Turbulent cascade and intermittency growth

Abstract

Most embodiments of Kolmogorov’s idea of turbulent cascade rest on heuristic modelling rather than mathematical treatment of the Navier-Stokes equations. However, low-order systematic approximation via renormalized perturbation theory also exists. It supports the -5/3 inertial-range spectrum law and yields quantitatively good predictions of both the Kolmogorov constant and the dissipation spectrum. Pressure plays an essential role in the inertial range. This is highlighted by the qualitatively different predictions of the same theory applied to Burgers turbulence. A recent, non-perturbative analytical approach via nonlinear mapping of gaussian fields may contribute to the resolution of questions of intermittency corrections to the inertial and dissipation ranges. Again, comparison with Burgers turbulence is illuminating. Whatever the tools, it is crucial that a theory be able to describe what happens at finite Reynolds numbers, both to understand the limit of infinite Reynolds number and to interpret correctly the data from existing experiments and simulations.