A Dynamic Multilevel Model for the Simulation of the Small Structures in Homogeneous Isotropic Turbulence

In turbulence simulations, the small scales of motion, even if they carry only a very small percentage of the whole kinetic energy, must be taken into account in order to accurately reproduce the statistical properties of the flows. This induces strong computational restrictions. In an attempt to understand and model the nonlinear interaction between the small and large scales, a dynamic multilevel procedure is proposed and applied to homogeneous turbulence. As in large eddy simulation, filtering operators are used to separate the different scales of the velocity field. In classical models (Smagorinsky), only the large scale equation is resolved. A different approach is proposed here. Indeed, by analyzing the nonlinear interaction term in the large scale equation, we show that they locally have a very small contribution to the whole dynamic of the flow. We then propose to treat them less accurately. Specific treatments for these terms are achieved by a space and time adaptative procedure; the cut-off value (filter width) which defines the scale separation varies as time evolves. Simulations at Re in the range of 60 to 150 have been performed until statistical steady states are reached, i.e. over long time period. Comparisons with direct simulations (DNS) show that this numerical modeling provides an efficient resolution of the nonlinear interaction term. The multilevel algorithm is shown to be stable; the corresponding simulated flows reach a statistically steady state very close to the DNS ones. The shape of the energy spectrum functions as well as the characteristic statistical properties of the velocity and its derivatives are accurately recovered.