Radiation-induced instability of a finite-chord Nemtsov membrane

Labarbe, Joris and Kirillov, Oleg (2022) Radiation-induced instability of a finite-chord Nemtsov membrane. Physics of Fluids, 34 (1). 014106. ISSN 1070-6631

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Official URL: https://doi.org/10.1063/5.0076892


We consider a problem of stability of a membrane of an infinite span and a finite chord length that is 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 placed this phenomenon into the general physical context of the anomalous Doppler effect. In the present work we derive a full nonlinear eigenvalue problem for an integro-differential equation in the case of 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 parameter in the system, to find explicit analytical expressions for the frequencies and the growth rates of the membrane modes coupled to the surface waves. This result reveals a new intricate pattern of instability pockets in the parameter space and allows for its analytical description. 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 appears to play the same paradigmatic role for understanding radiation-induced instabilities as the famous Lamb oscillator coupled to a string has played for understanding radiation damping.

Item Type: Article
Additional Information: Funding information: We thank the London Mathematical Society for supporting Prof. Overton’s visit to Northumbria through the Scheme 4 Research in Pairs grant No 41820. J. L. was supported by a Ph.D. Scholarship from Northumbria University that also provided him with an opportunity to run the code on the High Performance Cluster. The research of O. N. K. was supported in part by the Royal Society Grant No. IES\R1\211145.
Uncontrolled Keywords: radiation-induced instabilities, dissipation through dispersion, radiation damping, anomalous Doppler effect, flow-structure interaction, nonlinear eigenvalue problem
Subjects: F300 Physics
G100 Mathematics
Department: Faculties > Engineering and Environment > Mathematics, Physics and Electrical Engineering
Depositing User: John Coen
Date Deposited: 01 Apr 2021 07:52
Last Modified: 07 Jan 2023 08:00
URI: https://nrl.northumbria.ac.uk/id/eprint/45851

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