c
° TU Delft, The Netherlands, 2006
ON THE REQUIRED REYNOLDS-NUMBER DEPENDENCE
OF VARIATIONAL MULTI-SCALE SMAGORINSKY
MODELS
Johan Meyers∗1 and Pierre Sagaut†
Universit´e Pierre et Marie Curie, Laboratoire de Mod´elisation en M´ecanique Boˆıte 162, 4 place Jussieu, 75252 Paris cedex 05, France
∗e-mail: meyers@lmm.jussieu.fr; †e-mail: sagaut@lmm.jussieu.fr
Key words: turbulence, modelling, large-eddy simulation, variational multi scale, Smagorin-sky
Abstract. A theoretical analysis is presented on the dependence of the standard and
variational multi-scale Smagorinsky models on the proximity of the LES filter width ∆ to the integral length scale of turbulence L on the one hand, and to the Kolmogorov scale η on the other hand. Moreover modifications of the models are formulated, which respond better to ∆/η changes. Apart from a priori evaluations of L/∆ and ∆/η effects, the quality of our proposed modifications to the models is further evaluated and corroborated based on LES of decaying homogeneous isotropic turbulence.
1 INTRODUCTION
One of the pioneering contributions to the introduction of multi-scale modelling in the field of large-eddy simulations has been the work by Hughes et al.1–3 Here, it was proposed
that for large-edddy simulations (LES), subgrid-scale models should act only on a small-scale extraction of the turbulent field, and this idea was applied to the Smagorinsky model. It was shown in various studies,1–6 o.a., for homogeneous isotropic turbulence
and for plane channel flow, that the variational multi-scale (VMS) formulations of the Smagorinsky model provide improved results.
The standard Smagorinsky model7 has been very extensively studied in literature. A
classical contribution on the model has been presented by Lilly,8 who determined the
Smagorinsky constant based on the required subgrid-scale dissipation of a sharp cut-off filtered k−5/3spectrum. Using the same technique, Hughes et al.1provided expressions for
the VMS Smagorinsky constants. However, the assumptions underlying Lilly’s analysis are not relevant in a lot of effective LES cases. At moderate to high Reynolds numbers, the energy spectrum deviates from k−5/3 at low and high wavenumbers. Moreover, the
more recent studies,9,10 it has been demonstrated that the dependence of the (standard)
Smagorinsky coefficient on the ratio of the LES filter width and the Kolmogorov scale, is quite relevant.
In the current study, we present an analysis of the standard and variational multi-scale Smagorinky models based on a generalization of Lilly’s concept. We will show that the coefficients depend on three important effects, i.e., the ratio of the LES filter width to the Kolmogorov scale ∆/η, the ratio of the integral length scale to the LES filter width L/∆, and the shape of the LES filter. Moreover, for the VMS models, the coefficient further depends on the ratio of the LES filter width to the width of the high-pass filter used for the small-scale extraction ∆/∆0 (as also reported by Hughes et al.1), and on the shape of
this high-pass filter.
An a priori evaluation, which is based on the integration of Pope’s11 spectra, is
pre-sented on the L/∆ and ∆/η behavior of the coefficients. First of all, since large-scale turbulent effects are flow-case specific, no general conclusions or relations can be formu-lated on the L/∆ dependence of coefficients, and our analysis presumes homogeneous isotropic turbulence in the large scales. Therefore, we mainly evaluate trends, and fur-ther study the asymptotic convergence to L/∆-independent values for the different model coefficients. Interesting differences are observed between the standard and the VMS Smagorinsky models. Secondly, regarding the ∆/η dependence of the model coefficients (assuming L/∆ À 1), more general conclusions can be formulated, since these are based on the universal behavior of small-scale turbulence at high Reynolds numbers. It is ob-served that the various coefficients depend strongly on ∆/η, and the use of constant Lilly-coefficients can erroneously increase the subgrid-scale dissipation level. Therefore, modified models are proposed which account in a generic and easy-in-use manner for the ∆/η-dependence of the model coefficients.
Apart from theoretical a priori results, we further evaluate the standard and VMS Smagorinsky models, and our modified proposals in actual large-eddy simulations. De-caying homogeneous isotropic turbulence is investigated at different resolutions and com-parisons with DNS results12 are presented. It is shown, for two investigated VMS models,
that they perform consistently better than the standard Smagorinsky model. Moreover, the proposed modifications, which respond in theory better to ∆/η changes during a large-eddy simulation, consistently provide improved results.
2 A THEORETICAL A PRIORI ANALYSIS OF THE SMAGORINSKY AND VMS MODEL COEFFICIENTS
In this section we extend Lilly’s methodology to evaluate L/∆ and ∆/η effects on the model coefficients of the standard Smagorinsky model and two VMS Smagorinsky models. Moreover, modifications to the models are proposed. In §2.1, the different models are briefly introduced. Next, in §2.2 the methodology is elaborated for the standard Smagorinsky model. Subsequently, equivalent derivations are presented for two VMS models in §2.3. Finally, based on Pope’s formulation of the energy spectrum,11 an a priori analysis of the models and our proposed modifications is presented in §2.4.
2.1 Formulation of the models
The governing equations for large-eddy simulations are obtained by formally filtering the Navier–Stokes equations with a high-pass filter. Therefore, consider a one-dimensional low-pass filter operation G as
f (y) = Gf =
Z
KG(y − y0)f (y0) dy0, (1)
where KG(y) is the filter kernel. Further, G has the property that Gc = c, for every
constant function c. The Fourier transform of the above filter definition corresponds to
f (k) = G(k)f (k), with G(k) the transfer function associated with KG(y). Obviously, to
formally filter the Navier–Stokes equations, three-dimensional filters should be employed. However, in the current study, we will only consider spherical-symmetric filters, such that we can use an equivalent one-dimensional representation of the filter function.
Filtering the Navier–Stokes equations introduces a closure problem. A set of terms, i.e., the subgrid-scale stresses τij arise, which have to be approximated by a model mij.
Hence,
mij → τij = uiuj − uiuj. (2)
One of the most often employed formulations for mij is the Smagorinsky model,7 which
approximates the deviatoric part of τij as
mij = −2 (Cs∆)2|S|Sij, (3)
with Cs the Smagorinsky coefficient, ∆ the LES filter width, and |S| = (2SijSij)1/2 the
magnitude of the filtered strain-rate tensor Sij, where Sij = [∂ui/∂xj + ∂uj/∂xi]/2.
By employing a multi-scale framework to analyze subgrid-scale modelling in LES, Hughes et al.1 proposed to operate the Smagorinsky model on a small-scale extraction
of the turbulent field. For various test cases, Hughes et. al2,3,13 showed very good
these observations were further established and some further modifications have been proposed.4,5,14
In the present article, we will restrict our theoretical analysis and associated numerical evaluations to one VMS model, i.e. the small-small model, and to a proposed modification of this model for smooth filters.4
The original formulation of the small-small model corresponds to
mij = −
h
2Cs12 ∆2|S0|S0ij
i0
, (4)
where the operator [·]0 is a high-pass filter H0. Vreman4 proposed, for smooth filters, to
discard the outer filter [·]0 in (4), i.e.
mij = −2Cs22 ∆2|S 0
|S0ij. (5)
Though this proposal is formally not consistent with the original multi-scale framework proposed by Hughes et al., this model did produce qualitatively comparable results to (4), and requires less filter operations.4
2.2 The Smagorinsky coefficient for finite Reynolds numbers
One of the first theoretical analysis presented on the Smagorinsky coefficient was pre-sented by Lilly.8 His derivation assumes very high Reynolds numbers and high ratio’s L/∆, employing an idealized k−5/3 spectrum, and further presumes that the LES filter G is a sharp cut-off filter, and that the laminar viscosity is negligible. Lilly’s well known
expression for the Smagorinsky coefficient reads8 Cs,∞ = 1 π µ 2 3α ¶3/4 , (6)
where α ≈ 1.5 is the Kolmogorov constant. Later, Voke9 extended Lilly’s evaluation, and
showed that the Smagorinsky coefficient depends on the ratio ∆/η.
The analysis in the present section is a generalization of Lilly’s8 and Voke’s9 work,
adding effects of the filter shape and the ratio L/∆. Moreover, the presented derivations are general, and do not depend on particularities of the selected description of the three-dimensional energy spectrum which is needed for the analysis. As we will show in §2.3, this further allows to apply the methodology to the VMS Smagorinsky models.
Central to the derivation of an expression for the Smagorinsky coefficient Cs is the
expression of the total turbulent dissipation ε as
ε = εt+ εν = (Cs∆)2 |S|SijSij ® + 2νSijSij ® (7) ≈ (Cs∆)2 SijSij ®3/2 + 2νSijSij ® , (8)
where we employ Lilly’s approximation h|S|SijSiji ≈ hSijSiji3/2. Experience has shown
that this is an acceptable assumption. For instance, McMillan and Ferziger15were among
ε = (Cs∆)2 µ 2 Z ∞ 0 k2E(k) dk ¶3/2 + 2ν Z ∞ 0 k2E(k) dk, (9) = (Cs∆)2 µ 2 Z ∞ 0 k2(G(k))2E(k) dk ¶3/2 + 2ν Z ∞ 0 k2(G(k))2E(k) dk, (10)
with E(k) the three-dimensional energy spectrum.
For finite Reynolds numbers one has a finite separation between the large turbulent length scales and the Kolmogorov scale η = (ν3/ε)1/4. Hence, the inertial subrange
is limited in range, and deviations of the energy spectrum E(k) from the k−5/3 shape at
very large and very small scales cannot be neglected. By introducing a common definition for an integral length scale L = E3/2/ε, and a Reynolds number Re
L = E1/2L/ν, one can
formally express the separation between L and η as L/η = Re3/4L . Hence, a general way to express the three-dimensional energy spectrum for finite Reynolds numbers is
E(k) = αε2/3k−5/3F (x, Re
L). (11)
Here, x = kL is a dimensionless auxiliary variable. Further F (x, ReL) is a function which
equals one in the inertial subrange and adapts the spectrum for the large scales (x ∼ 1), and for the small scales (x ∼ Re3/4).
Obviously, the selection of F (x, ReL) should correspond to physical observations. For
high Reynolds numbers, it is commonly accepted, based on Kolmogorov’s theory,16 that
the high-wavenumber range of F (x, ReL) is universal. Though no ‘exact’ analytical
ex-pressions exist for this range, very good fits can be found in the literature for a wide range of experiments.11,17 For the large turbulent scales (x ∼ 1), no universal behavior
exists. For homogeneous isotropic turbulence, one can employ Von K´arm´an’s interpola-tion formula,11,18 which connects the low-wavenumber behavior of the spectrum to the k−5/3 inertial subrange, and corresponds well to experiments.
In order to further elaborate (10) into an elegant expression for Cs, we first introduce
an auxiliary function Φ¡L ∆, ReL ¢ = R∞ 0 xR1/3(G(x/L))2F (x, ReL) dx ∞ 0 x1/3(G(x/L))2 dx . (12)
One can readily verify that Φ = 1 if ∆ ¿ L and ∆ À η. In this case, the filter G cuts in the inertial subrange, and further ReL À 1. For sharp cut-off filters G, with cut-off kc = π/∆, one can verify that the normalization in the denominator of (12) corresponds
to (πL/∆)4/3. In order to come to a more useful normalization of (12) for a general class
of filters G, we will introduce a correction factor γ, such that Φ = 4
3 R∞
0 x1/3(G(x/L))2F (x, ReL) dx
Hence, the factor γ is given by γ = ¡4 3 R∞ 0 k1/3(G(k))2 dk ¢3/4 π/∆ , (14)
and γ clearly depends solely on the shape of the filter G. Obviously, for a sharp cut-off filter with cut off kc = π/∆, γ = 1. For a gauss filter, G(k) = exp(−k2∆2/24), one can
find that γ ≈ 1.02; while for a top-hat filter, G(k) = 2 sin(−k∆/2)/k/∆, γ ≈ 1.12. By combining equations (10), (11), and (13), an expression for Cs can be elaborated,
i.e. Cs = Cs,∞ γ Φ −3/4 s 1 − µ γL Cs,∞∆ ¶4/3 1 ReL Φ. (15)
Using ReL= (L/η)4/3, one further obtains Cs = Cs,∞ γ Φ −3/4 s 1 − µ γη Cs,∞∆ ¶4/3 Φ. (16)
Clearly, for ∆ ¿ L and ∆/η → ∞, one recovers Lilly’s Smagorinsky constant Cs,∞.
Equation (16) provides a formula which can be used to determine the Smagorinsky constant, and this relation will be employed to perform some a priori evaluations of Cs
in §2.4. A strong dependence of Cs on both ∆/η and L/∆ will be shown.
Evidently, for simulations, equation (16) is unpractical, since properties such as Φ and ReL, or ∆/η are not straightforwardly available during most effective large-eddy
simulations. Therefore, in the remainder of this section, we concentrate on the formulation of a modification to the Smagorinsky model (for L/∆ À 1 asymptotic simulation settings), which is easy to implement in any Navier–Stokes solver and which approximates the relation for Cs to a certain degree. We do not envisage to directly account for L/∆ effects
in a modification of the model, since these effects are flow specific, and cannot—in the authors opinion—be captured into a simple algebraic subgrid-scale model.
First of all, the required inertial-subrange behavior of Cs can be obtained, by
ap-proximating Φ by its inertial-subrange value, i.e. Φ = 1. Hence, one can express an approximation C× s to Cs as C× s = Cs,∞ γ v u u tmax ( 1 − µ γη Cs,∞∆ ¶4/3 , 0 ) , (17)
where the max-function is introduced to keep the term under the square-root strictly positive for cases where the filter width ∆ is not situated in the inertial subrange. If now the associated turbulent viscosity ν×
νt× = ¡Cs×∆¢22SijSij ®1/2 = max n (Cs,∞∆/γ)2 2SijSij ®1/2 − ν, 0 o = max {νLilly − ν, 0} . (18)
This relation is quite intriguing, since it suggests that the turbulent viscosity νt should
scale in the inertial subrange and for finite Reynolds numbers as νLilly− ν. This is clearly
in contrast to Lilly’s Smagorinsky model.
The assumption Φ = 1 to obtain (17) and (18) does not yield a good approximation to Cs when ∆ is approaching η (cf. 16), since in this range Φ 6= 1. However, inspired by
(18), we have formulated a different combination of νLilly and ν, which behaves better for
∆ → η, i.e. νt∗ = q (Cs,∞∆/γ)4 2SijSij ® + ν2− ν. (19)
In §2.4 we will illustrate the quality of this modification based on an a priori analysis, while in §3, this is further corroborated based on large-eddy simulations of homogeneous isotropic turbulence.
2.3 Analysis of the small-small model
If the turbulent dissipation associated with both formulations (4) and (5) is expressed, one obtains εt,s1 = 2Cs12 ∆2 ¿h |S0|S0ij i0 Sij À = 2Cs12 ∆2 D |S0|S0ijS0ij E , (20) εt,s2 = 2Cs22 ∆2 D |S0|S0ijSij E . (21)
The second equality in (20) is valid provided that h·i is an appropriate inner product, and that H0 is self-adjoint,14 such that for two variables f and g, hf0gi = hf g0i. For
spherical-symmetric high-pass convolution filters, this property can be trivially verified. First, we will concentrate on the original formulation of the small-small model. In order to further elaborate relations for the model coefficient Cs1, an approximation similar to
(8) has to be introduced. This allows to express the associated turbulent dissipation (20) as εt,s1 = (Cs1∆)2 D 2S0ijS0ij E3/2 = (Cs1∆)2 µ 2 Z ∞ 0 k2(H0(k))2E(k) dk ¶3/2 , (22)
with H0(k) the transfer function of the high-pass filter H0.
As for Φ, the function Ψ1 is normalized in such a way that Ψ1 = 1 if L À ∆ À η (i.e.
the LES filter G cuts in the inertial subrange). In (23), β = ∆/∆0 is the ratio of the LES
filter width ∆ to the high-pass filter width ∆0. Further, γ
1 is a correction factor, which
accounts for a deviation of the filters H0 or G from a cut-off filter. The coefficient γ 1 is given by γ1 = Ã 4 3 R∞ 0 k1/3(H0(k))2(G(k))2 dk ¡π ∆ ¢4/3 (1 − β4/3) !3/4 , (24)
and one can easily verify that γ1 = 1 when both G and H0 are spherical-symmetric sharp
cut-off filters with cut-off wavenumber kc= π/∆ and kc0 = πβ/∆.
Using (22), (11) , and (23), the expression for Cs1 now yields Cs1 = Cs,∞ γ1 Ψ−3/41 (1 − β4/3)3/4 s 1 − µ γL Cs,∞∆ ¶4/3 1 ReL Φ, = Cs,∞ γ1 Ψ−3/41 (1 − β4/3)3/4 s 1 − µ γη Cs,∞∆ ¶4/3 Φ. (25)
An a priori evaluation of this expression, using Pope’s spectra to integrate Ψ1 and Φ will
be presented in the next section.
The inertial-range behavior of Cs1can be obtained by replacing Ψ1 by its inertial-range
value. This results into
Cs1 ∼ Cs,∞/γ1 (1 − β4/3)3/4 s 1 − µ γη Cs,∞∆ ¶4/3 . (26)
If now the associated generalized viscosity is expressed, one obtains
ν0 t,s1 = (Cs1∆)2 D 2S0ijS0ij E1/2 ∼ (γ/γ1)4/3 1 − β4/3 × µ Cs,∞∆ γ ¶2 (γ/γ1) 4/3D2S0 ijS 0 ij E 1 − β4/3 1/2 − ν . (27) Here, ν0
t,s1 is the generalized viscosity, where the prime in the notation is added to
distin-guish it from a classical viscosity which is multiplied with a normal strain tensor instead of with a high-pass filtered strain tensor. In (27) various scaling factors appear, which allow to correctly balance the dissipation effects of parts in the expression which have a different dimensionality. First of all, one can see that ∆2hS0
ijS 0
iji1/2 is normalized with a
factor such that it can correctly be combined with the viscosity ν. Further, the full term in square brackets is multiplied with a second factor, such that the total result effectively has the dimensionality of ν0
Based on the form of (27), we now express a modification to the original small-small model, and we introduce an expression similar to (19), i.e.
νt,s10∗ = (γ/γ1) 4/3 1 − β4/3 v u u tµCs,∞∆ γ ¶4 (γ/γ 1)4/3 D 2S0ijS0ij E 1 − β4/3 + ν2− ν . (28)
The quality of this approximation will be established in the next sections.
The derivation of the second small-small formulation (5) can be elaborated in a similar way. A new auxiliary function has to be introduced, i.e.
Ψ2 ¡L ∆, ReL ¢ = 4 3 R∞ 0 x1/3H0(x/L)(G(x/L))2F (x, ReL) dx (γ2πL/∆)4/3(1 − β4/3) , (29) where γ2 = Ã 4 3 R∞ 0 k1/3H0(k)(G(k))2 dk ¡π ∆ ¢4/3 (1 − β4/3) !3/4 . (30)
Combination with equations (21) and (11) yields
Cs2 = Cs,∞ Ψ −1/4 1 Ψ −1/2 2 γ11/3γ22/3(1 − β4/3)3/4 s 1 − µ γη Cs,∞∆ ¶4/3 Φ. (31)
Moreover, in the next sections, we will further investigate a modification to the second
small-small formulation, which corresponds to ν0∗ t,s2 = (γ/γ2)4/3 1 − β4/3 v u u tµCs,∞∆ γ ¶4 (γ/γ 1)4/3 D 2S0ijS0ij E 1 − β4/3 + ν2− ν . (32) 2.4 A priori results
In this section, we will make some a priori evaluations of the different model coeffi-cients as function of L/∆ and ∆/η, and further investigate the quality of the proposed modifications to the models. To this end, we use Pope’s11 analytical expression for the
energy spectrum, which corresponds to
F (x, ReL) = µ x [x2+ c L]1/2 ¶11/3 exp à −cβ ÷ x4 Re3 L + c4 η ¸1/4 − cη !! . (33)
Here, cβ = 5.2, and cL, cη are positive parameters which depend on the Reynolds number.
In fact cL, cη are determined by ensuring that
R∞
0 E(k) dk = E and
R∞
100 101 102 103 10−4 10−3 10−2 10−1 100 L/∆ Cs L,∆ ∞ − Cs , ∞ Cs , ∞
Figure 1: Deviation of Cs, Cs1(×γ1(1 − β4/3)3/4) and Cs2(×γ1/31 γ 2/3
2 (1 − β4/3)3/4) from the asymptotic
value Cs,∞ as function of L/∆ for ReL= ∞. The deviations are normalized using Cs,∞. (—): Cs; (−·):
small-small models. Symbols further correspond to (/): Cs1; (×): Cs2.
with E the total enstrophy. More details on the determination of cL, cη and cβ can be
found in Pope11 and in Meyers and Baelmans.19
We now first turn to the evaluation of Cs, Cs1, and Cs2 as function of L/∆. To this
end, ReL = ∞ is selected, and for both VMS models, a gaussian high-pass filter with β = 1/2 is used. In figure 1 the deviation of the respective coefficients from Cs,∞ is
displayed. Further, Cs1 and Cs2 are respectively normalized with γ1(1 − β4/3)3/4 and γ11/3γ22/3(1 − β4/3)3/4, such that their level can be compared with each other and C
s.
Differences between the Smagorinsky model and the VMS models are clear. One might directly compare the level of the deviation at a selected level of L/∆. For instance, for
L/∆ = 10, one finds deviations which are 10% for the Smagorinsky model and about 1%
for the VMS models. Inversely, by looking at the required ratio for a deviation below an arbitrary selected level of 1%, one obtains L/∆ ≈ 55 for the standard Smagorinksy model, and L/∆ ≈ 10 for the two VMS models. Obviously, these evaluations are only valid for homogeneous isotropic turbulence. Nevertheless, we think differences between the models clearly illustrates the higher potential of the VMS Smagorinsky models, when compared to the Smagorinsky model. Figure 1 further shows that the VMS models converge faster to an asymptotically independent level of the coefficient than the standard Smagorinsky model. In fact, the slopes correspond to (L/∆)−2 for the VMS models and (L/∆)−4/3 for
the Smagorinsky model. This provides some interesting insight in the improved quality of VMS models when compared to a standard Smagorinsky model.
In figure 2, the Smagorinsky coefficient Cs (cf. equation 16) is displayed as function of
∆/η (taking L/∆ À 1, i.e. L/∆ = 200 is used). Further, Cs,∞ (cf. equation 6), Cs× (cf.
equation 17) and C∗
s (= νt∗/(∆2|S|)1/2, cf. equation 19) are also shown. In this figure,
first of all, the strong dependence of Cs as function of ∆/η can be seen, and the deviation
100 101 102 0 0.05 0.1 0.15 0.2 ∆/η Cs
Figure 2: Evolution of Cs (16), Cs× (17) and Cs∗
(induced from 19) as function of the ratio ∆/η (L/∆ À 1 is assumed). (—): Cs; (−−): Cs×;
(−·): C∗
s; and (· · · ): Lilly’s Smagorinsky constant.
100 101 102 103 −0.1 0 0.1 0.2 0.3 0.4 0.5 ∆/η δε [%]
Figure 3: A priori error on the total dissipation as function of ∆/η (L/∆ À 1 is assumed). (—): orig-inal formulations; (−−): modified formulations. Symbols correspond to (×): standard Smagorin-sky model; and (4): VMS model (respectively 5 and 32) employing a gaussian high-pass filter with
β = 1/2.
Cs can be appreciated. One observes that Cs∗ provides a very good approximation to Cs
for ∆/η ≥ 10. For 4 ≤ ∆/η ≤ 10 the fit deteriorates but seems still to provide acceptable results. For ∆/η < 4, C∗
s is clearly evolving slower to 0 than Cs. Obviously, this deviation
of C∗
s from Cs in this last range is occurring at low values of the model coefficient, such
that its importance in the total dissipation balance will be negligible. We will further establish this based on an analysis of the induced errors on the turbulent dissipation.
Apart from Csas function of ∆/η, one can also evaluate Cs1 and Cs2. However, similar
results as those in figure 2 are obtained, and we will turn to a direct a priori evaluation of dissipation errors as function of ∆/η for the different models. To this end, define
δε= εmodel
t − εt
ε , (34)
with εmodel
t the dissipation of the subgrid-scale model, εt the exact turbulent dissipation
and ε the (exact) total dissipation. Using Φ, Ψ1, Ψ2, and the ratios ∆/η, these terms can
be all expressed straightforwardly. For instance,
εLillyt ε = (Cs,∞∆/γ)2 2SijSij ®3/2 ε = Φ ¡L ∆, ReL ¢3/2 . (35)
In figure 3, δε is presented for the standard Smagorinsky model and for the second small-small model (i.e. equation 5). For both models, the (constant-coefficient)
Cs,∞ as coefficient) yields dissipation errors up to 50%. The standard VMS model
(us-ing Cs,∞/[γ11/3γ 2/3
2 (1 − β4/3)3/4] as coefficient) provides errors which are lower, but can
still amount up to 30%. The improvement of the proposed modifications is clear, with maximum a priori errors around 5%.
3 LES OF DECAYING HOMOGENEOUS ISOTROPIC TURBULENCE
Large-eddy simulations of decaying homogeneous isotropic turbulence are carried out at a number of resolutions. The initial Reynolds number in the current study corresponds to Reλ = 100 in terms of the Taylor-Reynolds number Reλ and DNS reference results are
used for comparison.12 The initial fields for the LES are generated by filtering the initial
DNS fields12 with a spherical sharp cut-off filter, with cut-off wavenumber k
c = π/∆.
Subsequently energy decays for about two eddy turn-over times, and approximatelly 70% of the energy decays in this period of time.
The LES equations are discretized based on a Fourier pseudo-spectral method. The modes in the solution are restricted to a sphere in Fourier space with radius kc= π/∆ = πN/L , with L ≈ 2L the size of the computational box in physcal space. Dealiasing is
performed with the 2/3rd dealiasing method, employing 3N/2 cube grid points for the
calculation of non-linear terms in physical space. Fourier transforms are implemented using the FFTW library.20,21 For integration in time, a classical four-stage fourth-order
accurate Runge–Kutta scheme is used.
In order to present surveyable comparisons between different models, we introduce a class of relative errors
δp(N) = "RT 0 Rkc 0 k2p{ELES(k, t) − EDN S(k, t)} 2 dk dt RT 0 Rkc 0 k2pEDN S2 (k, t) dk dt #1/2 . (36)
Here, E(k, t) is the resolved three-dimensional energy spectrum, which is a function of the wavenumber k and time t. Further, p is a parameter, which allows to emphasize different scales in the solution. Low values of p allow to identify the errors in the low-wavenumber range of the spectrum; high values of p are related to errors in the high-wavenumber range. In the current paper, errors δ−1, δ0, δ1, and δ2 will be presented.
Simulations are performed using six different models, and resolutions ranging from 243
to 963. First of all the standard Smagorinsky model and the two VMS models are used,
all operated with infinite-Reynolds-number constant coefficients. Hence, these coefficients correspond respectively to Cs,∞, Cs,∞/[γ1(1 − β4/3)3/4], and Cs,∞/[γ11/3γ22/3(1 − β4/3)3/4].
Next to the standard formulations, the modifications (19), (28), and (32) are also em-ployed. For the VMS models, a gaussian high-pass filter is used with β = 1/2. For this type of filter and value of β, one can find by numerically evaluating (24) and (30) that
γ1 ≈ 0.505 and γ2 ≈ 0.763.
In figure 4(a–d) the errors δ−1, δ0, δ1, and δ2 are presented for the six different models.
As is expected, one consistently finds that δ−1 < δ0 < δ1 < δ2, showing that large-scale
20 40 60 80 100 0 1 2 3 4 5 6 7 8 N δ −1 [%] (a) 20 40 60 80 100 0 2 4 6 8 10 12 14 16 N δ0 [%] (b) 20 40 60 80 100 0 5 10 15 20 25 30 35 N δ 1 [%] (c) 20 40 60 80 100 0 10 20 30 40 50 N δ 2 [%] (d)
Figure 4: Errors δ−1, δ0, δ1, and δ2as function of the simulation resolution N for different subgrid-scale
models. (—): Smagorinsky model (3); (−−): small-small model, equation (4); (· · · ): small-small model, equation (5). Symbols further correspond to (2): standard formulation (cf. equations 3, 4 and 5); (◦): proposed modifications (cf. equations 19, 28 and 32.)
We now first focuss on the standard models. From the figures, one can appreciate that both small-small formulations perform better than the standard Smagorinsky model. The only exception is at very low resolutions, i.e. 243, where δ
−1 and δ0 are somewhat lower for
the Smagorinsky model. However, for the 243 resolution, L/∆ ≈ 12, such that
difficult-to-predict large-scale effects affect the performance of the models. For N > 24, we clearly see that the errors of the VMS models decrease faster than those of the Smagorinsky model, and this could be related to the observed differences in asymptotic L/∆ convergence in figure 1. One can further see in figure 4 that the original small-small formulation (4) displays somewhat lower errors than the second small-small formulation (5).
Improvements can be seen for all errors, but certainly the gains in accuracy are highest for the errors related to the resolved small-scale part of the solution (i.e. δ1 and δ2).
4 CONCLUSIONS
By extending Lilly’s classical method8 for the determination of the Smagorinsky
co-efficient, we presented an analysis on the behavior of the Smagorinsky model and two
small-small VMS models as function of the ratios L/∆ and ∆/η.
Firstly, using an a priori analysis, it was demonstrated that the small-small VMS mod-els converge faster to a L/∆ independent regime than the Smagorinsky model, requiring overall lower ratios of L/∆. These trends further elucidate the quality of VMS models compared to the standard Smagorinsky model.
Secondly, we demonstrated that ∆/η changes can have a large effect on the required model coefficients. A priori errors on the turbulent dissipation of constant-coefficient Smagorinsky and VMS models amount respectively up to 50% and 35% of the total dissipation. Therefore, we proposed modifications to the models, which respond better to changes in ∆/η, and a first evaluation of the improvements was performed within the framework of the theoretical analysis.
In order to further establish the trends which were observed in the theoretical analysis, a set of large-eddy simulations were performed of homogeneous isotropic turbulence, and we assessed the quality of simulation results both based on large-scale flow predictions and on (resolved) small-scale predictions. The results confirmed the well known improved behavior of the small-small models over the Smagorinsky model. Furthermore, during grid refinement, errors related to the VMS models were seen to drop faster than those of the Smagorinsky models. Moreover, the testing of the proposed modifications to the standard and VMS Smagorinsky models revealed that these modified models yield consistently lower errors than their original counterparts.
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