Instabilities of a horizontal shear flow with a free surface

Abstract

The simple shear-flow model of Stern & Adam (1973), in which a layer of uniform vorticity and depth overlies an infinitely deep fluid, is here extended by the addition of an upper fluid layer of uniform thickness and constant velocity. In this way many experimentally observed velocity profiles can be approximated. The normal mode instabilities of such a model can be found analytically, and their properties calculated through the solution of a quartic polynomial equation. The dispersion relation is here determined and illustrated in its dependence on the Froude number and on the ratio H₁/H₂, where H₁ and H₂ denote the mean depths of the surface layer and the base of the shear layer, respectively. It is found that two branches of instability which are distinct when H₁/H₂ is moderate or small can become merged when H₁/H₂[ges ]0.4924. Also calculated are the fastest-growing modes, and their wavelengths. The results are applied to some examples of surface flows generated by towed bodies, and to steady spilling breakers.