A comparative study of subgrid scale models in homogeneous isotropic turbulence
C. Fureby, G. Tabor, H. G. Weller, and A. D. Gosman
Citation: Physics of Fluids (1994-present) 9, 1416 (1997); doi: 10.1063/1.869254
View online: http://dx.doi.org/10.1063/1.869254
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A comparative study of subgrid scale models in homogeneous
isotropic turbulence
C. Fureby,a) G. Tabor, H. G. Weller, and A. D. Gosman
Department of Mechanical Engineering, Imperial College, London, United Kingdom
~Received 30 September 1996; accepted 27 December 1996!
Recently, a number of studies have indicated that Large Eddy Simulation ~LES! models are fairly
insensitive to the adopted Subgrid Scale ~SGS! models. In order to study this and to gain further
insight into LES, simulations of forced and decaying homogeneous isotropic turbulence have been
performed for Taylor Re numbers between 35 and 248 using various SGS models, representative of
the contemporary state of the art. The predictive capability of the LES concept is analyzed by
comparison with DNS data and with results obtained from a theoretical model of the energy
spectrum. The resolved flow is examined by visualizing the morphology and by analyzing the
distribution of resolved enstrophy, rate of strain, stretching, SGS kinetic energy, and viscosity.
Furthermore, the correlation between eigenvalues of the resolved rate of strain tensor and the
vorticity is investigated. Although the gross features of the flow appear independent of the SGS
model, pronounced differences between the models become apparent when the SGS kinetic energy
and the interscale energy transfer are investigated. © 1997 American Institute of Physics.
@S1070-6631~97!01705-4#
I. INTRODUCTION
Direct numerical simulations ~DNS! and Reynoldsaveraged simulations ~RAS! represent two extremes of the
possible methods for simulating turbulent flows. DNS represents a brute-force approach: all scales of motion are simulated, which is expensive for any turbulent flow. In RAS, the
ensemble-averaged mean flow is solved for, with an appropriate model being used to describe the effect of the fluctuations of the flow around this mean. Large eddy simulations
~LES! lie between these two extremes: the rationale is that
only the large-scale motions are noticeably affected by the
geometry of the domain, while the small-scale motions are
similar or even self-similar in the bulk of the flow. The division into large- and small-scale motion elements, hereafter
denoted grid scale ~GS! and subgrid scale ~SGS! components, can be accomplished by convolving the dependent
variables with a predefined kernel. The GS motion is explicitly simulated whilst the average effect of the SGS motion on
the GS motion is accounted for by a SGS model.
In recent years studies have identified some inherent
limitations of the SGS models currently used in LES. For
example, it has been demonstrated that the coefficient in the
algebraic eddy-viscosity model of Smagorinsky1 has to be
fine tuned for different flows,2 and that it is strongly dependent on the Reynolds ~Re! number.3 Moreover, this model
assumes the existence of an inertial range, which severely
limits its usefulness. To overcome the deficiencies of this
model, one-equation eddy-viscosity models,4,5 scale similarity models, and linear combination models6 were developed.
More recently, Germano et al.7 have suggested a dynamic
procedure in which the model coefficient~s! of an arbitrary
a!
Corresponding author: Department of Mechanical Engineering, Imperial
College, Exhibition Road, SW7 2BX London, United Kingdom. Telephone: 144-171-5947109; Fax: 144-171-8238845; electronic mail:
c.fureby@ic.ac.uk
functional relationship, selected to represent the SGS stress
tensor, can be evaluated as part of the simulation. This procedure, applied to the Smagorinsky eddy-viscosity model,
has proven quite versatile, and results indicate that it can
overcome most deficiencies of the original model. Several
alternative formulations of this model have been described,
including the localized dynamic model8 and the Lagrangian
dynamic model.9
Although the main features of the GS flow appear independent of the SGS models,10,11 different SGS models predict the mean effect of the SGS motion on the GS motion
differently. In LES of flows with very high or low grid Re
numbers ~ReG5D2iD̄i / n , where D is the grid spacing, n the
viscosity, and iD̄i the rate of strain!, the SGS model must
incorporate the effects of the viscous subrange and full inertial range, respectively. Also, the SGS models become important if LES is applied to flows that involve other physical
processes affected by the detailed properties of the flow. A
relevant test case for examining SGS models is a cubic box
of fluid subjected to a random body force f at large scales.
This test case exhibits many of the basic features of turbulence and requires only limited computational effort. In addition, at ReT numbers ~ReT5 u vrmsu l T / n , where l T is the
Taylor length scale! below ReT'170, DNS data are available for comparison.
This paper is divided into two halves, one theoretical and
one computational. In the first part, the LES model and the
SGS models are described, together with the forcing scheme
and the numerical methods. The second part contains results
from simulations covering ReT'35– 248. In particular, we
focus on the cases ReT'94 and 248, which display a wide
range of interesting features. The resulting energy spectra are
compared with empirical models,12 and, when possible, also
with results from DNS data.13 The characteristic properties
of the different SGS models are compared and presented in
terms of energy spectra and probability density functions
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1070-6631/97/9(5)/1416/14/$10.00
© 1997 American Institute of Physics
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~PDF! for the SGS kinetic energy and viscosity. The spatial
structure of the GS flow is visualized and quantified using
cumulative distribution functions ~CDF! of the GS enstrophy, rate of strain, and stretching, which are compared with
CDFs from DNS data for the low ReT number case. Finally,
we examine the intrinsic coupling between the GS rate of
strain tensor and the GS vorticity, and the spatial correlations
between the GS enstrophy and various quantities related to
the SGS models.
TABLE I. Overview over the different SGS models investigated in the
present study.
Model
Definition
A1
~3!1~4!
A2
B1
B2
II. THE LARGE EDDY SIMULATION MODELS
B3
In this study we focus on homogeneous isotropic flows
that are described by the incompressible Navier–Stokes
equations.14 In LES it is assumed that the dependent variables can be divided into GS and SGS components v5v̄
1 v8 , where v̄5G * v5 * D G( j ,D) v ( j ,t)d 3 j . Here D is the
computational domain having the boundary ] D and the closure Dø ] D. The kernel G5G(x,D) is any function of x
and the filter width D that is endowed with the properties
* D G( j ,D)d 3 j 51,
G5G( u xu ,D),
G(x,D)>0,
limD→0 G(x,D)5 d (x), and G(x,D)PC n (R 3 ) having compact support. Convolving the Navier–Stokes equation ~NSE!
with G and assuming that @ G * , ] t # v50 and @ G * ,“ # v50,
gives the filtered NSE,
C
D
div~ v̄! 50,
] t ~ v̄! 1div~ v̄ ^ v̄! 52grad p̄1div~ S̄2B! 1f,
Constant coefficient algebraic eddy-viscosity model
~AVM!
~3!1~6,7! Dynamic coefficient algebraic eddy-viscosity
model ~AVM!
~3!1~5! Constant coefficient one-equation eddy-viscosity
model ~OEEVM!.
~3!1~8,9! Dynamic coefficient one-equation eddy-viscosity
model ~DOEEVM!
~3!1~10! Localized dynamic coefficient one-equation eddyviscosity model ~LDOEEVM!
~11!
Linear combination model ~LCM!
~12!
Monotone integrated large eddy simulation ~MILES!
that B is an isotropic function of its arguments. Also, the
modeled B should have the same mathematical and physical
properties as the exact B;15 this requirement leads to a set of
constraints referred to as the realizability constraints, and
implies that B should be a positive definite tensor and that
G(x,D)>0. Three classes of SGS models will be analyzed;
all models selected, see Table I, are frame indifferent, but the
realizability constraints are not always fulfilled. This is beyond the scope of this study, but will be addressed in a future
paper.15
~1!
where v is the velocity field, p the specific pressure,
S52 n D the viscous stress tensor, n the molecular viscosity,
D5 21(grad v1grad vT), the rate of strain tensor, while the
SGS stress tensor is
B5v ^ v2v̄ ^ v̄
5 ~ v̄ ^ v̄2v̄ ^ v̄! 1 ~ v̄ ^ v8 1v8 ^ v̄! 1 ~ v8 ^ v8 !
5L1C1R,
Features
~2!
where L is the Leonard stress tensor, C the cross stress, and
R the Reynolds stress. Observe that L can be calculated exactly while C and R, or alternatively B, must be modeled.
The domain D is a cubic box of fluid having cyclic boundaries discretized on a uniform computational grid to ensure
that @ G * , ] t # v50 and @ G * ,“ # v50; Ref. 15. Also, f is the
forcing function that will be discussed in some detail later.
Finally, D5 b 3 AP 3i51 Dx i where Dx i is the cell size in direction i and b > 1.
The GS components are morphologically dependent on
the geometry of the flow via the boundary conditions, while
the effects of the SGS components on the GS motion must be
separately modeled. The objective of the modeling is to represent B in terms of the GS velocity field in such a manner
that the modeled SGS stress tensor portrays, as closely as
possible, the exact stress tensor and the effects caused by the
kernel that may be nonuniform. The interscale energy transfer e(D)52B–D̄ must also be accurately predicted by the
SGS model in order to permit coarse grid solutions of ~1!.
Since the NSE are frame indifferent it is natural to require
that the filtered NSE have the same property,15 this implies
A. Subgrid scale models of eddy-viscosity type
SGS models of the eddy-viscosity type are based on the
hypothesis that the deviatoric part of the SGS stress tensor is
locally aligned with the filtered deviatoric part of the rate of
strain tensor, while the normal stresses are assumed to be
isotropic and are thus representable through a SGS kinetic
energy,
B5 32 kI1BD 5 32 kI22 n k D̄D ,
k[ 12 tr~ B! ,
D̄D 5D̄2 31 tr~ D̄! I,
~3!
where k is the SGS kinetic energy, I the unit tensor,
and n k the SGS viscosity. The functional form B
5B(D̄,k, n k ;x,t) implies that B is parametrized only by k
and n k , which then must be specified before the model is
complete. This particular form of B that is a special case of
a more elaborate functional form B5B( v̄ ;x2y,t2s),
where yPD and sP @ 0,` # , reflects the assumption that the
effect of the SGS motion on the GS motion is of dissipative
nature, since e(D)52 n k i D̄i 2 .
The best-known model of this class is probably the Smagorinsky model,1 which has been used in LES for many
years. It can be derived from a k 25/3 spectra assuming
ReG→`, and hence,
k5c I D 2 i D̄i 2 ,
n k 5c D D 2 i D̄i ,
~4!
where the model coefficients are given the values
c I 52/p 2 50.202 and c D 5(&/ p 2 )( 23c K) 23/250.042, where
c K'1.50 is the Kolmogorov constant. This model will hereafter be referred to as model A1.
Phys. Fluids, Vol. 9, No. 5, May 1997
Fureby et al.
1417
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An exact balance equation for k can be derived by contracting the exact balance equation for B that follows from
the exact and filtered Navier–Stokes equations. Redistribution effects do not contribute to the k equation in isochoric
flows, and it can be assumed that diffusion and dissipation
can be reasonably well modeled by terms of the form
div(nk grad k) and e 5c e k 3/2/D. Hence,
] t ~ k ! 1div~ kv̄! 52B–D̄1div~ n k grad k ! 2 e ,
n k 5c k Dk ,
1/2
e 5c e k D
3/2
21
,
~5!
where the dimensionless model coefficients are given the
values c k 50.05 and c e 51.00, Ref. 5. Observe that model
A1 can be recovered from ~5! if we assume that production
equals dissipation, i.e. if and only if B–D̄1 e 50. This model
will henceforth be referred to as model B1.
The dynamic model was first suggested by Germano7 to
rectify some deficiencies of models A1 and B1. The rationale
is to sample information on the GS level to evaluate the
model coefficients in model A1. This is achieved by filtering
~1! a second time with another kernel of width D̄ to obtain a
relation between the representations of B at the consecutive
filter levels, hence L5T2B̄ where L5(v̄ ^ v̄2v% ^ v% ),
% ), and B̄5(v ^ v2v̄ ^ v̄). By noticing that L
T5(v
%2v
^v % ^v
can be calculated explicitly and assuming the existence of a
scaling law allowing both T and B to be expressed in the
same functional form, ~3! and ~4!, we derive the overdetermined system,
% iD
% !,
LD 5L2 31 tr~ L! I52 ~ c D D 2 i D̄i D̄2c D D̄2 i D
1
2
% i 2 2c D 2 i D̄i 2 .
tr~ L! 5c I D̄2 i D
I
~6!
The model coefficients c I and c D cannot be removed from
the filtering; instead a variational formulation,8 can be used
in which the square of the residual errors of ~6!, F(c I ) and
F(c D ), is minimized. In homogeneous turbulence c I and
c D can be assumed independent of position, and thus,
F ~ c I ! 5 ^ @ 21 tr~ L!# 2 & 22 ^ 21 tr~ L! m & c I 1 ^ m 2 & c 2I ,
F ~ c D ! 5 ^ LD –LD & 22 ^ LD –M& c D 1 ^ M–M& c 2D ,
~7!
% iD
% , m5D̄2 i D
% i 2 2D 2 i D̄i 2 , and
where M5D 2 i D̄i D̄2D̄2 i D
^•& is the integral over D. The values of c I and c D that
minimize ~7! are c I 5 ^ 1/2 tr(L)m & / ^ mm & and c D
5^ LD –M& / ^ M–M& , as first suggested by Ref. 7, but with the
difference that the averaging ~necessary for stability reasons!
is not ad hoc, but follows from the variational formulation.
This model will be referred to as model A2.
Following Ref. 8, the dynamic model can also be extended to model B1. Again using the Germano identity
L5T2B̄ and assuming the existence of a scaling law, allowing T and B to be expressed in the same functional form,
~3! and ~5!, an overdetermined system of equations for c k
results,
% !,
LD 5L2 31 tr~ L! I52 ~ c k Dk 1/2D̄2c k D̄K 1/2D
1
2
tr~ L! 5K2k̄,
K5 21 tr~ T! .
~8!
As a consequence of the functional form used, the dynamic
procedure likewise implies the existence of a balance equation for the kinetic energy at the second filter level. Since the
balance equations for k and K are required to be consistent
with the identity 1/2 tr(L)5K2k̄, the compatibility condition,
% 2 ] @ 1 tr~ L!# 2div@ 1 tr~ L! v% #
z 5B–D̄2T–D
2
t 2
5c e K 3/2D̄21 2c e k 3/2D 21 ,
~9!
can be exploited to determine the model coefficient c e .
Again the model coefficients c k and c e cannot be removed
from the filtering and hence a variational formulation is used
to evaluate these. In a homogeneous flow c k and c e can be
assumed independent of position, thus c k 5 ^ LD •M& /
^ M–M& and c e 5 ^ z m & / ^ mm & , where M5Dk 1/2i D̄i
% i and m5K 3/2/D̄2k 3/2/D. This model will here22 DK 1/2i D
after be referred to as model B2. Models A2 and B2 can be
generalized to inhomogeneous flows,8 at the additional cost
of solving two further integral equations for c I and c D , or
c k and c e .
With no constraints imposed on the variational formulation leading to models A2 and B2, the coefficients, and
hence the SGS viscosity and e(D), can be locally negative.
Instead of exponential damping the negative viscosity causes
exponential amplification of the local perturbations and the
resulting reverse energy transfer is observed to have a very
long autocorrelation time. This does not correspond to the
real physics of reverse energy transfer observed in turbulent
flows and may also lead to numerical instabilities. The constrained variational formulation prevents these instabilities,
but precludes a dynamic model from possessing straightforward localization and the incorporation of nonlocal and hereditary effects. Another approach has been proposed by
Kim and Menon,16 in which the scale similarity assumption
between variables defined at consecutive filter levels is used
to derive expressions for the model coefficients. As long as
the cutoff is located inside the range in which the scale similarity assumption holds, the model coefficients are the same
for the consecutive filter levels. Applying this method to
model B1, the following expressions result:
%,
LD 5L2 31 tr~ L! I522c k D̄@ 21 tr~ L!# 1/2D
% –D
% ! 5c @ 1 tr~ L!# 3/2D̄21 ,
n ~ D̄–D̄2D
e 2
~10!
of which the first can be solved in a least-square sense by
% , while the second
contracting with M522D̄@ 1/2 tr(L) # 1/2D
can be solved directly. This model will hereafter be referred
to as model B3. Observe that the denominators in the equations for the model coefficients resulting from ~10! contain
the energy on the resolved scale, which is nonzero, thereby
preventing numerical instabilities.
B. Subgrid scale models of scale similarity and linear
combination type
Scale similarity and linear combination models are based
on the hypothesis that the interaction between GS and SGS
components takes place mainly between the smallest scales
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Fureby et al.
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of the GS components and the largest scales of the SGS
components. This feature is not included in eddy-viscosity
models, which are therefore purely dissipative. The scale
similarity model can be derived from ~2! by assuming that
the GS and SGS components are uncorrelated,17 resulting in
separate expressions for L, C, and R, which is appealing
from a physical point of view. A drawback is that the resulting model ~for R! is not dissipative and should therefore be
combined with a viscosity model; i.e., R is decomposed into
a scale similarity part and a dissipative part. Consequently,
B5 ~ v̄ ^ v̄2v% ^ v% ! 1 32 kI22 n k D̄,
k5c 1 D 2 i D̄i ,
n k 5c D D 2 i D̄i ,
~11!
where the last two terms originate from the eddy-viscosity
model. Here the values of the model coefficients are given by
c I 5 ^ 21 tr(L)m & / ^ mm & and c D 5 ^ LD –M& / ^ M–M& corresponding to those for the dynamic eddy-viscosity model A2.
The values c I 50.0066 and c D 50.012, resulting from comparison with DNS data, are often quoted.18 This model will
hereafter be referred to as model C.
C. Monotone integrated large eddy simulation models
(MILES)
Recently, results from three-dimensional, timedependent simulations of turbulent flows obtained without
explicit SGS stress models have been presented.19 It can be
argued,20 that the discretization errors of monotone convection algorithms provide implicit SGS models that do minimal
damage to the GS motion while still qualitatively incorporating most effects of the SGS motion. The implicit SGS flux
vector b resulting from the leading-order term in the truncation error can be derived by subtracting the discretized version of ~1! from another discretized version of ~1!, but using
a higher-order energy conserving scheme. Using Ref. 21 for
the convection terms we find that
b5 31 ~ Dt ! 2 ] 3t ~ v̄! 1div$ G@ grad~ grad v̄!#
2G~@ grad~ grad v̄! T! d# 2G~ grad d! T grad v̄% , ~12!
where G51/4v̄ ^ d ^ d, G5~1/91Y!v̄ ^ d, YP@0,1# is the flux
limiter, and d is the topology vector connecting neighboring
control volumes. Only the last term in ~12! is dissipative
introducing the tensorial viscosity G(grad d) T so that the effective viscosity becomes n̄ I1G(grad d) T. A scalar-valued
measure of the viscosity is i G(grad d) Ti 5&/8u v̄u l, where
l5 Atr@ (grad d) T(d ^ d)(grad d) # is a characteristic length
scale associated with the grid. A disadvantage of MILES is
that monotone convection algorithms, like all other algorithms that use knowledge of the grid relative to variations of
the solution, cannot be frame indifferent. Moreover, the filter
kernel G is unknown and may vary over the computational
domain, and the numerics and physics are closely coupled
together. MILES can therefore not be expected to mimic B
and e(D) in great detail, but can still produce accurate results for the GS motion. This model will hereafter be referred
to as model D.
D. Numerical methods and forcing functions
The filtered NSE ~1! are discretized using an unstructured finite volume method, while the computational domain
~a cubic box with periodic boundary conditions in all three
coordinate directions! is discretized using uniform Cartesian
grids with 163 , 323 , and 643 control volumes. Here we will
focus on the 323 simulations, although we will refer to the
other simulations as appropriate, Fig. 1~a!. The convective
terms are evaluated using linear interpolation between neighboring control volumes to derive a second-order accurate approximation. Similarly, the diffusive terms are evaluated
with a second-order accurate approximation to the inner gradient operator. The time integration is performed with a linear multistep method and the time step is derived from the
requirement that the maximum Courant number is below 0.2.
In order to decouple the pressure–velocity system, a Poison
equation, derived from the discretized version of the momentum equation, is solved for the quasipressure ( p̄1 32k). Since
this step of the algorithm is implicit the time integration is
performed using a second-order accurate three-level fully
implicit scheme. The resulting algorithm is second-order accurate both in space and time and the truncation error is
dominated by a dispersive term. The resulting nonlinear system of algebraic equations is solved by an incomplete
Choleski conjugate gradient method. For the MILES model
alone the convective terms are discretized using a secondorder accurate monotone algorithm21 that relies on a flux
limiter to cater for the additional implicit interscale energy
transfer necessary to stabilize the simulations ~12!.
Since the interest is in investigating the behavior of SGS
models in homogeneous isotropic turbulence the specific
body force f can be used to create random forcing of the
large-scale motion. The forcing must maintain a steady spectrum, and allow us to collect statistics of higher-order correlations. Numerous forcing schemes have been adopted for
this purpose in the literature,22,23 however, since we need to
represent particular length scale and velocity distributions
corresponding to the DNS data of Ref. 13, a more versatile
method is needed. Here the forcing scheme of Eswaran and
Pope24 is adopted, in which the constrained random body
force f is given by
f~ k,t ! 5P~ k! w~ k,t !@ Q ~ k! 2Q ~ k2kF !# ,
P~ k! 5I2 u ku 22 ~ k ^ k! ,
~13!
where w(k,t) is a vector-valued Uhlenbeck–Ornstein stochastic diffusion process, characterized by ^ w(k,t) & 50 and
^ w(k,t) ^ w* (k,t1s) & 52 s 2 exp(2s/t)I in the equilibrium
limit. Here an asterisk denotes the complex conjugate, k the
wave number vector, and ^•& indicates ensemble averages. In
this scheme, four parameters are introduced: the amplitude
s, the time scale t, and the wave numbers k L and k F . Spatially, the forcing is close to being white over the interval
@ k L ,k F# , i.e. it is uniform with ^ w(k) ^ w(q) & ' d (k2q).
Temporally, we have a forcing distribution peaked at the
frequency t 21 . It is possible that the statistical properties of
the turbulence are influenced by the features of the driving
mechanism. Nevertheless, we have taken values for k L and
k F so that only the largest scales are being driven, and the
Phys. Fluids, Vol. 9, No. 5, May 1997
Fureby et al.
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FIG. 1. Large-scale normalized energy spectra from LES compared with energy spectra from DNS, of Jimenés et al.,13 and from the modeled energy
spectra,12 at ~b! ReT'36, ~c! ReT'94, and ~d! ReT'248. With the exception of ~a! and Fig. 5, linetypes used throughout the paper are the following: ~gray
solid line! model A1, ~gray dashed! model A2, ~black solid! model B1, ~black dashed! model B2, ~black dash–dotted! model B3, ~black dotted! model C, ~gray
dash–dotted! model D on a 323 grid, and ~1! in-house DNS on a 643 grid. Moreover, ~s! is used for the DNS data of Ref. 13 and ~3! for the modeled energy
spectra. For ~a!, linetypes are identified in the graph.
accuracy of these scales is not of critical importance here.
The initial velocity is created by superimposing Fourier
modes having a prescribed energy spectrum but random
phases and projecting these onto the divergence-free space.
To ensure similar forcing on all simulations the 643 runs are
forced on a 323 grid; this is nontrivial since the flow is sensitive to the rate of energy addition and its distribution. For
the 163 simulations the forcing is carried out on the 163 grid,
which implies that a comparison between the 163 and the
323 or 643 simulations cannot be entirely rigorous. For the
statistical properties, however, this is not believed to be a
serious problem.
III. RESULTS AND DISCUSSION
In this section some results from LES of forced and decaying homogeneous isotropic turbulence in a cubic box of
fluid with periodic boundary conditions are presented and
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Fureby et al.
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TABLE II. Numerical and nominal flow parameters for the basic cases discussed in this paper. The estimates of k/ ^ k & and e/^e& are based on evaluating the
integrals defining k and e using DNS data or the DK spectra, depending on ReT , over the unresolved part of the wave number range. The upper and lower
limits for the integral properties are due to the different SGS models.
Grid
13
DNS
DNS
LES
3
64
1283
2563
5123
643
323
323
323
323
323
l I ~m!
l T ~m!
35
61
94
168
35
3562
6162
9464
17165
24867
1.97
1.76
1.37
1.65
1.95
1.9460.03
1.7360.02
1.4160.04
1.6560.03
1.7160.06
0.773
0.526
0.312
0.213
0.772
0.77160.002
0.52360.003
0.31460.004
0.24560.005
0.23260.006
discussed. To facilitate the presentation some characteristic
parameters will first be introduced. The total kinetic energy
is defined by ^ k & 51/2^ v2 & 5 * `0 E(k,t)dk51/2^ v̄2 & 1k,
where E(k,t) is the three-dimensional energy spectrum and
the total dissipation rate is defined by ^ e & 52 n ^ i Di 2 &
52 n * `0 k2 E(k,t)dk52 n ^ i D̄i 2 & 1 e . The rms-velocity scale
is defined by v rms5 u ^ (v2 ^ v& ) 2 & 1/2u , while the integral length
scale is defined in terms of E(k,t) as l I
5* `0 k21 E(k,t)dk/ * `0 E(k,t)dk; furthermore, the Taylor
length scale is defined by l T5( ^ v2 & / ^ (grad v) 2 & ) 1/2, and the
Kolmogorov length scale by l K5( n 3 / e ) 1/4. The large eddy
turnover time is defined by t I 5l I / v rms , and the Taylor Re
number is defined as ReT5l Tv rms / n . LES have been performed for four target Re numbers: ReT535, 94, 160, and
248, where the three first cases correspond to the DNS of
Jiménez et al.12 and in-house DNS at ReT'35. Some characteristic flow parameters are given in Table II. Note that the
ReT obtained vary slightly from their target values ~based on
the initial flow field! used to characterise the case; these differences are due to the effects of the SGS models on the final
flow field and are therefore indicative of the SGS model. The
following physical space analysis is performed using a posteriori methods that attempt to make qualitative observations
pertaining to possible relations between characteristic features of the SGS models and discernible flow structures that
may appear in the resolved flow.
A. Macroscopic effects of the SGS modeling—results
and discussion
The starting point for comparisons between the LES
models are the energy spectra, which can be compared with
each other, and with spectra obtained from DNS data or from
theoretical models of the spectra. For the low ReT number
cases, a comparison will be made both with DNS data and
the theoretical model of Driscoll and Kennedy12 ~hereafter
DK!, while for the higher Re number case a comparison will
be made with the DK spectrum only. The DK model is an
isotropic spectral model based on an infinite functional series
expansion using B u ku 4 (11 u ku 2 ) (2n217)/6 as basis functions,
i.e.,
`
E ~ u ku ,t ! 5 u ku
4
(
n50
u^vrms& u ~m/s!
ReT
B n ~ 11 u ku 2 ! ~ 2n217! /6,
~14!
0.092
0.236
0.605
1.592
0.093
0.09260.003
0.23660.004
0.56760.002
1.40260.006
1.66560.008
T ~s!
k/ ^ k &
e/^e&
21.65
7.45
2.26
1.04
20.97
21.0860.4
7.3360.1
2.4960.06
1.0960.07
1.0260.03
•••
•••
•••
•••
•••
0.005
0.02
0.04
0.12
0.18
•••
•••
•••
•••
•••
0.02
0.18
0.50
1.34
3.15
where the constants B n are evaluated by requiring the spectrum to have the correct functional form in the limit of large
k. This spectrum is specified by E5E(k, ^ k & , ^ e & , n ), and its
behavior toward large k is dictated by n and ^e&. The DK
model is intended to model adequately the entire energy
spectrum from large scales down to the Kolmogorov scales.
In particular, it is intended to provide a good fit even in the
absence of a 25/3 region of the spectrum, i.e., for low
ReT . Since the structure of the large-scale turbulence is
dominated by the method of forcing, we cannot expect a
precise fit between the LES spectra and the DK model. However, this is irrelevant for our purposes since we are mainly
concerned with the behavior of the small-scale structures for
which the DK model will provide an accurate model, since
the small-scale motion is considered independent of the
large-scale turbulence. The DK spectrum can be constructed
by first using n and ^e& as evaluated from a spectrum of the
form E5c K^ e & 2/3u ku 25/3, where c K'1.5, to describe the part
that lies within the inertial range. Second, ^ k & can be derived
by insisting that the area under the spectrum in logarithmic
space is the same for both cases. From the LES, ^ k & , ^e&, and
n are readily available and we can therefore construct DK
spectra for each of the simulations performed. Using the
model DK spectrum or the DNS data we can evaluate the
SGS kinetic energy and the dissipation rate to compare with
the predictions from the SGS models. Moreover, using the
DNS database ~for low ReT numbers! PDFs or CDFs for a
number of quantities can be compared with PDFs or CDFs
from LES.
Figure 1 shows the energy spectra resulting from the
different LES models together with the DK spectrum and the
DNS spectrum after about ten eddy turnover times. Wave
numbers are nondimensionalized with a grid wave number,
while nondimensionalization of the energy density involves
the mean energy dissipation rate computed in the simulation.
The energy spectra are based on averaging over 20 spectra
evaluated at instants well separated in time so that they can
be considered statistically independent. Moreover, it has
been verified that there is no monotonic trend in the total or
GS energy, so we may assume that a statistically steady state
has been reached.
A set of 323 simulations without SGS models was first
carried out as a control case in order to confirm that without
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SGS models the energy decay is not captured correctly. The
energy spectra for the ReT'35 case is identical to those for
the DNS data and the 643 LES. In this case, the contribution
of the SGS modeling to the GS flow is small, and for this Re
we could justifiably do underresolved DNS. The contribution
of the SGS model on the GS flow in the other two cases
~ReT'94 and 248! is more significant; the energy spectra for
the control case has the wrong slope in the inertial subrange.
This implies that the presence of the SGS model is important
to ensure the correct distribution of energy on the grid scales.
In the case of the ReT'248 case the error is highly significant. It also demonstrates that any numerical diffusion
caused by the difference scheme is negligible in comparison
to the explicit diffusion produced by the SGS models.
For the ReT'35 case, shown in Fig. 1~b! we have access
to in-house DNS on 643 as well as the DNS data.13 As expected, no inertial range exists for ReT'35, which thus provides a real challenge for the SGS models and the numerical
algorithm used. All LES models studied reproduce the energy spectra of the DNS and the DK model satisfactorily,
although models A1 and B1 result in slightly steeper spectra
at u ku .10 than the other models. Also, the kink in the DNS
spectrum is an artefact of the second-order discretization.
When the dissipative scales are not properly resolved the
physical dissipation mechanism is suppressed and numerical
dispersion becomes increasingly important. More precisely,
the aliased higher k contributions cause a buildup of energy
in the range u ku P @ 16,20# . However, this increase in E(k,t)
is not strong enough to cause severe numerical instabilities
since the physical dissipation regains control over dispersion
as the rate of strain rises. For the ReT'94 case @Fig. 1~c!#,
having only a short inertial range, the situation is different;
the resulting energy spectra are found to depend on the SGS
models only at the high wave number end of the inertial
range and into the viscous subrange. Models A1, B1, and D
reproduce the low wave number part and the inertial range
satisfactorily, but fail at higher wave numbers, since they
overestimate the SGS dissipation. Models A2, B2, B3, and C
result in improved spectra, in that they reproduce the entire
inertial range and the lower end of the viscous subrange satisfactorily. However, in the remaining part of the viscous
subrange these models somewhat underestimate the SGS dissipation, which results in the accumulation of energy close to
the cutoff wave number. The ReT'248 case is distinguished
by a well-developed inertial range and all LES models investigated reproduced the energy spectrum reasonably well.
Still, differences between the models can be identified. Models A1 and B1 tend to overpredict the dissipation, and model
A2 tends to underpredict the dissipation i.e., only models B2,
B3, and C can reproduce the spectra accurately. From the
DNS and DK spectrum, ^ k & , ^e&, l I , v rms , l T , k, and e can
be calculated and compared with the integral parameters
evaluated from the LES Table II. All integral quantities with
the exception of ^e& and e are well reproduced by LES. The
reason for the discrepancy in ^e& is that e is linear in E and
quadratic in u ku and is therefore very sensitive to the unresolved part of the spectrum.
Figure 2 shows the average variation of the model coefficients from the dynamic models A2 @ c I (t),c D (t) # and B2
FIG. 2. The variation of the dynamically evaluated model coefficients c I ,
c D , c k , ^ c k & , c e , and ^ c e & for models A2, B2, and B3 with Taylor Re
number, ReT , on a 323 mesh.
@ c k (t),c e (t) # and the localized dynamic model B3
@ ^ c k (x,t) & , ^ c e (x,t) # with ReT . All model coefficients go
through changes in the early stages of the simulations; however, as a realistic flow develops the coefficients reach a
quasiasymptotic non-negative state around which they fluctuate with an amplitude of about 10%–20% of the mean. As
observed from Fig. 2, the values of these coefficients are
strongly dependent on ReT when ReT is lower than ;190.
Above ReT'190 they stabilize around the values c I '0.40,
c D '0.04, c k '0.07, and c e '0.95. The asymptotic values of
these coefficients are in good agreement with previous results, in particular c D which is close to the value of 0.042
resulting from spectral analysis. However, c k is lower than
the corresponding value of 0.094, but is in excellent agreement with the value suggested by Ref. 5 using the direct
interaction approximation. Furthermore, c e is found to be
sensitive to both the grid resolution and the shape of the filter
function, which is a less attractive feature of models B2 and
B3.
To clarify the relative dependence of the SGS models on
the macroscopic flow structures, cumulative distribution
functions ~CDF! of the GS vorticity magnitude uvu, stretching s̄ 5 v̄–D̄v̄/ v̄2 , and rate of strain i D̄i , are shown in Fig.
3 for ReT'94 and ReT'248, respectively. For comparison,
these diagrams for ReT'94 also show the corresponding unfiltered variables resulting from the DNS of Jiménez et al.13
The CDFs represent the volume fraction occupied by values
of these variables above a given threshold level. In order to
enhance the relative difference in performance between the
SGS models, as well as the deviation from the DNS data,
Fig. 3 presents the CDFs in a semilogarithmic format. Note,
however, that the variable tails involve only small fractions
of the total volume. In fact, if these diagrams were presented
in a linear format hardly any difference would be observed
between the SGS models or between the DNS and the LES.
By inspecting the CDFs of u v̄u , s̄ , and i D̄i , together with
the corresponding unfiltered variables, it is obvious that the
distribution of these variables are far from Gaussian, although the LES data are more Gaussian than the DNS data,
and display few signs of converging to a limit distribution
for large ReT . Also, the DNS data have a more pronounced
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FIG. 3. Cumulative distribution functions of vorticity magnitude ~a!, ~b!, rate of strain ~c!, ~d!, and stretching ~e!, ~f! for the LES models investigated and from
the DNS database.13 The left column refers to the ReT'94 case while the right column refers to the ReT'248 case. The lines are as in Fig. 1.
tail at high values than the LES data; this is clearly due to the
lack of resolution in LES; hence, the mesh resolution in LES
does not support the resolution of the small-scale, highintensity, vortical structures embedded in a weaker vorticity
of lower intensity. To support this, we note that the 643 LES
display a longer tail than the 323 LES, but a shorter tail than
the DNS data. Hence, most of the GS values are accommodated in a comparatively weak background that would not
affect low-order statistical moments, although it will domi-
nate high-order moments. Also apparent from Figs. 3~a!–
3~c! is that the volume fraction of vorticity magnitude in the
unresolved structures is larger than in the corresponding volume fractions of s̄ and iD̄i. This suggests that the latter are
associated with larger spatial scales than the former. Accordingly, the concept of LES seems to be independent of the
SGS model if it can channel kinetic energy out of the wave
numbers close to the cutoff wave number to prevent aliasing;
hence the SGS model must be activated at the correct loca-
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formed on a 323 grid, such vortical structures can only be
properly resolved if l K. 41D, which is only the case for the
ReT'36 case: examining the flow for this case ~not shown!
we do indeed find tubular structures with the correct diameter. Moreover, Vincent and Meneguzzi28 have observed fat
worms similar to those predicted by LES using filtered DNS
results. Considering these results, it seems reasonable that
the observed presence of the coherent elongated vortices is a
realistic prediction based on the pre-filtered Navier–Stokes
equation. Consequently, v̄ appears in coherent filaments that
are stretched and intensified by D̄. From Fig. 4 it is also
evident that contours of s̄v̄2 are often correlated with the
presence of these filaments. The peaks of s̄v̄2 are located in
the neighborhood of the fat worms while less activity is observed in distant regions. More precisely, the peaks are often
located near the rim of the fat worms, but not inside them.
Frequently the peak values occur between two closely
spaced fat worms that are found to be regions experiencing
large strain and production of turbulence. However, when
compressed ~negative velocity gradient in the direction of the
filament! the filaments become buckled rather than weakened. In regions of strong turbulence, the vortex filaments
are broken up as a result of the buckling or compression.
In order to understand the basic mechanisms responsible
for the formation of these elongated vortical structures, the
balance equation for the GS enstrophy, z̄ serves as a starting
point,
FIG. 4. Visualization of high-intensity vorticity regions in LES of isotropic
forced turbulence at ReT'94 ~left column! and ReT'248 ~right column!
using different SGS models ~a! and ~b! refers to Model B3, ~c! and ~d! to
model C, and ~e! and ~f! to model D. Surfaces correspond to points at which
z̄ 51.5^ z̄ & / z̄ rms and on the parallel planes, selected to cut some of the most
visible vortical structures, isocontours of the GS vortex stretching are
shown.
tions in configuration space, and there enhance the local dissipation at the correct rate.
B. The resulting macroscopic flow field—results and
discussion
Figure 4 shows isosurfaces of z̄ 5 21v̄2 at a fixed level of
z̄ 51.5^ z̄ & / z̄ rms ~where ‘‘rms’’ refers to the root-mean-square
value! for models B3, C, and D at ReT'94 and 248, respectively. Weak vorticity is defined as that having z̄ , z̄ rms ; intense vorticity as that above a threshold covering 8% of the
computational volume; and background vorticity as that
above z̄ rms but still weaker than the threshold. From Fig. 4 it
is clear that coherent vortical structures of considerable diameter, sometimes referred to as ‘‘fat worms,’’ are present in
the resolved flow. Similar structures have also been observed
by Briscolini and Santangelo,25 and Meneveau et al.,9 in LES
of isotropic turbulence. The presence of similar structures
but of smaller diameter, so-called ‘‘worms,’’ has previously
been observed in DNS.13,26,27–29 A collective attribute of
these DNS is that they predict tubular vortical structures,
with a characteristic diameter of 4l K . Clearly, such small
structures cannot be captured in LES of high ReT number
flows due to the low spatial resolution. For simulations per-
] t ~ z̄ ! 1div~ z̄ v̄ ! 5div~ n grad z̄ ! 1 v̄–D̄v̄2 n ~ grad v̄! 2
2 v̄–curl~ div B! 1 21 v̄•curl f.
~15!
In Fig. 3, instantaneous CDFs of u v̄u , iD̄i, and s̄ 5 v̄–D̄v̄/
v̄2 are presented. The stretching rate is the part of the strain
aligned with the local vorticity, which thus stretches the vortex lines, while the strain rate is related to the dissipation but
does not appear explicitly in ~15!. Hence, from Fig. 3 we
conclude that the distributions of these variables are far from
Gaussian and show few signs of converging to a limit distribution for large ReT . Most of u v̄u is accommodated in a
comparatively weak background that would hardly affect the
low-order statistics, although it will dominate the high-order
statistics. Although the different SGS models result in
slightly different distributions, they all reproduce the correct
behavior, with models B2, B3, and D offering the best comparison with DNS at ReT'94. Hence, most of the volume is
occupied by a relatively weak vorticity with strong vortices
filling only a limited fraction of the domain; of these strong
vortices only a fraction are large enough to be resolved in
LES. The structure of the resolved vorticity implies that
weak and strong vortices have different structures; while
there is no evident structure in the low-intensity regions, the
high-intensity regions tend to be organized mostly in tubes.
Such structures, which have a Gaussian radial distribution,
are found to appear on the edges of regions that are almost
uniform in velocity. Also, similar to the situation in DNS,13
collisions between macroscopic flow structures seem to create shear layers that may roll up into vortex tubes. In Table
III, statistical moments of v̄ and L̄5grad v̄ are shown. We
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TABLE III. Higher-order statistical moments for the GS velocity and the velocity gradient distributions obtained from model B3. The nth-order flatness of v 1 is denoted by F vn and the nth-order flatness of L 11 and
12
L 12 is denoted by F 11
n and F n , respectively.
Grid and case
5123 DNS, ReT516813
2563 DNS, ReT59413
323 LES, ReT5248
323 LES, ReT594
Gaussian distribution
F v4
F v6
F 11
4
F 11
6
F 12
4
F 12
6
2.80
2.80
2.77
2.78
3.0
12.5
12.0
11.8
12.1
15.0
6.1
5.3
3.6
3.62
3.0
125
80
23.7
26.4
15.0
9.4
7.6
4.95
5.01
3.0
370
200
52.1
54.9
15.0
observe an increase in the kurtosis, and also a weaker but
concordant increase in the skewness, which is more apparent
for higher-order moments.
It is clear from ~15! that enstrophy can be produced either by stretching the vortex lines or by the presence of a
body force f. In these numerical experiments a forcing function with predefined statistics is used to induce the turbulence. To demonstrate that the features described above are
not artefacts of the forcing function f, this was removed and
the flow field was allowed to decay. After two eddy turnover
times, the fat worms were found to remain, and the statistics
associated with them remain largely unchanged. Hence, the
forcing is not essential to the development or to the maintenance of these vortical structures, and the fat worms are a
natural product of the evolution of turbulent flows, both
forced and decaying. Accordingly, only vortex stretching remains as a source for the enstrophy. From Figs. 5~a! and
5~b!, which show two-dimensional joint PDFs for iD̄i and
u v̄u , it is evident that the correlation between iD̄i and u v̄u is
weak. Intense vorticity is correlated with intense strain, either because strong vortices generate high strain or because
they are generated by the strain. Another alternative is addressed in Figs. 5~c! and 5~d!, collating u v̄u and s̄ . Thus, the
peak stretching rates are not associated with regions of high
vorticity, but with the background vorticity. In addition, the
stretching associated with the regions of peak vorticity is
low, with little evidence of self-stretching by the strongest
vortices. Figures 5~e! and 5~f! show some correlation between high iD̄i and s̄ . High strain does not imply strong
vortex stretching, and the orientation of the principal axis of
the strain rate tensor seems to be independent of the local
vorticity direction.
Spatial correlations between the rate of strain and the
vorticity have previously been reported by Kerr22 and Vincent and Meneguzzi,28 using DNS of isotropic turbulence. In
this paragraph we will briefly describe the corresponding
spatial correlations found between D̄ and v̄ in LES of a
similar flow field. To this end, let l i and di , i51,...,3, denote the eigenvalues and corresponding eigenvectors of D̄
ordered in size so that l 1 ,l 2 ,l 3 , and with the restriction
that they sum to zero for isochoric flow. Hence, when the
middle eigenvalue ~or strain! l 2 is negative then there are
FIG. 5. Joint probability density functions of ~a!, and ~b! rate of strain and
~c!,~d! stretching versus vorticity magnitude and ~e!,~f! stretching versus rate
of strain. The left column refers to the ReT'94 case and the right column
refers to the ReT'248 case. Linetypes are as follows: ~black solid line!
model B3, ~gray solid! model C, and ~black dashed! model D.
FIG. 6. Probability of the alignment of the resolved vorticity and the principal strain rate directions at ReT'94 ~left column! and ReT'248 ~right
column!. Eigenvalues are ordered as l 1 ,l 2 ,l 3 , and figures ~a! and ~b!
correspond to eigenvector d1 , ~c! and ~d! to eigenvector d2 , and ~e! and ~f!
to eigenvector d3 . The lines are as in Fig. 1.
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two different directions of compression. Moreover, define
the alignment angle u i between v̄ and di , so that
a 5cos ui5(v̄–di )/( u v̄i di u ). In Fig. 6 we present instantaneous PDFs of the alignment between v̄ and di in terms of a
for the cases ReT'94 and ReT'248 using the previously
described SGS models. The general features of these LESbased PDFs correspond well to the DNS-based PDFs of Vincent and Meneguzzi28 and Ashurst et al.30 For comparison,
note that Ashurst et al. use 20 bins, Vincent and Meneguzzi
use 250 bins, and we use 50 bins to sample cos ui when
forming the probability function ~PF!. Since this is probability, not probability density, a uniformly distributed quantity
corresponds to a probability of 1/N, where N is the number
of bins. It is clear that the ReT number does not affect the
shape of the PFs to any appreciable extent. The grid scale
vorticity is most likely to point in the d2 direction and least
likely in the most compressive direction d1 , while there is
hardly any correlation in the d3 direction. Moreover, we have
observed that the vorticity is aligned with l 2 , even when
l 2 is negative. Another interesting observation, pertinent to
the DNS runs, is that the strain field within the neighborhood
of the vortical structures has a quasi-two-dimensional property. To investigate if this feature is also captured by LES,
we have studied the PFs of the alignment of di with a truncated v̄ field containing only the vorticity vector field within
the fat worms. From these PFs it is clear that d1 and d3
frequently lie in a plane locally orthogonal to the fat worms;
in addition, the largest extensional strain is almost comparable to the compressive strain. Consequently, many of the
same conclusions with respect to the macroscopic flow structures arrived at from DNS can also be found by inspecting
LES results performed on a much coarser grid, independent
of the SGS model. There is hardly any difference between
the PFs produced by the different SGS models; only the
constant coefficient eddy-viscosity model ~model A1! shows
any deviation from the other models. In the case of model
A1, v̄ is more likely to be perpendicular to d1 and parallel to
d2 ; the correlation with d3 is also different from all other
models.
C. Microscopic effects of the SGS modeling—results
and discussion
The spatial distribution of k and n k is of interest in order
to differentiate between the SGS models and to increase our
present understanding of the interaction between the macroscopic flow and the SGS models. To this end, instantaneous
PDFs of k and n k are shown in Fig. 7 for ReT'94 and
ReT'248, respectively. For the linear combination model
~model C!, the viscosity is defined as the effective viscosity
appearing in the total dissipation; hence n k 52 21BD –D̄/D̄2 ,
where BD 5B 2 31tr(B)I. Since no explicit SGS model is incorporated in the MILES model ~model D!, it is not meaningful to compare quantities representative of the SGS models between explicit and implicit simulation concepts, and
therefore no reference to MILES will be made in the following discussion.
For the SGS kinetic energy k, normalized by the resolved kinetic energy @Figs. 7~a! and 7~b!#, we note that
models B1, B2, and B3 produce PDFs of similar shape, hav-
FIG. 7. Probability density functions of ~a!, ~b! k and ~c!,~d! n k for the LES
models investigated. The left column refers to the ReT'94 case and the
right column refers to the ReT'248 case. The lines are as in Fig. 1.
ing superficially different domains of dependence; Model B1
is shifted towards lower k while models B2 and B3 are
shifted toward higher k. For models A1 and A2 the situation
is different; the dynamic version generates a sharper PDF
shifted toward higher values of k than the constant coefficient model. For model C, k5 k 1c 1 D 2 i D̄i , which differs
from the k of the one equation or the algebraic models, as
can be concluded from Figs. 7~a! and 7~b!. For the SGS
viscosity n k , normalized with the molecular viscosity, @Figs.
7~c! and 7~d!# we observe that the one-equation models again
result in PDFs of similar shape having slightly different domains of dependence; model B1 is shifted toward higher n k
while models B2 and B3 are shifted toward lower n k ; also,
the PDF from model B3 contains negative viscosity, implying that this model supports backscatter. For the algebraic
models the situation is analogous, though the differences are
more pronounced. The linear combination model results in a
viscosity PDF containing negative viscosity, implying that
this model is capable of creating backscatter via the scalesimilarity submodel. The fraction of reverse energy transfer
can be estimated to be about 15% of the viscosity in the two
cases investigated, a reasonable amount compared to about
25%, as suggested by Leslie and Quarini.31 Referring to Fig.
2 these results comply with the observed behavior of the
model coefficients from the dynamic models A2
@ c I (t),c D (t) # and B2 @ c k (t),c e (t) # , and the localized dynamic model and B3 @ ^ c k (x,t) & , ^ c e (x,t) & # . The coefficients
c k and ^ c k & are lower than for the constant coefficient model,
thus reducing the dissipation of k through a reduced value of
n k ; also, c e and ^ c e & is allowed to vary, thereby influencing
the balance between production and dissipation, resulting in
an increase in the production of k in models B2 and B3
compared to model B1. For model A2 both c D and c I increase with increasing ReT number but reach asymptotic values close to, but just below, the constant coefficient values at
ReT > 350. Furthermore, the dynamic version of the algebraic
model seems to underestimate the SGS viscosity at high
ReT numbers, since the energy spectra @Fig. 1~c!# show signs
of energy building up at the high wave number end. So far
these results suggest that nonlocal effects partly included in
models A2 and B2, and fully included in models B3 and C
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~via multilevel filtering! and hereditary effects incorporated
in models B1, B2, and B3 ~through the solution of a separate
balance equation for k!, are essential in the development and
maintenance of realistic spatial and temporal distributions of
both k and n k .
In order to investigate the influence of the shape of the
filter kernel on the results, the simulations performed with
models A2, B2, B3, and C using a ~spherical! top-hat filter
were repeated using a finite-volume equivalent of a Gaussian
filter kernel. The results concerning the behavior of the macroscopic quantities produced by the three dynamic eddyviscosity models ~models A2, B2, and B3! were left unaffected by the shape of the kernel, while the results resulting
from the use of the linear combination model ~model D!
were only moderately influenced by the shape of the kernel.
For the microscopic quantities apparent differences are found
for all models; however, the differences cannot be substantiated further since these variables can only be evaluated from
DNS data by filtering, a process in which the shape of the
filter kernel is again influential.
By inspecting the left and right columns of Fig. 4 separately, it can be observed that the different SGS models affect the fat worms to a small extent. Focusing on the
ReT'94 case it is clear that models B2, B3, C, and D result
in vortical structures that are similar, although some differences occur. In general, the eddy-viscosity models generate
the longest vortical structures, model D results in shorter and
more slender, less space-filling vortical structures while
model C produces a less structured and more space-filling
vorticity field. For the ReT'248 case, many of the same
conclusions can be drawn with respect to the impact of the
SGS models on the vortical structures. It is also instructive to
analyze the CDFs of u v̄u , iD̄i, and s̄ shown in Fig. 3; all
models generate surprisingly similar CDFs, considering their
different nature. The constant coefficient eddy-viscosity
models, together with MILES, form one group in which the
CDFs have a shorter and less intense tail; the dynamic models and the linear combination models form one group with
more intense and longer tails, whereas the localized dynamic
models form a separate group with even more intense and
longer tails. The third group of models produce CDFs that
are closer to the DNS data than the other two groups of
models. The primary reason for this is believed to be that the
second and third groups of models can successfully incorporate information of nonlocal and hereditary nature via the
multilevel filtering procedure, and, in particular, the third
group and the linear combination model are successful at this
since their coefficients are functions of space and time.
The next question to be addressed is whether the fields
that are characteristic to the individual SGS models bear any
relationship to the vortical structures. There are three levels
at which comparisons can be made: ~i! the tensor level i.e.
between the modeled SGS stress tensor B and u v̄u , ~ii! the
vector level i.e. between the modeled div B and u v̄u , and ~iii!
the scalar level, i.e. between the modeled e(D) and u v̄u .
e(D) acts as a dissipation term in the filtered balance equation of mechanical energy and as a production term in the
balance equation of SGS kinetic energy. A positive e(D)
implies that energy is transferred from large resolved struc-
FIG. 8. Visualization of high-intensity vorticity regions in LES of isotropic
forced turbulence at ReT'94 ~left column! and ReT'248 ~right column!
using different LES models ~a! and ~b! refers to model B2, ~c! and ~d! refers
to model B3, and ~e! and ~f! refers to model C. Surfaces correspond to
points at which z̄ 51.5^ z̄ & / z̄ rms . On the parallel planes, selected to cut some
of the most visible vortical structures, isocontours of the interscale energy
transfer, e(D), are shown.
tures toward small unresolved structures via a cascade process ~outscatter! and a negative e(D) implies that energy is
transferred in the opposite direction by a reverse cascade
process ~backscatter!. Backscatter may become important
when a large amount of turbulent kinetic energy is present in
the unresolved scales, i.e. when the spatial resolution is marginal, as is usual when LES is applied to a more complex
flow. Moreover, in these circumstances anisotropy effects
may also become increasingly important. For an arbitrary
eddy-viscosity model e(D)'2 n k i D̄i 2 , and for a linear combination model e(D)'2 n k i D̄i 2 1L̄–D̄. Hence, backscatter
can only appear in eddy-viscosity models if n k is negative,
and only models B3 and C support this formalism without
violating the realizability constraints. Figure 8 shows isocontours of e(D) together with isosurfaces of z̄ 51.5^ z̄ & / z̄ rms .
For ReT'94 ~the left column!, models B2 and B3 yield similar distributions of e(D), but small localized regions of reverse energy transfer can be identified for model B3. An
interesting observation is that e(D) frequently peaks between closely spaced tubular structures, which are regions
experiencing large strain and vortex stretching. Model D produces a somewhat different spatial distribution with larger
regions of backscatter, but e(D) still has maxima between
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Fureby et al.
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the tubular structures. For model C ^ e(D) & is 10% lower
than models B2 and B3; this difference is related to the excess backscatter formed by model C. For ReT'248 ~the right
column!, the principal distribution of e(D) is similar, with
maxima or minima occurring between tubular structures in
regions of high strain and stretching. The main difference
between the two cases relates to the higher level of intermittency present in the ReT'248 case, and to the fact that the
main structures are broken up as a result of the intensified
buckling due to the increased probability of negative velocity
gradients aligned with the resolved vortical structures.
IV. CONCLUDING REMARKS
It is of significant interest to the future development of
LES to study various characteristic properties of SGS models. However, such an investigation, if performed in a more
realistic complex geometry, would be prohibitively expensive. Instead we have focused on forced and decaying homogeneous isotropic turbulence in a cubic box. Since the forcing scheme used generates homogeneous isotropic
turbulence, comparisons using the three-dimensional energy
spectrum can be carried out; here comparisons are made with
the DNS data of Jiménez et al.13 for ReT below 160 and for
all cases with the modeled energy spectrum.12 Eight different
SGS models have been selected, subdivided into three
classes; ~i! eddy-viscosity models that are further partitioned
into constant coefficient models, dynamical models, and localized dynamical models, ~ii! linear combination models,
and ~iii! monotone integrated large eddy simulation models.
Identical simulations have been carried out for each model
for a variety of flow conditions. From this, we are in a position to compare the results of LES with DNS data to establish the accuracy and versatility of the LES concept. We are
also able to compare and contrast the flow fields resulting
from LES using different SGS models, and to relate the
properties of these flow fields to the properties of the SGS
models. Finally, we are able to extend the modeling of turbulent flows to ReT numbers much higher than currently possible through DNS, demonstrating that the structural features
predicted by DNS at low or moderately high ReT numbers
are still present in the high ReT numbers flows.
The spectral comparisons and the macroscopic flow features suggest that the differences between LES with different
SGS models are small but not insignificant, in particular, on
coarse grids. Judging from the comparisons of time-averaged
energy spectra, and cumulative distribution functions of
u v̄u , i D̄i , and s̄ with DNS data at ReT'94 and ReT'35,
suggests that the choice of SGS model is not critical for
correctly reproducing the macroscopic flow if the spatial
resolution is adequate. For a LES performed on a marginal
coarse grid, so that the assumption of production equals dissipation is no longer valid ~the ReT'94 case and the 163 grid
or the ReT'248 case and the 323 grid!, the one-equation
models are superior to the algebraic models. The reason for
this is believed to be related to the assumption of ~local!
alignment between B and D̄, together with the fact that a
separate transport equation for k is solved so that both k and
n k experience nonlocal or hereditary effects. If nonlocal or
hereditary effects are believed to be important the linear
combination model or the dynamic or localized dynamic
one-equation eddy-viscosity model are found to be more accurate. We have also noticed that the linear combination
model decreased in performance more rapidly with a decrease in spatial resolution than all other models; see the
distribution of k or n k in Fig. 7. MILES is justified when
interest is focused on the macroscopic flow and when a relatively fine grid can be afforded to preclude large fractions of
k in the subgrid scales. Another aspect of SGS modeling is
the extra computational cost; models A2 and B2 increase the
cost by about 20% compared to their constant coefficient
counterpart, while model B3 introduces an additional cost of
20% above that for B2, and model C is about 15% more
costly than model A1.
High-intensity vortical structures are reproduced differently by different SGS models. In principle, those models
that incorporate nonlocal and hereditary effects, via multilevel filtering, seem to reproduce both the high wave number
region of the energy spectra and the tails of the cumulative
distribution functions of enstrophy better than other models.
This suggests that dynamic localization models and linear
combination models are the best candidates for further studies in more complex geometries. We also find evidence for
coherent elongated vortices of different sizes previously reported by various sources. Close examination of the rate of
strain and the vortex stretching demonstrates the physical
mechanisms responsible for this behavior. Contrasts between
the SGS models can be found when inspecting k and n k ;
although the explicit models predict similar levels for k and
n k , values that are in agreement with the DK spectra and the
DNS data, the PDFs for k and n k differ significantly. As can
be expected, SGS models that introduce nonlocal and hereditary effects ~such as filtering on two different length scales!
produce more homogeneous behavior. Clearly with this set
of data it is impossible to say which SGS models generate
the ‘‘correct’’ behavior, not least because the true behavior
of a turbulent flow may well change depending upon the
specific case. Nevertheless, the attempt to discover and investigate the different properties of the SGS models is valuable, and the differences we have been able to highlight are
likely to be of crucial importance in any flow that involves
significant small-scale physics.
ACKNOWLEDGMENTS
This work is supported by the EPSRC under Grants No.
43902 and No. K20910 and by European Gas Turbines. We
would like to thank Professor Jiménez for providing the DNS
data.
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