Some general solutions for linear Bragg-Hawthorne equation

TL;DR

This paper systematically analyzes linear solutions of the Bragg-Hawthorne equation for steady axisymmetric flows, deriving general solutions, extending known flows, and revealing relations between vortex solutions in a spherical coordinate framework.

Contribution

It introduces a new vorticity decomposition and provides a comprehensive set of general solutions for 16 linear cases, extending classical vortex flows.

Findings

Derived general solutions for 16 linear cases

Extended well-known vortex solutions like Hill's vortex

Revealed relations between different vortex flows

Abstract

Linear cases of Bragg-Hawthorne equation for steady axisymmetric incompressible ideal flows are systematically discussed. The equation is converted to a more convenient form in a spherical coordinate system. A new vorticity decomposition is derived. General solutions for 16 linear cases of the equation are obtained. These solutions can be specified to gain new analytical vortex flows, as examples in the paper demonstrate. A lot of well_known solutions like potential flow past a sphere, Hill's vortex with and without swirl, are included and extended in these solutions. Special relations between some vortex flows are also revealed when exploring or comparing related solutions.