Analysis of unsteady turbulent flows through Proper Orthogonal Decomposition

In many fields, like fluid dynamics, a large part of the physical systems is described by partial differential equations. They could have prohibitively high dimensions and involve a huge number of unknowns. This makes the computational costs and the memory storage capacity required unaffordable. Therefore, during the last years, some researchers worked at finding mathematical tools of Model Order Reduction (MOR), with the goal of reducing the dimension of the systems thus reducing the computational effort, without simplifying the underlying physics. The key idea is to replace generic approximation spaces by problem specific reduced basis such that they contain the main features of the dynamic systems. Then, the full order model can be projected onto the reduced basis to obtain a lower order dimension one. The present work, developed in collaboration with Sauber Motorsport, is focused on one of the most famous method to find the optimal reduced basis, the Proper Orthogonal Decomposition (POD). The goal of POD is to find an optimal orthogonal reduced basis, through finite-dimensional singular values problem, obtained by post-process of a set of numerical or experimental data available. In this thesis, POD is applied to two test cases performed in OpenFOAM, showing how POD modes can be used to obtain a compact representation of the solution. The first example is a simple laminar flow around a circular cross-section cylinder and, the second one, is a Detached Eddy Simulation (DES) on a typical Sauber F1 car. In both cases, the dataset is the fluctuating part of a flow field, obtained subtracting the time average from the instantaneous field. As methods like POD are based on Singular Values Decomposition (SVD) of a matrix that comes from the data, they can have, for large datasets, prohibitively expensive memory and floating-point operation costs. In this work, to overcome this limitation, a revisiting of the classic POD version has been developed in Python, named Incremental POD (IPOD). Based on Brand's incremental SVD algorithm, the key idea is to update an existing initial SVD, obtained with a small dataset, as far as new data are collected. This approach leads to many advantages. For example, the computational costs and the memory required are reduced, as there is no need to store the full-time history. Moreover, Incremental POD can be applied along the simulation updating POD modes and singular values at run time. Indeed, IPOD results could be used real time for decision on the simulation, for example stopping and convergence criteria. IPOD is applied as the simulation advances, considering one data at a time and, therefore, assuming that neither the previous nor the following snapshots are available, it is impossible to prior subtract the ensemble average in order to obtain the fluctuating flow fields. Therefore, it is shown that the method can be applied directly on the instantaneous fields, without affecting the calculation of the POD modes but only the interpretation. Finally, the attention is focused on simulation convergence analysis. Judging when a numerical solution of periodic unsteady flows is fully convergent could be challenging. Therefore, a new convergence criterion based on the most energetic POD modes and singular values has been presented and applied on the two test cases considered in this work.