Nonlinear Dynamics of Anisotropic Disturbances in Plane Couette Flow

Abstract

The dynamics of disturbances to a two-dimensional inviscid Couette flow is examined under the assumption that the disturbances have a cross-stream scale which is asymptotically smaller than their streamwise scale. Such anisotropic disturbances emerge naturally from isotropic perturbations after long times because of the shearing effect of the basic flow. An asymptotic equation describing the nonlinear evolution of anisotropic disturbances is derived using aregular-perturbation technique. The close relationship between this equation and those found in critical-layer problems is discussed. The case of doubly periodic disturbances is examined in detail, since it leads to a remarkable dynamics: formally, the evolution is discontinuous in time and is given by a sequence of jumps which take place when the time is a rational number (multiplied by a fixed geometric factor). The interpretation of such a dynamical system, in a sense intermediate between discrete and continuous systems, is discussed. The asymptotic model is used to study simple interactions between two sheared (Fourier) modes and to investigate the hydrodynamic echo effect. Analytical results are complemented by results of direct numerical simulations.