Deep neural network framework for modeling pressure hessian tensor in incompressible turbulent flows

The dynamics of the velocity gradient tensor is crucial in understanding various turbulent nonlinear processes. One of the unclosed terms in the velocity gradient evolution equation for incompressible flows is the anisotropic part of the pressure Hessian tensor. This research introduces a novel approach to model this tensor using machine learning.

Two different neural networks are trained using the DNS data of incompressible decaying turbulence. The first neural network is designed to predict the more universal behavior of the normalized tensor, focusing on the alignment tendencies of its eigen directions. The second neural network models the tensor's intermittent magnitude. This separation allows for the use of more optimal loss functions focused to the specific outputs required from each neural network.

The combined output from these two neural networks is evaluated across different Reynolds numbers and different kinds of flows. The model evaluation is further extended to the conditioned compressible flows where the locally conditioned fluid elements are behaving like incompressible flows. The performance of the proposed model is compared with that of existing conventional as well as the neural network based  models.  Indeed the predictions are better than the existing models.