CaltechCONF

McWilliams, James C.

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Czech

Wave-Averaged Effects in Marine Boundary Layers

Mitarai, Satoshi

Riley, James J.

Kosaly, George

A Priori Testing of Turbulent Mixing Models for Subgrid-scale
Mixing and Reaction ∗
Satoshi Mitarai, James J. Riley, and George Kosa´ly
Department of Mechanical Engineering,
University of Washington, Seattle, WA 98195, USA
Using the data obtained from direct numerical simulations (DNS), a priori testing
of turbulent mixing models for large-eddy simulations of non-premixed turbulent
combustion have been tested, in particular the IEM (Villermaux and Devillon[1]),
the modified Curl (Janicka et al.[2]) and the EMST (Subramaniam and Pope[3]).
The DNS is performed for single-step, non-premixed reactions developing in incom-
pressible, isotropic, decaying turbulence where local flame extinction and re-ignition
occur (cf. Figs. 1 and 2); the simulation domain is divided into subvolumes for test-
ing. In order to demonstrate the ”true” performance of the mixing models, the
temporal evolution of the fluid particle positions and the exact value of the mixing
frequencies within each subvolume are obtained from the DNS, and are utilized in
testing. Comparison of the model predictions with the DNS data shows that the
EMST mixing model yields reasonable filtered quantities, such as the filtered tem-
perature and filtered reaction rate, while the IEM and the modified Curl mixing
models tend to underestimate the filtered quantities (cf. Figs. 3 and 4).
A priori testing of a new mixing model proposed by the authors (Mitarai et al. [4])
will also be presented. Among other things, this new mixing model accounts for the
difference between the mixing of passive and of reactive scalars, while the presently
available mixing models do not distinguish between them. Given the Lagrangian
time series of the mixture fraction and its dissipation rate, this new model well
describes the qualitative behavior of the Lagrangian time series of reactive scalars,
and yields accurate quantitative statistics.
∗ Supported by NSF grants CTS-9415280 and CTS-01339250.
2FIG. 1: Temporal evolution of the mixture fraction
field on a two dimensional slice through the compu-
tational domain.
FIG. 2: Temporal evolution of temperature field on
the same two dimensional slice through the compu-
tational domain as in Fig. 1.
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
 time=0.0 
filtered T (DNS)
filt
er
ed
 T
 (m
od
el)
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
 time=0.3 
filtered T (DNS)
filt
er
ed
 T
 (m
od
el)
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
 time=1.0 
filtered T (DNS)
filt
er
ed
 T
 (m
od
el)
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
 time=2.0 
filtered T (DNS)
filt
er
ed
 T
 (m
od
el)
FIG. 3: The filtered temperatures computed from
the modified Curl mixing model[2]) plotted against
those from the DNS.
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
 time=0.0 
filtered T (DNS)
filt
er
ed
 T
 (m
od
el)
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
 time=0.3 
filtered T (DNS)
filt
er
ed
 T
 (m
od
el)
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
 time=1.0 
filtered T (DNS)
filt
er
ed
 T
 (m
od
el)
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
 time=2.0 
filtered T (DNS)
filt
er
ed
 T
 (m
od
el)
FIG. 4: The filtered reaction rate computed from
the EMST mixing model[3] plotted against those
from the DNS.
[1] J. Villermaux and J. C. Devillon, in 2nd International Symposia on Chemical Reaction Engi-
neering (ISCRE, Netherlands, 1972), p. B1.
[2] J. Janicka, W. Kolbe, and W. Kollmann, J. Non-Equilib. Thermodyn. 4, 47 (1979).
[3] S. Subramaniam and S. B. Pope, Combust. Flame 115, 487 (1998).
[4] S. Mitarai, G. Kosa´ly, and J. J. Riley (2003), submitted to Combust. Flame.


English

A Priori Testing of Turbulent Mixing Models for Subgrid-scale Mixing and Reaction

Schilling, Oleg

Cabot, William

Workshop on LES & SGS Modeling for Turbulent Mixing and Reactive Flows, 
8-9 December 2003, California Institute of Technology, Pasadena, CA 
 
Subgrid Scale Dynamics of Rayleigh-Taylor Turbulence and Mixing 
 
Oleg Schilling and William H. Cabot 
 
University of California, Lawrence Livermore National Laboratory 
 
The energy transfer process and the interaction of different scales in a flow induced by 
the variable-density Rayleigh-Taylor instability in miscible fluids is investigated using a 
three-dimensional DNS dataset with a resolution of 5122 × 2040. The method used to 
quantify the energy transfer between the supergrid and subgrid scales in the 
homogeneous planes, determined by partitioning the modes into resolved and unresolved 
scales defined by a two-dimensional, artificial cutoff wave number kc in Fourier space, is 
applied to the kinetic energy evolution equation. Using a sharp Fourier cutoff filter, the 
kinetic energy transfer is decomposed into the resolved part, a part corresponding to the 
interaction between resolved and unresolved scales, and a part corresponding to the 
interaction between unresolved scales. These z-dependent spectra are computed for kc = 
8, 16, and 32 to investigate the dependence of the transfer process on the range of scales 
contributing to the subgrid interactions. The kinetic energy transfer is further decomposed 
into its positive and negative components corresponding to the forward and backward 
cascades of energy, respectively. The decomposition into resolved and unresolved scales 
is used to define an effective eddy and backscatter viscosity. The principal conclusions 
are: (1) the transfer spectra and eddy viscosities exhibit a strong dependence on kc; (2) the 
contributions from the interaction between resolved and unresolved scales dominate the 
contribution to the total subgrid eddy viscosities and are responsible for the cusp at large 
k/kc; (3) the contributions from the interaction between unresolved scales dominates the 
contribution to the total subgrid eddy viscosities at small k/kc and are responsible for the 
small, negative contribution, and; (4) backscatter is strongest in the regions near the 
boundaries of the mixing layer. The implications of these results for subgrid-scale 
modeling in large-eddy simulation (LES) of Rayleigh-Taylor instability-induced 
turbulence are discussed. 
 
This work was performed under the auspices of the U.S. Department of Energy by the 
University of California Lawrence Livermore National Laboratory under contract No. W-
7405-Eng-48. 
 


Subgrid Scale Dynamics of Rayleigh-Taylor Turbulence and Mixing

Holm, Darryl D.

LAGRANGIAN AVERAGED NAVIER-STOKES-ALPHA
(LANS − α) EQUATIONS FOR MODELING TURBULENCE
D. D. Holma,b
aTheoretical Division and Center for Nonlinear Studies
Los Alamos National Laboratory
Los Alamos, NM 87545, USA
dholm@lanl.gov
bMathematics Department
Imperial College of Science, Technology, and Medicine
London SW7 2AZ, UK
LES & SGS Modeling For Turbulent Mixing and Reactive Flows December 8-9, 2003, at Caltech
Lagrangian averaging (LA) is producing new ideas and methods for the construction of novel subgrid
models for momentum, energy and scalar transport in turbulence. LA is applied at ﬁxed Lagrangian
ﬂuid label, rather than at ﬁxed spatial location, as in Eulerian averaging and ﬁltering for RANS and LES
equations, respectively. Instead, LA is performed in a frame of motion that follows along with the main
turbulent eddies. The LA process commutes with the advective time derivation; so LA ﬂuid equations have
their own circulation theorem, potential vorticity conservation, and other Lagrangian transport invariants
of importance in applications. Thus, LA produces a new class of turbulence models that preserve the
coherence of spatio-temporal ﬂow features.
Although, the LA turbulence equations are formally similar to LES turbulence equations, they arise from
diﬀerent principles. Formally, LA produces equations that embody a “regularization principle,” involving
an explicit ﬁlter and its inversion. This regularization principle allows a systematic derivation of the
implied subgrid-model, which resolves the closure problem. Hence, both the interpretation of LANS−α
predictions in terms of direct simulation results, as well as the corresponding subgrid closure are speciﬁed
by the ﬁlter appearing in these equations. However, the LANS−α equations cannot be obtained by directly
applying the LES ﬁltering method. This is because the theoretical basis for obtaining this alternative to
the LES approach is Lagrangian averaging, not spatial ﬁltering. We emphasize that Lagrangian averaging
is a closely related alternative to the LES and RANS approaches, not a subset of them.
We shall report the results of a variety of tests of the LANS−α equations. These tests include comparisons
with data from pipe and channel experiments, with results of direct numerical simulations of turbulence
decay and with results of competing LES models, such as the dynamic model, for shear-driven mixing.
References
[1] B. J. Geurts and D. D. Holm, Regularization modeling for large-eddy simulation.
Phys. Fluids , v. 15, L13-L16, 2003.
[2] C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of ﬂuid turbulence.
Physica D 152 (2001) 505-519.


Lagrangian Averaged Navier-Stokes-Alpha

Sutherland, James C.

Smith, Philip J.

Chen, Jacqueline H.

(1)
(2)
A priori Evaluation of Mixing and Reaction Models for LES of Combustion
James C. Sutherland and Philip J. Smith
Department of Chemical & Fuels
Engineering
University of Utah
Salt Lake City, UT, 84098
Jacqueline H. Chen
Combustion Research Facility 
Sandia National Laboratories
Livermore, CA, 94551
This work focuses on validation of Large Eddy Simulation (LES) submodels for scalar mixing and
reaction using Direct Numerical Simulation (DNS).  Previous attempts at validation of LES combustion
submodels using DNS have been limited to simple kinetics (often single-step) and transport.  The present
study considers DNS of CO/H2-Air and CH4/H2-Air jets using detailed chemical kinetics and
mixture-averaged transport.  The DNS code used for this study uses 8th order spatial discretization and a 4th
order Runge-Kutta integrator.  A priori tests are performed for several LES mixing and reaction models.
Beta and clipped-gaussian presumed PDF models are used in conjunction with equilibrium and steady
flamelet reaction models to obtain LES state variables such as temperature and species mass fractions.
Comparisons are made between filter-scale state variables predicted by the models and the filtered DNS
values.  The effects of filter size and differential diffusion on model performance are investigated.
In the most abstract sense, reaction models map the state of the system, Ni onto a small set of reaction
variables, 0j,  .  The reaction models considered in this study are the equilibrium model (EQ)
and steady laminar flamelet model (SLFM).  These are coupled with $ and clipped-gaussian presumed-shape
PDF mixing models, which are parameterized by the mean and variance of the reaction variables.  The
filtered state variables are then given by
The tildes in equation 1 indicate Favre-filtered quantities.  The EQ model is parameterized by the mixture
fraction, , while SLFM introduces the dissipation rate as an additional reaction variable,
. 
These models are tested by first selecting a filter width ()) relative to the DNS grid spacing and
extracting the Favre-filtered reaction variables (and their variances).  Next, the model-predicted state is
computed from equation 1, and then compared to the DNS-filtered state variables.  When )=1 (the DNS
scale), the PDF’s in equation 1 reduce to *-functions, and the reaction model alone is the only source of
modeling error.  As ) increases, the mixing model begins contributing to the modeling error.  Thus, by
examining the modeling error as a function of ), we may determine the relative error contribution from the
mixing and reaction models independently.  This is important since in some regions of the flow, the scalar
fields may be well-resolved (i.e. the variance of the reaction variables is low and the PDF becomes a *-
function).  The modeling error is quantified as the ensemble average of the percent error conditioned on the
mixture fraction,
Standard deviations of this quantity are also considered.
Filter widths up to )=16 are considered in evaluating the model performance.  For these filter widths,
the reaction models are the primary source of error for most state variables.  In all cases, SLFM performs
better than EQ in prediction of all state variables.  Significant shortcomings in SLFM are observed, however,
even in cases far from extinction.  Shortcomings are attributed to differential diffusion and transient/time-
Figure 1 - EQ model performance for CO /H2 jet. Figure 2 - SLFM model performance for CO /H2 jet.
Figure 4 - Conditional percent error on temperature for
CH4/H2 jet.  Red point indicates stoichiometric.Figure 3 - SLFM performance for CH4/H2 jet.
history effects, neither of which is treated by SLFM or EQ.  The attached figures show a brief summary of
model performance for temperature for two cases: a CO/H2 jet and a CH4/H2 jet.  Figures 1 and 2 show the
EQ and SLFM model performance at )=1 for the CO/H2 jet.  
Figure 3 shows temperature for the CH4/H2 jet.  Figure 4 shows conditional percent error as defined
by equation 2 for the CH4/H2 jet.  Interestingly, the largest error occurs away from the stoichiometric mixture
fraction.  Further analysis shows that this is likely due to differential diffusion.


A priori Evaluation of Mixing and Reaction Models for LES of Combustion

Kosovic, Branko

A Proposal for A Two-Parameter Spectral Turbulence Closure
Branko Kosovic´
Lawrence Livermore National Laboratory
The simplest spectral turbulence model is based on a spectral eddy viscosity. The first spectral eddy
viscosity model was developed by Heisenberg (1948). Spectral eddy viscosity, νe(k|kc, t) is defined as:
νe(k|kc, t) = −
Te(k|kc, t)
k2E(k, t)
Spectral eddy viscosity is a function of the wave number k, cut-off wave number kc, and time t. Spectral
energy transfer is denoted with Te(k|kc, t) and E(k, t) is spectral energy density. Here, it is implicitly assumed
that the spectral eddy viscosity depends only on the spectral energy density and its transfer.
Andre´ and Lesieur (1977) have recognized the importance of helicity in modulating energy transfer from
large to small scales in high Reynolds number isotropic turbulence. However, only recently has the role of
helicity in energy transfer been studied in greater detail e.g. Holm and Kerr (2002), Chen et al. (2003).
In order to correctly predict energy transfer in large eddy simulation of turbulent flows, the effect of
helicity on the spectral energy transfer must be accounted for in subgrid turbulence models. We therefore
develop a two-parameter spectral model that accounts for the helicity effects in homogeneous incompressible
turbulent flow. In such a flow, Fourier transforms of velocity and vorticity span locally two-dimensional
space perpendicular to a wave vector k. Velocity and vorticity vectors in Fourier space form a complete
basis that can be used to represent the Fourier transform of the divergence-free projection of the nonlinear
term and its components. We use the velocity-vorticity basis to develop a two-parameter spectral model.
By using the evolution equations for the two invariants of three-dimensional Euler equations, kinetic energy
and helicity, and assuming that the flow is isotropic and has reached statistical equilibrium, we derive two
equations for two unknown closure parameters. We finally arrive at a system of two algebraic equations with
two unknowns represented by two model parameters.
The incompressible Navier-Stokes equations in Fourier space are[
d
dt
+ νk2
]
uˆi (k, t) = −ζˆ
<
i (k, t)− ζˆ
>
i (k, t)
Here, ζˆ<
i
(k)+ ζˆ>
i
(k) represents the divergence-free projection of the Fourier transform of the nonlinear term.
Both resolved, ζˆ<
i
, and subgrid, ζˆ>
i
(k), components are divergence free and therefore perpendicular to the
local wave vector k.
The Fourier transform of the vorticity vector is ωˆi = i εimnkmuˆn. Notice now that vector pairs
[< (uˆi) ,= (ωˆi)] and [= (uˆi) ,< (ωˆi)] represent two orthogonal bases for a complex two-dimensional space
locally perpendicular to the wave vector k.
The subgrid component of the nonlinear term can be expressed as a linear combination of basis vectors
ζˆ>i = a < (uˆi) + b = (ωˆi) + i c < (uˆi) + i d = (ωˆi)
Here, coefficients a, b, c, and d are real and depend on the wave vector, k, and possibly time, t.
We now assume that the turbulence flow under consideration is isotropic and that a(k), b(k), c(k) and
d(k) are functions of the wave number magnitude only. Due to the realizability condition, which requires
that both spectral energy density and spectral helicity density are real-valued functions, parameters b(k)
and c(k) must vanish. Using the evolution equations for the spherically-integrated spectral densities and
assuming that the statistical equilibrium has been reached, the two equations above can be solved for closure
parameters a(k) and d(k)
a(k) =
(2νk2E + T<e )k
2E − (2νk2H + T<
h
)H
k4E2 −H2
1
      
	  

     
	 







  
  
  	
 


   ﬀﬂﬁ ﬃ 

  
ﬁ  
!
 
"
 
#
 
# $ %
#
 
$ %
 
&
 
$ %
'( )* ) +-,ﬂ. / 0
)
'( ) 0
. 12 13 14 1
4 5 6
4
1 5 6
1
7
1 5 6
Figure 1: Spectral eddy viscosity, νe(k|kc, t), model parameter d(k|kc, t).
d(k) =
(2νk2H + T<
h
)E − (2νk2E + T<e )H
k4E2 −H2
Here, H is the spectral helicity density, while T<
h
and T>
h
are resolved and subgrid helicity transfer functions,
respectively. Similarly, T<e and T
>
e are resolved and subgrid energy transfer functions.
While the one-parameter spectral eddy viscosity (Heisenberg, 1948) is not sufficient to model the subgrid
component ζ>
i
of the nonlinear term in the momentum equation, the two-parameter model is complete.
Various statistical theories of turbulence have been used to determine spectral eddy viscosity defined by
Heisenberg (1948). Kraichnan (1976) used the test field method and predicted a so called “cusp” in spectral
eddy viscosity at the wave number cut-off. The existence of such a cusp was also confirmed using direct
numerical simulation (DNS) of Navier-Stokes equations by Domaradzki et al. (1993). For comparison to the
spectral eddy viscosity, we compute parameters a(k) and d(k) of the proposed spectral turbulence model
using the results of DNS. We filtered DNS results obtained at 256 grid-point resolution using a wave cut-off
filter with cut-off wave number kc = 48 and computed the subgrid transfer of energy and helicity in addition
to energy and helicity spectra. Spectral eddy viscosity νe(k|kc, t), and two model parameters corresponding
to the new proposed spectral turbulence model are given in figures below 1, 2, and 3. The new two-parameter
model is complete and does not exhibit the “cusp.”
References
Andre´, J. C. and M. Lesieur, 1977: Influence of helicity on high Reynolds number isotropic turbulence.J.
Fluid Mech., 81, 187–207.
Chen, Q., S. Chen, and G. L. Eyink, 2003: The joint cascade of energy and helicity in three-dimensional
turbulence. Phys. Fluids, 15, 361–374.
Domaradzki, J. A., W. Liu, and M. E. Brachet, 1993: An analysis of subgrid-scale interactions in numerically
simulated isotropic turbulence. Phys. Fluids A, 5, 1747–1759.
Heisenberg, W., 1948: Zur statistischen theorie der turbulenz. Z. Phys., 124, 628–657.
Holm, D. D. and R. Kerr, 2002: Transient vortex events in the initial value problem for turbulence. Phys.
Rev. Lett., 88, 244501.
Kraichnan, R. H., 1976: Eddy viscosity in two and three dimensions. J. Atmos. Sci., 33, 1521–1536.
2


A Proposal for a Two-Parameter Spectral Turbulence Closure

Moser, Robert D.

Zandonade, P.

Vedula, P.

Is Optimal LES a Practical Approach to LES Modeling?
R. D. Moser, P. Zandonade and P. Vedula
Department of Theoretical and Applied Mechanics
University of Illinois at Urbana-Champaign
104 S. Wright St., Urbana IL, 61081
r-moser@uiuc.edu, 217-244-7728
The shortcomings of most current LES modeling techniques are well known. Two of the more serious
are near-wall modeling and the ambiguities introduced by practical (i.e. low-order) numerical discretization.
Optimal LES, in which stochastic estimation techniques are used to formally optimize the LES model, can
in princible address these problems. The modeling procedure yields an approximation to the ideal LES evo-
lution, which is guaranteed to reproduce the single-time statistics of the filtered turbulence, and to minimize
the expected difference between the evolution of a filtered turbulence and the LES. Using direct numerical
simulation (DNS) statistical data to perform the estimates, several such models have been formulated for
different turbulent flows and different LES filter definitions. These models perform remarkably well. They
also yield important insights into the required properties of good LES models. To make these models useful
however, it is necessary to eliminate the need for detailed statistical data from DNS. When the small scales
can be considered isotropic, this can be accomplished through a combination of Kolmogorov inertial range
scaling, the quasi-normal approximation and a dynamic procedure, and the resulting model is as accurate as
that based on DNS data. Near walls, a variety of other theoretical considerations significantly constrain the
required statistics, and using this, a formulation requiring minimal empirical input is being devised. In this
talk, we will examine the development of the optimal LES to date, as outlined above, and the prospects that
this technique will be of value in developing practical LES models that address outstanding LES modeling
problems.
1


Is Optimal LES a Practical Approach to LES Modeling?

Grinstein, Fernando F.

ON IMPLICIT SUBGRID SCALE MODELING FOR TURBULENT FLOWS
Fernando F. Grinstein
LCP&FD, Naval Research Laboratory, Washington, DC 20375-5344
Large Eddy Simulation (LES) is an effective intermediate approach between DNS and
RANS, capable of simulating flow features which cannot be handled with RANS such as signifi-
cant flow unsteadiness and strong vortex-acoustic couplings, and providing higher accuracy than
RANS at reasonable cost but still typically an order of magnitude more expensive. Desirable
modeling choices involve selecting an appropriate discretization of the flow problem at hand,
such that the LES cut-off lies within the inertial sub-range, and ensuring that a smooth transition
can be enforced at the cut-off. The main assumptions of LES are: (i) that transport is largely
governed by large-scale unsteady features and that such dominant features of the flow can be
resolved, (ii) that the less-demanding accounting of the small-scale flow features can be un-
dertaken by using suitable Sub Grid Scale (SGS) models. In the absence of an accepted universal
theory of turbulence, the development and improvement of SGS models are unavoidably
pragmatic and based on the rational use of empirical information. Classical approaches have
included many proposals ranging from, inherently-limited eddy-viscosity formulations, to more
sophisticated and accurate mixed models, e.g., [1].
For the sake of the discussion, we restrict ourselves to the incompressible regime, and
focus on the crucial LES closure issue of modeling the SGS stress tensor, 
† 
B = v ƒ v - v N ƒ v N ,
which needs to be prescribed in terms of the represented-filtered velocity v N , where the upper bar
is denotes filtered quantities. The SGS stress tensor can be conveniently recast in the form
† 
B = (v ƒ v - vN ƒ vN ) + (vN ƒ vN - v N ƒ v N ) = B1 + B2 , where B1 denotes the interaction between
represented and non-represented scales – which is not known a priori – and therefore must be
modeled, whereas B2 relates to the interaction between filtered and discretized represented scales
which can be approximated by prescribing an estimated vN  in the represented-velocity space
(i.e., the solution to the so-called soft deconvolution problem), [2]. In this framework, a basic
structural SGS model such as the scale-similarity model provides B2, and the eventual need of
mixed models results from the recognition that  B 2  is not dissipative enough and a secondary
regularization through B1 is needed, i.e., an approximation to v in physical-velocity space must be
prescribed (the hard deconvolution problem).
The main drawback of mixed models relates to their computational complexity, and
ultimately, to the fact that well-resolved (discretization-independent) LES is prohibitively expen-
sive for the practical flows of interest at moderate-to-high Re. In fact, because of the need to
distinctly separate (i.e. resolve) the effects of explicit filtering and SGS reconstruction models
from those due to discretization, carrying out such well-resolved LES can typically amount in
practice to performing a coarse DNS. As a consequence, it has been argued that the use of hybrid
RANS/LES models for realistic whole-domain complex configurations might be unavoidable in
the foreseeable future, e.g., [3]. This has recently led many researchers to abandoning the
classical LES formulations, shifting the focus directly to the SGS modeling implicitly provided by
non-linear stabilization achieved algorithmically, through use of a particular class of numerical
schemes, or based on regularization of the discretization of the conservation laws, [4].
The traditional approaches motivated by physical considerations on the energy transfer
mechanism from resolved to subgrid scales, express B1 in terms of an appropriate functional B1
(e.g. an eddy-viscosity SGS model), and seek sufficiently high-order discretization and grid
resolution to ensure that their effects are sufficiently small. However, it can also be argued that B1
might be implicitly provided by discretization if non-linear stabilization can be achieved al-
gorithmically, through use of a particular class of numerical schemes, or based on regularization
of the discretization of the conservation laws. In fact, most schemes can potentially provide built-
in or implicit SGS models enforced by the discretization errors, provided that their leading order
terms are dissipative. We are thus lead to the natural question: to what extent can we avoid the
(explicit) filtering and modeling phases of LES (i.e., B2≡0) and focus on the implicit B1  provided
by a suitably-chosen discretization scheme?
Not all implicitly implemented SGS models are expected to work for LES: the numerical
scheme has to be constructed such that the leading order truncation errors satisfy physically re-
quired SGS-model properties, and hence non-linear discretization procedures will be required.
The analogy to be recalled is that of shock-capturing schemes designed under the requirements of
convergence to weak solution while satisfying the entropy condition. Finite-volume versions of
such schemes can likewise be viewed as relevant for Implicit LES (ILES) if we focus on the
small-scale characteristic features of turbulence. In the Monotonically Integrated LES (MILES)
approach – see  Ref. [5] for a recent survey, the effects of the SGS physics on the resolved scales
are incorporated in the functional reconstruction of the convective fluxes using locally-monotonic
methods. Analysis based on the modified equations can be used to demonstrate an intriguing fea-
ture of MILES, namely that when based on a particular class of flux-limiting schemes, the
convection discretization implicitly generates a non-linear tensor-valued eddy-viscosity that acts
to stabilize the flow and suppress unphysical oscillations. The MILES performance has been
demonstrated in case studies including, 1) canonical flows (forced & decaying homogeneous
isotropic turbulence and turbulent channel flows), 2) complex free and wall-constrained flows
(mixing layers, jets, wakes, jets and flow past a prolate spheroid), and, 3) extremely complex
flows at the frontiers of current unsteady flow simulation capabilities (e.g., submarine
hydrodynamics and pollutant dispersion in urban areas).
MILES seeks to emulate the flow features in the high-wave-number end of the inertial
subrange region of turbulent flows – characterized by thin filaments of intense vorticity
embedded in a background of weak vorticity. We have proposed that emulation of the latter
feature be the requirement for implicit SGS models. MILES can thus extend to the more general
concept of non-linear ILES in which the functional reconstruction of the convective flux
functions is carried out using high-resolution non-linear numerical schemes incorporating a sharp
velocity-gradient capturing capability operating at the smallest resolved scales. By focusing on
the inviscid inertial-range dynamics and on regularization of the under-resolved flow, ILES thus
follows up very naturally on the historical precedent of using this kind of numerical schemes for
shock capturing. Challenges for ILES development include developing a common appropriate
mathematical and physical framework for its analysis and development, further understanding the
connections between implicit SGS model and numerical scheme, and in particular, addressing
how to build physics into the numerical scheme to improve on the global ILES performance, i.e.,
on the implicitly-implemented SGS dissipation & backscatter features. Moreover, additional
(explicit) SGS modeling might be needed to address inherently small-scale physical phenomena
such as scalar mixing and combustion – which are actually outside the realm of any LES
approach: how do we exploit the implicit SGS modeling provided by the numerics, to build
efficient "mixed" (explicit/implicit) SGS models ?
[1] Sagaut P.; 2002, “Large Eddy Simulation for Incompressible Flows”, Springer, New York.
 [2] Adams N.A. & Stolz S.; 2001, “Deconvolution Methods for Subgrid-Scale Approximation in
LES”, in Modern Simulation Strategies for Turbulent Flows, B.J. Geurts, ed., Edwards, Phila-
delphia, p.21.
[3] Spalart P.R., Jou, W.H., Strelets M. and All-maras S.R.; 1997, “Comments on the Feasibility
of LES for Wings, and on Hybrid RANS/LES Approach”, in Advances in DNS/LES, First
AFOSR International Conference in DNS/LES (Greyden Press, Columbus).
[4] Alternative LES and Hybrid RANS/LES; 2002, edited by F.F. Grinstein & G.Em Karniada-
kis, J. Fluids Engineering, 124, pp. 821-942.
[5] Grinstein, F.F. and Fureby; 2003 C., Implicit Large Eddy Simulation of High-Re Flows with
Flux-Limiting Schemes, AIAA Paper 2003-4100, AIAA CFD Conference, Orlando, June 2003.


On Implicit Subgrid Scale Modeling for Turbulent Flows

Domaradzki, Julian A.

Eﬀective eddy viscosities in implicit large eddy simulations of turbulent ﬂows
J. Andrzej Domaradzki
Department of Aerospace and Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089-1191, U.S.A.,
tel. 213-740-5357, FAX 213-740-7774
Solving Navier-Stokes equations using monotone algorithms often produces results that are consistent with the dy-
namics of high Reynolds number turbulence despite formally insuﬃcient numerical resolution to capture all physically
relevant scales of motion. It is frequently argued that the truncation error of a numerical scheme provides a mechanism
for the energy dissipation and serves as an implicit subgrid scale model. Such an approach to turbulence modeling,
known loosely as the Monotonically Integrated LES (MILES), was originally proposed by Boris et al. (1992) and
reviewed recently by Grinstein and Fureby (2002). The primary eﬀect of a numerical discretization in MILES, which
is similar to the eﬀects of explicit SGS models, is the dissipation of turbulent kinetic energy. Despite its importance,
very little is known about details of numerical dissipation in MILES methods and, more importantly, its relation (or
lack of it) to the actual dissipative eﬀects of turbulence. We describe a method for computing eﬀective numerical
eddy viscosity, useful for assessing numerical dissipation of such approaches. The method is evaluated on an example
of a speciﬁc nonoscillatory ﬁnite volume scheme MPDATA developed for simulations of geophysical ﬂows. A series of
numerical simulations of homogeneous, isotropic turbulence are performed and the numerical eddy viscosities are de-
termined. The detailed quantitative comparisons are made between the numerical eddy viscosities and the theoretical
eddy viscosity as well as intrinsic eddy viscosities computed exactly from the velocity ﬁelds by introducing artiﬁcial
wave number cutoﬀs. A typical example of such a comparison is shown in the enclosed ﬁgure.
[1] J. P. Boris, F. F. Grinstein, E. S. Oran, and R. L. Kolbe. New insights into large eddy simulation. Fluid Dynamics Research,
10:199–228, 1992.
[2] F.F. Grinstein and C. Fureby. Recent progress on MILES for high Reynolds number flows. J. Fluids Eng., 124:848–861,
2002.
k/k
c
10-2 10-1 100
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
νth
+(k/k
c
) , k
c
=30
ν
n
+(k/k
c
) , k
c
=30
νt
+(k/k
c
) , k
c
=15
Zero line
ν=0
FIG. 1: Eddy viscosities for homogeneous isotropic turbulence. Broken line: theoretical eddy viscosity; solid line with symbols:
intrinsic eddy viscosity for kc = 15; solid line: numerical eddy viscosity. The intrinsic and numerical eddy viscosities obtained
from an inviscid simulation using MPDATA on a 643 mesh.


Effective Eddy Viscosities in Implicit Large Eddy Simulations of Turbulent Flows

Bodony, Daniel J

Lele, Sanjiva K.

Subgrid Scale Noise Models for Aeroacoustic Prediction
Daniel J. Bodony and Sanjiva K. Lele∗
Department of Aeronautics and Astronautics
Stanford University, Stanford, California
Submitted to the Workshop on LES & SGS Modeling For Turbulent Mixing and Reactive Flows at
Caltech, December 8–9, 2003
There is significant interest within the aircraft engine industry to be able to accurately predict the
noise generated by a jet engine design. Current prediction methods are RANS-based (Reynolds-
averaged Navier-Stokes) where the entire sound generation process is semi-empirically modeled.
It has been found that for complex nozzle designs, such as nozzles with tabs or chevrons, the
RANS predictions are not sufficiently accurate. It is believed that the discrepancies arise due to the
modeling of the flow unsteadiness, whose characteristics are subtly changed by the chevrons/tabs.
It is hoped that a LES-based (large-eddy simulation) prediction, where the flow unsteadiness occurs
naturally, would be able to capture the observed changes in the jet exhaust plume and thus better
predict the far-field sound.
Two difficulties with the LES predictions are that the Reynolds numbers typically used are two
to three orders of magnitude below those found in practical applications and that the affordable
grid resolution may be quite modest. As a result, the frequency spectrum of the LES predicted
far-field sound is artificially truncated at the higher frequencies. This is due, in part, to the absence
of the sound generated by the interaction of fluid motions near the grid cut-off scale with the larger,
resolved motions.
To recover the higher frequency sound, two subgrid noise models are proposed that attempt to
estimate the ‘missing’ sound produced by the turbulent motion. The first noise model is kinematic
and uses the approximate deconvolution model of Stolz et al. [Phys. Fluids, vol. 13, no. 4, 2001,
pp. 997–1015] to extrapolate the near grid scale motions from the resolved field. Model tests
using a fully resolved simulation of a turbulent, spatially developing shear layer, which idealizes
the near-nozzle region of the exhaust plume, will be presented and discussed. The second model
is based on a WKB representation of the subgrid scale motions and aims to correctly capture the
acoustically important space-time dynamics of the near grid scales. At the same time the model
provides a comprehensive closure for the subgrid scale mass, momentum, and energy fluxes. The
WKB model’s formulation and preliminary results will be presented.
This work is sponsored by the Aeroacoustics Research Consortium of the Ohio Aerospace
Institute.
∗Also with the Department of Mechanical Engineering, Stanford University


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