Wave Theory Offshore Structures

Offshore Engineering and Technology

THREE DAY SHORT COURSE Course Presentation

An introduction on Offshore Engineering and Technology

INSTRUCTOR: DR. SUBRATA SUBRATA K. CHAKRABARTI CHAKRABARTI

University of Illinois at Chicago Offshore Structure Analysis Inc. Plainfield, ILLINOIS

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1


Offshore Engineering and Technology Overview of the 3-Day Course COURSE OUTLINE ---------- DAY 1 ----------

Morning

Afternoon

Introduction to Design

Environment &Construction: &Construction:

• History and current state of the art • Design cycle of offshore structures • Preliminary concept design

• Offshore environment • Construction and launching

---------- DAY 2 ----------

Morning

Afternoon

Design Tools and Evaluation


• static and dynamic stability • environmental forces

• structure response and sea-keeping ---------- DAY 3 ----------

Morning


• probabilistic design • extreme load & strength & fatigue • anchoring and mooring system

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Afternoon

Design validation and model testing

• • • •

riser system Scaling laws & Model testing Challenges in deepwater testing Assessment of validation methods

2


History and current state of the art of offshore structures

Definition of Offshore Structures No fixed access to dry land May be required to stay in position in all weather conditions

Functions of Offshore Structures

♦

Exploratory Drilling Structures: A Mobile Offshore Drilling Unit [MODU] configuration is largely determined by the variable deck payload and transit speed requirements.

♦

Production Structures: A production unit can have several functions, e.g. processing, drilling, workover, accommodation, oil storage and riser support.

♦

Storage Structures: Stores the crude oil temporarily at the offshore off shore site before its transportation to the shore for processing.

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Types of Offshore structures Fulmar jacket platform

Jack-up Drilling Unit with Spud Cans LEG

QUARTERS HELIDECK DERRICK

CANTILEVER BEAM

HULL JACKING UNIT

SPUD CAN

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Historical Development of Fixed Offshore Structures

Gravity Base Structures placed on the seafloor and held in place by their weight

Troll A gas platform, world's tallest concrete structure in North Sea Day1AM

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Structures piled to the seabed

500,000 barrel oil storage gravity & piled structure in Persian Gulf Compliant Tower Deck

Pile top weld connected to the tower at top

Fw (t)

Tower Frame Ungrouted pile sliding inside the tower leg or pile guides

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Floating Offshore Structures moored to the seabed Articulated Towers Ready for Tow in the 1970s

Floating structure types Tension Leg Platform: vertically (taut) moored (by tendons) compliant platform

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Semisubmersible Platform: Semisubmersibles are multi-legged floating structures

Marine 7000 4th generation semi-submersible Floating Production Storage and Offloading (FPSO) ship shaped with a turr et

Balder: first "permanent" FPSO in the North Sea

Internal bow turret Day1AM

Cantilevered bow turret 8


Deepwater floater types Ultra deepwater (>1500m) wells drilled in the Gulf of Mexico 56

60 d 50 e d d 40 u p S s l l 30 e W f 20 o . o N 10

37

12 7 1

1

1

36

11

3

0 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Year

Compliant

TLP

Semi

SPAR

FPSO

Tension leg platforms installed as of 2002

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Progression of Spars

5th generation drilling semi-submersible “Deepwater Nautilus”

SeaStar MiniTLP for Typhoon Field

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Failure of Offshore Structures Global failure modes of concern are - capsizing/sinking - structural failure - positioning system failure

Examples of accidents which resulted in a total loss.

Alexander Kielland platform --fatigue failure of one brace, overload in 5 others, loss of column, flooding into deck, and capsizing Ocean Ranger accident sequence -- flooding through broken window in ballast control room, closed electrical circuit, disabled ballast pumps, erroneous ballast operation, flooding through chain lockers and capsizing. Piper Alpha suffered total loss after: a sequence of accidental release of hydrocarbons, as well as escalating explosion and fire events.

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Structural damage due to environmental loads

Accident of P-36 semi-submersible

P-36 sank after 6 days after: an accidental release of explosive gas, burst of emergency tank, accidental explosion in a column, progressive flooding, capsizing and. Gulf Oil Platform -- ThunderHorse Saturday, July 30, 2005

(a) Before hurricane hit (b) After hit The BP Thunderhorse semi took a hit from Hurricane Dennis listing 20-30 degrees with the topsides almost in the water. The platform costs over a billion dollars to build. Day1AM

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Design cycle of offshore structures Design Spiral (API RP2T)

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Preliminary concept design Factors Affecting Field Development Concepts and Components

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Offshore environment

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Construction process, launching and operations Construction of Caisson

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Running Pile from Derrick Barge

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Construction of the Spar (Technip Offshore)

®

Seastar fabrication, Louisiana (SBM Atlantia)

Offloading of Spar Hull for Transport (Dockwise)

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Truss Spar in Dry Tow

Holstein Spar being towed out of site

Brutus TLP

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Spar Upending Sequence (Kocaman et al, 1997)

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Spar Mooring Line Hookup (Kocaman et al, 1997)

Derrick Barge Setting Deck on Spar Platform (Technip Offshore)

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Holstein Spar in place being outfitted

ThunderHorse being launched

ThunderHorse being Towed to place

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Installation of the lower turret on FPSO FP SO Balder

APL Buoy Turret System

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Rendition of the new bridge left of the existing bridge

Final Drafts of Caisson Showing the Mooring Lines Attached

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Current Flow past New Caisson behind the Roughened Pier

Location of the Immersed Tunnel of the Busan-Geoje Fixed Link

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Immersion of TE into trench

Physical Testing of Rigs Moored above the Trench

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OCEAN ENVIRONMENT 

Standing Wave vs. Progressive Wave



Linear Theory (also called Stokes First Order, Airy)



Stokes Second Order Theory



Stokes Fifth Order Theory



Stream Function Theory



Measured Wave Properties in Wave Tank



Offshore Applications of Wave Theory



Current -- Uniform and Shear



Wave-current Interaction



Wind and Wind Spectrum

1

NOTE: Chapter 3, Hydrodynamics of Offshore Structures

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Idealization of Waves Actual Ocean Wave

Simple Idealizations to Regular Waves Sine Wave 1

e d u t i l 0 p m A

-1 0

90

180

270

360

Angle, deg.

Nonlinear Regular Wave 1.25

e d u t i l 0.00 p m A

-1.25 0

90

180

270

360

Angle, deg.

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STANDING WAVE VS PROGRESSIVE WAVE Standing wave: Oscillation of waves generated in confined boundaries with no forward speed. Variation with space and time may be separated in a mathematical description. Examples include wave tank, liquid storage tanks, harbors, lakes etc.

Space and time are uncoupled: sin kx sin ω t – some points do not see any waves 1

e d u t i l 0 p m A

0

90

180

270

360

-1 Angle, deg.

Standing wave at times t 1 and t1 + T/2 Progressive wave: Waves have a forward speed in addition to oscillation. Variation with space and time is combined in a mathematical description. Waves generated in the open oceans are progressive waves. We discuss progressive waves here.

Space and time are dependent: sin ( kx -ω t ) – all points see maximum waves 1

e d u t i l 0 p m 0 A

90

180

270

360

-1

Angle, deg.

Progressive wave at two times t 1 and t2 Day1AM

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WAVE THEORIES

Three Independent Parameters describe all waves so that they must be specified for a wave. They are: water depth,

d

wave height,

H

wave period,

T

d:

uniform water depth based on flat bottom

T:

fixed interval of time at which a succession of wave crests passes a given point

L:

wave length i.e., length between two consecutive crests

c:

celerity or velocity of wave propagation

c = L/T

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WAVE THEORY DEVELOPMENT

The theory is described in terms of A Boundary Value Problem -- BVP consisting of A differential equation certain boundary conditions it will be shown that A complete solution of BVP is not possible without approximation. Approximate solution is obtained through simplified assumptions Two general types of approximate theory: (1)

wave height as perturbation parameter (deep water)

(2)

water depth as perturbation parameter

(shallow water)

The theory is formulated in terms of either Potential Function,

Φ

Stream Function,

Ψ

or

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Basic Assumptions

• Fluid is incompressible Fluid density = ρ . Fluid is incompressible if ρ is constant. Then the variation of density with time, t is zero:

∂ ρ =0 ∂t since we are dealing with water it is considered incompressible.

• Flow is continuous • Flow is irrotational Starting points for the BVP are the following

•

Equation of Continuity

•

Bernoulli's Equation

Definitions of following important variables to follow:

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•

Stream Function

•

Potential Function

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TWO CLASSES OF WAVE THEORIES A.

Closed Form Linear wave theory Stokes higher-order wave theory

B.

Numerical Solution Stream Function theory -- Regular -- Irregular

Properties sought: Fluid particle velocity/acceleration components Dynamic pressure of fluid particle

,v, j u, v , w = velocity components i, j, k = unit vectors along x, y, z

,u,i z ,w,k Equation of Continuity

2D

3D

v2

v2

u1

∂u ∂v ∂w + + =0 ∂ x ∂ y ∂ z

∆y

u2 ∆x

v1

w1 u1

u2 v1

If the above equation is satisfied, then the Continuity of flow is maintained Day1AM

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Stream Function

existence of

Ψ

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Ψ

⇔

continuity

= constant defines a streamline

34


Potential Function -- Irrotational Flow The concept of irrotationality is somewhat abstract/mathematical, but is one of the most important in hydrodynamics. Flow is called rotational, if each fluid particle undergoes a rotation, in addition to translation and deformation.

existence of

in terms of

⇔

Φ

Φ

irrotationality

⇒

Continuity Equation becomes

2 2 2 ∂ Φ ∂ Φ ∂ Φ 2 ∇ Φ= 2 + 2 + 2 =0 ∂ x ∂ y ∂ z This is the governing differential equation known as LaPlace’s equation Similarly,

irrotationality

⇒ ∇ Ψ=0 2

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The definition of potential function yields the wave particle kinematics

We still need a description for the wave particle dynamic pressure. It comes from the Bernoulli's Equation which is derived from the Navier Stokes equation under simplified assumptions . Assumptions are made of

•

Irrotational flow, and

•

Perfect fluid – no change in viscosity

Then the wave particle pressure has the expression 2 2 2  ∂Φ 1  ∂Φ   ∂Φ   ∂Φ   − ρ   +   +    p = − ρ gy − ρ ∂t 2  ∂x   ∂y   ∂z  

hydrostatic linear dynamic

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nonlinear dynamic

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Example: Consider Φ ( y) = A exp(ky ) with amplitude A = 10 ft, and wave number k = 0.01,. Compute the three pressure components for this wave. 0 p1

-20 f -40 , h t p e D-60

Hydrostatic

-80 -100 0

2000

4000

6000

8000

Hydrostatic Pressure, psf

0 p2

-20 t f -40 , h t p e D-60

Linear hydrodynamic pressures

-80 -100 -8.5

-6.5

-4.5

-2.5

-0.5

Linear Pressure, psf 0 p3 -20

Nonlinear hydrodynamic pressures

t f -40 , h t p e D-60

-80 -100 -0.015

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-0.01 -0.005 Nonlinear Pressure, psf

0

37


Two-dimensional wave motion

Note that the wave theory is developed over a flat bottom y is measured up from the SWL thus y is negative below the SWL to the ocean floor s is measured up from the floor so that s=y+d

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Boundary Value Problem In terms of velocity potential Φ ( x,y,z,t ), the following equations are obtained in the presence of a body in fluid: Differential Equation 2 ∇ Φ =0

-- Laplace's equation

Bottom boundary condition

∂Φ =0 ∂ y

at y = -d

Fixed body surface condition

∂Φ =0 ∂n

on the fixed surface

Moving body surface condition

∂Φ = V n ∂n Day1AM

on the moving surface

39


Linear Wave Theory Development The following assumption is made: Waves are two-dimensional Problem is posed as determining: either, the velocity potential or, the stream function Perturbation technique Express Φ as a power series in wave slope (i.e., wave height/ wave length)

ε =

Perturbation parameter

kH 2

=

π H L

Then, the velocity potential is expressed as a series

Φ =

∞

∑ ε Φ −− Φ is the n order potential n

n

n

n =1

th

similarly, the free surface profile

η =

∞

∑ n =1

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ε nη n

−−

η n is the nth order profile 40


Linear Wave Theory (solution is obtained in a closed form) Dispersion relationship (relates the wave length or wave number to the wave f requency, wave period or wave speed)

c

2

=

g k

tanh kd

2

ω

or

= gk tanh kd

or

L =

g

2π

T 2 tanh kd

Horizontal particle velocity

u

=

gkH cosh ks 2ω cosh kd

cos[ k ( x − ct )]

Vertical particle velocity

v=

gkH sinh ks

sin[k ( x − ct )] 2ω cosh kd

Horizontal particle acceleration

u& =

gkH cosh ks 2

cosh kd

sin[ k ( x − ct )]

Vertical particle acceleration

v& = −

gkH sinh ks 2

cosh kd

cos[k ( x − ct )]

Dynamic pressure

H cosh ks p = ρ g cos[k ( x − ct )] 2 cosh kd

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Deepwater Criterion and Wave Length Certain simplifications in the mathematical expressions describing properties of linear waves may be made if the water depth is very deep or very shallow. The following table describes these criteria: Depth of Water

tanh(kd )

Criterion

Deep water

d/L = > 1/2

1

Shallow water

d/L < = 1/20

kd

Intermediate water depth

1/20 < d/L < 1/2

Deep water

L0

= gT 2 / 2π

L = T gd

L = [ L0 tanh(2π d / L0 )]

1/ 2

For example, in deep water Horizontal particle velocity becomes

u

=

π H T

exp( ky ) cos[ k ( x − ct )]

Note the exponential decay term with respect to the vertical coordinate y. Similar expressions follow for the other quantities.

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Wave Particle Orbits and Velocity Amplitude Profiles

(c) SHALLOW WATER

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d

1

L

20 43


Sample Calculation of Linear Wave Properties Water depth, d = 100 ft Wave Height, H = 20 ft Wave Period = 10 sec Elevation, y = 0 ft Deep water wave length

L0

= gT 2 / 2π = 5.1248T 2 = 512.48 ft

k 0 = 0.01226 d / L0 = 0.1951 d / L = 0.2210 k = 0.01389 sinh kd = 1.88;

cosh kd = 2.13

um = π.20/10 x 2.13/1.88 = 7.12 ft/sec vm = π.20/10 = 6.28 ft/sec

u& m = 2π/10 x 7.12 = 4.47 ft2/sec v&m = 2π/10 x 6.28 = 7.12 ft2/sec Day1AM

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Time History of Linear Wave Properties Note the phase relationships among variables

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Exercise 1 Consider a regular wave of 20 ft and 10 sec. in 400-ft water depth. Using linear theory compute the amplitude of velocity and acceleration profiles with depth. Calculations are carried out on an excel spreadsheet constants 2π = g= ρ = dy =

unit 6.2832 32.2 1.98 -10

y ft 0.0 -10.0 -20.0 -30.0 -40.0 -50.0 -60.0 -70.0 -80.0 -90.0 -100.0 -110.0 -120.0 -130.0 -140.0 -150.0 -160.0 -170.0 -180.0 -190.0 -200.0 Day1AM

ft^2-s slug/ft^3 ft

finite depth 200 d= 20.0 H= 10 T= 0.6283 ω = 512.48 Lo = ko = 0.0122604 504.93 Lprev = 504.93 L= k = 0.0124436 0.3961 d/L = 6.377 gkH/2 ω = 1.052 Const Term =

amplitudes u v ft/s ft/s 6.38 6.29 5.64 5.54 4.99 4.88 4.42 4.30 3.92 3.78 3.48 3.32 3.09 2.91 2.75 2.55 2.46 2.22 2.20 1.93 1.98 1.67 1.78 1.44 1.62 1.23 1.48 1.04 1.36 0.86 1.26 0.70 1.18 0.55 1.13 0.40 1.08 0.26 1.06 0.13 1.05 0.00

unit ft ft s 1/s ft 1/ft ft ft 1/ft ft/s

amplitudes u dot v dot ft/s^2 ft/s^2 4.01 3.95 3.54 3.48 3.14 3.07 2.78 2.70 2.46 2.37 2.19 2.08 1.94 1.83 1.73 1.60 1.54 1.40 1.38 1.21 1.24 1.05 1.12 0.90 1.02 0.77 0.93 0.65 0.85 0.54 0.79 0.44 0.74 0.34 0.71 0.25 0.68 0.17 0.67 0.08 0.66 0.00 46


Kinematic amplitudes with depth 0

0

-20 -40

d = 200 H = 20 T = 10

-20 -40

-60

-60

u v

-80

-80

-100 y

y -100

-120

-120

-140

-140

-160

-160

-180

-180

-200

-200 0.0

2.0

4.0 u, v

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6.0

d = 200 H = 20 T = 10

8.0

u dot v dot

0.0

1.0

2.0

3.0

4.0

u, v

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Non-dimensional horizontal particle velocity

u0

=

u0T 2π ( H / 2)

=

cosh kd ( s / d )

=

sinh kd ( s / d )

=

cosh kd ( s / d )

sinh kd

Non-dimensional vertical particle velocity

v0

=

v0T 2π ( H / 2)

sinh kd

Non-dimensional dynamic pressure

p0

=

p0 ρ g ( H / 2)

cosh kd

1.5 s/d = 1.0 s/d = 0.9 s/d = 0.8 s/d = 0.7 s/d = 0.5 s/d = 0.2

1.0

r a b u

0.5

0.0 1.0

2.0

3.0

4.0

5.0

kd

horizontal particle velocity

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1.2 s/d = 1.0 s/d = 0.9 s/d = 0.8 s/d = 0.7 s/d = 0.5 s/d = 0.2

0.8

r a b v

0.4

0.0 1.0

2.0

3.0 kd

4.0

5.0

vertical particle velocity 1.2 s/d = 1.0 s/d = 0.9 s/d = 0.8 s/d = 0.7 s/d = 0.5 s/d = 0.2

0.8

r a b p

0.4

0.0 1.0

2.0

3.0

4.0

5.0

kd

linear dynamic pressure

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Comparison of Linear Wave Pressures with Tank Measurements Water depth = 10 ft Wave Periods = 1.25 – 3.0 sec

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Stokes Second Order Theory

Φ = ε Φ 1 + ε 2Φ 2 Wave profile

η =

+

H 2

cos( kx − ω t )

π H 2 cosh kd 3

8 L sinh kd

[ 2 + cosh 2kd ] cos 2( kx − ω t )

Horizontal particle velocity

π H cosh ks

u=

T sinh kd

cos(kx − ω t )

2

+

3  π H  cosh 2ks





4c  T  sinh 4 kd

cos 2(kx − ω t )

Vertical particle velocity

v=

π H sinh ks T sinh kd

π H  +    4c  T  3

2

sin( kx − ω t )

sinh 2ks 4

sinh kd

sin 2(kx − ω t )

Dynamic pressure

p = ρ g

H cosh ks 2 cosh kd

cos[kx − ω t )]

2

3 π H 1 cosh 2ks 1   + ρ g − cos 2(kx − ω t ) 4 L sinh 2kd  sinh 2 kd 3  1

π H

4

L

− ρ g Day1AM

2

1 sinh kd

[ cosh 2ks − 1] 51


First and second order components Wave profile over one cycle 1.25 1st 2nd 1st + 2nd e d u t i l 0 p 0 m A

90

180

270

360

-1.25 Angle, de g.

NOTE: The higher order components yield the vertical asymmetry in the wave profile.

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Example of First and Second Order Horizontal Particle Velocity Components Wave Height = 4 ft (1.22m) Wave Period = 2 sec Water Depth = 8 ft (2.44m)

The second order componet is less than 4 % of the first order velocity in this example.

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Stokes Fifth Order Theory

Φ = ε Φ 1 + ε Φ 2 + ε Φ 3 + ε Φ 4 + ε Φ 5 2

3

4

5

5

u

= ∑ un cosh nks cos n(kx − ω t ) n =1

first term:

cosh ks cos( kx − ω t )

second term:

cosh 2 ks cos 2( kx − ω t )

third term:

cosh 3ks cos 3( kx − ω t )

fourth term:

cosh 4 ks cos 4( kx − ω t )

fifth term:

cosh 5ks cos 5( kx − ω t )

Note the presence of five frequencies in the profile.

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Example Given d=100 ft. H= 50 ft. T = 10 sec. Calculated L=502.219 ft. θ = 0 20 η= 32.21 27.21

0 s

0 u

132.21 125.00 113.93 110.00 100.00 94.06 82.21 80.00 60.00 40.00 20.00 0.00

28.73 25.77

40 16.36

60 5.44

80 -2.36

45 u

90 u

180 u

45 v

90 v

0 p

90 p

180 p

1028 2426 3774 5088 6379

1621 3008 4351 5652

343 11.87

20.74 18.09

100 120 140 160 180 -8.36 -12.65 -15.26 -17.05 -17.79

15.4

10.29

1034 1511

12.14 -2.8

11.24 -10.37

14.07 11.35 9.6 8.63 8.31

8.44 7.1 6.19 5.67 5.5

-1.98 -1.23 -0.8 -0.59 -0.52

-8.92 -8.01 -7.46 -7.28

8.44 5.67 3.5 1.67 0

9.05 6.36 4.04 1.96 0

2518 3596 4738 5938 7192

Note: The wave crest is at 132.21 ft while the wave trough is at 82.21 ft. The asymmetry in the horizontal velocity between the crest and the trough is remarkably pronounced.

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STREAM FUNCTION THEORY Two Types: (1)

Symmetric Regular (theoretical)

(2)

Irregular (when the measured profile is known)

Coordinate System moves with wave speed, c in the direction of the wave propagation Horizontal velocity = u - c

If current is present, add the current velocity, U to the above Horizontal velocity = u – c + U

For irregular waves, The measured wave profile is used as an additional constraint

Solution taken in a series form:

Ψ ( x, y ) = cy +

N

∑ X (n) sinh nks cos nkx n =1

where N is the order of Stream Function wave theory The unknowns X (n) are solved by using the dynamic free surface boundary condition Day1AM

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Order of Stream Function NORDER Based on Wave Parameters d, H, T

This chart is useful in determining a suitable order of the stream function theory in a particular wave case.

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Nonlinear Regular Stream Function Wave Profile

compared with Linear Wave Profile Wave Height = 195.4 ft (59.6m) Wave Period = 15 sec Water Depth = 1152 ft (351.2m)

NOTE: in this figure h represents d = water depth

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Irregular Stream Function Wave vs. Experimental Pressure Amplitude d = 10 ft; T > 3.00 sec

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Comparison of Computed Velocities with Measurements Irregular Stream Function Theory Water depth = 2.5 ft H/d = 0.7

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Region of application of wave theories

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Wave Theory Offshore Structures

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