Numerical Investigation of the Non-Linear Mechanics of Wave Disturbances in Plane Poiseuille Flows

摘要

The response of a plane Poiseuille flow to disturbances of various initial wavenumbers and amplitudes is investigated by numerically integrating the equation of motion. It is shown that for very low amplitude disturbances the numerical integration scheme yields results that are consistent with those predictable from linear theory. It is also shown that because of non-linear interactions a growing unstable disturbance excites higher wavenumber modes which have the same frequency, or phase velocity, as the primary mode. For very low amplitude disturbances these spontaneously generated higher wavenumber modes have a strong resemblance to certain modes computed from the linear Orr- Sommerfeld equation. In general it is found that the disturbance is dominated for a long time by the primary mode and that there is little alteration of the original parabolic mean velocity profile. There is evidence of the existence of an energy equilibrium state which is common to all finite-amplitude disturbances despite their initial wavenumbers. This equilibrium energy level is roughly 3-5% of the energy in the mean flow which is an order of magnitude higher than the equilibrium value predicted by existing non-linear theories.