Journal of Turbulence Volume 7, No. 28, 2006
Subgrid models for large-eddy simulation using unstructured grids in a stabilized finite element framework V. LEVASSEUR†∗ , P. SAGAUT† and M. MALLET‡ †Laboratoire de Mod´elisation en M´ecanique, Universit´e Pierre et Marie Curie, 4 place Jussieu, Boite 162, 75252 Paris cedex 05, France ‡Dassault Aviation—DGT/DTIAE/AERAV, 78, quai Marcel Dassault 92552 Saint-Cloud, France We present subgrid closures for large-eddy Simulation (LES) likely to be implemented in stabilized finite element methods. Selection criterion, dynamic procedure and multiscale approach are compared within simulations of freely decaying isotropic turbulence. In all cases, the numerical dissipation coming from the least-squares stabilization dominates the subgrid model. Despite this large numerical dissipation, the LES model, whichever it is, provides a sufficient physical dissipation to have a clear and major effect on the results. In particular the dynamic procedures and the multiscale models turn out to be very efficient, high-lighting a self-adaptive behaviour of the turbulent viscosity and consequently predict the correct energy transfer mechanisms, by accounting for the numerical part of the total dissipation.
1. Introduction Large-eddy simulation has proven to be a valuable technique for simulating turbulent flows, which cannot be handled through direct numerical simulation. The key idea is to truncate the solution of the Navier–Stokes equations and compute a filtered velocity field. This leads to a scale separation between the energy containing ones, i.e. the largest eddies, and the mainly dissipative small scales, which are, in practice, the scales that are not represented by the mesh. Most LES researches are concentrated on incompressible or subsonic flows computed on structured grids. First steps using unstructured codes have been made by Jansen [1], while simulating the flow over a NACA 4412 airfoil. Chalot et al. [2] proposed a consistent finite element approach to LES, applied to the decay of isotropic turbulence and a mixing layer. Two main strategies of modelling exist: whether to explicitly add a source term to account for the effects of subgrid-scales, or to use numerical schemes exhibiting errors that play the role of the subgrid model (implicit large-eddy Simulation). The present paper shows comparisons between several LES closures (among them ILES) implemented in an unstructured compressible code, using a finite element method, through the simulation of a decaying homogeneous isotropic turbulence in the limit of an infinite Reynolds number. The Galerkin method, tending to be underdissipative, requires stabilization procedures. To handle this, we use the Galerkin/least-squares formulation introduced by Hughes and Johnson (see [3] for details on the method). The least-squares operator ensures good stability characteristics, while maintaining a high level of accuracy. A major point is then to differentiate the dissipation due to the stabilization procedure from the turbulent dissipation to account for the loss of the ∗ Corresponding
author. E-mail:
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Journal of Turbulence c 2006 Taylor & Francis ISSN: 1468-5248 (online only) http://www.tandf.co.uk/journals DOI: 10.1080/14685240600600352
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small scales in LES. This work assesses the effect of the numerical stabilization over the total dissipation, in order to extract some adaptivity property of the LES models. First the numerical method used here is briefly recalled in section 2. Then the filtered equations for LES and the different approaches investigated to close the system are reviewed in section 3. The results of simulations of freely evolving isotropic turbulence are shown in section 4 and conclusions are drawn in section 5.
2. Numerics The code used here is AETHER, developed by F. Chalot from Dassault-Aviation (St Cloud, France). It relies on a semi-discrete Galerkin/least-squares formulation of the symmetrized compressible Navier–Stokes equations (see [4] for a description of entropy variables used to symmetrize the equations). The time integration is carried out using an implicit second-order backward difference scheme and piecewise linear elements are used for space discretization. A detail of this method applied to LES is found in Chalot et al. [2], and in Levasseur et al. [5] with emphasis on the variational multiscale closures. In this section, we merely recall the Galerkin/least-squares formulation applied to the Navier–Stokes equations in conservation form, in order to point out the stabilization term we will focus on in section 4. In conservation form, the Navier–Stokes equation writes formally d U ,t + F i,i = F i,i
(1)
U ,t + Ai U ,i = (K i j U , j ),i .
(2)
or equivalently, in quasi-linear form
Let us consider the time interval I = [0, T ] divided into N intervals In = [tn , tn+1 ], and a domain with boundary . Denote Q n = × In
(3)
Pn = × In .
(4)
The nth space-time ‘slab’ Q n is then tiled into (n el )n elements Q en . Within each Q n , n = 0, . . . , N − 1, the Galerkin/least-squares statement of the Navier– Stokes equation reads: find U h ∈ Snh (the trial function space) such that for all W h ∈ Vnh (the weighting function space) the following variational equation is satisfied: − W ,th · U h − W ,ih · F i (U h ) + W ,ih · K i j U ,hj dQ + +
Qn
− − (W h (tn+1 ) · U h (tn+1 ) − W h (tn+ ) · U h (tn− )) d
n el Q en
e=1
+ Pn
L (W h ) τ (U ,t + L(U h )) dQ
(5)
W h · (F i (V h ) − K i j U ,hj ) n i dP = 0.
The first, second, and last integrals in equation (6) constitute the full space-time Galerkin formulation. In the second integral, the jump condition is added to the time boundary integral resulting from the integration-by-part of the time flux term, so that the time continuity is
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weakly enforced. This integrated-by-part form of the Galerkin methods results in conservation of fluxes under inexact quadrature rules. The third integral is the least-squares stabilization operator. L is defined as the steady compressible Navier–Stokes operator: L = Ai
∂ ∂ − ∂ xi ∂ xi
Kij
∂ ∂x j
.
(6)
The symmetric τ matrix is the intrinsic time scales whose definition, of great influence on the solution, can be found in Shakib et al. [3]. When applied to a monodimensional scalar advection–diffusion equation, this stabilization consists in adding an artificial viscosity κnum = τ |u|2 = f (Pe)|u|h/2, where f (Pe) depends on the Peclet number |u|h/2κ and controls the stability of the solution in advection-dominated regions of the flow. This has been extended to the Navier–Stokes equations so that the numerical viscosity vanishes when the Peclet number decreases.
3. The large-eddy simulation approach 3.1 Filtered equations The case of compressible flows is considered here. For nonhypersonic flows, and considering the local thermodynamic equilibrium hypothesis, the perfect gas relation is supposed to be a proper approximation. The equations are the Navier–Stokes equations for a Newtonian fluid. The filtered equations resolved in this paper for LES, using the mass-weighted change of variables defined as φ˜ = ρφ/ρ, ¯ can be written as ∂t ρ¯ + ∂ j (ρ¯ u˜ j ) = 0, ˜ − ∂ j τi j , ∂t (ρ¯ u˜ i ) + ∂ j (ρ¯ u˜ i u˜ j ) + ∂i p¯ = ∂ j [2µS ˜ i j (u)] ˇ + ∂ j (ρ¯ Eˇ u˜ j ) + ∂ j ( p¯ u˜ j ) = ∂ j [2µS ˜ u˜ i ] + ∂i [κ∂ ∂t (ρ¯ E) ˜ i j (u) ˜ i T˜ ] + ∂i Q i + ∂ j (τi j u˜ i ),
(7) (8) (9)
with Q i = (ρ Eu i − ρ¯ E˜ u˜ i ) + ( pu i − p¯ u˜ i ) − τi j u˜ j and τi j = ρ¯ u ¯ u˜ i u˜ j that both need to iu j − ρ be modelled, and Si j (u) = 1/2(∂ j u i + ∂i u j ) − 1/3τkk δi j . Note that this above system of equations (7)–(9) is not exact. If the filtering of the momentum equations appears quite classic, by the introduction of a subgrid tensor similar to the one introduced for incompressible flows, there remain several approaches for the filtered energy equation. One can cite for instance the works of Vreman [6] or Sreedhar and Raghab [7]. Following the results of Vreman et al. [8] within the context of a priori tests on direct numerical simulation of a temporal mixing layer, the nonlinearity associated with viscous stress and heat flux is assumed negligible. Furthermore, the trace of the subgrid tensor is also neglected, highlighting as proposed by Erlebacher et al. [9, 10] that it is related to the subgrid Mach 2 number by τkk = γ Msgs p¯ and Msgs is small when the considered infinite Mach number is not too large. More precisely the authors emphasized that the thermodynamic pressure will be much more important when the subgrid Mach number is lower than 0.4, or equivalently for turbulent Mach number up to 0.6. This latter condition incorporates a major part of supersonic flows. This was quite well verified by Ducros et al. [11]. Eventually, the filtered total energy equation is replaced by the transport equation of computable energy: ρ¯ Eˇ = p¯ /(γ − 1) + 1/2ρ¯ u˜ i u˜ i = ρ¯ E˜ − 1/2τii , as proposed by Lee [12].
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3.2 LES closures ˜ and The subgrid stress is modelled within a turbulent viscosity hypothesis: τi j = −2µt Si j (u) ˜ t /Prt the vector Q j is modelled with reference to the heat flux: Q i = −κt ∂i T˜ , κt = γ cv µ where Prt is the turbulent Prandtl number, set equal to 0.9. Improvements on the basis of the LES model proposed by Smagorinsky [13] are now assessed for the simulation of compressible flows on unstructured meshes. A selective model, dynamic procedures and a multiscale framework are presented hereafter. If some of these closures appear quite common, they have hardly been investigated on unstructured grids or in the case of compressible fluids. Smagorinsky model: Smagorinsky [13], through a local equilibrium hypothesis, which im˜ plies that the energy transfer rate is equal to the subgrid dissipation, ε = −τi j Si j (u), proposed the following eddy viscosity, as a function of the resolved velocity field: ˜ i j (u) ˜ νt = (CS )2 |S( u )| = (CS )2 2Si j (u)S (10) where CS is the Smagorinsky constant. Selective model: David [14] introduces a structural sensor in order to improve the prediction of intermittent phenomena by the use of a selective function. The criterion he uses consists in comparing the instantaneous vorticity ω with the local averaged vorticity ω. ¯ The selective function incorporated in the subgrid model is thus
ω(x, t) ∧ ω(x, t) π 1 if α ≥ α0 H (x, t) = with α(x, t) = arcsin , α0 = 0 ω(x, t) · ω(x, t) 9 (11) and the subgrid viscosity becomes νt (x, t) = H (x, t) (CS )2 | S| Dynamic: Germano’s procedure enables the adaptation of the turbulent viscosity to the local property of the flow by computing a time- and space-dependent Smagorinsky constant, [15]. This new computed pseudo-constant is not bounded because it appears in the form of a fraction, and can take negative values. To enforce the stability of the calculation, the constant can be statistically averaged in the directions of statistical homogeneity, in time or local in space, or else it can be arbitrarily bounded. Moreover, the averaging procedures can be performed in two non-equivalent ways, either by averaging the denominator and numerator separately, or by averaging the quotient [16]. These different ways of achieving numerical stability have been tested, and the retained process, here, is to average the constant on the whole domain (see some further details in section 4). Besides, the test filter used in the dynamic algorithm is a derivative-based filter. Note that the turbulent Prandtl number remains constant, equal to 0.9. When applied to the Smagorinsky model, this algorithm is historically referred to as the dynamic model. Kolmogorov scaling: Following Kolmogorov’s dimensional analysis a characteristic time for turbulence can be written as T = ( 2 /ε)1/3 , where ε is the energy transfer rate within the inertial range, and is the size of the smallest resolved scale. Thus, the turbulent viscosity is expressed as νt ∝ ε 1/3 4/3 .
(12)
Wong and Lilly [17] proposed to apply Germano’s algorithm to this scaling formulation of the turbulent viscosity, equation (12), assuming that the two levels of filtering are located in the inertial range. Thus, the dissipation ε remains the same for these two levels, and eliminates from the expression of the eddy viscosity. See Carati et al. [18] for a generalization of dynamic eddy viscosity models in physical space.
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Denoting by ‘ ˆ ’ the test filter, the eddy viscosity obtained from equation (12), which will be referred to as the Kolmogorov model, reads
u) L i j Si j ( 1 νe = − where α = / (13) 2(α 4/3 − 1) Si j ( u )Si j ( u) is a characteristic length scale of the test filter. The dynamic Kolmogorov where model no longer requires the local equilibrium hypothesis, because it is independent of the time scale 1/|S( u )|, and involves less filtering operations compared with the classic dynamic procedure. Note that, as the dynamic model seen previously, the quotient in equation (13) is averaged on the whole domain of computation. Variational multiscale: This multiscale model, proposed by Hughes [19] relies on the change of the dependence of the closure on the resolved scales. The distant triadic interactions are neglected. Instead of an orthogonal projection, a Gaussian filter is used to separate the large scales from the resolved small ones as already proposed by Vreman [20]. In this study, VMS methods are represented by the so-called small-small closure: τ = −2ρC ¯ 1 2 |S(u )|S(u ),
2 /24 ∇ 2 u, u = −
(14)
where C1 is a constant based on the Smagorinsky constant within an analogy with Lilly’s approach, and set equal to 0.22. For this study, the scale partitioning is chosen such that kc /kc = 0.35, as in [5], kc and kc being the cut-off wavenumbers corresponding respectively to the large scales and the small scales. 4. Numerical results The retained test case is the freely evolving isotropic turbulence. Computations are carried out on 513 and 813 grids, at zero molecular viscosity, in a 2π -length computational domain. The simulations are initialized with a random, divergence-free velocity field, constrained with a specified energy spectrum: E(k) = k 4 exp(−2k 2 /k02 ), where k0 is the initial peak of kinetic energy. Bruyn and Riley [21] highlighted numerical instabilities while low-wavenumber confinement occurs, and proposed the following condition to alleviate this problem: Lkmin ≤ 0.3, L being the integral scale, and kmin = 2π/L box = 1, the lowest wavenumber for the simulation. kmin being given by the computational domain size, the only free parameter to verify the latter condition is the integral scale, or equivalently the initial peak wavenumber k0 . Therefore, for both grids, the simulations have been initialized with k0 = 3. The parameters of the simulations are detailed in table 1. 4.1 Energy transfers Preliminary studies [5] show that our numerical formulation is unable to perform implicit largeeddy simulation. On a very coarse grid (with ten wavenumbers computed), the stabilization is almost irrelevant, and the simulations provide a k 2 law for the forward energy cascade as expected from the truncated Euler equations. However, on finer grids, which we are interested Table 1. Computation parameters. General parameters Prandtl number 0.9
Smagorinsky constant 0.18
Selective model α0 20◦
Small-Small VMS constant 0.22
Scale partition 0.35
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Figure 1. Turbulent kinetic-energy spectra. The models are Smagorinsky, dynamic, Kolmogorov, small-small and ILES (no model)—513 .
in for the study, the stabilization term becomes larger because of the higher gradients likely to be captured. The kinetic-energy spectrum then stabilizes around a k −1 law. Therefore our formulation must incorporate a subgrid model in addition to the numerical dissipation. The energy spectra obtained for the different closures, including ILES, are first presented in figure 1, for the 513 mesh. In this particular numerical framework, both Smagorinsky and selective models are overdissipative in large wavenumbers. In contrast, Germano’s algorithm (dynamic and Kolmogorov models) by adapting the constant, and the VMS approach because it reduces the support of the turbulent viscosity, correctly represent the kinetic-energy transfers in the whole inertial subrange. The Kolmogorov law E(k) = C K ε 2/3 k −5/3 is then well reproduced, because these models account for the numerical dissipation. The constant Cd computed with the dynamic model is drawn in figure 2. It tends to stabilize around 0.1 which must be compared with the 0.18 value used for the Smagorinsky calculation. The difference represents the ability of the dynamic procedure to deduce part of the numerical dissipation from the
Figure 2. Time history of the dynamic constant Cd .
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Figure 3. Time history of the kinetic energy and enstrophy for the dynamic closures. Dynamic: the whole constant is averaged. Dynamic 2: the denominator and numerator are averaged separately.
subgrid model. In order to emphasize the behaviour of Germano’s procedure, the time evolution of the dynamically computed constant Cd has been plotted for different numerical formulations. On the one hand, artificial numerical dissipation has been added by introducing a discontinuity capturing operator, and on the other hand, the stabilization term influence has been lowered by dividing the characteristic time-scale matrix by 4. It appears that the asymptotic value of the dynamic constant diminishes when the numerical dissipation increases, as expected. Different ways of stabilizing the dynamic model have also been tested. Figure 3 shows the time evolution of the kinetic energy and the enstrophy by averaging the whole constant, denoted ‘dynamic’, or by averaging the denominator and numerator separately, denoted ‘dynamic 2’. The results here are very close to each other. This is confirmed by figure 4, which displays the kinetic-energy spectra for these two different implementations, and presents only very
Figure 4. Turbulent kinetic-energy spectra for the dynamic closures, 513 . Dynamic: the whole constant is averaged. Dynamic 2: the denominator and numerator are averaged separately.
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Figure 5. Turbulent kinetic-energy spectra—813 .
slight differences. Eventually, these two processes aiming at stabilization lead to a difference of the asymptotic values of the computed pseudo-constant of about 1.5%. Therefore, in the following, only one dynamic model is investigated. Figure 5 shows the kinetic-energy spectra for the 813 grid. For the sake of clarity, figure 5 displays the results for the Smagorinsky, selective, dynamic and small-small models. This confirms our previous statement as for the good predictions of the forward energy cascade when simulating with the dynamic or multiscale techniques. Figure 6 represents the time evolution of the enstrophy, confirming a greater presence of small structures for the dynamic, Kolmogorov and small-small models. The dynamic model exhibits a peak of enstrophy of amplitude: Dmax /D(t = 0) = 6.3. The maximum enstrophy is obtained at tc = 13.56 while, in the inviscid case, the critical time when enstrophy blows up is equal to 5.9D(t = 0)−1/2 ≈ 13.83 for a constant skewness factor equal to 0.4. (see Lesieur [22] for details).
Figure 6. Time history of resolved enstrophy—513 .
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Figure 7. Comparison of GLS and SGS dissipation spectra for the different closures—513 .
4.2 Subgrid and least-squares dissipation The two sources of dissipation, namely the stabilization technique and the subgrid model, are compared and analysed in figures 7 and 8, respectively for the 513 and 813 computations. The figures represent the probability spectral density of both the subgrid dissipation and the dissipation due to the least-squares stabilization. The subgrid energy dissipation rate is calculated as εsgs = ρu i τi j, j , and the least-squares dissipation is εgls = ρu i LS(u i ) where LS(u i ) is the stabilization term added to the Galerkin formulation of the momentum equations (see [5] for details). The spectra are normalized by the mean over all the simulations of the root mean square of the total dissipation, so that comparing the shape of the spectra as well as the relative level of dissipation is meaningful. For all closures, the dissipation due to the
Figure 8. Comparison of GLS and SGS dissipation spectra—813 .
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Figure 9. Effective viscosity corresponding to the subgrid kinetic-energy dissipation rate—513 .
numerical formulation dominates the subgrid effects. Garnier et al. already underlined it with the generalized Smagorinsky constant concept [23]. However the subgrid dissipation cannot be neglected since it has been seen that the different closures yield different results in terms of kinetic-energy transfers. This is also coherent with the ILES computations, which were not dissipative enough, with a resolved kinetic-energy decay higher than −5/3. Besides the different levels of the dissipation rate, between the subgrid and least-squares dissipation, the shape of the spectra appears also rather different. The subgrid models give rise to a dissipation spectrum of the form k 2 E(k) ∝ k 1/3 for the 513 computation and k 4/3 for the 813 one, while the least-squares dissipation spectrum is of the form k 4/3 . This shows that the GLS term seems less mesh dependent than the subgrid model as far as it concerns its spectral behaviour (but not the global rms value of the corresponding dissipation . . .), and tends to overdissipate the smallest scales. Eventually, figure 7 particularly highlights the self-adaptivity property of the dynamic procedures and the VMS method. The corresponding subgrid dissipation is lower compared
Figure 10. Effective viscosity corresponding to the least squares stabilization—513 .
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Figure 11. Effective viscosity corresponding to the subgrid kinetic-energy dissipation rate—813 .
with the Smagorinsky and selective models because they incorporate some information about the numerical dissipation. Figures 9 and 10 represent the equivalent effective viscosity of both dissipations, respectively: νesgs = εsgs /2k 2 E(k) and νegls = εgls /2k 2 E(k), on 513 grid. The same results are displayed in figures 11 and 12 for the 813 simulations. This can be compared with the work of Domaradzki et al. [24] who proposed a method for computing effective numerical eddy viscosity, evaluated on the nonoscillatory finite volume scheme MPDATA. Although not fully obvious, the expected ‘plateau-peak’ behaviour of the subgrid effective viscosity can be seen in figures 9 and 11. More interesting is the spectrum of the effective viscosity corresponding to the stabilization term. On both grid resolutions, it seems to be rather monotonic, except for the very low wavenumbers, and displays a k law, and even a k 2 law with the dynamic and smallsmall closures on 813 . Moreover, the numerical viscosity seems to vanish slowly when the
Figure 12. Effective viscosity corresponding to the least squares stabilization—813 .
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Figure 13. Coherence of least-squares and subgrid dissipation—513 .
resolution increases compared with the subgrid viscosity, indicating that the predominance of the GLS term over the subgrid model becomes higher. Nevertheless, as previously mentioned, it remains unable to perform ILES, and the subgrid model still has a significant effect. The coherence between least-squares and subgrid dissipation is drawn in figures 13 and 14, showing the different spectral behaviours of each model especially in low wavenumbers. It is defined as ∗ k gls sgs dk C(gls , sgs) = (15) ∗ ∗ k gls gls dk k sgs sgs dk where gls and sgs are respectively the least-squares and subgrid dissipation in the Fourier space, and C(gls , sgs) is the coherence coefficient.
Figure 14. Coherence of least-squares and subgrid dissipation—813 .
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Figure 15. Probability density function of one velocity component normalized by the rms velocity—813 .
4.3 Statistics After the spectral point of view, it is interesting to observe the statistics of the flowfield. Figures 15 and 16 display, for each closure, the probability density functions (PDF) of velocity and vorticity components. Figure 15 shows that the Gaussian distribution is very well recovered for the velocity components. What is more, the four LES closures predict essentially the same PDF behaviour. In contrast, the vorticity presents a distribution proportional to exp −|x|, figure 16, which has been quite well known experimentally and numerically for a long time, and explained by a small-scale intermittency. We now examine the PDFs of the subgrid dissipation, figure 17, compared with the PDFs of the least-squares dissipation, figure 18. If the different closures raise very similar results as far as the velocity or vorticity PDFs is concerned, the subgrid dissipation behaves really
Figure 16. Probability density function of one vorticity component normalized by the rms vorticity—813 .
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Figure 17. Probability density function of the subgrid dissipation normalized by u 3rms /L box —813 .
differently. First of all, the subgrid dissipation is seldom negative, while Kang et al. [25] exhibit experimentally a significant fraction of the subgrid dissipation −τi j Si j which is negative, indicating the backscatter of energy. Nevertheless, whereas both dynamic closure and small-small models provide very similar kinetic-energy spectra, figures 1 and 5, the dissipation distributions are rather different. The mean dissipation is almost the same, leading, as previously seen, to a correct kinetic-energy budget prediction, but the root mean square of the dissipation is higher with the small-small model than with the dynamic one, with a more important fraction of backscatter. On the other hand, the distribution of the least-squares dissipation is noticeably less skewed. Moreover, it confirms that the stabilization effect tends to increase when the subgrid dissipation is lower. This was also clear looking at the dissipation spectra, figures 7 and 8. Indeed, if the subgrid model overdissipates the small scales, the computed gradients involved in the stabilization term are weaker. At last, figures 17 and 18 confirm the
Figure 18. Probability density function of the least-squares dissipation normalized by u 3rms /L box —813 .
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dominance of the numerical source of dissipation, a.k.a the stabilization effect, over the added source term corresponding to the LES model.
5. Conclusions As a conclusion, several LES closures have been investigated through one particular numerical framework: the Galerkin/least-squares formulation. This method incorporates a stabilization term, which is roughly a dissipative effect. In order to perform accurate large-eddy simulations, it is then necessary to account for this first and unavoidable source of dissipation and adapt the subgrid model, which must account for the loss of the smallest scales in the simulation. The comparison between both sources of dissipation shows that the numerical one, namely the stabilization effect, dominates the subgrid model. However, this latter term is not at all negligible and the different closures raise very different results! It turns out that the dynamic procedure, by locally adapting to the property of the flow, and the variational multiscale method, providing a reduced-support subgrid viscosity based upon the smallest scales, incorporate information on the numerical errors and then manage to recover the right energy transfers.
Acknowledgments This work was supported by the French Ministry of Defence through a D.G.A. fellowship (D´el´egation G´en´erale pour l’Armement) and the aircraft maker Dassault-Aviation (SaintCloud, France). Most of the computational resources were allocated by IDRIS, the French CNRS supercomputing center, under project 051773. References [1] Jansen, K., 1995, Preliminary large-eddy simulations of flow over a NACA 4412 airfoil using unstructured grids. In Annual Research Briefs, pp. 61–72 (Stanford, CA: Centre for Turbulence Research, NASA Ames/Stanford University). [2] Chalot, F., Marquez, B., Ravachol, M., Ducros F., Nicoud F. and Poinsot, Th., 1998, A Consistent Finite Element Approach to Large Eddy Simulation, AIAA paper 98-2652. [3] Shakib, F., Hughes, T. J. R. and Johan, Z., 1991, A new finite element formulation for computational fluid dynamics: X. Computational Methods in Applied Mechanics and Engineering, 89, 141–219. [4] Hughes, T.J.R., Franca, L.P. and Mallet, M., 1986, A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier–Stokes equations and the second law of thermodynamics. Computational Methods in Applied Mechanics and Engineering, 54, 223–234. [5] Levasseur, V., Sagaut, P., Chalot, F. and Davroux, A., 2006, An entropy-variable-based VMS/GLS method for the simulation of compressible flows on unstructured grids. Computational Methods in Applied Mechanics and Engineering, 195, 1154–1179. [6] Vreman, A.W., 1995, Direct and large eddy simulation of the compressible turbulent mixing layer, PhD Dissertation, University of Twente, Twente. [7] Sreedhar, M. and Raghab S., 1994, Large eddy simulation of longitudinal stationary vortices. Physics of Fluids, 6, 2501–2514. [8] Vreman, B., Geurts, B. and Kuerten, H., 1995, A priori tests of large eddy simulation of the compressible plane mixing layer. Journal of Engineering Mathematics, 29, 299–327. [9] Erlebacher, G., Hussaini, M.Y., Speziale, C.G. and Zang, T.A., 1987, Toward the large eddy simulation of compressible turbulent flow, ICASE Report No. 87-20. [10] Erlebacher, G., Hussaini, M. Y., Speziale, C. G. and Zang, T. A., 1992, Toward the large eddy simulation of compressible turbulent flow. Journal of Fluid Mechanics, 238, 155–185. [11] Ducros, F., Comte, P. and Lesieur, M., 1996, Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate. Journal of Fluid Mechanics, 326, 1–36. [12] Lee, S., 1992, Large-eddy simulation of shock turbulence interaction. In Annual Research Briefs, pp. 73–84 (Stanford, CA: Center for Turbulence Research).
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