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Interaction of continuous-spectrum waves with each other and with discrete-spectrum waves

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Abstract

The nonlinear problem of the evolution of an initial perturbation in Couette flow is solved in the quadratic approximation and it is shown that the energy of the initial perturbation is transmitted to the main flow so that its profile is somewhat modified. The evolution of the initial perturbation in a fluid with a very simple model flow profile which, in addition to continuous-spectrum waves, also admits the existence of a single neutral mode of the discrete spectrum is then investigated. It is shown that as a result of the linear resonant interaction of the discrete-spectrum and continuous-spectrum waves disturbances that grow linearly with time may be formed. A flow that does not contain exponentially growing modes will be unstable with respect to certain initial disturbances; this instability is called algebraic [6, 7]. A physical interpretation of this effect is given. From this interpretation it is clear that algebraic instability is possible in a fluid with flow profiles of a more general type, in which there are neutral or weakly damped discrete-spectrum modes having a critical layer.

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Authors and Affiliations

  1. Moscow

    I. A. Sazonov

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  1. I. A. Sazonov
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Additional information

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 116–123, July–August, 1989.

The author is grateful to G. I. Barenblatt, S. Ya. Gertsenshtein, M. A. Mironov, S. A. Rybak, O. S. Ryzhov, and E. D. Terent'ev for their interest and useful comments.

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Sazonov, I.A. Interaction of continuous-spectrum waves with each other and with discrete-spectrum waves. Fluid Dyn 24, 586–592 (1989). https://doi.org/10.1007/BF01052422

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  • DOI: https://doi.org/10.1007/BF01052422

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