Deterministic wave trains

Applying the described non-linear approach all kinds of waves can be generated in a model tank:

•wave packets (Fig。 9)

•extreme waves such as ”Three Sisters” (Figs。 7, 12)

•storm seas (Fig。 6)

•random seas with embedded high wave sequences (Figs。 8, 10,  11)

•regular waves with embedded high wave groups (Figs。 7, 12)

•realization of natural wave scenarios (Figs。 8, 11)

We call these wave trains ”deterministic wave trains” and give examples of each of them in the following appli- cations。 The first application example is the validation of numerical wave tanks by modelling non-linear wave propagation。 Two examples of numerical wave tanks are given here。

Validation of a numerical wave tank based on potential theory

The first numerical wave tank is based on a Finite El- ement Method discretization of the fluid domain。 The two dimensional non-linear free surface flow problem is solved in time domain using potential theory: the fluid is inviscid and incompressible, and the flow is irrota- tional。 Wave breaking is not considered。 The atmo- spheric pressure above the free surface is constant and surface tension is neglected。 Hence, the flow field can be described by a velocity potential which satisfies the Laplace equation。 At each time step the velocity poten- tial is calculated in the entire fluid domain, Clauss and Steinhagen (1999)。

To develop the solution in time domain the fourth-order Runge-Kutta formula is applied。  At  each time step   a

Fig。  6:  Comparison  of  numerical  (potential the- ory/ FEM) and experimental wave sequence generated by the same wave maker signal calculated by the mod- ified non-linear theory (TP = 14。6 s, Hs = 15。3 m): registrations at different positions。

Fig。 7: Regular wave with wave packet for the in- vestigation of large rolling and capsizing of ships (see Fig。 12)。 Numerical simulation based on potential the- ory/ FEM。 Experimental data provided by the modified non-linear approach (scale 1:29)。

new boundary-fitted mesh is created。 The procedure is repeated until the desired time step is reached, or the wave train becomes unstable and breaks。 The numerical wave tank is able to simulate wave generators of piston type, single flap and double flap (and combinations)。 A complete description of this numerical wave tank is found in Steinhagen (2001)。

Fig。 6 presents numerical results as well as experi- mental data generated by the modified non-linear ap- proach to validate the numerical wave  tank。  The storm sea  realization  (TP  =  14。6  s,  Hs  =  15。3  m) has been modelled at the Hamburg Ship Model Basin (HSVA, length 300 m, width 18 m, water depth 5。6 m, equipped with a double flap wave generator) at a scale of 1:34。

Fig。 7 presents a superposition of a regular wave train with a wave packet。 This tailored wave sequence is used for the investigation of large rolling and capsizing of ships, Clauss and Hennig (2003)。 As will be shown in Fig。 12 this irregular wave train turns out to become a rather regular wave with an integrated extreme wave if transformed to a moving reference frame。

Fig。 8 shows a simulation of the so-called New Year

Wave。 This rogue wave was reported from the jacket platform Draupner in the North Sea on January 1st, 1995, Haver (2000)。 The platform was hit by a giant wave with a wave height of 25。6 m (significant wave height Hs = 11。92 m) that caused severe damage。 The wave is modelled in the wave tank at scale 1:81 (tank dimensions: length 80 m, width 4 m, water depth 1。5 m, piston type wave generator)。

Figs。 6-8 document the universality of the numerical wave tank for the calculation of wave evolution for dif- ferent wave tanks with different water depth and types of wave generators。 Like the modified non-linear theory the numerical wave tank predicts the non-linear evolu- tion of wave trains and the wave/ wave interaction quite well。 As the potential field is calculated at each time step, also velocity, acceleration and pressure fields are known。 Only the modified non-linear theory is able to provide control signals both for generating determinis- tic wave trains in a model tank and as an input for the moving wall boundary of a numerical wave tank (Fig。 8)。 To overcome the limitations of wave breaking a numer- ical wave tank using a commercial computational fluid dynamics (CFD) solver is introduced in the next sec- tion。