Investigating the dependence of the pressure Hessian process on the Cauchy-Green tensor in turbulent flows

Tracking the evolution of the velocity gradient tensor following a fluid element in a turbulent flow provides fundamental understanding of various nonlinear processes like energy cascading, intermittency, scalar mixing etc. Thus, development of simple dynamical models of the velocity gradient tensor in turbulent flows has been an active area of research. The time evolution equation of velocity gradient tensor has two unclosed terms - the pressure Hessian tensor and viscous Laplacian process. In the past, some attempts have been made to model the pressure Hessian tensor in terms of the Cauchy-Green tensor. While the predictions of such models have been indeed encouraging, more improvements are still expected. With the motivation to develop deeper insights into the relationship between the two tensors (the pressure Hessian and the Cauchy-Green tensors), in this work, we employ direct numerical simulation (DNS) database of homogeneous turbulence to examine the interdependence between these tensors. Specifically, we focus on the alignment tendencies of the eigenvector-systems of the two tensors. In the light of these findings, subsequently, we identify the shortcomings of the existing models and attempt to suggest improvements. This study is performed for both incompressible and compressible flow fields.