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Subsea lifting operations

Hydrodynamic properties of subsea structures

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Hydrodynamic properties for subseastructures• Content

– Forces on structures during crossing of the splash-zone.• Landing on bottom

– Hydrodynamic phenomena.– Hydrodynamic coefficients.

• Theoretical and experimental methods• Actual values.

– Summary

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Phases of the installation process

• 1) Lift-off from deck.• 2) Lift in air• 3) Crossing splash zone.• 4) Lowering through the water column• 5) Landing the structure.

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3

4

5

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Natural periodsPendulum in air:

Pendulum in water:

( )111 2

m A LT

mg gVπ

ρ+

=−

( )222 2

m A LT

mg gVπ

ρ+

=−

( )333 2 Em A L

TEA

πα+

=

1,2 2 LTg

π=

3 2 EmLTEA

πα

=

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Water entry of TOGP template (2000)

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Water entry of TOGP template (2000)

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Water exit of ROV

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Water entry forces, calm water.

V

• Vertical hydrodynamic force:

2333 33h z z

dAF gV A U Udh

ρ= + +

z

xz = 0

h

Uz

Hydrostatic Added mass Slamming

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Water entry forces, including waves

• Vertical hydrodynamic force:

z

x

ζ

V

z = 0 h

( ) ( )

( ) ( )

2333 33

1 2

hdAF gV V Adh

B B

ρ ρ ς ς η ς η

ς η ς η ς η

= + + − + −

+ − + − −

”Slamming term”Position dependent added mass

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Interpretation of the terms

( ) ( )

( ) ( )

23 33 3 3

1 2

hd AF gV V Adh

B B

ρ ρ ς ς η ς η

ς η ς η ς η

= + + − + −

+ − + − −gVρ : Buoyancy force

Vρ ς : ”Froude Krilof” force. Effect dynamic pressure in incident wave

( )33A ς η− : Disturbance effect of body. Added mass times relative acceleration.

( )233dAdh

ς η− : ”Slamming term”, Effect of time dependent added mass.

( )1B ς η− ( )2B ς η ς η− − : Linear and quadratic damping effect.

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Water entry force, basis for derivationAppendix E, Lecture Notes

• Ideal, non-viscous, non-compressible fluid• Fast water entry

– Local fluid accelerations >>g.– Added mass coefficient for infinite frequency

• No viscous effects– Added as separate terms

• Body dimensions << wave length• Solved as a perturbation to the incident wave field • Similar expression valid for horizontal loads.

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Water entry – Key elements in derivation

Momentum in the fluid inside Ω :

Ω

M ρVdΩ= ∫∫∫ Here V is the velocity vector of the fluid inΩ . Time derivative of the momentum:

nS

dM Vρ d ρ VU dSdt tΩ

∂= Ω +

∂∫∫∫ ∫∫

NOTE:In derivation:

Water volume: ΩSubmerged body vol.: Ω*

Velocity: VAbove:

Submerged volume: VVelocity: η

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• Invoke Euler’s equation (Relation between pressure and velocity in incompressibel fluid)

• Use Gauss theorem (Relation between volume integral and integralover surface enclosing the volume)

• Show that the far field contribution from a source plus its image vanishes.

• Assume at free surface.• Evaluate the contribution of each term to the integrals • Finds

• Reformulate:

0ϕ =

( ) *3 33 ρdF A V g

dt= + Ω

( ) *3 33

*3333

2 *3333

ρ

ρ

ρ

dF A V gdt

dA dh dVV A gdh dt dt

dA V A V gdh

= + Ω

= + + Ω

= + + Ω

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• “Slamming term”: Always upward force.• Water exit: at free surface questionable.• “Slamming term” neglected in most water exit

implementations.

• Enclosed water / drainage very important during exit

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Water entry versus water exit

ϕ =

*3 33 ρExitF A V g+ Ω

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Flow during water entry vs water exit.(Greenhow & Lin 1983)

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Classical slamming solutions.

2R 2R

Von Karman (1929) Wagner (1932)

2 2

1 1 22 2

VonKarman : Wagner : 2

s s s z s z

s

s

F C AU C RLU

CC

ρ ρ

ππ

= =

==

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Added mass for simple structures.Horizontal circular cylinder

-1 -0.5 0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Added mass for horizontal 2D cylinder. φ = 0 at z = 0

Submergence h/r

A33

/ ρπ

r2

Analytical expressionsAsymptotic values

-1 -0.5 0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Derivative of A33 for horizontal 2D cylinder.

φ = 0 at z = 0

Submergence, h/r

(dA

33/d

h)ρπ

r Numerical differentiationAsymptotic value

h

2R

φ=0

3η

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Added mass for simple structures.Sphere

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

zc/R

Aii/ ρ

R3

Added mass for sphere, infinite frequency

A11A33

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Added mass for simple structures.Sphere

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5A

ii/ ρR

3

zc/R

Added mass for sphere, zero frequency

A11A33

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Close to bottom

• No action from waves• Modified ”slamming term”• Added mass for body close to ”fixed wall” (Zero frequency limit)

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Close to bottom233

3 33

1 2

0.5hdAF gV Adh

B B

ρ η η

η η η

= − −

− −

h

2R

3η

33 0dAdh<

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A33 2D cylinder close to bottom

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.5

1

1.5

2

2.5Added mass for horizontal 2D cylinder. dφ/dz = 0 at z = 0

Centre distance from wall h/r

A33

/ ρπ

r2

π2/3-1

23 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.5

2

2.5

3

3.5

4

4.5

5

h/R

A33

/ ρR

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WAMIT results for t/R=0.05Asymptotic results for h/R<<1Asymptotic results for h/R>>1

Circular disc close to bottom

• Far from bottom h/R >>1:• Close to wall (Vinje 2001):

333

83

A Rρ=

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Perforated plate, circular hollows, potential theory• Plate L*B= 15m*10m• (0) 2

33 0.625A LBρ=

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Suction anchor

• Fully submerged, no ventilation (a=0)

2R

2a

H2

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233

α , α = 0.6 - 1(plus enclosed water)

413

A R H

RA R HH

ρπ

ρπ

=

⎡ ⎤⎢ ⎥+⎢ ⎥⎣ ⎦

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Suction anchor

• Fully submerged, with ventilation

2R

2a

H

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• Partly submerged, with ventilation– Free air flow

– Restricted airflow?

Suction anchor

2R

2a

H

33 0A

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Subsea template w/protection grilleTroll

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Templates with cover and mudmats.Range of experimental results

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Template. Numerical results.Sensitivity to period and draft.

Heave Added Mass as a Function of Wave Period

-5000

50010001500200025003000

0 10 20 30 40

Wave Period [s]

Add

ed M

ass

[t]

D = 3.5 m

D = 4.0 m

D = 4.5 m

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Protection cover made from tubularmembers. (approx 14 * 19m)

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Experimental determination of added massand damping. Free oscillation tests Marintek

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Computed and measured added massProtection cover

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5KC

Ca

Measured

Perforated plate

Sum of cylinders w/ interaction

2 AXKCB

π=

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Effect of pressure drop. Protection cover

2

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: Porosity, open area/total area: Discharge coeff. ( 0.5 -1)

p v vτ ρμτ

τμ

−Δ =

∼Added mass

0

100

200

300

400

500

600

700

0 0.5 1 1.5 2 2.5 3

Amplitude (m)

A_3

3/10

00 (k

g)

MolinMeasured

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Protection cover – linearized damping

Linearized damping

0

200

400

600

800

1000

0 1 2 3Amplitude (m)

B1

(kN

s/m

)

MolinMeasured

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Effect of pressure drop.Sensitivity to discharge ratio

0

5

10

15

20

0 0.5 1 1.5 2

KC

A_3

3 (k

g)

mju = 0.75mju = 0.5mju = 1.0

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Summary• Proper added mass values crucial to find wave loads during

installation. • Water entry equations contain a ”slamming term”.• In splash zone: Added mass sensitive to submergence and frequency.• By landing on bottom an increased added mass may contribute to

softer landing.• Numerical and experimental tools available to find added mass and

damping.• Viscous effects important. • Depth dependent values difficult to establish.• Theoretical expressions exist for several simple shapes.