On the axisymmetric steady incompressible Beltrami flows

TL;DR

This paper analytically and numerically derives axisymmetric steady Beltrami flows in various coordinate systems, providing insights into vortex dynamics and potential applications in electromagnetics.

Contribution

It introduces a comprehensive method for modeling axisymmetric Beltrami flows in multiple coordinate systems, including visualization and a qualitative vortex breakdown model.

Findings

Analytical solutions for Beltrami flows in cylindrical, spherical, paraboloidal, and spheroidal coordinates.

Numerical solutions and contour visualizations of stream functions.

A qualitative model of vortex breakdown progression.

Abstract

In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate systems include those that appear amenable to modeling such flows: the cylindrical, spherical, paraboloidal, and prolate and oblate spheroidal systems. The usual Euler equations are reformulated using the Bragg--Hawthorne equation for the stream function of the flow, which is solved analytically or numerically in each coordinate system under the assumption of separability of variables. Many of the obtained flows are visualized via contour plots of their stream functions in the $rz$-plane. Finally, the results are combined to provide a qualitative quasi-static model for a progression of flows that one might see in the process of a vortex breakdown. The results in this paper are equally applicable in electromagnetics, where the equivalent concept is that of a force-free magnetic field.