Analysis of localized damping effects in channel flows with arbitrary rough boundary

We justify a modified Navier–Stokes system that includes the damping effect for a channel flow with an arbitrary irregular boundary. The effects of rough walls are modeled by a localized damping term in the momentum equation. We prove the existence and uniqueness of a solution to the modified Navier–Stokes system in a smooth extended domain using the fixed-point theorem and Saint-Venant technique. The proposed damping term yields an o(ε) quadratic approximation of real flow for a sufficiently small flux, under the assumption of the almost periodicity of the roughness profile ω. We further obtain O(ε^3/2) quadratic approximation of real flows by the damping model with a quasi-periodic function (Formula presented.) and the diophantine condition. Consequently, we confirm that the localized damping effect can provide an effective model to predict channel flows with an arbitrary irregular surface.