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Stability of the Kolmogorov flow and its modifications

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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.

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Authors and Affiliations

  1. Southern Federal University, Rostov-on-Don, 344006, Russia

    S. V. Revina

  2. 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, 362027, Russia

    S. V. Revina

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Original Russian Text © S.V. Revina, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 6, pp. 1003–1022.

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Revina, S.V. Stability of the Kolmogorov flow and its modifications. Comput. Math. and Math. Phys. 57, 995–1012 (2017). https://doi.org/10.1134/S0965542517020130

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