Sparsification techniques for reduced order models of turbulent flows

The complexity of scale interactions, arising from the increasing number of dynamically active flow structures, is a well-known problem for the numerical modelling of high Reynolds number flows. Without doubts, this complexity is the main obstacle to the development of computationally affordable and physically interpretable models of complex flows. This research focuses on the nonlinear energy interactions across modes in reduced order Galerkin models of turbulent flows demonstrating a novel approach to automatically identify relevant interactions. This work is motivated by the key observation that, in the dynamics of high Reynolds number flows, not all the interactions have the same contribution to the energy transfer between flow structures. With the proposed work, we aim to develop a set of techniques to systematically select the dominant interactions in Galerkin models of turbulent flows, therefore identifying dominant triadic interactions. In the present work, two different approaches have been developed. First, a regression-based approach where the relevant interactions are identified a posteriori according to their relative strength. Second, an a priori approach, where a new set of basis functions, encoding the sparsity features of the flow, is generated. The key aspect of the latter approach is that the reduced-order model obtained by Galerkin projection onto the subspace spanned by the basis has sparse matrix coefficients without the need for any a posteriori evaluation. Both approaches have been tested on a set of flow configurations of increasing complexity. Results show that both approaches can identify the subset of dominant interactions preserving their physics throughout the sparsification process. In addition, further analysis showed that the a priori sparsification method preserves better the physics of triadic interactions, resulting in a better long term time stability and, therefore, should be preferred. Looking into the future, to scale up the a priori methodology to a more complex configuration some aspects need to be further investigated such as the role of the initial guess on the uniqueness of the result and its properties.