The paper presents large-eddy simulation (LES) formalism, along with the various subgrid-scale models developed since Smagorinsky's model. We show how Kraichnan's spectral eddy viscosity may be implemented in physical space, yielding the structure-function model. Recent developments of this model that allow the eddy viscosity to be inhibited in transitional regions are discussed. We present a dynamic procedure, where a double filtering allows one to dynamically determine the subgrid-scale model constants. The importance of backscatter effects is discussed. Alternatives to the eddy-viscosity assumption, such as scalesimilarity models, are considered. Pseudo-direct simulations in which numerical diffusion replaces subgrid transfers are mentioned. Various applications of LES to incompressible and compressible turbulent flows are given, with an emphasis on the generation of coherent vortices.

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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 $k^{-\frac{5}{3}}$ 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 1283 collocation points. It is shown that the temperature spectrum tends to follow in the energetic scales a k−1 range, followed by a $k^{-\frac{5}{3}}$ 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 ki = 20 and ∝ k8 for k → 0. In the k−1 range, the temperature spectrum is found to collapse according to the law ET(k, t) = 0.1η(〈u2〉/ε) k−1, where ε 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 − X2), 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 Smagorinsky's, 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 Batchelor's quasi-normal law: $\alpha C^2_{\rm k}\epsilon^{\frac{4}{3}}k^{-\frac{7}{3}}$ (Ck is Kolmogorov's constant), with α ≈ 1.32.

Numerical simulations investigating the formation and stability of quasi-two-dimensional coherent vortices in rotating homogeneous three-dimensional flow are described. In a numerical study of shear flows Lesieur, Yanase & Métais (1991) found that cyclones (respectively anticyclones) with |ω2D| ∼ O(2Ω), where ω2D is the vorticity and Ω is the rotation rate, are stabilized (respectively destabilized) by the rotation. A study of triply periodic pseudo-spectral simulations (643) was undertaken in order to investigate the vorticity asymmetry in homogeneous turbulence. Specifically, we examine (i) the possible three-dimensionalization of initially two-dimensional vortices and (ii) the emergence of quasi-two-dimensional structures in initially-isotropic three-dimensional turbulence. Direct numerical simulations of the Navier—Stokes equations are compared with large-eddy simulations employing a subgridscale model based on the second-order velocity structure function evaluated at the grid separation and with simulations employing hyperviscosity.Isolated coherent two-dimensional vortices, obtained from a two-dimensional decay simulation, were superposed with a low-amplitude three-dimensional perturbation, and used to initialize the first set of simulations. With Ω = 0, a three-dimensionalization of all vortices was observed. This occurred first in the small scales in conjunction with the formation of longitudinal hairpin vortices with vorticity perpendicular to that of the initial quasi-two-dimensional flow. In agreement with centrifugal stability arguments, when 2Ω = [ω2D]rms a rapid destabilization of anticyclones was observed to occur, whereas the initial two-dimensional cyclonic vortices persisted throughout the simulation. At larger Ω, both cyclones and anticyclones remained two-dimensional, consistent with the Taylor—Proudman theorem. A second set of simulations starting from isotropic three-dimensional fields was initialized by allowing a random velocity field to evolve (Ω = 0) until maximum energy dissipation. When the simulations were continued with 2Ω = [ω · Ω]rms/Ω, the three-dimensional flow was observed to organize into two-dimensional cyclonic vortices. At larger Ω, two-dimensional anticyclones also emerged from the initially-isotropic flow. The consequences for a variety of industrial and geophysical applications are clear. For quasi-two-dimensional eddies whose characteristic circulation times are of the order ofder of Ω−1, rotation induces a complete disruption of anticyclonic vortices, while stabilizing cyclonic ones.

Direct and large-eddy simulations (DNS/LES) are performed to analyze the vortex dynamics and the statistics of bifurcating jets. The Reynolds number ranges from ReD=1.5×103 to ReD=5.0×104. An active control of the inlet conditions of a spatially evolving round jet is performed with the aim of favoring the jet spreading in one particular spatial direction, thus creating a bifurcating jet. Three different types of forcing, based on the information provided by a LES of a natural (unforced) jet, are superimposed to the jet inlet in order to cause its bifurcation. The different forcing types mimic the forcing methods used in experimental bifurcating jets (Lee and Reynolds, Parekh et al., Suzuki et al.), but using excitations with relatively low amplitudes, which could be used in real industrial applications. The three-dimensional coherent structures resulting from each specific forcing are analyzed in detail and their impact on the statistical behavior of bifurcating jets is explained. In particular we focus on the influence of the forcing frequency and of the Reynolds number on the jet control efficiency. Analysis of the coherent vortex dynamics shows that an inlet excitation which combines an axisymmetric excitation at the preferred frequency and a so-called flapping excitation at the subharmonic frequency is the most efficient strategy for jet control, even at high Reynolds numbers.