Solution of the Inviscid Initial Value Problem for an Infinitesimal Disturbance on a Piecewise Linear Approximation of Plane Poiseuille Flow

摘要

The time evolution of a small disturbance on a piecewise linear mean flow, approximating a parabolic profile, is calculated using Fourier transform methods. The solution is found to consist of two parts: one dispersive, incorporating the spreading of waves, and one convective, characterized by a convection of the disturbance with the local mean velocity. Two dispersive modes are found: one symmetric with respect to the channel center line and one antisymmetric. The symmetric mode is in fair agreement with the symmetric mode obtained for inviscid parabolic flow, whereas the anti-symmetric mode is misrepresented. The solution to the horizontal velocities is found to contain a purely three dimensional part. This results from fluid elements retaining part of their horizontal momentum as they are lifted up by the time integrated effect of the vertical velocity.