Dynamic Research Of The Point Vortex System

Vortex motion is a very common and important form of motion in fluid,and it is also the main content of vortex dynamics research.This paper mainly studies the dynamic of point vortex system.Firstly,we use the canonical transformation to simplify the three-vortex model,give the global phase diagram analysis of the three-vortex system on the plane,and study the dynamic properties of the three-vortex system.Secondly,we consider the restricted 5-vortices problem of the collinear relative equilibrium configuration formed by the primaries(non-zero vorticity),and discuss the relevant dynamic of the passive particle(zero vorticity).Finally,we study the global dynamics of the restricted(N+1)-vortices problem on the plane,which the N point vortices with equal vorticity forming a relative equilibrium configuration of a regular N-polygons.The main structure of the paper is as follows:In Chapter 1,we briefly introduce the research status,background and development trend of point vortex systems,and outlines the main research content and research characteristics of this paper.In Chapter 2,we study the relevant dynamic of the three-vortex system.First,the canonical transformation is constructed by using invariants to simplify the Hamiltonian of the three-vortex system to an ideal two-dimensional Hamiltonian.Second,we get the expression of the two-dimensional Hamiltonian in various cases,and its phase diagram is drawn numerically.Finally,the equilibrium and singularity points of the Hamiltonian system are obtained in each case,and their stability are analyzed.In Chapter 3,we study the dynamic of the passive particle in the advection of velocity field formed by four point vortices,of which four point vortices form a collinear relative equilibrium configuration.Divided into two cases according to the position of the primaries,and the dynamics of the restricted 5-vortex problem as a function of the parameter a are studied.For each case,the existence,location,and stability of the equilibrium points are specifically analyzed,and the distribution of the motion trajectories in the phase space is depicted.In Chapter 4,we study the restricted(N+1)-vortex problem on the plane,which the N point vortices with equal and non zero vorticity forming a regular N-polygons relative equilibrium configuration,and the vorticity of the last point vortex being zero.We use qualitative theory to describe its global dynamics and explore the existence types of orbits.Meanwhile,a numerical study is carried out on the motion trajectories of the passive particle under specific cases,and the phase diagrams are established.In Chapter 5,we summarize the research content of this paper and put forward further prospects for the next research.