A sparse optimal closure for a reduced-order model of wall-bounded turbulence.

In the present study, a set of physics-informed and data-driven approaches are examined towards the development of an accurate reduced-order model for a fully-developed turbulent plane Couette flow. Based on the utilisation of the proper orthogonal decomposition (POD) modes, a focus is given on the development of a reduced-order model where the number of the POD modes is not large enough to cover the full dynamics of the given turbulent state, the situation directly relevant to the reduced-order modelling for turbulent flows. Starting from the conventional Galerkin projection approach ignoring the truncation error, three approaches enhanced by both physics and data are examined: 1) sparse regression of the POD-Galerkin dynamics; 2) Galerkin projection with an empirical eddy viscosity model; 3) Galerkin projection with an optimal eddy viscosity obtained from a newly-proposed sparse regression. The sparse regression of the POD-Galerkin dynamics is found to result in an unsuccessful reduced-order model with the solution blow-up due to the too-small number of POD modes to resolve the given chaotic dynamics. While the unsatisfactory performance of the Galerkin-projection-based model with an empirical eddy viscosity is observed, the newly proposed approach, which combines the concept of optimal eddy-viscosity closure with a sparse regression, accurately approximates the chaotic dynamics. This is demonstrated with the mean and time scale of the POD mode amplitudes as well as with first- and second-order turbulence statistics.