To determine wave forces

on an offshore structure

we need to know fluid

velocities & accelerations

under the wave surface

2013 Short Course on Fundamentals of Offshore

Structures and Design of Fixed Offshore Platforms

April 15-26, 2013, UT Austin

Wave Theory and Applications

by

Spyros A. Kinnas

Ocean Engineering Group

UT Austin

2013 Short Course on Fundamentals of Offshore Structures and

Design of Fixed Offshore Platforms - April 15-26, 2013, UT Austin 1 Copyright 2013 Prof. Spyros A. Kinnas

An offshore structure is

designed to withstand

the 100-year storm

(wave/current/wind). A

mono-chromatic wave of

height Hmax is assumed.

Hmax/L is larger than

required for linear wave

theory to be valid.

Fundamentals of Offshore Structures and Design of Fixed

Platforms, April 9-20, 2012, UT Austin 2 Copyright: Prof. S.A. Kinnas, 2012

Wave velocities and accelerations from linear

wave theory, when used in Morison’s equation,

under-predict wave forces!

2013 Short Course on Fundamentals of Offshore Structures and

Design of Fixed Offshore Platforms - April 15-26, 2013, UT Austin Copyright 2013 Prof. Spyros A. Kinnas 3

What is the maximum height a wave in deep

water can have (before it breaks)?

L

120o

7breakermax

LHH

Sharp peak!

qcrest

C

Stokes’ Criterion for wave breaking: qcrest=C

Q:How well could linear theory handle waves at breaking?

A: Pretty BAD! (predicts Hmax=L/p in deep water!)

2013 Short Course on Fundamentals of Offshore Structures and

Design of Fixed Offshore Platforms - April 15-26, 2013, UT Austin Copyright 2013 Prof. Spyros A. Kinnas 4

Stokes’ 2nd order non-linear wave theory (1/2)

)(2cos2

)cos(2

),( 21 tkxH

tkxH

tx

)/4cosh(2)/2(sinh

)/2cosh(

4 and

3

2

21 LdLd

Ld

L

HHHH p

p

pp

1.0/

60/1/

sec 8.3T 60m,L

m1 m, 6d

Ld

LH

H

Case 1

1st order term 2nd order term

2013 Short Course on Fundamentals of Offshore Structures and

Design of Fixed Offshore Platforms - April 15-26, 2013, UT Austin Copyright 2013 Prof. Spyros A. Kinnas 5

Stokes’ 2nd order non-linear wave theory (2/2)

107/1/

107mL

1) Casein as (same 8.3T

1 water,deep

LH

mH

Case 2

60/1/

sec 6.2T

1) Case as (same m60

1 water,deep

LH

L

mH

Case 3

2013 Short Course on Fundamentals of Offshore Structures and

Design of Fixed Offshore Platforms - April 15-26, 2013, UT Austin Copyright 2013 Prof. Spyros A. Kinnas 6

Cases 1, 2, & 3

on the graph

by Le Mehaute

2013 Short Course on Fundamentals of Offshore Structures and

Design of Fixed Offshore Platforms - April 15-26, 2013, UT Austin Copyright 2013 Prof. Spyros A. Kinnas 7

A cleaner version

of the graph

by Le Mehaute

2013 Short Course on Fundamentals of Offshore Structures and

Design of Fixed Offshore Platforms - April 15-26, 2013, UT Austin Copyright 2013 Prof. Spyros A. Kinnas 8

Different types of waves (1/2):

2013 Short Course on Fundamentals of Offshore Structures and

Design of Fixed Offshore Platforms - April 15-26, 2013, UT Austin Copyright 2013 Prof. Spyros A. Kinnas 9

Different types of waves (2/2):

2013 Short Course on Fundamentals of Offshore Structures and

Design of Fixed Offshore Platforms - April 15-26, 2013, UT Austin Copyright 2013 Prof. Spyros A. Kinnas 10

Short-comings of linear wave theory:

Does not produce accurate wave profiles for some

combinations of d/gT 2 and H/gT 2, and cannot handle waves

just before breaking

Does not allow for mass transfer along the wave direction.

Particle trajectories are evolving tight spirals, rather than

closed, as shown here:

Provides inaccurate fluid velocities and accelerations which,

when used in the Morison’s equation (to be described later),

under-predict the forces acting on the elements of an

offshore structure

2013 Short Course on Fundamentals of Offshore Structures and

Design of Fixed Offshore Platforms - April 15-26, 2013, UT Austin Copyright 2013 Prof. Spyros A. Kinnas 11

Detailed formulas for the wave profile and the

corresponding flow-field for a non-linear wave are given

in the provided document:

NOTES ON FIFTH-ORDER GRAVITY WAVE THEORY