Numerical computations to the point vortices in Euler flows

Tatsuyuki Nakaki

Faculty of Mathematics, Kyushu University, Hakozaki Fukuoka 812-8581, Japan

nakaki@math.kyushu-u.ac.jp

Contributed talk

In this talk we consider the motion of assembly of point vortices in two-dimensional flows governed by Euler equation. Let $z_j(t)$ be the complex coordinate of $j$th point vortex, then our problem can be described by

$${d\over dt}\overline{z_j}={1\over2\pi i}\sum_{k\ne j}{\Gamma_k\over z_j-z_k} \quad(j=1,2,\ldots,n),$$

where $\Gamma_k$ is a given real constant, which implies the strength of $k$th vortex ( $k=1,2,\ldots,n$), and $n$ is the number of vortices in the flow. When $n=2$, this problem is easily solved (see the textbook on fluid dynamics). Aref [1] treats the case when $n=3$ and gives a qualitative analysis. For $n=4$ many researchers obtain interesting results (see [2], [3], for examples).

In this talk, we consider the special case where $n=5$, $z_1(0)=-z_3(0)$, $z_2(0)=-z_4(0)$, $z_5(0)=0$, $\Gamma_1=\Gamma_3$ and $\Gamma_2=\Gamma_4$. The strength $\Gamma_5$ is determined so that the five point vortices are in the relative equilibrium, that is, $\{e^{-i\Omega t}z_j(t)\}$ is equilibrium for some real constant $\Omega$. The purpose of this talk is to show the following:

• The configuration of vortices is stable only on the narrow parameter range;
• Some unstable configuration exhibits the relaxation oscillation.