Application of Sum-of-squares of Polynomials Technique in Fluid Dynamics

这个博士项目有两个主要目标。首先，我们的目标是改进UODESys工具箱。特别是，我们的目标是提高工具箱的效率，并在X项上获得更紧的界，它出现在2中，因此在不确定动力系统中。提出了研究X项边界的不同方法。UODESys的主要缺点之一是使用低效的嵌套循环来实现某些计算。我们将尽可能用更有效的向量化代码取代嵌套循环，以加快不确定系统的推导。本研究的第二个主要目标是将SOS优化技术应用于特定的流体流动。这将通过首先推导出nse的有限维（截断的）伽辽金近似来完成。有限伽辽金基的选择将基于物理考虑。也就是说，有限维模型应该捕捉实际流体流动的所有物理特性。然后可以分析得到的ODE系统。我们将集中精力利用李雅普诺夫稳定性理论和SOS优化相结合的方法来确定ODE系统稳定解的最大雷诺数，从而使我们能够获得高于能量稳定极限的雷诺数的稳定极限。此外，对于各种流动，我们希望推导出湍流状态下流动特性的严格边界。如果需要，在这两个应用程序中都将使用UODESys来派生简化的和不确定的odessystem，以降低计算复杂性。

This PhD project will feature two primary objectives. Firstly, we aim to improve the UODESys toolbox. Inparticular, we aim to improve the efficiency of the toolbox and obtain tighter bounds on the X term, whichappears in 2 and hence in the uncertain dynamical system. It is proposed to investigate different methods ofbounding the X term.One of the main shortcomings of UODESys is that inefficient nested loops are used to implement certaincomputations. We will replace nested loops with more efficient vectorised code wherever possible, in order tospeed up the derivation of the uncertain system.The second primary objective of this research is to apply the SOS optimisation technique to specific fluidflows. This will be done by first deriving a finite-dimensional (truncated) Galerkin approximation to the NSEs.The finite Galerkin basis ui will be chosen based on physical considerations. Namely, the finite-dimensionalmodel should capture all the physical characteristics of the actual fluid flow. The resulting ODE system canthen be analysed. We will focus our efforts on determining the maximum Reynolds number for which a steadysolution of the ODE system is stable using a combination of Lyapunov stability theory and SOS optimisation,thus enabling us to obtain a stability limit on the Reynolds number that is higher than the energy stability limit.In addition, for various flows, we would like to derive rigorous bounds on flow characteristics in the turbulentregime. If required, in both of these applications UODESys will be used to derive a reduced and uncertain ODEsystem to reduce computational complexity.