# Incorporating Antisymmetric Vorticity into Two-Dimensional Incompressible Symmetric Fluid Systems: A Study of Merging of Two Identical Vortices

Michael Skytte Petersen & Anders Rømer Gylling

Studenteropgave: Speciale

## Abstrakt

The core growth model and numerical solutions of the vorticity transport equation has similar merging times of fluid systems with two identical vortices, when Re < 10. When Re > 10, the vorticity transport equation has lower merging times compared to the core growth model. Andersen et al.  showed that the dimensionless merging time for the core growth model is \tau = 2 under certain conditions. Cerretelli and Williamson  splits the vorticity into a symmetric and antisymmetric vorticity, where the antisymmetric vorticity is responsible for the reduced merging time. We proved that the core growth model does not contain antisymmetric vorticity under the assumption that the individual terms are equal in size and placed as roots of unity. In this project we have studied two vortices with equal vortex strength and how the merging time changes by including antisymmetric vorticity to symmetric fluid systems. One-dimensional and two-dimensional systems are studied. We found that adding an odd-functioned and a centered Gaussian vorticity distribution to a one-dimensional symmetric vorticity system reduces the merging time. Adding an antisymmetric vorticity to the core growth model in a two-dimensional system does not. We tried to emulate the advection term in the vorticity transport equation by adding vorticity distributions with time-dependent strength, \zeta(x,t) in one-dimensional systems and in two-dimensional systems. In the two-dimensional system, we let \zeta(x,t) be six different configurations of antisymmetry, and three different configurations of asymmetry. Adding the antisymmetric distributions to the diffusion equation did not reduce the merging times compared to the core growth model, while adding the asymmetric did in two of the three cases, although the topology of all the systems were unlike those produced by the vorticity transport equation. We found that the velocity field generated by the added vorticity did not affect the merging of the system. The vorticity and velocity field did not seem to be coupled in our models. We assume that the reduced merging times are caused by the \zeta(x,t)-function producing more vorticity in a neighborhood of the position of merging.

Uddannelser Fysik, (Bachelor/kandidatuddannelse) KandidatMatematik, (Bachelor/kandidatuddannelse) Kandidat Engelsk 10 sep. 2018 Jesper Schmidt Hansen

### Emneord

• core growth model
• vortex
• vorticity
• Lamb-Oseen model
• diffusion equation
• modelling
• vortices