The Navier wall law at a boundary with random roughness

Abstract

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size $\eps \ll 1$. In a parent paper, we derived a homogenized boundary condition of Navier type as $\eps \to 0$. We show here that for a large class of boundaries, this Navier condition provides a $O(\eps^{3/2} |\ln \eps|^{1/2})$ approximation in $L^2$, instead of $O(\eps^{3/2})$ for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables.