Modelling Extreme Wave Sequences for the Hydrodynamic Analysisof Ships and Offshore Structures

Gunther F. Clauss, Janou Hennig, Christian E. SchmittnerTechnische Universitat Berlin

Berlin, Germany

Abstract

For the analysis of non-linear processes such as largerolling and capsizing of ships as well for the evaluationof forces and motion behavior of offshore structuresin extreme sea states, experimental investigations arestill indispensable, both for the validation of numer-ical simulation tools and for basic insights into theunderlying mechanism. Especially in ship design nu-merical simulation tools have improved significantlyand are already considered routinely within the de-sign process but are still under development and re-quire further experimental confirmation.

One decisive point in such experimental investiga-tions is the generation of deterministic wave se-quences tailored for the individual test. This requiresmodelling of the non-linear wave propagation in or-der to know the wave evolution in space and timewhich allows the analysis of the non-linear process asa cause-reaction chain.

In this paper methods of analyzing non-linear wavepropagation are presented and compared to results oflinear wave theory as well as to corresponding mea-surements from wave tank experiments. A discussionof various practical applications closes the paper.

Keywords

Non-linear wave propagation, deterministic wavetrains, moving reference frame wave trains,RANSE/ VOF, numerical wave tank, wave gen-eration, model tests, intact stability

Introduction

The experimental investigation of extremewave/structure interaction scenarios puts highdemands on wave generation and calculation. Forthe deterministic analysis of motions and forcesof ships and offshore structures wave excitationdenotes the beginning of a complex cause reaction

chain. The problem gets complex when the structurewhich has to be analyzed is moving at constant ornon-constant speed since measurements in a modeltank cannot provide the wave train at the positionof the investigated model or in the moving referenceframe of a cruising ship.This paper recommends the use of an approach tak-ing into account both analytical models and empiri-cal terms for modelling non-linear wave propagation.This modified non-linear method uses linear wave the-ory as a backbone for non-linear wave description andis developed at each time step. The main advantageof the proposed method is the representation, synthe-sis, and generation of an arbitrary wave train at anyposition in time and space. Thus, standard modelseas as well as special wave scenarios can be realizeddeterministically in a model tank and transformedeither to other stationary positions or to the movingreference frame of a cruising ship.The above wave generation technique is used as a val-idation tool for two numerical wave tanks. The firstnumerical wave tank uses a time stepping methodbased on potential theory. The simulation procedurecalculates the free surface elevation and potential fieldof the entire fluid domain from which the pressure,velocity, and acceleration fields can be derived. Thesecond method is a commercial RANSE solver whichsolves the conservation equations for mass and mo-mentum. For capturing the free surface the volumeof fluid (VOF) approach is applied. As a consequence,breaking phenomena can be considered.A comparison with measurements discloses the ad-vantages and disadvantages of each method.The modified non-linear approach is applied to gen-erate the ”New Year Wave” which has been measuredin the North Sea in the wave tank. With this roguewave sequence the motions and forces of a semisub-mersible are investigated experimentally.The second application presented here addresses theexperimental investigation of intact stability. The rollmotion of a RoRo vessel due to deterministic wavetrains is given in the moving reference frame. The

test result is used for the practical design of shipsby validating numerical tools for the evaluation ofcapsizing risk.

Modified non-linear theory for model-ling wave propagation

The modified non-linear approach combines empiri-cal and analytical wave models to allow a fast andprecise prediction of non-linear wave propagation.It can be applied both to ”forward” (downstream)and ”backward” (upstream) prediction of wave trains.The ”forward” prediction at arbitrary positions of themodel tank (similar to the below presented numeri-cal wave tanks) includes the representation of wavetrains in the moving reference frame of a cruisingvessel. The ”backward” calculation is used for thetransformation of given target wave trains to the lo-cation of the wave maker: This is a unique feature ofthe proposed procedure.

The method starts with a linear wave train ζ0(t),either measured close to the wave maker or knownfrom calculating the control signal of the wave maker.Thus, as a first step the wave train is checked with re-gard to linearity H

L0< 0.05 over the entire wave length

range. As a further step in pre-processing the wavetrain is written as Fourier series and time mappedwith respect to the Shannon theorem:

ζ0(ti) =n/2∑j=0

Aj cos(ωjti + ϕ0j), i = 0, 1, . . . n− 1 (1)

where

Aj = |Fj |4ω = 4ω|n0∑

i0=0

ζ0(ti0)e−iωjti04t|, (2)

j = 0, 1, . . . n0/2

is the Fourier spectrum of ζ0(t) with 4ω = 2πn4t , ωj =

j4ω, and i0 = 0 . . . n0 denotes the initial time map-ping. The corresponding initial phase spectrum isalso calculated by Fourier transform of the initial lin-ear wave train:

ϕ0j = arctan(=(Fj)<(Fj)

), j = 0, 1, . . . n/2. (3)

The Hilbert transform of a function f is defined as

H(f) := IFFT(√

(FFT(f))2 + (FFT(f)eiπ/2)2) (4)

where ”(I)FFT” is the abbreviation of the (inverse)Fourier Transform (Eq. 2), calculated by the Fast

40 60 80 100 120 140 160 180

−0.2

−0.1

0

0.1

0.2

ζ [m

]

time [s]

measured Hilbert transform

Fig. 1: Transient wave packet measured close to thewave board at x = 8.82 m: Linear wave theory is stillacceptable for its description.

Fourier Transform algorithm. The inverse FFT gives:

ζ(ti) =12π

n/2∑j=0

Fjeiωjti4ω, i = 0, 1, . . . n. (5)

As a test case we chose a transient wave packet mea-sured at two positions — the first location close tothe wave board (x = 8.82 m) where the wave trainis linear and the second position where the waves arealready steeper and cannot be calculated by lineartransform anymore (x = 85.03 m). Fig. 1 shows thelinear wave train and its envelope.According to Airy wave theory a wave train at anarbitrary position xl is transformed to another posi-tion xl+k by linear phase shift (the Fourier spectrumremains the same):

ζ(ti, xl+k) =12π

∑j

F (ωj , xl)ei(ωjti−k(xl+k−xl))4ω.

(6)Propagation of higher waves cannot be described byAiry theory since the propagation velocity increaseswith the instantaneous wave height. Also wave asym-metry and mass transport are introduced as consid-erable quantities. Fig. 2 shows the wave train fromFig. 1 transformed to x = 85.03 m by means of linearwave theory. Note that Airy theory is not adequateanymore. Especially, the higher frequencies deviateobviously since they propagate faster than predictedby linear wave theory. Also the shape does not cor-respond with the measured wave train (flat troughs,steep crests).Our non-linear semi-analytical approach is based onStokes III. It can be replaced by other terms fromdifferent theories as well.Adapting Eq. 6, the phase Cij is adjusted to thenon-linear wave celerity cij . For each step l in spacethe following iteration scheme for the non-linear wavetrain has to be run:

Cij = ϕ0j − (4x)lkij (7)

where ϕ0j is the initial phase spectrum from Airytheory and Cij is the modified phase calculated from

140 145 150 155 160 165 170

−0.2

−0.1

0

0.1

0.2

ζ [m

]

t [s]

measured linear calculation

Fig. 2: Transient wave packet at x = 85.03 m: Com-parison of registration with calculated data (lineartransformation from x = 8.82 m — see Fig. 1) provesthat linear wave theory gives inaccurate results.

the theory adequate to the investigated case. Herethe following equations have to be solved to calculatethe kij (see e. g. Kinsman (1965), Skjelbreia (1959)):

1. deep water d/L0 ≥ 0.5:

ω2j = gkij(1 + (kijai)2) (8)

(Stokes III) — solved by Cardan formulae

2. intermediate water depth 0.04 < d/L0 < 0.5:

ω2j = gkij tanh(kijd)(1+(kijai)2

cosh(4kijd) + 88 sinh4(kijd)

)

(9)(Stokes III) — solved by fix point iteration

3. shallow water d/L0 ≤ 0.04:

ω2j = gkij tanh(kijd) (10)

(linear wave theory)

Our test case is a transient wave packet measured atthe Hamburg Ship Model Basin with a water depthof d = 5.6 m. Thus deep water limit frequency isω = 2.34 rad/s, the shallow water limit frequencyω = 0.44 rad/s.

kij is subject to the temporary envelope ai = a(ti) =H(ζi). Thus the required Hilbert transform for theparticular xl is calculated at each time step ti since itrepresents the instantaneous wave height at a particu-lar point in time and space. It also considers the factthat the wave height increases on the way throughthe tank and non-linearities gain more and more in-fluence. Fig. 3 gives an impression of the iteration ofthe kij .

In accordance with Stokes III wave theory the corre-sponding wave components at xl are:

ζl,1(ti) =n/2∑j=0

Aj cos(ωjti + Cij), (11)

ζl,2(ti) =n/2∑j=0

12aiAj cos(2ωjti + 2Cij), (12)

0 2 4 6 8 10 120

5

10

15

ωj [rad/s]

k ij [rad

/m]

i = 1i = 35i = 36i = 37i = 38i = 39i = 40

Fig. 3: Iteration of wave numbers kij(ωj , a(ti)) asfunction of the instantaneous wave envelope a at timestep ti. Propagation velocity cij = ωj/kij increaseswith ”wave amplitude” ai (see Eqs. 8-10).

ζl,3(ti) =n/2∑j=0

38a2

i Aj cos(3ωjti + 3Cij). (13)

After summation of these components, ζl =∑3

k=1 ζlk,the preliminary instantaneous wave train at the posi-tion xl is given. Note that the phase velocity dependsnot only on frequency but also on wave elevationwhich is represented by the instantaneous envelopeand its linear amplitude distribution. The correctshape is also composed of higher order components(bounded waves — Eq. 12 and 13).The calculation of Cij , Eq. 7-13, is repeated twiceto average kij from the first and second step. The(4x)l are chosen such that they decrease with in-creasing non-linearity. In our example the iterationis done with 2× 105 steps in space and 1024 steps intime. Fig. 4 presents some iteration steps. The resultof the calculation procedure is shown in Fig. 5 andcompared to the measured wave train. Agreementwith the measured time series is good. Compared toFig. 2 the higher frequency terms show the adequatepropagation speed and a pronounced non-linear shapewith steep crests and flat troughs.

Wave generation for model tests

For model tests defined wave trains are generated in amodel tank - preferable as deterministic wave groupsat defined target locations which allows to correlatewave excitation to structural response. The wave gen-eration process can be divided into four steps:

40 60 80 100 120 140 160 180

−0.2−0.1

00.10.2

ζ [m

]

measured

40 60 80 100 120 140 160 180

−0.2−0.1

00.10.2

ζ [m

]

step 20

40 60 80 100 120 140 160 180

−0.2−0.1

00.10.2

ζ [m

]

step 50

40 60 80 100 120 140 160 180

−0.2−0.1

00.10.2

ζ [m

]

step 90

40 60 80 100 120 140 160 180

−0.2−0.1

00.10.2

ζ [m

]

t [s]

step 105measured

Fig. 4: Non-linear transformation of wave train inFig. 1 to downstream positions (showing selected it-eration steps): Comparison with measured data atx = 85.03 m is satisfactory (see also Fig. 5).

140 145 150 155 160 165 170

−0.2

−0.1

0

0.1

0.2

ζ [m

]

t [s]

measured non−linear calculation

Fig. 5: Wave train from Fig. 1 is transformed toposition x = 85.03 m using the described non-linearcalculation procedure (iteration step 105) and com-pared to measurements.

1. definition of the target wave train

2. transformation of the target wave train to theposition of the wave maker

3. calculation of wave maker control signal withregard to the characteristic RAOs of the wavemaker

4. performance of model test

Definition of the target wave train

For the definition of an appropriate target wave traindifferent methods are available. One procedure is todefine target parameters like a typical ”Three Sisters”wave sequence Hs − 2Hs −Hs in terms of the signif-icant wave height Hs (Wolfram et al. (2000)). Anoptimization routine is applied to get a target wavetrain satisfying the selected parameter set, see Claussand Steinhagen (2000). A target wave is also definedby full scale measurements or as output of numericalsimulations.

Transformation of the target wave to the wavemaker

The second step in wave generation is the transfor-mation of the given target wave train to the locationof the wave maker. This makes use of the non-linearcalculation scheme introduced above. Note that onlythe semi-empirical method allows the upstream calcu-lation of a (non-linear) wave sequence from the targetlocation back to the wave board.

Another approach to obtain the wave train at thewave board according to given target parameters isthe combination of a numerical wave tank with anoptimization method (Steinhagen (2001)).

Calculation of control signals

Knowing the wave train at the wave board it is easyto calculate the appropriate control signal(s) usingthe different characteristic transfer functions of thewave maker (Clauss and Hennig (2003)) which allowsto generate the desired wave train for the model testat the target location.

Calculation of non-linear wave trains inthe moving reference frame of cruisingstructures

For the deterministic analysis of motions and forcesof ships and offshore structures the wave excitationdenotes the beginning of a complex cause reactionchain. Dealing with a linear system the required timeseries are gained from frequency domain calculations.For a non-linear system, these models become inad-equate, and more sophisticated methods have to beused. The problem gets even more complex whenthe structure is moving at constant or non-constantspeed, as stationary measurements in a model tankdo not provide the wave train at the position of theinvestigated model in the moving reference frame ofa cruising ship. Wave probes can be installed at de-fined positions, but usually not at the position of themodel (due to relative motions and disturbances). Es-pecially when the ship is sailing at non-constant speedit is not a state of the art task to determine the waveexcitation with regard to a moving reference point.

Therefore the measured wave train has to be trans-formed to a reference position of the model (bothstationary and moving reference frame). The linearcalculation scheme for moving models (wave in themoving reference frame of a ship) can be derived as

follows:

ζ(ti, xl+k(ti)) =12π

∑j

F (ωj , xl)ei(ωjti−kj4xi)4ω

(14)where 4xi stands for the time varying distance be-tween both locations. Considering the non-linearkj and the adequate F (ωj , xl) from the empirical-analytical approach the wave train has to be calcu-lated at the position of the model reference point x(ti)at each time step in order to get the non-linear mo-ving reference frame wave train.

Deterministic wave trains

Applying the described non-linear approach all kindsof 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 ap-plications. The first application example is the valida-tion of numerical wave tanks by modelling non-linearwave propagation. Two examples of numerical wavetanks are given here.

Validation of a numerical wave tankbased on potential theory

The first numerical wave tank is based on a Finite El-ement Method discretization of the fluid domain. Thetwo dimensional non-linear free surface flow problemis solved in time domain using potential theory: thefluid is inviscid and incompressible, and the flow isirrotational. Wave breaking is not considered. Theatmospheric pressure above the free surface is con-stant and surface tension is neglected. Hence, theflow field can be described by a velocity potentialwhich satisfies the Laplace equation. At each timestep the velocity potential is calculated in the entirefluid domain, Clauss and Steinhagen (1999).

100 110 120 130 140 150 160 170 180 190 200−0.5

0

0.5

ζ (t

)

x = 108.740 m

100 110 120 130 140 150 160 170 180 190 200−0.5

0

0.5

ζ (t

)

x = 114.115 m

100 110 120 130 140 150 160 170 180 190 200−0.5

0

0.5

ζ (t

)

time [s]

x = 139.785 m

numerical wave tankexperiment

Fig. 6: Comparison of numerical (potential the-ory/ FEM) and experimental wave sequence gener-ated by the same wave maker signal calculated bythe modified non-linear theory (TP = 14.6 s, Hs =15.3 m): registrations at different positions.

To develop the solution in time domain the fourth-order Runge-Kutta formula is applied. At each timestep a new boundary-fitted mesh is created. Theprocedure is repeated until the desired time step isreached, or the wave train becomes unstable andbreaks. The numerical wave tank is able to simulatewave generators of piston type, single flap and dou-ble flap (and combinations). A complete descriptionof this numerical wave tank is found in Steinhagen(2001).

Fig. 6 presents numerical results as well as exper-imental data generated by the modified non-linearapproach to validate the numerical wave tank. Thestorm 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 scaleof 1:34.

Fig. 7 presents a superposition of a regular wave trainwith a wave packet. This tailored wave sequence isused for the investigation of large rolling and capsiz-ing of ships, Clauss and Hennig (2003). As will beshown in Fig. 12 this irregular wave train turns outto become a rather regular wave with an integratedextreme wave if transformed to a moving referenceframe.

Fig. 8 shows a simulation of the so-called New YearWave. This rogue wave was reported from the jacketplatform Draupner in the North Sea on January 1st,1995, Haver (2000). The platform was hit by a giantwave with a wave height of 25.6 m (significant waveheight 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

120 130 140 150 160 170 180 190 200−0.5

0

0.5

ζ (t

)

x = 108.740 m

120 130 140 150 160 170 180 190 200−0.5

0

0.5

ζ (t

)

x = 114.115 m

120 130 140 150 160 170 180 190 200−0.5

0

0.5

ζ (t

)

time [s]

x = 139.785 m

numerical wave tankexperiment

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

depth 1.5 m, piston type wave generator).

Figs. 6-8 document the universality of the numer-ical wave tank for the calculation of wave evolu-tion for different wave tanks with different waterdepth and types of wave generators. Like the mod-ified non-linear theory the numerical wave tank pre-dicts the non-linear evolution of wave trains and thewave/ wave interaction quite well. As the potentialfield is calculated at each time step, also velocity, ac-celeration and pressure fields are known. Only themodified non-linear theory is able to provide controlsignals both for generating deterministic wave trainsin a model tank and as an input for the moving wallboundary of a numerical wave tank (Fig. 8). To over-come the limitations of wave breaking a numericalwave tank using a commercial computational fluiddynamics (CFD) solver is introduced in the next sec-tion.

Validation of a numerical wave tankbased on a RANSE solver

The second numerical wave tank presented here is setup using the commercial state of the art CFD solverFLUENT (Fluent (2003)). For all flows, FLUENTsolves the conservation equations for mass and mo-mentum (Navier-Stokes equations). For simulatingthe free surface the Volume of Fluid method (VOF)is used which can deal with wave breaking phenom-ena. For simulating the wave board motion of thepiston type wave generator a dynamic mesh approach(dynamic layering) is introduced. The motion of thewave board is simulated by moving the boundary for-wards and backwards like the wave board in the ex-

450 500 550 600 650 700 750

−10

0

10

20Target wave train (New Year Wave)

ζ [m

]

t [s]

recorded in the North Sea

0 10 20 30 40 50 60 70 80−2

−1

0

1

2

t [s]

U [V

]

Wave maker control signal

50 55 60 65 70 75 80

−0.1

0

0.1

0.2

ζ [m

]

t [s]

recorded in the North Seagenerated in the wave tanknumerical wave tankmodified non−linear theory

Fig. 8: Generation and analysis of the New YearWave: applying the modified non-linear approach thetarget wave train (top) is transformed upstream tothe position of the wave maker. The resulting con-trol signal is shown here (second graph). Both themodified theory and the numerical wave tank basedon potential theory are able to calculate the down-stream wave train at the target position — the nu-merical tank only from the control signal which isprovided by the modified theory. The correspondingwave field characteristics are provided by the numer-ical wave tank (scale 1:81).

periment. Therefore, cells have to be added to ordeleted from the fluid domain as the size of the cal-culation domain changes with time. Further detailson the numerical wave tank are given in Clauss et al.(2004a).

Fig. 9 presents the comparison between calculations

60 70 80 90 100 110 120−0.2

−0.1

0

0.1

0.2

x = 9.5m

ζ [m

]

60 70 80 90 100 110 120−0.2

−0.1

0

0.1

0.2

x = 15.0m

ζ [m

]

60 70 80 90 100 110 120−0.2

−0.1

0

0.1

0.2

x = 19.5m

time [s]

ζ [m

]

numerical wave tankexperiment

Fig. 9: Simulation of a wave packet (Hmax = 0.35 m)based on RANSE/ VOF in comparison to measure-ment (same JONSWAP spectrum as in Fig. 10). Gen-eration of the wave maker control signal for bothmodel tank and numerical wave tank data using themodified theory.

and measurement of a wave packet with a JONSWAPspectrum at different positions in the wave tank (tankdimensions: length 80 m, width 4 m, water depth1.5 m, piston type wave generator). It can easily beseen, that the phases and amplitudes are well pre-dicted by the numerical wave tank. Wave packetscan e.g. be used to simulate extreme waves (Kuhn-lein et al. (2002)) or to model high wave groups withina natural sea state like in Fig. 10: It shows the sim-ulation of the wave packet from Fig. 9 integrated toirregular seas.

As a RANSE solver is used for this wave tank theduration of the calculation is significantly higher ascompared to the numerical wave tank based on poten-tial theory and FEM. To make use of the benefits ofboth methods a combined approach is recommended:potential theory as long as no wave breaking occurs,and RANSE code if breaking is encountered.

Both numerical wave tanks are not capable to trans-form a given wave train backwards to the positionof the wave generator. In order to generate a pre-determined wave sequence at a target location thenumerical wave tank has to be combined with themodified approach or optimization routines (Claussand Steinhagen (2000)).

20 30 40 50 60 70 80 90 100 110 120−0.2

−0.1

0

0.1

0.2

x = 9.5m

ζ [m

]

20 30 40 50 60 70 80 90 100 110 120−0.2

−0.1

0

0.1

0.2

x = 15.0m

ζ [m

]

20 30 40 50 60 70 80 90 100 110 120−0.2

−0.1

0

0.1

0.2

x = 19.5m

time [s]

ζ [m

]

numerical wave tankexperiment

Fig. 10: Irregular sea (JONSWAP spectrum, TP =4.2 s, Hs = 0.1 m) with integrated wave packet(Hmax = 0.35 m) at different positions in the wavetank. Numerical simulation based on RANSE/ VOF.Generation of the wave maker control signal for bothmodel tank and numerical wave tank data using themodified theory.

Experimental investigation of offshorestructures

For the experimental investigation of wave-structureinteraction of stationary offshore structures it is im-portant to know the exact wave train at the modelposition. Measurements close to the model are dis-turbed by radiation and diffraction. Using the pre-sented modified non-linear approach the wave traincan be calculated at any position of the model tank.

Fig. 11 (top) presents the wave train of the so calledNew Year Wave recorded in the North Sea and simu-lated in the wave tank using the above introducedmodified approach. This rogue wave with an un-usual Hmax/Hs ratio of 2.15 is applied to both theexperimental and numerical investigation of roguewave impact on the structural (splitting) forces ofa semisubmersible (Fig. 11, Clauss et al. (2003b)).Wave sequences calculated by the modified non-linearapproach have also been successfully applied to theinvestigation of rogue wave impacts on the verticalbending moments of a stationary crane vessel (Clausset al. (2003a)) and an FPSO ship (Clauss et al.(2004b)).

200 250 300 350 400 450 500

−10

0

10

20New Year Wave

surf

ace

elev

atio

n ζ

[m]

recorded in the North Seagenerated in the wave tank

200 250 300 350 400 450 500−50

0

50corresponding splitting force

time [s]

split

ting

forc

e [1

06 N]

calculatedmeasured

Fig. 11: Top: comparison of recorded New YearWave and wave tank simulation (scale 1:81). Bot-tom: measured and calculated splitting forces of thesemisubmersible due to the rogue wave impact (alldata presented as full scale data).

Experimental investigation of intactstability

For the experimental investigation of intact stabilitywith regard to both extreme and resonance phenom-ena the wave train as the beginning of the cause re-action chain can be directly compared to the reactionof the cruising ship since all time series are calculatedresp. measured in the moving reference frame of theship model.

For controlled capsizing tests (Clauss and Hennig(2003)) we generate a regular wave with an embedded”Three Sisters wave” sequence at a moving referenceframe. Fig. 12 shows a wave packet within a regularwave measured at a stationary wave probe close tothe wave board (x = 297.8 m, model scale 1:34). It istransformed to the position of the cruising ship. Asshown in Fig. 12 this resulting wave sequence is quiteregular and contains the target ”Three Sisters wave”at the location of interaction with the cruising ship.Thus, high roll angles (lower diagram) can be inducedby generating tailored moving reference frame wavetrains deterministically.

Numerical predictions can also be directly comparedto model tests applying the following scheme: Thewave train used in the numerical simulation for as-sessing ship safety is given as full scale target wavetrain (Fig. 13 top) and transformed to model scale(1:34). Now the modified non-linear approach is ap-plied to obtain the wave train at the position of thewave maker. Thus the corresponding control signalfor driving the wave maker (signals for upper andmain flap of double flap wave maker at Hamburg ShipModel Basin). The generated wave train is registeredat a stationary wave probe close to the wave maker

Fig. 12: Roll motion of a multipurpose vessel (GM =0.44 m, v = 14.8 kn und µ = ±20◦) in a regular wavefrom astern (λ = 159.5 m, ζcrest = 5.8 m) with pro-ceeding high transient wave packet (compare Fig. 7).

and transformed to x = 125 m (compare target wavetrain). The ship position is measured during the test.Thus, the stationary wave train is transformed to themoving reference frame of the ship model to obtainthe wave as experienced by the ship. The resulting(measured) roll motion can be directly compared tothis wave train (Fig. 13 bottom). The same test datais used for a visual comparison of numerical simula-tion and model test results in Cramer et al. (2004).

Discussion, conclusions and perspective

In this paper a modified non-linear approach for mod-elling wave propagation is presented which provides

• transformation of arbitrary wave trains from sta-tionary positions to other arbitrary positions, es-pecially

• upstream transformation to the wave board toget the corresponding wave maker signal, and

1200 1300 1400 1500 1600 1700−10

0

10

t [s]

ζ [m

]

wave train used in numerical simulation (full scale)

190 200 210 220 230 240 250 260 270 280 290 300

−0.2

0

0.2

t [s]

ζ [m

]

target wave train for model test at x = 125 m (model scale 1:34)

50 100 150 200 250 300

−0.2

0

0.2

t [s]

ζ [m

]

target wave train transformed to position of wave maker

50 100 150 200 250 300−10

0

10

t [s]

U [V

]

control signal for upper flap of wave maker

50 100 150 200 250 300−10

0

10

t [s]

U [V

]

control signal for main flap of wave maker

50 100 150 200 250 300

−0.2

0

0.2

t [s]

ζ [m

]

registration at stationary wave probe close to wave maker (x = 5.19 m)

190 200 210 220 230 240 250 260 270 280 290 300

−0.2

0

0.2

t [s]

ζ [m

]

wave train at x = 125 m, calculated from registration at x = 5.19 m

180 190 200 210 220 230 2400

50

100

t [s]

x [m

] ship position in x−direction

180 190 200 210 220 230 240−0.3−0.2−0.1

00.10.20.3

t [s]

ζ [m

]

wave at ship (moving reference frame)

180 190 200 210 220 230 240−50

−25

0

25

50

t [s]

φ [°

]

capsizing of the RoRo vessel

Fig. 13: Experimental realization of dangerous wavesequences from numerical capsize simulations: Start-ing with the target wave train the wave at the positionof the wave maker is calculated using the modifiednon-linear approach to get the corresponding controlsignals. From registration at a stationary wave probeclose to the wave maker the stationary wave train atx = 125 m is given by the modified theory (comparetarget wave) and transformed to the position of theship model (scale 1:34) to obtain the moving referenceframe wave train which can be compared to the rollmotion of the RoRo ship which subsequently capsizes.See also Fig. 3 in Cramer et al. (2004).

• generation of a all kinds of deterministic wavetrains,

• transformation of arbitrary wave trains from sta-tionary positions to a moving reference frame ofa cruising structure.

The modified non-linear approach is fast and preciseand applicable in day-to-day use for experimental in-vestigations. The method can be adapted easily fornew requirements as the implementation of differentwave theories is possible.

Compared to the numerical wave tanks the methodis capable of backward transformation of wave trainsand is therefore ideally suited for generating waveboard control signals. As target signal, measurementsfrom full scale or arbitrary synthesized wave trains,e.g. from optimization processes can be used. Thepotential of the procedure is demonstrated by simu-lating different wave scenarios like wave packets, ir-regular and regular seas with embedded wave packets.

Furthermore, the method allows the transformationof given wave sequences into the moving referenceframe of a cruising vessel. With this technique wavescenarios can be analyzed from the point of view of asailing ship.

A wide range of applications of the presented non-linear wave calculation procedure is given:

• Validation of a numerical wave tank based on po-tential theory in combination with a Finite Ele-ment Method

• Validation of a numerical wave tank using aRANSE code and a dynamic mesh approach

• Experimental investigation of seakeeping charac-teristics of offshore structures

• Experimental investigation of intact stability andcomparison with numerical simulations

Both numerical wave tanks provide detailed knowl-edge of the pressure, velocity and acceleration fieldsin the entire fluid domain. This information can beused in a next step to investigate wave/structure in-teraction.

Recapitulating, the modified non-linear theory is apowerful tool to face the complex tasks related tothe experimental investigation of extreme structurebehaviour such as large rolling, capsizing, and roguewave impacts.

In conclusion, the modified non-linear theory is an ex-cellent tool to generate deterministic wave trains atarbitrary positions (even in a moving reference frame)

as this is the only procedure which can calculate up-stream to determine wave board motions and the as-sociated control signals. In calculating the wave el-evation downstream, the above method agrees wellwith the numerical wave tanks based on potentialtheory/ FEM and RANSE/ VOF (Fluent). Thus,the additional capabilities of numerical wave tanks,i. e. the determination of wave field characteristics(velocity, acceleration and pressure fields) and of theRANSE/ VOF method (consideration of wave break-ing) can be ideally combined with the above method.

As a consequence, the combination of the proposedprocedures is an innovative tool kit to analyze theinteraction of wave and structure, and to investigatethe structure behaviour in all kinds of wave scenarios.

Acknowledgment

The authors are indebted to the German Fed-eral Ministry of Education, Research, and Tech-nology, BMBF, for funding the project SinSee(FKZ 03SX145). The authors also want to expresstheir gratitude to the German Research Foundation(DFG) for funding the research project ”Extreme See-gangsereignisse” and to the European Union for fi-nancing the research project MAXWAVE (contractnumber EVK-CT-2000-00026). Many thanks go toMr. Michael Alex for preparing so many figures.

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