Viscous potential free-surface flows in a fluid layer of finite depth

TL;DR

This paper develops a simplified viscous potential flow model for free-surface flows over finite depth, incorporating a new nonlocal viscous term without correction procedures, and derives related long wave equations.

Contribution

It introduces a novel viscous potential flow formulation using Helmholtz-Leray decomposition and Fourier-Laplace transforms, avoiding previous correction methods.

Findings

Derived a new nonlocal viscous term in the bottom boundary condition

Expressed vortical velocity component solely in terms of potential and free-surface elevation

Formulated long wave equations based on the new viscous potential flow model

Abstract

It is shown how to model weakly dissipative free-surface flows using the classical potential flow approach. The Helmholtz-Leray decomposition is applied to the linearized 3D Navier-Stokes equations. The governing equations are treated using Fourier--Laplace transforms. We show how to express the vortical component of the velocity only in terms of the potential and free-surface elevation. A new predominant nonlocal viscous term is derived in the bottom kinematic boundary condition. The resulting formulation is simple and does not involve any correction procedure as in previous viscous potential flow theories [Joseph2004]. Corresponding long wave model equations are derived.