Machine-learning energy-preserving nonlocal closures for turbulent fluid flows and inertial tracers

We formulate a data-driven, physics-constrained closure method for coarse-scale numerical simulations of turbulent fluid flows. Our approach involves a closure scheme that is non-local both in space and time, i.e. the closure terms are parametrized in terms of the spatial neighborhood of the resolved quantities but also their history. The data-driven scheme is complemented with a physical constraint expressing the energy conservation property of the nonlinear advection terms. We show that the adoption of this physical constraint not only increases the accuracy of the closure scheme but also improves the stability properties of the formulated coarse-scale model. We demonstrate the presented scheme in fluid flows consisting of an incompressible two-dimensional turbulent jet. Specifically, we first develop one-dimensional coarse-scale models describing the spatial profile of the jet. We then proceed to the computation of turbulent closures appropriate for two-dimensional coarse-scale models. Training data are obtained through high-fidelity direct numerical simulations (DNS). We also showcase how the developed scheme captures the coarse-scale features of the concentration fields associated with inertial tracers, such as bubbles and particles, carried by the flow but not following the flow. We thoroughly examine the generalizability properties of the trained closure models for different Reynolds numbers, as well as, radically different jet profiles from the ones used in the training phase. We also examine the robustness of the derived closures with respect to the grid size. Overall the adoption of the constraint results in an average improvement of 26% for one-dimensional closures and 29% for two-dimensional closures, being notably larger for flows that were not used for training.