Spectral Methods in Polar Coordinates with AN Application to the Stability of a Trailing Vortex.

In cylindrical and spherical coordinates the coordinate singularity can decrease the accuracy or computational efficiency of the spectral method. The problem arises due to the increased resolution near the coordinate singularity. For time dependent advection problems it becomes a stiffness problem which forces one to choose an unusually small time step compared to other spectral methods which do not have coordinate singularities. In this dissertation we present two sets of basis functions which inherently do not suffer from any stiffness problem. The first set of basis functions consists of the eigenfunctions of a singular Sturm-Liouville equation and is suited for a spectral method on the unit disk in polar coordinates. The basis functions are polynomials. The second set of basis functions is algebraically mapped associated Legendre functions whose domain extends from the origin to infinity in the radial direction of a cylindrical coordinates. The basis functions are rational functions. These basis sets satisfy simple recurrence relations for important operations such as the multiplication of some elementary functions, differentiation, and the application of the Laplacian and Helmholtz operators. The forward and backward application of these recurrence relations can be made very efficient. We illustrate these new methods by some examples. The examples include the treatment of a vector field by its toroidal and poloidal components. As an application of practical importance, we apply the rational basis function method to simulate the nonlinear development of linearly unstable modes of an airplane trailing vortex with axial flow. It is found that the nonlinear development of the linearly unstable modes depends strongly on the swirl parameter q which is the ratio of the magnitude of swirling motion to that of the axial flow of the vortex. For q = 0.2 and q = 0.6, the vortex core breaks up significantly and the mean core radius becomes a few times larger than the unperturbed core radius. If q = 1.0, the mean core radius does not become larger than the unperturbed case even though instability develops initially.