CaltechCONF

Pitsch, Heinz

Large-Eddy Simulation of Turbulent Combustion
Heinz Pitsch
Department of Mechanical Engineering
Stanford University
Stanford, CA 94305-3030
H.Pitsch@stanford.edu
In recent years, Large Eddy Simulation (LES) has been successfully applied to non-premixed and premixed
turbulent combustion problems [1, 2, 3]. In most technical combustion applications, the pure non-premixed
or premixed combustion models are no longer valid, since partially premixed combustion has to be taken
into account. An example is the stabilization region of a lifted non-premixed flame. To overcome this
problem, a Combined Conserved Scalar/Level-Set Flamelet model has been proposed [4, 5], which allows
for the computation of reactive flows, where both premixed and non-premixed combustion occurs. In this
formulation, the G-equation method is used to describe partially premixed flame propagation and to distin-
guish between burned and unburned regions. The reacting gases in the post-flame region are described by
a laminar diffusion flamelet approach. The principal roadmap for the further development and validation of
these models is to develop and validate the appropriate methods for the pure premixed and non-premixed
combustion situations, and combine the resulting models for a more general formulation. Much of this work
has been done in the past. We have developed unsteady flamelet models for LES of non-premixed turbulent
combustion [1, 2], and a level-set method for LES of premixed turbulent combustion [3]. These models have
been validated with premixed and non-premixed turbulent jet flame experiments. Preliminary simulations
with a Combined Conserved Scalar/Level-Set Flamelet Model have already been performed [6, 5]. Results
from these simulations are shown in Figs. 1 and 2. Recently, we have developed a new consistent formu-
Figure 1: Left: Axial velocity (top) and mixture fraction (bottom) from the cold flow simulation. The blue
lines are zero axial velocity and stoichiometric mixture fraction in the velocity and the mixture fraction
figure, respectively. Right: Axial velocity (top) and temperature (bottom) from the fired case simulation.
Blue lines represent the instantaneous flame surface, red lines are zero axial velocity and stoichiometric
mixture fraction in the velocity and the temperature figure, respectively
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2
U
/U
0
r/R
Mean
RMS
-1
-0.5
0
0.5
1
1.5
0 0.5 1 1.5 2 2.5 3
U
/U
0
r/R
Mean
RMS
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 0.5 1 1.5 2 2.5 3
U
/U
0
r/R
Mean
RMS
-0.2
0
0.2
0.4
0.6
0.8
0 0.5 1 1.5 2 2.5 3
U
/U
0
r/R
Mean
RMS
Figure 2: Axial velocities from the fired simulation (lines) compared with experimental data (symbols) at
x/R = 0.08, x/R = 1.8, x/R = 3.3, and x/R = 6.3 from left to right
lation of the G-equation approach for LES [7]. The solution of the instantaneous unfiltered G-equation has
physical meaning only at the instantaneous flame front location. The definition of the remaining G-field is
arbitrary. Hence, in a filtering procedure, applied to derive the appropriate LES equation, only states on the
1
instantaneous unfiltered flame surface can be considered, which excludes the use of conventional volumetric
LES filters. A new filter kernel has been developed, which averages along surfaces. Using this filter, the
G-equation for the filtered flame front location can be derived This equation has two unclosed terms. Since
the new filter kernel only considers states at the instantaneous flame front, the convection velocity appearing
in this equation is an average conditioned on the location of the flame surface. To relate this conditional
velocity to the unconditionally Favre-filtered velocity v˜, which is known from the solution of the momentum
equations, a model for this quantity has also been developed. The resulting equation for the filtered flame
front location is
∂Gˇ
∂t
+ v˜ · ∇Gˇ =
ρu
ρ
sT,u
∣∣∇Gˇ
∣∣ . (1)
The second unclosed term involves the turbulent burning velocity. A model for this term has also been
developed. A length scale equation for the sub-grid flame brush thickness has been derived. An analytic
expression for the turbulent burning velocity follows from the assumption that production equals dissipation
in this equation. Interestingly, because of the use of the new filtering procedure, a propagation term propor-
tional to the curvature of the mean front, which appeared in earlier formulations of the filtered G-equation,
is no longer present in Eq. (1). This is an important difference, since the curvature term has a stabilizing
effect on the flame front and therefore leads to a decreased resolved flame wrinkling. The model is validated
by simulations of three turbulent premixed Bunsen flames at different Reynold numbers. Earlier simulations
for the lower Reynolds number case using a different level set formulation resulted in an underprediction of
the flame brush thickness. The results of the present simulations are compared with the earlier simulations
and experimental data for temperature, velocity field, and turbulent flame brush thickness.
References
[1] H. Pitsch and H. Steiner. Large-eddy simulation of a turbulent piloted methane/air diffusion flame
(Sandia flame D). Phys. Fluids, 12(10):2541–2554, 2000.
[2] H. Pitsch. Improved pollutant predictions in large-eddy simulations of turbulent non-premixed combus-
tion by considering scalar dissipation rate fluctuations. Proc. Combust. Inst., 29:1971–1978, 2002.
[3] H. Pitsch and L. Duchamp de Lageneste. Large-eddy simulation of premixed turbulent combustion using
a level-set approach. Proc. Combust. Inst., 29:2001–2008, 2002.
[4] N. Peters. Turbulent Combustion. Cambridge University Press, 2000.
[5] H. Pitsch, P. Trouillet, C. D. Pierce, E. Tribbett, C. M. Sipperley, C. F. Edwards, and C. T. Bowman.
A joint experimental/large-eddy simulation study of a model gas turbine combustor. Western States
Section Meeting of the Combustion Institute, 2002, San Diego, CA, pages WSSCI 02S–59, 2002.
[6] H. Pitsch and L. Duchamp de Lageneste. Large-eddy simulation of lifted turbulent diffusion flames. 2nd
Joint meeting of the US sections of the Combustion Institute, Oakland, CA, page paper 263, 2001.
[7] H. Pitsch and L. Duchamp de Lageneste. A g-equation formulation for large-eddy simulations of premixed
turbulent combustion. CTR Annual Research Briefs, pages 3–14, 2002.
2


English

Large-Eddy Simulation of Turbulent Combustion

Pantano, Carlos

Pullin, Dale I.

Large Eddy Simulation of a Turbulent Reacting Compressible Jet
C. Pantano   and D.I. Pullin
Graduate Aeronautical Laboratories,
California Institute of Technology 105-50,
1200 E. California Blvd., Pasadena CA 91125
September 10, 2003
We report results of a Large Eddy Simulation (LES) effort of turbulent compressible nonpremixed reactive jets. It
is well known that there are several difficulties associated with compressible reactive flows in comparison with incom-
pressible flows. In incompressible flows, the energy and mass conservation (for constant density flows) equations are
decoupled from the velocity field. Moreover, the subgrid terms that are required in these flows need only provide with
the deviatoric part of the subgrid contributions. The isotropic part can be absorbed in the pressure term. In reacting
incompressible flows, one can also avoid the need to integrate the conservation equations for mass and energy by using
a conserved scalar approach. In this case, both temperature and density are determined from a state relationship that
is known beforehand. On the other hand, for compressible reacting flows one is not allowed to discard the energy and
mass conservation equation since acoustics and in some cases shocks coexist with heat release. In the present investi-
gation we use the totally conservative formulation in terms of total sensible energy. Other forms of the energy equation
require closure for pressure velocity correlations, [1]. This is not requited in the total sensible energy form, although
some other correlations are needed, notably, those between enthalpy and mass fractions of species. Furthermore, we
are interested in using local subgrid models. The advantages of this kind of approach are self evident, specially in
complex geometries. The results discussed in this abstract are obtained with the stretched vortex model for turbulent
transport [2], scalar transport [4] and subgrid scalar variance [3]. Here, we consider a model problem of a turbulent
planar nonpremixed reacting jet. The modeling assumptions are described next, followed by some preliminary results.
1 Formulation and Results
For variable density flows, the LES governing equations are formulated using Favre filtered variables, defined as

 
	
, where the “conceptual” filter operator is denoted by the overbar. Conservation of mass, momentum and
energy are expressed in the usual form, [5], in terms of filtered density, 	 , velocity,  , and pressure 	 and the total
sensible energy conservation equation is

	




	


	
ﬀ


ﬂﬁ


ﬃ




 
!
ﬁ


"

#
$!


!%ﬁ&
')(

+* (1)
where


is the resolved thermal conductivity. The resolved Newtonian viscous stress tensor is given as usual with ,
as the resolved shear viscosity. The dependence on temperature of , and


is typically not small for combusting flow,
primarily, due to the large temperature variations in the flow. They are approximated here by ,

,
-


 

 
-
ﬀ/. and

01
-


 

 
-
ﬀ2.
, where 3
54
*76 . Finally, the total sensible energy,
	

is given by
	

8	:9

;
=<
>





@?A&
	


	

BC (2)
where

BD
FE
G
&



Hﬀ

> is the subgrid kinetic energy. The Favre filtered enthalpy is decomposed into a resolved
part and a subgrid part through,

;0JI
K
MLON
;



 
ﬀ

P


;RQ@C ;QﬃJI
K
SLON

E
;

P

&
;



 
ﬀ

P

ﬁ
C (3)
T
e-mail:cpantano@galcit.caltech.edu
1
Figure 1: Mixture fraction and subgrid kinetic energy distribution.
where U is the number of species,
;
 is the enthalpy of component V , given by
;



 
ﬀ
5;
-

XWZY
[
[]\_^$`


 _a
ﬀ2b
 _a
C (4)
and the system of equations is closed by the equation of state,
	

c	ed
-

 fI
K
SLON

P

g



Q
C

Q
8	hd
-
I
K
SLON
E
 iP

&

 

P

g

(5)
^
`
 is the specific heat of component V ,
;
-
 is the enthalpy of formation of the corresponding species,
d
- is the gas
constant and g  is the molecular weight of species V .
Combustion is handled through the conserved scalar approach, where a a mixture fraction field, j , is used to obtain
the mass fraction of the different species. The quantities that need to be modeled are the subgrid momentum stresses,
scalar transport and subgrid total energy transport, defined as
'

!
8	
E



!"&





!kﬁ
C
'l
!
8	

E
j

!"&

j


!%ﬁ
C
'
(
!
8	

E
;

!:&

;


!@ﬁ

	
>
ZE





!"&
E






!%ﬁ
* (6)
The triple correlations in
'
(
!
of Eq. (6) are neglected and the other terms are modeled using the stretched vortex
model. The combustion related terms,
;
Q
and 
Q
, are closed through an assumed beta pdf model, where the subgrid
scalar variance needed in the assumed pdf model is obtained from the approximate reconstruction model of [3]. The
governing equations are integrated using a compact Pade´ scheme in space of sixth order and a third-order low-storage
Runge-Kutta scheme in time. The chemistry is modeled as an infinitely fast one step methane-air reaction. This
determines a state relation of the form
P


P


j
ﬀ
. The stoichiometric mixture fraction was set to 0.2. The cold
Reynolds number was set equal to 20000 and the jet Mach number was equal to 0.6. Figure 1 shows mixture fraction
isosurfaces (a) and subgrid kinetic energy (b) at one instant in time. Further details will be provided in the presentation.
References
[1] A.W. Cook and J.J. Riley. Subgrid-scale modeling for turbulent reacting flows. Combustion and Flame,
112(4):593–606, 1998.
[2] A. Misra and D.I. Pullin. A vortex-based subgrid model for large-eddy simulation. Phys. Fluids, 9:2243, 1997.
[3] C. Pantano and S. Sarkar. A subgrid model for nonlinear functions of a scalar. Phys. Fluids, 13(12):3803–3819,
2001.
[4] D.I. Pullin. Vortex-based model for subgrid flux of a passive scalar. Phys. Fluids, 12:2311, 2000.
[5] F. A. Williams. Combustion Theory. Benjamin/Cummings, Menlo Park, CA, 1985.
2


Large Eddy Simulation of a Turbulent Reacting Compressible Jet

McDermott, Randy

Schmidt, Rodney C.

Kerstein, Alan R.

Smith, P.

Subgrid closure for large eddy simulation using one-dimensional
turbulence
R. McDermott, R. Schmidt, A. Kerstein and P. Smith
September 15, 2003
1 Abstract
In this talk we describe the development of a new subgrid closure model for large eddy simulation (LES)
based on the one-dimensional turbulence model (ODT) of Kerstein [JFM, 2001], and present a comparison of
computed results with the experimental data of Comte-Bellot and Corrsin [JFM, 1971] for decaying isotropic
turbulence. We also discuss unique advantages of this approach when extended to turbulent reacting flows.
For LES of constant density flows, the subgrid physics to be modeled include energy dissipation, turbulent
stresses (both near-wall and bulk flow), backscatter, anisotropy, intermittency, and transition. At present,
the “mixed model,” which consists of a linear combination of the dynamic Smagorinsky model [Germano
et al., Phys. Fluids A, 1991] and the scale similarity model of Bardina et al. [AIAA, 1980-1357], might be
considered the current state of the art. However, although this approach has been shown to adequately treat
energy dissipation and bulk flow stresses, most of the other challenges just mentioned remain problematic.
In particular, wall bounded flows have been the Achilles heel of LES owing to anisotropy and the dramatic
length scale reduction near the wall. A recent study analyzed in detail the requirements for accurate closure
for turbulent channel flow [S. Volker et al., Phys. Fluids, 2002], where it was concluded that it is insufficient
solely to capture the transfer of energy from grid resolved to subgrid scales. At a minimum, the closure must
also capture the subgrid transport (especially wall normal transport in near-wall flow), subgrid stresses, and
subgrid intercomponent transfer due to pressure effects. To the best of our knowledge, no closure approach
currently being pursued, except LES/ODT, can demonstrably capture these effects in principle, let alone
represent them accurately.
In reacting flows, scalar mixing, thermochemical state history, buoyancy, and multi-component transport
effects can be important. An LES must therefore include transport equations for energy and species, which
introduces still more subgrid terms that must be closed. Often in practice these terms are handled with the
gradient diffusion hypothesis, with diffusivity based on a turbulent Schmidt number. It is important to note,
however, that there is nothing which requires the subgrid flux vectors to be aligned with scalar gradient
vectors (a key assertion in gradient diffusion). In all cases the subgrid term definitions are decompositions of
the filtered non-linear advective term, which is entirely inviscid in nature. It seems desirable, therefore, to
seek a model which can address the previously mentioned issues, and enforces a tight coupling of the subgrid
scalar field to the subgrid velocity field.
ODT is a novel approach to modeling turbulence that has characteristics uniquely suited to addressing
many of the subgrid closure issues associated with LES. As a stand alone model, ODT has found use in
several classes of flows, including: compressible shear layers, boundary layers, oceanographic flows, reacting
jets, and even astrophysical flows such as Rayleigh-Benard convection in stars. In ODT, the fields defined
on its 1d domain evolve by two mechanisms: (1) molecular diffusion, and (2) a sequence of instantaneous
transformations, called “eddy events,” which represent turbulent stirring. Each eddy event may be inter-
preted as the model analog of an individual turbulent eddy. In this regard, ODT resembles its predecessor,
the linear-eddy model (LEM) [Kerstein, JFM, 1991] for turbulent scalar mixing, making it readily applicable
to subgrid scalar transport in reacting flows. In ODT, however, the dependence of the event frequency on
1
eddy location and length scale evolves dynamically, governed by a probabilistic model, rather than being
specified by an assumed frequency distribution.
Recently, Schmidt et al. [JCP, 2003] used ODT as the basis for a near-wall LES closure model and tested
the approach in turbulent channel flow. This model introduced a system of equations similar to the Prandtl
boundary layer equations, with the added feature of a stochastic rearrangement process to simulate wall
normal subgrid advection. The more general LES closure proposed here is an outgrowth of ideas developed
during that effort, but has important conceptual and procedural differences.
The LES/ODT algorithm can be summarized as follows. A 3d ODT lattice is used as a support for the
fully resolved field (a DNS surrogate). The filtered values of the ODT field in each of the three directions
are forced to match the LES cell averages through a low wave number adjustment procedure, which does
not disturb the high wave number (subgrid) structure. The subgrid stresses are not computed directly.
Rather, the time integrated divergence of the stress is modeled with the ODT evolution procedure. This
procedure consists of stochastic instantaneous eddy events (which models inviscid subgrid advection), and
viscous transport. The difference in the ODT cell average from before and after the evolution procedure
provides a measure of the integrated stress divergence (subgrid force). This term shows up on the right hand
side of the LES equations. The LES velocity field is corrected for continuity via a projection method; and
these corrected velocities become the new targets for the low wave number ODT adjustment, hence closing
the cycle.
The new closure method was tested by comparing simulation results against the experimental data of
Comte-Bellot and Corrsin (Reτ = 72) [JFM, 1971] for decaying isotropic turbulence (see figure below). A
direct numerical simulation of this experiment is obtainable using a 5123 grid; therefore, each ODT line
contained 512 points. For an N3 LES, ODT requires 3N2 lines (one line for each direction and each column
of LES control volumes), or 3N2 × 512 points, for this case.
Overall, satisfactory performance was obtained with the ODT closure model for the Reynolds numbers
tested, Reτ = 72 and 720. [Although not shown here, we have successfully tested this method against the
data of Kang et al. (Reτ = 720) [JFM, 2003] using the same model parameters.] Having only tested one
canonical flow, a major question facing this method is the constancy of the model parameters (there are
two: an eddy rate constant, and a viscous cut-off) for other flows. This issue is not resolved. Despite this
question mark, LES/ODT has found success where many synthetic field models have not, namely: correctly
predicting energy transfer between the resolved and subgrid scales without the use of an eddy viscosity.
Figure 1: 3d spectra of 323 LES (left) and 1d spectra of 512 ODT (right) from a coupled LES/ODT simulation
(3(32)2 × 512 ODT points). The Comte-Bellot and Corrsin data are the solid lines. The simulation results
are the dots (left) and circles (right) connected by lines. The “dip” in the ODT spectra is caused by forcing
the ODT field to match the LES field up to LES Nyquist limit.
2


Subgrid Closure for Large Eddy Simulation Using One-Dimensional Turbulence

Martin, Pino

Abstract Submitted to the Workshop on LES & SGS Modeling for Turbulent Mixing and Reactive Flows 
California Institute of Technology, December 8-9, 2003 
 Page 1 of 2   
Status of the CROCCO Laboratory: 
Understanding and Predicting Turbulent Hypersonic Flows  
Pino Martin 
Mechanical and Aerospace Engineering Department 
Princeton University 
Princeton, NJ 08544 
Phone: 609-258-7318; Fax 609-258-1993 
Email: pmartin@princeton.edu 
 
Abstract 
Progress is being made in the advancement of basic knowledge and predictive 
capabilities for turbulent hypersonic flows. For the first time, experiments and 
computations are being performed at the exact same conditions for a one-to-one 
comparison.  As part of the research, an open database of traceable, standardized 
hypersonic turbulent flows is being established.   A modular and efficient infrastructure is 
being developed so that researchers outside the Crocco Lab may be able to access the 
database and simulation tools to advance knowledge, data-mining techniques, and 
simulation procedures in fluid mechanics and other related fields.  In the proposed talk, I 
will describe how these techniques are being integrated to develop accurate turbulence 
models. 
 
Progress Report 
I founded the Crocco Laboratory in Princeton University the summer of 2002.  The 
laboratory is composed of four Ph.D. students, an undergraduate student, a lab technician, 
a dedicated PC cluster with 0.5 teraflop capability, a RAID system for data storage, in-
house developed simulation procedures, and myself.  The on-going research includes the 
development of basic knowledge and predictive capabilities for turbulent flows in 
chemical nonequilibrium.  Such a capability does not currently exist and is essential for 
the design of efficient hypersonic systems.  In this regard, the research in the Crocco 
Laboratory includes theory, simulation and collaboration with experimentalists to 
develop a basic understanding of the physics behind the interaction of chemical reactions 
and turbulent flows.  Based on the new understanding, the group develops efficient 
simulation models that accurately represent the flow physics.  In addition, we are 
establishing an open database of traceable, standardized hypersonic turbulent flows.  The 
database includes experimental and computational data.  For the firs time, these 
experiments and computations are being performed under the exact same conditions for a 
one-to-one comparison.  The data and predictive capabilities are being integrated into a 
Modular and Efficient Scientific Computing Infrastructure (MESCI) that will be remotely 
accessible. The first version of MESCI was successfully implemented this summer (E. 
Deckter, K. Schreer, B. Kernighan, and P. Martin). Accordingly, the ongoing research 
will provide:  
1. Improved knowledge of hypersonic aerodynamics and aerothermodynamics in the 
turbulent regime.  
Abstract Submitted to the Workshop on LES & SGS Modeling for Turbulent Mixing and Reactive Flows 
California Institute of Technology, December 8-9, 2003 
 
 Page 2 of 2 
2. Accurate physical models for the prediction of turbulent hypersonic flows in 
dynamic and thermo-chemical nonequilibrium.  
3. Efficient simulation capabilities to predict the characteristics of the flows of 
interest.  
4. A well-documented experimental/computational database of turbulent hypersonic 
flows.  
The goals of the on-going research are being accomplished via accurate data analysis and 
simulation of canonical flow configurations at typical hypersonic conditions using direct 
numerical and large-eddy simulation, DNS and LES respectively. These configurations 
include turbulent boundary layers with and without adverse pressure gradient; 
shockwave/turbulent boundary layer interaction configurations; and flow around 
axisymmetric and elliptical cross-section cones. Effects of Mach number, wall-
temperature conditions, and chemical reactions, as well as the effect of boundary 
conditions, numerical treatment of the governing equations, and simulation procedures 
are being studied. Also being conducted are analyses of the mean flow, flow structure, 
turbulence fields, and energy transfer mechanisms. A key feature of these simulations is 
validation against experimental data under the exact same conditions. Data is being 
obtained from experimentalists in Princeton University and California Institute of 
Technology.  In this regard, data-mining and post-processing techniques are being 
developed to make accurate comparisons between the simulation and experimental data. 
 
During my talk, I will use representative canonical flow configurations to illustrate the 
physical phenomena that we are studying.  I will also give a progress report on how the 
results from experiments, DNS and LES are being integrated to gain understanding and to 
develop accurate predictive tools and how these will be accessible to the scientific 
community. 
 
 


Status of the CROCCO Laboratory: Understanding and Predicting Turbulent Hypersonic Flows

Rider, William J.

Numerically Lucid Modeling
William J. Rider
Continuum Dynamics Group (CCS-2)
Computer and Computational Science Division
Los Alamos National Laboratory
What is numerically lucid modeling? It is acknowledging that numerical
integration itself involves considerable modeling as a matter of course
and accounting for its impact in conducting any additional modeling.
When modeling subgrid effects in a physical system, the effects of
numerically integrated the system must be accounted for.  In the vast
majority of cases, the subgrid modeling is done without regard for the
numerical integration.  This is only valid under extremely restrictive
circumstances.  These circumstances are virtually absent from the set of
applications that interest the Department of Energy.  For problems with
discontinuities such as shock waves or material interfaces methods
adaptively dissipative must be used.
Fortunately numerical methods used in most applications are built upon
our physical understanding rather than simply being numerical
approximations.  In fact, the best numerical approximations encode a
substantial amount of physics directly into the integration procedures.
Examples of this are found in upwind and Godunov methods, but the
archetypical example is the venerable artificial viscosity.
The question that must be answered is how to characterize the effects of
the numerical integration.  The technique of modified equation analysis
is a means to this end.  Even where the numerical solution is (nearly)
discontinuous this method is remarkably successful in uncovering the
effective subgrid model in a numerical method.  In most cases, the
numerical integration produces effects that are of the same order as the
subgrid effects.  Thus numerical integration must be considered hand-in-
hand with the modeling.


Numerically Lucid Modeling

Stevens, David E.

Compressible multiphase flows in an ALE framework
David Stevens
Center for Applied Scientific Computing
Lawrence Livermore National Labortory
dstevens@llnl.gov
Multiphase flows are of great interest in many applications. These are flows where particles
(or droplets) are embedded as one or multiple phases in a continuous medium that might
be chemically evolving. The number concentration of particles is prohibitively high in many
situations to track them in a Lagrange manner. In these situations, an Eulerian treatment
of the particles is necessary in which particle effects are parameterized. This approach uses
coupled sets of Euler equations for both the medium and particle phases. The Eulerian
approach is very challenging from a subgrid-scale modeling perspective in that the issues
associated with unresolved fluid mixing are compounded by the unresolved interactions of
the particles between themselves and the fluid.
This presentation describes a new approach that uses an Arbitrary Lagrangian Eulerian
(ALE) technique to simulate these flows in domains that are also allowed to evolve. This
method is being developed in conjunction with “DNS” direct numerical simulations at the
particle scale O(1mm). This allows the interactions of O(10-100) particles to be analyzed
and their interactions to be parameterized in the macro scale of the ALE method.
1


Energetic Multiphase Flows in an ALE Framework

Cook, Andrew W.

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Danish

Effective Subgrid-Scale Models for High-Resolution LES

Mattner, T.W.

Dimotakis, Paul E.

Large eddy simulations of miscible Rayleigh-Taylor instability
T. W. Mattner and P. E. Dimotakis
Introduction
Rayleigh-Taylor (RT) instability occurs when fluids
of different density are subject to an acceleration-
induced pressure gradient opposite in direction to the
density gradient. The instability is driven by buoyancy
forces due to local density variations. Direct numeri-
cal simulations (DNS) of miscible RT instability are
unable to settle basic questions concerning growth-
rate because resolution limitations prevent sufficiently
high Reynolds numbers or self-similarity from being
attained. The purpose of this paper is to apply the
stretched-vortex model to large eddy simulation (LES)
of RT instability, and compare these with DNS results.
Governing equations
The equations of motion are obtained by Favre-
filtering those used in the direct numerical simulations
of Cook & Dimotakis1. The Favre-filter is defined by
f˜ ≡
ρf
ρ
, (1)
where f is any field, ρ the density,
f ≡
∫
G(x− x′)f(x′)dx′, (2)
x the spatial coordinate vector, and G the filter func-
tion. The fluids are assumed to be incompressible and
the density variable due to changes in mixture com-
position. The Favre-filtered heavy-fluid mass-fraction,
Y˜ , and the filtered heavy-fluid mole-fraction, X, are
related to the filtered density, ρ, by
1
ρ
=
Y˜
ρ2
+
1− Y˜
ρ1
and X =
ρ− ρ1
ρ2 − ρ1
, (3)
where ρ1 and ρ2 are the light- and heavy-fluid den-
sities, respectively. The Favre-filtered continuity,
species-transport, and momentum equations are
∂ρ
∂t
+
∂ρu˜j
∂xj
= 0, (4a)
∂ρY˜
∂t
+
∂ρu˜j Y˜
∂xj
=
∂
∂xj
(
ρD
∂Y
∂xj
)
−
∂ρqj
∂xj
, (4b)
∂ρu˜i
∂t
+
∂ρu˜iu˜j
∂xj
= −
∂p
∂xi
+
∂τ ij
∂xj
−ρgδi3−
∂ρTij
∂xj
, (4c)
where D is the diffusivity, p the pressure, τij the vis-
cous stress tensor, g the acceleration, qj the subgrid-
scale (SGS) scalar flux, and Tij the SGS stress tensor.
Equations (3), (4a), and (4b) imply
∂u˜j
∂xj
= −
∂
∂xj
(
D
ρ
∂ρ
∂xj
)
−
(
1
ρ2
−
1
ρ1
)
∂ρqj
∂xj
(5)
and this constraint is used instead of (4b).
Equations (4a), (4c), and (5) are closed using the
stretched-vortex SGS stress model of Misra & Pullin 2
and the stretched-vortex SGS passive scalar mixing
model of Pullin3. These uniform-density models are
adpated to the variable-density case following Kosovic
et al.4. For a single subgrid vortex, the SGS stress is
Tij = K(δij − eiej) (6)
where K is the subgrid kinetic energy per unit mass,
and e = (e1, e2, e3) is the unit vector of the subgrid
vortex axis. The SGS scalar flux is
qj = −
1
2
∆K1/2(δij − eiej)
∂Y˜
∂xi
(7)
where ∆ is the local mesh size. Together with (3) and
(5), this implies
∂u˜j
∂xj
= −
∂
∂xj
(
D
ρ
∂ρ
∂xj
)
−
∂
∂xj
[
1
2
∆K1/2(δij − eiej)
1
ρ
∂ρ
∂xi
]
. (8)
The diffusive (and viscous stress) term is written such
that the original unfiltered equation is recovered when
contributions from the model are small. In our numeri-
cal scheme, the divergence of the Favre-filtered velocity
is used in the solution of a variable-coefficient ellip-
tic equation for the filtered pressure. K is estimated
from a prescribed energy spectrum, either E(k) =
K0
2/3k−5/3 (Kolmogorov spectrum without viscous
cutoff) or E(k) = K0
2/3k−5/3 exp[−2k2ν/(3|a|)]
(Lundgren spiral spectrum), where k is the wavenum-
ber, K0 the Kolmogorov prefactor,  the local cell-
averaged dissipation rate, ν the kinematic viscosity,
and |a| the axial strain along the subgrid vortex
axis.4,5 The group K0
2/3 is estimated from the local
second-order velocity structure function using circular
averages in the homogeneous plane.5 The proportion
of subgrid vortices aligned with the principal exten-
sional eigenvector of the resolved rate-of-strain tensor,
S˜ij , and the resolved vorticity vector, ω, is given by
λ and (1− λ), respectively, where λ = λ3/(λ3 + ||ω||)
and λ3 is the principal extensional eigenvalue (model
1b2).
Results
Figure 1 shows a constant mole-fraction isosurface
from a 1282 × 512 LES of miscible RT instability, at a
1 of 2
Fig. 1 X = 1/2 isosurface at a density ratio ρ2/ρ1 =
3. Red indicates heavy-fluid (X = 1) and blue light-
fluid (X = 0).
density-ratio of 3. The acceleration is in the vertical di-
rection. The top and bottom of the domain are no-slip
walls, and the transverse homogeneous plane is peri-
odic. This simulation uses the unbounded Kolmogorov
spectrum for the subgrid energy. The Reynolds num-
ber is about 18,000, where the Reynolds number is
defined by the thickness of the mixing zone, its rate
of growth, the arithmetic mean density and the vis-
cosity. For comparison, the highest Reynolds number
obtained by DNS is around 5,500 on a 5122 × 2040
grid.6
In this example, the density field exhibits
small-amplitude (∼ 1% ρ1) high-wavenumber un-
der/overshoots (ρ < ρ1 or ρ > ρ2) which are, however,
unphysical. In the implmentation described here, the
excursions are controlled by means of a circular spec-
tral filter in the homogeneous plane, and a compact
Pade filter in the vertical inhomogeneous direction.
The filters are used to control wavenumbers in the
range k > kc, where kc = pi/∆ is the model cutoff
wavenumber. Without such filtering, the amplitude of
the excursions may grow to as large as 10% ρ1.
Figure 2 shows mixing-zone growth as measured
by the bubble and spike heights, hB and hS, respec-
tively. These are defined in terms of the lateral av-
erage filtered mole-fraction, 〈X〉, which is a function
of the vertical coordinate, z, as 〈X〉(z = hB) = 0.99
and 〈X〉(z = hS) = 0.01. In the absence of exter-
nally imposed length-scales, and at sufficiently large
Reynolds number, classic dimensional analysis sug-
gests the growth will be quadratic. In this example,
bubble growth is almost linear.
Performance of the model is assessed by comparison
of coarse resolution LES against fully-resolved DNS
t / τ
h B
,
h S
0 1 2 3 4 5-1
-0.5
0
0.5
1
Fig. 2 Mixing-zone growth. hB > 0, hS < 0.
τ =
√
L/Ag, where L is the lateral dimension of the
computational domain and A = (ρ2 − ρ1)/(ρ2 + ρ1).
at three density-ratios: 5/3, 3, and 7. The initial
conditions for the LES are obtained by filtering the
DNS data at a time when the mixing-zone has grown
sufficiently thick to be represented on the coarse LES
grid. The Reynolds numbers for these tests are limited
to less than 4,000. Statistics for comparison include
mixing-zone width, mole-fraction mean and variance,
kinetic energy and dissipation rate, and spectra.
Acknowledgements
We gratefully acknowledge the help of D. I. Pullin
in implementing the subgrid vortex model and A. W.
Cook for supplying the DNS code on which the current
LES implementation is based.
References
1 A. W. Cook and P. E. Dimotakis. Transition stages
of Rayleigh-Taylor instability between miscible flu-
ids. J. Fluid Mech., 443:69–99, 2001.
2 A. Misra and D. I. Pullin. A vortex-based subgrid
stress model for large-eddy simulation. Phys. Fluids,
9:2443–2454, 1997.
3 D. I. Pullin. A vortex-based model for the subgrid
flux of a passive scalar. Phys. Fluids, 12:2311–2319,
2000.
4 B. Kosovic, D. I. Pullin, and R. Samtaney. Subgrid-
scale modeling for large-eddy simulations of com-
pressible turbulence. Phys. Fluids, 14:1511–1522,
2002.
5 T. Voekl, D. I. Pullin, and D. C. Chan. A physical-
space version of the stretched-vortex subgrid-stress
model for large-eddy simulation. Phys. Fluids,
12:1810–1825, 2000.
6 A. W. Cook and Y. Zhou. Energy transfer in
Rayleigh-Taylor instability. Phys. Rev. E, 66, 2002.
2 of 2


Large Eddy Simulations of Miscible Rayleigh-Taylor Instability

Hill, David J.

Pullin, Dale I

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Large-Eddy Simulations of a Richtmyer-Meshkov Instability With Re-Shock

Winckelmans, Gregoire S.

Georges, L.

A link between energy conserving schemes and LES models of
tensor-diffusivity type
G. Winckelmans and L. Georges, Universite´ catholique de Louvain (UCL), 1348 Louvain-la-
Neuve, Belgium
Large-eddy simulation (LES) mainly consists of a “projection”: an effective “truncation” to much
fewer modes than what is required to perform a direct numerical simulation (DNS). This can be done
in different ways: using a reduced set of discrete field values (formally equivalent to sampling, on the
coarse grid, of the continuous DNS field), or using a reduced set of continuous functions, etc. One
can also first assume a filtering of the DNS field (using an regular filter of effective width ∆; this is
basically the view in the method of finite volumes, where the filter is a top hat of width ∆ = h), but
this step is necessarily followed by the projection step.
The main property of a projector is that applying the projector twice is like applying it once.
Letting u˜ be the LES projection of u (i.e., the u field as “seen” on the coarse LES grid), one indeed
has that ˜˜u = u˜: projection is idempotent. The product of the LES fields is also captured on the LES
grid. Thus, in LES, one really solves a “tilde” equation (as all terms have the same status):
∂tu˜i + ∂j ˜˜uiu˜j + ∂iP˜ + ∂j τ˜ij = ν ∂j∂j u˜i (1)
with τ˜ij = u˜i uj − ˜˜uiu˜j the effective subgrid-scale stress (SGS): the product of DNS quantities minus
the product of LES quantities, projected on the LES grid. The effect of this term must be modelled.
As mentioned previously, one can do LES with regular filtering of effective width ∆ before the
projection step, such filtering being explicitely added or implicitely assumed (as in finite volumes)
and not idempotent (u 6= u). One then solves LES equations for u˜: the projection/truncation of
the filtered field u. There is a then a subfilter-scale stress (SFS) and a subgrid-scale stress (SGS).
All terms in the effective LES equation still have a “tilde” status (see, e.g., JFM (2001), vol. 441,
pp. 119-138). This added filtering concept can also be used in spectral methods (see e.g., Phys.
Fluids (2001), vol. 13(5), pp. 1385-1403), yet with little added value (due to the almost perfect
numerical behavior of spectral methods). It appears that the potential interest of filtering lies more
in grid-based methods. Once the filter is characterized (not an easy step with implicit filtering),
one can use “deconvolution tools” to recover the SFS stress: high level deconvolution using van
Cittert iterative method or the direct SFS reconstruction series (limited to second order filters); low
level deconvolution using the tensor-diffusivity model (TDM, which is the first term of the direct
reconstruction series, and is same for all second order filters of same effective width, see e.g., JFM
2001), or the scale-similarity model, or one van Cittert iteration.
Second, it is also important to recognize that the numerics can be seen as a filter acting on
the scales of the LES grid (from k = 0 to k = pi/h): the concept of modified wavenumber. The
numerical evaluation of a derivative can be expressed as effective filtering of the exact derivative or,
equivalently, as the exact derivative of a filtered function. The effective numerical filter then depends
on the order of the derivative and on the method used (order and type). Recall also that spectral
methods have a perfect wavenumber behavior: no numerical filter.
Third, the equations being non-linear, the discrete energy conservation (in absence of viscosity
and/or SGS model, of course) is strongly affected by the discretization of the non-linear convective
term. Dealiased spectral methods are also efficient in that respect, as they lead to discrete energy
conservation. The same property is desirable for the simulation of turbulent flows (DNS and LES)
using grid-based methods: it is important to have schemes with good discrete energy conservation
properties. For LES, this also guarantees to better distinguish between the effects of the added SGS
model and the numerics.
We here first consider the simplest grid-based method: second order finite differences. We consider
the different ways to discretize the non-linear convective term (advective, divergence and mixed
1
forms), and we focus on the discrete energy conservation.
We first consider the case of Burger’s equation in 1-D. The energy conservation properties of
second order finite differences are then shown to be related to the discrete skewness which is negative
in the “turbulent” regime (as for the 3-D Navier Stokes equations). One then obtains that:
• the advective form produces energy;
• the divergence form evaluated using a compact stencil produces energy (yet four times less
than the advective form). We here call it the “compact divergence form”;
• the divergence form evaluated using a wide stencil dissipates energy. We here call it the “wide
divergence form”.
For each form, a correction term is developed that leads to an enhanced scheme with good energy
conservation properties (exact conservation in the cases of the enhanced advective and compact
divergence forms). It is noted that the added term is similar to various forms (advective or divergence)
of the “non-linear deconvolution model” (i.e., TDM when in 3-D) used in some LES approaches with
explicit filtering to partially recover the SFS stress (e.g., as in Phys. Fluids 2001).
The correction term that appears in the enhanced compact divergence form (also the most natural
discretization form) is seen to be formally equivalent to the TDM with ∆ = h: the filter width ∆
to use in the TDM correction term is thus equal to the grid size h. In the present context, this
supports the view that this correction term essentially amounts to partial deconvolution of the SFS
stress corresponding to an implicit filter of effective width equal to the grid size.
We also notice that one must remain limited to the lowest order correction term: the TDM
term; using more terms of the SFS reconstruction series would degrade the performance of those
corrected schemes. We also remark that deconvolution methods that are based on “refiltering the
LES field” (such as the van Cittert iteration, or the scale-similarity model) cannot be used here, as
any filter applied in physical space necessarily has an effective width greater than the grid spacing.
The TDM-like approach is thus also special in that respect.
An new link is also made between the proper numerical evaluation of the non-linear convective
derivative (i.e., proper evaluation of the “product rule” for derivatives) and the TDM-like term. This
also relates to the concept of numerical filter (effect of modified wavenumber behavior on the product
rule). We believe that this new formalism constitutes a tool for better understanding and/or deriving
energy conserving scheme. Its extension to higher order finite difference methods is ongoing work.
The case with added (explicit) filtering, thus of effective width ∆ > h, is considered next. So is
the case of LES: thus with added truncation and SGS modelling.
Supporting numerical results are presented, using the forced Burger’s equation (white noise forc-
ing) with fixed mesh and for different viscosities: DNS ranging from very well-resolved to much
under-resolved, and also LES. Cases with added (explicit) filtering are also presented. The results
are compared to solutions obtained using the reference method: the spectral method with dealiasing.
Finally, the case of the 3-D Navier Stokes equations is also considered, with respect to the existing
second order schemes used to discretize the non-linear convective term. Starting from non energy
conserving schemes, it is shown that one can again recover energy conserving schemes by adding
TDM-like elements. The results obtained so far are presented.
2


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