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

Abstract

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.