Modelling and simulation of a wave energy converter

In this work we present the mathematical model and simulations of a particular wave energy converter, the so-called oscillating water column. In this device, waves governed by the one-dimensional nonlinear shallow water equations arrive from offshore, encounter a step in the bottom and then arrive into a chamber to change the volume of the air to activate the turbine. The system is reformulated as two transmission problems: one is related to the wave motion over the stepped topography and the other one is related to the wave-structure interaction at the entrance of the chamber. We finally use the characteristic equations of Riemann invariants to obtain the discretized transmission conditions and we implement the Lax-Friedrichs scheme to get numerical solutions.

Application of particle swarm optimization in optimal placement of tsunami sensors

Rapid detection and early warning systems demonstrate crucial significance in tsunami risk reduction measures. So far, several tsunami observation networks have been deployed in tsunamigenic regions to issue effective local response. However, guidance on where to station these sensors are limited. In this article, we address the problem of determining the placement of tsunami sensors with the least possible tsunami detection time. We use the solutions of the 2D nonlinear shallow water equations to compute the wave travel time. The optimization problem is solved by implementing the particle swarm optimization algorithm. We apply our model to a simple test problem with varying depths. We also use our proposed method to determine the placement of sensors for early tsunami detection in Cotabato Trench, Philippines.

Long Wave Run-Up Resonance in a Multi-Reflection System

Wave reflection and wave trapping can lead to long wave run-up resonance. After reviewing the theory of run-up resonance in the framework of the linear shallow water equations, we perform numerical simulations of periodic waves incident on a linearly sloping beach in the framework of the nonlinear shallow water equations. Three different types of boundary conditions are tested: fully reflective boundary, relaxation zone, and influx transparent boundary. The effect of the boundary condition on wave run-up is investigated. For the fully reflective boundary condition, it is found that resonant regimes do exist for certain values of the frequency of the incoming wave, which is consistent with theoretical results. The influx transparent boundary condition does not lead to run-up resonance. Finally, by decomposing the left- and right-going waves into a multi-reflection system, we find that the relaxation zone can lead to run-up resonance depending on the length of the relaxation zone.

A mathematical formulation for estimating maximum run-up height of 2018 Palu tsunami

Run-up is defined as sea wave up-rush on a beach. Run-up height is affected by many factors, including the shape of the bay. As an archipelagic country, Indonesia consists of thousands of islands with bays of diverse profiles, including Palu Bay, which is a well-known example of a bay with a drastically-increasing wave run-up height. In the case of the 2018 Palu tsunami, scientists found that the incident wave was amplified by the shape of the bay. The amplifying wave played a large role in the significant increase of run-up height. The run-up in question caused severe inundation, which led to a high number of casualties and damages. Therefore a mathematical model will be constructed to investigate the wave run-up. The bay's geometry will be approximated using three linearly-inclined channel types: one of parabolic cross-section, one of triangular cross-section, and a plane beach. We use the generalized nonlinear shallow water equations, which is then solved analytically using a hodograph-type transformation. As a result, the nonlinear shallow water equation system can be reduced to a one-dimensional linear equation system. Assuming the incident wave is sinusoidal, we can obtain a simple formula for calculating maximum run-up height on the shoreline.

Mathematical Model of Geophysical Fluid Flow over Variable Bottom Topography

In this paper, the bottom topography of a geophysical fluid flow is modelled in the presence of Coriolis force by the nonlinear shallow water equations. These equations, which are a system of three partial differential equations in two space dimensions, are solved using the perturbation method. The Effects of the Coriolis force and the bottom topography for particular initial flows on the velocity components and different kind of flow patterns possible in geophysical fluid flow have been studied and the results illustrated graphically.

Extreme Inundation Statistics on a Composite Beach

The runup of initial Gaussian narrow-banded and wide-banded wave fields and its statistical characteristics are investigated using direct numerical simulations, based on the nonlinear shallow water equations. The bathymetry consists of the section of a constant depth, which is matched with the beach of constant slope. To address different levels of nonlinearity, time series with five different significant wave heights are considered. The selected wave parameters allow for also seeing the effects of wave breaking on wave statistics. The total physical time of each simulated time-series is 1000 h (~360,000 wave periods). The statistics of calculated wave runup heights are discussed with respect to the wave nonlinearity, wave breaking and the bandwidth of the incoming wave field. The conditional Weibull distribution is suggested as a model for the description of extreme runup heights and the assessment of extreme inundations.