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Numerical and Mathemtical Analysis of the motion of vortices

涡流运动的数值和数学分析

基本信息

批准号：
10640120
负责人：
NAKAKI Tatsuyuki
金额：
$ 1.73万
依托单位：
KYUSHU UNIVERSITY (1999)Hiroshima University (1998)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640120/
关键词：
point vortices finite vortices relaxation oscillation heteroclinic orbit periodic motion advected particle

项目摘要

(1)We consider five point vortices in the two-dimensional Euler fluid. Let α and κ be parameters which indicate the initial configulation of the vortices and the strength of a vortex, respectively. When α=1 and κ<-0.5, our numerical simulation show the rotating motion of vortices with relaxation oscillations appears. Mathematically we prove the existence of the heteroclinic orbits, which induces such a motion. We also prove that the rotating motion is stable against some perturbation for α=1 and κ>-0.5. When α≠1, we find that the vortices behave periodic or quasi-periodic. Under certain situation, we prove that periodic motion occurs. By numerical simulations, we indicate the values of α and κ under which periodic motion occurs. We also analyze the shape of the periodic motion.(2)We make numerical simulations for the motion of five finite vortices by the contour dynamics method. For some value of the area, our simulations display that the finite vortices begin to deform and rotate rapidly. For large value of the area, as many researchers are already reported, the coalescence of vortices is observed. We also make simulations for finite and point vortices are on the fluid.(3)We consider the motion of passively advected particles in the flow induced by five point vortices which behave periodic. Our analysis is based on the numerical simulations of the motion of particles on the Poincare section. We find that, according to the initial position of particle, the following cases occurs. (1)The particle stays near the vortex. (2)The particle moves the far area. (3)Chaotic behavior of the paricle occurs. (4)The particle on the section is concentrated at some area.

(1)We考虑二维欧拉流体中的五点涡流。设α和κ分别为表示涡的初始振荡和涡的强度的参数。当α = 1，κ <-0.5时，数值模拟结果表明，涡的旋转运动出现了弛豫振荡。在数学上，我们证明了异宿轨道的存在性，它导致这样的运动。我们还证明了当α = 1和κ>-0.5时，旋转运动对某些扰动是稳定的.当α ≠ 1时，我们发现涡旋表现出周期性或准周期性。在一定条件下，证明了周期运动的存在。通过数值模拟，我们指出了发生周期运动的α和κ值。我们还分析了周期运动的形状。(2)We用轮廓动力学方法对五个有限长旋涡的运动进行了数值模拟。对于某些面积值，我们的模拟显示，有限的旋涡开始变形和旋转迅速。正如许多研究人员已经报道的那样，对于大的面积值，观察到涡旋的合并。我们还对流体上的有限涡和点涡进行了模拟。(3)We考虑在由表现为周期性的五点涡引起的流动中被动平流粒子的运动。我们的分析是基于粒子在庞加莱截面上运动的数值模拟。我们发现，根据粒子的初始位置，会出现以下情况。（1）粒子停留在旋涡附近。（2）粒子移动远区。（3）粒子发生混沌行为。（4）断面上的颗粒在某一区域集中。