A hybrid SPH model of the Navier-Stokes and Boussinesq equations for simulating wave propagation and deformation

A two-way hybrid model based on the smoothed particle hydrodynamics (SPH) method is proposed in this paper to simulate wave propagation and deformation effectively. The hybrid model is composed of two submodels: the first submodel is based on the Boussinesq equations (B-SPH) for simulating wave propagation, and the second submodel is based on the Navier-Stokes (N-S) equations (NS-SPH) for simulating wave deformation caused by interactions with structures. The exchange of information between the two submodels is realized through an open-boundary method: the results derived from the B-SPH submodel are transferred to the open boundaries of the NS-SPH submodel through a relaxation function; the information derived from the NS-SPH submodel is matched to the open boundaries of the B-SPH submodel through the continuity equation. To obtain a stable pressure field in the NS-SPH submodel, the particle shifting technique (PST) is implemented. The subsequent validation of the hybrid model includes four typical numerical examples related to solitary and regular waves. The agreement between the numerical results and the experimental data indicates that the proposed hybrid model is suitable for simulating the propagation and deformation of shallow water waves. Compared with the traditional NS-SPH model, which uses N-S equations to describe the whole fluid domain, the hybrid model has greater computational efficiency.