direct coupling by means such as combining the conditional filtered di↵usion
with the chemical source term for modeling purposes but little progress has been made.5
This remains
somewhat of an unresolved issue within the community. Nevertheless, the limitations of the existing mixing
models are well-recognized.77,136–141
However, as discussed above modeling the conditional di↵usion has
been experiencing rapid improvement in recent years making the future of LES-FDF promising.
Another issue requiring future address involves the “DNS limit” of turbulent combustion closures. While
perhaps primarily of interest to the academic community, it is certainly desirable that a LES closure converge
to the solution of the “exact” underlying governing equations in the limit as the filter width approaches zero.
While true of many hydrodynamic LES closures, this is not generally the case with any of the three LES
turbulent combustion closures discussed in this work. If the DNS limit is defined as the direct resolution of
all length and time scales for the full complete chemical manifold then any model that makes use of reduced
manifolds will not, by definition, converge to the DNS limit. As discussed above, all three approaches
generally make some level of manifold reduction assumption.
In LES-Flamelet models, the LES transport equations for the filtered mixture fraction and/or progress
variables may converge to their exact equations; however, the combustion will always be dictated by the
flamelet library (not by the solution of the detailed coupled chemistry). The LES-Flamelet approach is always
bound by a very strict assumption of a very low dimensional manifold reduction. In contrast, LES-LEM and
LES-FDF are not technically bound by any manifold reduction assumptions, despite the fact they are often
15 of 20
American Institute of Aeronautics and Astronautics

16. used for computational convenience. Despite this, the limiting behavior for LES-LEM is poorly defined. As
the filter width shrinks so does the 1D domain upon which all of the chemistry and di↵usion occur. In the
limit, the 1D domain, being defined as the length of the filter width, becomes a point. Such a point could
retain the chemistry but all molecular di↵usion of species would vanish. Furthermore, the fate of the large
scale advection is not well defined since the nonlinear advection term is treated by a Lagrangian splicing of
1D domains. This stochastic splicing procedure for 1D domains is ill defined for point sized domains and
certainly does not recover the formal advection terms in the scalar and energy transport equations.
Finally, for LES-FDF the standard modeled terms in the Eulerian LES equations certainly can recover their
DNS forms in the limit of zero filter width and infinite particles. However, the convergence properties of
terms modeled by particles are less clear; with the conditional filtered di↵usion term being the primary issue
(other terms are typically modeled to reduce to standard LES models when the FDF equation is integrated
and therefore have the same convergence criteria). In contrast, models for the conditional filtered di↵usion
do not necessarily converge to the standard DNS di↵usion term. In one exception, Popov and Pope142
show
that the IEM model with the mean drift term [Eq. (24)] does converge in the DNS limit as the particles all
approach their filtered values and the mean drift term recovers the standard (Fickian) di↵usion form. The
newer shadow position model discussed above takes a similar form to that of the IEM. As such, it is likely
to be coupled with a comparable mean drift term and to recover the DNS limit. However, this has yet to be
proven to the authors’ knowledge.
Acknowledgments: The authors gratefully acknowledge the support of funding for this project provided
by Dr. Chipping Li under an AFOSR STTR AF13-AT12 grant.
References
1L.Y.M. Gicquel, G. Sta↵elbach, and T. Poinsot, “Large Eddy Simulations of Gaseous Flames in Gas Turbine Combustion
Chambers,” Prog. Energ. Combust. 38, 782–817 (2012).
2R.O. Fox, Computational Methods for Turbulent Reacting Flows (Cambridge University Press, Cambridge, United
Kingdom, 2003).
3P. Givi, “Filtered Density Function for Subgrid Scale Modeling of Turbulent Combustion,” AIAA J. 44, 16–23 (2006).
4H. Pitsch, “Large-Eddy Simulation of Turbulent Combustion,” Annual Rev. Fluid Mech. 38, 453–482 (2006).
5D.C. Haworth, “Progress in Probability Density Function Methods for Turbulent Reacting Flows,” Prog. Energy Comb.
Sci. 36, 168–259 (2010).
6T. Poinsot and D. Veynante, Theoretical and Numerical Combustion, 3rd ed. (R. T. Edwards, Inc., Philadelphia, PA,
2011).
7S.B. Pope, “Small Scales, Many Species and the Manifold Challenges of Turbulent Combustion,” Proc. Combust. Inst.
34, 1–31 (2013).
8S.L. Yilmaz, N. Ansari, P.H. Pisciuneri, M.B. Nik, C.C. Otis, and Givi. P., “Applied Filtered Density Function,” J.
App. Fluid Mech. 6, 311–320 (2013).
9J. Foster and R.S. Miller, “A Priori Analysis of Subgrid Mass Di↵usion Vectors in High Pressure Turbulent Hydrogen-
Oxygen Reacting Shear Layer Flames,” Phys. Fluids 24, 075114 (2012).
10J. Foster, Doctoral Dissertation, Clemson University, Department of Mechanical Engineering, 2012.
11C.H. Sohn and S.H. Chung, “E↵ect of Pressure on the Extinction, Acoustic Pressure Response, and NO Formation in
Diluted Hydrogen-Air Di↵usion Flames,” Combust. Flame 121, 288–300 (2000).
12T.S. Wang, “Thermophysics Characterization of Kerosene Combustion,” J. Thermo. Heat Trans. 15, 140–147 (2001).
13R.C. Reid, J.M. Prausnitz, and B.E. Poling, The Properties of Gases and Liquids (McGraw Hill, Boston, Massachusetts,
1987).
14S. Palle and R.S. Miller, “Analysis of High Pressure Hydrogen, Methane, and Heptane Laminar Di↵usion Flames:
Thermal Di↵usion Factor Modeling,” Combust. Flame 151, 581–600 (2007).
15J. Foster and R.S. Miller, “Fundamentals of High Pressure Combustion,” in High Pressure Processes in Chemical
Engineering, M. Lackner, ed., (2010), pp. 53–75.
16S. Ghosal, T. S. Lund, P. Moin, and K. Akselvoll, “A Dynamic Localization Model for Large-Eddy Simulation of
Turbulent Flows,” J. Fluid Mech. 286, 229–255 (1995).
17S.M. deBruynKops, J.J. Riley, and G. Kosal, “Toward Large Eddy Simulations of Non-premixed, Turbulent Reacting
Flows,” Western States Section of the Combustion Institute Spring Meeting Paper No.97S-051 (1997).
18U. Piomelli and J. Liu, “Large Eddy Simulation of Rotating Channel Flows Using a Localized Dynamic Model,” Phys.
Fluids 7, 839–848 (1995).
19A. Scotti, C. Meneveau, and M. Fatic, “Dynamic Smagorinsky Model on Anisotropic Grids,” Phys. Fluids 9, 1856–1868
(1997).
20M. Martin, U. Piomelli, and G.V. Candler, “Subgrid-Scale Models for Compressible Large-Eddy Simulations,” Theoret-
ical and Computational Fluid Dynamics 13, 361–376 (2000).
16 of 20
American Institute of Aeronautics and Astronautics

17. 21B. Chaouat and R. Schiestel, “A New Partically Integrated Transport Model for Subgrid-Scale Stresses and Dissipation
Rate for Turbulent Developing Flows,” Phys. Fluids 17, 17–22 (2005).
22A.W. Cook and J.J. Riley, “Subgrid-scale Modeling for Turbulent Reacting Flows,” Comb. Flame 112, 593–606 (1998).
23F.A. Jaberi and S. James, “A Dynamic Similarity Model for Large Eddy Simulation of Turbulent Combustion,” Phys.
Fluids 10, 1775–1777 (1998).
24J. A. Van Oijen and L.P.H. De Goey, “Modeling of Premixed Laminar Flames Using Flamelet-Generated Manifolds,”
Comb. Sci. Tech. 161, 113–137 (2000).
25J. A. Van Oijen, R.M. Bastiaans, G.A. Groot, and L.P.H. De Goey, “Direct Numerical Simulations of Premixed Turbulent
Flames with Reduced Chemistry: Validation and Flamelet Analysis,” Flow, Turb. and Comb. 75, 67–84 (2005).
26J. A. Van Oijen, R.M. Bastiaans, and L.P.H. De Goey, “Low-Dimensional Manifolds in Direct Numerical Simulations
of Premixed Turbulent Flames,” Proc. Comb. Inst. 31, 1377–1384 (2007).
27L. Verhoeven, W.S. Ramaekers, J. A. Van Oijen, and L.P.H. De Goey, “Modeling Non-Premixed Laminar Co-Flow
Flames Using Flamelet-Generated Manifolds,” Comb. Flame 159, 230–241 (2012).
28O. Gicquel, N. Darabiha, and D. Thevenin, “Laminar Premixed Hydrogen/Air Counterflow Flame Simulations Using
Flame Prolongation of ILDM,” Proc. Comb. Inst. 28, 1901–1908 (2000).
29B. Fiorina, R. Baron, O. Gicquel, D. Thevenin, and S. Carpentier, “Modelling Non-adiabatic Partially Premixed Flames
Using Flame-Prolongation of ILDM,” Comb. Theory Model. 7, 449–470 (2003).
30B. Fiorina, O. Gicquel, L. Vervisch, S. Carpentier, and N. Darabiha, “Approximating the Chemical Structure of Partially
Premixed and Di↵usion Counterow Flames Using FPI Flamelet Tabulation,” Comb. Flame 140, 147–160 (2005).
31C. D. Pierce and P. Moin, “Progress-Variable Approach for Large-Eddy Simulation of Non-Premixed Turbulent Com-
bustion,” J. Fluid Mech. 504, 73–97 (2004).
32M. Ihme, C.M. Cha, and H. Pitsch, “Prediction of Local Extinction and Re-Ignition E↵ects in Non-Premixed Turbulent
Combustion Using a Flamelet/Progress Variable Approach,” Proc. Comb. Inst. 30, 793–800 (2005).
33M. Ihme and H. Pitsch, “Prediction of Extinction and Reignition in Nonpremixed Turbulent Flames Using a
Flamelet/Progress Variable model 1. A Priori Study and Presumed PDF Closure,” Comb. Flame 155, 70–89 (2008).
34A. Parente and J. C. Sutherland, “Prinicipal Component Analysis of Turbulent Combustion Data: Data Pre-Processing
and Manifold Sensitivity,” Comb. Flame 160, 340–350 (2012).
35Yang. Y., S. B. Pope, and J. H. Chen, “Empirical Low-Dimensional Manifolds in Composition Space,” Comb. Flame
160, 1967–1980 (2013).
36J. C. Sutherland and A. Parente, “Combustion Modeling Using Principal Component Analysis,” Proc. Comb. Inst. 32,
1563–1570 (2009).
37A. Biglari and J. C. Sutherland, “A Filter-Independent Model Identification Technique for Turbulent Combustion
Modeling,” Comb. Flame 159, 1960–1970 (2012).
38N. Peters, Turbulent Combustion (Cambridge University Press, Cambridge, UK, 2000).
39Law C.K and C. J. Sung, “Structure, Aerodynamics, and Geometry of Premixed Flamelets,” Prog. Energ. Combust.
Sci. 26, 459–505 (2000).
40J.M. De Swart, R.M. Bastiaans, J. A. Van Oijen, L.P.H. De Goey, and R.S. Cant, “Inclusion of Preferential Di↵usion in
Simulations of Premixed Combustion of Hydrogen/Methane Mixtures with Flamelet Generated Manifolds,” Flow, Turb. and
Comb. 85, 473–511 (2010).
41J. F. Driscoll, “Turbulent Premixed Combustion: Flamelet Structure and its E↵ect on Turbulent Burning Velocities,”
Prog. Energ. Combust. Sci. 34, 91–134 (2008).
42E. Knudsen, S. H. Kim, and H. Pitsch, “An Analysis of Premixed Flamelet Models for Large Eddy Simulation of
Turbulent Combustion,” Phys. Fluids 22 (2010).
43G. Lodier, L. Vervisch, V. Moureau, and P. Domingo, “Composition-Space Premixed Flamelet Solution with Di↵erential
Di↵usion for In-Situ Flamelet-Generated Manifolds,” Comb. Flame 158, 2009–2016 (2011).
44V. Mittal and H. Pitsch, “A Flamelet Model for Premixed Combustion Under Variable Pressure Conditions,” Proc.
Comb. Inst. 34, 2995–3003 (2013).
45N. Peters, “Laminar Di↵usion Flamelet Models in Non-Premixed Turbulent Combustion,” Prog. Energ. Combust. 10,
319–339 (1984).
46L. Vervisch and T. Poinsot, “Direct Numerical Simulation of Non-Premixed Turbulent Flames,” Annu. Rev. Fluid Mech.
30, 655–691 (1998).
47D. Carbonell, C. Perez-Segarra, P J Coelho, and A. Oliva, “Flamelet Mathematical Models for Non-Premixed Laminar
Combustion,” Comb. Flame 156, 334–347 (2009).
48E. Knudsen and H. Pitsch, “A General Flamelet Transformation Useful for Distinguishing Between Premixed and
Non-Premixed Modes of Combustion,” Comb. Flame 156, 678–696 (2009).
49P. Nguyen, L. Vervisch, V. Subramanian, and P. Domingo, “Multidimensional Flamelet-Generated Manifolds for Par-
tially Premixed Combustion,” Comb. Flame 157, 43–61 (2010).
50E. Knudsen and H. Pitsch, “Capabilites and Limitations of Multi-Regime Flamelet Combustion Models,” Comb. Flame
159, 242–264 (2012).
51Y. Xuan, G. Blanquart, and M. Mueller, “Modeling Curvature E↵ects in Di↵usion Flames Using a Laminar Flamelet
Model,” Comb. Flame 161, 1294–1309 (2014).
52Y. Niu, L. Vervisch, and P. Tao, “An Optimization-Based Approach to Detailed Chemistry Tabulation: Automated
Progress Variable Definition,” Comb. Flame 160, 776–785 (2013).
53G. Lacaze and J.C. Oefelein, “A Non-Premixed Combustion Model Based on Flame Structure Analysis at Supercritical
Pressures,” Comb Flame 159, 2087–2103 (2012).
17 of 20
American Institute of Aeronautics and Astronautics

18. 54P. Domingo, L. Vervisch, and D. Veynante, “Large-Eddy Simulation of a Lifted Methane Jet Flame in a Vitiated
Coflow,” Combust. Flame 152, 415–432 (2008).
55M. Ihme and Y. See, “Prediction of Autoignition in a Lifted Methane/Air Flame Using an Unsteady Flamelet/Progress
Variable Model,” Comb. Flame 157, 1850–1862 (2010).
56S. Menon and A.R. Kerstein, “The Linear Eddy Model,” in The Linear-Eddy Model, T. Echekki and E. Mastorakos,
eds., (Springer-Verlag, Cambridge, 2011), Chap. 10, pp. 221–247.
57A. R. Kerstein, “A Linear Eddy Model of Turbulent Scalar Transport and Mixing,” Combust. Sci. Tech. 60, 391–421
(1988).
58T. Echekki, A. R. Kerstein, J. Y. Chen, and T. Dreeben, “One-Dimensional Turbulence Simulation of Turbulet Jet
Di↵usion Flame,” Combust. Flame 125, 1083–1105 (2001).
59Wm. T. Ashurst and A. R. Kerstein, “One-Dimensional Turbulence: Variable-Density Formulation and Application to
Mixing Layers,” Phys. Fluids 17, 1–25 (2005).
60W.H. Calhoun, Doctoral Dissertation, Georgia Institute of Technology, Department of Aerospace Engineering, 1996.
61V. Sankaran, Doctoral Dissertation, Georgia Institute of Technology, Department of Aerospace Engineering, 2003.
62V. Sankaran and S. Menon, “Subgrid Combustion Modeling of 3-D Premixed Flames in the Thin-Reaction-Zone Regime,”
Proceedings Comb. Inst. 30, 575–582 (2005).
63A. Sen, E.R. Hawkes, and S. Menon, “Large Eddy Simulation of Extinction and Reignition with Artificial Neural
Networks Based Chemical Kinetics,” Comb. Flame 157, 566–578 (2010).
64J. S. Ochoa and A.S. Fueyo, “Subgrid Linear Eddy Mixing and Combustion Modeling of a Turbulent Nonpremixed
Piloted Jet Flame,” Flow Turb. Combust. 89, 295–309 (2012).
65S. B. Pope, “Computationally E cient Implementation of Combustion Chemistry Using In Situ Adaptive Tabulation,”
Technical Report No. FDA 96-02, Cornell University, Ithaca, NY (1996) .
66V. Hiremath, Z. Ren, and S.B. Pope, “Combined Dimension Reduction and Tabulation Strategy Using ISAT-RCCE-
GALI for the E cient Implementation of Combustion Chemistry,” Combust. Flame 158(11), 2113–2127 (2011).
67V. Hiremath, S.R. Lantz, H. Wang, and S.B. Pope, “Large-Scale Parallel Simulations of Turbulent Combustion Using
Combined Dimension Reduction and Tabulation of Chemistry,” Proc. Comb. Inst. 34, 205–215 (2013).
68Y. Yang, H. Wang, S.B. Pope, and J.H. Chen, “Large-Eddy Simulation/Probability Density Function Modeling of a
Non-Premixed CO/H2 Temporally Evolving Jet Flame,” Proc. Comb. Inst. 34, 1241–1249 (2013).
69H. Wang and S.B. Pope, “Large Eddy Simulation/Probability Density Function Modeling of a Turbulent CH4/H2/N2
Jet Flame,” Proc. Comb. Inst. 33, 1319–1330 (2011).
70P.J. Colucci, F.A. Jaberi, P. Givi, and S.B. Pope, “Filtered Density Function for Large Eddy Simulation of Turbulent
Reacting Flows,” Phys. Fluids 10, 499–515 (1998).
71L. Y. M. Gicquel, P. Givi, F. A. Jaberi, and S. B. Pope, “Velocity Filtered Density Function for Large Eddy Simulation
of Turbulent Flows,” Phys. Fluids 14, 1196–1213 (2002).
72M. R. H. Sheikhi, P. Givi, and S. B. Pope, “Velocity-Scalar Filtered Mass Density Function for Large Eddy Simulation
of Turbulent Reacting Flows,” Phys. Fluids 19, 095196 (2007).
73M.R.H. Sheikhi, T.G. Drozda, P. Givi, and S.B. Pope, “Velocity-Scalar Filtered Density Function for Large Eddy
Simulation of Turbulent Flows,” Phys. Fluids 15, 2321–2337 (2003).
74M.R.H. Sheikhi, P. Givi, and S.B. Pope, “Frequency-Velocity-Scalar Filtered Density Function for Large Eddy Simulation
of Turbulent Flows,” Phys. Fluids 21, 075102 (2009).
75M.B. Nik, Doctoral Dissertation, University of Pittsburgh, Department of Mechanical Engineering, 2012.
76R. McDermott and S.B. Pope, “A Particle Formulation for Treating Di↵erential Di↵usion in Filtered Density Function
Methods,” J. Computational Phys. 226, 947–993 (2007).
77R. O. Fox, Computational Models for Turbulent Reacting Flows (Cambridge University Press, Cambridge, UK, 2003).
78V. Raman, H. Pitsch, and R. Fox, “Eulerian Transported Probability Density Function Sub-Filter Model for Large-Eddy
Simulations of Turbulent Combustion,” Comb. Theory Modeling 10, 439–458 (2006).
79J. Lee and Y. Kim, “Transported PDF Approach and Direct-Quadrature Method of Moment for Modeling Turbulent
Piloted Jet Flames,” J. Mech. Sci. Tech. 25, 3259–3265 (2011).
80H.S. Koo, P. Donde, and V. Raman, “A Quadrature-Based LES/Transported Probability Density Function Approach
for Modeling Supersonic Combustion,” Proc. Comb. Inst. 33, 2203–2210 (2011).
81P.P. Donde, Doctoral Dissertation, University of Texas Austin, Department of Mechanical Engineering, 2012.
82S.B. Pope, “PDF Methods for Turbulent Reacting Flows,” Prog. Energ. Combust. 11, 119–192 (1985).
83J.P. Minier, S. Chibbaro, and S.B. Pope, “Guidelines for the Formulation of Lagrangian Stochastic Models for Particle
Simulations of Single-Phase and Dispersed Two-Phase Turbulent Flows,” Phys. Fluids 26, 113303 (2014).
84C. Dopazo and E.E. O’Brien, “An Approach to the Autoignition of a Turbulent Mixture,” Acta Astronaut 1, 1239
(1974).
85D. A. Shetty, A. J. Chandy, and S.H. Frankel, “A New Fractal Interaction by Exchange with the Mean Mixing Model for
Large Eddy Simulation/Filtered Mass Density Function Applied to a Multiscalar Three-Stream Turbulent Jet,” Phys. Fluids.
22, 025102 (2010).
86S. Liu and C. Tong, “Subgrid-Scale Mixing of Mixture Fraction, Temperature, and Species Mass Fractions in Turbulent
Partially Premixed Flames,” Proc. Combust. Inst. 34, 1231–1239 (2013).
87S.B. Pope, “A Model for Turbulent Mixing Based on Shadow-Position Conditioning,” Phys. Fluids 25, 110803 (2013).
88S.B. Pope, “On the Relationship between Stochastic Lagrangian Models of Turbulence and Second Order Closures,”
Phys. Fluids 6, 973–985 (1994).
89R.O. Fox, “On Velcoity-Conditioned Scalar Mixing in Homogeneous Turbulence,” Phys. Fluids 8, 2678–2691 (1996).
18 of 20
American Institute of Aeronautics and Astronautics

19. 90H. Wang, P.P. Popov, and S.B. Pope, “Weak Second Order Splitting Schemes for Lagrangian Monte Carlo Particle
Methods for the Composition PDF/FDF Transport Equations,” J. Comput. Phys. 229, 1852–1878 (2010).
91D. Meyer and P. Jenny, “A Mixing Model for Turbulent Flows Based on Parametrized Scalar Profiles,” Phys. Fluids
18, 035105 (2006).
92R.L. Curl, “Dispersed Phase Mixing: I. Theory and E↵ects in Simple Reactors,” AIChE J. 9, 175–181 (1963).
93S.B. Pope, “An Improved Turbulent Mixing Model,” Combustion Science and Technology 28, 131–135 (1982).
94P.A. Nooren, H.A. Wouters, T.W.J. Peeters, D. Roekaerts, U. Mass, and D.Schmidt, “Monte Carlo PDF Modelling of
a Turbulent Natural-Gas Di↵usion Flame,” Combustion Theory and Modelling 1, 79–96 (1997).
95S.B. Pope, “Mapping Closures for Turbulent Mixing and Reaction,” Theor. Comp. Fluid Dyn. 2, 255–270 (1991).
96A.K. Klimenko and S.B. Pope, “The Modeling of Turbulent Reactive Flows Based on Multiple Mapping Conditioning,”
Phys. Fluids 15, 1907–1925 (2003).
97S. Subramaniam and S.B. Pope, “A Mixing Model for Turbulent Reactive Flows Based on Euclidean Minimum Spanning
Trees,” Comb. Flame 115, 487–514 (1998).
98Richardson E.S. and Chen J.H., “Application of PDF Mixing Models to Premixed Flames with Di↵erential Di↵usion,”
Comb. Flame 159, 2398–2414 (2012).
99A. Krisman, J.C.K. Tang, E.R. Hawkes, D.O. Lignell, and J.H. Chen, “A DNS Evaluation of Mixing Models for
Transported PDF Modelling of Turbulent Nonpremixed Flames,” Comb. Flame 161, 2085–2106 (2014).
100R. S. Barlow and J. H. Frank, “E↵ects of Turbulence on Species Mass Fractions in Methane/Air Jet Flames,” Proc.
Combust. Inst. 27, 1087–1095 (1998).
101M.R.H. Sheikhi, T.G. Drozda, P. Givi, F.A. Jaberi, and S.B. Pope, “Large Eddy Simulation of a Turbulent Nonpremixed
Piloted Methane Jet Flame (Sandia Flame D),” Proc. Combust. Inst. 30, 549–556 (2005).
102M. B. Nik, S. L. Yilmaz, P. Givi, M. R. H. Sheikhi, and S. B. Pope, “Simulation of Sandia Flame D Using Velocity-Scalar
Filtered Density Function,” AIAA J. 48, 1513–1522 (2010).
103B. B. Dally, A. R. Masri, R. S. Barlow, and G. J. Fiechtner, “Instantaneous and Mean Compositional Structure of
Blu↵-Body Stabilized Nonpremixed Flames,” Combust. Flame 114, 119–148 (1998).
104B. B. Dally, A. R. Masri, R. S. Barlow, and G. J. Fiechtner, “Two-Photon Laser-Induced Fluorescence Measurement of
CO in Turbulent Nonpremixed Blu↵ Body Flames,” Combust. Flame 132, 272–274 (2003).
105Y.-C. Chen, N. Peters, G. A. Schneemann, N. Wruck, U. Renz, and M. S. Mansour, “The Detailed Flame Structure of
Highly Stretched Turbulent Premixed Methane-Air Flames,” Combust. Flame 107, 223–226 (1996).
106T. G. Drozda, M. R. H. Sheikhi, C. K. Madnia, and P. Givi, “Developments in Formulation and Application of the
Filtered Density Function,” Flow Turbul. Combust. 78, 35–67 (2007).
107S. L. Yilmaz, M. B. Nik, P. Givi, and P. A. Strakey, “Scalar Filtered Density Function for Large Eddy Simulation of a
Bunsen Burner,” J. Propul. Power 26, 84–93 (2010).
108X. Y. Zhou and J. C. F. Pereira, “Large Eddy Simulation (2D) of a Reacting Planar Mixing Layer Using Filtered Density
Function,” Flow Turbul. Combust. 64, 279–300 (2000).
109S. Heinz, “On Fokker-Planck Equations for Turbulent Reacting Flows. Part 2. Filter Density Function for Large Eddy
Simulation,” Flow Turbul. Combust. 70, 153–181 (2003).
110V. Raman and H. Pitsch, “Large-Eddy Simulation of a Blu↵-Body-Stabilized Non-premixed Flame Using a Recursive
Filter-Refinement Procedure,” Combust. Flame 142, 329–347 (2005).
111E. van Vliet, J. J. Derksen, and H. E. A. van den Akker, “Turbulent Mixing in a Tubular Reactor: Assessment of an
FDF/LES Approach,” AIChE J. 51, 725–739 (2005).
112M. D. Carrara and P. E. DesJardin, “A Filtered Mass Density Function Approach to Modeling Separated Two-Phase
Flows Using LES I: Mathematical Formulation,” Int. J. Multiphas. Flow 32, 365–384 (2006).
113R. Mustata, L. Vali˜no, C. Jim´enez, W. P. Jones, and S. Bondi, “A Probability Density Function Eulerian Monte Carlo
Field Method for Large Eddy Simulations: Application to a Turbulent Piloted Methane/Air Di↵usion Flame (Sandia D),”
Combust. Flame 145, 88–104 (2006).
114W. P. Jones, S. Navarro-Martinez, and O. R¨ohl, “Large Eddy Simulation of Hydrogen Auto-Ignition with a Probability
Density Function Method,” Proc. Combust. Inst. 31, 1765–1771 (2007).
115W. P. Jones and S. Navarro-Martinez, “Large Eddy Simulation of Autoignition with a Subgrid Probability Density
Function Method,” Combust. Flame 150, 170–187 (2007).
116S. James, J. Zhu, and M. S. Anand, “Large Eddy Simulations of Turbulent Flames Using the Filtered Density Function
Model,” Proc. Combust. Inst. 31, 1737–1745 (2007).
117J. Y. Chen, “A Eulerian PDF Scheme for LES of Nonpremixed Turbulent Combustion with Second-Order Accurate
Mixture Fraction,” Combust. Theor. Model. 11, 675–695 (2007).
118V. Raman and H. Pitsch, “A Consistent LES/Filtered Density Function Formulation for the Simulation of Turbulent
Flames with Detailed Chemistry,” Proc. Combust. Inst. 31, 1711–1719 (2007).
119A. Afshari, F. A. Jaberi, and T. I. P. Shih, “Large-Eddy Simulations of Turbulent Flows in an Axisymmetric Dump
Combustor,” AIAA J. 46, 1576–1592 (2008).
120T.G. Drozda, G. Wang, V. Sankaran, J.R. Mayo, J.C. Oefelein, and R.S. Barlow, “Scalar Filtered Mass Density Functions
in Nonpremixed Turbulent Jet Flames,” Combust. Flame 155, 54–69 (2008).
121J. R’eveillon and L. Vervisch, “Subgrid-Scale Turbulent Micromixing: Dynamic Approach,” AIAA J. 36, 336–341
(1998).
122C. M. Cha and P. Troullet, “A Subgrid-Scale Mixing Model for Large-Eddy Simulations of Turbulent Reacting Flows
Using the Filtered Density Function,” Phys. Fluids 15, 1496–1504 (2003).
123S. Heinz, “Unified Turbulence Models for LES and RANS, FDF and PDF Simulations,” Theor. Comp. Fluid Dyn. 21,
99–118 (2007).
19 of 20
American Institute of Aeronautics and Astronautics

20. 124C. Tong, “Measurements of Conserved Scalar Filtered Density Function in a Turbulent Jet,” Phys. Fluids 13, 2923–2937
(2001).
125D. Wang and C. Tong, “Conditionally Filtered Scalar Dissipation, Scalar Di↵usion, and Velocity in a Turbulent Jet,”
Phys. Fluids 14, 2170–2185 (2002).
126D. Wang, C. Tong, and S. B. Pope, “Experimental Study of Velocity Filtered Joint Density Function For Large Eddy
Simulation,” Phys. Fluids 16, 3599–3613 (2004).
127D. Wang and C. Tong, “Experimental Study of Velocity-Scalar Filtered Joint Density Function for LES of Turbulent
Combustion,” Proc. Combust. Inst. 30, 567–574 (2005).
128N. Ansari, G.M. Goldin, M.R.H. Sheikhi, and Givi. P., “Filtered Density Function Simulator on Unstructured Meshes,”
J. Comp. Phys. 230, 7132–7150 (2011).
129N. Ansari, P.H. Pisciuneri, P.A. Strakey, and Givi. P., “Scalar-Filtered-Mass-Density-Function Simulations of Swirling
Reacting Flows on Unstructured Grids,” AIAA J. 50, 2476–2482 (2012).
130P.H. Pisciuneri, L. Yilmaz, A. Strakey, and Givi. P., “An Irregularly Proportioned FDF Simulator,” SIAM J. Sc. Comput.
35, 438–452 (2013).
131S. B. Pope, Turbulent Flows (Cambridge University Press, Cambridge, UK, 2000).
132R.W. Bilger, “Future Progress in Turbulent Combustion Research,” Prog. Energ. Combust. 26, 367–380 (2000).
133N. Peters, Turbulent Combustion (Cambridge University Press, 2000).
134Handbook of Numerical Heat Transfer, 2nd ed., W. J. Minkowycz, E. M. Sparrow, and J. Y. Murthy, eds., (John Wiley
& Sons, Inc., New York, NY, 2006).
135N. Ansari, F. A. Jaberi, M. R. H. Sheikhi, and P. Givi, “Filtered Density Function as a Modern CFD Tool,” in
Engineering Applications of Computational Fluid Dynamics, A. R. S. Maher, ed., (International Energy and Environment
Foundation, www.IEEFoundation.org, 2011), Chap. 1, pp. 1–22.
136R.S. Miller, S.H. Frankel, C.K. Madnia, and P. Givi, “Johnson-Edgeworth Translation for Probability Modeling of
Binary Mixing in Turbulent Flows,” Combust. Sci. Tech. 91, 21–52 (1993).
137C. K. Madnia, S. H. Frankel, and P. Givi, “Reactant Conversion in Homogeneous Turbulence: Mathematical Modeling,
Computational Validations and Practical Applications,” Theor. Comp. Fluid Dyn. 4, 79–93 (1992).
138P. A. McMurtry and P. Givi, “Direct Numerical Simulations of Mixing and Reaction in a Nonpremixed Homogeneous
Turbulent Flow,” Combust. Flame 77, 171–185 (1989).
139R. O. Fox, “Computational Methods For Turbulent Reacting Flows in Chemical Process Industry,” Revue De L’Institut
Francais Du Petrole 51, 215–246 (1996).
140S. Subramaniam and S. B. Pope, “A Mixing Model for Turbulent Reactive Flows Based on Euclidean Minimum Spanning
Trees,” Combust. Flame 115, 487–514 (1998).
141S. Subramaniam and S. B. Pope, “Comparison of Mixing Model Performance for Nonpremixed Turbulent Reactive
Flow,” Combust. Flame 117, 732–754 (1999).
142P.P. Popov and S.B. Pope, “Implicit and Explicit Schemes for Mass Consistency Preservation in Hybrid Particle/Finite
Volume Algorithms for Turbulent Reactive Flows,” J. Comput. Phys. 257, 352–373 (2014).
20 of 20
American Institute of Aeronautics and Astronautics