Extreme waves in directional wave fields traversing uniform

currents

NTNU Marintek Ocean Basin

NTNU Ocean Basin Extreme waves

EC contract no. 261520

Status: final

Date: Jul 2009

Infrastructure NTNU Marintek Ocean Basin

Project Extreme waves in directional wave fields traversing uniform currents

Campaign HyIII-NTNU-25

Title NTNU Ocean Basin Extreme waves

Lead Author Jaak Monbaliu Email jaak.monbaliu@bwk.kuleuven.be

Contributors Alessandro Toffoli Email toffoli.alessandro@gmail.com

Date

Campaign

Start

27/07/2009 Date

Campaign

End

07/08/2009

Date Final

Completion 02/11/2009

Contents:

Heading:

Contents:

1 Scientific aim and background

2 User-Project Achievements and difficulties encountered (max 250 words)

3 Highlights important research results (max 250 words)

4 Publications, reports from the project

5 Description

5.1 General description, including sketch

5.2 Definition of the coordinate systems used

5.3 Instruments used

5.4 Definition of time origin and instrument synchronisation

6 Definition and notation of the experimental parameters

6.1 Fixed parameters

6.2 Variable independent parameters

6.3 Derived parameters and relevant non-dimensional numbers

7 Description of the experimental campaign, list of experiments

8 Data processing

9 Organisation of data files

10 Remarks about the experimental campaign, problems and things to improve

1. Scientific aim and background:

Extreme waves represent a serious threat for marine structures. An accurate description of the

statistical properties of the surface elevation can contribute to improvement of warning criteria for

extreme waves in wave forecasting and to enhancing safety at sea in general [1]. There are many

mechanisms that can lead to the formation of large amplitude waves and hence to a different shape

of the probability density function of the surface elevation: for example, nonlinear processes such as

the modulational instability [2-6], and wavecurrent interaction [7,8]. This laboratory campaign

intends to investigate the combined role of the modulational instability and wave-current interaction

on the formation of extreme waves in two horizontal dimensions, when waves cross a steady

current.

In absence of ocean currents, numerical and theoretical works [2-6] have demonstrated that

nonlinear processes such as the modulational instability have a relevant role in the formation of

extreme waves, provided waves are sufficiently steep and narrow banded. Under these

circumstances, large amplitude waves may occur within a fairly short scale of tens of wavelengths

[4-6]. However, strong deviations from Gaussian statistics are only observed if waves are rather

long crested i.e. the spectral energy is concentrated on a narrow range of directions. For short

crested seas (broad directional distributions), the effect of modulational instability becomes less

prominent and, as a result, the occurrence of extreme waves does not exceed predictions from linear

or second-order theory [9,10]. Recently, a comprehensive analysis on the effect of directionality on

the modulational instability has been carried out at the Marintek's directional wave basin (the same

facility that we used for the present research). Experimental results confirmed the existence of a

transition region from strongly to weakly non-Gaussian wave fields as short crestedness increases

[11]. This transition is determined by a balance between nonlinearity (which promotes non-

Gaussian behavior) and directionality (which suppresses non-Gaussian behavior). Thus, if there are

circumstances when the nonlinearity is locally enhanced, we can expect that non-Gaussian behavior

will persist at broader directional spreads.

In region of strong currents (for example, the Gulf Stream, the Agulhas Current and the Kuroshio

Current), large amplitude waves can also be expected as a result of wave-current interaction;

interesting, in this respect, are a number of ship accidents reported near the Agulhas Current, off the

South African coast [7,8]. One possible mechanism that may lead to the formation of extreme wave

events can be triggered when a current flows with opposite or oblique direction to incident wave

trains. In these circumstances, if the current is strong enough, the wave energy is forced to coalesce

in certain areas [8] with a consequent formation of large waves (caustic theory). In a random wave

field, this may result in a substantial modification of the statistical distribution of the surface

elevation [12]. Depending on the strength of the current, however, deviations from Gaussian

statistics might require scales of thousands of wavelengths before appearing [12]. It is important to

mention that previous studies have only considered the interaction between a linear wave field and a

current, excluding therefore contributions from nonlinear mechanisms. According to the linear

dispersion relation, however, the wavenumber increases in the process of the wave-current

interaction for current opposing waves (see, e.g., [8]), even in the presence of relatively mild current

conditions. Under adverse currents, waves become shorter with a consequent increase in wave

steepness, making nonlinear processes, e.g. the modulational instability mechanism, more likely.

Nonetheless, the role of nonlinear mechanisms on the statistical properties of the wave field, and

hence the possible increase in the probability for the formation of extreme waves, is not yet clear

under these circumstances. In the light of results of [11], however, it is reasonable to assume that, in

such environment, non-Gaussian behavior may persist at broader directional spreads.

A number of laboratory experiments have been undertaken to verify the behavior of regular and

irregular waves when opposing a strong current (see, for example, [13]). Experimental results,

though, have been obtained in wave flumes, where only long crested waves can be considered (i.e.,

one dimensional problem). Fewer experiments have dealt with waves crossing oblique currents [14]

but these have been confined to a study of the kinematics and linear properties. For the present

study, we accessed the directional wave basin facility at Marintek in order to address the more

general two dimensional problem, where a multi directional wave field propagates obliquely over a

uniform current in partial opposition. Our goal was to verify experimentally the role of increasing

wave steepness due to wave-current interaction on the modulational instability mechanism, and

hence the formation of extreme waves, within a wide range of wave directional spreading. In

particular, we intended to verify whether the presence of a current could lead to the modulational

instability of broad directional wave trains, which are expected to be stable in absence of a current.

References

[1] Toffoli A., Lefevre J.M., Bitner-Gregersen E., Monbaliu J., 2006. Towards the identification of

warning criteria: analysis of a ship accident database. Applied Ocean Research 27 (2005), 281-291.

[2] Onorato M., Osborne A.R., Serio M., Bertone S., 2001. Freak waves in random oceanic sea

states. Phys. Rev. Lett. 86, 5831–5834.

[3] Onorato M., Osborne A.R., Serio M., Cavaleri L., 2005. Modulational instability and non-

Gaussian statistics in experimental random water-wave trains. Phys. Fluids 17, 078101.

[4] Onorato M., Osborne A.R., Serio M., Cavaleri L., Brandini C., Stansberg C., 2006. Extreme

waves, modulational instability and second order theory: wave flume experiments on irregular

waves. Europ. J. Mech. B/Fluids 25, 586–601.

[5] Janssen P.A.E.M., 2003. Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr.

33, 863–883.

[6] Babanin, A.V., D. Chalikov, I.R. Young, and I. Savelyev, 2007: Predicting the breaking onset of

surface water waves, Geophysical Research Letters, 34, L07605, doi:10.1029/2006GL029135, 6p

[7] Lavrenov I.V., 1998. The wave energy concentration at the Agulhas current of South Africa.

Natural Hazards 17, 117–127.

[8] White B.S., Fornberg B., 1998. On the chance of freak waves at the sea. J. Fluid Mech. 255,

113–138.

[9] Onorato M., Osborne A. R., Serio M., 2002. Extreme wave events in directional random oceanic

sea states. Physics of Fluids 14 (4), 25–28.

[10] Socquet-Juglard, H., Dysthe, K., Trulsen, K., Krogstad, H., Liu, J., 2005. Distribution of

surface gravity waves during spectral changes. J. Fluid Mech. 542, 195–216.

[11] Onorato M., Cavaleri L., Fouques S., Gramstad O., Janssen P.A.E.M., Monbaliu J., Osborne

A.R., Pakozdi C., Serio M., Stansberg C.T., Toffoli A., Trulsen K., 2009. Statistical properties of

mechanically generated surface gravity waves: a laboratory experiment in a 3D wave basin. J. Fluid

Mech., 627, 235-257.

[12] Heller, E.J., Kaplan L., Dahlen A., 2008. Refraction of a Gaussian Seaway. J. Geophys. Res.,

113, C09023, doi:10.1029/2008JC004748.

[13] Chawla A., Kirby J.T., 2002. Monochromatic and random wave breaking at blocking points. J.

Geophys. Res., VOL. 107, C7, doi:10.1029/2001JC001042.

[14] MacIver, R. D., R. R. Simons, and G. P. Thomas, 2006, Gravity waves interacting with a

narrow jet-like current, J. Geophys. Res., VOL. 111, C03009, doi: 10.1029/2005JC003030.

2. User-Project Achievements and difficulties encountered:

Two types of experiments were conducted. The first one tested the evolution of deterministic

(tailor-made) waves over a transverse current; wave trains were generated using monochromatic

waves with two side band perturbations. The second set of experiments tested the evolution of

irregular waves characterized by a JONSWAP spectrum and a cos N(θ) directional distributions

over a transverse current. Two incident angles between the wave field and the current as well as

several directional distributions for the irregular wave fields were tested by generating waves with a

(mean) direction forming an angle of 110 and 120 degrees with the current so that waves partially

opposed the current. Note that for the tailor-made experiments we also tested a condition of 130

degrees. The deterministic tests verified that the current is able to trigger the instability of initially

stale wave packets and hence generate large amplitude waves. Thus, we thentested whether the

interaction between waves and current can increase the probability of occurrence of extreme waves

in random sea states. Here we observed an increase of kurtosis. This was more notable for large

directional distribution when waves are generally stable without interacting with a current.

However, the increase of kurtosis was rather weak. The main difficulties that we encountered in

these tests were related to side-wall reflection. This limited the angle of incidence between waves

and current. In this respect, it would have been interesting to test angular configurations of 130 or

140 degrees.

3. Highlights important research results:

The main research results can be summarized as follows:

a) deterministic (tailor-made) tests:

I) stable wave packets becomes unstable in the present of a current.

II) current-induced instability leads to the formation of large amplitude waves as waves propagates

along the tank.

III) the maximum wave amplitude is a function of the incident angle between waves and currents.

b) random tests:

I) in a random wave field, the presence of a partially opposing current results in a weak increase of

the probability of occurrence of extreme waves (i.e. kurtosis). This was mainly observed in rather

short crested sea states.

4. Publications, reports from the project:

The results of these experiments will be presented in a number of international journal and

conference publications.

Toffoli A, Onorato M, Cavaleri L, Gramstad O, Janssen P, Monbaliu J, Osborne AR, Serio M,

Stansberg ST, Trulsen K (2010) Statistical properties of mechanically generated surface gravity

waves: a laboratory experiment in a 3D wave basin. In Proc. HydralabIII Joint User Meeting, 2-4

February, 2010, Hannover, Germany

Toffoli A, Babanin AV, Ardhuin F, Benoit M, Bitner-Gregersen EM, Cavaleri L, Monbaliu J,

Onorato M, Osborne AR, Fouques S, Moe V, Pakodzi C, Satnsberg CT (2010) Extreme waves in

sea states crossing an oblique current. Proc. Int. Conf. Offshore Mech. Artic Eng., June 6-12,

Shanghai, China (OMAE2010-20619)

Toffoli A, Ardhuin F, Babanin AV, Benoit B, Bitner-Gregersen EM, Cavaleri L, Monbaliu J,

Onorato M, Osborne AR (2010) Extreme waves in directional wave fields traversing uniform

currents. In Proc. HydralabIII Joint User Meeting, 2-4 February, 2010, Hannover, Germany

5. Description:

5.1. Description:

The methodology of the experiment was fairly simple. The idea was to monitor the spatial

evolution of regular and irregular wave fields propagating over a uniform current. Regular fields

were characterized by a monochromatic wave (carrier wave) and two side band perturbations

(see details in Table 7.1). Note that some tests were also performed with simple monochromatic

waves to test the facility. On the other hand, irregular wave fields were characterized by different

JONSWAP-type spectra with different angular spreading distributions, ranging from long to

short crested conditions. The initial (input) spectrum at the wave maker was determined as

follows:

· the JONSWAP spectrum was used to describe the frequency domain. The peak period was set

to 1s, Phillips parameter α was 0.016 and peak enhancement factor γ was 6. This configuration

define a significant wave height H s = 0.08 m and an wave steepness (kp Hs / 2, where kp is the

wavenumber associated to the peak period) equal to 0.16 (without current).

· the directional distribution was described using a cosN(θ) function, where the frequency-

independent coefficient N assumes the following values: 840; 200; 90; 50; 24 (from long to short

crested waves).

In the directional basin, the current flows in the longitudinal basin direction, so that it crosses

waves generated by the multi-flap wave maker. A specified mean wave direction was imposed

on the wave field to cross the current obliquely with waves and current in partial opposition. We

mention, however, that this angle was carefully chosen in order to limit side wall reflection

(there is no beach on the left side of the basin where the waves are directed to) and reduce

shadow zones. In this respect we used two angular configurations: one with an incident angle of

110 degrees (previous experiments with similar mean wave direction were already performed at

Marintek without detecting significant reflection from the side wall); and a second one with an

angle of 120 degrees. For the latter configuration, preliminary tests indicated that the side wall

reflection did not affect the wave field in the measurement area significantly. For regular waves,

moreover, we also tested an angular configuration of 130 degrees, despite the side-wall

reflection. A sketch of the experiment is presented in Fig. 5.1. Note that we ran the current at its

maximum speed of 0.2 m/ s. However, the current speed was observed to be slightly lower in the

part of the tank near the directional wave maker. Thus, this generated a small gradient in the

current speed along the mean wave direction of propagation. Note that the group velocity for 1s

waves is approximately 0.78 m/s.

Along the mean wave direction, a number of wave and current probes were deployed (see Fig.

5.1.1 and 5.2.1) in order to trace the surface elevation and the current velocity. Sampling

frequency was set to 75Hz. The deterministic experiments were rather simple to plan as they

only required to trace the spacial evolution of a wave train along the basin. The random

experiments, on the other hand, required a more detailed planning in order to collect enough

samples. As a matter of fact, the aim of the random tests was to collect statistical results on the

probability of occurrence of extreme waves. In this respect, it is important to record a large

number of waves. Therefore, we performed 4 tests with different random amplitudes and phases

for each spectral configuration. Because each experiment lasted 30 minutes, we were able to

measure about 7500 individual waves at each probe.

Figure 5.1.1 A schematic representation of the experiment.

5.2. Definition of the coordinate systems used:

The coordinate system is centered in the middle of the directional tank. The x-axis is parallel to

the directional wavemaker, while the y-axis is perpendicular to the directional wavemaker. A

scheme of the coordinate system is presented in Fig. 5.2.1 (upper panel).

Throughout the experiments we have measured the surface elevation and the current speed. To

this end, we used 28 resistance wave probes and 7 current probes. A scheme of their location is

presented in Fig. 5.2; the coordinates of all probes are provided in Table 5.2.1.

In order to measure directional wave spectral, 15 of the aforementioned wave probes were

deployed as three 5-probe arrays. Each array was arranged as a square with a probe in the centre.

Details of these arrays can be found in the lower panels of Fig. 5.2. The current probes measure

the particle velocity (in the x and y direction) at a depth of 0.35 m. However, because

measurements of the velocity profile were not available, we located one current probe, probe

number 7, at a depth of 1.5 m. Measurements at probe 7 combined with measurements at probe 4

provide a rough idea of the vertical velocity profile. Note that current probe number 3 did not

work properly during most of the tests.

Table 5.2.1 Probes' coordinates.

Figure 5.2.2 Coordinate system and probes locations.

5.3. Instruments used:

5.4. Definition of time origin and instrument synchronisation:

6. Definition and notation of the experimental parameters:

6.1. Fixed parameters:

During the experiments, time series of the surface elevation η and current speed along the x (u)

and y (v) directions were measured. A list of relevant parameters involved in the present

experiments is presented in Table 6.1.

Table 6.1 Relevant parameters

6.2. Variable independent parameters:

Notation Name Unit Definition Remarks

Table 6.2.1

6.3. Derived parameters and relevant non-dimensional numbers:

Notation Name Unit Definition Remarks

Table 6.3.1

7. Description of the experimental campaign, list of experiments:

Experiment Name Experiment Date Remarks

Table 7.1

A list of all runs is presented in Table 7.1. Note that the following tests were performed after

August 6th: 1301, 1004, 1211, 1013, 1014, 2003, 4101, 4102, 4103, 4104, 4111, 4112, 4113, 4241,

4142, 4045, 7131, 7132, 7133, 7211,7212, 7013, 7032, 7033.

8. Data processing:

During the experiments, some statistical analysis on the recorded data were performed to monitor

the quality of the measurements. Standard statistical parameters as the standard deviation, skewness

and kurtosis of the recorded signal were calculated. This information were archived together with

the direct measurements. The raw data were also post-processed following a quality control

procedure. In this respect, all current records were low-pass filtered at 3.75Hz (ramp 0.5Hz).

Furthermore, calibration coefficients were applied to correct the time series of the surface elevation

for some specific probes. This mainly applied for those cases which were re-ran (see part 10). The

initial zero has been corrected for tests number 1014, 1211, 4011, 4024.

9. Organisation of data files:

· SetUP_TestProgram: miscellaneous documentation regarding the set up of the experiment

· Photo: photos of set-up

· Timeseries:

o Curr: measurements of current velocity (current only)

o Irr: measurements of surface elevation and current velocity in random sea states (waves only and

waves & current)

o Reg: measurements of surface elevation and current velocity for regular waves (preliminary tests)

o Tailor_Made : measurements of surface elevation and current velocity for deterministic

experiments

· Analysis:

o Curr: preliminary statistical analysis for current velocity (current only)

o Irr: preliminary statistical analysis for surface elevation (waves only and waves & current)

10. Remarks about the experimental campaign, problems and things to improve:

During the first part of the experiment, two of the 5-point wave staff arrays were not perfectly

arranged, such that the spacing piece between the single staffs have made some disturbances on the

time series around -0.02 m level. This was discovered after test 5021 and corrected on Thursday the

6th August. With the spacers at the -0.02 m level, measurements of the positive elevation (and

hence wave crests) were not disturbed. Unfortunately, this problem produced a deformation of the

wave trough (knee-like shape), which resulted in an unusual high skewness. The other (single)

staffs were not affected by this, so that there are reliable data for a good range of locations in the

tank also from these first day test runs. (Note that the problem was eliminated for tests performed

after the 6th August, i.e. random tests ending with 3, 4 and 5 – e.g. 4043, 4044, 5023, 5024, etc...).

In order to have complete sets of reliable data with the 5-point arrays for at least some test

conditions, Marintek has kindly performed some extra runs after the official end of the project. It

was decided to re-run irregular wave tests in 110 deg with N=840 and N=24. In addition, one

regular wave test, a current-only test, and some Tailor-made (deterministic) tests have also been re-

ran. It is important to mention that the extra runs also provide reference records for a possible

correction of the erroneous data. We mention that during the repetitions of the aforementioned tests

current probe number 2 as well as current probe number 3 were out of order. For the purpose of

these repetitions, however, the absence of these two current probes has been regarded as negligible;

previous measurements of current speed in such conditions were in fact reliable. During the post

processing of the wave records, we also noted that wave probe number 1 provides unusually high

values of significant wave height; unusual values are also recorded for the kurtosis. An example is

presented in Fig. 5.3, where the significant wave height is plotted as a function of the dimensionless

distance from the wavemaker for record 1211.

Figure 10.1 Significant wave height as a function of the dimensionless distance from the

wavemaker for record 1211.

Window size: x

Viewport size: x