Onset of self-oscillations upon the loss of stability of spatially periodic two-dimensional viscous fluid flows relative to long-wave perturbations

Melekhov, A. P.; Revina, S. V.

doi:10.1134/s0015462808020051

Cited by 7 publications

(1 citation statement)

References 5 publications

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“…In the former case, this is , which, as shown in [2][3][4], is stable at l x < 2π. The longwave instability (l x → ∞) was obtained at ; for the cells of finite length l x > 2π the oscillatory instability was found in [15] and for a more general case, at nonzero flow rate Re x in [14]. The stationary ZPK96 solution [12] at l x = l y = 2π takes the form:…”

Section: The Known Stationary Solutions and Their Stabilitymentioning

confidence: 99%

Instabilities, Bifurcations, and Nonlinear Dynamics in Two-Dimensional Generalizations of Kolmogorov Flow

et al. 2022

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A family of two-dimensional flows of viscous incompressible fluid in a plane rectangular region with periodic boundary conditions (two-dimensional torus) is considered. The flows are induced by a force, periodic in the two spatial variables and independent of time. In the particular case of the harmonic dependence of the force on one coordinate and in the absence of average flow the well-known Kolmogorov flow is realized. In the general two-dimensional case restructurings of the stationary solutions of the Navier–Stokes equations are investigated numerically and the stability domains are determined in the space of governing physical and geometric parameters, namely, Reynolds numbers, force amplitudes, and spatial dimensions of periodicity cells. It is found that in a square region, whose side is equal to the spatial period of the external force, the main stationary flow preserves its stability against variation in the force amplitude and the Reynolds number. Contrariwise, in the cells, whose sides include several force periods, the variation in the parameters destabilizes the stationary flow. Stationary and self-oscillatory nonlinear secondary flows are considered. The effect of nonstationarity on the Lagrangian dynamics is discussed: the mechanisms of transition to the chaotic advection of passive particles depend on the commensurability of the Reynolds numbers characterizing the average flow in mutually perpendicular directions.

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Section: The Known Stationary Solutions and Their Stabilitymentioning

confidence: 99%

Instabilities, Bifurcations, and Nonlinear Dynamics in Two-Dimensional Generalizations of Kolmogorov Flow

et al. 2022

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Long Wavelength Asymptotics of Self-Oscillations of Viscous Incompressible Fluid

Revina

2021

Trends in Mathematics

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Stability and nonlinear secondary modes of double-periodic flows with pumping

Wertgeim

Zaks

Sagitov

et al. 2020

J. Phys.: Conf. Ser.

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We investigate, both qualitatively and numerically, the family of forced doubly periodic plane flows of viscous fluids. These flows generalize the seminal Kolmogorov flow to the case of two-dimensional periodicity of the driving force and to the presence of pumping in two spatial directions. The dimensionless parameters of the problem are the flow rates in two perpendicular directions, the forcing intensity, and two spatial periods of the force. The Kolmogorov flow itself corresponds to the particular case when the force depends on a single coordinate and the mean drift is absent. When the forcing amplitude is increased, the basic stationary flow pattern, described by the explicit solution of the Navier-Stokes equations, displays structural rearrangements: isolated vortices appear on the background of the global flow. In the present study, we consider destabilization of the basic flow pattern: there, onset of time dependence can influence the previously reported unusual spectral and transport properties of Lagrangian dynamics. Analysis of possible stationary states, of their stability and forms of the secondary oscillatory and stationary modes is performed. Equations of fluid dynamics are solved numerically through the spectral and finite-difference methods. Stability of explicit stationary solutions with respect to small hydrodynamical perturbations is investigated.

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Onset of self-oscillations upon the loss of stability of spatially periodic two-dimensional viscous fluid flows relative to long-wave perturbations

Cited by 7 publications

References 5 publications

Instabilities, Bifurcations, and Nonlinear Dynamics in Two-Dimensional Generalizations of Kolmogorov Flow

Instabilities, Bifurcations, and Nonlinear Dynamics in Two-Dimensional Generalizations of Kolmogorov Flow

Long Wavelength Asymptotics of Self-Oscillations of Viscous Incompressible Fluid

Stability and nonlinear secondary modes of double-periodic flows with pumping

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