Resumen
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.