Spilling breakers : applications to Favre waves and to the shoaling and the breaking of the solitary wave

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

Numerical Optimization of PID-Regulator for Object with Distributed Parameters

The control of dynamic objects is very important for technologies and technical sciences. The difficulty of the control depends on the complexity of object mathematical model. Objects with distributed parameters are most difficult for control with feedback loop. The mathematical model of such object is much more complicated. There are two method of calculating regulator. The first method is analytical calculation of the coefficients. It can be used only for relatively simple object. The second method is numerical optimization. It can be used even with complex object. But object with distributed parameters has such complex model that in this case the use of the method of numerical optimization is a difficult problem. Only approximate model of such object can be used for the optimization with the help of the simulation program. Newest results in numerical optimization are based on some specific cost functions. Various composite cost functions are effective tools for regulator calculation. But even these tools are not sufficient. The paper researches possibilities of the use of approximate mathematical model of objects with distributed parameters and various cost functions, including new ideas, for the calculation of the regulator for system with such object. With all known methods, it was impossible to reduce the overshooting less than 22%. The proposed new cost function allows reducing of the overshoot to a value of about 11%, which is be preferred for many applications. The proposed method adds an arsenal of techniques of control of complex dynamic objects

INTERACTION OF A SUBSURFACE BUBBLE LAYER WITH LONG INTERNAL WAVES

International audienceA two-layer model describing the interaction of a shear bubble layer formed by breaking waves and an underlying potential layer is derived in shallow water approximation. A non-hydrostatic formulation taking into account the entrainment effects in shear flows is proposed. Time and space periodic solutions are found, and some basic problems (the formation of bores and periodic structures from a uniform flow) are numerically solved