A two-layer long wave approximation of the homogeneous Euler equations for a free surface flow over a rigid bottom is derived. The upper layer is turbulent and is described by depth averaged equations for the layer thickness, the average fluid velocity inside the layer, and the fluid turbulent energy. The lower layer is almost potential and can be described by Serre–Su–Gardner–Green–Naghdi equations (second order shallow water approximation with respect to the parameter H/L where H is a characteristic water depth, and L is a characteristic wave length). The interaction between the layers is due to the turbulent mixing. The dynamics of the interface separating two layers is governed by the turbulent energy of the upper layer. Stationary supercritical solutions to this model are first constructed , containing, in particular, a local subcritical zone at the forward slope of the wave. Such a local subcritical zone corresponds to an intense increasing of a turbulent layer thickness and can thus be associated with the spilling breaker formation. Non-stationary model was then numerically solved and compared with experimental data for the following two problems. The first one is the study of surface waves resulting from the interaction of a uniform free surface fluid flow with an immobile wall ('the water hammer problem with a free surface'). These waves are sometimes called 'Favre waves' in homage to Henry Favre and his contribution to the study of this phenomena. When the Froude number is between 1 and approximately 1.3, the undular bore appears. The turbulent energy of the flow is localized at the wave crests. The characteristics of the leading wave are in good agreement with the experimental data by Favre (1935) and Treske (1994). When the Froude number is between 1:3 and 1:4, the transition from the undular bore to the breaking (monotone) bore occurs. In the breaking bore, the turbulent energy is localized at the front of the bore. The shoaling and the breaking of the solitary wave propagating in a mild - slope (1/60) long channel (300 m) was then studied. Comparisonwith the experimental data by Hsiao et.al. (2008) show a very good agreement of the wave profile evolution.
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