FrequencyFrequency--Domain Calculations of Moored Domain Calculations of Moored Vessel Motion Including Low Frequency EffectVessel Motion Including Low Frequency Effect

OMAE 2008OMAE 2008Estoril, PortugalEstoril, PortugalJune 15June 15--20, 200820, 2008

Cedric Le Cunff Cedric Le Cunff PrincipiaPrincipiaSam Ryu Sam Ryu SOFECSOFECJ.J.--M. HeurtierM. Heurtier PrincipiaPrincipiaArun DuggalArun Duggal SOFECSOFEC

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Outline

1. Background

2. Summary of TD and FD Analyses

3. Case Studies

4. Comments on Modeling

5. Summary

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Where We Are

1. Background

2. Summary of TD and FD Analyses

3. Case Studies

4. Comments on Modeling

5. Summary

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Background“Derivation of CALM Buoy Coupled Buoy RAOs in FrequencyDomain and Experimental Validation” (Le Cunff et al., 2007)

deepwater CALM buoy

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CALM Buoy System with Tanker Connected

FPSOTanker Buoy

Flowlines (OOLs)

Buoy Mooring

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Validation with Model Test Results

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Comparisons between TD and FD (Pitch Motion)

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Where We Are

1. Background

2. Summary of TD and FD Analyses

3. Case Studies

4. Comments on Modeling

5. Summary

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Time-Domain Analysis• Hydrodynamic Loads on the Bodies

– hydrodynamic calculations via BEM method

– added mass, radiation damping, first order wave force

• Loads on the Lines– FEM-based computer program (DeepLines)

– lines characteristics + Morison’s formulation

• Coupled System– extra node with six DOF for the floating body

– solve the coupled equation:

{ }0

(1) (2)

( ) ( )τ τ∞

+ ∞ + − +

= + + +

∫r r r&& &

r r r ra

d w w line

M M X R t Xd KX

F F F F

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Frequency-Domain Analysis

• Equation of motion:

• Assuming that the position at time t is given by:

with

– Static equilibrium :

– Imposed displacement :

FxMxBKx =++ &&&

( ) ( ) ( )∑ ∑=

−

= ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡++=

n

i

tjif

Nimp

impimpiimpstat e ixxaXtX

1 1

Re)(ω

ωω

0=impstatxK

statstatstat FxK =

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(cont’d)

• Loads on body from hydrodynamic calculations

• Dependency of matrices on frequency

• Linearization of quadratic viscous damping (both body and lines)

– Linearization Coefficients

• Regular waves : (A: norm of velocity)

• Irregular waves: (σ: standard deviation)

( ) ( )( ) ( )[ ] ( ){ } { })()(2 ωωωωωωω FXMMBBjK aa =+−+−

relrelrelrel vvvv )(Ω≈

Aπ38=Ω

σπ 22=Ω

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• wave elevation:

• low-frequency (LF) force, Molin (2002)

• LF force spectrum

Low-Frequency Wave Loading

[ ])expRe(1∑=

+−=n

llll jtjA θωη

( ) ( )[ ]))(exp,Re(1 1

)2()2( ∑∑= =

−+−−=n

l

n

mmlmlmlmlwave jtjfAAF θθωωωω

( ) ( ) ( ) ( ) ωωωωω dfSSSF

2

0

)2()2( ,8∫∞

−ΩΩ+=Ω

QTF:

Direct calculation

2) Newman’s approximation

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Low-Frequency Wave Loading (QTF)

• Newman’s approximation

• Second order wave potential (x-direction for example)

( ) ( ) ( )mdldml fff ωωωω ×=2)2( ,

( ) ( )[ ] 2

2)2(

)()()(

)()(1,

mlmlml

mlmlmlxmlx hkkthkkg

kkVCmf

ωω

ωωωωρωω

−−−−

−−+=

( )( )[ ]( )[ ] βcos

hkkchhzkkch

jl

jl

−

+−×

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• Wind speed driven from wind spectrum:

• Wind force:

• Evaluation of wind coefficients based on mean direction (wind w.r.t. body)

Wind Loading

[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛+−= ∑

=

n

llllwind jtjAV

1

expRe θω

2, 2

1vesselwindxairwindx VVSCF −= ρ

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Where We Are

1. Background

2. Summary of TD and FD Analyses

3. Case Studies

4. Comments on Modeling

5. Summary

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FD/TD Calculation Results Comparison

0°

45°

90°

135°

180°

225°

270°

315°

Wave Direction (e.g. 180 deg)

earth-fixed+ X (North)

earth-fixed+ Y (West)

vessel-fixed+y (sway)

vessel-fixed+x (surge)

+Roll: Starboard Down+Pitch: Bow Down+Yaw: Bow to Port+Z: Upwards

Spread Moored FPSOComparison FD/TD:

FD - spectrum directly computed

TD - FFT of time series

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Time-Domain Calculation

• 3-hour simulation

• LF and WF components

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Case 1 (waves only; no wind)Results – FPSO Surge

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Results – FPSO Sway (cont’d)

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Results – FPSO Yaw (cont’d)

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Case 2 (waves + wind)Results – FPSO LF Motions

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Where We Are

1. Background

2. Summary of TD and FD Analyses

3. Case Studies

4. Comments on Modeling

5. Summary

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1. Second-Order Wave Potential

• Effect negligible

• Induces a very small force spectrum at low frequency range

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2. Reduction of LF Contribution by WF Components

Comparisons– first order waves (WF) only

– low frequency (LF) only

– LF + WF

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3. Low Stiffness Mooring

• Wave + wind aligned: OK

• Otherwise TD required

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Where We Are

1. Background

2. Summary of TD and FD Analyses

3. Case Studies

4. Comments on Modeling

5. Summary

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Summary

1. Freq-domain analysis methodology is presented as a tool of motion estimate even including LF components.

2. Classical linearization methods applied to quadratic drag term

3. Case studies carried out with a spread moored FPSO

4. Comparison b/w FD and TD calculations shows that the results are in good agreement.

5. Second-order wave potential is not significant for the relatively low frequency range.

6. Unstable time-varying yaw motion can only be analyzed by using a TD analysis.

7. Comments on modeling presented.

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Future Work: Validation against Model Test Results

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Thank you very much!