发表机构
School of Physics, Nanjing University of Science and Technology（南京理工大学物理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文基于实Schur形式提出速度梯度张量的拉格朗日动力学理论，推导特征涡量模态及交换涡量算子的演化方程，揭示压力Hessian在涡量模态转化中的新作用。
AI 中文摘要
本文针对速度梯度张量（VGT）$\bm{A}\equiv\bm{\nabla}\bm{u}$，基于其在判别式为正的传统涡旋区域内的规范实Schur形式，提出了一种新颖的拉格朗日动力学理论。从可压缩牛顿流体的Navier-Stokes方程出发，我们推导了实Schur形式中六个旋转不变量的通用演化方程，其中Schur框架的角速度沿任意拉格朗日轨迹被显式确定。这些结果进而使我们能够获得导出的VGT不变量的演化方程，这些方程也涵盖了三个主要的VGT不变量及其Schur表示。值得注意的是，或许这是首次，特征涡量模态$(\bm{R}_{N},\bm{S}_{N})$的演化方程在Schur框架内以本征形式和分量形式被推导出来，并随后扩展到最近提出的交换涡量算子对$(\bm{\Psi}_{R},\bm{\Gamma}_{S})$。我们发现，拉伸应变率张量$\bm{D}_{EL}$仅用于拉伸或压缩刚体旋转涡量$\bm{R}_{N}$的积分线，而剪切涡量$\bm{S}_{N}$与剪切应变率张量$\bm{D}_{SH}$之间的相互作用则在旋转轴法平面内改变$\bm{S}_{N}$。该理论进一步揭示了压力Hessian张量（更准确地说，对于可压缩流动为焓Hessian张量）在两种涡量模态/算子的相互转化和重新分布中的新物理作用。所提出的理论可能有助于理解各种涡旋流动中VGT组成部分和本征涡量模态的动力学。
英文摘要
The present work proposes a novel Lagrangian dynamical theory for the velocity gradient tensor (VGT) $\bm{A}\equiv\bm{\nabla}\bm{u}$, formulated on the basis of its canonical real Schur form in the traditional vortex region with positive discriminant. Starting from the Navier-Stokes equations for compressible Newtonian fluids, we derive the general evolution equations for the six rotational invariants in the real Schur form, in which the angular velocity of the Schur frame is explicitly determined along any Lagrangian trajectory. These results then enable us to obtain the evolution equations for the derived VGT invariants, which also encompass the three principal VGT invariants and their Schur representations. Notably, and perhaps for the first time, the evolution equations for the characteristic vorticity modes $(\bm{R}_{N},\bm{S}_{N})$ are derived in both intrinsic and component forms within the Schur frame, and are subsequently extended to the recently proposed commutative vorticity-operator pair $(\bmΨ_{R},\bmΓ_{S})$. We find that the straining strain-rate tensor $\bm{D}_{EL}$ acts solely to stretch or contract the integral lines of the rigid-rotation vorticity $\bm{R}_{N}$, whereas the interaction between the shear vorticity $\bm{S}_{N}$ and the shear strain-rate tensor $\bm{D}_{SH}$ alters $\bm{S}_{N}$ within the rotation-axis-normal plane. The theory further unveils a new physical role of the pressure Hessian tensor (more precisely, the enthalpy Hessian tensor for compressible flow) in the mutual transformation and redistribution of the two vorticity modes/operators. The proposed theory could be useful for understanding the dynamics of VGT constituents and intrinsic vorticity modes in a variety of vortical flows.
