On the Prandtl-Kolmogorov 1-Equation Model

Turbulence modeling in practice requires predicting averages of solutions of the Navier-Stokes equations. We examine eddy viscosity URANS models based on the 1-equation model of Prandtl and Kolmogorov. Many of these models fail due to over dissipation in the near wall region. For general eddy viscosity models, we show that the ratio of the near wall average viscosity to the effective global viscosity is the key parameter. 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. We propose a modification to traditional choices of l: away from walls, interpreting an early suggestion of Prandtl, we set

where τ= selected time scale. In the near wall region analysis suggests replacing the traditional l=0.41d (d= wall normal distance) with l= 0.41d√(d/L)giving, e.g.,

This l(⋅) results in a simpler model with correct near wall asymptotics. Its energy dissipation rate scales no larger than the physically correct O(U³/L), balancing energy input with energy dissipation.