AI 中文总结
本文引入关联细长涡面几何的坐标系，建立通用数学框架推导涡运动方程组，用于计算任意轴向流动的细长涡环运动，所得结果与早期研究一致，该框架可用于开发涡运动与相互作用的简化方程。
AI 中文摘要
理想不可压缩流体中细长涡的运动是流体力学中的经典问题。涡环速度公式可追溯至19世纪开尔文（Kelvin）、亥姆霍兹（Helmholtz）、希克斯（Hicks）与戴森（Dyson）的研究，而近年来已开发出多种模型用于模拟任意形状细长涡度管的运动。对于局域在曲线C(t)附近的涡管或涡丝，衡量其细长度的相关小参数为涡管半径与曲线C曲率半径的比值。本文通过引入与细长涡中涡面几何紧密关联的坐标系，重新研究这一系列经典问题。该坐标系中流体微元的运动具有作用-角形式，据此涡度方程可大幅简化；但与此同时，涡的形状或演化的大部分特性被纳入坐标描述，尤其是对应的度量与体积形式。由于该坐标系非正交且随时间变化，微分几何工具最适合描述涡度与坐标系的结构，本文建立了通用数学框架。所得方程组在仅保留涡细长性假设的前提下推导，允许涡核与曲线C(t)发生任意运动和畸变。该建模方法被应用于计算具有任意轴向流动的细长涡环运动，直接求解受扰涡面的形状，所得结果与早期研究一致。本文建立的通用框架适用于未来研究中开发涡运动与相互作用的简化方程。
英文摘要
The motion of a slender vortex in ideal incompressible fluid is a classic problem in hydrodynamics. Formulae for the velocity of a vortex ring go back to work of Kelvin, Helmholtz, Hicks and Dyson in the nineteenth century, while more recently a number of models have been created for simulating the motion of slender tubes of vorticity of general shape. For a vortex tube or filament localised near to a curve C(t), the relevant small parameter to measure slenderness is the tube radius divided by the radius of curvature of C. The present paper revisits this range of classic problems by introducing a coordinate system closely linked to the geometry of vortex surfaces in a slender vortex. The motion of fluid elements in this coordinate system has an action--angle form, and with this the vorticity equation simplifies radically. At the same time, however, most aspects of the shape or evolution of a vortex are thrown into the description of the coordinates, and in particular the corresponding metric and volume form. As the coordinate system is non-orthogonal and time-dependent, tools of differential geometry are most easily used to describe the structure of both vorticity and coordinate system, and a general mathematical framework is set out. This resulting system of equations is taken as far as possible with only the assumption of vortex slenderness in place, but allowing arbitrary motions and distortions of the vortex core and of the curve C(t). The modelling is applied to calculate the motion of a slender vortex ring with arbitrary axial flow, solving directly for the shape of perturbed vorticity surfaces and giving results in agreement with earlier studies. The general framework set up in this paper is suitable for the development of simplified equations for vortex motion and interaction in future studies.
