Three-dimensional exponential asymptotics and Stokes surfaces for flows past a submerged point source

TL;DR

This paper explores the exponential asymptotics and Stokes surfaces in three-dimensional fluid flows over a submerged point source, revealing how waves are 'switched-on' across Stokes surfaces and extending previous linearized results.

Contribution

It develops a method to compute Stokes surfaces in three-dimensional flows, generalizing prior linearized analyses to nonlinear bodies and complex geometries.

Findings

Derived the structure of Stokes surfaces in 3D flows.

Extended previous linear results to nonlinear cases.

Provided a framework for computing wave regions in fluid dynamics.

Abstract

When studying fluid-body interactions in the low-Froude limit, traditional asymptotic theory predicts a waveless free-surface at every order. This is due to the fact that the waves are in fact exponentially small---that is, beyond all algebraic orders of the Froude number. Solutions containing exponentially small terms exhibit a peculiarity known as the Stokes phenomenon, whereby waves can 'switch-on' seemingly instantaneously across so-called Stokes lines, partitioning the fluid domain into wave-free regions and regions with waves. In three dimensions, the Stokes line concept must extend to what are analogously known as 'Stokes-surfaces'. This paper is concerned with the archetypal problem of uniform flow over a point source---reminiscent of, but separate to, the famous Kelvin wave problem. In theory, there exist Stokes surfaces i.e. manifolds in space that divide wave-free regions from regions with waves. Previously, in Lustri & Chapman (2013) the intersection of the Stokes surface with the free-surface, z=0, was found for the case of a linearised point-source obstruction. Here we demonstrate how the Stokes surface can be computed in three-dimensional space, particularly in a manner that can be extended to the case of nonlinear bodies.