INSTABILITIES OF A HORIZONTAL SHEAR-FLOW WITH A FREE-SURFACE

The simple shear-flow model of Stern & Adam (1973), in which a layer o f uniform vorticity and depth overlies an infinitely deep fluid, is he re extended by the addition of an upper fluid layer of uniform thickne ss and constant velocity. In this way many experimentally observed vel ocity profiles can be approximated. The normal mode instabilities of s uch 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 H-1/H-2, where H-1 and H-2 denote the m ean depths of the surface layer and the base of the shear layer, respe ctively. It is found that two branches of instability which are distin ct when H-1/H-2 is moderate or small can become merged when H-1/H-2 gr eater than or equal to 0.4924. Also calculated are the fastest-growing modes, and their wavelengths. The results are applied to some example s of surface flows generated by towed bodies, and to steady spilling b reakers.