2005 Anomalous scaling of structure functions and dynamic constraints on turbulence simulations

The connection between anomalous scaling of structure functions (intermit-tency) and numerical methods for turbulence simulations is discussed. It is argued that the computational work for direct numerical simulations (DNS) of fully developed turbulence increases as Re4, and not as Re3 expected from Kolmogorov’s theory, where Re is a large-scale Reynolds number. Various rela-tions for the moments of acceleration and velocity derivatives are derived. An infinite set of exact constraints on dynamically consistent subgrid models for Large Eddy Simulations (LES) is derived from the Navier–Stokes equations, and some problems of principle associated with existing LES models are high-lighted. KEY WORDS: Hydrodynamic turbulence; anomalous scaling; direct numerical simulations; large eddy simulations. 1. BACKGROUND The theory of turbulence and the development of calculation methods for high-Reynolds-number flows became an active research topic around the beginning of the twentieth century. This effort yielded many important results of general interest in statistical physics. For instance, Kolmogorov’s work(1−3) on turbulence theory formulated the scaling ideas for the first time, and Kraichnan(4) proposed the mode coupling approach. However, the “turbulence problem, ” lacking a small parameter characterizing the strong nonlinear interactions, has turned out to be remarkably difficult— and it remains so today