The Navier wall law at a boundary with random roughness

TL;DR

This paper analyzes the Navier-Stokes equation in domains with random rough boundaries, showing that a Navier boundary condition approximates the true boundary behavior with an error depending on the randomness scale.

Contribution

It extends previous homogenization results by quantifying the approximation error for random boundary roughness using probabilistic methods.

Findings

The Navier boundary condition approximates the true boundary behavior with an error of order $O( ext{eps}^{3/2} | ext{ln eps}|^{1/2})$.

Decay properties of the boundary layer are established via a central limit theorem for dependent variables.

The analysis applies to a large class of random boundary irregularities, broadening the scope of homogenization results.

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.