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VORTEX LATTICE THEORY: A LINEAR ALGEBRA APPROACH
by
George C. Chamoun
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(AEROSPACE AND MECHANICAL ENGINEERING)
August 2008
Copyright 2008 George C. Chamoun
Object Description
| Title | Vortex lattice theory: a linear algebra approach |
| Author | Chamoun, George C. |
| Author email | chamoun@usc.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Aerospace & Mechanical Engineering (Dynamics & Control) |
| School | Viterbi School of Engineering |
| Date defended/completed | 2008-05-28 |
| Date submitted | 2008 |
| Restricted until | Unrestricted |
| Date published | 2008-07-28 |
| Advisor (committee chair) | Newton, Paul |
| Advisor (committee member) |
Kukavica, Igor Browand, Fred Friedlander, Susan Ghanem, Roger |
| Abstract | Vortex lattices are prevalent in a large class of physical settings that are characterized by different mathematical models. We present a coherent and generalized Hamiltonian fluid mechanics-based formulation that reduces all vortex lattices into a classic problem in linear algebra for a non-normal matrix A. Via Singular Value Decomposition (SVD), the solution lies in the null space of the matrix (i.e., we require nullity(A) > 0) as well as the distribution of its singular values. We demonstrate that this approach provides a good model for various types of vortex lattices, and makes it possible to extract a rich amount of information on them. The contributions of this thesis can be classified into four main points. The first is asymmetric equilibria. A "Brownian ratchet" construct was used which converged to asymmetric equilibria via a random walk scheme that utilized the smallest singular value of A. Distances between configurations and equilibria were measured using the Frobenius norm . F and 2-norm . 2, and conclusions were made on the density of equilibria within the general configuration space. The second contribution used Shannon Entropy, which we interpret as a scalar measure of the robustness, or likelihood of lattices to occur in a physical setting. Third, an analytic model was produced for vortex street patterns on the sphere by using SVD in conjunction with expressions for the center of vorticity vector and angular velocity. Equilibrium curves within the configuration space were presented as a function of the geometry, and pole vortices were shown to have a critical role in the formation and destruction of vortex streets. The fourth contribution entailed a more complete perspective of the streamline topology of vortex streets, linking the bifurcations to critical points on the equilibrium curves. |
| Keyword | vortex; Hamiltonian; lattice; SVD; fluid mechanics |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Type | texts |
| Legacy record ID | usctheses-m1418 |
| Rights | Chamoun, George C. |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | http://www.usc.edu/isd/libraries/services/ask_a_librarian/email/ |
| Filename | etd-Chamoun-20080728 |
| Archival file | uscthesesreloadpub_Volume40/etd-Chamoun-20080728.pdf |
Description
| Title | Page 1 |
| Full text | VORTEX LATTICE THEORY: A LINEAR ALGEBRA APPROACH by George C. Chamoun A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (AEROSPACE AND MECHANICAL ENGINEERING) August 2008 Copyright 2008 George C. Chamoun |
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