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A complex analysis approach to the motion of uniform vortices

机译：均匀涡旋运动的复杂分析方法

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摘要

A new mathematical approach to kinematics and dynamics of planar uniform vortices in an incompressible inviscid fluid is presented. It is based on an integral relation between Schwarz function of the vortex boundary and induced velocity. This relation is firstly used for investigating the kinematics of a vortex having its Schwarz function with two simple poles in a transformed plane. The vortex boundary is the image of the unit circle through the conformal map obtained by conjugating its Schwarz function. The resulting analysis is based on geometric and algebraic properties of that map. Moreover, it is shown that the steady configurations of a uniform vortex, possibly in presence of point vortices, can be also investigated by means of the integral relation. The vortex equilibria are divided in two classes, depending on the behavior of the velocity on the boundary, measured in a reference system rotating with this curve. If it vanishes, the analysis is rather simple. However, vortices having nonvanishing relative velocity are also investigated, in presence of a polygonal symmetry. In order to study the vortex dynamics, the definition of Schwarz function is then extended to a Lagrangian framework. This Lagrangian Schwarz function solves a nonlinear integrodifferential Cauchy problem, that is transformed in a singular integral equation. Its analytical solution is here approached in terms of successive approximations. The self-induced dynamics, as well as the interactions with a point vortex, or between two uniform vortices are analyzed.
机译：提出了一种新的数学方法来研究不可压缩的无粘性流体中平面均匀涡的运动学和动力学。它基于涡旋边界的Schwarz函数与感应速度之间的积分关系。该关系首先用于研究具有Schwarz函数且在变换平面中具有两个简单极点的涡旋的运动学。涡旋边界是通过共轭图的共形图获得的单位圆的图像，该共形图是通过共轭其Schwarz函数获得的。结果分析基于该图的几何和代数性质。此外，表明均匀涡旋的稳定构型，可能在存在点涡旋的情况下，也可以通过积分关系来研究。涡旋平衡分为两类，这取决于边界上的速度行为，该速度是在随该曲线旋转的参考系统中测量的。如果消失，则分析非常简单。但是，在多边形对称的情况下，也研究了相对速度不变的涡旋。为了研究涡旋动力学，然后将Schwarz函数的定义扩展到Lagrangian框架。拉格朗日Schwarz函数可解决非线性积分微分柯西问题，该问题可转换为奇异积分方程。在此，按照逐次逼近法来求解其解析解。分析了自感应动力学以及与点涡旋或两个均匀涡旋之间的相互作用。

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