WIAS Preprint No. 2545, (2018)

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



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Naumann, Joachim

2010 Mathematics Subject Classification

  • 76F05 76D05 35Q30 47H05

Keywords

  • Navier--Stokes equation, mean turbulent kinetic energy, energy estimates, upper and lower pointwise bounds, pseudo-monotone operators, defect measure

DOI

10.20347/WIAS.PREPRINT.2545

Abstract

This paper is concerned with Kolmogorov's two-equation model for free turbulence in space dimension 3, involving the mean velocity u, the pressure p, an average frequency omega, and a mean turbulent kinetic energy k. We first discuss scaling laws for a slightly more general two-equation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's two-equation model under space-periodic boundary conditions in cubes with positive side length l. To this end, we provide new a priori estimates and invoke existence result for pseudo-monotone operators.

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