On the application of selected wave group spectra for the experimental of low frequency motions of a moored structure

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ABSTRACT

A moored structure shows both wave and low frequency mo-tions in waves. Wave frequency momo-tions are related to the wave elevation and wave power spectrum of the sea state while low fre-quency motions are driven by wave groups and the correspond-ing wave group spectrum. Wave power spectra can be calibrated

for model tests. The corresponding wave group spectrum

fol-lows from the wave power spectrum together with the applied

wave seed or phasing of the wave train. Thus, in common prac-tice (both in simulations and tnodel tests), the wave group

spec-trum follows from the arbitrary choice of a random seed. This

can lead to an under- or overestimation of the resulting low fre-quency motions of the moored object as compared to the theoreti-cal group spectrum. As an alternative approach, the seeds which give the highest and lowest wave group spectra can be applied in the tests. In this paper, first results of model tests with a moo red tanker based on an intentional choice of wave group spectra are presented.

'Address all correspondence to this author.

Proceedings of the ASME 2009 28th International Conference on Ocean, Offshore and Arctic Engineering OMAE2009 May 31 - June 5, 2009, HonoIuu, Hawaii, USA

Deift University of Technology

Ship Hydromechanics Laboratory

Library

Mekelweg 2

2628 CD DeIft

Phone: +31 (0)15 2786873

E-mail: p.w.deheer@tudelft.nl

ON THE APPLICATION OF SELECTED WAVE GROUP SPECTRA FOR THE

EXPERIMENTAL INVESTIGATION OF LOW FREQUENCY MOTIONS OF A MOORED

STRUCTURE

INTRODUCTION

A moored structure shows both wave and low frequency mo-tions in waves. Wave frequency momo-tions are related to the wave elevation and wave power spectrum of the sea state while low fre-quency motions are driven by wave groups and the corresponding wave group spectrum. Wave power spectra can be calibrated for

model tests. The corresponding wave group spectrum follows

from the wave power spectrum together with the applied wave

seed or phasing of the wave train. Thus, in common practice

(both in simulations and model tests), the wave group spectrum follows from the arbitrary choice of a random seed. This can lead to an under- or overestimation of the resulting low frequency mo-tions of the moored object as compared to the theoretical group spectrum. As an alternative approach, the seeds which give the highest and lowest wave group spectra can be applied in the tests.

One year ago, we presented the "Worst Sea - Best Sea"

con-cept in [1], where we also mentioned model tests with a moored tanker in waves in MARIN's Shallow Water Basin. The model

tests were carried out to investigate the effects of variations in

water depth, wave seed and wave steepness on the properties of the generated waves and the motion response of a moored floater.

In this paper, results of these model tests based on an intentional choice of wave group spectra are presented which also include

OMAE2009-801 27

Janou Hennig Antonio Carlos Fernandes

MAR IN LabOceano. COPPEIUFRJ

Netherlands Brazil

j.hennig@marin.nI acfernandes@alternex.com.br

Hans Cozijn Marcio Maia Domingues Jr.

MAR IN Petrobras AS

Netherlands Brazil

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Table 1. OVERVIEW OF WAVE PROBES POSITIONS IN THE BASIN (AT MODEL SCALE 1:85).

the tanker motions due to these wave group spectra.

TEST SETUP

The model tests were carried out at a scale of 1:85 in

MARIN's Shallow Water Basin ("BT"), which is 220 m long, 15.8 m wide and has a maximum water depth of 1.1 m. The

basin is equipped with a new piston type wave maker. The water depth was varied between intermediate (85 m) and shallow (25 m). Irregular waves at one peak period and different significant wave heights and seeds were generated.

The wave elevation was measured by resistance type wave probes located at two reference positions in the wave tank. The positions are given in the drawing in Fig. 1. The reference wave probe WAVE 8 is located at 44 m distance from the wave maker (at model scale). At this position, the wave power spectra were calibrated and also the center of gravity of the model tanker was installed in order to measure the tanker motions due to the cal-ibrated waves. WAVE 23 is located another 88 m downstream to investigate the effect of different positions. At both positions, WAVE 8 and WAVE 23, the wave probes were removed for the tests with the tanker. Therefore, for reference purposes, wave probes WAVE 14 and WAVE 24 were mounted closer to the side wall of the basin, which were also present during the tests with the tanker. The corresponding setup of the soft moored tanker is given in Fig. 2 (centre of the tanker at the position of wave probe WAVE 8).

The distances of the wave probes from the piston type wave maker are given at model scale 1:85 in Table 1.

CHOICE OF TEST CONDITIONS AND EXECUTION OF TESTS

The following spectrum was calibrated for the model tests

(see Fig. 12): Tp = 8.5 s, H = 2.0 m, y= 3.3, where Tp is the

peak period, H, the significant wave height and 'ythe peakedness

WAVEMAKEM CH MID A VAVE 23 WAV 24 A,800' ASIN

Figure 1. WAVE PROBE POSITIONS DURING WAVE CALIBRATION AND MEASUREMENT AS WELL AS TANKER LOCATIONS.

Probe name Distance [ml Wave maker from Mid tank WAVE 1 9.65 0.00 WAVE 8 44.00 0.00 WAVE 14 44.00 6.08 WAVE 23 132.00 0.00 WAVE 24 132.00 6.08 WAVES WAVE 14 6.AOO

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543OOm CLi RF WAVE REF .-. 516.ECUn 85( DOn 14

Figure 2. SETUP OF THE SOFT MOORED TANKER IN THE BASIN.

factor of the JONSWAP spectrum. First, the wave spectrum (per water depth) was calibrated. Then, different seeds were applied to the same spectrum. From the results, the highest wave group spectrum, called the Worst Sea, the lowest, called the Best Sea,

and one in between (and closest to the theoretical group

spec-trum), called the Mean Sea, was chosen. For these choices, the wave height was scaled up to the two higher values of H, = 4.0 m

and H = 6.0 m to investigate the effect of the significant wave

height on the resulting wave group spectra. For the Worst, Best

and Mean wave condition, the model tests with a soft moored

tanker were carried out.

The variation of wave conditions discussed in this paper is summarized in Table 2.

Table 2. OVERVIEW OF WAVE CONDITIONS DISCUSSED IN THIS

PAPER(Tp = 8.5 s).

WAVE GROUP SPECTRA

The wave group spectra discussed in the following are cal-culated by different approaches (see also [11, [21 and [3]):

Derived from low frequency part of squared wave record Derived theoretically based on spectrum of measured wave Derived theoretically from spectrum of theoretical wave

The wave power and group spectra discussed in the following are all given in Fig. 11 though 19 at the end of the paper. The

wave group spectra are given as power spectra of the envelope of the wave train in m4s in terms of the circular wave frequency in radls. The group spectra determined by the first two methods are compared to the last theoretical one in order to classify the

spectra due to the different wave seeds and to chose the "Worst" and "Best" Sea regarding the low frequency wave energy content

and the "Mean" Sea which is closest to the theoretical graph.

Wave seed H, [ml Classification Water depth d = 85 m 3 2.0 "Mean Sea (MS)" 4 2.0 "Best Sea (BS)" 20 2.0 "Worst Sea (WS)" 3 4.0 "MS" 4 4.0 "BS" 20 4.0 "WS" 3 6.0 4 6.0 "BS" 20 6.0 "WS" Water depth d = 25 m 3 2.0 "Worst Sea (WS)" 4 2.0 "Mean Sea (MS)" 7 2.0 "Best Sea (BS)" 3 4.0 "WS" 4 4.0 "MS" 7 4.0 "BS" 3 6.0 "WS" 7 6.0 "BS"

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First, the Worst, Mean and Best Sea at a water depth of 85 m are selected from a series of sea states with 20 different wave seeds. Then, the influence of wave height, water depth and location in the tank is discussed.

RESULTS

In the following, the results of the wave and tanker motion measurements are discussed. In the analysis, the full time trace including the part in the beginning where the wave is still zero (due to the time it takes for propagation), and also the additional starting up time of half an hour duration is not to be considered. Thus, e. g. at wave probe 8, the time window of the signal only between t = 1899 sand t = 12741 s was analyzed in order to get the correct spectrum.

For a better interpretation of the results, it is noted that the

tanker has the following natural periods (corresponding water

depths are given in brackets):

Surge: 94.8 s (25 m) and 90.5 s (85 m) Sway: 91.0 s (25 m) and 76.6 s (85 m) Yaw: 34.6 s (25 m) and 30.4 s (85 m)

Reference Case at Intermediate Water Depth

In Fig. 11, the measured wave group spectra with the small-est and largsmall-est deviation from the theoretical wave group spec-trum are shown for a water depth of 85 m. On the left hand side, the largest wave group spectrum is given, on the right hand side

the smallest group spectrum. The mid picture shows the wave

group spectrum which is closest to the theoretical one.

In Fig. 12, the corresponding wave power spectra are shown.

They are based on one calibrated spectrum, which is

subse-quently run with different wave seeds without further calibration.

Thus, the power spectra vary in height depending on the wave

seed. In this case, the highest wave group spectrum corresponds

to the highest wave power spectrum. However, since the

spec-tral shape varies within some confidence intervals anyway, this is not the only reason for the variation of the group spectra, and a slightly higher wave power spectrum does not necessarily result

in a significantly higher wave group spectrum. Therefore, the

three wave conditions represented by Fig. 11 will serve as a base case to examine the influence of further parameter variations.

Reference Case at Shallow Water

In shallow water of 25 m depth, we can identify similarly the Worst, Mean and Best wave group spectra for different seed

numbers, although not as pronounced as for our deeper water

case. This is illustrated in Fig. 13.

Figure 3. MOTION TIME TRACES FOR WORST AND BEST WAVE

SEEDS AT WAVE PROBE POSITION 8 (WATER DEPTH OF 25 m)

Tanker Behavior for the Reference Cases

In Fig. 3 the time traces of the motions of the tanker in surge and sway direction as well as the yaw motions are shown for the Worst and Best Sea at the reference position in 25 m water depth.

Visually, it seems that the high surge motions are occurring in

groups more often for the Worst Sea (red) than for the Best Sea (blue). Sway and yaw appear to be similar for both seas. Looking at the response of the tanker in frequency domain, the full picture

is revealed (Fig. 4, 5, and 6). The measured tanker behavior

shows a clear trend: Surge and sway are significantly larger for the Worst Sea and yaw is larger for the Worst Sea in the lower frequency range.

The mooring loads show a similar trend; the forces in X

direction are much higher than those in Y direction (Fig. 7) and both are larger for the Worst Sea (Fig. 8). This can be expected

as the tanker encounters head waves.

Influence of the Wave Height

For the Worst Sea, Best Sea and Mean Sea seeds chosen above, the significant wave height is increased to 4.0 m. The resulting wave power spectra are shown in Fig. 14. The related

wave group spectra are shown in Fig. 15. For both the wave

power and group spectra, the difference between the graphs from

theoretical calculations and those based on the measurements have changed as compared to the wave spectra due to a wave height of 2.0 m: The wave power spectra are almost the same

whereas the wave group spectrum of the Best Sea has changed to a higher wave group spectrum and the Worst Sea has a much

higher group spectrum. This change can be explained by the 2000 3000 4000 5000 6003 7000 6000 9000 10000 11000 12000 0.5 S 0 0 (C 5. -0.5 2000 3000 4000 5000 5000 7000 8000 9000 10000 11003 12000 Iir,Ce [sJ 0,2 _BS 0.1 ws > -0.1 -0.2 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 Sr,,o 51

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0.015

2

- BS

ws

Figure 4. SURGE RESPONSE OF TANKER DUE TO BEST AND

WORST WAVE SEEDS AT WAVE PROBE POSITION 8 (WATER DEPTH OF 25 m).

Figure 5. SWAY RESPONSE OF TANKER DUE TO BEST AND WORST

WAVE SEEDS AT WAVE PROBE POSITION 8 (WATER DEPTH OF

25 m).

fact that althouh the seed remains the same, the higher wave train develops differently on its way through the basin due to

non-linear interaction between the wave components. Thus, the resulting seed at the reference position is not the same as for the

lower case. It can be expected that consequently also the wave

0.9- 0.8-0. 0.8-a 0.5-8 > 0.4-0.3 0.1

Figure 6 YAW RESPONSE OF TANKER DUE TO BEST AND WORST

WAVE SEEDS AT WAVE PROBE POSITION 8 (WATER DEPTH OF

25 m). 1000 500 a -500 1000 2000 3000 4000 5000 6000 7000 8000 BS ws 5000 6000 7000 8005 8000 10000 11000 12000 tin,e [a] BS ws

Figure 7. MOORING FORCE TIME TRACES FOR WAVE SEEDS 7

(BS) AND 4 (WS) AT WAVE PROBE POSITION 8 (WATER DEPTH OF 25 m).

grouping is changed.

The same explanation can be applied to the variation of wave

group spectra due to the parameter variations discussed in the

following. En Fig. 16, the wave height is further increased to 6.0 m and the resulting group spectra are much closer to each

other, i. e. to the Worst Sea.

0 0.1 02 03 0,4 0 5 0.6 5.7 0.8 09

wave frequency (red/a]

as - -.ws 0.01 0 0.005 0 0.2 0,4 0.6 0.5 1 12 1.4 16

wece frequency red/a]

0.2 04 0.6 0.8 1 1.2 1.4 1.6 1.8 2 wave frequency(fad/si

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z S 2.5 0.5 0 0 BS wo 0.8 1 1.2

wave frequency [rad/s]

wave frequency [rad/sJ

as ws

Figure 8. MOORING FORCES FX AND FY DUE TO WAVE SEEDS 4

(BS) AND 20 (WS) AT WAVE PROBE POSITION B (WATER DEPTH OF 25 m).

Looking at the tanker surge response, we find that the tanker motions is highest for the Worst Sea as given in Fig. 9: However, this is not the case for the wave height of 4.0 m (Fig. 10). It is not yet understood why it is.

Similarly, for the wave height increased to 4.0 m in the

shal-lower water case, the difference between the Worst Sea, Mean Sea and Best Sea group spectra decreases due to a change of wave seed at the reference position, see Fig. 17. A clear

dis-tinction between Worst and Best Sea seems impossible for those three group spectra.

0.9 0.8 0.7 0.6 E 5 05 0.4 C 'I, 0.3 0.2 0.1 0.35 03 0.25 E 0.2 0 15 01 0.05

WORST WAVE SEEDS AT A WAVE HEIGHT OF 6.0 M AND WAVE

PROBE POSITION 8 (WATER DEPTH OF 25 m).

0.2 04 0.6 08 1 1.2 1.6 1.8 wave frequency [rod/n]

WORST WAVE SEEDS AT A WAVE HEIGHT OF 4.0 M AND WAVE

PROBE POSITION 8 (WATER DEPTH OF 25 m).

The phasing of the wave train is also different for the same

initial seed and power spectrum but different positions in the

wave tank. Therefore, group spectra for the Worst Sea are

much lower than the theoretical group spectrum, both closer to

the wave generator and further downstream which is shown in

Fig. 18. This is also reflected in Fig. 19 where the power spectra

0.2 0.4 05 00 I 1.2 1.4 1.6 1.8

wave frequency Irad/el

0.2 0.4 0.6 1,4 1.6 1.8

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at the different probe positions are compared.

OBSERVATIONS

The following observations can be summarized from the

model tests with varied wave group spectra.

If another wave seed is applied to the same calibrated -wave power spectrum, the spectral density and significant

wave height might differ significantly. In Fig. 12, the signif-icant wave height deviates by 3.5 %, 6.5 % and 11.5 % from the targeted value of 2.0 m.

The highest wave group spectrum is related to the wave

power spectrum with the highest resulting significant wave height for the considered cases.

The differences between the wave group spectra observed

for the shallower water depth of 25 m are not as significant as for the deeper water case. However, a Worst, Mean and Best Sea could be identified, which was considered most im-portant for this study. For future work, a closer look to the reasons for the less significant differences would be interest-ing.

If the wave power spectra for the chosen Worst, Mean and Best wave group spectra are increased to a significant wave height of 4.0 m the resulting significant wave height and the

variation between the resulting wave power spectra at the

same reference position become less pronounced up to neg-ligible. This can be explained by another phasing developed at the reference position due to non-linear wave-wave inter-action on the way through the wave tank.

When the wave height is further increased up to 6.0 m, the

differences between the three wave group spectra become

even less significant.

A similar effect can be observed for the increased wave

height at the shallower water depth of 25 m.

The calibration of the wave power spectrum is usually only

valid for a relatively small area around the reference

posi-tion. Comparing the wave power spectra of the positions of

44 m upstream and 88 m downstream (at model scale), it

can be noted that the significant wave height is much lower

(about 15 %) there. This results in a very low wave group spectrum at these positions as compared to the high group

spectrum at the reference location.

Also the wave group spectra depend on the test position due to both different wave power spectra and phasing.

In general, the surge motions and the mooring load in surge direction is highest for the Worst wave condition. However,

this is not the case for all the considered variations. This

inconsistency has to be further investigated.

PERSPECTIVES

The present tests support our idea of choosing a wave seed such that the Worst Sea with respect to low frequent structure be-havior as well as the Best Sea can be chosen purposely to include

a range of test conditions in the model tests. This range has to

be determined by calculations based on estimation theory prior to model tests (see [11).

The biggest challenge will be the generation of deterministic wave seeds in the basin. This will be further developed in future projects.

REFERENCES

[I] Fernandes, A. C., Hennig, J., Domingues Maia, Jr., M..

Co-zijn, H., and Sena Sales, Jr., J., 2008. "Worst Sea - Best

Sea Wave Group Spectra from Random Sea States". In Pro-ceedings of OMAE 2008 27th International Conference on Offshore Mechanics and Arctic Engineering.

OMAE2008-5782 1.

Wichers, J., 1988. "Simulation Model for a Single Point

Moored Tanker". Doctoral Thesis. Delft University of Tech-nology.

Pinkster, J., 1980. "Low Frequency Second Order Wave Ex-citing Forces on Floating Structures". Doctoral Thesis. Delft University of Technology.

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/

Figure 12. RESULTING WAVE POWER SPECTRA FOR DIFFERENT SEEDS AT A WATER DEPTH OF 85 m, SEE FIG. 11. FROM LEFT TO RIGHT: WORST SEA, MEAN SEA, BEST SEA.

Figure 11. WAVE GROUP SPECTRA FOR DIFFERENT SEEDS BASED ON SAME WAVE POWER SPECTRUM AT A WATER DEPTH OF 85 m. FROM LEFT TO RIGHT: WORST SEA, MEAN SEA, BEST SEA.

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I

Figure 13. WAVE GROUP SPECTRA FOR DIFFERENT SEEDS AND SAME WAVE POWER SPECTRUM AT A WATER DEPTH OF 25 m. FROM LEFT TO RIGHT: WORST SEA, MEAN SEA. BEST SEA.

C.,,,.

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Figure 14. WAVE POWER SPECTRA FOR DIFFERENT SEEDS AT A WATER DEPTH OF 85 m, WAVE HEIGHT OF 4.0 in INSTEAD OF 2.0 m. FROM LEFT TO RIGHT: WORST SEA, MEAN SEA, BEST SEA.

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Figure 15. WAVE GROUP SPECTRA FOR DIFFERENT SEEDS AND SAME WAVE POWER SPECTRUM AT A WATER DEPTH OF 85 m, WAVE

HEIGHT OF 4.0 m INSTEAD OF 2.0 m. FROM LEFT TO RIGHT: WORST SEA, MEAN SEA, BEST SEA. NOTE DIFFERENT SCALE FOR DIAGRAM OF WORST SEA ON LEFT HAND SIDE.

w.-D- rwwoIw_._a.p.w1

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HEIGHT OF 4.0 m INSTEAD OF 2.0 m, FROM LEFT TO RIGHT: WORST SEA, MEAN SEA, BEST SEA.

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Figure 18. WAVE GROUP SPECTRA FOR WAVE SEED 20 AT DIFFERENT WAVE PROBE POSITIONS (WATER DEPTH OF 85 m). FROM LEFT TO RIGHT: WAVE 1 WAVE 8. WAVE 23.

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Figure 19. WAVE POWER SPECTRA FOR WAVE SEED 20 AT DIFFERENT WAVE PROBE POSITIONS (WATER DEPTH OF 85 ni). FROM LEFT TO RIGHT: WAVE 1 WAVE 8, WAVE 23.