Stability of leapfrogging vortex pairs: A semi-analytic approach
Stability of leapfrogging vortex pairs: A semi-analytic approach
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蛙跳涡对的稳定性：半解析方法
DOI：
10.1103/physrevfluids.4.124703
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发表时间：
2019
影响因子：
2.7
通讯作者：
Roy H. Goodman
中科院分区：
文献类型：
--
作者：
Brandon M. Behring;Roy H. Goodman
我们研究了被称为“跳跃”轨道的四涡问题的单参数周期解族的稳定性。W. Grobli（1877）和a . E. H. Love（1883）已经知道这些由两对相同但符号相反的漩涡组成的解，并且可以通过与初始构型几何相关的无量纲参数$\alpha$来参数化。Acheson（2000）的模拟和Tophoj和Aref（2012）的数值Floquet分析都表明，在许多数字中，分叉发生在$1/\alpha=\phi^2$，其中$\phi$是黄金比例。本研究旨在解释这一显著价值的起源。利用引力二体问题中的一个技巧，我们改变变量，使Floquet问题以更易于分析的显式形式呈现。然后，我们利用计算机代数实现了g.w.h ill的高阶调和平衡方法，构造了一个分岔值的渐近逼近的快速收敛序列，证实了前面发现的值。
We investigate the stability of a one-parameter family of periodic solutions of the four-vortex problem known as `leapfrogging' orbits. These solutions, which consist of two pairs of identical yet oppositely-signed vortices, were known to W.\ Grobli (1877) and A.\ E.\ H.\ Love (1883), and can be parameterized by a dimensionless parameter $\alpha$ related to the geometry of the initial configuration. Simulations by Acheson (2000) and numerical Floquet analysis by Tophoj and Aref (2012) both indicate, to many digits, that the bifurcation occurs when $1/\alpha=\phi^2$, where $\phi$ is the golden ratio. This study aims to explain the origin of this remarkable value. Using a trick from the gravitational two-body problem, we change variables to render the Floquet problem in an explicit form that is more amenable to analysis. We then implement G. W. Hill's method of harmonic balance to high order using computer algebra to construct a rapidly-converging sequence of asymptotic approximations to the bifurcation value, confirming the value found earlier.