Modelling The Pressure Hessian in Turbulence Through Tensor Function Representation Theory

One of the main challenges in Lagrangian modelling of velocity gradients in isotropic turbulence is to capture the non-local effects of the pressure Hessian and viscous diffusion. In statistical models of the velocity gradient tensor, the non-local effects give rise to unclosed conditional averages, making the statistical framework ideal for tensor function representation theory. By employing data from direct numerical simulation (DNS) of incompressible, statistically steady and isotropic turbulence, we analyze the mean non-local pressure Hessian conditional on the local velocity gradient tensor configuration. We define a basis consisting of velocity gradients products and investigate the trend of the associated Hessian components as functions of the velocity gradient invariants. Results show that some of the Hessian components display a pronounced dependence on the gradient invariants, undergoing sharp variations in specific phase space regions. Thus, our analysis lays the foundation for improved dynamical velocity gradient models that are accurate across the whole phase space