An exact axisymmetric spiral solution of incompressible 3D Euler equations

TL;DR

This paper derives exact axisymmetric spiral solutions to the 3D Euler equations, revealing new vortex structures and classical solutions, with applications to natural phenomena like typhoons and tornadoes.

Contribution

It provides a set of new exact analytical solutions to the 3D Euler equations describing spiral vortex structures, including typhoon-like and multi-layer solutions, expanding understanding of complex axisymmetric flows.

Findings

Derived new exact spiral solutions of the 3D Euler equations.

Identified vortex structures with applications to natural phenomena.

Connected solutions to classical vortex models like Rankine and Batchelor.

Abstract

Spiral structure is one of the most common structures in the nature flows. A general steady spiral solution of incompressible inviscid axisymmetric flow was obtained analytically by applying separation of variables to the 3D Euler equations. The solution, depending on 3 parameters, describes the spiral path of the fluid material element on the Bernoulli surface, whereas some new exact solutions were obtained to be bounded within the whole region. The first one is a continued typhoon-like vortex solution, where there are two intrinsic length scales. One is the radius of maximum circular velocity $r_m$, the other is the radius of the vortex kernel $r_k=\sqrt{2}r_m$. The second one is a multi-planar solution, periodically in $z$-coordinate. Within each layer, the solution is a umbrella vortex similar to the first one. The third one is also a multi-planar solution in $z$-coordinate. In each layer, it is a combination of two independent solutions like the Rankine vortex, which is also finite but discontinued for either vertical or horizontal velocity. The fourth one is a multi-paraboloid vortex solution finite for $z$-coordinate but infinite for $r$-coordinate. Besides, some classical simple solutions (Rankine vortex, Batchelor vortex, Hill spherical vortex, etc.) were also obtained. The above explicit solutions can be applied to study the radial structure of the typhoon, tornados and mesoscale eddies. Both the solutions and approaches used here could also be applied to other complex flows by the Navier-Stokes equations.