On modeling of undular bores based on the second approximation of the shallow water theory

Abstract

It is studied the possibility of modeling of undular bores on the basis of the second approximation of the shallow water theory. The classical differential Green-Naghdi model cannot be used for correct numerical simulation of wave flows with undular bores. The reason for this is that this model is derived within the framework of the long-wave approximation, by virtue of which the characteristic depth of the stream is much less than the characteristic length of the surface waves, which is not performed in the undular bore front. An integro-differential Green-Naghdi model is proposed for numerical simulation of undular bores. In this model we used the divergent differential form of the continuity equation and the integral conservation law of horizontal momentum. This model is derived from two-dimensional integral conservation laws of mass and momentum, describing a plane-parallel flow of an ideal incompressible fluid over a horizontal bottom. The basis of this conclusion is the concept of a local hydrostatic approximation, which generalizes the concept of the long-wave approximation and is used to justify the applicability of shallow water models to describe wave flows with the hydraulic bores.