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Numerical and mathematical analysis to low-dimensional solutions in. mathematical fluid problems

数学流体问题低维解的数值和数学分析

基本信息

批准号：
15340034
负责人：
NAKAKI Tatuyuki
金额：
$ 8万
依托单位：
HIROSHIMA UNIVERSITY (2005)Kyushu University (2003-2004)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340034/
关键词：
point vortex vortex ring, vortex tube one- or two-phase fluid problems relaxation oscillation instability weakly nonlinear stability topological chaos flux-free finite element method 渦糸の緩和振動球面上の渦糸渦輪の非線形不安定性らせん渦理想MHDの数値解析法移動・

项目摘要

(1) We consider problems of point vortices in a plane, which are described by the two-dimensional Euler equation. (i) Stability of relative equilibria for five point vortices is treated. We found that some unstable configurations exhibit the relaxation oscillation. Four and three point vortices which exhibit the relaxation oscillation are also found. Mathematical justifications are made. (ii) When vortices come close each other, numerical difficulty occurs. To overcome this difficulty, we propose a numerical scheme.(2) We consider the point vortices in a sphere. When two vortices are fixed at the poles of sphere, the stability of some configuration and reduction to a center manifold are shown.(3) Three-dimensional linear instability of vortex flow is considered. By using spectral theory with Hamiltonian, we found a primary factor of instability for some cases.(4) To analyze the instability of vortex ring in the nonlinear region, we introduce a weakly nonlinear system by multi-scale methods. By using this system, we found the mechanism of the Windall instability and unstable modes.(5) By singular limit methods, we propose and analyze the numerical methods to the classical one-phase Stefan problems and the flow through porous media.(6) For immiscible two-phase flow, we consider a flux-free finite element method, which shows good conservation of mass. We obtain error estimates and convergence of numerical solution.(7) We apply a numerical verification method to the driven cavity problem, which is one of famous two-dimensional fluid problems. We succeeded in the verification of some steady solutions.

(1)我们考虑平面中点涡的问题，它是由二维欧拉方程描述的。(i)本文讨论了五点涡相对平衡点的稳定性。我们发现一些不稳定的组态表现出弛豫振荡。还发现了四点涡和三点涡的弛豫振荡，并给出了数学证明。(ii)当旋涡彼此靠近时，数值计算就困难了。为了克服这个困难，我们提出了一个数值方案。(2)我们考虑球中的点涡。当两个涡固定在球的极点时，给出了某些位形的稳定性和中心流形的约化。(3)考虑了三维线性涡不稳定性。利用谱理论和哈密顿量，我们找到了某些情况下的主要不稳定因素。(4)为了分析涡环在非线性区域的不稳定性，我们采用多尺度方法引入了一个弱非线性系统。利用该系统，我们发现了Windall不稳定性和不稳定模式的机制。(5)利用奇异极限方法，提出并分析了经典单相Stefan问题和多孔介质渗流问题的数值解法。(6)对于不互溶两相流，我们考虑了一种无通量有限元方法，它显示出良好的质量守恒。得到了误差估计和数值解的收敛性。(7)本文对二维流体力学中著名的驱动空腔问题进行了数值验证。我们成功地验证了一些稳定的解决方案。