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Analyzing the fluid velocity gradients in a Lagrangian reference frameprovides an insightful way to study the small-scale dynamics of turbulentflows, and further insight is provided by considering the equations in theeigenframe of the strain-rate tensor. The dynamics of the velocity gradienttensor is governed in part by the anisotropic pressure Hessian, which is anon-local functional of the velocity gradient field. This anisotropic pressureHessian plays a key role in the velocity gradient dynamics, for example inpreventing finite-time singularities, but it is difficult to understand andmodel due to its non-locality and complexity. In this work a gauge symmetry forthe pressure Hessian is introduced to the eigenframe equations of the velocitygradient, such that when the gauge is added to the original pressure Hessian,the dynamics of the eigenframe variables remain unchanged. We then exploit thisgauge symmetry to perform a rank reduction on the three-dimensional anisotropicpressure Hessian, which, remarkably, is possible everywhere in the flow. Thedynamical activity of the newly introduced rank-reduced anisotropic pressureHessian is confined to two dimensional manifolds in the three dimensional flow,and exhibits striking alignment properties with respect to the strain-rateeigenframe and the vorticity vector. The dimensionality reduction, togetherwith the strong preferential alignment properties, leads to new dynamicalinsights for understanding and modelling the role of the anisotropic pressureHessian in three-dimensional flows.

Gauge symmetry and dimensionality reduction of the anisotropic pressure Hessian