On the existence of global-in-time weak solutions and scaling laws for Kolmogorov's two-equation model for turbulence

Abstract

This paper is concerned with Kolmogorov's two-equation model for turbulence in (Formula presented.) involving the mean velocity u, the pressure p, an average frequency (Formula presented.), and a mean turbulent kinetic energy k. We consider the system with space-periodic boundary conditions in a cube (Formula presented.), which is a good choice for studying the decay of free turbulent motion sufficiently far away from boundaries. In particular, this choice is compatible with the rich set of similarity transformations for turbulence. The main part of this work consists in proving existence of global weak solutions of this model. For this we approximate the system by adding a suitable regularizing r-Laplacian and invoke existence result for evolutionary equations with pseudo-monotone operators. An important point constitutes the derivation of pointwise a priori estimates for ω (upper and lower) and k (only lower) that are independent of the box size a, thus allow us to control the parabolicity of the diffusion operators.