My research interests involve permutations of these topics: point vortices, hollow vortices, compressible flows, free boundary problems, and exact solutions of the Euler equations. Brief descriptions of some of the research projects follow, with links to corresponding publications. If you see anything that interests you, feel free to get in touch!

For these and other projects, I collaborate with (in no particular order) Miles H. Wheeler, Adrian Constantin, Darren G. Crowdy, Mark A. Stremler, Rhodri Nelson, Stefan G. Llewellyn Smith and Takashi Sakajo.

Point vortex equilibria

Point vortex equilibria are patterns of point vortices which do not move relative to one another. We have discovered a transformation between rational functions that takes a given stationary point vortex equilibrium and creates a new, distinct stationary point vortex equilibrium [10, DOI ]. Selected examples of equilibria obtained via this transformation are shown here, as (continuous) functions of an arbitrary parameter. Blue diamonds and black disks are positive and negative point vortices; their size gives the vortex strength.

This procedure can sometimes be iterated to produce infinite hierarchies of equilibria. The first few stages for two such hierarchies are shown below.

(Use 'right click' and "View Image" for larger images.)

Compressible vortex dynamics

The Kármán vortex street is a widely observed feature of fluid flows. Various solutions, including the classic point vortex street model, are available for incompressible flows. For a compressible flow on the other hand, a point vortex is especially problematic due to negative densities appearing above a certain fluid speed. We can still consider a steadily translating Kármán point vortex street in a compressible flow and, after clarifying the meaning of a point vortex in equilibrium in such a flow, an exact expression for the speed of the street can be obtained [1, DOI ]. We also consider finite-area hollow vortex streets, and find that it can speed up or slow down relative to an incompressible street depending on the area of the vortices. The results depend on the geometric arrangement of the vortices, encoded in κ which is the horizontal to vertical separation of the vortices [2, DOI ].

Point vortex dynamics

Taking a geometrical point of view of three point vortex motion, we consider the evolution of the angles and circumradius of the vortex triangle. An autonomous system of equations for these variables can be derived [3, DOI ]. In the case of finite-time collapse of the vortices, this allows us to obtain formulas linking the Hamiltonian energy of the system with the time of collapse (or expansion) [4, DOI ].