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Symmetry breaking and Hamiltonian bifurcation theory for three-dimensional instability of a vortex tube

涡流管三维不稳定性的对称破缺和哈密顿分岔理论

基本信息

批准号：
14540379
负责人：
FUKUMOTO Yasuhide
金额：
$ 1.41万
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：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

项目摘要

1.Three-dimensional instability of an elliptic vortex and a vortex ring was investigated from the viewpoint of the Hamiltonian spectral theory.(1)A pure shear breaks the SO(2)xO(2)-symmetry of the Rankine vortex and deforms the circular core into an ellipse.We have succeeded in explicitly writing down the infinitesimal disturbances in terms of the Bessel functions.The relation with the elliptical instability is clarified.(2)For a vortex ring, symmetry-bresking perturbation is the curvature effect. The infinitesimal disturbances on Kelvin's vortex ring are written out in a closed form.We have found a possibility of parametric resonance between pairs of Kelvin waves whose azimuthal wavenumbers are separated by two.A local stability analysis is also made, using the WKB method It is clarified that the structure of unstable modes on a Gaussian core is very different from that on Kelvin's vortex ring.(3)We have derived weakly nonlinear evolution equations of amplitudes of several unstable mo … More des on a vortex tube subjected to a pure shear, with the aid of equivariant vector fields associated with Z2xO(2) symmetry.2.When a pulse of excimer laser of a few ten nanoseconds is irradiated on a Co-coated substrate, vortex filaments are created, quenched and frozen at once.By estimating the time scales from a stability analysis in the localized induction approximation, we have extracted a dynamical picture from the micrographs of the frozen pictures.(2)By repeating the irradiation, of pulses several times, unstable vortex rings are obtained.3.For the side-band instability of a plane wave, it is proven that, near the linear critical point, the methods based on envelope equations, on amplitude equations and on a secondary-instability analysis are all equivalent.4.For the Rayleigh-Benard convection, the fifth-order amplitude equations are derived, using the center-manifold reduction, and their bifurcation analysis is made near the degeneracy point with quadratic dependence on temperature of the density being an unfolding parameter It is clarified that right-hexagonal, right-triangle and patch-work quilt patterns are unstable as a primary bifurcation solution of the generic normal form equations. Less

1.从哈密顿谱理论的角度研究了椭圆涡和涡环的三维不稳定性。(1)纯剪切破坏了朗肯涡的SO(2)xO(2)对称性，使圆形核变形为椭圆。我们成功地用贝塞尔函数明确地写出了无穷小扰动。 (2)对于涡环，破坏对称性摄动是曲率效应。开尔文涡环上的无穷小扰动以封闭形式写出。我们发现方位角波数相隔二的开尔文波对之间存在参量共振的可能性。还使用WKB方法进行了局部稳定性分析。阐明了高斯核上的不稳定模态结构与开尔文涡旋上的不稳定模态结构有很大不同。 (3)借助与Z2xO(2)对称性相关的等变矢量场，我们推导出了受到纯剪切作用的涡流管上几个不稳定模的振幅的弱非线性演化方程。2.当几十纳秒的准分子激光脉冲照射在Co涂层基底上时，会产生涡旋丝，通过局域感应近似中的稳定性分析估计时间尺度，从冻结图像的显微照片中提取出动态图像。(2)通过脉冲的多次重复照射，得到不稳定的涡环。3.对于平面波的边带不稳定性，证明了在线性临界点附近，基于包络的方法 4.对于Rayleigh-Benard对流，采用中心流形约简推导了五阶振幅方程，并在简并点附近进行了分岔分析，以密度温度的二次相关性为展开参数阐明了直角六边形、直角三角形和拼凑结构被子模式作为一般范式方程的一次分岔解是不稳定的。较少的