1990 Wave loads on offshore structures_Faltinsen

Prévia do material em texto

Annu. Rev. Fluid Meeh. /990. 22: 35--56 
_ Copyright © 1990 by Annual Reviews Inc. All rights reserved 
WAVE LOADS ON OFFSHORE 
STRUCTURES 
O. M. Faltinsen 
Division of Marine Hydrodynamics, Norwegian Institute of Technology, 
University of Trondheim, N-7034 Trondheim - NTH, Norway 
INTRODUCTION 
Knowledge of wave-induced loads is essential in both the design and 
operation of offshore structures. In hostile areas like the North Sea, the 
significant wave height (mean of the highest one third of the waves present 
in a sea) can be larger than 2 m 60% of the time. The most probable largest 
wave height in 100 years can be more than 30 m. The mean wave period 
can be from 15 to 20 s in extreme weather situations, and it is seldom 
below 4 s. Environmental loads due to current and wind are also important, 
and in some cases the interactive effect between waves and current is 
significant. Current velocities of 1-2 m S-l and extreme wind velocities of 
40-45 m S-l must be used in designing offshore structures in the North 
Sea. 
Figure 1 shows five examples of offshore structures. Two of them, the 
jacket type and the gravity platform, penetrate the seafloor. At present, 
fixed structures have been built for water depths of up to 312 m. Two of 
the structures, the semi submersible and the drilling ship, are free floating. 
The tension-leg platform (TLP) is restrained from oscillating vertically by 
tethers, which are vertical anchor lines that are tensioned by the platform 
buoyancy being larger than the platform weight. Both the ship and the 
semi submersible are kept in position by a spread mooring system. An 
alternative would be to use thrusters and a dynamic positioning system. 
Pipes (risers) are used as connections between equipment on the seafloor 
and the platform. 
Motions of floating structures can be divided into wave-frequency 
motion, high-frequency motion, slow-drift motion, and mean drift. The 
rigid-body translatory motions are referred to as surge, sway, and heave, 
35 
0066-4189/90/0115-0035$02.00 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
Isabella
Realce
Isabella
Realce
Isabella
Realce
Isabella
Realce
Isabella
Realce
36 FALTINSEN 
Figure 1 Five types of offshore structures. From left to right, they are the jacket, gravity 
platform, semisubmersible, floating production ship, and tension-leg platform (TLP) struc­
tures. (Diagram partly based on a figure provided by Veritec A(S.) 
with heave being the vertical motion. The angular motions are referred to 
as roll, pitch, and yaw, with yaw being rotation about a vertical axis. For 
a ship, surge is the longitudinal motion and roll is the angular motion 
about the longitudinal axis. 
The wave-frequency motion is mainly linearly excited motion in the 
wave-frequency range of significant wave energy. High-frequency motion 
is only significant for TLPs and is often referred to as "ringing." These 
motions are due to resonance oscillations in heave, pitch, and roll of the 
platform. The restoring forces are due to the tendons, and the mass forces 
are due to the platform. The natural periods of these motion modes are 
typically 2--4 s, which is less than most wave-spectrum frequencies. They 
are excited by nonlinear interaction mechanisms between the wave and 
the structurally induced fluid motion. Similar nonlinear effects cause slow 
drift and mean motions in waves and current. Wind can also cause slow 
drift and mean motion. Slow-drift motion arises from resonance oscil­
lations. For a moored structure it occurs in surge, sway, and yaw. The 
restoring forces are due to the mooring system, and the mass forces are 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
Isabella
Realce
Isabella
Realce
Isabella
Realce
Isabella
Realce
Isabella
Realce
Isabella
Realce
WAVE LOADS ON OFFSHORE STRUCTURES 37 
due to the structure. Typical resonance periods are of the order of 1-2 min 
for conventionany moored systems. 
Both viscous effects and potential-flow effects may be important in 
determining the wave-induced motions and loads on marine structures. 
Included in the potential flow is the wave diffraction, or scattering, around 
the structure. A general trend is that viscous effects are important when 
the ratio between the wave height and a characteristic cross-sectional 
dimension D of the structure is large. This is true for a jacket structure, 
as shown in Figure 1. Wave diffraction is important when the ratio between 
a representative wavelength and D is small. This is true for ships and 
often for TLPs. These types of structures are referred to as large-volume 
structures. 
In this review I analyze in more detail some of the topics mentioned 
above. Linear wave-induced motions and loads are only briefly discussed. 
Previous review articles by Mei (1978), Newman (1978), and Yeung (1982) 
should be studied for more details. 
LINEAR WAVE-INDUCED MOTIONS AND LOADS 
There exist practical numerical tools to predict linear wave-induced 
motions and loads on large-volume structures like ships and TLPs. A wave 
spectrum is used to describe a sea state, and results for an irregular sea 
can be obtained by linear superposition of results from regular incident 
waves. Panel methods are the most common techniques to analyze the 
linear steady-state response of large-volume structures in regular waves. 
An example of the paneling of a TLP is shown in Pigure 2, where a total 
of 12,608 panels are used for the whole structure. In general about 1000 
elements would be sufficient for engineering calculations. However, more 
elcmcnts may be needed for calculating loads at small wavelengths and in 
multi body flows. 
The boundary-value problem for the fluid motion around a floating 
structure in regular incident waves is split into seven parts. A radiation 
problem is associated with forced harmonic oscillations in each of six 
modes of rigid-body motion. In addition, one finds the diffraction potential 
when the structure is restrained from oscillating in incident regular waves. 
In each subproblem a velocity potential <P is found as the solution of the 
Laplace equation subject to a body boundary condition, a sea-bottom 
condition, the classical linear free-surface condition, and a radiation con­
dition requiring that the body generates only outgoing waves. If a panel 
method (boundary-element method) is used to solve the problem, there 
are different methods of solution. One way is to distribute sources over 
the mean wetted body surface. Each source satisfies the free-surface con-
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
Isabella
Realce
38 FALTINSEN 
Figure 2 Submerged portion of one quadrant of a six-column TLP, discretized with 3 I 52 
panels per quadrant (12,608 panels for the total structure) (Korsmeyer et al. 1988). 
dition, the radiation condition, and the sea-bottom condition (Wehausen 
& Laitone 1960). The source density is found by requiring no flow through 
the body. Another way to solve the boundary-value problem is to use a 
mixed distribution of both sources and normal dipoles distributed over 
the mean wetted body surface. Efficient ways of calculating the sources 
have been derived by Newman (1985). Irregular frequencies may in special 
cases cause problems if boundary-element methods are used (see Ursell 
1981). The work by Lee (1988) shouldbe noted as a practical way to avoid 
such problems. 
Source methods have been used in practical calculations of wave loads 
on large-volume offshore structures for about 20 years. However, recent 
comparative studies performed by the ISSC (International Ship and Off­
shore Structures Congress) and the ITTC (International Towing Tank 
Conference) have shown large variations in results from different computer 
programs with the same theoretical foundation (see also Takagi et a1. 
1985). These differences can be due to grid shape, size, and distribution; 
geometry approximation; singularity density distribution over each panel; 
Green's-function calculation; and how singularities are integrated over 
panels. 
A panel method is based on potential theory and can only predict 
damping due to radiation of surface waves. Thus a panel method does not 
satisfactorily predict rolling motion of a ship close to the roll resonance 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
W A VB LOADS ON OFFSHORE STRUCTURES 39 
period. The reason is that (a) the wave radiation damping moment due to 
roll is small and (b) viscous damping effects due to flow separation are 
important. Another significant case where panel methods fail and viscous 
effects matter is in predicting vertical forces on a TLP in extreme wave 
situations in a frequency domain, where the excitation loads become small 
owing to cancellation effects. In general, this happens in a period domain 
between 15 and 20 s and is important in establishing design loads for the 
tethers of a TLP. 
The common panel methods for wave loads on large-volume structures 
do not account for interaction effects between waves and current. This is 
presently an area of active research. Potential theory is used, but it is likely 
that flow separation is more important with current than without it. The 
reason is that typical offshore structures have a blunt shape, causing the 
flow to separate in a current. One important exception to this is the flow 
in the longitudinal direction of a slender ship. The situation is more 
complex in combined current and waves than in current alone. This can 
be illustrated by experimental results for a hemisphere presented by Zhao 
et al. (1988). In calm water the center of the hemisphere was in the mean 
free surface. A vertical cylinder was mounted on top of the hemisphere to 
avoid sharp corners on the submerged surface in waves. Stated simply, the 
flow past the hemisphere always separates when the current velocity is 
larger than the maximum velocity of ambient flow due to wave action. If 
the latter dominates, the flow will not separate for small ratios between 
the wave amplitude and the diameter of the sphere. For flow around bodies 
with sharp corners, the flow will always separate. 
If we neglect the effect of flow separation, the combined effect of waves 
and current can be solved as outlined in Zhao et al. (1988). Close to the 
body it is important to account for the interaction between the local current 
and the unsteady flow. In the formulation of the body boundary condition, 
terms are introduced because the steady potential satisfies the body bound­
ary condition on the mean body-surface position . The resulting correction 
terms due to the steady potential cannot be applied for sharp corners. The 
interaction effect far from the body can be illustrated by the waves gen­
erated by a harmonic oscillatory source (Wehausen & Laitone 1960). There 
are several wave systems. An important parameter is r = wUe/g, where w 
is the circular frequency of oscillation, Ue the velocity of the current, and 
9 the acceleration due to gravity. When r > 1/4 there is no effect straight 
ahead of a translatory harmonic oscillatory source. For environmental 
conditions normally encountered in the North Sea, r would be less than 
1/4, and the most important wave system will be "ring" waves propagating 
in all directions from the source. The wavelength is a function of the local 
wave propagation direction relative to the current direction. The local 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
40 PAL TINSEN 
wavelength becomes longest when the local wave propagation direction 
coincides with thc current direction and shortest when the local wave 
propagation direction is opposite to the current direction. Grue & Palm 
(1985) have analyzed the two-dimensional problem for a submerged cir­
cular cylinder for any r-value. 
NONLINEAR WAVE-INDUCED MOTIONS AND 
LOADS 
The theoretical methods outlined in the previous section assume linearity. 
This means that one considers incident regular sinusoidal waves of small 
amplitude 'a and assumes that hydrodynamic variables like velocity, pres­
sure, forces, and moments are linearly proportional to 'a. The limitation 
of linear theory can be illustrated by considering regular deep-water waves 
without the presence of the structure. Schwartz (1974) has provided a 
solution in terms of a series expansion that satisfies the exact nonlinear 
free-surface condition within potential theory. For waves of steepness 
HI A = 0.1 the exact theory predicts a 20% higher maximum wave elevation 
than the linear approximation. For a wave period of 12 s (i.e. a wavelength 
of A � 225 m), this steepness corresponds to a wave height H of 22.5 m. 
This is not an unrealistic combination of wave height and period and 
illustrates, for instance, that linear theory can give a significant under­
prediction of the air gap between the waves and a platform deck. 
The solution by Schwartz (1974) assumes periodicity in space. It also 
assumes that the waveform is symmetric about a vertical axis through the 
wave crest. This means that it cannot he used for studying plunging 
breakers and irregular steep waves. Dommermuth et al. (1988) have pre­
sented interesting numerical studies of plunging breakers. They applied a 
boundary-element method and a mixed Eulerian-Lagrangian solution 
scheme to study the exact nonlinear free-surface potential-flow problem. 
Good agreement was shown between theory and experiments for a "two­
dimensional" plunging breaker produced by a wave maker. If we wanted 
to use a similar technique to obtain statistical estimates for the complete 
nonlinear three-dimensional motion problem for offshore structures, a 
direct extrapolation of the requisite computational time suggests that the 
total effort will be prohibitive and infeasible even for modern super­
computers. Computer time is, obviously, a function of the numerical 
accuracy we are aiming for; but this illustrates the present "impossibility" 
of solving the completely nonlinear motion problem for offshore structures 
by rational methods. Finite-difference (volume-discretization) methods 
with a resolution comparable to that of a boundary-element method result 
in similar time estimates. 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
WAVE LOADS ON OFFSHORE STRUCTURES 41 
In addition to obvious limitations in available computer power, there 
are unsolved physical problems, such as (a) reentry of breaking waves, (b) 
intersection problems between the body and the free surface, (c) initial­
value conditions, and Cd) far-field conditions. 
I address some of these problems in detail. The "state-of-the-art" in 
describing numeri cal ly what happens during the reentry of breaking wavesis not satisfactory. The in viscid potential-flow solution breaks down when 
the jet from the overturning wave hits the water. Peregrine (1983) has 
discussed what happen s physically after the jet hits the free surface. The 
most common phenomenon is that the plunging jet encloses a volume of 
air. Associated with the flow there is a significant circulation around the 
air pocket. The plunging jet may "rebound" or penetrate into the water. 
In both cases it cau ses splash-up, which may be higher than the original 
overturning wave. The air pocket enclosed by the plunging jet will finally 
collapse. After the plunge there is strong mixing of air and water. 
The intersection between the free surface and the wave maker did not 
cause any practical problems in the numerical study by Dommermuth et 
al. (1988), who used the numerical scheme described by Lin et al. (1984). 
This is a further development of the work by Vinje & Brevig (1981). Lin 
et al.'s procedure assumes that the solution is regular at the intersection 
points. What the local solution should look like at the intersection point 
is not completely clear. D. H. Peregrine's (unpublished note, 1972) analysis 
for an impulsive velocity motion of a vertical wave maker shows a log­
arithmic singularity at the intersection point. Lin et al.'s numerical simu­
lation agrees with Peregrine's local solution except for the intersection point 
and its immediate neighborhood. The same is true for Greenhaw & Lin's 
(1983) experimental results, except in the immediate neighborhood of the 
intersection, where a jet is ejected. 
The numerical scheme of Lin et al. (1984) has been applied to many exit 
and entry problems of two-dimensional structures. For entry of a wedge 
into calm water Greenhow (1987) had to modify the procedure to handle 
the intersection point. If the pressure on the wedge becomes negative, the 
intersection point is regarded as an ordinary free-surface point and a new 
intersection point is introduced where the pressure is zero. The reason 
for doing this is not completely clear. Severe numerical problems were 
encountered for deadrise angles less than 45°. (Deadrise angle is the angle 
between the wedge side and a horizontal line through the wedge apex.) 
One reason for these problems is that the jet flow created when the wedge 
enters the water becomes more difficult to follow numerically. Greenhaw 
(1988) had difficulties in starting the solution for flow around a circular 
cylinder when the cylinder axis was above the mean free surface and the 
angle between the mean free surface and the tangent of body surface was 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
42 FALTlNSEN 
significantly different from 90° at the intersection point. This complication 
arises from the initial free-surface condition that the velocity potential c/> 
is zero on the mean free surface. This is also approximately true in a 
subsequent small time interval. The free-surface condition ¢ = 0 implies 
a singularity at the intersection point if the cylinder axis is above the mean 
free surface. The singular behavior vanishes when the body-surface tangent 
is orthogonal to the free surface and is strongest when the body-surface 
tangent is parallel to the free surface. When the cylinder axis is initially 
below the free surface, there is no singularity at the intersection point. The 
occurrence of a singularity implies that the formulation of the boundary 
conditions is wrong, but it is not known how to avoid the singularity 
problems with the starting condition. 
When a flat-bottomed body enters the free surface, it is known that the 
air flow between the body and the free surface is important for times when 
the air gap between the body surface and the free surface is small relative 
to the transverse dimensions of the body (Verhagen 1967). The free-surface 
elevation depends on the pressure in the air above the free surface, and 
the pressure in the air depends on the gap between the free surface and 
the bottom of the body. The water elevation is at first noticeable at the 
edges of the body. This causes the air to be entrapped between the free 
surface and the body. lf the body has a small deadrise angle (2-3°) (Koehler 
& Kettleborough 1977), the air will not be entrapped. The entrapped air 
will have a pronounced cushioning effect, and the impact pressure on the 
hull is therefore sensitive to small changes in deadrise angle. This indicates 
that it is difficult to create deterministic flow situations in impact problems. 
Entry problems have several practical applications in naval architecture 
and ocean engineering. An example is impact loads (slamming) on ship 
bows caused by large relative motions between the ship and the waves. 
Another case is crane operation when a body js lowered through the free 
surface. For practical calculations of forces on bodies during entry, one 
has to rely on simplified formulas based on momentum consideration. 
This means that the vertical hydrodynamic force F3 on a body entering 
calm water with a downward velocity V is often calculated by 
(1) 
where p is the mass density of the water, t is the time variable, V is the 
submerged volume, and A33 is the added mass in heave for the body as 
a function of submergence relative to the calm-water level. The free-surface 
condition c/> = 0 implies that fluid accelerations should be much larger 
than the gravitational acceleration g, and it is used in calculating A33• The 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
WAVE LOADS ON OFFSHORE STRUCTURES 43 
formula breaks down in the case of a flat-bottomed body entering the free 
surface. The mathematical reason is that dA33/dt is infinite at the instant 
of contact between the body and the free surface. A physical reason is that 
the free-surface boundary condition does not account for the air flow and 
for the free-surface deformation. 
The formula cannot be used for exit problems. One reason is that the 
free-surface condition <p = 0 is a poor approximation. Greenhow (1988) 
has used a potential-flow method based on the exact nonlinear free-surface 
condition in a similar way as he did for entry problems. It was shown that 
the numerical method is able to predict the thin layer of water that can 
form on the top of the cylinder. A drawdown of the water may happen 
close to the body as a consequence of a strong negative pressure under the 
cylinder. This will eventually cause ventilating bubbles. The numerical 
method is able to predict, but not describe, the rapid breakdown of the 
free surface. Depending on the distance the cylinder has moved, vortices 
can also be observed in the fluid. The potential-flow method referred to 
above is not able to predict this effect. 
In long-time numerical simulations of large-amplitude motions of off­
shore structures, there are problems connected with a proper formulation 
of far-field conditions. The computational domain must be limited, and 
one has to formulate how the fluid behaves far away from the body. In 
the case of small-amplitude motion, where linear theory applies, one can 
use a classical radiation condition. In the cases of zero and small current 
velocities, this implies that all waves generated by a body propagate away 
from the body. For large-amplitude motions the formulation of the far­
field condition is more difficult. For instance, there is the possibility that 
waves may break. This may reduce the waves, which may then be approxi­
mated by a perturbation method far away from the body with linear theory 
as afirst approximation. However, there exists no theory that predicts 
how much energy is dissipated during the breaking process. If the waves 
are not breaking, one could claim from energy-balance arguments that the 
waves have to be small in the far field of a three-dimensional flow problem 
and can be approximated by a perturbation solution (Dommermuth & 
Yue 1987). In a two-dimensional flow problem it is not possible to say 
this. Since many numerical studies of steep surface waves interacting 
with bodies are two dimensional, there is a need to formulate far-field 
conditions. In the case of large-amplitude forced-heave motions of a two­
dimensional circular cylinder, Faltinsen (1977) approximated the far-field 
behavior with a two-dimensional time-dependent source satisfying the 
time-dependent linearized free-surface condition. The source point was 
inside the body, and the far-field domain was chosen outside of where the 
linear wave front would be. This implies that the computational domain 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
44 FALTINSEN 
increases with time and thus becomes impractically large as time goes on . 
In such a formulation, one has not accounted for the possibility that 
nonlinear wave components can propagate faster than the linear wave' 
front. If the computational domain stays constant and one allows wave 
disturbances at the border of the computational domain, one cannot be 
sure that there are no wave components representing incident waves due 
to nonlinear interaction. There does not exist any accepted rational way 
to formulate a far-field condition in this case. If the nonlinearities are 
small, a perturbation scheme can be used to solve nonlinear problems. 
This is discussed in the following section. 
SECOND-ORDER NONLINEAR PROBLEMS 
The normal way to solve nonlinear wave-structure problems in ship and 
offshore hydrodynamics is to use perturbation analysis with the wave 
amplitude/wavelength as a small parameter. Potential theory is assumed, 
and the problem is solved to second order in the incident wave amplitude. 
The solution of the second-order problem results in mean, difference­
frequency, and sum-frequency forces. 
In a potential-flow model, mean drift forces are due to a structure's 
ability to create waves. The consequence of this is that drift forces are 
small in a potential-flow model when mass forces dominate. This occurs 
for semisubmersibles. Viscous effects may also contribute to drift forces, 
but they are third-order effects and are not studied in any detail in this 
context. Let us examine why one obtains a mean wave force on a structure 
in regular, incident, harmonically oscillating waves. For a surface-piercing 
body, a major contribution is due to the relative vertical motion between 
the structure and the waves. This causes some of the body surface to spend 
part of the time in the water and part of the time out of the water. 
Examining thc pressure on one of the points in this surface zone, and 
realizing that the relative vertical motion is not the same around the 
structure in Figure 3, one sees that the result is a nonzero mean force even 
in regular harmonically oscillating waves. However, there are also other 
Pressure 
Partly in 
L J 
or out -v.��t---
of water Non-zero 
/2 � =(:> mean force 
.. II Time 
Out of the 
water 
Figure 3 One contribution to mean wave forces on offshore structures. 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
WAVE LOADS ON OFFSHORE STRUCTURES 45 
causes. Onc of them arises from averaging the quadratic term in Bernoulli's 
equation. Other contributions arise because we have to correct the linear 
first-order problem, which evaluates pressures on the mean position of the 
body, and integrate pressure forces and moments on the body when it is 
in its mean position. For a small-volume structure, where the incident wave 
system is not modified by the structure, the relative motion will be the same 
around the structure in Figure 3 and result in a zero-mean-force contribution. 
The procedure outlined above is referred to in the literature as a direct 
pressure-integration method. All three force components and all three 
moment components can be obtained. However, to avoid inaccuracies 
care should be shown in using the method. For instance, if a boundary­
element method is used to calculate the linear flow motion and one assumes 
constant velocity potential over each element, then one has implicitly 
assumed that the tangential velocity is zero over the element. This gives 
an error in the calculation of the quadratic velocity term in Bernoulli's equa­
tion. There are ways to avoid this difficulty-for instance, by extrapolat­
ing calculated values of fluid velocities from points along the normal to the 
element that are further away than a characteristic length of the element. 
Another way to obtain mean wave forces in regular waves is to use the 
equations for conservation of momentum in the fluid. In the case of 
horizontal forces and yaw moments, one only needs information on the 
linear fluid behavior at a vertical cylindrical control surface encompassing 
the body and far away from the body. The latter approach may be com­
bined with the equation for conservation of energy in the fluid. It is then 
more evident that drift forces and moments in a potential-flow model are 
due to a structure's ability to create waves (Maruo 1960). The equations 
for conservation of momentum can also be used to calculate vertical drift 
forces and mean pitch and roll moments. In addition to contributions 
from the vertical cylindrical control surface far away from the body, there 
are, for instance, contributions along the frce surface inside the vertical 
control surface. 
The effect of current is often neglected in the calculations of mean wave 
forces and moments. Zhao et al. (1988) pointed out that the current 
velocities of 2 knots may very well represent a 50% increase in drift forces 
on large-volume structures. This conclusion was based on both numerical 
and experimental investigations. Qualitatively, one can understand that 
current must have an effect on mean wave forces if one considers that drift 
forces are due to a structure's ability to create waves, and that current has 
an influence on the wave picture. 
Since mean wave forces and moments are small, a high degree of accu­
racy is needed both in the calculations and in the experiments. The results 
are sensitive to wave heading, body form, body motion, wavelength, and 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
46 FALTINSEN 
wave height. Even if the problem is nonlinear, it is possible to add results 
from regular waves to obtain wave drift loads in irregular seas. It can be 
shown that mean wave loads can be sensitive to the sea state. 
Let us now study the second-order sum-frequency and difference-fre­
quency forces. Consider an idealized sea state consisting of two linear wave 
components of circular frequencies W I and W2. By simple algebra it follows 
(from, for instance, the quadratic velocity term in Bernoulli's equation) 
that there is a pressure term that oscillates with the difference frequency 
WI -W2. For a more realistic representation of the seaway, and consider­
ing the waves as the sum of N components of different circular frequen­
cies Wi, we findpressure terms with difference frequency Wj-Wk (k, j = 
1, . . . , N). These nonlinear interaction terms produce slow-drift excita­
tion forces and moments that may cause resonance oscillations in the 
surge, sway, and yaw motions of a moored structure. 
It also follows that there are nonlinear effects that can create excitation 
forces with higher frequencies than the dominant frequency components 
in a wave spectrum. This is due to terms with oscillation frequencies 2wl, 
2Wb and (WI +w2). These may be important for exciting the resonance 
oscillations in the heave, pitch, and roll of a tension leg platform (TLP). 
However, it can be shown that the contribution from the quadratic velocity 
term in Bernoulli's equation to the sum-frequency forces in heave for a 
tension leg platform is normally small at the natural period in heave. This 
follows from the exponential depth decay of the term, and from the fact 
that it is the pontoons (see Figure 1) that normally contribute to the 
vertical force on a TLP. The most important contribution to the vertical 
sum-frequency force on a TLP comes from the second-harmonic part of 
the second-order potential (Kim & Vue 1988). The second-order potential 
4>2 follows from solving a boundary-value problem with the inhomo­
geneous free-surface condition 
a 2 2 2 I a 4htt+g4>2z = - a/4>lx+¢IY+¢IZ)+ g¢lt az (¢ltt+g¢lz) on z = o. 
(2) 
Here z is the vertical axis, positive upward, and z = 0 corresponds to the 
mean free surface. The effect of current is neglected in formulating (2). 
Equation (2) states that the first-order potential ¢ I affects ¢2 in a way that 
can be physically interpreted as an imposed pressure on the free surface. 
The existence of second-harmonic pressure at large depth was explained 
by Newman (1990). A similar phenomenon exists for the second-order 
oscillatory wave field when there are two linear wave fields propagating in 
opposite directions. We can show this and write the linear (first-order) 
potential as 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
WAVE LOADS ON OFFSHORE STRUCTURES 47 
alwl k z • • a2w2 .k z . ( k 5; ) <PI = - Te ' sm (wlt - k lx + {) I) - JZ;C2 sm w2t-A 2X+U2, 
(3) 
where A = ± 1, depending on the propagation direction of the second wave 
in (3). A particular solution that satisfies (2), the deep-water condition, and 
the Laplace equation can be written as 
<P -
2aja2Wjwiwj-(2) Ik,-k2Iz 
2 - -(Wj-W2)2+glkl-k21 
e 
x sin [(w ,- w2)t-(k,- k2)X+b ,-b2] for A = 1, (4) 
A. _ � 2a1a2wlwz(wl +(2) Ik,-k2lz '/'2 - 2 e 
-(WI +(2) +glkl-k21 
x sin[(wl+w2)t�(kl-k2)X+(jI+(j2] for A = -1. (5) 
Equations (4) and (5) show that it is only when the waves propagate 
in opposite directions that there are sum-frequency effects. If the two 
frequencies WI and Wz are equal, we note that the second-harmonic term 
is uniform in space. Also, when W I is close to W2, we note that the sum­
frequency effects have a slow depth decay. 
In the case of regular waves incident on a two-dimensional body, the 
reflected waves will interact with the incident waves in the way represented 
by Equation (5). In this case, WI = Wz. For the transmitted-wave system, 
the linear waves generated by the body propagate in the same direction as 
the incident waves and there is no effect from the second-order potential 
in the far field. For a three-dimensional flow problem, the waves generated 
by the body along a ray opposite to the incident-wave direction create a 
similar second-order depth effect as in the two-dimensional case with the 
reflected-wave system. Since what is happening along the ray opposite to 
the incident-wave direction is not independent of what is happening along 
other rays, the second-order effects are not equal in the two-dimensional 
and three-dimensional cases. By assuming that the dominant contribution 
to the second-order velocity potential is from the far-field part of the 
forcing function in the free-surface boundary condition, Newman (1988) 
showed that the second-harmonic potential is algebraically, and not expo­
nentially, decaying for large depth. This is in agreement with Kim & Yue's 
(1988) numerical results. In the case of two linear frequency components 
WI and Wb the slow depth decay occurs for the 2wI and 2Wl terms. The 
(WI +(2) term has a slow depth decay if the two frequency components 
are close to each other. This is similar to the two-dimensional case [see 
Equation (4)]. This discussion also illustrates that the nonlinear far-field 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
48 FALTINSEN 
behavior cannot be described only by outgoing waves. This has relevance 
to the discussion in the last section on what far-field conditions one should 
use in solving nonlinear wave-induced motions and loads on offshore 
structures. 
Second-order slow-drift solutions have been extensively studied in the 
literature. Ogilvie (1983) and Faltinsen (1988) have given state-of-the-art 
reports on the subject. The traditional way to set up the equations of slow­
drift motion is to estimate second-order excitation forces without the 
effect of slow-drift motion and to use a linear response model with mass, 
damping, and spring terms. Nonlinearities are sometimes accounted for 
in the restoring forces due to the anchor lines. Since slow-drift oscillations 
are resonant oscillations, damping plays an important role. In the simplest 
damping model, the damping is set empirically to be a fraction of the 
critical damping. A more rational way is to include wave-drift damping 
(Wichers 1982), anchor-line damping, and huB damping due to skin friction 
and eddy making. Generally, wave-drift damping and anchor-line damping 
are most important. This is particularly true for higher sea states, but in 
cases where the slow-drift motion causes significant flow separation, eddy­
making damping should be accounted for. The presence of wave-drift 
damping can be explained by interpreting the slowly varying velocity as a 
quasi-steady speed of the structure, and by realizing that mean wave forces 
are functions of the speed of the structure. The term in the mean wave 
forces that is linearly dependent on the speed of the structure can be 
interpreted as a damping term. 
The traditional way of hydrodynamically analyzing slow-drift motion 
has weak points. The necessity of introducing wave-drift damping indicates 
that the slow-drift motion should be hydrodynamically analyzed in com­
bination with the slow-drift excitation forces. Owing to the nonlinear 
nature of the problem, one cannot, in a simple way, separate the slow­
drift motion response from the slow-drift excitation forces. Faltinsen 
(1988) made an attempt to analyze the combined effect of slow-drift motion 
and excitation. It was shown that time-dependent terms additional to . 
Wichers' wave-drifting damping terms occurred. Zhao & Faltinsen (1988) 
showed that the time-dependent damping terms had an important effect 
on the extreme values but not on the standard deviation and mean values 
of the slow-drift motion. The analysis by Faltinsen (1988) did not account 
for viscous effects. In practice, such effects must be accounted for by 
empirical drag formulas. This is discussed more in the next section. 
TriantafyBou (1982) has pointed out that the first-order motions cannot 
be decoupled from the second-order slow-drift translatory and angular 
motions. The slowly varying angular motions cause a quasi-steady change 
in wave heading, and the slow-drift motions create a slowly varying fre-
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
990.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
w AVE LOADS ON OFFSHORE STRUCTURES 49 
quency-of-encounter effect between the waves and the structure. The latter 
effect may be particularly important when a lightly damped roll resonance 
oscillation is excited. The roll response will depend strongly on the value 
of the slowly varying frequency of encounter relative to the roll natural 
frequency when the roll motion is lightly damped. Tn practice, it is not 
normal to account for the influence of the slow-drift motion on the first­
order motions. 
Resonance roll motion also causes other difficulties in the analysis. 
Nonlinear viscous effects are significant in the estimation of roll at reso­
nance. This implies that a linear first-order roll motion does not exist at 
resonance. The difficulty of accurately predicting roll resonance response 
causes important inaccuracies in the prediction of second-order hydro­
dynamic forces for frequency domains where roll resonance matters. This 
is demonstrated by Faltinsen (1988) in the prediction of mean second­
order wave forces. 
In engineering calculations Newman's (1974) approximate model for 
slow-drift excitation forces in long-crested irregular seas is popular. The 
hydrodynamic problem is significantly simplified and reduced to finding 
mean wave loads in regular waves. The second-order potential does not 
matter in determining the mean wave loads when one regular wave system 
is studied. This is also evident from Equation (4), where the second-order 
potential disappears when WI = W2. As pointed out earlier, this is not true 
for the sum-frequency part of the second-order potential. If the second­
order potential should be accounted for in slow-drift interaction between 
two wave components with frequencies WI and W2, it is evident from the 
slow-drift depth decay demonstrated in (4) that the finite-depth effect 
matters much more in the solution of the second-order slow-drift potential 
than in the first-order potential. 
Second-order slow-drift motion may also occur in heave, pitch, and roll 
of large-volume structures with low waterplane area. These problems have 
not been so extensively studied as slow-drift horizontal motions of moored 
structures. Nobody seems to have analyzed the combined effect of slow­
drift motions in heave, pitch, and roll and the second-order excitation in 
a similar way as outlined for horizontal slow-drift motions. 
VISCOUS LOADS 
Viscous-flow phenomena are of importance in several problems related to 
wave loads on offshore structures. Examples are wave- and current­
induced loads on jackets, risers, tethers, and pipelines; roll damping of 
ships and barges; slow-drift oscillation damping of moored structures in 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
Isabella
Realce
50 FAL TINSEN 
irregular seas and winds; and anchor-line damping and "ringing" damping 
ofTLPs. 
The main factors influencing the flow can be characterized in terms of 
the Reynolds number = UD/v (where U is the characteristic free-stream 
velocity, D the characteristic length of the body, and v the kinematic 
viscosity coefficient), the roughness number = kiD (where k is the charac­
teristic dimension of the roughness on the body surface), the Keulegan­
Carpenter number KC = UMT/D for ambient oscillatory planar flow with 
velocity UM sin (2mIT+£) past a fixed body, the relative current number 
[= UelUM when the current velocity Uc is in the same direction as the 
oscillatory-flow velocity UM sin (2m1T +8)]. Also important are body form, 
free-surface effects, seafloor effects, the nature of ambient flow relative to 
the structure's orientation, and the reduced velocity ( = UlfnD for an 
elastically mounted cylinder with natural frequency fn)' Detailed dis­
cussions on many of the factors mentioned above can be found in Sarpkaya 
& Isaacson (1981). All cases of interest involve very high Reynolds number 
flow. For instance, for a survival condition of a jacket, the Reynolds 
number may be up to 107• The free-surface effect is particularly important 
for roll damping. Seafloor proximity is important for pipelines, and the 
reduced velocity has relevance for structures where vortex-induced 
vibrations ("lock-in") are a possibility. 
Generally speaking, the flow separates whenever viscous effects are 
significant for offshore structures, which means that pressure forces due 
to separated-flow effects are more important than shear forces. There is 
some confusion as to what is precisely meant by separation in unsteady 
flow. There is obviously no disagreement that the flow is separated if 
vortices can be clearly observed in the fluid. However, it is difficult to set 
a precise criterion for when one would call a flow separated or not, but 
there is a need for such a criterion if approximate methods are used to 
describe the flow. Telionis (1981) has discussed criteria for flow separation 
in unsteady flow. One definition is that "unsteady separation occurs when 
a singularity develops in the classical boundary layer equations" (van 
Dommelen & Shen 1980). Even if the boundary-layer equations do not 
describe the physical fluid behavior where the flow is separating from the 
body, one may argue that the occurrence of a singularity in the math­
ematical solution is an indication that the flow strongly breaks away from 
the body. On the other hand, one may question if this is a sufficient 
description of separation in unsteady flow. The flow may also be called 
separated if it does not break strongly away from the body (for instance, 
if there is a thin recirculating wake close to the body). A necessary require­
ment for a recirculating wake to be present in a two-dimensional flow is 
that the shear stress is zero at a point on the body surface, i.e. that the 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
Isabella
Realce
W A VB LOADS ON OFFSHORE STRUCTURES 51 
shear stress is changing sign. This is the normal separation criterion for 
steady incident flow. For unsteady flow, Telionis (1981) prefers to call the 
point of zero skin friction the "detachment" point. 
The flow around blunt-shaped marine structures with no sharp comers 
will not separate in oscillatory fluid motion at low Keulegan-Carpenter 
numbers. The presence of a current will affect the occurrence of flow 
separation. When there is only current and no waves, the flow will always 
separate around blunt-shaped marine structures. 
Sarpkaya (1986) has examined experimentally circular cylinders at low 
KC-numbers for different Reynolds numbers, with no current present, 
and reported that separation occurred for KC-numbers as low as 1.25. 
Bearman (1985) has discussed how the flow behaves for different KC­
numbers of practical interest for offshore structures. Most of the results 
showing the effect of KC-number are based on ambient planar oscillatory 
motion. This cannot always be applied to wave-induced motions and loads 
on offshore structures. For instance, the "wake" behaves differently for a 
horizontal cylinder in waves than for a vertical cylinder in waves with the 
same KC-number (Chaplin 1984). 
The state of the art in calculating the viscous loads on offshore structures 
is not satisfactory. Traditionally, Morison's equation (Morison et a1. 1950) 
has been used in the offshore industry to calculate wave and current loads 
on cylindrical shapes appliedin structural work. For those who are not 
acquainted with this formula I briefly discuss it here. For simplicity, let us 
consider a fixed vertical circular cylinder penetrating the free surface. The 
horizontal force dF per unit length on a strip of the cylinder is written as 
(6) 
The force direction is in the wave-propagation direction. Furthermore, p 
is the mass density of the water, D is the cylinder diameter, and u and ax 
are the horizontal undisturbed fluid velocity and acceleration, respectively, 
at the midpoint of the strip at the cylinder's center position. The mass and 
drag coefficients (CM and CD, respectively) have to be empirically 
determined and depend on parameters mentioned in the beginning of this 
section. Morison's equation cannot predict the oscillatory forces due to 
vortex shedding at all. Bearman et a1. (1984) have presented an empirical 
force equation that gives reasonable predictions for transverse forces on 
cylinders. 
It is easy to criticize Morison's equation, but there is nothing better 
from a practical point of view. The reason for this is the very complicated 
flow picture that occurs for separated flow around marine structures. 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
Isabella
Realce
Isabella
Realce
Isabella
Realce
Isabella
Realce
52 FALTINSEN 
Research on applying more rational methods is stilI going on, using, for 
example, the single-vortex method, the discrete-vortex method, the local­
vortex model, the vortex-sheet model, Chorin's method, the vortex-in-cell 
method, and Navier-Stokes solvers. A general feature is that a simple 
model will have deficiencies, which often lead to a desire for more com­
plicated numerical models. The consequence of such a trend is that one 
ends up with wanting to solve numerically Navier-Stokes equations for 
turbulent flow. At the present state of computer and software development, 
this is impossible to do for separated flow around marine structures in 
waves and current. Instead of following this trend with a more and more 
complicated model for the separated flow, one may be tempted to go the 
opposite way and try to find a very simple model that can explain important 
features of separated flow around marine structures. 
The simplest model to use is a single-vortex model. This is an inviscid 
method that models flows with thin vortex sheets at high Reynolds number. 
It has been used for low-KC-number separated flow from sharp corners. 
Faltinsen & Sortland (1987) used the method to study drag coefficients for 
arbitrary two-dimensional bodies with sharp corners. The strength and 
the position of the separated vortex followed from the Brown & Michael 
(1955) solution. This means that the Kutta condition was satisfied at each 
separation point, and that a zero-force condition was satisfied on the sum 
of a single vortex and the cut joining the single vortex and the adjacent 
separation point. Faltinsen & Sortland showed satisfactory agreement 
with experimental results for two-dimensional ship sections without bilge 
keels, but the single-vortex method could not explain the effect of bilge 
keels satisfactorily. 
A discrete-vortex model uses many discrete vortices to represent the 
separated flow. The problem is solved as an initial-value problem, and at 
every time instant in the numerical time integration a new vortex is intro­
duced into the fluid from a separation point. This method is obviously 
mote advanced and time consuming than the single-vortex method. For a 
body with sharp corners, it has been applied for arbitrary KC-number. 
Care must be taken when a discrete vortex comes close to the body or to 
another vortex. For a circular cylinder in steady incident flow, Sarpkaya 
& Shoaff (1979) have coupled the discrete-vortex method with a laminar 
steady boundary-layer method. They found satisfactory results for both a 
stationary cylinder and an elastically mounted cylinder with "lock-in" 
behavior. 
The discrete vortices represent thin free shear layers that separate from 
the body. The vorticity outside the boundary layers is concentrated in the 
free shear layers. For a continuously curved surface, the discrete-vortex 
method represents the separated vortex sheet in the vicinity of a separation 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
WAVE LOADS ON OFFSHORE STRUCTURES 53 
point rather poorly. The vortex tracking method proposed by Faltinsen & 
Pettersen (1987) is better able to handle such details. They used a con­
tinuous representation of the free shear layers and distributed sources and 
dipoles over boundaries and free shear layers. For bodies with sharp 
corners it has been applied with satisfactory results for a wide variety of 
KC-numbers. Free-surface effects have been accounted for. In general, 
parts of the free shear layers and body surfaces can come very close without 
creating problems. This is partly due to the rediscretization of the free 
shear layers that is done at selected time instants. On the other hand, one 
cannot exclude the possibility that the rediscretization procedure arti­
ficially dampens small-wavelength instabilities of the free shear layers. 
There also exists a problem in very long time simulations. This is partly 
due to the detailed numerical description of the free shear layers. The 
computer time increases approximately as N2, where N is the number of 
free-shear-layer elements. If no "dumping" of vorticity into discrete vor­
tices is done, it means that N increases by two for each time instant if 
there are two separation points. When vorticity is returned to the body, 
complications may occur if a vortex sheet comes close to a separation 
point. 
For small KC-values and separation from continuously curved surfaces, 
there are difficulties in combining a vortex-sheet method or a discrete­
vortex method with a boundary-layer calculation. The flow is then charac­
terized by rapid motions of the separation points. The nature of the 
boundary layer may change from laminar to transitional to turbulent. The 
same is true in the critical-flow regime for any KC-number. There are also 
uncertainties regarding what separation criteria should be used. 
When the incident flow is steady and the boundary layer is either laminar 
or turbulent, Aarsnes et al. (1985) have combined the vortex-sheet method 
with boundary-layer calculations. The triple-deck method was generalized 
and found necessary in order to predict properly the separation points for 
laminar flow. The point of zero skin friction was used as a separation 
criterion. 
Sarpkaya & Shoaff (1979) had to introduce an empirical time-dependent 
circulation reduction of the discrete vortices in order to predict satis­
factorily the forces and flow characteristics. Aarsnes et a1. (1985) argued 
that it is more important to introduce secondary separation. In doing 
so, they had difficulties in predicting the secondary separation points. 
Furthermore, a vortex sheet from a secondary separation point moved 
slowly and was difficult to follow in time. Diffusion is probably more 
important than convection in some of the flow close to a secondary sepa­
ration point. This conclusion is neglected in the theoretical model. 
One way to handle laminar diffusion in an approximate way is to use 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
5 4 FALTINSEN 
Chorin's (1973) method, in which convection is separated from diffusion 
and a Gaussian random walk with a standard deviation (2vM) 1/2 and zero 
mean is used to simulate diffusion. Here /).t is the time step in the numerical 
time integration. The vorticity is modeled by discrete vortices. The pro­
cedure may require the number of discrete vortices to be on the order of 
the Reynolds number to work properly. Stansby & Dixon ( 1983) therefore 
introduced the vortex-in-cell method as an efficient way to evaluate the 
convection of the discrete vortices. No separate boundary-layer method is 
necessary with this method, i.e. one avoids difficulties in predicting primary 
and secondary separation points. Smith & Stansby ( 1988) showed satis­
factory results for impulsively started flow around a circular cylinder. 
Models based on Chorin's method do not model the turbulent wake 
properly. 
Successful attempts to solve Navier-Stokes equations for laminar flow 
with a Reynolds number less than 1000 have been demonstrated by 
Lecointe & Piquet ( 1988). The solution of Navier-Stokes equations with 
turbulent flow is still unrealistic (Rodi 1985). There is obviously a need to 
treat incoming turbulent flow more satisfactorily, since it occurs in revers­
ing flow, in interaction between bodies, and for pipelines on the seafloor. 
There have been few attempts to address three-dimensional effects. A 
major obstacle is the lack of available computer power. Three-dimensional 
effects are important. For instance, it is well known that the flow around 
a long fixed cylinder exposed to incident transverse uniform steady flow is 
badly correlated in the length direction of the cylinder. An exception to 
this is for "lock-in," which occurs when the vortex shedding frequency is 
close to a natural frequency of the structure. It may be easier to treat three­
dimensional effects for smalI-KC-number flow. One possibility, which 
remains to be elaborated, is to combine a local two-dimensional flow 
orthogonal to the separation line with a three-dimensional flow simulation. 
CONCLUSIONS 
Practical numerical tools currently exist to predict linear wave-induced 
motions and loads on large-volume offshore structures. The interaction 
effect between waves and current is presently an area of active research. 
Strong nonlinear effects may be important. There exists no rational theo­
retical approach to long-time simulations of strongly nonlinear motions 
and loads. For weakly nonlinear systems, there exist practical methods 
that are used to study mean forces and slowly varying motions of moored 
offshore structures. However, there are still unsolved problems associated 
with the interaction between linear and second-order motions. The state 
of the art in calculating viscous loads on offshore structures is not satis-
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
WAVE LOADS ON OFFSHORE STRUCTURES 55 
factory. The simple and semiempirical Morison's equation is still the most 
accepted approach. Encouraging results have been demonstrated with 
vortex methods, Chorin's method, and Navier-Stokes solvers, but these 
methods are either not robust enough or too time consuming to be 
recommended for engineering use. 
ACKNOWLEDGMENTS 
Critical remarks and suggestions by K. S. Eckhoff, J. M. R. Graham, M. 
Greenhow, J. Grue, P. McIver, J. N. Newman, M. Ohkusu, and D. K. P. 
Yue are appreciated. 
Literature Cited 
Aarsnes, J. V., Faltinsen, O. M., Pettersen, 
B. 1 985. Application of a vortex tracking 
method to current forces on ships. Proc. 
Con! Sep. Flow Around Mar. Struct., 
Trondheim, Nor., pp. 309-46. Trondheim: 
Nor. Inst. Techno!. 
Bearman, P. W., Graham, J. M. R., Obasaju, 
E. D. 1 984. A model equation for thc 
transverse forces on cylinders in oscil­
latory flows. Appl. Ocean Res. 6: 1 66-72 
Bearman, P. W. 1985. Vortex trajectories in 
oscillatory flow. Proc. Can! Sep. Flow 
Around Mar. Struet., Trondheim, Nor., pp. 
1 3 3-54. Trondheim: Nor. Inst. Techno!. 
Brown, C. E., Michael, W. H. 1 955. On slen­
der delta wings with leading edge sepa­
ration. NACA Tech. Note 3430 
Chaplin, J. R. 1 984. Nonlinear forces on a 
horizontal cylinder beneath waves. J. 
Fluid Meeh. 147: 449-64 
Chorin, A. J. 1973. Numerical study of 
slightly viscous flow. J. Fluid Meeh. 57: 
785-96 
Dommermuth, D. G., Yue, D. K. P. 1 987. 
Numerical simulations of nonlinear axi­
symmetric flows with a free surface. J. 
Fluid Meeh. 1 78: 1 95-2 1 9 
Dommermuth, D. G., Yue, D. K . P . , Lin, 
W. M . , Rapp, R. J . , Chan, E. S., Melville, 
W. K. 1 988. Deep water plunging break­
ers: a comparison between potential 
theory and experiments. J. Fluid Mech. 
1 89: 423-42 
Faltinscn, O. M. 1977. Numerical solutions 
of transient nonlinear free surface motion 
outside or inside moving bodies. Proe. 
Int. Can! Numer. Ship Hydrodyn., 2nd, 
Berkeley, Calif., pp. 347-57 
Faitinsen, O. M., Pettersen, B. 1987. Appli­
cation of a vortex tracking method to sep­
arated flow around marine structures. J. 
Fluids Struct. I : 2 1 7-37 
Faitinsen, O. M ., Sortland, B. 1987. Slow 
drift eddymaking damping of a ship. App/. 
Ocean Res. 9(1) : 37-46 
Faltinsen, O. M. 1 988. Second order non­
linear interactions between waves and low 
frequency body motion. lUTAM Symp. 
Non-linear Water Waves, ed. K. Horikawa, 
H. Maruo, pp. 29-43. Berlin/Heidelberg: 
Springer-Verlag 
Greenhow, M., Lin, W. M. 1 983. Non-linear 
free surface effects: experiments and 
theory. Rep. No. 83-19, Dep. Ocean Eng., 
Mass . Inst. Techno!. , Cambridge 
Greenhow, M. 1 987. Wedge entry into 
initially calm water. Appl. Ocean Res. 9(4) : 
2 14--23 
Greenhow, M. 1988. Water-entry and -exit 
of a horizontal circular cylinder. Appl. 
Ocean Res. 1 0(4) : 1 9 1 -98 
Gme, J., Palm, E. 1985. Wave radiation and 
wave diffraction from a submerged body 
in a uniform current. J. Fluid Mech. l S I : 
257-78 
Kim, M. H., Vue, D. K. P. 1988. The non­
linear sum-frequency wave excitation and 
response of a tension-leg platform. Int. 
Can! Behav. oj Offshore Struct. (BOSS), 
5th, Trondheim, Nor., pp. 687-704 
Koehler, B. R., Kettleborough, C. F. 1977. 
Hydrodynamic impact of a falling body 
upon a viscous incompressible fluid. J. 
Ship Res. 2 1(3) : 1 65-�H 
Korsmeyer, F. T., Lee, C. H., Newman, J. 
N., Sclavonous, P. D. 1988. The analysis 
of wave effects on tension-leg platforms. 
OMAE '88, Can!, Houston, pp. 1-14 
Lecointe, Y., Piquet, J. 1988. Computation 
of the characteristics of the wake past an 
oscillating cylinder. Int. Corif'. Behav. of 
Off.�hore Struct. (BOSS), 5th, Trondheim, 
Nor., pp. 457-70 
Lee, C. H. 1988. Numerical methods Jar the 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
56 FALTINSEN 
solution of three-dimensional integral equa­
tions in wave-body interactions. PhD thesis. 
Mass. Inst. Techno!., Cambridge 
Lin, W. M., Newman, J. N., Yue, D. K. P. 
1 984. Non-linear forced motions of float­
ing bodies. Proc. Symp. Nav. Hydrodyn., 
Hamburg, pp. 33-49. Washington, DC: 
Natl. Acad. Press 
Maruo, H. 1 960. The drift of a body floating 
in waves. J. Ship Res. 4(3): 1-10 
Mei, C. C. 1978. Numerical methods in 
water-wave diITraction and radiation. 
Annu. Rev. Fluid Mech. 10: 393--416 
Morison, J. R., O'Brien, M. P. , Johnson, J. 
W., Schaaf, S. A. 1950. The force exerted 
by surface waves on piles. Pet. Trans. 189: 
1 49-54 
Newman, 1. N. 1 974. Second order, slowly 
varying forces on vessels in irregular 
waves. Int. Symp. Dyn. of Mar. Veh. and 
Struct. in Waves, pp. 1 82-86. London: 
Inst. Mech. Eng., UniversityCoIJ. 
Newman, 1. N. 1978. The theory of ship 
motions. Adv. Appl. Mech. 18: 221-83 
Newman, J. N. 1985. Algorithms for the free­
surface Green function. J. Eng. Math. 19: 
57-67 
Newman, 1. N. 1990. Second-harmonic wave 
diffraction at large depths. J. Fluid Mech. 
In press 
Ogilvie, T F. 1983. Second order hydro­
dynamic effects on ocean platforms. Int. 
Workshop Ship and Platform MOlions, 
Berkeley, Calif., pp. 205---65 
Peregrine, D. H. 1983. Breaking waves on 
beaches. Annu. Rev. Fluid Mech. 1 5: 149-
78 
Rodi, W. 1985. Turbulence modelling. Proc. 
Conf. Sep. now Around Mar. Struct., 
Trondheim, Nor., pp. 261-88. Trondheim: 
Nor. Inst. Techno!. 
Sarpkaya, T., Shoaff, R. L. 1979. A discrete­
vortex analysis of flow about stationary 
and transversely oscillating circular cylin­
ders. Tech . Rep. NPS-69SL79011. Nav. 
Postgrad. Sch., Monterey, Calif. 
Sarpkaya, T, Isaacson, M . 1 98 1 . 111echanics 
of Wave Forces on Offshore Structures. 
New York: Van Nostrand Reinhold 
Sarpkaya, T 1986. Force on a circular cylin­
der in viscous oscillatory flow at low 
Keulegan-Carpenter numbers. J. Fluid 
Meeh. 1 65: 61-71 
Schwartz, L. W. 1 974. Computer extension 
and analytic continuation of Stokes' ex­
pansion for gravity waves. J. Fluid Mech. 
62: 553-78 
Smith, P. A., Stansby, P. K. 1 988. Impul­
sively started flow around a circular cylin­
der by the vortex method. J. Fluid Mech. 
1 94: 45-77 
Stansby, P. K., Dixon, A. G. 1 983 . Simu­
lation of flows around circular cylinders by 
a Lagrangian vortex scheme. Appl. Ocean 
Res. 5(3): 1 67-78 
Takagi, M., Arai, S., Takezawa, S., Tanaka, 
K., Takerada, N. 1 985. A comparison of 
methods for calculating the motions of a 
semi-submersible. Ocean Eng. 1 2(1): 45--
97 
Telionis, D. P. 1 98 1 . Unsteady Viscous Flows. 
New York: Springer-Verlag. 408 pp. 
Triantafyllou, A. W. 1982. A consistent 
hydrodynamic theory for moored and 
positioned vessels. J. Ship Res. 26: 97-
105 
UrseH, F. 1 98 1 . Irregular frequencies and the 
motion of floating bodies. J. Fluid Mech. 
105: 143--56 
van Dommelen, L L, Shen, S. F. 1 980. The 
spontaneous generation of the singularity 
in a separating laminar boundary layer. J. 
Comput. Phys. 38: 1 25--40 
Verhagen, J. H. G. 1 967. The impact of a flat 
plate on a water surface. J. Ship Res. 1 1 (4): 
2 1 1-23 
Vinje, T., Brevig, P. 1 98 1 . Numerical cal­
culations of forces from breaking waves. 
Proc. Conf Hydrodyn. in Ocean Eng., 
Trondheim, Nor., pp: 547--66. Trondheim: 
Nor. lnst. Techno!. 
Wehausen, J. V., Laitonc, E. V. 1 960. Sur­
face waves. In Handbueh der Physik, 9: 
446-778. Berlin: Springer-Verlag 
Wichers, 1. E. W. 1982. On the low­
frequency surge motions of vessels 
moored in high seas. Annu. Offshore Tech­
nol. Conf., 14th, Houston. O TC 4437, 4: 
7 1 1 -34 
Yeung, R. W. 1982. Numerical methods in 
free-surface flows. Annu. Rev. Fluid Mech. 
14: 395--442 
Zhao, R., Faltinsen, O. M. 1 988. A com­
parative study of theoretical models for 
slowdrift sway motions of a marine struc­
ture. OMAE '88 Conf., Houston, pp. 1 53-
58 
Zhao, R., Faltinsen, O. M., Krogstad, 1. R., 
Aanesland, V. 1988. Wave-current inter­
action effects on large-volume structures. 
Int. Conf Behav. of Offshore Struct. 
(BOSS), 5th, Trondheim, Nor., pp. 623-
38 
A
nn
u.
 R
ev
. F
lu
id
 M
ec
h.
 1
99
0.
22
:3
5-
56
. D
ow
nl
oa
de
d 
fr
om
 w
w
w
.a
nn
ua
lr
ev
ie
w
s.
or
g
by
 T
ex
as
 A
&
M
 U
ni
ve
rs
ity
 -
 C
ol
le
ge
 S
ta
tio
n 
on
 0
9/
09
/1
4.
 F
or
 p
er
so
na
l u
se
 o
nl
y.
	Annual Reviews Online
	Search Annual Reviews
	Annual Review of Fluid Mechanics Online
	Most Downloaded Fluid Mechanics Reviews
	Most Cited Fluid Mechanics Reviews
	Annual Review of Fluid Mechanics Errata
	View Current Editorial Committee
	ar: 
	logo: