Run-up of non-breaking waves — A finite-element approach

Abstract

An important aspect of the study of wave-structure interaction is the extent to which the wave-induced motion carries water up the structures. This is commonly referred to as the wave run-up. Several studies in the past have produced numerical solutions to this problem under restrictive assumptions. One of the important restrictions in those studies is that the bottom is gently sloping — such as slopes of 1 to 10 or flatter. The traditionally used shallow-water equations cannot be applied to regions where the vertical accelerations become significant. Such accelerations are significant when water moves over dykes or levees with fairly steep slopes. In the present study an approach has been made to develop the appropriate momentum equation which will account for the vertical acceleration and hence replace the shallow-water equation for the situation mentioned above. The newly developed equation is also non-linear in nature. This, along with the continuity equation, forms a system of coupled non-linear time-dependent partial differential equations for the problem of wave run-up. In order to solve this system the Finite-Element Technique is adopted for the space domain. The Finite Difference Euler Predictor-Corrector is used to step up the variables in the time domain. Such a combination, known as the semi-discrete method, is found to produce a powerful model for the wave run-up. An important aspect of this problem is that the problem domain changes with time. In other words, it is a moving boundary problem and gives rise to peculiarities in handling the boundary conditions. In the present model this aspect is handled using a time-dependent end-element. Numerical results are obtained using sinusoidal waves, solitary waves (tsunamis) and surges as the forcing functions. Single-slope faces as well as composite-slope faces (as is often the case in levees) are included in the study. The results of these cases (i.e. particle velocities and heights of run-up) are given in the form of charts. In the design of coast protective structures with sloping sides, this model will be found particularly useful in predicting the run-up and particle velocities.