Modelling turbulence generation in solitary waves on shear shallow water flows

We derive a dispersive model of shear shallow water flows which takes into account a non-uniform horizontal velocity. This model generalizes the Green-Naghdi model to the case of shear flows. Besides the classical dispersion term in the Green-Naghdi model related to the acceleration of the free surface, it also contains a new dispersion parameter related to the flow structure. This parameter is related to the second moment of the velocity fluctuation with respect to the vertical coordinate. The distinction between shearing and turbulence based on the scale of variation of the velocity fluctuation is proposed. In particular, an equation for the turbulence generation is derived. Solitary waves for this model are obtained in explicit form. Comparison of solitary wave profiles with experimental ones is also performed. The agreement is very good apart from the small region near the top of the wave.