The Inviscid Initial Value Problem for a Piecewise Linear Mean Flow

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; 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 dispersivity of the symmetric mode is in fair agreement with the symmetric mode obtained for inviscid parabolic flow, whereas the antisymmetric mode is misrepresented. One of the parts of the solution to the horizontal velocities is found to be purely three‐dimensional. 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. Calculations of the development of a particular disturbance modeling two vortex pairs are also made. The results show that the dispersive part, although decaying, is largest for the vertical velocity. For the horizontal velocity the three‐dimensional lift‐up effect provides the largest amplitudes. This part does not show any sign of decay, in agreement with earlier analysis by Gustavsson [8] and Landahl [16]. This last effect partly explains the sensitivity to three‐dimensional disturbances seen in transition experiments and calculations. Comparison of the solution to a full numerical simulation using the Navier‐Stokes equations shows good agreement for short times.