Oscillating sources in a shear flow with a free surface

TL;DR

This paper develops a linearized theory for oscillating sources in a shear flow with a free surface, revealing a new critical layer-like flow and vortex street phenomena in rotational flow, extending potential theory to more realistic conditions.

Contribution

It introduces the first linearized solutions for oscillating sources in rotational shear flows, uncovering new flow structures absent in irrotational models.

Findings

Discovery of a critical layer-like flow drifting downstream at the source depth.

Identification of a vortex street phenomenon in 3D at every depth.

Extension of potential theory to include rotational shear flows.

Abstract

We report on progress on the free surface flow in the presence of submerged oscillating line sources (2D) or point sources (3D) when a simple shear flow is present varying linearly with depth. Such sources are in routine use as Green functions in the realm of potential theory for calculating wave-body interactions, but no such theory exists in for rotational flow. We solve the linearized problem in 2D and 3D from first principles, based on the Euler equations, when the sources are at rest relative to the undisturbed surface. Both in 2D and 3D a new type of solution appears compared to irrotational case, a critical layer-like flow whose surface manifestation ("wave") drifts downstream from the source at the velocity of the flow at the source depth. We analyse the additional vorticity in light of the vorticity equation and provide a simple physical argument why a critical layer is a necessary consequence of Kelvin's circulation theorem. In 3D a related critical layer phenomenon occurs at every depth, whereby a street of counter-rotating vortices in the horizontal plane drift downstream at the local flow velocity.