Do irrotational water waves remain irrotational in the limit of a vanishing viscosity?

TL;DR

This study uses numerical simulations to examine whether irrotational water waves remain irrotational as viscosity approaches zero, revealing that vortex shedding occurs and the limit is singular under no-slip boundary conditions.

Contribution

It demonstrates through simulations that irrotationality is not preserved in the vanishing viscosity limit with no-slip boundaries, challenging common assumptions in water wave theory.

Findings

Vortex pairs shed from the bottom boundary layer into the flow.

Perturbations from vortex shedding do not vanish as Reynolds number increases.

The vanishing viscosity limit is singular under no-slip boundary conditions.

Abstract

Theoretical results on water waves almost always start by assuming irrotationality of the flow in order to simplify the formulation. In this work, we investigate the well-foundedness of this hypothesis via numerical simulations of the free-surface Navier-Stokes equations. We show that, in the presence of a non-flat bathymetry, either angular or smooth, a gravity wave of finite amplitude can shed vortex pairs from the bottom boundary layer into the bulk of the flow. As these eddies approach the free surface they modify the shape of the wave. It is found that this perturbation does not vanish as the Reynolds number is increased. The vanishing viscosity limit of water waves is therefore singular when no-slip boundary conditions are enforced on the bottom.