hydromechanics linear theory offshore

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


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:


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


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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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


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 )


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:


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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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


1. Flow due to Undisturbed wave

kza0 = e sin cos sin

gt kx ky

2. Flow due to Diffract ion

Flow superposition


0 7 0n n

00 7


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


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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Pressure due to undisturbed incoming waveIntegration of this pressure gives F FK


Hydromechanic forcedepends on motion

Wave Forceindependent ofmotion

( ) FK D W m a z b z c z F F F

Recap: Motion equation


Moving ship in waves:Not in air but in water!


S

S

F p n dS

M p r n dS

p

t


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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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:


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


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


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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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


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


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


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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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


Response in Irregular Waves


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

hydromechanics linear theory offshore

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