Stokes waves in rotational flows: internal stagnation and overhanging profiles

TL;DR

This paper develops a numerical method to compute periodic Stokes waves with vorticity, capturing complex features like internal stagnation and overhanging profiles, and presents new solutions for various vorticity distributions.

Contribution

It introduces a conformal mapping-based finite difference scheme that handles internal stagnation points and overhanging profiles, advancing the modeling of rotational flows with arbitrary vorticity.

Findings

Validated method against known solutions for zero and constant vorticity.

Produced new solutions for affine vorticity functions.

Generated solutions for a two-layer constant vorticity case.

Abstract

Periodic travelling waves at the free surface of an incompressible inviscid fluid in two dimensions under gravity are numerically computed for an arbitrary vorticity distribution. The fluid domain over one period is conformally mapped from a fixed rectangular one, where the governing equations along with the conformal mapping are solved using a finite difference scheme. This approach accommodates internal stagnation points, critical layers, and overhanging profiles, thereby overcoming limitations of previous studies. The numerical method is validated through comparisons with known solutions for zero and constant vorticity. Novel solutions are presented for affine vorticity functions and a two-layer constant vorticity scenario.