Marco Klein

Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test

Under suitable assumptions, the nonlinear dynamics of surface gravity waves can be modeled by the one-dimensional nonlinear Schrödinger equation. Besides traveling wave solutions like solitons, this model admits also breather solutions that are now considered as prototypes of rogue waves in ocean. We propose a novel technique to study the interaction between waves and ships/structures during extreme ocean conditions using such breather solutions. In particular, we discuss a state of the art sea-keeping test in a 90-meter long wave tank by creating a Peregrine breather solution hitting a scaled chemical tanker and we discuss its potential devastating effects on the ship.

Simulations and experiments of short intense envelope solitons of surface water waves

The problem of existence of stable nonlinear groups of gravity waves in deep water is considered by means of laboratory and numerical simulations with the focus on strongly nonlinear waves. Wave groups with steepness up to Acrωm2/g ≈ 0.30 are reproduced in laboratory experiments (Acr is the wave crest amplitude, ωm is the mean angular frequency, and g is the gravity acceleration). We show that the groups remain stable and exhibit neither noticeable radiation nor structural transformation for more than 60 wavelengths or about 15-30 group lengths. These solitary wave patterns differ from the conventional envelope solitons, as only a few individual waves are contained in the group. Very good agreement is obtained between the laboratory results and numerical simulations of the potential Euler equations. The envelope soliton solution of the nonlinear Schrodinger equation is shown to be a reasonable first approximation for specifying the wave-maker driving signal. The short intense envelope solitons possess ver...

FORMATION OF EXTRAORDINARILY HIGH WAVES IN SPACE AND TIME

The existence of freak waves is indisputable due to observations, registrations, and severe accidents. The occurrence of extreme waves, their characteristics and their impact on offshore structures is one of the main topics of the ocean engineering research community. Real sea measurements play a major role for the complete understanding of this phenomenon. In the majority of cases only single point registrations of real sea measurements are available which hinders to draw conclusions on the formation process and spatial development in front of and behind the respective registration points. One famous freak wave is the “New Year Wave”, recorded in the North Sea at the Draupner jacket platform on January 1st, 1995. This wave has been reproduced in a large wave tank and measured at different locations, in a range from 2163 m (full scale) ahead of to 1470 m behind the target position — 520 registrations altogether. Former investigations of the test results reveal freak waves occurring at three different positions in the wave tank and these extreme waves are developing mainly from a wave group. The possible physical mechanisms of the sudden occurrence of exceptionally high waves have already been identified — superposition of (nonlinear) component waves and/or modulation instability (wave-current interaction can be excluded in the wave tank). This paper presents experimental and numerical investigations on the formation process of extraordinarily high waves. The objective is to gain a deeper understanding on the formation process of freak waves in intermediate water depth such as at the location of the Draupner jacket platform where the “New Year Wave” occurred. The paper deals with the propagation of large amplitude breathers. It is shown that the mechanism of modulation instability also leads to extraordinarily high waves in limited water depth. Thereby different carrier wave length and steepnesses are systematically investigated to obtain conclusions on the influence of the water depth on the modulation instability and are accompanied by numerical simulations using a nonlinear potential solver.Copyright

Influence of the Bow Shape on Loads in High and Steep Waves

To ensure survival of floating structures in rough seas, a precise knowledge of global and local loads is an inevitable integral part for safe design. One of the key parameters is the vertical bending moment. Not only vertical forces but — as previous investigations revealed — also longitudinal forces significantly contribute to the vertical wave bending moment. Three segmented ships, equipped with force transducers, are investigated systematically in high and steep regular waves and in harsh wave environments at several cruising speeds to identify the structural loads. The model tests are carried out in the seakeeping basin of the Technical University Berlin at a scale of 1:70. To cover possible influences of the bow geometry, three different types of vessels are chosen, a bulk carrier with a full bow, a Ro/Ro vessel and a container vessel with a V-shaped frame design. For identifying the influence of the wave height and steepness on the vertical bending moment, model tests in regular waves with different crest/trough asymmetries are performed with the Ro/Ro vessel and the bulk carrier. The program can be subdivided into three parts, each characterized by the same wave lengths with varying wave steepness. The first test series includes regular waves with small amplitudes, thus linear wave theory can be applied. In the second part the same (regular) wave lengths have been generated with increased wave heights, i.e. increasing crest/trough asymmetries and wave profiles within Stokes II domain. During the last part of the experimental program the wave heights are further increased to reach wave profiles within Stokes III domain. For the evaluation of the test results in regular waves — in particular in high steep waves — the results are compared to the respective Response Amplitude Operator determined by the transient wave package technique. Here the focus lies on the asymmetry of the hogging and sagging loads with respect to the wave steepness and the bow geometry of the investigated ship models. Furthermore, the influence of the freeboard height on the vertical bending moment is analysed. For this purpose a container vessel is investigated with two different freeboard configurations in a harsh wave environment.Copyright

Time Domain Comparison With Experiments for Ship Motions and Structural Loads on a Container Ship in Abnormal Waves

The paper presents experimental results from model tests with a containership advancing in abnormal wave conditions and comparisons with numerical simulations. A nonlinear time domain method based on strip theory is used for the calculation of vertical ship responses induced by abnormal waves. This code combines the linear diffraction and radiation forces with dominant nonlinear forces associated with vertical response arising from Froude-Krylov forces, hydrostatic forces and shipping of green water. The time domain simulations are compared directly with experimental records from tests with a model of a container ship in deterministic waves for a range of Froude numbers. Extreme sea conditions were replicated by the reproduction of realistic abnormal waves like the New Year Wave and abnormal wave from North Alwyn. Head sea condition is considered and the comparisons include the wave elevation, the vertical motions of the ship and the vertical bending moment at midship.Copyright

INFLUENCE OF WAVE GROUP CHARACTERISTICS ON LOADS IN SEVERE SEAS

The precise knowledge of loads and motions in extreme sea states is indispensable to ensure reliability and survival of ships and floating offshore structures. In the last decades, several accidents in severe weather with disastrous consequences have shown the need for further investigations. Besides the sea state behavior and the local structural loads, one key parameter for safe ship design is the vertical bending moment. Previous investigations revealed that different ship design criteria, such as bow geometry and wave board height, affect the global loads significantly. Investigations in regular waves as well as in single high waves of vessels with different bow flares and freeboard heights show that the vertical bending moment increases significantly with increasing bow flare and freeboard height. Furthermore it became apparent that critical loads and motions do not have to come along with the highest wave which results in the main question of this paper:

Generation of Rogue Waves With Predefined Steepness

For the deterministic investigation of cause-reaction relationships and for analyzing structure responses due to special wave scenarios methods for the precise generation of tailored wave sequences are required. In this paper we present an optimization approach for the experimental generation of wave sequences with defined characteristics, in particular with predefined wave steepness. Firstly, target parameters, such as wave crest front steepness, wave height and wave period at a particular time are defined for a wave group. Starting with a random phase distribution in frequency domain the phases are optimized in order to satisfy the target parameter in time domain. From this optimized target wave sequence a control signal based on linear wave theory for the wave generator is derived and the wave sequence is registered in a physical wave tank. As nonlinear effects like wave-wave interaction and wave breaking are insufficiently covered by the generation process the registered wave train may differ from the target parameters defined. To improve the accuracy of the measured wave sequence at target location the control signal is iteratively improved by a fully automated optimization process, controlling the wave generator, measuring and analyzing the created waves and modifying the control signal. As a result of the optimization process, a control signal for a wave train, satisfying all target parameters is obtained. The method is applied to generate a breaking rogue wave with defined wave crest front steepness superimposed to irregular seas.

Laboratory and numerical study of intense envelope solitons of water waves: Generation, reflection from a wall, and collisions

The investigation of dynamics of intense solitary wave groups of collinear surface waves is performed by means of numerical simulations of the Euler equations and laboratory experiments. The processes of solitary wave generation, reflection from a wall, and collisions are considered. Steep solitary wave groups with characteristic steepness up to kAcr ≈ 0.3 (where k is the dominant wavenumber and Acr is the crest amplitude) are concerned. They approximately restore the structure after the interactions. In the course of the interaction with the wall and collisions, the maximum amplitude of the wave crests is shown to enhance up to 2.5 times. A standing-wave-like structure occurs in the vicinity of the wall, with certain locations of nodes and antinodes regardless the particular phase of the reflecting wave group. A strong asymmetry of the maximal wave groups due to an anomalous setup is shown in situations of collisions of solitons with different frequencies of the carrier. In some situations of head-on col...

Application of Higher Order Spectral Method for Deterministic Wave Forecast

This paper addresses the Higher Order Spectral (HOS) method as very fast and accurate non-linear method for deterministic wave forecast. The focus of the paper lies on wave propagation, with the objective to draw conclusions on the applicability of the HOS method for deterministic wave forecast. Systematic numerical and experimental investigations are conducted. The investigations comprise exact solutions of the nonlinear Schr ¨ odinger equation (NLS) as special non-linear wave groups, which are used as initial conditions for the subsequent simulations. Different parameters such as relative water depth, wave steepness and propagation distance are varied to evaluate the applicability of the HOS method. The results obtained are validated by experiments as well as fully non-linear simulations. It is shown that the HOS method is capable for non-linear wave forecast due to high accuracy and fast calculation time at once.

Application of Breather Solutions for the Investigation of Wave/Structure Interaction in High Steep Waves

During the design process of floating structures, different specifications have to be aligned such as the range of application, the warranty of economical efficiency as well as the reliability and are an inevitable integral part of the evaluation process during the design stage. The validation of the performance by means of model tests in terms of sea state behavior and the associated local and global structural loads are an important milestone within this process. Therefore it is necessary to determine an adequate test procedure which covers all essential areas of interest. Thereby one field of interest are limiting criteria of the design such as maximum local and global loads as well as maximum accelerations due to the impact of extraordinarily high waves, at which the floating structure has to survive. Different alternatives are available to conduct model tests in high, steep waves — transient wave packages, regular waves, irregular waves with random phases or more sophisticated deterministic tailored irregular wave sequences such as reproductions from numerical simulations and real-world measurements.This paper introduces a new approach for the systematic investigation of wave/structure interaction in high, steep waves. Exact solutions of the nonlinear Schrodinger equation — the so called breather solutions — are implemented for the generation of extraordinarily high waves. Three types of breather solutions are investigated in the seakeeping basin and to cover the full range of interest, each solution has been used to generate freak waves at certain frequencies.To evaluate the applicability of breather solutions for model tests two types of ships — a LNG Carrier and a Chemical Tanker — are investigated in the seakeeping basin. The ships are segmented and connected with strain gauges to detect the vertical wave bending moment. Furthermore, green water probes are installed on deck to evaluate the local impact on the bow of the freak waves. The obtained results are compared to investigations in regular waves with certain frequency and steepness as well as in real-world freak wave reproductions.Copyright