On non-linear Stokes problems with viscosity depending on the distance to the wall

Abstract

In fluid mechanics, the RANS modeling (Reynolds Averaged Navier–Stokes equation) assumes that the period of the averaged solutions to the Navier–Stokes equations is several orders of magnitude larger than the turbulent fluctuations. A type of simple model often used by engineers is a mixing-length model called “Smagorinsky modeling”. In this paper, we present some theoretical and numerical results on a mixing-length model in which the eddy viscosity is depending on the strain tensor and on the distance to the wall of the fluid flow domain. In particular, we show that the so-called von Karman model becomes an ill-posed problem when the laminar viscosity tends to zero.