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 $${\varepsilon \ll 1}$$ . In the parent paper [8], we derived a homogenized boundary condition of Navier type as $${\varepsilon \rightarrow 0}$$ . We show here that for a large class of boundaries, this Navier condition provides a $${O(\varepsilon^{3/2} |\ln \varepsilon|^{1/2})}$$ approximation in L 2, instead of $${O(\varepsilon^{3/2})}$$ 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.