This research is concerned with the stability of a two-dimensional, electromagnetically forced, zonal flow on a circular domain. Flows like these are found in nature (e.g. shear flow in the atmosphere, Jovian disk) and experiment (e.g. plasma flow in a Fusion reactor) and a requirement for experiments is often that these types of flows remain stable and axi-symmetric. A numerical method is developed based on a spectral expansion into an infinite system of ordinary differential equations for velocity functions resulting from a Stokes eigenvalue problem. The system is truncated to gain a finite-dimensional system which is useful for computations of both equilibrium flows and strongly disturbed flows. Numerical results are compared to both finite difference method results and analytical results for the equilibrium basic flow. Both linear and nonlinear stability are explored for the Navier-Stokes equations on the circular domain and for the system of ordinary differential equations. Differences in stability and the evolution of perturbations are explained on the basis of discrepancies between infinite-dimensional partial differential equations like the Navier-Stokes equations and a finite-dimensional system of ordinary differential equations resulting from a Galerkin truncation. On the basis of both stability analyses a control system is developed which stabilizes the system of ordinary differential equations to stay in a desired equilibrium. It is argued that this control system is also usable for the control of the Navier-Stokes equations.
|Date of Award
|31 Jul 2013
|Hans J. Zwart (Supervisor 1), L.P.J. Kamp (Supervisor 2) & M. Lauret (External coach)