The Vorticity Topology of the Core Growth Model

Marc John Bordier Dam

Studenteropgave: Speciale

Abstrakt

It is important to understand vortex merging to be able to reduce drag when vortices are present. Vortex merging in two dimensions is usually studied using the point vortex model, which is a simple non-viscous model of vortices. In this project we have studied the core growth model, which is an extension of the point vortex model that takes viscosity into account, and therefore displays some viscous phenomena, which are missing in the point vortex model. Our focus has been on classifying the topology of the vorticity contours through the critical points of the vorticity. This is inspired by [Andersen et al.] where the classification is carried out for the case of two vortices. This project tries to extend the results to the case of three vortices. In the symmetric case, where the three vortices are identical and evenly spaced on a circle, we determine the trajectories of the critical points and thereby the topology of the vorticity contours. In the asymmetric case, where one of the vortices is of a different vortex strength than the other two, we determine the trajectories of the critical points along the axis of symmetry. We do not attempt to investigate the case of three vortices of three different vortex strengths or the case where the vortices are not evenly spaced on a circle. Along the way we present a result on bifurcations of the vorticity at the center of vorticity in the symmetric case of $N \geq 3$ identical vortices, evenly spaced on a circle. This results states that the center of vorticity is a critical point for all $t > 0$, and that the only bifurcation of the center of vorticity happens at $\tau =\frac{4 \nu}{d^2} t = 1$, where $\nu$ is the viscosity and $d$ is the radius of the circle. We also briefly consider the case of four identical vortices evenly spaced on a circle, for which we determine the trajectories of the critical points. Finally, we compare the above results about the core growth model with numerical solutions of the vorticity transport equation, where we see excellent agreement at lower Reynolds numbers, and deviations at higher Reynolds numbers.

Uddannelser Fysik, (Bachelor/kandidatuddannelse) MasterMatematik, (Bachelor/kandidatuddannelse) Kandidat Engelsk 2018 Morten Andersen