Hydromechanics linear theory Offshore
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8/10/2019 Hydromechanics linear theory Offshore
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1OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Relation between Motions and WavesHow to calculate RAO’s and phases ?
FloatingStructure
Input: regular wave, ω Output: regular motion
ω , RAO, phase
phase
R A O
2OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Mass-Spring system:
z
c b
m
cosaF F t
F m z
waves w FK diff
hydrostatic restoring
hydrodynamic reaction radiation
F F F F
F c z
F F a z b z
wmz bz cz F
Newton’s 2 nd law:
3OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Mass-Spring system: Assumptions Excitation F w • Not affected by motions • Resulting from harmonic pressures
integrated over average submergedhull surface
Excitation fromwaves is harmonic:
,coswa F mz bz cz F t
4OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Solution Mass-Spring system:
0 20
2
2
2 22
cos
sin 1
2
cos
tan
a
t t t
a
aa
mz bz cz F t
z t A e t
b
mc
z t z t
ba
m c
F z
m c b
Transient solution
Damping ratio
Steady state solution:
measure amp ofmotion and phaseshift, we already havewave phase andamplitude, devidefirst by later amp,gives RAO.
Froude Krylov due toundisturbed wave anddiffraction due to factthat ship reflects anddiverge the waveswhich hit it.
If excitation is harmonic, response is also harmonic
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5OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Solution Mass-Spring system:
, 2
2
2 22
cos
cos
tan
a
a
z F
a
a
mz bz cz F t
z t z t
ba
m c
F z m c b
, , , z z F F
6OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Moving ship in waves:
( ) wm a z b z c z F
Wave force is calculated forrestricted ship in meanposition: No motions
Hydromechanicreaction forces
motions
Wave forces
motionsWaves
Hydromechanic reaction forces(reaction on calculatedwave force)are calculatedfor flat water ( NO WAVES )
7OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Right hand side of m.e.:Wave Exciting Forces
• Incoming: regular wave with given frequency andpropagation direction
• Assuming the vessel is not moving
Let’s describe flow due to the incoming wavesaround the vessel using potential theory:
8OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Water Particle Kinematicstrajectories of water particles in infinitewater depth
0 ( , , , ) sin( cos sin )kza g x y z t e kx ky t
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9OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Water Particle Kinematicstrajectories of water particles in finitewater depth
0cosh( ( ))
( , , , ) sin( cos sin )cosh( )
a g k h z x y z t kx ky t
kh
10OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
1. Flow due to Undisturbed wave
kza0 = e sin cos sin
gt kx ky
2. Flow due to Diffract ion
Flow superposition
( ) wm a z b z c z F
0 7 0n n
00 7
11OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Pressure in the fluid can be found using Bernouilli equationfor unsteady flow:
2 212
2 212
( ) 0
( )
pu w gz
t
p u w gz
t
1 st order fluctuatingpressure
Hydrostatic pressure(Archimedes) constantin time, not relevantfor dynamics
2nd order (smallquantitysquared=small enoughto neglect)
Pressure
12OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Potential Theory
From the velocity potential we can derive:• Pressure• Forces and moments can be derived from pressures:
S
S
F p n dS
M p r n dS
xb
zb
( 0, 0 , 1)n dS , ,( ,0, )b P b Pr x y
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13OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Pressure due to undisturbed incoming waveIntegration of this pressure gives F FK
14OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Hydromechanic forcedepends on motion
Wave Forceindependent ofmotion
( ) FK D W m a z b z c z F F F
Recap: Motion equation
15OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Moving ship in waves:Not in air but in water!
( ) wm a z b z c z F
S
S
F p n dS
M p r n dS
p
t
16OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
3 3 3 3
3 3
r w d s
w d
m z F F F F F
m a z bz cz F F
Calculating hydrodynamic coeffiecients and diffractionforce
p7-4 course notes
• Hydrostatic buoyancy: 3c z • Diffraction• Incoming undisturbed wave (= F FK )
• Radiation: 3 3a z b z
As long as motions are veryslow only hydrostatics willcount
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17OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
3 3 3 3r w d sm z F F F F F
Calculating hydrodynamic coeffiecients and diffractionforce
Radiation Force: 3 3 3r F a z b z
To calculate force: first describe fluid motions dueto given heave motion by means of radiation
potential:
18OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Potential theory Radiation potential
Radiation potential heave
= flow due to heave motion
Knowing the potential, ca lculating resulting force is straight forward:
3 , , , x y z t
( ) w d m a z b z c z F F
S
S
F p n dS
M p r n dS
pt
S
S
F n dS t
M r n dS t
19OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Potential theory
0
0
0
0
0
0
33 3 3 0
33 3 3 0
13 3 1 0
13 3 1 0
23 3 3 0
23 3 3 0
S
S
S
S
S
S
a n dS
b n dS
a n dS
b n dS
a n dS
b n dS
3resulting from heave motions ,
0
0
0
0
0
0
43 3 01
43 3 01
53 3 02
53 3 02
63 3 03
63 3 03
S
S
S
S
S
S
a r n dS
b r n dS
a r n dS
b r n dS
a r n dS
b r n dS
Forces Moments
20OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Potential theory Radiation potential
Radiation potential heaveBoundary condition:
3 , , , x y z t
3 , , , , , ,n x y z t v x y z t n
At , , hull surface x y z
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25OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
0
3 3 01 ˆ ˆ ˆ ˆˆ ˆ, , , , , , , , ,
4 S
x y z x y z G x y z x y z dS
Green’s function : influence onpotential at (x,y,z) by sourceat
• Sat isfies the boundarycondition at the free surface
• Sat isfies the boundarycondition at the sea bed
ˆ ˆ ˆ, , x y z
Relaxed!
P 7-42 formulae for G
Numerical solving source strengths for radiation potential
26OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
12 3 3 3
1
14
N mn
m n n mn
GS n
n
This equation must be solvedfor every panel m
Taking into account sources on all otherpanels
11 1 1,3 3,1
1 ,3 3,
3
3
1 (influence of source at panel n on at its own collocation point)
2
1 (influence of source at panel n on4
N
N NN N N
nn
mnmn n
A A n
A A n
An
G A S n n
,3
at collocation point m)
unknown source strength of heave radiation potential at panel nn
Numerical solving source strengths for radiation potential
27OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
0 7 0n n
0 7, , , , 0 x y z t x y z t
n n
0 7, , , , 0 x y z x y zn n
Numerical solving source strengths for diffraction potential
Normal velocitydue to incomingwave
Normal velocitydue to diffractedwave
Diffraction potentialBoundary condition:
7 , , , x y z t
At , , hull surface x y z
28OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
012 7 7
1
14
N mn
m n nn m
GS
n n
This equation must be solved for every panel m
Taking into account sources on all other panels
0
111 1 1,7
1 ,70
71 (influence of source at panel n on at its own collocation point)
21
(influence of4
N
N NN N
N
nn
mnmn n
n A A
A A
n
An
G A S
n
7
,7
source at panel n on at collocation point m)
unknown source strength of diffraction potential at panel nn
n
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29OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
What is ‘linear’ ???
1. Linear waves:• ‘nice’ regular harmonic (cosine shaped) waves
• Wave steepness small: free surface boundary condition
satisfied at mean still water level
• Pressures and fluid velocities are proportional to wave elevation and have same
frequency as elevation
2. linearised wave exciting force:
• Wave force independent of motions
• Wave force only on mean wetted surface
3. Motion amplitudes are small
• Restoring force proportional to motion amplitude
• Hydrodynamic reaction forces proportional to motion amplitude
R
L
Motions are
proportional to
wave height !
Motions have samefrequency as
waves
30OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Response in Irregular Waves
31OE4630 2012-2013, Offshore Hydromechanics, Part 2 SESSION IDMarine Engineering, Ship Hydromechanics Section
Motions in Irregular waves
2
a z
a
S z
S
0
2
4
6
0 0.5 1.0 1.5
T2 = 8.0 s
0
0.5
1.0
1.5
2.0
0 0.5 1.0 1.5
0
3
6
9
0 0.5 1.0 1.5
za1/3 = 1.92 mT
z2 = 7.74 s
SHIP: RAOWaves:spectrum