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We formulate a data-driven, physics-constrained closure method forcoarse-scale numerical simulations of turbulent fluid flows. Our approachinvolves a closure scheme that is non-local both in space and time, i.e. theclosure terms are parametrized in terms of the spatial neighborhood of theresolved quantities but also their history. The data-driven scheme iscomplemented with a physical constrain expressing the energy conservationproperty of the nonlinear advection terms. We show that the adoption of thisphysical constrain not only increases the accuracy of the closure scheme butalso improves the stability properties of the formulated coarse-scale model. Wedemonstrate the presented scheme in fluid flows consisting of an incompressibletwo-dimensional turbulent jet. Specifically, we first develop one-dimensionalcoarse-scale models describing the spatial profile of the jet. We then proceedto the computation of turbulent closures appropriate for two-dimensionalcoarse-scale models. Training data are obtained through high-fidelity directnumerical simulations (DNS). We also showcase how the developed scheme capturesthe coarse-scale features of the concentration fields associated with inertialtracers, such as bubbles and particles, carried by the flow but not followingthe flow. We thoroughly examine the generalizability properties of the trainedclosure models for different Reynolds numbers, as well as, radically differentjet profiles from the ones used in the training phase. We also examine therobustness of the derived closures with respect to the grid size. Overall theadoption of the constraint results in an average improvement of 26% forone-dimensional closures and 29% for two-dimensional closures, being notablylarger for flows that were not used for training.

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