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

对于在相对平坦的表面上高速流动的流体，我们能否预测放置在表面附近的小规模三维结构/扰动的下游响应？例如，注入高速流动的类似流体的窄喷流通过表面或窄区域的“粗化”表面。在这些情况下的新颖之处在于，为了模拟高速流动的这些情况，需要在物理推导的控制方程中重新包含传统上在较大长度尺度上被忽略的项，类似地，在表面附近产生的流动结构可以（在某些情况下）有利于某些应用（例如，通过延迟局部流动分离）。虽然过去已经对这种结构的周期阵列的高速流动进行了一些关注，但在这里，我们解决了更具挑战性的问题，即空间孤立的扰动区域。以前已经考虑过类似问题的直接数值模拟，但我们的方法是研究计算成本低得多的边界层/区域公式及其预测最终流动响应的能力。这些方程的数学特征与传统的公式有些不同，从扰动点到自由流动的代数衰减，需要一些小心/分析才能获得精确的结果流及其稳定性的公式。学生将制定并解决一组简化的偏微分方程，这些方程在高速极限下模拟这些流动，通过应用于计算结果的渐近方法分析和分类行为。流的稳定性将通过应用于（双全局）特征值问题的计算和分析来解决。

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.