عنوان مقاله [English]
The propagation and run-up of long waves are interesting hydrodynamic phenomena that have received extensive attention from coastal engineering communities over the past few decades. Among various types of long wave, solitary waves are of crucial importance, due to their capability of being reasonably representative of giant waves, like tsunamis. This highlights the importance of investigating the nearshore behavior of solitary waves, as well as their run-up features.In this study, non-linear shallow water (NLSW) equations are considered as the mathematical tool governing the propagation and run-up of long waves. The analytical solutions to this system of hyperbolic conservation laws are limited to non-breaking waves. This shortcoming is mainly attributed to large gradients of flow shocks in the vicinity of the breaking wave front, which can collapse analytical attempts. Hence, numerical modeling is inevitable for breaking waves.To develop an efficient numerical model, the conservative form of NLSW was first discretized in space by the FORCE-MUSCL scheme and, then, integrated in time by the optimal third-order TVD Runge-Kutta method. The mentioned shocks were efficiently captured using the non-linear slope limiter SUPERBEE, which, not only controls these large gradients, but, also, preserves the smooth regions of the flow field in a truly physical manner.In this shock-capturing finite volume model, the steady state preserving property (C-property) was achieved with the aid of a surface gradient method. This property becomes important when modeling wave propagation over uneven bathymetries, and avoids the free surface being spuriously disturbed in such situations. A minimum water depth was also introduced to overcome the challenges of wave propagation over an initially dry beach, and to identify the moving shoreline boundary during the run-up process.Non-breaking and breaking solitary waves propagating in a region of constant depth and climbing up a plane sloping beach were considered as benchmarks to assess the performance of the present numerical model. The experimental data and analytical solution for this run-up problem have been reported in the literature.For non-breaking waves, the observed agreement between the computed results, experimental data and analytical solution, was acceptable. Interestingly, the numerical results appear to be closer to experiments than the analytical counterpart. This trend can be related to numerical dissipation of the scheme, which resembles the real fluid effects arising in experimental data.In the case of breaking waves, though the overall agreement with experimental data was satisfactory, the position of wave breaking was not well predicted by the model. This inaccuracy is mainly due to lack of dispersive effects in the governing equations. Nevertheless, the model provides a more accurate wave profile at the instance of bore collapse. This superior result should be ascribed to the conservation property of the scheme, which avoids the mass being lost after wave breaking. This property was also confirmed by a negligible mass loss error.