Offshore
Hydromechanics
Part 2
Ir. Peter Naaijen
1. Introduction and Ship Motion
OE 4630 2012 - 2013
Offshore Hydromechanics, lecture 1
Teacher module II:
• Ir. Peter Naaijen
• p.naaijen@tudelft.nl
• Room 34 B-0-360 (next to towing tank)
Book:
• Offshore Hydromechanics, by J.M.J. Journee & W.W.Massie
Useful weblinks:
• http://www.shipmotions.nl
• Blackboard
[1]
[2]
Take your laptop, i- or whatever smart-phone and go to:
www.rwpoll.com
Login with session ID
OE4630 2012-2013, Offshore Hydromechanics, Part 2
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Marine Engineering, Ship Hydromechanics Section
OE4630 module II course content
• +/- 7 Lectures
• Bonus assignments (optional, contributes 20% of your exam grade)
• Laboratory Excercise (starting 30 nov)
•
1 of the bonus assignments is dedicated to this exercise
•
Groups of 7 students
•
Subscription available soon on BB
• Written exam
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Learning goals Module II, behavior of
floating bodies in waves
•
Lecture notes:
Definition of ship motions
Motion Response in regular waves:
• How to use RAO’s
• Understand the terms in the equation of motion: hydromechanic reaction forces, wave exciting forces
• Disclaimer: Not everything you (should) learn is in the lecture notes (lees: niet alles
• How to solve RAO’s from the equation of motion
Motion Response in irregular waves:
wat op het tentamen gevraagd kan worden staat in diktaat…) •How to determine response in irregular waves from RAO’s and wave spectrum without forward speed
3D linear Potential Theory
•How to determine hydrodynamic reaction coefficients and wave forces from Velocity Potential
Make personal notes during lectures!!
•How to determine Velocity Potential
Motion Response in irregular waves:
• How to determine response in irregular waves from RAO’s and wave spectrum with forward speed
• Don’t save your questions ‘till the break Ch. 8
• Make down time analysis using wave spectra, scatter diagram and RAO’s
Structural aspects:
• Calculate internal forces and bending moments due to waves
Ask if anything is unclear
Nonlinear behavior:
Ch.6
• Calculate mean horizontal wave force on wall
• Use of time domain motion equation
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Marine Engineering, Ship Hydromechanics Section
Introduction
Introduction
Offshore → oil resources have to be explored in deeper water → floating
structures instead of bottom founded
[4]
[3]
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Introduction
Introduction
Reasons to study waves and ship behavior in waves:
• the dynamic loads on the floating structure, its cargo or its equipment:
Reasons to study waves and ship behavior in waves:
• the dynamic loads on the floating structure, its cargo or its equipment:
• Inertia forces on sea fastening due to accelerations:
• Direct wave induced structural loads
[5]
Minimum required air gap to avoid wave damage
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Introduction
Introduction
Reasons to study waves and ship behavior in waves:
• Determine allowable / survival conditions for offshore operations
Decommissioning / Installation / Pipe laying Excalibur / Allseas ‘Pieter Schelte’
• Motion Analysis
[7]
[5]
[5]
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Introduction
Introduction
Reasons to study waves and ship behavior in waves:
• the dynamic loads on the floating structure, its cargo or its equipment:
Floating Offshore: More than just oil
• Forces on mooring system, motion envelopes loading arms
Floating wind farm
[9]
[10]
OTEC
[8]
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Introduction
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Introduction
Floating Offshore: More than just oil
[11]
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Floating Offshore: More than just oil
[12]
Wave energy conversion
Mega Floaters
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Introduction
Reasons to study waves and ship behavior in
waves:
Real-time motion prediction
Using X-band radar remote wave observation
• Determine allowable / survival conditions for offshore
operations
• Downtime analysis
WinterDataofAreas8,9,15and16oftheNorthAtlantic(Global WaveStatistics)
T2 (s)
Hs(m)
14.5
13.5
12.5
11.5
10.5
9.5
8.5
7.5
6.5
5.5
4.5
3.5
2.5
1.5
0.5
3.5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
4.5
0
0
0
0
0
0
0
0
0
0
0
0
4
40
350
5.5
0
0
0
0
0
0
0
0
1
7
31
148
681
2699
3314
6.5
0
0
0
0
1
4
12
41
138
471
1586
5017
13441
23284
8131
7.5
2
3
7
17
41
109
295
818
2273
6187
15757
34720
56847
47839
5858
8.5
30
33
72
160
363
845
1996
4723
10967
24075
47072
74007
77259
34532
1598
9.5
154
145
289
585
1200
2485
5157
10537
20620
36940
56347
64809
45013
11554
216
10.5
362
293
539
996
1852
3443
6323
11242
18718
27702
33539
28964
13962
2208
18
11.5
466
322
548
931
1579
2648
4333
6755
9665
11969
11710
7804
2725
282
1
12.5
370
219
345
543
843
1283
1882
2594
3222
3387
2731
1444
381
27
0
13.5
202
101
149
217
310
432
572
703
767
694
471
202
41
2
0
Total
1586
1116
1949
3449
6189
11249
20570
37413
66371
111432
169244
217115
210354
122467
19491
Total
5
394
6881
52126 170773 277732 256051 150161
61738
19271
4863 999995
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Irregular wind waves
Definitions & Conventions
Regular waves
Ship motions
apparently irregular but can be
considered as a superposition of a
finite number of regular waves, each
having own frequency, amplitude and
propagation direction
[5]
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Regular waves
regular wave propagating in direction µ:
ζ (t, x) = ζ a cos (ωt − kx cos µ − ky sin µ )
Regular waves
k = 2π / λ
ω = 2π / T
(Ch.5 revisited)
Linear solution Laplace equation
y
λ
x
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• Regular waves
Co-ordinate systems
• regular wave propagating in direction µ
Definition of systems of axes
Earth fixed: (x0, y0, z0)
Following average ship position: (x,y,z)
wave direction with respect to ship’s exes system: µ
ζ (t, x) = ζ a cos (ωt − kx cos µ − ky sin µ )
Phase angle ε
y0
Wave direction = x0
µ
Phase angle wave at black dot
with respect to wave at red dot:
ε ζ ,ζ = −k ( x − x ) cos µ − k ( y − y ) sin µ
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X0
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Behavior of structures in waves
Behavior of structures in waves
Ship’s body bound axes system (xb,yb,zb) follows all ship motions
Definition of translations
NE
EN
1 x
Schrikken
Surge
2 y
Verzetten
Sway
3
Dompen
Heave
z
xb
z
Z: heave
zb
x
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Behavior of structures in waves
Definition of rotations
NE
EN
How do we describe ship motion response?
Rao’s
Phase angles
Z: yaw
4
x
Slingeren
Roll
Y: pitch
5
y
Stampen
Pitch
6
z
Gieren
Yaw
X: roll
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Motions of and about COG
Mass-Spring system:
Amplitude
mz + bz + cz = Fa cos ( ωt )
Sway(verzetten) :
z ( t ) = za cos (ω t + ε )
Phase angle
Surge(schrikken) : x = xa cos ( ωt + ε xζ
Motion equation
Heave(dompen) :
m
Steady state solution
c
z = za cos ( ωt + ε zζ
phi φ = φa cos (ωt + ε φζ )
Roll (rollen) :
b
y = ya cos ( ωt + ε yζ
)
)
)
Pitch(stampen) :
theta θ = θ a cos (ωt + εθζ )
psi ψ = ψ a cos ( ωt + εψζ )
Yaw( gieren) :
Phase angles ε are related to undisturbed wave at origin of steadily
translating ship-bound system of axes (→COG)
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Motions of and about COG
Motions of and about COG
Phase angles ε are related to undisturbed wave at origin of steadily
Surge(schrikken) : x = xa cos ( ωt + ε xζ
translating ship-bound system of axes (→COG)
Sway(verzetten) :
Heave(dompen) :
)
y = ya cos (ωt + ε yζ
z = za cos ( ωt + ε zζ
)
:
)
phi φ = φa cos (ωt + ε φζ )
Roll (rollen) :
Pitch(stampen) :
theta θ = θ a cos (ωt + εθζ )
psi ψ = ψ a cos ( ωt + εψζ )
Yaw( gieren) :
x
RAOSurge
RAOSway :
a
ζa
ya
ζa
(
ωµ
,
)
(ω , µ )
za
(ω , µ )
ζa
φa
RAORoll :
(ω , µ )
ζa
θa
RAOPitch :
(ω , µ )
ζa
ψa
RAOYaw :
(ω , µ )
ζa
RAOHeave :
RAO and phase depend on:
•
•
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Wave frequency
Wave direction
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Example: roll signal
Motions of and about COG
1 Surge(schrikken) : x = xa cos (ωe t + ε xζ
2 Sway(verzetten) :
3 Heave(dompen) : z = za cos ( ωe t + ε zζ
4 Roll(rollen) :
6 Yaw( gieren) :
Velocity…
Acceleration…
φ = φa cos (ωet + ε φζ )
φ = −ω φe asin (ω te + ε φζ) = ω φe acos (ω te + ε φζ + π / 2 )
φ = −ω 2φ cos (ω t + ε ) = ω 2φ cos (ω t + ε + π )
e
a
e
φζ
e
a
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)
)
theta θ = θ a cos ( ωe t + ε θζ
psi ψ = ψ a cos (ωet + εψζ
)
)
•
Frequency of input (regular wave) and output (motion) is ALWAYS THE SAME !!
•
Phase can be positive ! (shipmotion ahead of wave elevation at COG)
•
Due to symmetry: some of the motions will be zero
•
Ratio of motion amplitude / wave amplitude = RAO (Response Amplitude Operator)
•
RAO’s and phase angles depend on wave frequency and wave direction
•
RAO’s and phase angles must be calculated by dedicated software or measured by experiments
•
Only some special cases in which ‘common sense’ is enough:
φζ
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Consider Long waves relative to ship dimensions
Local motions (in steadily
What is the RAO of pitch in head waves ?
•
•
•
•
)
phi φ = φa cos ( ωe t + ε φζ
5 Pitch(stampen) :
Displacement
)
y = ya cos (ωe t + ε yζ
•
Only variations!!
•
Linearized!!
translating axes system)
 xP ( t )   x ( t )   0
−ψ ( t ) θ ( t )   xbP 

 
 
 

t
=
y
t
+
ψ
t
0
−φ ( t )  ⋅  ybP 
y
(
)
(
)
(
)
 
 P  

 
 
0   zbP 
 Pz ( t)  z ( t)  −θ ( t ) φ ( t )
Phase angle heave in head waves ?...
RAO pitch in head waves ?...
Phase angle pitch in head waves ?...
Phase angle pitch in following waves ?...
6 DOF Ship motions
Location considered point
xP ( t ) = x ( t ) − ybPψ ( t ) + zbPθ ( t )
yP ( t ) = y ( t ) + xbPψ ( t ) − zbPφ ( t )
z P ( t ) = z ( t ) − xbPθ ( t ) + ybPφ ( t )
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Complex notation of harmonic functions
Local Motions
The tip of an onboard crane, location:
xb,yb,zb = -40, -9.8, 25.0
Example 3: horizontal crane tip motions
1 Surge(schrikken) : x = xa cos ( ωet + ε xζ )
(
(
i ωt +ε xζ
= Re xa e
P
(
= Re xa e
ε xζ
⋅e
)
)
iωt
)
Complex motion amplitude
= Re ( x̂a ⋅ eiωt )
For a frequency ω=0.6 the RAO’s and phase angles of the ship motions are:
SURGE
RAO
1.014E-03
SWAY
phase
degr
3.421E+02
RAO
5.992E-01
phase
degr
2.811E+02
HEAVE
RAO
9.991E-01
phase
degr
3.580E+02
ROLL
RAO
deg/m
2.590E+00
phase
degr
1.002E+02
PITCH
RAO
phase
deg/m degr
YAW
RAO
phase
deg/m
degr
2.424E-03
2.102E-04
1.922E+02
•
:
5.686E+01
Calculate the RAO and phase angle of the transverse horizontal motion (y-direction)
of the crane tip.
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Mass-Spring system:
Relation between Motions and Waves
How to calculate RAO’s and phases ?
Forces acting on body:
F = Fa cos ( ωt )
…?
z
Input: regular wave, ω
m
Output: regular motion
ω, RAO, phase
b
RAO
Floating
Structure
c
phase
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Mass-Spring system:
Moving ship in waves:
mz + bz + cz = Fa cos ( ωt )
Transient solution
z ( t ) = At e −ζω t sin
0
F = Fa cos ( ωt )

b 
ζ =

2 mc 

z
(
1 − ζ 2 ω0 t + ϕt
F
m
)
b
Damping ratio
m
c
[14]
m3  z + b3 z + c3 z = Fa 3 cos (ωt )
Steady state solution:
b
z
c
z ( t ) = za cos (ωt + ε )
( −bω )
 −ω 2 ( m ) + c


ε = a tan 
za =
(
Fa
m4ϕ + b4ϕ + c4ϕ = Fa 4 cos ( ωt )


)
Restoring coefficient for heave ?
Restoring coefficient for roll ?
m for roll ?
2
(( − ( m)ω + c ) + (bω ) )
2
2
2
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What is the hydrostatic spring
coefficient for the sway motion ?
Non linear stability issue…
m2 ⋅ y + b2 ⋅ y + c2 ⋅ y = Fa 2 cos (ωt )
z
y
x
Alat
Awl
x
33%
33%
33%
A. c2=Awl ρ g
B. c2=Alat ρ g
C. c2=0
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Moving ship in waves:
Floating stab.
‫ܯ‬
Stability moment
m4ϕ + b4ϕ + c4ϕ = Fa 4 cos ( ωt )
Restoring coefficient for roll ?
M s = ρ g∇ ⋅ GZϕ = ρ g∇ ⋅ GM sin ϕ ≈ ρ g∇ ⋅ GM ⋅ ϕ
φ
G
G
G
φ
‫ܤ‬
Rotation around COF
‫ܤ‬஦
ρ
F
F
஦
‫ܩ‬
Rotation around COG
=Rotation around COF
dz = FG − FG cos ϕ ≈ 0
+vertical translation
+horizontal translation dy = FG sin ϕ ≈ FGϕ
ߘ
‫ܭ‬
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Moving ship in waves:
Not in air but in water!
46
Moving ship in waves:
F
m
b
z
[14]
c
z
(m + a) ⋅ z + b ⋅ z + c ⋅ z = Fw
[14]
F = m ⋅ z
Fw
−c ⋅ z
−b ⋅ z
−a ⋅ z
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Marine Engineering, Ship Hydromechanics Section
Hydromechanic
reaction forces
motions
(Only potential / wave
damping)
Waves
motions
Wave forces
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Moving ship in waves:
Solutions
2.
Back to Regular waves
(m + a) ⋅ z + b ⋅ z + c ⋅ z = Fw
(Wave Force effected by motions and other way around)
Hydromechanic reaction forces
wave force)are calculated
for flat water (NO WAVES)
Waves
ζ (t, x) = ζ a cos (ωt − kx cos µ − ky sin µ )
Linear solution Laplace equation
In order to calculate forces on immerged bodies:
What happens underneath free surface ?
reaction forces
motions
regular wave propagating in direction µ
motions
Wave forces
Wave force is calculated for
restricted ship in mean
position: No motions
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Back to Regular waves
Potential Theory
regular wave propagating in direction µ
ζ (t, x) = ζ a cos (ωt − kx cos µ − ky sin µ )
What is potential theory ?:
way to give a mathematical description of flowfield
Linear solution Laplace equation
In order to calculate forces on immerged bodies:
What happens underneath free surface ?
Most complete mathematical description of flow is
viscous Navier-Stokes equation:
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→
Navier-Stokes vergelijkingen:
ρ
ρ
ρ
∂u
∂t
∂v
∂t
+ ρu
+ ρu
∂p ∂ 
∂u  ∂   ∂v ∂u  ∂   ∂u ∂w
∂u
∂u
∂u
+ ρv
+ ρw = − + λ∇⋅V + 2µ  + µ +  + µ + 
∂x  ∂y   ∂x ∂y  ∂z   ∂z ∂x 
∂x ∂x 
∂x
∂y
∂z
∂v
∂x
+ ρv
∂v
∂y
+ ρw
∂v
∂z
=−
Apply principle of continuity on control volume:
∂p ∂   ∂v ∂u  ∂ 
∂v  ∂   ∂w ∂v 
+ µ +  + λ∇⋅V + 2µ  + µ + 
∂y  ∂z   ∂y ∂z 
∂y ∂x   ∂x ∂y  ∂y 
∂w
∂w
∂w
∂w
∂w
∂p ∂   ∂u ∂w ∂   ∂w ∂v  ∂ 
= − + µ +  + µ +  +  λ∇⋅V + 2µ 
+ ρu + ρv + ρw
∂z 
∂z ∂x   ∂z ∂x  ∂y   ∂y ∂z  ∂z 
∂t
∂x
∂y
∂z
(A-relaxed !)
Continuity: what comes in,
must go out
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If in addition the flow is considered to be irrotational and non
viscous →
This results in continuity equation:
∂u ∂v ∂w
+
+
=0
∂x ∂y ∂z
Velocity potential function can be used to describe
water motions
Main property of velocity potential function:
for potential flow, a function Φ(x,y,z,t) exists whose derivative in a
certain arbitrary direction equals the flow velocity in that
direction. This function is called the velocity potential.
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Summary
From definition of velocity potential:
u=
•
∂Φ
∂Φ
∂Φ
,v =
,w =
∂x
∂y
∂z
Potential theory is mathematical way to describe flow
Important facts about velocity potential function Φ:
• definition: Φ is a function whose derivative in any direction equals the
flow velocity in that direction
Substituting in continuity equation:
• Φ describes non-viscous flow
∂u ∂v ∂w
+
+
=0
∂x ∂y ∂z
• Φ is a scalar function of space and time (NOT a vector!)
Results in Laplace equation:
∂ 2Φ ∂ 2Φ ∂ 2Φ
+
+
=0
∂x 2 ∂y 2 ∂z 2
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trajectories of water
particles in infinite water
depth
Velocity potential for regular wave is obtained by
•
Solving Laplace equation satisfying:
1.
Seabed boundary condition
2.
Dynamic free surface condition
Φ( x, y, z, t ) =
ζ a g kz
⋅ e ⋅ sin(kx cos µ + ky sin µ − ωt )
ω
Φ( x, y, z, t ) =
ζ a g cosh(k (h + z))
⋅
⋅ sin(kx cos µ + ky sin µ − ω t )
ω
cosh(kh)
3.
58
Water Particle Kinematics
Summary
•
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Kinematic free surface boundary condition results in:
Dispersion relation =
relation between wave
frequency and wave length
ω 2 = kg tanh(kh)
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Water Particle Kinematics
Pressure
trajectories of water
particles in finite water
depth
Pressure in the fluid can be found using Bernouilli equation
for unsteady flow:
p
∂Φ 1 2
+ 2 (u + w2 ) + + gz = 0
∂t
ρ
∂Φ 1
p = −ρ
− ρ (u 2 + w2 ) − ρ gz
∂t 2
1st order fluctuating
pressure
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2nd order (small
quantity
squared=small enough
to neglect)
Hydrostatic pressure
(Archimedes)
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Wave Force
Potential Theory
Determination Fw
From all these velocity potentials we can derive:
• Pressure
•
Froude Krilov
•
Diffraction
• Forces and moments can be derived from pressures:
Spar
Barge
F = − ∫∫ ( p ⋅ n ) dS
S
M = − ∫∫ p ⋅ ( r × n ) dS
zb
S
r = ( xb,P , 0, yb ,P )
dS
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xb
[15]
n = (0, 0, −1)
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Flow superposition
(m + a) ⋅ z + b ⋅ z + c ⋅ z = F
1.
w
Considering a fixed structure (ignoring the motions) we will
try to find a description of the disturbance of the flow by
the presence of the structure in the form of a velocity
potential. We will call this one the diffraction potential and
added to the undisturbed wave potential (for which we
have an analytical expression) it will describe the total flow
due to the waves.
Flow due to Undisturbed wave
Φ0 = −
ζag
ω
2.
Flow due to Diffraction
Exciting force due to waves
(m + a) ⋅ z + b ⋅ z + c ⋅ z = Fw
= FFK + FD
1.
⋅ e ⋅ sin( ω t − kx cos µ − ky sin µ)
kz
Has to be solved. What is boundary
condition at body surface ?
Undisturbed wave force (Froude-Krilov)
2.
ζ g
Φ 0 = − a ⋅ e kz ⋅ sin (ωt − kx cos µ − ky sin µ + ε )
ω
Φ7
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Has to be solved. What is boundary
condition at body surface ?
Φ7
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Diffraction force
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Wave Forces
Wave force acting on
vertical wall
Force on the wall
0
F = − ∫ p ⋅ ndz
z
−∞
ζ g
ζ g
Φ 0 = a e kz sin ( kx − ωt ) , Φ 7 = − a e kz sin ( kx + ωt )
ω
ω
p = −ρ
=∂Φ = − ρ ∂ ( Φ
∂t
+ Φ7 )
=
∂t
n = (−1,0, 0)
0
x
 ζ g kz

∂  − 2 a ⋅ e sin ( ωt ) cos ( kx ) 
ω


−ρ
=
∂t
kz
2 ρζa g ⋅ e cos ( kx ) cos ( ωt )
n = ( −1, 0, 0 )
x=0
0
F x=
−∞
2ρ
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ζag
k
 ζ g

g ⋅ ekz cos ( ωt ) dz = 2 ρ a ⋅ ekz cos ( ωt )  =
k

 −∞
0
∫2ρζ
a
⋅ cos ( ωt ) − 0
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17
Sources images
[1] Towage of SSDR Transocean Amirante, source: Transocean
[2] Tower Mooring, source: unknown
[3] Rogue waves, source: unknown
[4] Bluewater Rig No. 1, source: Friede & Goldman, LTD/GNU General Public License
[5] Source: unknown
[6] Rig Neptune, source: Seafarer Media
[7] Pieter Schelte vessel, source: Excalibur
[8] FPSO design basis, source: Statoil
[9] Floating wind turbines, source: Principle Power Inc.
[10] Ocean Thermal Energy Conversion (OTEC), source: Institute of Ocean Energy/Saga
University
[11] ABB generator, source: ABB
[12] A Pelamis installed at the Agucadoura Wave Park off Portugal, source: S.Portland/Wikipedia
[13] Schematic of Curlew Field, United Kingdom, source: offshore-technology.com
[14] Ocean Quest Brave Sea, source: Zamakona Yards
[15] Medusa, A Floating SPAR Production Platform, source: Murphy USA