experimental study of the effect of second order wavemaker

Experimental study of the effect of second order wavemaker theory on the response of a large diameter monopile in irregular sea

Fatemeh H. Dadmarzi1,Matthias Tonnel2, Maxime Thys3,Erin E. Bachynski1,

Trygve Kristiansen1

(1) Norwegian University of Science and Technology, NTNU , Trondheim, Norway(2) Leopold-Franzens Universität Innsbruck, Innsbruck, Tyrol, Austria(3) SINTEF Ocean, Trondheim, Norway

(WAS-XL: Wave loads and soil support for extra large monopiles)

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Background/Objectives

• Larger offshore wind turbines -> longereigenperiods and increased dynamic response tononlinear wave loads

• Model testing:– Understanding hydrodynamic loads and structural

responses– Validating numerical models

• Need to investigate uncertainties and effects oftesting techniques

• How important is the second order wavemakercorrection for the measured responses?

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Outline

• Second order wavemaker correction– Theory– Implementation and experimental assessment

• Monopile testing – Experimental setup– Repeatability– Statistical results– Sample events of extreme responses

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Wavemaker theory (piston-type wavemaker)

Second-order wavemaker theory for irregular waves, Hemming A. Schäffer, Ocean Engng, Vol. 23, No. 1, pp. 47-88. 1996

Bottom

Wave board𝑋𝑋(𝑡𝑡)

ℎ

𝑥𝑥

𝑧𝑧𝜂𝜂 𝑥𝑥, 𝑡𝑡

∇2Φ = 0

Φ𝑧𝑧 = 0

Φ𝑥𝑥 = 𝑋𝑋𝑡𝑡(𝑡𝑡)

−𝜕𝜕Φ𝜕𝜕𝑡𝑡

−12∇Φ ⋅ ∇Φ − 𝑔𝑔𝑧𝑧 = 0

𝜕𝜕Φ𝜕𝜕𝑧𝑧

−𝜕𝜕Φ𝜕𝜕𝑡𝑡

− ∇Φ ⋅ ∇𝜂𝜂 =1𝜌𝜌𝑝𝑝𝑎𝑎 DFSBC

KFSBC

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Second-order wavemaker theory for irregular waves, Hemming A. Schäffer, Ocean Engng, Vol. 23, No. 1, pp. 47-88. 1996

First and second order wavemaker motion

• First order wavemaker motion– Sum of harmonic components

• This motion results in generation of – Desired first order waves– Desired second order bound waves– Undesired second order spurious

waves– And even higher order waves

• Second order wavemaker motion is added to counter the spurious waves

𝑋𝑋0(1) = −𝑖𝑖�

𝑛𝑛=1

𝑀𝑀

𝑋𝑋𝑛𝑛𝑒𝑒𝑖𝑖𝜔𝜔𝑛𝑛𝑡𝑡

𝑋𝑋0(2) = �

𝑛𝑛=1

𝑀𝑀

�𝑚𝑚=2

𝑀𝑀

−𝑖𝑖𝐹𝐹𝑛𝑛𝑚𝑚+𝐴𝐴𝑛𝑛𝐴𝐴𝑚𝑚ℎ

𝑒𝑒𝑖𝑖 𝜔𝜔𝑛𝑛+𝜔𝜔𝑚𝑚 𝑡𝑡

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Dispersion relation

First order waveSecond order bound wave

Circular frequency 𝜔𝜔

Wav

e nu

mbe

r 𝑘𝑘

Spurious wave

Free wave dispersion relation

Bound wave dispersion relation

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Experimental setup wave probes

• Wave elevation measurement along 7m length of the tank centerline with a step of 0.08m

• Acceptable resolution for calculating 2D FFT

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Dispersion relation: Gaussian spectrum Tp = 12s, Hs = 4m

Circular frequency 𝜔𝜔

Wav

e nu

mbe

r 𝑘𝑘

Circular frequency 𝜔𝜔W

ave

num

ber 𝑘𝑘

No correction With correction

Free wave dispersion relation

3rd order spurious wave (pink)

3nd order bound wave

2nd order bound wave (red)

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Monopile testing: experimental setup• Instrumentation: 11 wave probes, water temp, wavemaker

position, strain gauges and 6DOF force transducer• Water depth: 27 m• Monopile diameter: 9 m• Automated test setup for increased statistics

10

0 20 40 60 80 100

Time [s]

-2

-1

0

1

2

Bend

ing

mom

ent [

Nm

]

10 8

0 20 40 60 80 100

Time [s]

-1

-0.5

0

0.5

1

Bend

ing

mom

ent [

Nm

]

10 8

Decay tests

First mode: 𝒇𝒇𝟏𝟏 = 𝟎𝟎.𝟐𝟐𝟐𝟐 (Hz)Damping ratio: 1.1%

Second mode:𝒇𝒇𝟐𝟐 = 𝟏𝟏.𝟐𝟐𝟓𝟓 (Hz)Damping ratio: 0.4%

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Repeatability of bending moment response at two largest events from 10 test repetitions

Second mode responseFirst mode responseQuasi-static response

Second mode responseFirst mode responseQuasi-static response

Repeatability of the phase of the responses at second natural frequency.

Time [s] Time [s]Hs 8.6 m, Tp 11 s

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Coefficient of variation (COV) for the 10 largest bending moment maxima in 10 repetitions of one seed

Contributions calculated at the exact time of the maximum total moment.

Hs 8.6 m, Tp 11 s

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Effect of second order correction on the wave elevation

Hs 9.0 m, Tp 12.5 s

Mean spectrum of the wave

10 realizations of the same sea state

Little change in wave elevation statistics using this correction

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Mean spectrum of the mud-line bending moment response

Mean spectrum around the second natural frequency

Hs 9.0 m, Tp 12.5 s

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Mean spectrum of the mud-line bending moment response

Mean spectrum around the second natural frequency

Hs 9.0 m, Tp 12.5 s

𝑓𝑓𝑃𝑃

2𝑓𝑓𝑃𝑃

𝐹𝐹1

𝐹𝐹2

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Exceedance probability of bending momentresponse for 10 realizations of the testswithout and with second-order correction

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Largest excitation in second mode within 10 seeds

Bending moment response in the tests without and with wavemaker second-order correction

Hs 9.0 m, Tp 12.5 s

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Without correction With correction

Snapshots of the wave at the model for the tests without and with wavemaker second-order correction

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Bending moment response in the tests without and with wavemaker second-order correction

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Frequency content of 20 extreme responses from a single realization

Without correction

With correction

Response at second natural frequencyResponse at first natural frequency Quasi-static response

Event

Event

Hs 9.0 m, Tp 12.5 s

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Conclusions

• Second order wavemaker theory (developed by Schäffer) for piston-type wavemaker motion in irregular waves

• Experimental observation of reduction in the second order spurious wave component for Gaussian spectra

• Experimental study of the effect of the presence/absence of this spuriouswave on the response of a monopile

• Statistically, minor effect on response measurements

• However, affects individual breaking wave events and slamming loads

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Thank you!

This work is part of the Wave Loads and Soil Support for Extra Large Monopiles (WAS-XL) project, funded by NFR grant 26818 and industry partners

b

experimental study of the effect of second order wavemaker

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