Reappraisal of the Kelvin–Helmholtz problem. Part 1. Hamiltonian structure
Reappraisal of the Kelvin–Helmholtz problem. Part 1. Hamiltonian structure
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DOI：
10.1017/s0022112096004272
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发表时间：
1997-02
影响因子：
3.7
通讯作者：
T. Benjamin;T. Bridges
中科院分区：
文献类型：
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作者：
T. Benjamin;T. Bridges
本文和第二部分报告了对经典的Kelvin-Helmholtz问题的各种新的见解，该问题模拟了平面涡面的不稳定性及其引起的复杂运动。充分的非线性版本的流体动力学问题进行处理，与津贴的重力和表面张力，和帐户处理精确的方式与几个固有的特有性质的数学模型。在§3中给出的这篇论文的主要成就是证明了这个问题有一个正则哈密顿公式，它代表了一个新的表示动能扰动的泛函的变分定义。由此揭示的哈密顿结构然后被用来系统地解释对称性和守恒律之间的关系，而这些被研究的关系似乎都是以前没有注意到的。在§4中，当涡面折叠时，一个广义的非正则哈密顿结构被证明是适用的，因此需要一个参数表示，这是众所周知的，发生在开尔文-亥姆霍兹不稳定性演化的后期阶段。在这种情况下，证明了进一步的不变性质。最后，在第5节中，我们根据哈密顿结构重新评价了这个问题的线性化版本（在第2.1节中作了简要回顾），并说明了如何将开尔文-亥姆霍兹不稳定性解释为波模的重合，波模的特征分别是代表总能量扰动的哈密顿泛函的正值和负值。
This paper and Part 2 report various new insights into the classic Kelvin–Helmholtz problem which models the instability of a plane vortex sheet and the complicated motions arising therefrom. The full nonlinear version of the hydrodynamic problem is treated, with allowance for gravity and surface tension, and the account deals in precise fashion with several inherently peculiar properties of the mathematical model. The main achievement of the paper, presented in §3, is to demonstrate that the problem admits a canonical Hamiltonian formulation, which represents a novel variational definition of a functional representing perturbations in kinetic energy. The Hamiltonian structure thus revealed is then used to account systematically for relations between symmetries and conservation laws, and none of those examined appears to have been noticed before. In §4, a generalized, non-canonical Hamiltonian structure is shown to apply when the vortex sheet becomes folded, so requiring a parametric representation, as is well known to occur in the later stages of evolution from Kelvin–Helmholtz instability. Further invariant properties are demonstrated in this context. Finally, §5, the linearized version of the problem – reviewed briefly in §2.1 – is reappraised in the light of Hamiltonian structure, and it is shown how Kelvin–Helmholtz instability can be interpreted as the coincidence of wave modes characterized respectively by positive and negative values of the Hamiltonian functional representing perturbations in total energy.