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Zh. Vychisl. Mat. Mat. Fiz., 2017, Volume 57, Number 6, Pages 1003–1022 (Mi zvmmf10550)  

This article is cited in 1 scientific paper (total in 1 paper)

Stability of the Kolmogorov flow and its modifications

S. V. Revinaab

a Southern Federal University, Rostov-on-Don, Russia
b Southern Mathematical Institute, Vladikavkaz Research Center, Russian Academy of Sciences and the Government of the Republic of North Ossetia-Alania, Vladikavkaz, Republic of North Ossetia-Alania, Russia

Abstract: Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for general two-dimensional viscous incompressible shear flows. It is shown that the eigenvalues of the linear eigenvalue problem are odd functions of the wave number, while the critical values of viscosity are even functions. If the velocity averaged over the long period is nonzero, then the loss of stability is oscillatory. If the averaged velocity is zero, then the loss of stability can be monotone or oscillatory. If the deviation of the velocity from its period-average value is an odd function of spatial variable about some $x_0$, then the expansion coefficients of the velocity perturbations are even functions about $x_0$ for even powers of the wave number and odd functions about for $x_0$ odd powers of the wave number, while the expansion coefficients of the pressure perturbations have an opposite property. In this case, the eigenvalues can be found precisely. As a result, the monotone loss of stability in the Kolmogorov flow can be substantiated by a method other than those available in the literature.

Key words: stability of two-dimensional viscous flows, Kolmogorov flow, long wavelength asymptotics.

DOI: https://doi.org/10.7868/S0044466917020144

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English version:
Computational Mathematics and Mathematical Physics, 2017, 57:6, 995–1012

Bibliographic databases:

UDC: 519.634
Received: 15.02.2016
Revised: 19.05.2016

Citation: S. V. Revina, “Stability of the Kolmogorov flow and its modifications”, Zh. Vychisl. Mat. Mat. Fiz., 57:6 (2017), 1003–1022; Comput. Math. Math. Phys., 57:6 (2017), 995–1012

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Kazarnikov, S. V. Revina, “Bifurkatsii v sisteme Releya s diffuziei”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:4 (2017), 499–514  mathnet  crossref  elib
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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