Effects of boundary roughness and inertia on the fluid flow through a corrugated pipe and the formula for the Darcy–Weisbach friction coefficient

Abstract

We consider an incompressible viscous fluid flowing through a cylindrical pipe with rough wall. Motivated by the applications, we assume the periodicity of the roughness in the longitudinal direction and that the flow is governed by the prescribed pressure drop between pipe’s ends. The goal of the paper is to investigate the effects of the corrugated boundary on the fluid flow by taking into account the effects of inertia as well. Starting from the Navier–Stokes system and employing the average boundary roughness as the small parameter, we propose a higher-order correction of the standard Hagen–Poiseuille velocity via boundary layer analysis. Using the obtained result we propose the new formula for the Darcy–Weisbach friction coefficient.