On the Prandtl-Kolmogorov 1-equation model of turbulence

Abstract

We prove an estimate of total (viscous plus modelled turbulent) energy dissipation in general eddy viscosity models for shear flows. The ratio of the near wall average viscosity to the effective global viscosity is the key parameter in the estimate. This result is then applied to the 1-equation, URANS model of turbulence for which this ratio depends on the specification of the turbulence length scale. The model, which was derived by Prandtl in 1945, is a component of a 2-equation model derived by Kolmogorov in 1942 and is the core of many unsteady, Reynolds averaged models for prediction of turbulent flows. Let τ denote a selected time scale. Away from walls, interpreting an early suggestion of Prandtl, we set [Formula: see text]In the near-wall region analysis suggests replacing the traditional [Formula: see text] ([Formula: see text] wall normal distance) with [Formula: see text] giving [Formula: see text]This specification of [Formula: see text] results in a simpler model with correct near wall asymptotics. Its energy dissipation rate scales no larger than the physically correct [Formula: see text], balancing energy input with energy dissipation. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Keywords: URANS model; energy dissipation; turbulence NS model.