A High-Order Navier-Stokes Solver for Viscous Toroidal Flows

This thesis details our work in the development and testing of highly efficient solvers for the Navier-Stokes problem in simple toroidal coordinates in three spatial dimensions. In particular, the domain of interest in this work is the region occupied by a fluid between two concentric toroidal shells. The study of this problem was motivated in part by extensions of the study of Taylor-Couette instabilities between rotating cylindrical and spherical shells to toroidal geometries. We note that at higher Reynolds numbers, Taylor-Couette instabilities in cylindrical and spherical coordinates are essentially three dimensional in nature, which motivated us to design fully three-dimensional solvers with a OpenMP parallel numerical implementation suitable for a multi-processor workstation.

We approach this problem using two different time stepping procedures applied to the so-called Pressure Poisson formulation of the Navier-Stokes equations. In the first case, we develop an ADI-type method based on a finite difference formulation applicable for low Reynolds number flows. This solver was more of a pilot study of the problem formulated in simple toroidal coordinates. In the second case - the main focus of our thesis - our main goal was to develop a spectral solver using an explicit fourth order Runge-Kutta time stepping, which is appropriate to the higher Reynolds number flows associated with Taylor-Couette instabilities.

Our spectral solver was developed using a high order Fourier representation in the angular variables of the problem and a high order Chebyshev representation in the radial coordinate between the shells. The solver exhibits super-algebraic convergence in the number of unknowns retained in the problem. Applied to the Taylor-Couette problem, our solver has allowed us to identify (for the first time, in this thesis) highly-resolved Taylor-Couette instabilities between toroidal shells. As we document in this work, these instabilities take on different configurations, depending on the Reynolds number of the flow and the gap width between shells, but as of now, all of these instabilities are essentially two dimensions. Our work on this subject continues, and we confident that we will uncover three-dimensional instabilities that have well-known analogues in the cases of cylindrical and spherical shells.

Lastly, a separate physical problem we examine is the flow between oscillating toroidal shells. Again, our spectral solver is able to resolve these flows to spectral accuracy for various Reynolds numbers and gap widths, showing surprisingly rich physical behaviour. Our code also allows us to document the torque required for the oscillation of the shells, a key metric in engineering applications. This problem was investigated since this configuration was recently proposed as a mechanical damping system.