Multi-scale dynamics and subgrid-scale modeling of turbulence in two and three dimensions.

摘要

In a sharp contrast to three-dimensional (3D) fluid turbulence which gives rise a joint cascade of kinetic energy and helicity all way down to the smaller and smaller scales, in two-dimensional (2D) turbulence, however, there exist a direct cascade of enstrophy (half the mean-square vorticity) and an inverse cascade of kinetic energy . The mechanisms of 2D cascades have been the matter of debate since the leading paper of Kraichnan (1967), and remain unclear. In this thesis, a novel filtering approach [42] is employed to study the dual cascade picture in 2D hydrodynamic turbulence in a statistically steady situation by direct numerical simulation (DNS) of the 2D Navier-Stokes equations.;For 2D enstrophy cascade, we observed an energy spectrum steeper than k-3 law predicted by Kraichnan (1967), but in a perfect agreement with his log-correction [57]. It was found that the 2D enstrophy cascade is strongly ultraviolet (UV) local. Physically, the 2D enstrophy cascade originates from steepening of inertial-range vorticity gradients due to compression of vorticity level-sets by the large-scale strain field.;For 2D inverse energy cascade, a similarity wavenumber range was observed in which the mean spectral energy flux is a negative constant and the energy spectrum scales as k-5/3, consistent with the prediction of Kraichnan (1967). We found that the inverse energy cascade is scale-local, but that the strongly local contribution vanishes identically, as argued by Kraichnan (1971). In particular, we examined a Multi-Scale Gradient (MSG) expansion developed by Eyink (2006a) for the turbulent stress, which up to second-order in space gradients predicts very well the magnitude, spatial structure and scale distribution of the local energy flux. Our findings give strong support to vortex-thinning [59, 101, 106] as the fundamental mechanism of 2D inverse energy cascade.;Another topic in this thesis is the subgrid-scale (SGS) stress modeling for large eddy simulation (LES) of turbulent flows. We propose to impose physical constraints in the dynamic procedure and to calculate the SGS model coefficients using a constrained variation. An SGS dissipation constraint is deduced based on the scale-invariance of fluid turbulence in the inertial range. Numerical simulations of forced and decaying isotropic turbulence demonstrate that the constrained dynamic mixed model predicts the energy evolution and the SGS dissipation more accurately than the non-constrained model does. The constrained SGS model also shows a strong correlation with the "actual" stress from a priori test and is capable of predicting the energy backscatter, manifesting a desirable feature of combining the advantages of dynamics Smagorinsky and mixed models.