The mathematical description of the motion of fluids such as water and air, when the effects of viscosity can be neglected, is given by the Euler equation. A powerful and completely equivalent description of the fluid motion is given by the vorticity equation, which describes the evolution of the vorticity field, and is mathematically simpler due to the absence of a complicated pressure term. We may mathematically define vortices inspired by coherent structures in real-world flows, and use the vorticity equation to study their dynamics. An important form of vortex dynamics theory is reduced-order modeling in two spatial dimensions (2D), which considers finite dimensional solutions of the vorticity equation while simultaneously preserving the essential physics of the flow.
The theory of compressible vortex dynamics is much less developed when compared to incompressible vortex dynamics. A hollow vortex is a vortex with a constant pressure core and a non-zero circulation around it, with a finite velocity field everywhere. While the lack of any singularities in its velocity field allows its use in the study of compressible vortex dynamics, the problem is mathematically challenging because of the presence of free boundaries. The modern theory of analytic functions defined on multiply connected circular domains, allied with conformal mapping methods, makes possible exact solutions for such free-boundary problems.
In this project, we will study compressible vortex wakes by modeling the vortices as hollow vortices: A vortex wake is a periodic arrangement of vortices that forms behind an object moving in 2D, over a wide range of Reynolds numbers. The properties of the vortex wake determine important physical properties of the flow, such as the drag and lift forces acting on the body producing the wake. The effect of compressibility can be determined by using perturbation theory on a corresponding incompressible solution which is known. Special functions called Schottky-Klein prime functions, that are defined on multiply connected domains, will be utilized for the development of the mathematical theory.