Flow Structures Beneath Stationary Waves with Constant Vorticity Over Variable Topography

TL;DR

This study explores how variable bottom topography affects flow structures beneath stationary waves with constant vorticity, revealing new features like saddle points and pressure minima not seen in flat-bottom cases.

Contribution

It introduces a numerical analysis of flow beneath stationary waves over variable topography, extending previous flat-bottom studies with a modified conformal mapping approach.

Findings

Existence of saddle points beneath wave crests

Presence of a pressure global minimum on the bottom boundary

Distinct flow features due to variable topography

Abstract

The flow structures beneath waves have received significant attention from both theoretical and numerical perspectives. Most studies on this topic assume a flat bottom, leading to questions about the effects of variable bottom topography. To address this gap, we investigate the flow structures beneath stationary waves with constant vorticity, considering the influence of variable topography. Specifically, we numerically analyze the role of vorticity in the emergence of stagnation points and the pressure distribution within the fluid in two bottom topography scenarios: a bump and a hole. Our numerical approach is based on a variation of the classical Dyachenko, Zakharov, and Kuznetsov conformal mapping technique for free-boundary water wave problems. Our results reveal the existence of saddle points beneath wave crests and center beneath depression solitary waves. Additionally, we observe that the pressure can exhibit distinctive features, such as a global minimum on the bottom boundary -- behavior that is markedly different from the usual flat-bottom case.