V.P. Reutov - Dr.Sc. (Phys.-Math.), Leading Research Scientist, Department of Nonlinear Dynamics, Federal State Budgetary Scientific Institution «Federal Research Center The Institute of Applied Physics of the Russian Academy of Sciences» (IAP RAS) (Nizhny Novgorod). E-mail: reutov@appl.sci-nnov.ru G.V. Rybushkina - Ph.D. (Phys.-Math.), Senior Research Scientist, Department of Nonlinear Dynamics, Federal State Budgetary Scientific Institution «Federal Research Center The Institute of Applied Physics of the Russian Academy of Sciences» (IAP RAS) (Nizhny Novgorod). E-mail: ryb@appl.sci-nnov.ru

Classical hydrodynamic flows (jets, mixing layers, boundary layers, etc.) are referred to the non-equilibrium systems (media) the cha-racteristic feature of which is an instability of wave (or wave-like) motions. It is well known that the nonlinear restriction of wave in-stability leads usually to the onset of self-oscillations («auto-oscillations»). The transition to turbulence in the non-equilibrium systems is associated with the occurrence of dynamical chaos that exhibits as self-oscillations of a complicated form indistinguishable from the random process. The transitional states from the regular to chaotic self-oscillations (turbulence) plays an important part in different applications. This paper is concerned with the onset of dynamical chaos in quasi-two-dimensional jet flows created in thin liquid layers thereby the horizontal velocity component is essentially large compared to the vertical one. In such flows the three-dimensional effects due to the formation of the near-bottom boundary layer can be described by supplementing an external friction in the purely two-dimensional hydrodynamic equations. Similar quasi-two-dimensional models incorporating the rotation effects are used for the description of large-scale structures in zonal (directed along the parallels) flows in the low atmosphere and in the ocean. As the con-sidered processes do not described within the weak nonlinearity theory, we employ the numerical solution of the basic equation written in the terms of the vorticity and stream function. The plane-parallel jet flow is confined by the channel walls and the periodic boundaty conditions are specified. The velocity profiles in the form of the Bickley jet and asymmetric binary jet modified to meet the non-slip conditions at the walls are examined. The pseudospectral method with the discreet Fourier transform in the longitudinal coordinate (including 64 harmonics) and finite differences in the transversal direction (having 200 grid point) is employed. Evolution of the flow regimes due to the maximal jet velocity increase was studied. The supercriticality parameter was defined taking into account the linear instability theory and was changed by small steps every time after achievement of stationary generation. The one-fold mode alternation and the hysteresis phenomenon were detected for the Bickley jet while the continuous alteration of the main mode was obtained in the case of the asymmetric jet. The double vortex street appeares in the jets at very beginning and, with increasing flow velocity, the vortex waves rapidly become nonlinear (containing large multiple harmonics). The longitudinal velocity perturbations and their time derivative were used for drawing of the phase plane and the frequency spectra. The transition to chaos in the Bickley jet was found to be due to destruction of a quasi-periodic regime (the Ruelle and Takens scenario). The satellites in the frequency spectrum are proofed to be produced by a modulation instability of nonlinear vortex wave. The occurrence of the dynamical chaos is confirmed by the calculation of the maximal Lyapunov exponent. It is shown that the transition to the dynamical chaos in the asymmetric binary jet also obeys the Ruelle and Takens scenario, but it starts at smaller supercriticality compared to the Bickley jet. Besides, the transition becomes complicated due to appearance of an interval of frequency synchronization inside of which the multifrequency regimes of different complexity alternate.

 

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