Gauge symmetry and dimensionality reduction of the anisotropic pressure Hessian

TL;DR

This paper introduces a gauge symmetry to analyze the anisotropic pressure Hessian in turbulent flows, enabling a rank reduction that reveals new alignment properties and simplifies understanding of its complex, non-local dynamics.

Contribution

A gauge symmetry is introduced to reduce the anisotropic pressure Hessian's dimensionality, providing new insights into its role in turbulence dynamics.

Findings

The anisotropic pressure Hessian can be reduced to a rank-2 tensor everywhere in the flow.

The reduced Hessian's activity is confined to two-dimensional manifolds.

The reduced Hessian exhibits strong alignment with strain-rate eigenframe and vorticity.

Abstract

Analyzing the fluid velocity gradients in a Lagrangian reference frame provides an insightful way to study the small-scale dynamics of turbulent flows, and further insight is provided by considering the equations in the eigenframe of the strain-rate tensor. The dynamics of the velocity gradient tensor 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 role in the velocity gradient dynamics, 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 gauge symmetry for the pressure Hessian is introduced to the eigenframe equations of the velocity gradient, such that when the gauge is added to the original pressure Hessian, the dynamics of the eigenframe variables remain unchanged. We then exploit this gauge symmetry to perform a rank reduction on the three-dimensional anisotropic pressure Hessian, which, remarkably, is possible everywhere in the flow. The dynamical activity of the newly introduced rank-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 flows.