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）我们考虑二维Euler流体中的五个涡流。令α和κ为参数，分别指示涡旋的初始配置和涡旋的强度。当α= 1和κ<-0.5时，我们的数值模拟显示出具有松弛振荡的涡流的旋转运动。从数学上讲，我们证明了杂斜轨道的存在，从而引起这种运动。我们还证明，对于α= 1和κ> -0.5的某些扰动，旋转运动是稳定的。当α≠1时，我们发现涡旋的表现是周期性或准周期性的。在某些情况下，我们证明会发生定期运动。通过数值模拟，我们指出了发生周期性运动的α和κ的值。我们还分析了周期性运动的形状。（2）我们通过轮廓动力学方法对五个有限涡度运动进行数值模拟。对于该区域的某些值，我们的模拟显示有限的涡旋开始变形并迅速旋转。对于许多研究人员已经报道了该地区的大量价值，观察到涡流的合并。我们还对有限的涡流进行了模拟。（3）我们考虑了由五个点涡流引起的流动中的被动先进颗粒的运动，这些涡流的表现是定期定期的。我们的分析基于对Poincare部分中粒子运动的数值模拟。我们发现，根据粒子的初始位置，发生以下情况。（1）粒子停留在涡旋附近。（2）粒子移动远处。（3）发生刻度的混乱行为。（4）截面上的粒子集中在某些区域。