# Vortex cusps

## TL;DR

This paper models vortex sheet cusps using a reduced Birkhoff-Rott equation, revealing conditions for their existence, and highlights the importance of accurate numerical schemes to avoid spurious results.

## Contribution

It derives a reduced model for vortex cusp formation, providing formulas for cusp exponents and identifying conditions for their existence based on similarity exponent and strain.

## Key findings

- Piecewise quadratic vortex sheet approximations agree with the model.
- Linear schemes produce spurious results and violate mass conservation.
- Vortex cusps only exist under certain conditions related to circulation and strain.

## Abstract

We consider pairs of self-similar 2d vortex sheets forming cusps, equivalently single sheets merging into slip condition walls, as in classical Mach reflection at wedges. We derive from the Birkhoff-Rott equation a reduced model yielding formulas for cusp exponents and other quantities as functions of similarity exponent and strain coefficient. Comparison to numerics shows that piecewise quadratic and higher approximation of vortex sheets agree with each other and with the model. In contrast piecewise linear schemes produce spurious results and violate conservation of mass, a problem that may have been undetected in prior work for other vortical flows where even point vortices were sufficient. We find that vortex cusps only exist if the similarity exponent is sufficiently large and if the circulation on the sheet is counterclockwise (for a sheet above the wall with cusp opening to the right), unless a sufficiently positive strain coefficient compensates. Whenever a cusp cannot exist a spiral-ends jet forms instead; we find many jets are so narrow that they appear as false cusps.