AI 中文总结
本研究基于高斯最小约束原理提出不可压缩涡量方程的变分公式，建立螺旋度与涡量、压力与速度的对偶关系，验证了该原理在Burgers涡中的有效性，为流体动力学提供统一变分框架。
AI 中文摘要
我们基于高斯最小约束原理，利用高斯约束泛函$\boldsymbol{Z}_ω$，提出了不可压缩涡量方程的一种变分公式。核心结果是欧拉-拉格朗日方程$\boldsymbol{Z}_ω= -\nabla h$，其中$h = \boldsymbol{u} \boldsymbol{\nabla} \times \boldsymbol{u}$是螺旋度密度。这表明螺旋度梯度$\nabla h$起到了维持涡量场无散性的约束力作用，这与Taha等人提出的压力梯度最小化原理中压力梯度$\nabla p$维持不可压缩性的作用完全一致。螺旋度密度自然地作为拉格朗日乘子出现，用于强制满足$\nabla \boldsymbol{\nabla} \times \boldsymbol{u}=0$，并且在解处流动使螺旋度梯度的范数$\nabla h$最小化。这建立了精确的对偶关系：螺旋度之于涡量，正如压力之于速度。我们将该变分原理应用于Burgers涡，并验证了欧拉-拉格朗日方程。该变分原理与Moffatt的螺旋度守恒定理、Arnold的理想流体流动几何表述以及Kambe的规范理论表述相关联。这项工作为流体动力学提供了一个统一的变分框架，横跨经典力学、几何力学和拓扑场论。我们讨论了这项工作如何为理解螺旋度梯度在边界层动力学中的作用提供理论基础，这对相干结构的形成和转捩的起始具有潜在意义，这些应用留待未来研究。
英文摘要
We present a variational formulation of the incompressible vorticity equation based on Gauss's principle of least constraint using the Gauss constraint functional $\Zvec_ω$. The central result is the Euler--Lagrange equation $\mathbf{Z}_ω= -\nabla h$, where $h = \mathbf{u} \cdot\boldsymbol{omega}$ is the helicity density. This reveals that the helicity gradient $\nabla h$ acts as the constraint force maintaining the solenoidality of the vorticity field, exactly as the pressure gradient $\nabla p$ maintains incompressibility in Taha et al.'s pressure-gradient minimization principle. The helicity density naturally emerges as the Lagrange multiplier enforcing $\nabla \cdot \boldysmbol{omega}=0$, and at the solution the flow minimizes the norm of the helicity gradient $\|\nabla h\|^2$. This establishes the exact duality: helicity is to vorticity as pressure is to velocity. We apply the variational principle to the Burgers vortex and verify the the Euler-Lagrange equation. The variational principle connects to Moffatt's helicity conservation theorem, Arnold's geometric formulation of ideal fluid flow, and Kambe's gauge-theoretic formulation. This work provides a unified variational framework for fluid dynamics that spans classical mechanics, geometric mechanics, and topological field theory. We discuss how this work may provide a theoretical foundation for understanding the role of helicity gradients in boundary layer dynamics, with potential implications for the formation of coherent structures and the onset of transition. These applications are reserved for future work.
