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

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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.