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UBC Theses and Dissertations

The application of vorticity methods to rotating fluid flows McMillan, David G.


Studies of Earth's core dynamics often require a computational method which can account for non-linear effects and non-periodic time dependence. One important application involves the large scale, possibly chaotic, fluid motion associated with a past resonant tidal forcing of the free core nutation. This thesis explores the utility of vorticity methods for modelling contained rotating fluids, such as the Earth's core. Establishing this method in the general context of rotating fluids is the first step in the development of a non-linear, three dimensional, time dependent model of Earth's fluid core. Conventionally, vorticity methods have been applied to incompressible fluid flow in infinite domains with small, finite regions of non-zero vorticity. The capability of approximating solutions to non-linear, three dimensional, time dependent, inviscid fluid flow problems suggests that these algorithms may also be well suited to flows in finite domains under uniform rotation. Physically, the method under consideration involves grid-free tracking of fluid particles which carry distributions, or 'blobs,' of vorticity, each with a prescribed strength. The strengths and positions of the blobs determine the velocity field through the Biot-Savart law. The fluid particles, along with their vorticities, are advected by the velocity field. In addition, the vorticity of each blob is altered by the strain associated with the velocity field and must be updated at each time step. The method of solution reduces to a set of first order, ordinary differential equations in time for the Lagrangian displacement of the fluid particles and their corresponding vorticity strengths. The ODEs are advanced by standard integration routines. This work is a presentation of the development of the vorticity method algorithm for rotating fluids. In addition to the development of the algorithm, several new computational methods are described. The concept of effective vorticity is introduced as a means of relating known physical quantities to computational results. In order to satisfy inviscid boundary conditions in an efficient manner, approximate methods using image particles are developed and incorporated into two of the test cases. Quick initialization of vorticity fields is made possible by the formulation of an approximate linear system of equations. The modelling of a uniformly rotating fluid requires the initialization of a uniform vorticity field. This procedure is decomposed into two steps, increasing efficiency while creating a reasonable initial vorticity field. The implementation of the algorithm is verified by comparison with theoretical approximations. The first test cases involve the self advection of a thin vortex ring and the interaction of two vortex rings. These examples confirm the operation of the algorithm without introducing the issue of boundary conditions. Solid boundaries are introduced in the problem of standing waves on a bounded vortex filament. Finally, the problem of a contained fluid with a uniform vorticity distribution is examined by modelling inertial modes in a rotating cylinder.

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