Radiation-induced instability of a finite-chord Nemtsov membrane

Abstract

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 [“Flutter effect and emission in the region of anomalous and normal Doppler effects,” Radiophys. Quantum Electron. 28(12), 1076–1079 (1985)] has shown that an infinite-chord membrane is susceptible to flutter instability due to the 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 integrodifferential 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 unprecedented wide range of a 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.