有限弦Nemtsov膜的辐射诱导不稳定性
Radiation-induced instability of a finite-chord Nemtsov membrane
摘要：
本文研究了无限翼展、有限弦长、淹没在有自由表面的有限水深均匀流中的薄膜的稳定性问题。在浅水近似中，Nemtsov（1985）已经表明，如果流速超过波的相速度，无限弦薄膜由于自由表面上长重力波的激发而容易发生颤振不稳定性，并将这种现象与反常多普勒效应联系起来。在目前的工作中，我们推导出一个完整的非线性特征值问题的积分微分方程的有限弦Nemtsov膜在有限的深度流。在浅水和深水的限制，我们开发了一个小的附加质量比参数作为一个有效的耗散扰动理论，找到明确的解析表达式耦合到表面波的膜模式的频率和增长率。我们发现了一个复杂的模式的不稳定口袋中的参数空间，并描述它的分析。任意水深自由表面流动的情况下，需要一个新的非多项式非线性特征值问题的数值解。我们提出了一个原始的方法相结合的复分析和留数演算，Galerkin离散化，牛顿法和并行化技术在MATLAB中实现的系统的参数范围内产生前所未有的高精度稳定图。我们相信，涅姆佐夫膜起着同样的范式作用，理解辐射引起的不稳定性的兰姆振荡器耦合到一个字符串发挥了理解辐射阻尼。
英文摘要：
We consider a problem of stability of a membrane of an infinite span and a finite chord length, submerged in a uniform flow of finite depth with free surface. In the shallow water approximation, Nemtsov (1985) has shown that an infinite-chord membrane is susceptible to flutter instability due to excitation of long gravity waves on the free surface if the velocity of the flow exceeds the phase velocity of the waves and related this phenomenon with the anomalous Doppler effect. In the present work we derive a full nonlinear eigenvalue problem for an integro-differential equation for the finite-chord Nemtsov membrane in the finite-depth flow. In the shallow- and deep water limits we develop a perturbation theory in the small added mass ratio parameter acting as an effective dissipation, to find explicit analytical expressions for the frequencies and the growth rates of the membrane modes coupled to the surface waves. We find an intricate pattern of instability pockets in the parameter space and describe it analytically. The case of an arbitrary depth flow with free surface requires numerical solution of a new non-polynomial nonlinear eigenvalue problem. We propose an original approach combining methods of complex analysis and residue calculus, Galerkin discretization, Newton method and parallelization techniques implemented in MATLAB to produce high-accuracy stability diagrams within an unprecedentedly wide range of system's parameters. We believe that the Nemtsov membrane plays the same paradigmatic role for understanding radiation-induced instabilities as the Lamb oscillator coupled to a string has played for understanding radiation damping.
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