Gravitational and Shear Instabilities in Compound and Composite Channels

Linear analysis of gravitational instabilities in the presence of a shear layer and shear instabilities in the presence of a free surface is performed. This study is relevant to shallow mixing layers, such as flow in compound and composite channels and inflows at channel junctions. The variations of the channel bed, velocity profile, Froude number, and friction coefficients with the transverse (lateral) coordinate are considered. It is found that there is a threshold Froude number above which the flow is unstable with respect to gravity waves and below which the flow is unstable with respect to shear waves for a certain range of the bed friction number. For values of Froude number larger than the threshold value, the influence of the shear layer and channel walls on the characteristics of the gravitational instability is strong when the channel and the shear layer are of comparable width. This influence reduces as the channel becomes wider and disappears in the limit when the channel width becomes infinite. When the Froude number is below the threshold value, free surface deformation in the form of gravitational waves exerts a strong stabilizing influence on the shear instability. In particular, the value of the critical bed friction number decreases when either the Froude number of the fast stream (main channel) or the slow stream (flood plain) increases. That is, shallow mixing layers become more stable as the Froude number increases. Comparisons of the linear stability calculations with experimental data show reasonable agreement.