Modeling the pressure-Hessian tensor using deep neural networks

The understanding of the dynamics of the velocity gradients in turbulent flows is critical to understanding various nonlinear turbulent processes. Several simplified dynamical equations have been proposed earlier that model the Lagrangian velocity gradient evolution equation. A robust model for the velocity gradient evolution equation can ultimately lead to the closure of the system of equations in the Lagrangian probability distribution function method. The pressure Hessian and the viscous Laplacian are the two important processes that govern the Lagrangian evolution of the velocity gradients. These processes are nonlocal in nature and unclosed from a mathematical point of view. The recent fluid deformation closure model (RFDM) has been shown to retrieve excellent statistics of the viscous process. However, the pressure Hessian modeled by the RFDM has various physical limitations. In this work, we first demonstrate such limitations of the RFDM. Subsequently, we employ a tensor basis neural network (TBNN) to model the pressure Hessian using the information about the velocity gradient tensor itself. Our neural network is trained on high-resolution data obtained from direct numerical simulation (DNS) of isotropic turbulence at the Reynolds number of 433. The predictions made by the TBNN are evaluated against several other DNS datasets. Evaluation is made in terms of the alignment statistics of the pressure-Hessian eigenvectors with the strain-rate eigenvectors. Our analysis of the predicted solution leads to the finding of ten unique coefficients of the tensor basis of the strain-rate and the rotation-rate tensors, the linear combination of which is used to accurately capture key alignment statistics of the pressure-Hessian tensor.