Characterizing two-dimensional incompressible flows through traveling wave and symmetry solutions of the vorticity transportation equation

TL;DR

This paper derives exact solutions for two-dimensional incompressible flows using the vorticity transportation equation, revealing new symmetry solutions and complex flow patterns, including traveling waves and vortex structures.

Contribution

It introduces novel symmetry solutions to the vorticity equation, expanding understanding of flow dynamics beyond existing potential flow results.

Findings

Traveling wave solutions include continuous and solitary waves.

New symmetry solutions exhibit complex and diverse flow patterns.

Symmetry solutions to vortex and doublet cases show intricate dynamics.

Abstract

We use the vorticity transportation equation as the start point--with the help of stream function for two-dimensional planar incompressible flows--to obtain exact solutions that characterize evolution and dynamics of the flows. These traveling wave solutions, including both continuous and solitary waves, also recover the results existing in the theory of potential flows. We further analyze the flow patterns regarding symmetry solutions generated by Lie transformation groups. Some of the symmetry solutions have not been discussed in the current literature. They could display flow patterns very different from that of the original exact solutions, suggesting the symmetry solutions are not trivial. Particularly, the symmetry solutions to Lame-Oseen vortex and doublet exhibit more complicated dynamics.