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

Abstract

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 $o(\epsilon )$ quadratic approximation of real flow for a sufficiently small flux, under the assumption of the almost periodicity of the roughness profile $\omega$. We further obtain $O({\epsilon }^{3/2})$ quadratic approximation of real flows by the damping model with a quasi-periodic function $\omega$ 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.

Keywords:

• Damping effects
• Navier–Stokes equations
• rough boundary
• channel flows

AMS Subject Classifications:

• 35Q30
• 76D10

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korean government(MSIP) [grant number NRF-20151009350]; in part by the Yonsei University Future-leading Research Initiative of 2014.