https://doi.org/10.1140/epjp/s13360-024-04857-1
Regular Article
Nonlinear oscillations in a two-dimensional spatially periodic flow
1
A.M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Pyzhevsky 3, 119017, Moscow, Russia
2
Scientific and Production Association “Typhoon”, Obninsk, Kaluga Region, Russia
3
Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia
4
Lomonosov Moscow State University, Moscow, Russia
Received:
29
September
2023
Accepted:
2
January
2024
Published online:
1
February
2024
The stability of a two-dimensional spatially periodic flow of a homogeneous fluid, named the Kolmogorov flow, is studied in the absence of viscosity, that is, for very large values of the Reynolds number. To solve the stability problem, the Galerkin method with three basis Fourier harmonics was used. A system of ordinary differential equations for the amplitudes of the Fourier harmonics is formulated. The solution of the nonlinear system for amplitudes is obtained in terms of Jacobi elliptic functions. It is shown that the exponential growth of linear perturbations of the inviscid Kolmogorov flow is replaced by the regime of stable nonlinear oscillations or vacillations. Such vacillations were first observed experimentally in R. Hide's well-known laboratory studies. An analytical expression is obtained for the period of arising nonlinear oscillations. It is shown that, as a result of the instability, a system of closed vortex cells is formed in a periodic flow.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.