A numerical study of the run-up and the force exerted on a vertical wall by a solitary wave propagating over two tandem trenches

The propagation and transformation of water waves over varying bathymetries is a subject of fundamental interest to ocean, coastal and harbor engineers. The specific bathymetry considered in this paper consists of one or two, naturally formed or man-made, trenches. The problem we focus on is the transformation of an incoming solitary wave by the trench(es), and the impact of the resulting wave system on a vertical wall located after the trench(es). The maximum run-up and the maximum force exerted on the wall are calculated for various lengths and heights of the trench(es), and are compared with the corresponding quantities in the absence of them. The calculations have been performed using the fully nonlinear water-wave equations, in the form of the Hamiltonian coupled-mode theory, recently developed by Papoutsellis et al. (Eur J Mech B/Fluids 72:199–224, 2018). Comparisons of the calculated free-surface elevation with existing experimental results indicate that the effect of the vortical flow, inevitably developed within and near the trench(es) but not captured by any potential theory, is not important concerning the frontal wave flow regime. This suggests that the predictions of the run-up and the force on the wall by means of nonlinear potential theory are expected to be nearly realistic. The main conclusion of our investigation is that the presence of two tandem trenches in front of the wall may reduce the run-up from (about) 20 to 45% and the force from 15 to 38%, depending on the trench dimensions and the wave amplitude. The percentage reduction is greater for higher waves. The presence of only one trench leads to reductions 1.4–1.7 times smaller.