Mathematical and numerical analysis to the assembly of vortices in fluid

流体中涡流聚集的数学和数值分析

基本信息

批准号：
12640130
负责人：
NAKAKI Tatsuyuki
金额：
$ 2.05万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

(1) We consider five point vortices on the vertices and center of a rectangular. We already known that there exists a periodic motion of vortices for some parameters on the initial configuration and strength of vortices. We study the periodic motion for another value of parameters. As a result, we find that there exists a family of periodic motions. By numerical simulations, these periodic motions have a simple structure.(2) We consider five point vortices on the vertices and center of a diamond. For some values of strength of vortices, the five vortices is in a relative equilibrium, that is, it is in a equilibrium with respect to a rotation coordinate with uniform angular velocity. There are two parameters in this problem. We study the stability of the relative equilibrium. By numerical simulations, we find that the equilibria are stable in a narrow parameter range. To show the stability, we construct the Lyapunouv function and apply the computer-assisted proof using interval computations. As a result, under the same signature of strengths, we find that many elliptic equilibria are stable. Until now we do not succeed in proving stability for all elliptic equilibria.(3) Under the same situation stated in (2), we numerically find that some unstable equilibria exhibit the relaxation oscillation. We do not know whether or not all unstable equilibria exhibit the oscillation, however, our numerical simulations suggest that an unstable equilibrium which does not exhibit the oscillation exists. We already know that there are square-shaped five point vortices which exhibit the oscillation, however, the oscillation we find belongs to a different kind category. The mathematical reason why such a oscillation occurs is one of our future works.

（1）我们考虑矩形的顶点和中心上的五个涡流。我们已经知道，在初始配置和涡旋强度上，涡流的定期运动是针对某些参数的。我们研究了另一个参数值的周期性运动。结果，我们发现存在一系列周期性动议。通过数值模拟，这些周期性动作具有简单的结构。（2）我们考虑钻石的顶点和中心上的五个涡旋。对于某些涡旋强度值，五个涡旋处于相对平衡，也就是说，相对于旋转坐标的旋转坐标的平衡均匀。这个问题有两个参数。我们研究相对平衡的稳定性。通过数值模拟，我们发现平衡在狭窄的参数范围内是稳定的。为了显示稳定性，我们构建Lyapunouv函数，并使用Interval Computation应用计算机辅助证明。结果，在强度的相同特征下，我们发现许多椭圆形平衡是稳定的。到目前为止，我们还没有成功证明所有椭圆形平衡的稳定性。（3）在（2）中指出的相同情况下，我们实际上发现某些不稳定的平衡表现出松弛振荡。我们不知道所有不稳定的平衡是否表现出振荡，但是，我们的数值模拟表明，不稳定的平衡不存在振荡。我们已经知道，有振荡的方形五点涡旋，但是，我们发现的振荡属于另一种类别。这种振荡发生的数学原因是我们未来的作品之一。