Marcel Lesieur

Spectral large-eddy simulation of isotropic and stably stratified turbulence

We first recall the concepts of spectral eddy viscosity and diffusivity, derived from the two-point closures of turbulence, in the framework of large-eddy simulations in Fourier space. The case of a spectrum which does not decrease as

Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate

k^{-\frac{5}{3}}

Parameterization of Small Scales of Three-Dimensional Isotropic Turbulence Utilizing Spectral Closures

at the cutoff is studied. Then, a spectral large-eddy simulation of decaying isotropic turbulence convecting a passive temperature is performed, at a resolution of 128 3 collocation points. It is shown that the temperature spectrum tends to follow in the energetic scales a k −1 range, followed by a

A numerical investigation of the coherent vortices in turbulence behind a backward-facing step

k^{-\frac{5}{3}}

Coherent structures in rotating three-dimensional turbulence

inertial–convective range at higher wavenumbers. This is in agreement with previous independent calculations (Lesieur & Rogallo 1989). When self-similar spectra have developed, the temperature variance and kinetic energy decay respectively like t −1.37 and t −1.85 , with identical initial spectra peaking at k i = 20 and ∝ k 8 for k → 0. In the k −1 range, the temperature spectrum is found to collapse according to the law E T ( k, t ) = 0.1η(〈 u 2 〉/e) k −1 , where e and η are the kinetic energy and temperature variance dissipation rates. The spectral eddy viscosity and diffusivity are recalculated explicitly from the large-eddy simulation: the anomalous ∝ ln k behaviour of the eddy diffusivity in the eddy-viscosity plateau is shown to be associated with the large-scale intermittency of the passive temperature: the p.d.f. of the velocity component u is Gaussian (∼ exp − X 2 ), while the scalar T , the velocity derivatives ∂ u /∂ x and ∂ u /∂ z , and the temperature derivative ∂ T /∂ z are all close to exponential exp - | X | at high | X |. The pressure distribution is exponential at low pressure and Gaussian at high. For stably stratified Boussinesq turbulence, the coupling between the temperature and the velocity fields leads to the disappearance of the ‘anomalous’ temperature behaviour ( k −1 range, logarithmic eddy diffusivity and exponential probability density function for T ). These are the highest-resolution calculations ever performed for this problem. We also split the eddy viscous coefficients into a vortex and a wave component. In both cases (unstratified and stratified), comparisons with direct numerical simulations are performed. Finally we propose a generalization of the spectral eddy viscosity to highly intermittent situations in physical space: in this structure-function model , the spectral eddy viscosity is based upon a kinetic energy spectrum local in space. The latter is calculated with the aid of a local second-order velocity structure function. This structure function model is compared with other models, including Smagorinskys, for isotropic decaying turbulence, and with high-resolution direct simulations. It is shown that low-pressure regions mark coherent structures of high vorticity. The pressure spectra are shown to follow Batchelors quasi-normal law:

The mixing layer and its coherence examined from the point of view of two-dimensional turbulence

\alpha C^2_{\rm k}\epsilon^{\frac{4}{3}}k^{-\frac{7}{3}}

Large- and small-scale stirring of vorticity and a passive scalar in a 3-D temporal mixing layer

( C k is Kolmogorovs constant), with α ≈ 1.32.

Direct and large-eddy simulations of transition in the compressible boundary layer

It is well known that subgrid models such as Smagorinskys cannot be used for the spatially growing simulation of the transition to turbulence of flat-plate boundary layers, unless large-amplitude perturbations are introduced at the upstream boundary: they are over-dissipative, and the flow simulated remains laminar. This is also the case for the structure-function model (SF) of Metais & Lesieur (1992). In the present paper we present a sequel to this model, the filtered-structure-function (FSF) model. It consists of removing the large-scale fluctuations of the field before computing its second-order structure function. Analytical arguments confirm the superiority of the FSF model over the SF model for large-eddy simulations of weakly unstable transitional flows. The FSF model is therefore used for the simulation of a quasi-incompressible ( M ∞ = 0.5) boundary layer developing spatially over an adiabatic flat plate, with a low level of upstream forcing. With the minimal resolution 650 × 32 × 20 grid points covering a range of streamwise Reynolds numbers Re x 1 e [3.4 × 10 5 , 1.1 × 10 6 ], transition is obtained for 80 hours of time-processing on a CRAY 2 (whereas DNS of the whole transition takes about ten times longer). Statistics of the LES are found to be in acceptable agreement with experiments and empirical laws, in the laminar, transitional and turbulent parts of the domain. The dynamics of low-pressure and high-vorticity distributions is examined during transition, with particular emphasis on the neighbourhood of the critical layer (defined here as the height of the fluid travelling at a speed equal to the phase speed of the incoming Tollmien–Schlichting waves). Evidence is given that a subharmonic-type secondary instability grows, followed by a purely spanwise (i.e. time-independent) mode which yields peak-and-valley splitting and transition to turbulence. In the turbulent region, flow visualizations and local instantaneous profiles are provided. They confirm the presence of low- and high-speed streaks at the wall, weak hairpins stretched by the flow and bursting events. It is found that most of the vorticity is produced in the spanwise direction, at the wall, below the high-speed streaks. Isosurfaces of eddy viscosity confirm that the FSF model does not perturb transition much, and acts mostly in the vicinity of the hairpins.

Effects of spanwise rotation on the vorticity stretching in transitional and turbulent channel flow

Abstract A spectral equation derived from two-point closures applied to three-dimensional isotropic turbulence is studied from the subgrid-scale modeling point of view, with a cutoff wavenumber kc located in the inertial range of turbulence. Ideas of Kraichnan concerning eddy viscosities are then used to evaluate the parameterized subgrid-scale transfer. This, together with a suitable boundary condition at kc, allows us to predict statistically the large scales (k kc). A k−5/3 energy spectrum extending to kc is recovered without any artificial dissipation range in the neighborhood of kc. This procedure is valid both for forced stationary turbulence and for freely decaying turbulence. The same eddy-viscosity is then introduced in a direct numerical simulation of three-dimensional homogeneous isotropic turbulence without external forcing. Again, the energy spectrum, evaluated by averaging on a spherical shell of radius k, follows the Kolmogorov law u...

Statistical Predictability of Decaying Turbulence

This paper presents a statistical and topological study of a complex turbulent flow over a backward-facing step by means of direct and large-eddy simulations. Direct simulations are first performed for an isothermal two-dimensional case. In this case, shedding of coherent vortices in the mixing layer is demonstrated. Both direct and large-eddy simulations are then carried out in three dimensions. The subgrid-scale model used is the structure-function model proposed by Metais & Lesieur (1992). Lowstep computations corresponding to the geometry of Eaton & Johnstons (1980) laboratory experiment give turbulence statistics in better agreement with the experimental data than both Smagorinskys method and K -e modelling. Furthermore, calculations for a high step show that the eddy structure of the flow presents striking analogies with forced plane mixing layers: large billows are shed behind the step with intense longitudinal vortices strained between them.