Nonlinear instability of a fluid layer flowing down a vertical wallunder imposed time-periodic perturbations

We investigate the nonlinear instability of a fluid layer flowing down a vertical wall and subjected to a continuously imposed oscillatory pressure gradient. In each case the frequency ω of oscillation is fixed. The flow instability is investigated numerically by means of the Benney equation with a space- and time-dependent inhomogeneous term. The wave evolution is followed in space and time for different Reynolds numbers. It is found that for a range of wave numbers near critical, saturation occurs only for frequencies ω smaller than a critical value. For larger frequencies, the waves grow unboundedly everywhere. For ω smaller than that value, subharmonics occur between the curves of criticality and subcriticality. An increase in Reynolds number leads the instability to the region of subcriticality where wave subharmonics appear for all the frequencies investigated in this paper. Larger Reynolds numbers give additional subharmonics until a magnitude is reached, at which the flow becomes chaotic. As the frequency increases above a critical value, subharmonics are more difficult to find in the supercritical region.