E. van Groesen

Qualitative comparisons of experimental results on deterministic freak wave generation based on modulational instabilty

A number of qualitative comparisons of experimental results on unidirectional freak wave generation in a hydrodynamic laboratory are presented in this paper. A nonlinear dispersive type of wave equation, the nonlinear Schrodinger equation, is chosen as the theoretical model. A family of exact solutions of this equation the so-called Soliton on Finite Background describing modulational instability phenomenon is implemented in the experiments. It is observed that all experimental results show an amplitude increase according to the phenomenon. Both the carrier wave frequency and the modulation period are preserved during the wave propagation. As predicted by the theoretical model, a phase singularity is also observed in the experiments. Due to frequency downshift phenomenon, the experimental signal and spectrum lose their symmetric property. Another qualitative comparison indicates that the Wessel curves for the experimental results are the perturbed version of the theoretical ones.

Accurate modelling of uni-directional surface waves

This paper shows the use of consistent variational modelling to obtain and verify an accurate model for uni-directional surface water waves. Starting from Lukes variational principle for inviscid irrotational water flow, Zakharovs Hamiltonian formulation is derived to obtain a description in surface variables only. Keeping the exact dispersion properties of infinitesimal waves, the kinetic energy is approximated. Invoking a uni-directionalization constraint leads to the recently proposed AB-equation, a KdV-type of equation that is also valid on infinitely deep water, that is exact in dispersion for infinitesimal waves, and that is second order accurate in the wave height. The accuracy of the model is illustrated for two different cases. One concerns the formulation of steady periodic waves as relative equilibria; the resulting wave profiles and the speed are good approximations of Stokes waves, even for the Highest Stokes Wave on deep water. A second case shows simulations of severely distorting downstream running bi-chromatic wave groups; comparison with laboratory measurements show good agreement of propagation speeds and of wave and envelope distortions.

Localization in Spatial-Spectral Method for Water Wave Applications

In the description of water waves, dispersion is one of the most important physical properties; it specifies the propagation speed as function of the wavelength. Accurate modelling of dispersion is essential to obtain high-quality wave propagation results. The relation between speed and wavelength is given by a non-algebraic relation; for finite element/difference methods this relation has to be approximated and leads to restrictions for waves that are propagated correctly. By using a spectral implementation dispersion can be dealt with exactly above flat bottom using a pseudo-differential operator so that all wavelengths can be propagated correctly. However, spectral methods are most commonly applied for problems in simple domains, while most water wave applications need complex geometries such as (harbour) walls, varying bathymetry, etc.; also breaking of waves requires a local procedure at the unknown position of breaking. This paper deals with such inhomogeneities in space; the models are formulated using Fourier integral operators and include non-trivial localization methods. The efficiency and accuracy of a so-called spatial-spectral implementation is illustrated here for a few test cases: wave run-up on a coast, wave reflection at a wall and the breaking of a focussing wave. These methods are included in HAWASSI software (Hamiltonian Wave-Ship-Structure Interaction) that has been developed over the past years.

Variational Boussinesq model for simulation of coastal waves and tsunamis

In this paper we describe the basic ideas of a so-called Variational Boussinesq Model which is based on the Hamiltonian structure of gravity surface waves. By using a rather simple approach to prescribe the profile of vertical fluid potential in the expression for the kinetic energy, we obtain a set of dynamic equations extended with one additional elliptic equation for the amplitude of the vertical profile. All expressions in the energy contain at most first order derivatives, which makes a numerical implementation with finite elements relatively easy. The applicability of the code is illustrated for two different applications in this paper. One application deals with tsunami simulations, for which we show the phenomenon of tsunami waveguiding before the coast of Lampung in Indonesia. Another application deals with simulations of coastal waves entering the small harbour of Cilacap on the south-coast of Java, Indonesia; we will show that the simulations indicate a resonance phenomenon in the small inner harbour.

Numerical solution for Laplace equation with mixed boundary condition for ship problem in the sea

One interesting phenomena is investigating the movement of ships at the sea. To start with the investigation in modelling of this problem, we will assume that the ship is only a one-dimensional object that is floating on the sea surface. Similarly, we assume that the water flow is uniform in parallel directions to the ship. Therefore, we simply use the two-dimensional Laplace equation in this problem. In the section that describes the surface of sea, Neumann boundary condition is imposed in part related to the ship and the Dirichlet boundary condition for others. Then on the other three boundaries, we imposed the Neumann boundary condition by assuming that the water does not flow on the bottom, and both end. The model is solved by numerical solution using the finite element method. Velocity potential solution on the whole domain is demonstrated as a result of the implementation of the finite element method. In this paper, we initiate an investigation with assuming that the ship is on the water so that the domain of the Laplace equation is rectangular. Then we assume the drift ship. Furthermore, we also study the dependence of width and depth of the domain to the velocity potential.

Simulations for design and reconstruction of breaking waves in a wavetank

To determine forces on fixed and flexible structures such as wind mills and oil platforms, experiments in wave tanks are useful to investigate the impacts in various types of environmental waves. In this paper we show that the use of an efficient simulation code can optimize the experiments by designing the influx such that waves will break at a predefined position of the structure. The consecutive actual measurements agree well with the numerical design of the experiments. Using the measured elevation close by the wave maker as input, the software recovers the experimental data in great detail, even for rather short (up to L/D=1) and very steep breaking waves with steepness parameter (ak) till 0.4. n nThe experiments were carried out in the TUD-wavetank and the simulation is done by HaWaSSI-AB, a spatial-spectral implementation of a Hamiltonian Boussinesq model with an eddy-viscosity breaking mechanism that is initiated by a kinematic breaking condition.

Hamiltonian Boussinesq simulations for waves entering a harbor with access channel

Numerical simulations are often used to predict the deformation of waves and their impact on structures in harbors and access channels. An underprediction of these waves could lead to structural failure or ship accidents, so accurate numerical models should be capable of capturing the complex physical processes correctly. In this contribution, the authors consider wave penetration into a harbor with a complex bathymetry from an access channel using numerical simulations with wave models in a software program. The wave models were derived based on consistent modeling using a variational principle for water waves to produce the dynamic equations in Hamiltonian form. By approximating the Hamiltonian, the so-called analytic Boussinesq (AB) model was used and discretized into a global spatial-spectral numerical method. Numerical results compared with laboratory experiments show better performance than in other publications on the same application.

Freak Wave Formation from Phase Coherence

This paper describes freak waves by a (pseudo-)maximal wave proposed in [1]. The freak wave is a consequence of a group event that is present in a time signal at some position and contains successive high amplitudes with different frequencies. The linear theory predicts the position and time of the maximal amplitude wave quite well by minimizing the variance of the total wave phase of the given initial signal. The formation of the freak wave is shown to be mainly triggered by the local interaction of waves evolving from the group event that already contained large local energy. In the evolution, the phases become more coherent and the local energy is focussed to develop a larger amplitude. We investigate two laboratory experimental signals, a dispersive focussing wave with harmonic background and a scaled New Year wave. Both signals generate a freak wave at the predicted position and time and the freak wave can be described by a pseudo-maximal wave with specific parameters.

Wave run-up of a possible Anak-Krakatau tsunami on planned and optimized Jakarta Sea Dike

The infrastructural plans in the Jakarta Bay to reduce risks of flooding in Jakarta city comprise a large Sea Dike that encloses a retention lake. Part of the planned dike has the shape of the iconic Garuda bird. This paper shows that if in the future an explosion of Anak Krakatau will occur with strength 1/4th of the original Karkatau 1883 explosion, wave crests of 11m and troughs of 6m may collide against the bird’s head. As an alternative example, a more optimized design of the dike is constructed that reduces the maximal wave effects considerably.