The Conservative Pressure Hessian and the Free Fluid Particle Model

In this work we aim to understand the interplay between deformation and energy of a fluid particle in a turbulent flow, focusing on the role of the pressure Hessian. A new decomposition of the Hessian is proposed, defining a conservative Hessian which contains the stabilizing effects of the full Hessian on energy and deformation of fluid elements. This conservative Hessian is then used to construct models for the velocity gradient which allow controlling alignments and singularities.