Numerical and Mathemtical Analysis of the motion of vortices
Project/Area Number  10640120 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  KYUSHU UNIVERSITY(1999) Hiroshima University(1998) 
Principal Investigator 
NAKAKI Tatsuyuki Kyushu University, Graduate School of Mathematics, Associate Professor, 大学院・数理学研究科, 助教授 (50172284)

CoInvestigator(Kenkyūbuntansha) 
KIMURA Masato Hiroshima University, Department of Mathematical and Life Science, Assistant Professor, 理学部, 講師 (70263358)
TOMOEDA Kenji Osaka Institute of Technology, Department of General Education, Professor, 工学部, 教授 (60033916)
FUKUMOTO Yasuhide Kyushu University, Graduate School of Mathematics, Associate Professor, 大学院・数理学研究科, 助教授 (30192727)

Project Period (FY) 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥2,700,000 (Direct Cost : ¥2,700,000)
Fiscal Year 1999 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1998 : ¥1,900,000 (Direct Cost : ¥1,900,000)

Keywords  point vortices / finite vortices / relaxation oscillation / heteroclinic orbit / periodic motion / advected particle 
Research Abstract 
(1)We consider five point vortices in the twodimensional 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 quasiperiodic. 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.

Report
(3results)
Research Output
(10results)