Three-dimensional nonlinear stability theory of vortices from the view point of Hamiltonian dynamicalsysytems

哈密​​顿动力系统视角下的涡三维非线性稳定性理论

• 批准号: 16540345
• 负责人: FUKUMOTO Yasuhide
• 金额: $ 1.47万
• 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: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540345/
• 关键词: Kelvin's vortex ring; curvature instability; Krein theory; weakly nonlinear instability; three-wave resonance; laser-matter interactions; helical vortex tube; energy balance of magnetic eddies; Dysonの方法; ノーマルフォーム; 振幅方程式; らせん渦; ハミルトン的スペクトル理論

A new instability mechanism "curvature instability", originating from the curvature of vortex lines, is found for Kelvin's vortex ring. The eigenfunction is explicitly writing out and thereby asymptotic form of the growth rate at large wavelengths is derived. We have elucidated structure of the spectrum from the viewpoint of the Krein theory for Hamiltonian systems. By repeating irradiation of pulses of excimer laser on a Co-coated substrate, unstable vortex rings are created. By analyzing the micrographs of their frozen pictures. we identified four types of instability modes. We made a weakly nonlinear stability analysis of Kelvin's vortex ring. It is found from numerical analysis of the derived amplitude equations that the system exhibits a chaotic behavior. We also conducted a direct numerical simulation of a vortex ring, and clarified a detailed structure of the Widnall instability.We exploited the Hamiltonian normal form, associated with the SO(2) x O(2)-symmetry, to select possib … More le form of the weakly nonlinear evolution equations of amplitude of the bending modes (azimuthal wavenumber m=1,-1) of Kelvin waves on an elliptically strained vortex tube. The coefficient of the amplitude equations are evaluated for non-rotating modes. The nonlinear terms make the elliptical instability saturate. However, it is pointed out that three-wave resonance of modes of m=3,4 and a bending mode derives a secondary instability.By extending Dyson's technique to three dimensions, we developed a method of asymptotic expansions of the Biot-Savart integral, which takes account of the influence of finite core thickness on a vortex tube. This method is applied to calculation of velocity field around a helical vortex tube. In the neighborhood of the core, we cannot ignore the dipoles arranged on the tube center line which reflects the distribution of vorticity in the core.A formulation of contour dynamics is given to the magnetohydrodynamic evolution of axisymmetric magnetic eddies. A family of exact solutions is found. Their significance in energy balance is clarified by both numerical implementation of the contour dynamics and a direct numerical simulation. Less

开尔文涡环发现了一种源于涡线曲率的新的不稳定机制“曲率不稳定性”。特征函数被显式写出，从而导出大波长下生长速率的渐近形式。我们从哈密顿系统的 Kerin 理论的角度阐明了谱的结构。通过在Co涂层基底上重复照射准分子激光脉冲，会产生不稳定的涡环。通过分析他们冷冻照片的显微照片。我们确定了四种类型的不稳定模式。我们对开尔文涡环进行了弱非线性稳定性分析。对导出的振幅方程进行数值分析发现，系统表现出混沌行为。我们还对涡环进行了直接数值模拟，并阐明了 Widnall 不稳定性的详细结构。我们利用与 SO(2) x O(2) 对称性相关的哈密顿范式，选择了开尔文波弯曲模式（方位角波数 m=1,-1）振幅弱非线性演化方程的可能形式 椭圆形应变涡流管。针对非旋转模式评估振幅方程的系数。非线性项使椭圆不稳定性饱和。然而，有人指出，m=3,4模式和弯曲模式的三波共振会产生二次不稳定性。通过将戴森技术扩展到三维，我们开发了一种Biot-Savart积分的渐近展开方法，该方法考虑了有限核心厚度对涡流管的影响。该方法适用于螺旋涡管周围速度场的计算。在核心附近，我们不能忽视排列在管中心线上的偶极子，它反映了核心中涡量的分布。给出了轴对称磁涡的磁流体动力学演化的轮廓动力学公式。找到了一系列精确解。通过轮廓动力学的数值实现和直接数值模拟阐明了它们在能量平衡中的重要性。较少的

项目成果

Reconnexion of vortex and magnetic tubes subject to an imposed strain : an approach by perturbation expansion

受到外加应变的涡旋和磁管的重新连接：一种微扰膨胀方法

• 发表时间: 2005
• 期刊: Fluid Dynamics Research (in press)
• 作者: Y.Hattori;H.K.Moffatt
• 通讯作者: H.K.Moffatt

Reconnexion of vortex an magnetic tubes subject to an imposed strain : an approach by perturbation expansion

受到外加应变的涡流与磁管的重新连接：微扰膨胀的方法

• 发表时间: 2005
• 期刊: Fluid Dynamics Research Vol.36,No.4-6
• 作者: Y.Hattori;H.K.Moffatt
• 通讯作者: H.K.Moffatt

Linear and nonlinear Instability of a vortex ring

涡环的线性和非线性不稳定性

• 发表时间: 2006
• 期刊: Proc. of IUTAM Symposium on Elementary Vortices and Coherent Structures (eds. S. Kida, Kluwer) (in press)
• 作者: Y.Fukumoto;Y.Hattori
• 通讯作者: Y.Hattori

The velocity field induced by a helical vortex tube

• DOI: 10.1063/1.2061427
• 发表时间: 2005-10
• 期刊: Physics of Fluids
• 影响因子: 4.6
• 作者: Y. Fukumoto;V. Okulov

Hierarchicalin stability of a vortex ring array in multipulse laser-matter interactions

多脉冲激光-物质相互作用中涡环阵列的分级稳定性

• 发表时间: 2005
• 期刊: Fluid Dynamics Research Vol.36,No.4-6
• 作者: Y.Saito;K.Takeuchi;A.Suzuki;B.-F.Feng;T.Sakajo;T.Sakajo;E.Kin;Y.Fukumoto;S.Lugomer
• 通讯作者: S.Lugomer