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 k^-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 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 k^-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 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 〉/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 − 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 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: C² ₊^4{3}k^-7{3} (C k is Kolmogorov's constant), with α ≈ 1. 32.