Sivaguru Sritharan, Ph.D. (Committee Co-Chair); Qun Li, Ph.D. (Committee Co-Chair); Qingbo Huang, Ph.D. (Committee Member); Mohammed Sulman, Ph.D. (Committee Member)
Doctor of Philosophy (PhD)
This work contains the derivation and type analysis of the conical Euler and Ideal Magnetohydrodynamic equations. The 3 dimensional Euler equations and the Ideal MHD equations with Powell source terms, subject to the assumption that the solution is conically invariant, are projected onto a unit sphere using tools from tensor calculus. Conical flows provide valuable insight into supersonic and hypersonic flow past bodies, but are simpler to analyze and solve numerically. Previously, work has been done on conical inviscid flows governed by the compressible Euler equations with great success. It is known that some flight regimes involve flows of ionized gases, and thus there is motivation to extend the study of conical flows to the case where the gas is electrically conducting. This thesis shows that steady conical flows for these cases do exist mathematically and that the governing systems of partial differential equations are of mixed type. Throughout the domain they can be either hyperbolic or elliptic depending on the solution. A numerical scheme is also developed to solve the conical Euler and Ideal Magnetohydrodynamic equations. Special care had to be taken in developing the method because these equations contain geometric source terms which account for the fact that they are defined on a curved surface. In order for a numerical method to accurately capture the behavior of the system it is solving, any source terms must be discretized in a way which preserves the appropriate behavior. For a partial differential equation which has been formulated on a curved manifold using tensor calculus, it is desirable for the discretization to preserve the tensorial transformation relationships. Such discretizations are presented in this work, and a numerical method involving them is developed and demonstrated.
Department or Program
Department of Mathematics and Statistics
Year Degree Awarded
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