Growth of unsteady wave groups by shear flows

Abstract: A weakly nonlinear theory has been proposed and developed for calculating the energy- transfer rate to individual waves in a group. It is shown what portion of total energy- transfer rate, over the envelope of wave group, affects individual waves in the group. From this an expression for complex phase speed of individual waves is calculated. It is deduced that each wave in a group does not grow at the same rate. It is shown that the critical layer is no longer symmetrical compared with the ideal monochromatic waves. This asymmetry causes the critical layer height to be lower over the downwind part. Therefore the positive growth of the individual waves on the upwind part of the wave group exceeds the negative growth on the downwind part (which would not be true if $z_c$, where the mean flow $U$ is equal to the speed of the wave propagation, was the same over the whole group). This leads to the critical layer group effect producing a net horizontal force on the waves, in addition to the sheltering effect. Computational simulations over a non-growing wave group is also presented, which confirms the above postulation made by Sajjadi, Hunt and Drullion (2014) (SHD).