Symmetry transformation and dimensionality reduction of the anisotropic pressure Hessian

The dynamics of the velocity gradient tensor in turbulence is governed in part by the anisotropic pressure Hessian, which is a non-local functional of the velocity gradient field. This anisotropic pressure Hessian plays a key dynamical role, for example in preventing finite-time singularities, but it is difficult to understand and model due to its non-locality and complexity. In this work a symmetry transformation for the pressure Hessian is introduced, such that when the transformation is applied to the original pressure Hessian, the dynamics of the invariants of the velocity gradients remains unchanged. We then exploit this symmetry transformation to perform a dimensional reduction on the three-dimensional anisotropic pressure Hessian, which, remarkably, is possible everywhere in the flow except on zero-measure sets. The dynamical activity of the newly introduced dimensionally reduced anisotropic pressure Hessian is confined to two-dimensional manifolds in the three-dimensional flow, and exhibits striking alignment properties with respect to the strain-rate eigenframe and the vorticity vector. The dimensionality reduction, together with the strong preferential alignment properties, leads to new dynamical insights for understanding and modelling the role of the anisotropic pressure Hessian in three-dimensional turbulent flows.

Duke Scholars

Author Andrew Bragg Civil and Environmental Engineering