A physics-informed deep learning closure for Lagrangian velocity gradient evolution

The pressure Hessian tensor is entangled with the inherent nonlinearity and nonlocality of turbulence; thus, it is of crucial importance in modeling the Lagrangian evolution of the velocity gradient tensor (VGT). In the present study, we introduce the functional modeling strategy into the classic structural modeling strategy to model the pressure Hessian tensor based on deep neural networks (DNNs). The pressure Hessian tensor and its contributions to the VGT evolution are set as, respectively, the structural and functional learning targets. An a priori test shows that the present DNN-based model accurately establishes the mapping from the VGT to the pressure Hessian tensor and adequately models the physical effect of the pressure Hessian tensor on VGT invariants. An a posteriori test verifies that the present model reproduces well the principal features of turbulence-like skewness and vorticity strain-rate alignments obtained via direct numerical simulations. Importantly, the flow topology is accurately predicted, particularly for the strain-production-dominant regions in the invariant space. Moreover, an extrapolation test shows the generalization ability of the present model to higher Reynolds number flows that have not been trained.