Asymptotic and numerical approaches to three-dimensional flows and their stability

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For high-speed fluid flows over a relatively flat surface, can we predict the response downstream of a small scale three-dimensional structure/disturbance placed near the surface? For example, the injection into a high speed flow of a narrow jet of similar fluid through the surface, or a narrow region of "roughened" surface. The novelty in these cases is that to model these situations for high speed flow requires the re-inclusion of terms in the physically-derived governing equations that are traditionally ignored at larger length scales, similarly the flow structures produced near the surface can (in some cases) be beneficial to some applications (for example, by delaying flow separation locally). Although some attention has been applied to high-speed flow over periodic arrays of such structures in the past, here we tackle the more challenging problem of spatially isolated regions of disturbance. Direct numerical simulation of similar problems has been considered previously, but our approach is to investigate the computationally much less expensive boundary-layer/region formulation and its ability to predict the resulting flow response. The mathematical character of these equations is somewhat different from the traditional formulation, with algebraic decay away from a disturbance site into the free stream flow and some care/analysis is needed to achieve an accurate formulation of the resulting flow and its stability properties.The student will formulate and solve a reduced set of partial-differential equations that model these flows in the high-speed limit, analyse and classify the behaviour through asymptotic methods applied to computational results. The stability of the flow will be addressed by computation and analysis applied to (bi-global) eigenvalue problems.