各向异性压力Hessian的规范对称性与降维
Gauge symmetry and dimensionality reduction of the anisotropic pressure Hessian
AI总结：
本研究针对湍流中难以建模的各向异性压力Hessian，提出其在速度梯度本征标架下的规范对称性，借此实现三维压力Hessian的秩约简，发现其动力学局限于二维流形且有强对齐特性，为相关动力学研究与建模提供新视角。
摘要：
在拉格朗日参考系中分析流体速度梯度为研究湍流的小尺度动力学提供了一种富有洞见的途径，而通过考虑应变率张量本征标架下的方程可获得进一步的认识。速度梯度张量的动力学部分受各向异性压力Hessian支配，该量是速度梯度场的非局域泛函。这种各向异性压力Hessian在速度梯度动力学中发挥着关键作用，例如在阻止有限时间奇点方面，但由于其非局域性与复杂性，人们很难对其进行理解和建模。本研究将压力Hessian的一种规范对称性引入速度梯度的本征标架方程中，使得当规范被添加到原始压力Hessian时，本征标架变量的动力学保持不变。随后我们利用这一规范对称性对三维各向异性压力Hessian进行秩约简，值得注意的是，该操作在流场的任意位置都可实现。新引入的秩约简各向异性压力Hessian的动力学活动被限制在三维流场的二维流形上，并且相对于应变率本征标架和涡量矢量表现出显著的对齐特性。这种降维效应连同强烈的优先对齐特性，为理解和建模各向异性压力Hessian在三维流场中的作用提供了新的动力学洞见。
英文摘要：
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.
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