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A study of built-in filter for some eddy viscosity
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2. A study of built-in filter for some eddy viscosity models in large-eddy simulation
Jean-Christophe Magnient, Pierre Sagaut, and Michel Deville
Citation: Physics of Fluids (1994-present) 13, 1440 (2001); doi: 10.1063/1.1359186
View online: http://dx.doi.org/10.1063/1.1359186
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3. A study of built-in filter for some eddy viscosity models
in large-eddy simulation
Jean-Christophe Magnient and Pierre Sagaut
ONERA, 92322 Chaˆtillon cedex, France
Michel Deville
Laboratory of Fluid Mechanics EPFL, CH-1015 Lausanne, Switzerland
͑Received 6 January 1999; accepted 7 February 2001͒
Traditional large-eddy simulation ͑LES͒ provides good results on standard benchmark turbulent
flows, but engineering applications involving more complex geometries are restricted. This is
mainly due to the physical hypotheses of homogeneity, isotropy, or local equilibrium of the
subgrid-scale ͑SGS͒ models, which are questionable in more realistic configurations. It appears
essential to qualify the built-in filters of LES models before performing coarse mesh/high Reynolds
number computations. A study of the Smagorinsky and the Schumann models with a
mesh-independent filtering length larger than the mesh size is presented in this paper. The forms of
the associated built-in filters are identified, and a control of the filter length through the SGS model
is performed. Finally an asymptotic behavior of the filter is found to be independent of the spatial
scheme for large filter length. This work is partly an assessment of previous theoretical studies on
LES filters and kinetic energy spectra. © 2001 American Institute of Physics.
͓DOI: 10.1063/1.1359186͔
I. INTRODUCTION
The large-eddy simulation method was developed in
view of performing computation on high Reynolds number
flows. Whereas direct numerical simulation ͑DNS͒ of such
turbulent flows requires a prohibiting number of grid points
͑scaling like ϰRe9/4
, where Re denotes the Reynolds num-
ber͒, the LES uses a spatial filtering operation upon the fluid
equations in order to remove the small scales ͑high wave
number components͒ and model their behavior indepen-
dently of the numerical method. The remaining filtered
scales should be large enough to loosen the constraint upon
the mesh size and enable numerical simulation, but should
also account for most of the turbulent kinetic energy so that
the energy-containing scales are resolved.
Traditional LES assumes that the filtering relies on the
grid discretization, mainly for computational cost consider-
ations, so that the filter length is usually directly scaled using
the grid spacing. In this approach the filter is explicitly de-
pendent on the grid. But problems arise with anisotropic,
inhomogeneous, or unstructured grids for which a definition
of the mesh size is unclear. Some authors, such as
Deardorff,1,2
and more recently Scotti et al.,3
have proposed
different methods to evaluate a local characteristic mesh size
with such grids. Another strategy proposed by Bardina et al.4
is the use of a local length tensor linked to the geometry of
the mesh, and therefore defining an eddy viscosity tensor
͑see also Carati and Cabot5
for an anisotropic eddy viscos-
ity͒. But in these cases the global filter operation on the
variables and the nature of the solution is still unknown so it
is difficult to compare the ‘‘real’’ flow or a DNS with the
solution obtained from a LES. Furthermore, it has been
shown by Ghosal6
that the numerical errors are usually of the
same order or greater than the SGS terms for most of the
wave numbers.
The prefiltering method applies an explicit analytical fil-
ter on the variables during a LES computation at a filter
length larger than the grid spacing, in order to ensure both a
control upon the filtering length and a preponderance of the
SGS terms over numerical errors. Ghosal6
argues that a com-
bination of prefiltering and a high-order accurate numerical
scheme can be a good compromise to minimize numerical
errors. But filtering explicitly the Navier–Stokes equation
increases significantly the computational cost ͑see also
Cabot7
and Lund et al.,8
who used an explicit grid filtering
together with a dynamic model͒. Mason and his
co-workers,9,10
and Muschinsky11
have developed the notion
of a built-in filter in the models ͑see also Ref. 12͒. They
studied the influence of the constant eddy-viscosity models
and the relationship with the kinetic energy spectrum. A dis-
cussion is also presented by Schumann.13
It appears that a
new challenge is to control the filtering process via grid-
independent LES models only. But the modeling-terms influ-
ence on the filtering process and on the flows dynamics is not
well described yet. So, following the lead of Mason:10
‘‘Since the Smagorinsky model only involves a single scalar
variable l0 ͓¯͔, only the characteristic scale of a filter can
be represented. A key, but yet unanswered, question is the
following: what particular shape of filter operation does the
Smagorinsky model correspond to?,’’ we will give some in-
sight into the built-in filter of two different SGS models.
The identification of the built-in filter associated to the
SGS model, which appears as the only source term in the
discretized set of equations containing an information on the
filter and its length scale ͑if the prefiltering technique is not
PHYSICS OF FLUIDS VOLUME 13, NUMBER 5 MAY 2001
14401070-6631/2001/13(5)/1440/10/$18.00 © 2001 American Institute of Physics
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4. employed͒, appears also as a key point for understanding the
behavior of very large-eddy simulation- ͑VLES͒ type meth-
ods such as Spalart’s detached-eddy simulation ͑DES͒
method14
or the Speziale model.15
This is also true for the Reynolds-averaged Navier–
Stokes ͑RANS͒ equations that rely on a statistical theory of
turbulence and do not depend specifically on the grid size but
theoretically allow only computations of statistically aver-
aged steady-state flows. It can also be noted that the mono-
tonic integrated LES ͑MILES͒ method, introduced by Boris
et al.,16
does not use an explicit model, but the subgrid-
scales effects are included in the intrinsic numerical dissipa-
tion of the spatial schemes, yielding the definition of an
equivalent filter.17
The paper is organized as follows. In Sec. II the founda-
tions of the LES theory are summarized. We first describe
the filtering operation, which can be defined mathematically,
and introduce the notion of filter length and filter function.
The resulting LES equations are presented. Then we will
show the characteristics of two models that have been stud-
ied: the Smagorinsky18
and the Schumann19
models. A simi-
larity theory of homogeneous turbulence for the Smagorin-
sky model is due to Muschinsky.11
In Sec. III the results of
the simulations are analyzed. After presenting the numerical
method to integrate the Navier–Stokes equations, we focus
on the effective filter rendered by the simulation and the
built-in filter of the models, by changing a characteristic
length in the models. Expressions for these filters are found
and the influence of the ratio of the filter length to the mesh
size is analyzed. The behavior of a LES equivalent of the
Kolmogorov constant is compared to the theory. The influ-
ence of the spatial scheme is also studied. Finally, we discuss
the results in Sec. IV and draw concluding remarks in Sec.
V.
II. THEORETICAL BASIS
A. Equations of a LES fluid
1. The filtering process
In order to allow an analytical study of LES, the filter
operation is mathematically defined in the physical space as
a convolution integral that is similar to a multiplication in the
spectral space:
a¯ϭa*Gϭ ͵a͑y͒G͑xϪy͒dy,
a¯ˆ ϭGˆ aˆ ,
where G is the weight or filter function that has to be real,
positive, and normalized (͐Gϭ1) and is usually symmetric
and homogeneous. The Fourier transform of G is the transfer
function Gˆ . This low-pass filter is characterized by a band-
width ⌬¯ , which is the typical length scale that separates the
physically resolved scales and the modeled scales or subfilter
scales. The corresponding wave number kcϭ␲/⌬¯ is the filter
cutoff wave number ͑Fig. 1͒. In a LES this cutoff wave num-
ber should be located in the inertial range of the Kolmogorov
energy spectrum of a turbulent flow. Our concern is to define
and to calculate the cutoff wave number for some LES mod-
els, and more generally to determine G or Gˆ .
Now, if we consider a numerical simulation based on a
uniform grid, the largest numerically resolvable wave num-
ber kmaxϭ␲/⌬ is related to the grid spacing ⌬. The scales
smaller than ⌬ are defined as the subgrid scales ͑Fig. 1͒. In
practice, the LES filter operation is usually associated by the
grid discretization, so it is considered that ⌬¯ ϭ⌬ and no dis-
tinction is made between subgrid and subfilter scales. But in
the present approach the LES models, terms, or variables
will be qualified as subfilter, as proposed by Mason.10
2. The LES equations
When the filter is applied to the Navier–Stokes equa-
tions for an incompressible Newtonian fluid, we get a set of
LES equations for the filtered variables ͑with the symbol
denoting the tensor product͒:
“u¯ϭ0,
͑1͒
‫ץ‬u¯
‫ץ‬t
ϩ“͑u¯‹u¯͒ϭϪ“p¯ϩ“͑2␯S¯Ϫ␶LES͒,
where S¯ϭ 1
2 (“u¯ϩt
“u¯) is the resolved rate of deformation
tensor ͑the superscript t indicates the transpose͒, ␯ the mo-
lecular kinematic viscosity, p the pressure field, and ␶LES the
subfilter stress tensor that has to be parametrized in terms of
filtered velocity field u¯ generated by the LES. In order to
study the effects of the subfilter model alone, the hypotheti-
cal case ␯ϭ0 is considered. Here Muschinsky11
introduces
the notion of a ‘‘LES fluid,’’ which is the theoretical fluid
governed by the system of equations ͑1͒. The advantage of
this concept is to allows similarities with a Newtonian fluid
so as to get a better idea of the physical properties and the
dynamics of the LES equations. Comparisons will be made
between characteristic variables or parameters of the fluids,
and a similarity theory for LES turbulence, regarding the
kinetic energy spectrum, will be recalled further in Sec. II C.
Most of the LES models are based on the eddy viscosity
assumption:
FIG. 1. Definition of the different scales for the case ⌬¯ Ͼ⌬. The LES
filtering process is characterized by a cutoff wave number kc in the inertial
range of the turbulent kinetic energy spectrum.
1441Phys. Fluids, Vol. 13, No. 5, May 2001 A study of built-in filter for some eddy viscosity
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5. ␶LESϭϪ2␯LESS¯, ͑2͒
where ␯LES is a subfilter viscosity representing the dissipa-
tion effect of the nonresolved small eddies, and varying in
time and/or space. In this case the LES equations are similar
to equations of motion of a generalized Newtonian fluid with
a nonconstant viscosity ␯LES . Note that no specific filter is
necessary at this point and that the subfilter stress ␶LES con-
tains the full information on the filter. Defining a model for
the subfilter stress tensor is equivalent to defining a filter
function.
B. Subfilter models
We will first consider the simple and widely used
Smagorinsky18
model, which proposes an eddy viscosity de-
rived from the local shear rate and a length scale:
␯LESϭl0
2
͉S¯͉ ͑3͒
where ͉S¯͉ϭͱ2S¯:S¯ is the second invariant of S¯ and l0 is a
characteristic length. Equations ͑1͒, ͑2͒, and ͑3͒ define a
‘‘Smagorinsky fluid.’’ 11
In analogy to the Kolmogorov
length ␩, we can define the dissipation length of a LES fluid,
␩LESϭ͗␯LES͘3/4
⑀Ϫ1/4
, ͑4͒
⑀ being the energy dissipation rate, and ͗•͘ represents for the
volume average. For the Smagorinsky model it can be shown
that ␩LESϭl0 , using approximations of ␯LES and ⑀ in terms
of the kinetic energy spectrum ͑see Ref. 11͒.
Since l0 determines the magnitude of the subfilter vis-
cosity and considering that ␶LES defines the effective spatial
filter, the filter length ⌬¯ is defined by l0 . Assuming a sharp
spectral cutoff and an energy spectrum of the form E(k)
ϭK0⑀2/3
kϪ5/3
, where K0 is the Kolmogorov constant, one
can obtain Lilly’s evaluation of l0 :20,10,11
l0 /⌬¯ ϭCSϭ␲Ϫ1
͑ 3
2 K0͒Ϫ3/4
. ͑5͒
But this result should depend on the filter shape, and this will
be discussed in Sec. III C. We consider the standard value of
K0ϭ1.5 like Muschinsky11
and Mason,10
and as obtained by
Zhou21
or Champagne22
͑for a review of some published
values of K0 , see Refs. 23 and 24͒. However, the Smagor-
insky model can be written as
␯LESϭ͑CS⌬¯ ͒2
͉S¯͉. ͑6͒
As ⌬¯ ͑or equivalently ␩LESϭl0ϭCS⌬¯ ͒ and ␩ are the char-
acteristic lengths of the smallest scales of the turbulent fluid,
for, respectively, the LES and the Newtonian fluid, it is clear
that they play the same physical role. But while in the
Navier–Stokes equations the viscosity ␯ is a material prop-
erty of the fluid, in the LES equations ⌬¯ is a flow property ͑a
fixed parameter of the simulation͒ and ␯LES is an
⑀-dependent variable.
The second model that has been used in a more elabo-
rated model proposed by Schumann,19,13
which is a one-
equation model based on the subfilter ͑SF͒ kinetic energy
qSF
2
,
␯LESϭC⌬¯ ͑qSF
2
͒1/2
, ͑7͒
dqSF
2
dt
ϭ␶LES :S¯ϪC1
qSF
2 3/2
⌬¯
ϩC2“͓⌬¯ ͑qSF
2
͒1/2
“qSF
2
͔
ϩ␯ٌ2
qSF
2
, ͑8͒
with the constants Cϭ(3K0/2)Ϫ3/2
/␲ϳCS
4/3
, C1
ϭ␲(3K0/2)Ϫ3/2
͑see Ref. 13͒ is taken as 1 and C2ϭ0.1 ͑see
Refs. 25 and 26͒. The subfilter viscosity depends on the same
characteristic length ⌬¯ as the Smagorinsky model, and we
will verify that it is again the filter length. As the constants of
the models are evaluated by calculation of the mean subfilter
viscosity, assuming a Kolmogorov spectrum in the inertial
range and a sharp spectral filter at kc , they should produce
the same mean subfilter viscosity and dissipation rate. There-
fore Eqs. ͑4͒ and ͑5͒ are also a good evaluation of, respec-
tively, the LES dissipation length and the dissipation length
over the filter length ratio for the Schumann model. But note
that, unlike the Smagorinky model, changing ⌬¯ is not simply
equivalent to changing linearly the subfilter viscosity ampli-
tude because of the transport equation. This is confirmed by
the results of computations shown hereafter.
C. Analysis of isotropic turbulence in a LES fluid
The following theoretical analysis is based on the previ-
ous work of Muschinsky11
͑see also Mason10
͒. It is a study of
the widely used Smagorinsky model18
that introduces the
concept of a ‘‘Smagorinsky fluid,’’ and presents a physical
interpretation of the LES equations in analogy to the Kol-
mogorov theory of homogeneous turbulence. We will
present it from a filtering point of view.
According to Kolmogorov’s similarity theory,27,28
the
turbulent kinetic energy ͑TKE͒ spectrum of a high Reynolds
number flow has the form E(k)ϭK0⑀2/3
kϪ5/3
in the inertial
range. The Kolmogorov coefficient K0 is a quasiuniversal
constant characteristic of homogeneous turbulence. We can
assume a LES with a cutoff wave number in the inertial
range and far from the dissipation range ͑⌬¯ or ␩LESӷ␩͒,
which is obviously the case when ␯ϭ0 (Reϭϱ), ␩ being
zero. Then we can derive an expression of the rendered/
filtered TKE spectrum: E¯ (k)ϭK0⑀2/3
kϪ5/3
f, where f is a
nondimensional damping function that can be identified as
the square transfer function Gˆ 2
. Muschinsky11
proposes
three similarity hypotheses for a Smagorinsky-type LES that
can actually be postulated for any LES fluid:
First: E¯ (k) depends on ⑀, ␩LES , ⌬ in the most general case;
second: E¯ (k) is determined by ⑀ and ⌬ only, for wave num-
bers kӶ␩LES
Ϫ1
;
third: E¯ (k) is determined by ⑀ and ␩LES if ⌬Ӷ␩LES .
The third hypothesis reflects the fact that LES is a nu-
merical simulation of the discretized LES equations, and that
it is expected to be asymptotically independent of the grid
size. The criterion is similar to the one applied in DNS for
which the grid size has to be smaller than the dissipation
length.
1442 Phys. Fluids, Vol. 13, No. 5, May 2001 Magnient, Sagaut, and Deville
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6. A dimensional analysis shows that the TKE spectrum
depends on the nondimensional variables k␩LES and
␩LES /⌬. For a ‘‘Smagorinsky fluid’’ or any other LES fluid
based on a model statistically equivalent to the Smagorinsky
model, which produces the same mean dissipation rate and
subfilter viscosity ͑e.g., the Schuman model͒, ␩LES is propor-
tional to ⌬¯ . In this case we prefer to use the nondimensional
ratio rϭ⌬¯ /⌬ and k/kc instead, kc being the cutoff wave
number ␲/⌬¯ . These considerations lead to the following
form of the TKE spectrum:
E¯ ͑k͒ϭFLES͑k/kc ,r͒⑀2/3
kϪ5/3
, ͑9͒
FLES͑k/kc ,r͒ϭK0LES
͑r͒fLES͑k/kc ,r͒, ͑10͒
where the LES equivalent of the Kolmogorov coefficient
K0LES
is the value of the function FLES for kϭ0, and fLES is
the LES damping function. Note that in LES this coefficient
is not expected to be a constant, but the third similarity hy-
pothesis implies an asymptotical behavior of the functions
K0LES
and fLES for large values of r: the Kolmogorov simi-
larity theory probably holds for a LES-generated turbulence
too, in this limit. The effective filter transfer function is de-
fined as
Gˆ 2
͑k͒ϭ
E¯ ͑k͒
E͑k͒
ϭ
K0LES
͑r͒
K0
fLES͑k/kc ,r͒. ͑11͒
A clear distinction between the effective and the built-in
filter has to be made. The effective filter defined above con-
tains the combined effects of the SGS model, of the numeri-
cal differential schemes errors, and of the discretization ͑i.e.,
the projection of the continuous solution on the discrete finite
basis͒. The representation of the solution on a grid implies a
numerical cutoff at ␲/⌬. Therefore, in practice, the effective
filter is evaluated with E¯ (k) being the spectrum rendered by
the LES simulation, and E(k) being the spectrum of a DNS
filtered with a sharp cutoff filter at wave number ␲/⌬, where
⌬ is the grid spacing of the LES. The built-in filter is the
filter associated with the SGS model only. It can be evalu-
ated only if the numerical errors are negligible.
III. NUMERICAL RESULTS
A. Numerical implementation
The computations have been carried out on a homoge-
neous isotropic turbulent and freely decaying flow. The ini-
tial random velocity field has a Gaussian kinetic energy spec-
trum of the form k4
exp(Ϫ2k2
/k0
2
), centered at wave number
k0ϭ2, and is solenoidal. The discretization mesh has N3
points in a 2␲-periodic cube. Different simulations have
been performed with Nϭ60 and Nϭ120. The simulations
have been carried out by solving the full unsteady compress-
ible Navier–Stokes equations with a low Mach number, in
order to avoid undesirable compressible effects and to re-
main in the same previous theoretical frame. The PEGASE
code, developed at ONERA, uses the conservative variables
(␳,␳ui ,E) with a finite difference scheme on structured uni-
form grids. Here ␳ is the volumic mass, and E is the total
energy field. Convection terms are written in skew-
symmetric form, and along with diffusion terms are dis-
cretized using a fourth-order accurate centered scheme. In
Sec. III E a second-order and compact sixth-order accurate
schemes are also considered. Time integration is performed
with a third-order low-storage explicit Runge–Kutta
scheme29
͑see also Spalard et al.30
for a more general formu-
lation͒.
In order to simulate only the effect of the subfilter
model, computations are done at zero molecular viscosity,
i.e., infinite Reynolds number. The initial Mach number M0
is set equal to 0.1 for which previous studies ͑Sarkar et al.31
͒
have demonstrated that the compressibility has no significant
effect. As expected, Fig. 2 shows that the compressible part
of the resolved kinetic energy is negligible compared to its
solenoidal part, so that compressible effects can be ne-
glected. Consequently we use the approximation ␳aϭ␳ a
ϩO(M0
2
) for all variables and the system of equations ob-
tained for the leading terms is
‫␳ץ‬¯
‫ץ‬t
ϩ“͑␳¯ u¯ ͒ϭ0, ͑12͒
‫␳ץ‬¯u¯
‫ץ‬t
ϩ“͑␳¯ u¯ ‹u¯͒ϭϪ“p¯ϩ“␶LES , ͑13͒
‫ץ‬E¯
‫ץ‬t
ϩ“͑u¯E¯ ͒ϭ“͑p¯ u¯ Ϫq¯͒ϩ“ALES . ͑14͒
Along with these equations we have the perfect-gas law and
an expression for the filtered internal energy E¯ I :
p¯ϭ
␳¯T¯
␥M0
2 , ͑15͒
E¯ IϭE¯ Ϫ
␳¯
2
͉u¯͉2
ϭ
p¯
␥Ϫ1
, ͑16͒
where q¯ is the filtered heat flux, T the temperature, and ␥ the
specific heat ratio. All variables are nondimensionalized in
terms of a reference density ␳0 , velocity u0 , length l0 , and
temperature T0 . There is no fluid stratification or heating:
the initial temperature and volumic mass fields are set to 1 in
the whole volume. To model the subfilter term in the energy
FIG. 2. At Mach 0.1 the compressible part of the kinetic energy is negligible
compared to the incompressible part ͑the ratio is about 10Ϫ3
͒. Compressible
effects are neglected. The number of points is Nϭ34.
1443Phys. Fluids, Vol. 13, No. 5, May 2001 A study of built-in filter for some eddy viscosity
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7. equation ͑14͒ we use a first gradient-type closure. The turbu-
lent diffusivity is proportional to the eddy viscosity ͑2͒ with
the subfilter Prandtl number Prt set equal to 0.5, a compro-
mise between the values 0.4 used by Speziale et al.32
and 0.7
used by Zang et al.33
and Kida and Orszag.34
The resulting
term reads as
ALESϭ
1
Prt
␳¯ ␯LES“T¯ . ͑17͒
B. Analysis of the effective filters
As our purpose in this study is to determine the function
fLES of Eqs. ͑9͒ and ͑10͒, several simulations are performed
for different values of the ratio rϭ⌬¯ /⌬ for which the TKE
or the compensated TKE spectrum E¯ (k)k5/3
⑀Ϫ2/3
are plotted.
The dissipation rate of the energy ⑀ is calculated as
⑀ϭϪ͗␶LES :S¯͘. ͑18͒
At the beginning of the computation the flow is in a transi-
tional period for about five nondimensional time units, while
the energy cascades from the large scales containing the ini-
tial energy to the small scales. The energy is redistributed to
all scales through nonlinear interactions. The flow is then
fully developed, and the results presented further are ob-
tained at time tϭ10l0 /u0 .
For any model, the case rϽ1 is not relevant because the
subfilter model would not take into account the wave num-
bers that are not resolved, kmaxϽkϽkc , and the lack of dis-
sipation from the eddy viscosity leads to a growth of energy
in the small scales which usually ends up in an unstable
computation. Therefore only value of rу1 are investigated.
1. The Smagorinsky model
For rу1 three different cases are identified.
Let us first consider rϭ1 ͑i.e., the usual value of ⌬¯
ϭ⌬͒. Taking the value of K0ϭ1.5, Eq. ͑5͒ gives an estimate
of the Smagorinsky constant CSӍ0.174. We used CSϭ0.18
as this value yields the appropriate kϪ5/3
TKE spectrum up
to kmax ͑Fig. 3͒. Therefore the damping function associated
to the filter applied to the energy spectrum is
fLES͑x,rϭ1͒ϭ1, ͑19͒
where x is the nondimensional wave number k/kc .
Second, in the case rу2 the computed flow displays a
spectrum that is clearly damped in the small scales, and the
damping effect increases with the value of r ͑Fig. 3͒. This
suggests an increase of the filtering length. In counterpart,
we note that the energy in the large scales increases with r
identically with both resolutions. An explanation of this en-
ergy pile-up is the rapid fall of the energy slope near the
cutoff wave number kc , where the eddy viscosity becomes
effective. This can be assimilated to the ‘‘bottleneck’’ effect
in which the energy cascade is inhibited by a change of the
nature of the triadic interactions between the inertial and the
dissipation ranges.35
Muschinsky11
proposes the Pao36
and
the Heisenberg37
functions as possible forms of the damping
function. Figure 4 show that the TKE spectrum actually has
the form of the Heisenberg function fHϭ͓1
ϩ(3/2K0)3
(k␩LES)4
͔Ϫ4/3
as the plots of K0LES
(r) using
fLESϭfH in Eqs. ͑9͒ and ͑10͒ exhibit values independent of
the wave number at small scales, which is to be expected.
This is not the case with the Pao function fP
ϭexp͓Ϫ3/2K0(k␩LES)4/3
͔. Figure 4͑a͒ shows the decreasing
function K0LES
obtained with fLESϭfP for rϭ2. The filter
associated with the Heisenberg function will be called a
Heisenberg filter. As the Smagorinsky constant verifies ͑5͒
the Heisenberg function can be simplified and
FIG. 3. Spectra obtained at tϭ10 with a Smagorinsky model for different
values of the filter length ⌬¯ at a fixed number of grid points N. The TKE
spectrum displays an increasing damping with r. ͑a͒ Nϭ60. r ranges from 1
to 4 with a 0.5 step. ͑b͒ Nϭ120 and rϭ1,2,3,4,8.
FIG. 4. Plots of E(k)⑀Ϫ3/2
k5/3
fLES
Ϫ1
with the Smagorinsky model for different
values of r: ͑a͒ 603
simulations. fLESϭfH ͑solid line, rϭ1,2,2.5,3,3.5,4͒ and
fLESϭfP ͑symbols, rϭ2͒; ͑b͒ 1203
simulations (rϭ1,2,3,4).
1444 Phys. Fluids, Vol. 13, No. 5, May 2001 Magnient, Sagaut, and Deville
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8. fLES͑x,rу2͒ϭfHϭ͑1ϩx4
͒Ϫ4/3
. ͑20͒
It is easy to verify that fLES(xϭ1)ϭfH(kc)Ӎ1/e, which is
the definition for the cutoff wave number proposed by
Muschinsky.11
This validates ⌬¯ ϭ␲/kc as the filter length.
Eventually, 1ϽrϽ2 is a transitional range and the damping
function remains undetermined.
Similar results have been obtained for different test cases
with Nϭ60 ͑Fig. 5͒. We used different values of the energy
peak wave number k0 in the expression of the initial field. In
all cases, after an initial transient period the most energetic
scale shifts to k0ϭ2 and at a quasiequilibrium state the TKE
spectra are the same. Another test was done by preceeding
the free decay of turbulence by a forcing period using a
random external force without any noticeable change. As
proposed by Kida and Orszag,34
the forcing term is Fi(x,t)
ϭAij(t)sin xjϩBij(t)cos xj , where A(t) and B(t) are statisti-
cally independent Gaussian variables with zero mean. A pure
solenoidal forcing ͑the compressible component is zero͒ is
applied at mode kϭ2. These two tests show that the final
spectrum, i.e., a steady form, is independent of the initial
conditions.
A forced turbulent flow was also computed. Although
presenting a slope a little bit steeper than kϪ5/3
for rϭ1, the
time-averaged TKE spectrum of the forced turbulence re-
mains identical to the freely decaying case at rϭ2, for in-
stance. Then we used an x-independent form for ␯LES , i.e.,
the value of the eddy viscosity is constant on the overall
volume: ␯LES(t)ϭ(CS⌬¯ )2
͉͗S¯͉͘. Again, the results for the
spectrum are comparable to those obtained with the local
expression ␯LES(x,t) used in all the previous calculations. So
it appears that the spatial distribution of the eddy viscosity is
not at stake for a homogeous flow with the Smagorinsky
model.
Finally, to test the influence of the resolution and the
validity of our results, computations were performed on a
finer 1203
grid ͓Fig. 4͑b͔͒. As for the 603
simulations the
form of the damping function fLES is found to correspond to
the Heisenberg function ͓Fig. 4͑b͔͒. A more detailed analysis
of these results is given in Sec. III D.
2. The Schumann model
The TKE spectrum of isotropic turbulence with the
Schumann model ͓Eqs. ͑7͒ and ͑8͔͒ is different from the one
obtained with a Smagorinsky model, despite the similarity
for small values of the ratio rϭ⌬¯ /⌬ ͑Fig. 6͒. The Schumann
model is also based on an eddy-viscosity assumption for the
Reynolds stress, but, in addition, it integrates a transport
equation for the subfilter kinetic energy chosen as the param-
eter in the expression of ␯LES . With this model the results
are qualitatively the same but quantitatively different
whether we use the local expression ͑7͒ of the eddy viscosity
␯LES(x,t) or the constant form ͗␯LES͘(t)ϭC⌬¯ ͗qSF
2 1/2
. This
means that the form of the function FLES is the same, but the
slopes and coefficients differ. For the constant form the sub-
filter kinetic energy is still a function of x, so that Eq. ͑8͒ is
unchanged but the eddy viscosity is averaged on the whole
domain at each time step. Conclusions of the simulations are
the following.
For rϭ1, with Cϭ0.09, the simulation achieves a sharp
cutoff filtering, identical to the Smagorinsky model ͓Eq.
͑19͔͒, and fLESϭ1.
For rу1.5, Fig. 6 shows that ln͓E¯(k)⑀Ϫ3/2
k5/3
͔ is a linear
function of k, apart from the large scales, of the form ␣(r)
ϩb(r)k. Therefore the attenuation function FLES is an expo-
nential. In terms of nondimensional variables we can write
FLES͑x,r͒ϭexp͓␣͑r͔͒•exp͓Ϫx␤͑r͔͒, ͑21͒
where rϭ⌬¯ /⌬, xϭk/kc , ␤(r)ϭb(r)kc , and kcϭ␲/(r⌬).
The values of the ␣ and ␤ have been determined by a linear
regression from the computed results ͑Fig. 7͒.
For rу2 the slope of ln(FLES),b(r), linearly depends on
r, and this function is also determined by linear regression.
We then deduce an expression for ␤(r). Furthermore, ␣(r)
FIG. 5. A comparison of K0LES
(rϭ2) with a Smagorinsky model for freely
decaying or forced isotropic turbulence ͑time-averaged͒, an initial field
spectrum centered on k0ϭ4 instead of 2, a space-independent eddy viscos-
ity ͓␯LESϭ͗␯LES͘(t)͔, and a freely decaying turbulence after removal of the
forcing term (ftϩfd).
FIG. 6. The Schumann model: E(k)⑀Ϫ3/2
k5/3
for r ranging from 1 to 4 for a
local ␯LES(x,t) ͑solid line͒, and an x-independent form ␯LES(t) ͑dashes͒. For
comparison the Smagorinsky model is also plotted ͑dots͒.
1445Phys. Fluids, Vol. 13, No. 5, May 2001 A study of built-in filter for some eddy viscosity
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9. has an asymptotic behavior similar to ␤(r). In the case of a
local eddy viscosity and in the case of a constant eddy vis-
cosity, we find, respectively,
␤͑r͒ӍϪ4.4/rϩ5, ͑22͒
␤͑r͒ӍϪ3.5/rϩ4. ͑23͒
We note that this FLES function ͑21͒ is close to the form
of the energy spectrum in the dissipation range previously
published, E(k)ϳ(k␩)␣
Ј exp͓Ϫ␤Ј(k␩)͔. This exponential
decay has been suggested by Kraichnan’s direct interaction
approximation38
͑DIA͒. Martı´nez35
found the values ␣Јϳ
Ϫ 5
3 and ␤Јϳ5 in the near dissipation range from a DNS of a
forced isotropic turbulence ͑as kmax␩LESϷ0.5r with rр4, the
present LES would correspond to the near-dissipation range͒.
Considering that ͗␯LES͘ and ͗␩LES͘ are analogous to, respec-
tively, ␯ and ␩ this corresponds to a slope ␤ϭ␲CS␤Ј
ϳ2.8. This is within the range ͓2.45–3.5͔ of values obtained
for the homogeneous form of the subfilter viscosity with r
у2. From Eq. ͑11͒ we deduce Gˆ 2
(x)ϭFLES(x)/K0 and the
filter transfer function is Gˆ (x)ϭexp͓x␤/2͔.exp͓␣/2͔/K0 .
This is a Cauchy filter and the associated filter function is
͑see Pope12
͒
G͑y͒ϭ
a
␲⌬¯
ͫͩy
⌬¯ ͪ
2
ϩa2
ͬ
e␣/2
K0
, ͑24͒
with aϭ␤/2(␲).
C. The Kolmogorov constant
On the one hand, with the knowledge of the analytical
expression of fLES for the Smagorinsky model and using
approximations of the ensemble average ⑀ and ͗␯LES͘, we
can derive the theoretical expression K0LES
ϭ1/2
͐0
␲rCs
x1/3
fLESdx ͑see Muschinsky11
͒. Note that fLESϭfH is
actually valid for all k since fHϳ1 at large scales, unlike the
expression ͑21͒ found for the Schumann model. This allows
us to integrate the spectrum over the entire wave number
range and to get the analytical expression of K0LES
.
For rϭ1 we retrieve Lilly’s relation K0LES
ϭ2/3(␲CS)Ϫ4/3
ϭK0 .
For rу2, using the Heisenberg function that was previ-
ously found to be the filter shape, it is found that K0LES
ϭK0H
(K0 ,r,CS)ϭK0 ͓1ϩ(3K0/2)Ϫ3
(␲rCS)Ϫ4
͔1/3
. This
relation shows that K0LES
goes to K0 as r tends to infinity, but
note that K0H
(rу2)ӍK0 . However, the constant CS being
used satisfies ͑5͒ and we can simplify K0H
:
K0LES
ϭK0H
͑r͒ϭK0͑1ϩrϪ4
͒1/3
. ͑25͒
On the other hand, the LES equivalent of the Kolmogorov
constant can be calculated from simulations ͓see Fig. 4͑a͔͒
and compared to the theoretical values presented above ͑Fig.
8͒. As proposed by Muschinsky, we need to take into ac-
count the turbulent Reynolds number Retϭ͓N/(rCS)͔4/3
, in
order to explain the discrepancies between theory and simu-
lation. If this number is too low, the LES fluid is not fully
turbulent and cannot generate an inertial-range spectrum re-
sulting in a decrease of K0LES
. Considering this effect, Fig. 9
shows qualitatively the expected behavior of K0LES
for dif-
ferent resolutions ͑different values of N͒, indicating a critical
FIG. 7. Coefficients ␣ ͑a͒ and ␤ ͑b͒ of the FLES function (21) calculated by
linear regression of the logarithm of the compensated energy, with
␯LES(x,t) (solid line) and ␯LES(t) (dashes).
FIG. 8. The LES-equivalent Kolmogorov constant: theoretical values ͑black
symbols͒ ͑the effect of Ret is not taken into account͒ compared to simula-
tions ͑white symbols͒: Smagorinsky 603
͑circles͒ and 1203
͑squares͒, Schu-
mann 603
with ␯LES(x,t) ͑gradients͒ and with ␯LES(t) ͑deltas͒.
FIG. 9. Qualitative behavior of K0LES
for different resolutions N1ϽN2
ϽN3Ͻ¯ showing the effect of Ret .
1446 Phys. Fluids, Vol. 13, No. 5, May 2001 Magnient, Sagaut, and Deville
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10. value of r for each number of grid points N1 ,N2 ,..., corre-
sponding to a critical turbulent Reynolds number below
which K0LES
is smaller than K0ϭ1.5.
The same behavior is obtained with Nϭ120 and N
ϭ60, which corresponds approximately to the case N1 in
Fig. 9. As suggested by Muschinsky it appears that the reso-
lution NϷ100 is probably too low to display the ‘‘plateau-
behavior,’’ which is expected for larger values of N as for the
case N4 in the figure. The results obtained with rϭ4 for N
ϭ60 could be questionable, as the corresponding wave num-
ber is located at the beginning of the inertial range. However,
the 1203
simulation with rϭ4 gives a similar value of K0LES
,
and the behavior of this variable is consistent in both cases,
which demonstrates that the results are relevant for the in-
vestigated values of r.
The LES equivalent of the Kolmogorov constant has
also been evaluated for the Schumann model. For rϭ1,
K0LES
is simply deduced from the plot of the compensated
spectrum. For rу2 we can assume the property for the func-
tion FLESϭexp(␣Ϫ␤)ϭK0LES
/e at xϭ1 ͑at the cutoff wave
number͒, as verified with the Smagorinsky model. Therefore
with this assumption the values of K0LES
are calculated as
exp(␣Ϫ␤ϩ1) for rу2. Figure 8 shows the curves obtained
for the local and the homogeneous formulations of the
model. These values are somewhat different from those of
the Smagorinsky model, as they are close to 1.5 for all r
greater than 2. We need to consider that they are not calcu-
lated the same way and we note that the evaluation of K0LES
is quite sensitive to the difference (␣Ϫ␤) that is estimated
from the plots. This is probably the reason why in this case
the values of the LES-equivalent Kolmogorov constant are
not lower than 1.5, as observed with the Smagorinsky model.
D. Simulations at constant filter length
In order to test the influence of the resolution on the
solution independently of the model, it is interesting to com-
pare the results of simulations on different grid resolutions
with a fixed filter length ⌬¯ . In Fig. 10 the compensated ki-
netic energy spectra obtained with the Smagorinsky model
are plotted for three different values of the cutoff wave num-
ber kcϭN/(2r): 30, 15, and 7.5. These values correspond,
respectively, to rϭ1, 2, 4 for the 603
grid and to rϭ2, 4, 8
for the 1203
grid.
It is clear that the two simulations at kcϭ30 are not of
the same nature because of the value of the parameter r. In
the 603
simulation, rϭ1 and the transfer function is fLES
ϭ1: in the inertial range the compensated spectrum is con-
stant. In the 1203
simulation rϭ2 and the compensated spec-
trum is damped: the effective filter is a Heisenberg filter.
Therefore, the value of r, through the presence or absence of
the subfilter scales, has a drastic influence on the nature of
the LES and the effective filter. For rϭ1 the model is influ-
enced by fact that the mesh imposes a cutoff at the wave
number kmax , preventing any energy transfer toward higher
wave numbers. The hypothesis of a sharp spectral cutoff
used in the evaluation of the constant CS by Lilly ͓Eq. ͑5͔͒,
which also set the dissipation rate of the model, is actually
satisfied because of the discretization cutoff at wave number
␲/⌬. Obviously, the Smagorinsky model itself cannot pro-
vide a spectral cutoff, as shown by the cases rϾ1.
For the two other cutoff wave numbers, the influence of
r ͑or equivalently of the resolution͒ is more tenuous and the
form of the spectra are identical. The discrepancy between
two different resolutions at the end of the spectra reflects the
dependence of the LES-equivalent Kolmogorov constant on
r: when r increases ͑corresponding to a greater resolution͒
K0LES
decreases. These results prove that the simulation be-
comes independent of the mesh as the resolution is increased
and that the filter FLES has a quasiuniversal behavior. This
also validates the previous calculations with the 603
grid.
We can also note that the energy pile-up that has been
observed is not influenced by r but is due to the decrease of
the cutoff wave number in a consistent manner with its ex-
planation as a result of a bottleneck effect.
E. The influence of the spatial scheme
The numerical error is the sum of the ‘‘finite-
differencing’’ and ‘‘aliasing’’ errors. According to
Kravchenko and Moin,39
the skew-symmetric form of the
convective term in the Navier–Stokes equations has the
smallest aliasing error compared to other forms, and these
errors can usually be neglected. They also showed that with
low-order schemes and at low r the contribution of the sub-
filter model is small compared to numerical errors. Ghosal6
has shown that the finite-differencing errors are sensible to
the parameter r and recommends the use of rϾ1. These er-
rors are reduced as r increases or when the order of the
spatial scheme is increased. Vreman40
states that when r in-
creases the resolved variables contain less information but
are more accurate because the discretization errors are
smaller than the modeling errors. Consequently, if the value
of r is high enough there is no need of high-order schemes:
he used rϭ2 for LES of mixing layers. This value seems a
good compromise between the accuracy of the result and the
loss of information implied by rϾ1.
The results presented in the previous sections are ob-
tained with a fourth-order centered spatial scheme. Figures
FIG. 10. Compensated spectra for Nϭ120 and Nϭ60 for different values of
kcϭN/(2r) ͑Smagorinsky model͒: kcϭ30 ͑line͒, 15 ͑black symbols͒, and
7.5 ͑white symbols͒.
1447Phys. Fluids, Vol. 13, No. 5, May 2001 A study of built-in filter for some eddy viscosity
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11. 11 show the same computations with the Smagorinsky model
for two other finite difference schemes: a second-order cen-
tered and a sixth-order tridiagonal compact scheme ͑Lele41
͒.
These simple finite difference schemes have been chosen as
they are similar to the those used in more complex geom-
etries. From these simulations we can see that both schemes
produce similar behavior of the TKE spectrum in the inertial
range, which means that the filter is a Heisenberg filter inde-
pendently of the spatial scheme. However, for rϭ1 the val-
ues of K0LES
differ significantly and are about 1 for the sixth-
order scheme and 2.7 for the second-order scheme ͓Fig.
11͑a͔͒. When r increases this difference is reduced and be-
comes negligible at rϭ4 ͓Fig. 11͑b͔͒ ͑cases rϭ2 and 3 are
not shown but are consistent with this behavior͒.
In agreement with Ghosal,6
the influence of the numeri-
cal scheme is strong for small values of r and almost negli-
gible at rϭ4 and larger values, the numerical errors being
dominated by the subfilter terms amplitude. Although the
computed effective filter is a combination of the model filter
and a numerical filter, for large values of r the numerical
influence can be neglected. Therefore the filter forms found
for both the Smagorinsky and the Schumann model with r
у2 are specific to the model itself: they are the built-in filter
associated to the models.
IV. DISCUSSION
The damping function fH found for the Smagorinsky
model is derived from the Heisenberg’s theory of
turbulence37,42
and is based on the assumption of eddy vis-
cosity, which implies statistical independence between the
large and the small scales. It is also based on the assumption
of statistical equilibrium of the smallest eddies and the qua-
siequilibrium of the large eddies. Thus, the ‘‘Smagorinsky
fluid’’ probably has these properties.
The fact that the two models presented in this study do
not have the same built-in filter ͑independently of the nu-
merical scheme͒ proves that the form of the filter is not a
property of the eddy-viscosity assumption for the Reynolds
stress but that the transport equation plays a crucial role: it
takes into account more physical phenomena such as nonlo-
cal effects.
As the computed flow is homogeneous, one could expect
that the spatial distribution of the eddy viscosity has no sig-
nificant effect from a statistical point of view and that only
the average viscosity is meaningful. From the simulations
with a local and a constant eddy viscosity it appears that the
spatial variations of the viscosity does play a role with the
one-equation model.
For a steady-state isotropic turbulence with a constant
eddy viscosity, the LES equations at infinite Reynolds num-
ber are expected to be similar to the Navier–Stokes equa-
tions with a molecular viscosity ␯ϭ͗␯LES͘. Although the
simulations presented in this paper deal with decaying turbu-
lence, it is interesting to compare the asymptotic filtering
functions of the models ͑for the larger r limit, or mesh size
small compared to the viscosity length ␩LES͒ with the form
of the energy spectrum of a Newtonian fluid in the dissipa-
tion range. Here the ratio r can be seen as a resolution coef-
ficient similar to kmax␩ used to define a ‘‘well-resolved’’
DNS. Studies of the dissipation range spectrum are also done
in order to try to improve turbulence models that are sup-
posed to model the effect of the dissipative small scales.
Theoretical energy spectrum functions have been pro-
posed by Heisenberg,37
Pao,36
or Kraichnan.38
Some experi-
ments ͑Saddoughi and Veeravalli43
͒ and computations ͑Mar-
tf´inez et al.35
͒ tend to confirm an energy spectrum of the
form proposed by Kraichnan but, beside the inertial range,
the near and the far dissipation range should be distinguished
͑transition at k␩ϳ4, according to Martf´inez et al.35
͒. How-
ever, it appears that there is no definitive form of the spec-
trum yet ͑see Schumann13
for a discussion͒ and no analytical
function valid for the entire wave number range.
Finally, we have seen that for large values of r the ef-
fective filter has a ‘‘quasiuniversal’’ behavior, because both
K0LES
and fLES become independent of r ͑i.e., of the numeri-
cal cutoff at ␲/⌬͒ and of the numerical scheme, as shown,
respectively, in Secs. III D and III E. In this case the filter is
entirely determined by the model, and for rу2 we can con-
sider that the effective filter is the built-in filter of the model.
However, ⌬¯ cannot be too large as the cutoff wave num-
ber must remain in the inertial range. From the previous
results it appears that for the 603
grid the values of r ranging
from 2 to 4 are satisfactory regarding the validity of the
simulation. Whether the grid independence for large r can
also be observed with anisotropic, nonhomogeneous, or un-
structured grids remains to be verified. Moreover, the draw-
back of large filter width is a possible inhibition of transi-
tions in more complex flows due to high subfilter viscosity
and low turbulent Reynolds number.
FIG. 11. Influence of the numerical scheme on the TKE spectrum with a
Smagorinsky model for rϭ1 and rϭ4: centered second-order scheme
͑black symbols͒ centered fourth-order scheme ͑line͒, and a sixth-order tridi-
agonal compact scheme ͑white symbols͒.
1448 Phys. Fluids, Vol. 13, No. 5, May 2001 Magnient, Sagaut, and Deville
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12. V. CONCLUDING REMARKS
It appears that rϭ⌬¯ /⌬ is an important parameter regard-
ing the effective filter of LES. We found the form of the
built-in filter function for two models whose bandwidth can
be controlled through the characteristic length ⌬¯ appearing
in the expression of the Reynolds stress. This study should
be extended to other LES models in which this parameter is
not explicit and should be determined. It would be also in-
teresting to consider more complex flows ͑mixing layers, ro-
tating flows, boundary layers, ...͒ involving more complex
physical processes.
ACKNOWLEDGMENTS
J.-C. Magnient and P. Sagaut are grateful to Dassault-
Aviation for their support and thank the reviewers for their
valuable comments. The authors also appreciated the help of
S. B. Pope for providing a draft copy of his book. The com-
putations were carried out on a NEC SX-5 at ONERA.
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