Gauge symmetry and dimensionality reduction of the anisotropic pressure Hessian

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.

References in corpus (5)

• Lagrangian dynamics and statistical geometric structure of turbulence
• Modeling the pressure Hessian and viscous Laplacian in Turbulence: comparisons with DNS and implications on velocity gradient dynamics
• Deformation statistics of sub-Kolmogorov-scale ellipsoidal neutrally buoyant drops in isotropic turbulence
• Large-Scale Pattern Formation in the Presence of Small-Scale Random Advection
• Turbulence on hyperbolic plane: the fate of inverse cascade

Cited by in corpus (7)

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• Exploring the turbulent velocity gradients at different scales from the perspective of the strain-rate eigenframe
• Capturing Velocity Gradients and Particle Rotation Rates in Turbulence
• Role of pressure in generation of intense velocity gradients in turbulent flows
• Understanding the effect of Prandtl number on momentum and scalar mixing rates in neutral and stably stratified flows using gradient field dynamics
• Extraction of invariant manifolds and application to turbulence with a passive scalar
• Physics-Informed Machine Learning of the Lagrangian Dynamics of Velocity Gradient Tensor