Modelling pressure-Hessian from local velocity gradients information in an incompressible turbulent flow field using deep neural networks

The understanding of the dynamics of the velocity gradients in turbulent flows is critical to understanding various non-linear turbulent processes. The pressure-Hessian and the viscous-Laplacian govern the evolution of the velocity-gradients and are known to be non-local in nature. Over the years, several simplified dynamical models have been proposed that models the viscous-Laplacian and the pressure-Hessian primarily in terms of local velocity gradients information. These models can also serve as closure models for the Lagrangian PDF methods. The recent fluid deformation closure model (RFDM) has been shown to retrieve excellent one-time statistics of the viscous process. However, the pressure-Hessian modelled by the RFDM has various physical limitations. In this work, we first demonstrate the limitations of the RFDM in estimating the pressure-Hessian. Further, we employ a tensor basis neural network (TBNN) to model the pressure-Hessian from the velocity gradient tensor itself. The neural network is trained on high-resolution data obtained from direct numerical simulation (DNS) of isotropic turbulence at Reynolds number of 433 (JHU turbulence database, JHTD). The predictions made by the TBNN are tested against two different isotropic turbulence datasets at Reynolds number of 433 (JHTD) and 315 (UP Madrid turbulence database, UPMTD) and channel flow dataset at Reynolds number of 1000 (UT Texas and JHTD). The evaluation of the neural network output is made in terms of the alignment statistics of the predicted pressure-Hessian eigenvectors with the strain-rate eigenvectors for turbulent isotropic flow as well as channel flow. Our analysis of the predicted solution leads to the discovery of ten unique coefficients of the tensor basis of strain-rate and rotation-rate tensors, the linear combination over which accurately captures key alignment statistics of the pressure-Hessian tensor.