HSE
Health & Safety
Executive
Nonlinear potential flow forcing:
the ringing of concrete gravity
based structures
A summary report
Prepared by Ocean Wave Engineering Ltd
for the Health and Safety Executive 2006
RESEARCH REPORT 468
HSE
Health & Safety
Executive
Nonlinear potential flow forcing:
the ringing of concrete gravity
based structures
A summary report
Peter Tromans*†, Chris Swan† & Stephen Masterton†
*Ocean
Wave Engineering
Liss
Hampshire
GU33 7AT
†Imperial
College
London
SW7 2AZ
This report represents a summary of a larger report that describes a substantial programme of work
concerning the nonlinear potential flow loads acting on single and multiple column structures. In particular,
the report addresses the issue of structural ringing, or transient structural deflections occurring at frequencies
well above the incident wave frequencies. The work has involved both theoretical modelling and experimental
observations.
The key findings of the work are that both single and multiple column structures may be subjected to
unexpected high-frequency forces. These are entirely dependent upon conditions at the water surface, and are
associated with the unexpected scattering of high-frequency waves that cannot be predicted by existing
diffraction solutions. The nonlinear forcing includes significant force components up to the fifth harmonic of the
incident wave. However, with the scattering of high-frequency waves in large part controlled by the movement
of fluid around the circumference of the structure, a harmonic analysis based solely on the characteristics of
the incident waves neglects important time-scales and is unlikely to be universally applicable. The magnitude of
the nonlinear forces is critically dependent upon both the steepness and the period of the incident waves, with
the largest forces occurring in peak periods around 12-14 seconds. This suggests that the effects may
represent important local design conditions, but are unlikely to be significant for maximum global loading
conditions. The link between nonlinear forcing and scattering also suggests that localised vertical jetting is a
related phenomenon; the latter having implications for the setting of an effective air-gap.
This report and the work it describes were funded by the Health and Safety Executive (HSE). Its contents,
including any opinions and/or conclusions expressed, are those of the authors alone and do not necessarily
reflect HSE policy.
HSE BOOKS
© Crown copyright 2006
First published 2006
All rights reserved. No part of this publication may be
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Applications for reproduction should be made in writing to: Licensing Division, Her Majesty's Stationery Office, St Clements House, 2-16 Colegate, Norwich NR3 1BQ or by e-mail to [email protected]
ii
Table of Contents
EXECUTIVE SUMMARY.................................................................................... V
1. INTRODUCTION............................................................................................ 1
2. BACKGROUND ............................................................................................. 3
2.1 DESCRIPTION OF APPLIED FORCES ..................................................................................... 3
2.2 WAVE MODELLING .................................................................................................................... 6
2.3 DYNAMIC RESPONSE.............................................................................................................. 13
3. APPLICATION OF A SPECTRAL RESPONSE SURFACE METHOD........ 17
3.1 MODEL DESCRIPTION ............................................................................................................. 17
3.2 WAVE CONDITIONS ................................................................................................................. 19
3.3 NUMERICAL RESULTS AND PHYSICAL INTERPRETATION ............................................ 21
3.3.1 Single-column calculations ................................................................................................. 21
3.3.2 Directionality in the incident wavefield............................................................................... 29
3.3.3 Multiple-column calculations .............................................................................................. 29 3.3.4 Spectral evolution and accurate kinematic predictions .................................................... 33
4. LABORATORY DATA: COMPARISONS WITH EXISTING MODELS ........ 38
4.1 SINGLE-COLUMN DATA .......................................................................................................... 39 4.1.1 Forcing in uni-directional regular waves ............................................................................ 39
4.1.2 Forcing in uni-directional focussed wave groups.............................................................. 46
4.1.3 Effects of directionality ........................................................................................................ 49
4.1.4 Dynamic Response ............................................................................................................. 59
4.2 MULTIPLE COLUMN DATA ..................................................................................................... 60
4.2.1 Forcing in uni-directional regular waves ............................................................................ 60
4.2.2 Forcing in uni-directional focussed wave groups.............................................................. 68
4.2.3 The Effects of Directionality................................................................................................ 72
4.2.4 Dynamic Response ............................................................................................................. 78
5.0 NONLINEAR WAVE SCATTERING AND RELATED ISSUES ................. 87
5.1 BACKGROUND: EARLIER OBSERVATIONS OF WAVE SCATTERING ........................... 87
5.2 HIGH FREQUENCY WAVE LOADING: A PHYSICAL EXPLANATION ............................... 92
5.3 WAVE SLAMMING .................................................................................................................... 93
6. DISCUSSION OF RESULTS: PRACTICAL IMPLICATIONS .................... 100
6.1 NONLINEAR FORCING .......................................................................................................... 100
iii
6.2 WAVE CONDITIONS AND WAVE MODELLING.................................................................. 102
6.3 DYNAMIC RESPONSE............................................................................................................ 103
6.4 WAVE SLAMMING .................................................................................................................. 103
6.5 RELATED ISSUES AND MODELLING PROCEDURES...................................................... 105
7. CONCLUDING REMARKS ........................................................................ 106
REFERENCES ................................................................................................. 107
iv
EXECUTIVE SUMMARY
This report concerns the nonlinear potential flow loads acting on single and multiple column
structures, the diameters of which are large but not sufficiently large that the structure lies
within the linear diffraction region. In particular, the report addresses the issue of structural
ringing, or transient structural deflections occurring at frequencies well above the incident wave
frequencies. Although the report specifically concerns concrete Gravity Based Structures (or
GBS's), since these are well known to suffer from a number of nonlinear wave and wavestructure interaction effects, many of the key findings are relevant to a much wider class of
offshore structure. The study has considered the forces acting on these structures; their
implications for the onset of dynamic response; the wave conditions in which the most
problematic forcing is likely to arise; and related issues concerning the loss of an effective airgap and the occurrence of impact forces due to wave slamming, both on the horizontal deck
structure and the vertical columns. The key findings of the study are as follows:
1. Both single and multiple column structures may be subject to unexpected highfrequency forces. This corresponds to nonlinear potential flow loading and may be
expected to arise on structures where surface-piercing columns are of sufficient
diameter, D, to ensure that the fluid loading is dominated by the inertial or potential
flow loads, but not so large that linear diffraction becomes significant. This flow regime
may be characterised by small Keulegan-Carpenter numbers (KC=UT/D5, where U is
the velocity amplitude and T is the wave period) and a diameter to wavelength ratio
such that D/<0.2.
2. With column diameters of 10-20m, most concrete gravity based structures lie within
this regime. If these structures are relatively stiff with natural periods less than two
seconds (Tn2s), as is the case for many traditionally designed structures, there are
unlikely to be any significant problems associated with dynamic response. However, if
they are more flexible with Tn >4s, the high-frequency forcing may well provoke a
transient dynamic response.
3. The nonlinear or high-frequency forces are entirely dependent upon conditions at the
water surface. The flaring of the legs with depth and the existence of large storage
caissons, essentially providing a local reduction in the water depth, play no role in the
generation of these forces. The only exception to this occurs in a number of shallow
water structures where the height of the storage caissons and their plan area are
sufficient to cause a local steepening of the wave profile. In these cases both the
nonlinear forcing and the related effects (see (5), (13) and (14) below) will be much
enhanced.
4. The nonlinear force components cannot be predicted by state-of-the-art force models;
even where these are based upon highly accurate descriptions of the wave-induced
water particle kinematics. Evidence of this is provided by the existence of a secondary
loading cycle which occurs shortly after the passage of a large wave crest. However,
this is only part, albeit the most visible part, of several high-frequency force
oscillations, the magnitude of which can be as large as 20% of the maximum linearly
predicted load, where the latter is based on the integral of Morison's inertia force.
5. The generation of these forces is associated with the movement of fluid around the
circumference of an individual column, close to the instantaneous water surface. This
effect is closely associated with the unexpected scattering of high-frequency, or short
v
wavelength, waves that cannot be predicted by existing diffraction solutions and have
been shown to be critically important in the assessment of air-gap provision.
6. The present study has established a new link between the occurrence of high-frequency
forcing and scattering and, in so doing, has provided a physical basis for the origins of
the force components.
7. Fourier analysis of the nonlinear forcing has confirmed that it includes significant force
components up to the fifth-harmonic of the incident wave. Although the absolute
frequencies associated with these components are correct, their interpretation in terms
of harmonics of the incident wave is misleading. This arises because the force
components are based upon a different set of time-scales: these are dependent upon the
time taken for the fluid to move around the column, a process which is largely
independent of the incident wave frequencies. Indeed, this inability to account for
alternative time-scales explains why the existing solutions, based solely on a harmonic
expansion of the incident wave frequencies, provide a poor description of the applied
forces.
8. The magnitude of the nonlinear forces is critically dependent upon both the steepness
and the period of the incident waves, with the largest forces generated by relatively
small peak periods, 12sTp14s for columns of D=10m. This suggests that these effects
are unlikely to contribute to the 1 in 100 or 1 in 10,000 year design loads since these are
associated with much larger waves (Tp=18s) of lower steepness.
9. The directionality of the wavefield is also significant, with increases in the directional
spread leading to a notable reduction in the nonlinear force components. This is
believed to be due to a reduction in the in-line wave front steepness (point (8)) and
suggests that earlier model test results, undertaken in uni-directional seas states, may
yield excessively large nonlinear force components.
10. Contrary to linear predictions the largest nonlinear loads, and hence the highest risk of
dynamic response, occurs in wave groups produced by the focussing of wave crests
rather than wave slopes; where the latter might be expected to produce the steepest
wave profiles. This is entirely dependent on the phasing of the nonlinear force
components, which is critically important in determining the nature of the highfrequency loads.
11. The prediction of the nonlinear loads is dependent upon both the nonlinearity of the
incident wavefield and the wave-structure interaction. This is at odds with the
methodology outlined by Newman (1996) who argued that a linear representation of the
wave conditions is adequate. Comparisons with laboratory data confirm that this is not
the case. Indeed, the single most important advantage of the Rainey (1995) model is that
the derivation of the slender body loads is based on an energy argument, thereby
allowing the introduction of exact (nonlinear) descriptions of the water particle
kinematics.
12. The key issue in terms of the wave modelling is the inclusion of the rapid and local
evolution of a wave spectrum in the vicinity of an extreme wave event (Gibson & Swan,
2005. This has much wider implications in terms of hydrodynamic calculations, not
least providing a possible explanation for the occurrence of freak or rogue waves.
However, in the present context, it has a significant effect on that part of the applied
loading that can be explained. In deep water the spectral evolution involves a reduction
in the directional spread and a movement of energy to the higher frequencies. This leads
vi
to both increased 'linear' loading in the high frequencies and a very large increase in the
nonlinear loads; both being relevant to dynamic excitation.
13. Having established a link between the nonlinear forcing and scattering (item (5) above)
it follows that the occurrence of vertical jetting, leading to a loss of air-gap and wave
impacts on the underside of the deck structure, is a closely related phenomenon. It
occurs in very similar wave conditions, the only difference being that it requires more
than one large isolated wave event so that the first can provoke the nonlinear wave
scattering and the second can interact nonlinearly with it to produce large local
increases in the water surface elevation. This suggests that the relevant sea states are
characterised by a small directional spread and a large peak enhancement factor (
=5.0). The issue of air-gap is particularly important in relation to structures at which
significant sea bed subsistence has occurred (or is predicted to occur) due to the gradual
de-pressurisation of the underlying reservoir; the latter effect commonly associated with
gas production.
14. In investigating the air-gap issues noted in item (13) above, a second contribution to the
onset of structural ringing was identified. With small changes in the phasing of the
wave components, the nonlinear interactions between the high-frequency scattered
waves and the subsequent incident waves may not result in vertical jetting but rather a
very steep (near-breaking) local wave event that is subsequently swept backwards into
the structure, with the potential to cause significant impact loads at high elevations on
the columns.
15. Finally, it is important to stress that whilst the occurrence of high-frequency loading,
the loss of an effective air-gap, or the occurrence of impact forces due to wave
slamming may provoke significant local loading, these events occur in steep but
relatively short waves; perhaps characterised by 1 in 10 or 1 in 50 year events. As a
result, although they may represent important design criteria, they are unlikely to
contribute to the maximum global loading and hence threaten the overall integrity of a
typical GBS. However, this will not necessarily be the case for other types of structures
having higher natural periods or being subject to steeper shallow water waves.
vii
viii
1. INTRODUCTION This report concerns the problem of structural ringing, or the transient structural deflections that
are believed to occur at frequencies much higher than those of the incident wavefield. Although
ringing-like events can potentially effect a broad range of structural forms, this report
specifically concerns concrete Gravity Based Structures (GBS). The motivation behind this lies
in the fact that some field observations, together with some laboratory-scale model tests, have
hinted that the size and spacing of the legs of a typical GBS make this class of structure
particularly susceptible to transient dynamic response. The purpose of this report is, in effect, to
quantify the practical importance of ringing type processes to GBS's, subject to wave conditions
representative to those arising in the UK sector of the North Sea.
A ringing event involves the excitation of transient structural deflections at, or close to, the
natural frequency of the structure. Some evidence suggests that such motions may be due to
large nonlinear force components arising at the third-harmonic of the incident wavefield. As
such, it is distinct from, but perhaps closely related to, the more common occurrence of
springing. This latter effect involves the excitation of vertical motions in Tension Legged
Platforms (TLP's) and is thought to be associated with nonlinear forcing arising at the secondharmonic of the incident waves. Irrespective of the details of the applied loading, the occurrence
of high-frequency motions necessarily involves the development of significant nonlinearity.
With the broadest possible perspective, there are three possible sources for this nonlinearity: the
wave motion, the applied forcing, and the dynamic response.
In simple terms, these separate contributions can be distinguished in the following way. If the
wave motion incident to a structure is modelled using a linear wave theory, typically a linear
random wave theory, and the applied forcing is assumed to be dominated by the potential inertia
load, represented by the linear term in Morrison's equation, the calculated forces will be entirely
linear. However, if a nonlinear wave model is applied to calculate the unsteady water particle
accelerations, u / t , as should be the case if the incident wave is steep, their substitution into
the standard Morrison's inertia term will result in the prediction of nonlinear force components.
These contributions are defined as arising due to the nonlinearity of the wave motion.
Alternatively, early work by Lighthill (1979) together with more recent contributions from
Rainey (1989, 1995a,b), Faltinsen et al. (1995), Newman (1996), Eatock Taylor & Hung (1987)
and Malenica & Molin (1995) have confirmed that the linear Morrison's inertia term represents
only a first approximation, with other potentially important nonlinear force components arising
at higher orders of the wave steepness. This has been confirmed for both the limiting case of a
slender column (often described as a line with hydrodynamic properties) and for columns of
finite diameter. In this second case, which is obviously of more practical importance, the
nonlinear force components must be based upon a description of both the incident waves and
the scattered wavefield, where the latter arises from the results of a diffraction solution. If the
incident wave motion is described using a linear wave model, the additional nonlinear force
components clearly arise due to the nonlinearities in the description of the applied forcing.
The third possibility concerns the nature of the dynamic response model. If this is itself
nonlinear it may, in part, explain the occurrence of high-frequency oscillations. In reality, this is
unlikely to be important. Nevertheless, it should be investigated to ensure that it does not play a
significant role.
The methodology underpinning the present study is to isolate these separate effects and, in each
case, provide detailed comparisons between calculations based upon the best available methods
and laboratory observations. The purpose of this approach is to fully establish whether
1
significant high-frequency forces, leading to the possibility of transient dynamic excitation, are
anticipated on a typical GBS subject to representative wave conditions. In addressing this issue
it will establish, as far as is possible, the origins of any significant nonlinear loads, provide
recommendations as to the best or most effective modelling procedures and identify the wave
conditions, or structural configurations, in which they are most likely to be significant.
2
2. BACKGROUND
2.1 DESCRIPTION OF APPLIED FORCES
As far as potential-flow loading is concerned, much of the interest in the appropriateness of
Morrison's inertia load and the possible importance of additional nonlinear potential flow loads
was stimulated by Lighthill (1979). In this paper he sought to establish the inadequacy of
Morrison's equation by accounting for the physical origins of the nonlinear potential flow loads
that it neglects.
To a first approximation Lighthill noted that the linear potential, , describing the local
irrotational reactions produced by a vertical column of radius a must include both the effects of
the fluctuating horizontal velocity, u, and the fluctuating extensional motion, E; where one
component of the latter is the horizontal gradient of the horizontal velocity, u/x.
Using this result, Lighthill first confirmed that the integral of the unsteady pressure, -/t,
over the surface of the column (up to the SWL, or z=0) gives the total force corresponding to
the Morrison's inertia load
F1 = 2 gA a 2 cos (t ) ,
(0.1)
where A is the wave amplitude and the wave frequency or 2/T, where T is the wave period.
In contrast, the integral of the dynamic pressure, -()2, over the surface of the column
(again up to the SWL) yields an additional quadratic (or second-harmonic) force
1 Fd = kA gA a 2 sin (2t ) ,
(0.2)
4 where k is the wavenumber or 2/, where is the wavelength. This force is additional to those
included in Morrison's equation and clearly increases in importance with the steepness of the
wave form, kA.
In addition, Lighthill also noted that whilst the hydrostatic pressure beneath SWL gives no net
force, the sum of the hydrostatic pressure and the unsteady pressure, -gz-/dt, integrated
around the column and from the SWL up to the instantaneous water surface yields a further
quadratic contribution which he termed the waterline force
Fw = (kA ) gA a 2 sin (2t ) .
(0.3)
In the limit of small ka, it is interesting to note that the waterline force, Fw, is four times larger
than the dynamic force, Fd. This hints at the importance of forces arising close to the water
surface.
The last contribution to the nonlinear or quadratic loading considered by Lighthill relates to the
effects of the quadratic potential, q . He notes that if one is solely interested in the applied
forces, the direct calculation of q can be avoided and the forces arising defined by
C 1
Fq = 16 log 3 k 2 a 2 (kA ) gA a 2 sin (2t ) ,
ka 2
3
(0.4)
where the constant C is taken as 0.28.
In the limit of small ka, in which Lighthill was principally interested, this term becomes very
small. As a result, the nonlinear potential flow loading (additional to the Morrison's inertia load)
is dominated by the sum of Fd and Fw, given in Equations (2.2) and (2.3) respectively. However,
as the size of the column increases Fq will clearly add to the applied loading. Indeed, Lighthill
concludes by arguing that nonlinear diffraction effects may become more significant than
previously expected. Further discussion of this force is provided by Eatock Taylor & Hung
(1987).
Building directly on the work of Lighthill (1979), Faltinsen et al. (1995), hereafter referred to as
FNV, also considered the case of a vertical surface-piercing column subject to regular waves.
In their analysis they argue that a conventional perturbation expansion on which diffraction
theories are typically based (Eatock Taylor & Hung, 1987), Chao & Eatock Taylor (1992) and
Malencia & Molin (1995)) is inappropriate to the study of ringing because it assumes that the
wave amplitude is asymptotically small in relation to all other relevant length-scales, including
both the wavelength, , and the diameter of the structure, D=2a. They suggest a more
appropriate regime is one in which the wavelength is assumed to be long, the so-called long
wavelength regime, but the wave amplitude may be comparable to the column radius. This
implies kA<<1, ka<<1, but A/a=0(1).
Within this flow regime, FNV describes the forces arising from the linear potential, denoted by
the superscript (I), as
(I )
Fx1 = 2 ga 2 Acos (t )
(I ) 5
Fx2 = gka 2 A2 sin (2t )
4
(I )
Fx3 = gk 2 a 2 A3 cos (3t )
(0.5)
(0.6)
(0.7)
At this point it is important to note that Equation (0.5) corresponds to the Morison's inertia load;
Equation (0.6) represents the sum of the dynamic force, Fd, and the waterline force, Fw, given in
Equations (0.2) and (0.3) respectively; while Equation (0.7) describes an additional third-order
point load representing a nonlinear contribution from the linear potential.
In addition to these loads, FNV argue that nonlinear loads arising from the nonlinear potential,
, solved within an inner domain close to the body, must also be included:
( )
Fx3 = gk 2 a 2 A3 (cos (t ) cos (2t )),
(0.8)
Combining Equations (0.7) and (0.8) gives a total third-order force:
(I + )
Fx3
= gk 2 a 2 A3 (cos (t ) 2 cos (3t ))
(0.9)
In considering the nature of these loads, it is significant to note that the third-harmonic
contribution (at third-order) arises from two separate but equal contributions. Neglecting the
nonlinear potential, , would therefore lead to a 50% underestimate of this force component.
In an extension of the method proposed by FNV, Newman (1996) sought to describe the forces
acting on a vertical surface-piercing column subject to uni-directional irregular waves. The
4
proposed solution again concerns the long wave regime, with the corresponding forces defined
by:
0
(I )
Fx1 = 2 a 2 ut d
z
(0.10)
d
0
(I)
Fx2 = a2 (2wwx + uu x )d
z + 2a u2 t1
(0.11)
d
(
)
ut 2
(I )
Fx3 = a2 1 utz1 + 2wwx + uu x ut wt u 2 + w2 g
g
(0.12)
2
( ) 4 a 2
u ut
Fx3 =
g
(0.13)
where 1 , is the first-order surface profile.
At this stage it is important to note that in calculating the nonlinear potential, , both FNV and
Newman (1996) argue that in deep water the linear incident wave potential is exact up to and
including terms of order A3, provided the dispersion relation 2=gk is replaced by
2=gk[1+k2A 2]. Indeed, they argue that the higher-order correction to the incident wave is
insignificant. This assertion will be further discussed in the light of recent developments in
wave modelling and is shown to be incorrect.
In a separate line of enquiry, but again building on the work of Lighthill (1979), Rainey (1989,
1995a,b) considered the potential flow forces acting on a cylindrical column for the limiting
case of small diameter and argued that the potential flow loading may be represented by the sum
of three terms corresponding to the Morrison's inertia load, the axial divergence load and free
surface intersection load:
u uu wu (s ) FI = 2 a 2 +
+
z
x
z t
d
w
(s ) FAD = a 2u
z
z
d
2
(s ) 1
FSI = a 2
u
2
x
(0.14)
(0.15)
(0.16)
where the superscript (s) denotes the fact that they are based upon a slender body
approximation. Unlike the results discussed previously, Rainey's slender body forces were not
based on a formal perturbation expansion, but were instead derived from energy considerations.
This approach is highly un-intuitive and provides little physical understanding of the origins of
the force components1. Nevertheless, Rainey argues that the over-riding advantage of this
approach is that its application is not restricted by the convergence of some series solutions and,
as a result, if the water surface elevation and the associated water particle kinematics are
predicted by an appropriate nonlinear wave theory it can be applied to practically important
cases involving large/steep waves.
Considering each of these force terms in turn, F(I)(s) in Equation (0.14) is a distributed load
representing the Morrison's inertia force, with Cm=2.0. However, this includes two important
1
In a separate contribution Manners & Rainey (1992) re-derived the axial divergence force, FAD(s), from first
principles using pressure integration and argued that it appears to be related to the rate of change of added mass.
5
changes: (i) it is based on the total water particle acceleration, both unsteady and convective;
and (ii) the integration is extended to the instantaneous water surface, (t). These changes lead
to the inclusion of nonlinear force components not present in the earlier values of the Morrison's
load given in Equations (0.1) and (0.5).
The axial divergence term, FAD(s), in Equation (2.15) is again a distributed load. Applying a
linear wave model for u and and integrating to the still water level (z=0), it is easily shown
that for d=-, FAD(s) is identical to Lighthill's dynamic force, Fd in Equation (0.2), derived from
the integral of the dynamic pressure. Similarly, the integral of the unsteady acceleration from
the SWL to (t) (the first term in Equation (0.14)) yields Lighthill's waterline force, Fw in
Equation (0.3). To this extent, up to the second-harmonic terms arising at second-order, there
appears to be agreement between Lighthill, FNV and Rainey. However, this only remains true
because the inclusion of the convective acceleration terms in Equation (2.14) provides no net
contribution to the forces at second-order (i.e. up to the SWL).
The surface intersection force, FSI(s), in Equation (2.16) represents a point load, acting at the
water surface which first arises at a third-order of wave steepness. Adopting a linear wave
description, the third-order third-harmonic contribution arising from this term is given by
1
FSI( s ) ~ gk 2 a 2 A3 cos (3t ),
8
(0.17)
which is eight times smaller than the corresponding term identified by FNV. The difference
between these results are directly attributable to the assumptions inherent in slender body
theory. Indeed Rainey (1995b) considers this point and argues that if the rate of change of the
energy associated with the surface distortion, or the disturbance of the surface caused by the
presence of the column, is included an additional surface distortion force, FSD, arises. This
corresponds to an additional point load defined by
FSD = 7 a 2 2 u
u
.
t
2g
(0.18)
With the application of linear theory, it may be shown that FSD is seven times larger than FSI. As
a result, the total third-harmonic load arising at third-order becomes consistent with the earlier
FNV solution. However, the range of wave conditions for which the various force components
are valid remains a point of on-going discussion.
2.2 WAVE MODELLING
It is clear from both field observations and laboratory data that the onset of structural ringing is
associated with the generation of high-frequency forcing and hence with the occurrence of steep
incident waves. Earlier studies of the wave-induced water particle kinematics have
demonstrated that accurate representations of both the fluid velocities and the accelerations
associated with a large wave event can only be achieved using a fully nonlinear wave model
(Swan et al., 2002). This is particularly important close to the water surface where the nonlinear
free surface boundary conditions dominate, and where the kinematic predictions define the
surface point loads identified in Section 2.1. In the absence of any clear understanding as to
what wave events provoke a ringing response, the present study has considered a wide range of
wave conditions. In each case two approaches to the description of the water particle kinematics
will be adopted:
6
Figure 2.1 Water surface profiles describing an extreme wave event in a uni-directional
JONSWAP spectrum: comparisons between fully nonlinear calculations based upon
BST and linear theory with second- and third-order bound wave corrections. (a) The
overall wave group; (b) Details of the largest wave crest
7
Figure 2.2 The evolution of the amplitude spectrum describing the freely propagating
wave components in a uni-directional JONSWAP spectrum. (a) Comparisons between
spectra at various times; (b) Comparisons between BST and the third-order resonant
terms calculated using ZE at the position of the extreme event.
8
(a) Wave solutions consistent with the applied force models. In this case the expansion or
ordering of the results will be preserved, but the description of the kinematics known to
be of limited accuracy. An important example of this is the application of Newman's
model, in which the force components were formulated on the assumption that a linear
random wave model was adequate. Whilst this may be appropriate for the description of
small amplitude random waves, it is not true in extreme wave events.
(b) The application of state-of-the-art wave models providing the best possible physical
realisation of extreme wave events. Although these kinematic predictions are
inappropriate to certain force models, they can be employed within the description of
the Rainey force terms, notably the axial divergence force (Equation (2.15)) and the
free-surface intersection force (Equation (2.16)), and can also be employed in ad-hoc
engineering calculations in which the nonlinear contributions from the Morrison's
inertia load (Equation (2.14)) based on exact kinematic predictions are assessed.
Comparisons between these approaches, (a) and (b) above, will begin to establish the principle
cause of the ringing response: whether it arises due to the nonlinear kinematics associated with
steep waves; the nonlinear force components, possible accounting for some wave-structure
interaction; or whether it is due to some as yet undefined mechanism presumably associated
with a highly nonlinear wave-structure interaction.
Within the present study a wide range of wave conditions have been addressed. From the outset
it should be stressed that the small number of regular wave cases considered are in no sense
believed to be representative of large waves arising in realistic sea states. Uni-directional regular
waves are steady, propagating without change of form; for a given wave height and wavelength
they exhibit reduced crest-trough asymmetry; and, most importantly, they are less steep than an
irregular wave of equivalent size (H, ), implying reduced maximum water particle acceleration
occurring close to the water surface. Furthermore, laboratory observations of wave-structure
interaction in regular wave trains (commonly adopted in model test programmes) can
potentially be misleading, particularly in respect of the scattered wavefield. Nevertheless,
despite these significant short-comings, regular wave tests do have a role to play, not least
because they are consistent with some of the established force models, notably FNV, and they
provide a simplified wave environment in which detailed harmonic analysis of the applied
forcing is easily undertaken. Within the present study a limited number of regular wave cases
are investigated and, in relation to part (b) above, the nonlinear water particle kinematics are
based on a high-order stream function solution based on Sobey et al. (1987). Much of this work
concerns a possible explanation of the secondary loading cycle first identified by Grue et al.
(1993) and further discussed in Sections 3 and 4.
In deep water the largest or steepest waves arise in random and directionally spread wavefields
and evolve as isolated events that rapidly disperse in both space and time. The formation of such
waves is predominantly, but not exclusively, driven by linear dispersion. Within a given sea
state there are a large number of freely propagating wave components, of differing frequency,
travelling at different speeds and in different directions. If the phasing of these components is
such that a large number of wave crests (or wave slopes) are superimposed at one point in space
and time, a large (or steep) wave event arises. This process is commonly referred to as wave
focussing and leads to isolated wave events that are characterised as being: unsteady or
transient, short-crested or directionally spread, and, in the case of the largest waves, highly
nonlinear.
In the broadest possible sense large wave events can be modelled, both experimentally and
numerically, using one of two approaches: (a) long time-domain simulations; and (b) focussed
wave events. In the present study we have concentrated on the latter approach. The reasons for
9
this are three-fold. First, this approach is consistent with the application of the best-available
wave models. Fully nonlinear computations are inevitably computationally intensive and
therefore inappropriate to long time-domain simulations. Second, the application of focussed
wave groups avoids many of the difficulties associated with the imposition of unrepresentation
boundaries; the extreme event having formed and dispersed before any unwanted reflections can
contaminate the area of interest. This is particularly relevant to laboratory studies in facilities of
finite dimensions, but is also relevant to numerical computations. Third, and perhaps most
important, the extreme ‘bursting’ behaviour typical of a ringing response suggests that these
events are provoked by very short but energetic wave groups; making the focussed wave
approach entirely appropriate.
Recent research concerning the description of extreme waves (notably, Baldock et al. (1996),
Johannessen & Swan (2001) & (2003), Swan et al. (2002) and Bateman & Swan (2005)) has
shown that if models are to be effective they must incorporate the three underlying
characteristics of any realistic sea: unsteadiness, directionality and nonlinearity. On the
assumption that most realistic seas are broad-banded, in both frequency and direction, the wave
models proposed by Bateman et al. (2001) & (2003) may be taken together and represent one in
a very limited number of appropriate wave models. In the first of these papers Bateman et al.
(2001) provide a highly efficient wave model in which a spatial description of the water surface
elevation, (x,y), and the velocity potential on that surface, (x,y,) are time-marched using the
fully nonlinear free-surface boundary conditions coupled with a Taylor series expansion of the
Dirichlet-Neumann operator. In a follow-up paper, Bateman et al. (2003) apply a related
approach to accurately describe the internal water particle kinematics based on the previous
solution for (x,y) and (x,y,). An essential element of both of these solutions lies in their
computational efficiency. This is not sought for its own sake, but is an absolute requirement
necessary to achieve the very high resolution in both the wave number and the directional
domains, without which realistic wavefields cannot be successfully modelled. Taken together
these two wave models, hereafter referred to as BST, provide a complete solution of highly
nonlinear transient wave events arising in irregular or random wavefields, involving significant
directional spreads. Further details concerning the application of these models to realistic ocean
spectra are given in Bateman & Swan (2005)2.
Wave calculations, based upon the BST model, have allowed the systematic investigation of
extreme waves in realistic wave spectra. These results (see Gibson & Swan, 2005) have shown
that in some sea states the formation of the largest or steepest waves may be accompanied by
the rapid and local evolution of the wave spectrum; including the movement of energy into the
higher frequencies. Indeed, evidence of unexpected local changes in the wave spectrum first
arose as a possible explanation for freak or rogue waves (Johannessen & Swan, 2003); where
the latter may be thought of as waves that are higher or occur more often than is statistically
expected given the underlying linear characteristics of the sea state in which they arise. Given
that such waves are by definition high and steep, they will also be associated with severe
loading events and may, perhaps, be associated with the onset of structural ringing. However,
these effects have not previously been taken into account in force predictions, and are at odds
with the assumptions under-pinning Newman's theory.
Within a general description of a realistic wavefield both the water surface elevation, (x,y,t),
and the velocity potential, (x,y,z,t), describing the wave-induced water particle kinematics can
be described by:
2
It is important to note that several aspects of this work, concerning the practical applications of BST to realistic
wave spectra, arose as a result of the present study.
10
or = f ( Ai )
i
(
+ g Ai A j
i j
Underlying linear components
)
(
)
3rd-order bound waves
(
)
3rd-order resonant waves
+ hb Ai A j Ak
i j k
+ hr Ai A j Ak
i j k
2nd-order bound waves
+ lb ( Ai A j Ak Am )
4th-order bound waves
+ lr ( Ai A j Ak Am )
4th-order resonant waves.
i j k m
i j k m
(2.19)
Adopting this description, the first term corresponds to the linear sum of the linear wave
components. These are all assumed to be freely propagating and hence satisfy the linear
dispersion relation (0.19)
2 = gk tanh dk(
).
(0.20)
It is these terms that provide the kinematic input into Newman's force model. In contrast, the
second set of terms describe the second-order, 0(A2), bound waves representing the interactions
arising due to two wave couplings, unless i=j in which case they correspond to the well-known
second-order Stokes terms, or self-interaction terms. These waves do not satisfy the dispersion
equation (2.20), their phase velocity being dependent upon the interacting free waves. They
include both the low-frequency or frequency-difference terms and the high-frequency or
frequency-sum terms; the latter being important in the context of ringing since any highfrequency motions might also be expected to produce high-frequency forces. An analytic
solution of these wave components is provided by Sharma & Dean (1981).
The third term in Equation (2.19) describes a similar set of bound waves arising at third-order,
due to three-wave couplings. Within a traditional perturbation expansion one would expect
these terms to be significantly smaller than the second-order bound waves. In contrast, the
fourth term describes the so-called resonant interactions, again arising at third-order. In this
case the interaction between three wave components produces a fourth which satisfies, or
nearly-satisfies, the dispersion Equation (2.20), thereby forcing a wave mode that propagates
freely. Under these conditions energy is exchanged between the wave components and the
amplitude of the resonant wave mode can grow in time. In a mathematical sense, these
interactions take the form of a linear resonator, Phillips (1960), hence their name. Most
significantly, the transfer of energy associated with the growth of these terms can lead to the
modification of the underlying linear spectrum defining the freely propagating wave
components described in the first term. Such effects can only be produced by resonant or nearresonant interactions and are entirely independent of the associated bound waves. The final
terms, 5 and 6, included in Equation (2.20) represent the bound and resonant interactions
occurring due to four wave couplings, 0(A4); their magnitude expected to be small in
comparison to the corresponding 0(A3) terms noted above.
The importance of these terms in respect of the water surface elevation is noted on Figure 2.1.
This concerns the description of a large focussed wave event arising within a unidirectional
JONSWAP spectrum having a peak period of Tp =12.8s and a peak enhancement factor of
=5.0. In this example the linearly predicted crest elevation, corresponding to the linear sum of
11
the component wave amplitudes defining the initial JONSWAP spectrum, is max=9.0m. The
second-order bound wave interactions increase this elevation to 9.9m; whilst the third-order
bound waves increase the elevation to 10.1m. In contrast, the fully nonlinear calculations based
on BST suggest a maximum crest elevation of 11.9m. Results of this type confirm that the
largest ocean waves cannot be modelled assuming a constant regime of freely propagating wave
components coupled with their associated bound waves. Unfortunately, this approach is central
to most, if not all, design wave solutions; with the bound waves calculated on the basis of a
second-order model (Sharma & Dean, 1981).
Although the BST model provides an accurate representation of both the water surface elevation
and the underlying kinematics, it provides little by way of physical understanding and no
guidance as to which, if any, of the interactions noted in Equation (2.19) account for the
significant increases in crest elevation and wave steepness. To overcome this difficulty the
Zakharov equation (Zakharov, 1968), hereafter referred to as ZE, is very useful. This is a
nonlinear evolution-equation that can be applied to realistic broad-banded, directionally spread,
wave spectra and has been derived in Hamiltonian form up to fourth-order by Krasitskii (1994).
Using this approach it is possible to isolate the various interactions identified in Equation (2.19)
and hence, the physical processes that occur during the formation of an extreme wave event can
be identified. By utilizing both BST and ZE it is possible to account for the full nonlinearity of
the problem and to gain a physical understanding of the process involved.
Figure 2.2 again concerns a JONSWAP spectrum (Tp=12.8, =5.0) and describes the evolution
of the freely propagating spectrum during the formation of an extreme wave event (Figure 2.1).
The results presented on Figure 2.2(a) were calculated using BST and highlight very significant
changes in the amplitudes of the freely propagating wave components during the evolution of a
large wave. These changes, which must be due to resonant interactions, are surprising in two
respects: first, they produce a significant broadening of the free-wave regime, with energy
transferred to the higher frequencies; and, second, they occur very rapidly with significant
changes to the spectrum occurring in the ten wave periods prior to the extreme event. This latter
result is very different to the earlier work of Hasselmann (1962) who showed that the resonant
evolution of random seas was very slow, occurring over hundreds or, perhaps, thousands of
wave cycles. Figure 2.2(b) concerns the spectrum at the focus position and shows that the
predicted evolution can be fully explained using ZE in which only the third-order resonant or
near-resonant interactions have been included, corresponding to the fourth term in Equation
(2.19).
The significance of these results are two-fold. First, an increase in the bandwidth of the freely
propagating wave components leads to an increase in the sum of the amplitudes of the wave
components, A=1 Nai, allowing the possibility of larger maximum crest elevations. This occurs
because the total energy, which must remain constant, is proportional to the sum of the squares
of the amplitudes of the wave components, INai2. If the amplitude sum, A, is spread evenly over
N wave components, ai=A/N, it follows that A=NE. The amplitude sum of the spectrum is
thus proportional to the square root of the number of components over which the amplitude is
spread. It therefore follows that as a spectrum becomes more broad-banded (N increases), its
amplitude sum increases. Evidence of this effect is given in Figure 2.3 which describes the
changes in the amplitude sum of the freely propagating wave components as a large wave
evolves in the previous JONSWAP spectrum.
Secondly, the broadening of the free wave spectrum involves a transfer of energy into the higher
frequency and higher wavenumber components. This effect is considered in Figure 2.4 which
provides a contour plot of the energy distribution in the wave frequency-wavenumber domain at
the time and location of the extreme wave event. These results are obtained by applying a time­
12
frequency analysis, based on the Stockwell transform (Stockwell et al., 1996), to numerical
results generated by BST. At the start of the model run, defined by t=t0, the sea state is fully
dispersed, and the energy was distributed according to a JONSWAP spectrum (Tp =12.8s and
=5.0) consisting of freely propagating wave components. As a result, all of the energy was
distributed along the dashed line representing the linear dispersion equation. As the wave group
evolves towards the extreme event a number of clearly identifiable ridges become apparent.
From the top of Figure 2.4 downwards, these represent: the third-order sum components
(bound), the second-order sum components (bound), the linear components (free), and the
second-order difference components (bound); the latter being very small in deep water.
Figure 2.4 also highlights changes to the linear components: in particular, the growth of a spread
of wave energy that almost, but not quite, satisfies the linear dispersion relation. This energy
corresponds to the third-order resonant interactions that alter both the amplitude and the phasing
of the freely propagating wave components; the latter effect leading to the apparent change in
the phase velocity. Both the growth of the bound wave amplitudes and, in particular, the
widening of the free wave regime involves a movement of energy into the higher frequencies
and hence the possibility of high-frequency forcing.
All of the data presented above relates to one unidirectional JONSWAP spectrum. Several other
cases are presented in the full report, with a full interpretation provided in Gibson & Swan
(2005). In these cases the directionality of the wavefield is shown to be significant, although it
is important to note that the local and rapid spectral evolution typically leads to a steeper wave
form.
The rapid and local evolution of the wave spectrum in the vicinity of a large or steep wave event
may potentially be very significant in terms of the applied nonlinear forcing. If it is incorrectly
assumed that a wavefield may be defined in terms of a constant spectrum of freely propagating
wave components, together with their associated bound waves, there may be some justification
for arguing that a linear kinematics model is appropriate for the force predictions up to thirdorder. However, given the nature and extent of the spectral changes arising in the vicinity of a
large wave event, this approach is clearly inappropriate. The transfer of energy to wave
components of higher frequency will undoubtedly contribute additional high-frequency forcing.
Furthermore, changes in the spectral bandwidth defining the freely propagating wave
components leading to higher and steeper waves, coupled with changes in the directional spread
leading to more uni-directional waves, will tend to increase the inline wavefront steepness. This,
in turn, implies larger nonlinear water particle accelerations and, hence, larger nonlinear
potential flow forces. These effects clearly need to be taken into account if the onset of
structural ringing is to be predicted.
2.3 DYNAMIC RESPONSE
For the purpose of the present study the dynamics of the structure, whether it is a single vertical
column or a multiple column arrangement, will be modelled by a single degree of freedom
system. Although this represents a significant simplification of the dynamic behaviour exhibited
by a real structure, it is appropriate to the present study; being consistent with both the
development of the Spectral Response Surface method (Section 3) and the comparisons with
laboratory model test data (Section 4). Indeed, the purpose of the present work is not to exactly
reproduce the dynamics of a particular structure, but to determine if and when dynamics
becomes a problem.
13
Elevation or Amplitude Sum (m)
12
11.5
11
10.5
10
9.5
9
8.5
8
ï1000
ï800
ï600
ï400
ï200
0
Time (s)
Figure 2.3 Changes in the amplitude sum of the freely propagating wave components
as a large wave evolves in a uni-directional JONSWAP spectrum.
Figure 2.4 Energy distributions in the wave frequency - wavenumber domain at the
time and location of an extreme wave event in a uni-directional JONSWAP spectrum.
14
Figure 2.5 Modelling the structure as a single degree of freedom system.
A single degree of freedom system subject to a rotational displacement, s, is outlined in Figure
2.5 The lumped mass, m, the rotational stiffness, ks, and the damping coefficient, Cs, define the
characteristics of the body; whilst the over-turning moment, M(t), represents the applied load
due to the wave motion, combining the effects of both the distributed forces acting along the
length of the column(s) and the point loads acting at the water surface. For a statically
responding structure, the structural response would simply be the over-turning moment, M(t), or
perhaps the total horizontal force, F(t), defining the base sheer. However, for a dynamically
responding structure, the structural response is defined by the reaction moment, RM(t),
transferred through the spring.
The equation of motion for this system is defined by
m&&s (t ) + Cs&s (t ) + ks s (t ) = M (t ) ,
(0.21)
where the over-dot indicates a time derivative, /t, so that if s defines the rotational
displacement, & s is the angular velocity and && S the angular acceleration. If the natural
frequency of the system, n, and the relative damping, , are defined by
n2 = ks / m
and
=
Cs
,
2 ks m
(0.22)
Equation (2.21) can be re-written in a more convenient form
&&s (t )
n2
+ 2
&s (t )
M (t )
.
+ s (t ) =
n
ks
(0.23)
If the excitation is harmonic
M (t ) M̂ sin(t)
=
,
ks
ks
15
(0.24)
where M̂ is the amplitude of the applied moment and it's angular frequency, the solution of
Equation (2.23) is of the form
s =
M̂
D( )sin t + s ( ) ,
ks
(0.25)
where D() is the dynamic amplification factor and s is the phase shift between the applied
moment, M(t), and the angular displacement, s. Solutions of this form are widely applied.
However, it is important to stress that they are only valid for harmonic excitation, Equation
(2.24). If a significant part of the applied moments are nonlinear and therefore not harmonic, as
would be the case in a ringing event, the applied moments must be decomposed into Fourier
components before applying Equation (2.25).
16
3. APPLICATION OF A SPECTRAL RESPONSE SURFACE
METHOD
The purpose of this section is to apply the spectral response surface method, hereafter referred
to as the SRS method, to quantify both the maximum wave-induced fluid loading and the
maximum dynamic response. Two specific sea states will be considered corresponding to a 5­
year and a 100-year storm arising in the northern North Sea. In each case the wave conditions
are specified using a JONSWAP spectrum, S (), with appropriate values for the peak period,
Tp, and the significant wave height, Hs. No prior assumptions are made concerning the nature of
the individual wave events, or wave groups, arising within these sea states. Indeed, these
represent an important part of the required solution: the water surface elevations corresponding
to the wave events responsible for the maximum applied loading and the maximum dynamic
response, the two not necessarily being the same, form part of the output arising from the
application of the SRS method. For each sea state, two structural configurations are considered:
the first is the simple case of a single, vertical, surface-piercing column; while the second is
representative of a typical 3-legged GBS.
With calculations undertaken in the probability domain, the SRS method eliminates the need for
long time-domain simulations providing a highly efficient method of determining the timehistory of the maximum dynamic response, RM(t), the applied forcing associated with it, F(t) or
M(t), and the water surface elevations or wave group, (t), necessary to generate it. In
undertaking this modelling activity the principle goal is to determine whether the existing force
models, notably that due to Newman (1996), are able to provoke a ringing -type response and, if
so, to identify the force components on which it principally depends and the wave conditions
under which it arises. In addition, the effects of directionality in the incident wavefield and the
layout of the structure, particularly the leg spacing, will be addressed. The purpose of this latter
work being to identify those parameters that contribute to the onset of dynamic response.
3.1 MODEL DESCRIPTION
The SRS-method is not, as yet, commonly applied as a design tool and therefore a brief
overview of the method is appropriate. In its present form, the method involves the application
of a conventional first-order reliability method (FORM), Madsen et al. (1986) and Melchers
(1987), to problems involving a spectral representation. In a traditional reliability analysis,
commonly adopted in structural engineering, a relatively small number of variables are
considered and the method seeks to establish a design point and some associated probabilities.
This point arises on a line or surface, referred to as the limit-state surface, defining the boundary
between the safe region and the failure region noted on Figure 3.1. Within this space the design
point, denoted by (x1*, x2*), corresponds to the point on the limit-state surface which is closest
to the origin and hence has the highest probability of occurrence. Indeed, the distance , from
(x1*, x2*) to the origin 0, defines the reliability index; the greater the value of the smaller the
probability of failure and, hence, the safer the structure.
The SRS method is closely related to this process. It can be applied to any problem capable of a
spectral representation and, typically, involves the consideration of many more variables than is
the case with traditional reliability methods. It also differs from the latter in that rather than
seeking to locate the most probable event on a line or surface linking events of equal response,
the SRS method seeks to determine the maximum response on a line or surface linking events
having an equal probability of occurrence. The identification of this maximum involves the
solution of a constrained optimisation problem. With the introduction of a large number of
variables, typically one for each spectral component of the sea state, this task represents a
significant hurdle, not least because the response function is nonlinear and may also involve
17
several local maxima. Furthermore, to determine the probability of exceedence of a given
response, some assumption must be made concerning the shape of the response surface: in a
first-order reliability method (FORM) this is assumed to be linear; whereas in a second-order
reliability method (SORM) a parabolic distribution is adopted.
In a linear sense the water surface elevation may be defined by the sum of a large number (N) of
random, narrow-banded, frequency components, each of which is normally distributed and may
be assumed independent and uncorrelated. If Aj is the random amplitude of the jth component
(1jN), j the circular frequency, kj the wavenumber vector (kxj, kyj), j the random phase
angle and t the time, the water surface elevation is given by
(
)
( ) (t) = j = A j sin j t k j x + j ,
1
N
N
j =1
j=1
(0.26)
where the superscript (1) indicates the first-order or linear approximation and x defines the
horizontal spatial location, (x,y). Individual frequency components, j, can be transformed into
standardised variables, having zero mean and unit variance, by subtracting its mean, j, and
dividing by its standard deviation, j,
j j
%j j
,
(0.27)
and
x% j =
xj =
j
j
where the superscript ~ indicates a Hilbert transform which, in the present context, may be
thought of as the value of a variable with its phase shifted by /2. Accordingly, using the
definition given in Equation (3.1).
(
)
(3.3a)
(
)
(3.3b)
j = A j sin j t k j x + j
% j = A j cos j t k j x + j
and % j=j. With zero mean values, j= % j=0, it follows that the amplitude and phasing of the
frequency components can be defined by
(0.28)
(
A j = j x j
12
) + ( j x% j ) 2
2
(0.29)
and
x j j = tan 1 ,
x% j (0.30)
where the latter is based on the assumption that the point of interest is defined by x=0 and the
time, t, is also set to zero at the instant the system evolves through the design point. If, at this
stage, we were interested in the statistics of linear crest elevation, Equation (3.1) becomes the
response function, the SRS method provides solutions for a given probability of exceedence in
terms of the standardised variables (xj, x% j), and Equations (3.4) and (3.5) are used to define the
corresponding local water surface profile and hence the crest elevation. In a linear calculation of
this type the SRS method would simply reproduce the results of the NewWave model (Tromans
18
et al., 1991). Similarly, if second-order crest statistics are required the water surface elevation
predicted by Sharma & Dean (1981) would need to be written explicitly in terms of the
underlying linear wave components to provide an alternative to Equation (3.1) and hence a new
response function. With this defined, the process is simply repeated. In the present case we are
interested in the optimisation of both the applied load and the dynamic response. To achieve
this, the required response function, defining the total base shear, F(t), the over-turning moment,
M(t), or the dynamic response expressed in terms of the reaction moment RM(t)=Kss, must be
re-cast in terms of the standardised variables xj and x% j representing the underlying linear wave
components using the force model proposed by Newman (1996). Full details of how this is
achieved and, in particular, how nonlinear terms involving products of the standardised
variables are dealt with using a Fourier decomposition are given in the full report.
3.2 WAVE CONDITIONS
Within this section two sea states will be considered, details of which are given on Table 3.1. In
each case the sea state is defined in terms of a JONSWAP spectrum (Hasselmann (1973) so that
4p
g2
S ( ) = 5 exp 4
(
p
exp
2 2p 2
)
2
(0.31)
with
= 0.07, p
= 0.09, > p
where =0.0081, =1.25, p is the frequency of the spectral peak, or 2/Tp where Tp is the
peak period, and is the peak enhancement factor. This latter parameter defines the spectral
bandwidth; the higher the value of the more narrow-banded the spectrum. This is an important
parameter because large wave groups arising in a narrow-banded spectrum are characterised as
being less compact than their equivalent broad-banded cases and therefore consist of more than
one large isolated wave event. Earlier work, Swan & Sheikh (2004), has shown that this is a
critical factor in determining the occurrence of wave impacts on the deck structure of a multicolumn GBS. Given the possible linkage between nonlinear scattering and nonlinear forcing,
the present study has concentrated on narrow-banded spectra. Field observations suggest that for
a typical North Sea storm, the peak enhancement factor lies within the range 2.8 3.3.
However, this parameter is notoriously variable, with Hasselmann (1973) noting than an upper
limit of =5.5 was not unrealistic. In light of the earlier results, this upper-limit has been
adopted in the present calculations.
In the later stages of this section the effects of directionality in the incident wavefield will also
be considered. Throughout this report directionality is modelled using a wrapped-normal
distribution, applied uniformly to all frequency components, so that
( ) =
2 exp 2 ,
2 (0.32)
where is a normalising coefficient, defines the direction of wave propagation relative to the
mean and is the standard deviation.
Having defined the wave spectra corresponding to the storm conditions, the application of the
SRS-method requires the specification of a short-term exceedence probability. This defines the
19
probability that an individual response maxima exceeds some chosen response level. In the
present calculations the exceedence probability was set to be 10-4, corresponding to a reliability
index of =4.29 and giving (approximately) a 10% probability of occurrence within a 3 hour
storm. These values were chosen for two reasons. First, in the context of ringing we are clearly
interested in nonlinear effects and therefore need to consider some extreme wave events.
Second, with the parameters so chosen, the maximum wave heights, Hmax, correspond very
closely to the design criteria specified on Table 3.1.
Table 3.1: Typical Met-Ocean design criteria for the northern North Sea.
Table 3.1 Typical Met-Ocean
Return Period
Wave Characteristics
5 – year
100 - year
Hs (3hr)
12.9
15.6
Hmax (3hr)
23.9
28.7
Tz (Central)
11.1
12.2
Tp (Central)
15.1
16.6
Wave Steepness (1/2 Hmaxkp)
0.214
0.208
Failure region
x2
( x1* , x 2* )
lines of
constant
probability
limit-state surface
design point
x1
0
tangent
Safe region
Figure 3.1 Sketch of SRS method in two-dimensional space, x1, and x2. Note: A
traditional first order reliability analysis considers the limit-state surface, separating the
safe and failure regions, with the design point corresponding to the point on this
surface with the highest probability of occurrence. In contrast, the present SRS method
considers lines of constant probability, denoted by the concentric circles, and searches
for the point of maximum response.
20
Figure 3.2: Layout of single column geometry. F(t) is the total horizontal force acting
on the column, due to the sum of the distributed and point loads, M(t) is the applied
over-turning moment, and RM(t) is the reaction moment necessary to keep the system
in equilibrium (See also Figure 4.1)
3.3 NUMERICAL RESULTS AND PHYSICAL INTERPRETATION
The results that follow are sub-divided into four parts. The first concerns a single vertical
column subject to uni-directional waves and seeks to identify whether the available nonlinear
loading formula can account for the occurrence of transient structural deflections and, if so, to
identify the force components responsible. The second part also concerns the single column
case and assesses the effect of directionality in the incident wavefield. The third part considers a
number of multi-column layouts and addressees the significance of the leg spacing. In
particular, it determines whether the phase-lag in the forcing applied to adjacent legs is able to
reinforce the dynamic response. Both uni-directional and multi-directional waves are
considered, with the effects associated with the former shown to be more interesting. Finally,
the fourth part concerns the rapid and local evolution of the wavefield in which spectral changes
(Section 2.2) may lead to the formation of higher and steeper waves. This section will consider
the importance of the nonlinearity of the wavefield, with accurate kinematics predictions based
on the BST model used to re-evaluate the force components predicted by the Rainey (1995)
model and the subsequent dynamic response.
3.3.1 Single-column calculations
The results presented in this section concern a single column with a uniform diameter of
D=12.2m (a=6.1m). The water depth is 140m and the column extends from the top of the
storage casings, at z=~h= – 60m, up through the water surface. A diagram indicating the layout
of this structure is given on Figure 3.2. For this arrangement, and for each of the two sea states
noted above Table 3.1 the SRS-method has been applied in three separate sets of calculations to
optimise:
(a) The total horizontal force or base shear, F
(b) The total over-turning moment, M
(c) The dynamic response, RM, indicative of a rotational displacement, , about the column
base.
21
6000
(a)
4000
M(1)(t)
2000
0
ï2000
ï4000
ï6000
ï30
ï20
ï10
0
10
20
30
ï20
ï10
0
10
20
30
ï20
ï10
0
t
10
20
30
6000
(b)
M(1)(t) + M(2)(t)
4000
2000
0
ï2000
ï4000
ï6000
ï30
6000
M(1)(t) + M(2)(t) + M(3)(t)
(c)
4000
2000
0
ï2000
ï4000
ï6000
ï30
Figure 3.3 Optimisation of the over-turning moment, M(t), acting on a single column:
(a) M(1)(t); (b) M(1)(t)+M(2)(t); (c) M(1)(t)+M(2)(t)+M(3)(t). - - - 5-year storm and _____ 100-
year storm
22
4
x 10
1 (a)
M
R(1)(t)
0.5
0
ï0.5
ï1
ï30
ï20
ï10
0
10
20
30
ï20
ï10
0
10
20
30
ï20
ï10
0
t
10
20
30
4
x 10
0.5
M
R(1)(t) + R(2)(t)
1 (b)
M
0
ï0.5
ï1
ï30
4
x 10
M
0.5
M
R(1)(t) + R(2)(t) + R(3)(t)
1 (c)
0
M
ï0.5
ï1
ï30
Figure 3.4 Optimisation of the dynamic response function, RM(t), for a single column in
a 100-year storm: (a) RM(1)(t); (b) RM(1)(t) + RM(2)(t); (c) RM(1)(t) + RM(2)(t) + RM(3)(t). - - positive dynamic response and _____ negative dynamic response
23
Table 3.2 Maximum values of (a) Forces, F(t); (b) Moments, M(t); and (c) Dynamic
response, RM(t), for a single vertical column of diameter D=12.2m subject to the 5-year
and 100-year storms defined in Table 1
(a)
Forces
(1)
(F )max
(1)
(2)
(F +F ) max
(1)
(2)
(3)
(F +F +F ) max
5-Year Storm
32.53kN
35.49kN
39.19kN
100-Year Storm
39.34kN
42.87kN
47.32kN
(b)
Moments
(1)
(M )max
(1)
(2)
(M +M )max
(1)
(2)
(3)
(M +M +M )max
5-Year Storm
2970kNm
3548kNm
4416kNm
100-Year Storm
3220kNm
3965kNm
5107kNm
(c)
Dynamic Response
(1)
(RM )max
(1)
(2)
(RM + RM )max
(1)
(2)
(3)
(RM + R M + R M )max
5-Year Storm
4128kNm
5593kNm
6539kNm
100-Year Storm
-4128kNm
-7033kNm
-11531kNm
In each of these cases separate optimisations were undertaken including all terms up to first-,
second-, and third-orders of the incident wave steepness, Ak. By comparing the magnitude of
the responses obtained, (F, M and RM) and by contrasting the wave events responsible for these
optimal responses, it is possible to gauge the importance of the various nonlinear effects.
Figure 3.3 describes the optimised moment; part (a) concerns the linear moment M(1), part (b)
the sum of the first- and second-order moments, M(1)+M(2), and part (c) all the moments up to
third-order, M(1)+M(2)+M(3). In the linear case (Figures 3.3(a)) the applied moment corresponds
to the integral of Morrison's inertia force and is shown to be symmetric about its maxima. The
wave group necessary to generate this corresponds to the summation of wave slopes at the focus
location (t=0) such that, in a linear sense, the maximum wave crest and the preceding wave
trough are of equal magnitude. The inclusion of second-order moments, indicated on Figure
3.3(b), disrupts this pattern. With the difference in the phasing of the first- and second-order
moments, the SRS-method optimises the phasing and amplitude of the wave components so that
although there is a small reduction in the maximum sum of the linear moments, this is more
than offset by the additional second-order moments. This produces a non-symmetric moment
record, with a 23% increase in the maximum moment relative to that predicted by linear theory
for the 100-year storm. The changes in the applied moments are further accentuated by the
inclusion of third-order terms, as indicated on Figure 3.3(c). For example, for the 100-year
storm the maximum moment is further increased, achieving a value 59% larger than that
predicted by linear theory. Full details of the maximum horizontal force, F(t), and the maximum
over-turning moment, M(t), in both the 5-year and the 100-year storms are given in Tables
3.2(a) and 3.2(b) respectively.
An explanation for the nonlinear change in the applied moment lies in the fact that part of the
second-order forcing and all of the third-order forcing (at least in terms of Newman's force
model) corresponds to point loads acting at the water surface. Although these may be relatively
small in magnitude, their associated moment-arm in relation to moments about the base of the
column is large. The importance of the nonlinear terms is also demonstrated by the change in
the time-history of the applied moments, M(t). Comparisons between Figures 3.3 (a) and 3.3(c)
highlight the nature of this change: the rapid reversal in the applied moment in the latter case
providing exactly the type of conditions in which excitation of the structure at it's natural
frequency might be expected to occur. Furthermore, there appears to be some evidence of a
24
high-frequency loading cycle, perhaps akin to the secondary-loading cycle discussed identified
by Grue et al. (1993), occurring soon after the maximum applied moment (Figure 3.3(c)).
The data presented on Figure 3.3 represent the results of a static analysis and cannot, therefore,
contribute directly to a study of nonlinear ringing. However, having established the importance
of the nonlinear loading, the results arising from the optimisation of the dynamic response
function, RM(t), are more easily explained. Figure 3.4 describes the optimised dynamic response,
at various orders, and the associated wave profiles; whilst Table 3.2(c) quantifies the maximum
value of the dynamic response functions, RM(t). These results all relate to the 100-year storm,
and provide data describing both the maximum positive dynamic response, corresponding to an
anti-clockwise moment, and the maximum negative response. In the linear case these responses
(Figure 3.4(a)), as well as the surface elevations responsible for them, are of equal magnitude
but opposite sign. However, with the inclusion of second- and third-order responses it becomes
unclear whether the maximum positive or the maximum negative response will be larger.
Accordingly, the SRS-method was run twice, to identify the optimal or maximum response in
each case. In this example, the inclusion of second-order terms leads to a slightly larger negative
response (Figure 3.4(b)); while the inclusion of third-order terms leads to a significantly larger
negative response (Figure 3.4(c)).
More importantly, the data presented on Figure 3.4 confirms that the inclusion of the nonlinear
terms provokes a significant increase in the dynamic response. With the frequency of the
nonlinear moments approaching the natural frequency of the structure, this result is not
unexpected. For example, the linearly predicted response (Figure 3.4(a)) has a maximum value
(see Table 3.2(c)) some 39% larger than the linearly predicted static moment (Figure 3.3(a)).
The time-history presented on Figure 3.4(a) suggests that the linear response is governed by two
frequencies: the first corresponds to the incident wave frequencies, is characterised by the peak
spectral frequency, wp, and is dominant; while the second is significantly smaller in amplitude
and corresponds to the natural frequency of the structure. This result is entirely expected since
the product of the input spectrum and the dynamic amplification factor, D(), highlights exactly
these two frequencies. In contrast, the magnitude of the increase in the dynamic response due to
the inclusion of the higher-order terms is surprisingly large, reflecting the significant changes to
the applied moments identified in Figure 3.3. For example, in the 100-year storm, the response
function calculated using the first- and second-order terms, RM(1)(t)+RM(2)(t) on Figure 3.4(b), is
very different to the linear predictions (Figure 3.4(a)). Most notably, the maximum value of the
response is 74% larger than the linearly predicted value and, although a significant component
of the response occurs at the incident wave frequencies, the dominant response is at the natural
frequency of the structure, wn. These changes are further apparent in Figure 3.4(c), where the
response based on the first three terms, RM(1)(t)+RM(2)(t)+RM(3)(t), leads to a 179% increase over
linear predictions, and is clearly dominated by dynamic effects arising at n.
The data generated by the SRS-method is further considered in Figures 3.5, 3.6 and 3.7 These
respectively concern the base shear, F, the over-turning moment, M, and the dynamic response
function, RM, in the 100-year storm; identifying the individual contributions arising at increasing
orders: first and second in the case of the second-order optimisations and first, second and third
in the third-order optimisations. In respect of the static calculations, F(t) and M(t) on Figures 3.5
and 3.6, it is interesting to note that although the linear loading remains dominant, the optimal
phasing of the wave components leads to the constructive super-position of the second- and
third-order loads. This explains the significant overall change in both F(t) and M(t).
Furthermore, it is also important to note that the second- and third-order static contributions are
of comparable size; indeed, in the case of the applied moments the third-order terms are larger
due to the moment-arm effects noted above.
25
40 (a)
F(t)
20
0
ï20
ï40
ï30
ï20
ï10
0
10
20
30
ï20
ï10
0
t
10
20
30
40 (b)
F(t)
20
0
ï20
ï40
ï30
Figure 3.5 Force components, F(t), acting on a single column in a100-year storm: (a)
Second-order optimisation, F(1)(t) + F(2)(t); (b) Third-order optimisation, F(1)(t) + F(2)(t) +
F(3)(t). _____ F(1)(t), - - - F(2)(t) - - - F(3)(t), _____ FTOTAL
6000
4000
(a)
M(t)
2000
0
ï2000
ï4000
ï6000
ï30
ï20
ï10
0
10
20
30
ï20
ï10
0
t
10
20
30
6000
4000
(b)
M(t)
2000
0
ï2000
ï4000
ï6000
ï30
Figure 3.6 Components of the over-turning moment, M(t), acting on a single column in
a 100-year storm: (a) Second-order optimisation, M(1)(t) + M(2)(t); (b) Third-order
optimisation, M(1)(t) + M(2)(t) + M(3)(t). _____ M(1)(t), - - - M(2)(t), - - - M(3)(t), _____ MTOTAL
26
4
x 10
1 (a)
RM(t)
0.5
0
ï0.5
ï1
ï30
ï20
ï10
0
10
20
30
ï20
ï10
0
t
10
20
30
4
x 10
1 (b)
RM(t)
0.5
0
ï0.5
ï1
ï30
Figure 3.7 Components of the dynamic response, RM(t), for a single column in a 100year storm: (a) Second-order optimisation, RM(1)(t)+RM(2)(t); (b)Third-order optimisation,
RM(1)(t)+RM(2)(t)+RM(3)(t). _____ RM(1)(t), - - - RM(1)(t), - - - RM(1)(t), _____ RMTOTAL.
With regard the dynamic response function, RM(t), Figure 3.7 confirms that the nonlinear
contributions are very significant: in the second-order optimisation, RM(2) is at least comparable
to RM(1); whilst in the third-order optimisation, RM(3), provides the largest single contribution.
The magnitude and frequency of various contributions arising in this latter case are further
considered in Figure 3.8. This provides separate descriptions of the applied moments, M(t), and
the dynamic response function, RM(t), arising at each order. These results demonstrate that the
first-order moments (Figure 3.8(a)) primarily generate a first-order response (Figure 3.8(b));
whereas the second-and third-order moments provoke a response at the natural frequency of the
structure, n. This is particularly true of the third-order moment which provides the largest
contribution to both the applied high-frequency moment and the resulting dynamic response at
wn.
The nature of this latter response, Figure 3.8, shows many of the characteristic features of a
transient ringing event:
(i)
No response until the occurrence of a near-impulsive loading event, involving the
rapid reversal of the applied moment.
(ii)
The rapid build-up of significant dynamic response, with energy transferred to the
natural frequency of the structure.
(iii)
A gradual decay, entirely dependent upon the damping of the structural system.
27
4000
6000
(a)
(b)
4000
2000
R(1)(t)
0
M
M(1)(t)
2000
0
ï2000
ï2000
ï4000
ï4000
ï30
ï20
ï10
0
10
20
ï6000
ï30
30
4000
ï20
ï10
0
10
20
30
ï20
ï10
0
10
20
30
ï20
ï10
0
t
10
20
30
6000
(c)
(d)
4000
2000
R(2)(t)
0
M
M(2)(t)
2000
0
ï2000
ï2000
ï4000
ï4000
ï30
ï20
ï10
0
10
20
ï6000
ï30
30
4000
6000
(e)
(f)
4000
2000
R(3)(t)
0
M
M(3)(t)
2000
0
ï2000
ï2000
ï4000
ï4000
ï30
ï20
ï10
0
t
10
20
ï6000
ï30
30
Figure 3.8 Contributions to the applied moment, M(t), and the resulting dynamic
response, RM(t), for a single column in a 100-year storm optimised for the third-order
dynamic response, RM(1)(t) + RM(2)(t) + RM(3)(t): (a) M(1)(t), (b) RM(1)(t), (c) M(2)(t), (d)
RM(2)(t), (e) M(3)(t), (f) RM(3)(t)
28
These results indicate that a ringing-type event can arise in a severe sea state and suggest that
it's occurrence, or at least our ability to predict it, is critically dependent on the description of
the nonlinear loads. Furthermore, the SRS-method also suggests that the wave event most likely
to provoke such a response, (t), is very different to that expected from linear theory.
3.3.2 Directionality in the incident wavefield
The results presented above all relate to uni-directional sea states. However, the formulation of
the SRS-method can be extended to incorporate the effects of directional spreading. Figure 3.9
concerns the same single column arrangement (D=12.2m on Figure 3.2) subject to the 100-year
storm (Table 3.1) and describes the optimised values of F(t), M(t) and RM(t) for three directional
spreads, corresponding to wrapped-normal distributions with standard deviations of =0° (or
uni-directional), 15° and 30°. In adopting these distributions the directional spread, a() in
Equation (3.7), is applied uniformly to all frequency components. Following the analysis of
field data gathered at the Tern platform in the northern North Sea, Jonathan & Taylor (1995)
found that severe storms were characterised by =30°. More recently, Ewans (1998) has
suggested that wave frequencies in the vicinity of the spectral peak, accounting for a large
proportion of the total wave energy, may be less directionally spread, or more uni-directional.
Accordingly, the values of adopted in Figure 3.9, 0° 30°, are appropriate to engineering
calculations. Furthermore, the results presented on this figure suggest that for the present single
column arrangement, representative values of the directional spreading have no practical impact
on the horizontal forces, F(t) on Figure 3.9(a); the over-turning moment, M(t) on Figure 3.9(b);
or the dynamic response, RM(t) on Figure 3.9(c). Indeed, with =30°. The maximum value of
the dynamic response, RMmax, is only reduced by 7% relative to the equivalent uni-directional
result.
3.3.3 Multiple-column calculations
To investigate the importance of the spatial layout of a multiple-column arrangement, two
configurations are considered. The first, indicated on Figure 3.10(a), considers uni-directional
waves incident to a two-column structure, where the center-to-center spacing of the column is l.
As far as the applied forcing and hence the dynamic response is concerned the relevant lengthscale is lcos, where is the angle separating the alignment of the columns and the direction of
wave propagation, since this determines the phase-lag between the forces applied at the
individual columns. In the second case, indicated on Figure 3.10(b), multi-directional waves
characterised by the standard deviation, , are normally incident, =90°, to two columns again
separated by a distance l. In this case the key issue is to see how the short-crestedness of the
incident wavefield alters the combined load on the two columns and how this varies with the
column spacing.
Although these two cases appear rather abstract, they provide information essential to the
description of the applied loads acting on all multiple-column structures. At this point it is
important to stress that the applied forces, F(t), and the over-turning moments, M(t), are
calculated using the existing force models (Section 2.1). These are appropriate to a singlecolumn structure and do not therefore include any of the interactions that may occur due to the
presence of an adjacent column. In applying the SRS method both columns are included within
the optimisation, with the phasing of the wave components adjusted to produce the maximum
total response. However, the forces or moments acting on an individual column are calculated as
if the other was not there.
29
50
F (t)
(a)
0
ï50
ï10
ï8
ï6
ï4
ï2
0
t
2
4
6
8
10
ï8
ï6
ï4
ï2
0
t
2
4
6
8
10
ï8
ï6
ï4
ï2
0
t
2
4
6
8
10
6000
(b)
4000
M (t)
2000
0
ï2000
ï4000
ï6000
ï10
5000
(c)
M
R (t)
0
ï5000
ï10000
ï10
Figure 3.9 The effect of directionality on the response of a single-column structure,
D=12.2m, subject to a 100-year storm: optimised values of (a) horizontal force, F(t); (b)
over-turning moment, M(t); (c) dynamic response, RM(t). _____ Uni-directional waves
=30°
=0, - - - =15°,
30
Figure 3.11 concerns the first case noted above (Figure 3.10(a)) and describes the variation in
the maximum dynamic response RMmax with the effective leg spacing, lcos. Within this figure
three curves are presented corresponding to the optimisation of RM(1), RM(1) + RM(2) and RM(1) +
RM(2) + RM(3). In all cases the negative response was found to be largest and consequently it is
the magnitude of this term that is presented on Figure 3.11. For lcos=0 the maximum
responses are twice those calculated for the single-column case (Section 3.3.1).With increasing
values of lcos, the magnitude of the maximum response reduces. However, this reduction is
not monotonic; at certain values of the leg spacing there is clearly some constructive
interference between the loads applied to individual legs leading to an increased response. This
is particularly noticeable with the inclusion of the second- and third-order responses; although
the mechanism responsible for it is identical in all three cases.
A rigorous explanation of these effects is provided in the complete report. However, to
summarise our findings it appears that that the gradual decline in the responses noted on Figure
3.11 is due to the wave-frequency response becoming progressively out of phase; whilst the
higher frequency oscillations are due to the natural frequency responses moving into (point A)
and out of (point B) phase; the latter effects becoming more significant with the inclusion of the
second- and third-order response terms, RM(2) and RM(3).
The second case considered on Figure 3.10(b) concerns the role of directionality and
specifically the variation in the response with increased column separation due to the short­
crestedness of a large wave group. Figure 3.12 concerns a large directional spread, =30°, and
describes the variation in the dynamic response occurring at increasing orders, RM(1), RM(1) +
RM(2) and RM(1) + RM(2) + RM(3). These results indicate that there is a gradual reduction in the
maximum response. However, the overall reduction is relatively limited: for example, a column
spacing of l=50m leads to a 19% reduction in the total response based on RM(1) + RM(2) + RM(3).
This is consistent with the fact that relative to typical column spacings, the crest length in even
highly directional sea states remains long; or, alternatively, the reduction in crest elevations
and/or wave steepness is relatively small over length-scales comparable to the column spacings.
Table 3.3: Maximum values of the horizontal forces, F, the over-turning moments, M,
and the response moment, RM, for the wave profiles indicated on Figure 3.13. Note:
Where two values are given for the Rainey solution, the first corresponds to the slender
body solutions (Raineyslender), which excludes the contribution from the surface
distortion force, FSD, while the second is the total solution (Raineytotal)
(1)
F
(2)
F
(3)
F
(4)
F
(1)
M
(2)
M
(3)
M
M
RM
Newman force model: Equations 2.102.13
BST
BST
Linear Input
(d /dx)max
max
(d /dx)max
39.3
37.8
36.9
7.1
8.95
9.73
4.0
7.6
9.77
43.6
42.8
41.1
3196
2857
2795
1036
1209
1330
698
1065
1368
1057
4160
4399
-6420
-8204
-9382
31
Rainey force model: Equations 2.14-2.18
Linear Input
(d /dx)max
44.1
2.18
1.1/6.1
45.6/48.6
4654
279
168/947
4940/5600
-6975/11000
BST
(d /dx)max
43
2.78
1.62/19.28
47.2/63.6
5182
371
254/3030
5784/8547
-9893/15516
BST
max
36.5
3.1
2.1/24.5
41.1/61.4
4502
425
328/3900
5203/8686
-11400/18800
(a)
(b)
Figure 3.10 Sketch of multiple-column layouts. (a) Uni-directional waves incident at an
angle to a 2-column arrangement. (b) Multi-directional waves incident to a 2-column
arrangement.
4
2.5
x 10
2
|RM|max
1.5
1
A
•
B
•
0.5
0
0
10
20
30
40
50
l cos r
60
70
80
90
100
Figure 3.11 The variation in the maximum dynamic response, |RM|max, with the effective
RM(1)+RM(2), - - - RM(1)+RM(2)+RM(3). Note: A and B
leg spacing lcos. ----- RM(1),
correspond to points of interest.
32
4
2.5
x 10
2
|R |
M max
1.5
1
0.5
0
0
5
10
15
20
25
l
30
35
40
45
50
Figure 3.12 Maximum dynamic response, |RM|max, in directional seas, =30°, with
varying leg spacing, _____ RM(1);
RM(1)+ RM(2); - - - RM(1) + RM(2) + RM(3)
3.3.4 Spectral evolution and accurate kinematic predictions
In the previous sections applications of the SRS-method have been based upon a linear
representation of the wavefield; with the applied loads and response functions calculated using
Newman's force model. In this section we will assess the importance of: (i) the local evolution
of the wave spectrum, first discussed in Section 2.3, and (ii) the additional nonlinear loads
arising from an improved representation of the nonlinear wave profile and the associated water
particle kinematics. In essence, both of these effects concern additional nonlinear effects; the
former leading to the unexpected growth of new freely propagating wave components, and the
latter contributing to a further improvement in the description of the wavefield. Within the
present calculations, the first effect can be incorporated within the Newman force model by
employing a fully nonlinear wave model to determine the spectrum of the freely propagating
wave components in the vicinity of a large wave event. In contrast, the second effect requires
the implementation of the nonlinear water particle kinematics within the Rainey (1995) force
model.
In undertaking these approaches, it is important to understand that the fully nonlinear wave
model (BST) is computationally demanding and cannot easily be implemented within the SRSmethod. As a result, no new optimisations are presented within this section. Rather, the earlier
optimisations based upon a linear description of the sea state are adopted, and the nonlinearities
noted above introduced. Although this approach is inevitably conservative in terms of
estimating the importance of the nonlinear terms 3, the results are instructive and in stark contrast
to Newman's original claim that a linear representation of a wavefield is adequate for nonlinear
force predictions.
Figure 3.13(a) concerns the 100-year wave event considered previously (Table 3.1) and
provides three alternative time-histories of the water surface elevation, (t). The smallest wave
record, having the lowest crest elevation, corresponds to a linear representation of the original
3
A fully nonlinear optimisation would ensure that all the force components arising from the nonlinear predictions of
the BST model come into phase at one point in space and time, thereby producing the maximum nonlinear response.
33
JONSWAP spectrum and defines the wave profile having the steepest wave slope, (d/dx)max.
The other two wave profiles are defined by the fully nonlinear wave model (BST). In both these
cases the input to the BST model was based upon the original JONSWAP spectrum; with the
wave profiles describing the steepest, (d/dx)max, and the highest, max, wave crests arising
within the computational domain. Within these calculations, the local and rapid evolution of the
wave spectrum in the vicinity of these large wave events, highlighted on Figure 3.13(b),
involves the movement of wave energy to the higher frequencies allowing higher and steeper
waves to evolve.
To investigate the importance of the nonlinear changes in these wave forms, the forces and
moments acting on the earlier single-column structure (D=12.2m) and the dynamic response
generated will be considered. Figure 3.14 concerns the two steepest wave forms, one linear and
the other nonlinear, and contrasts the predictions based upon both the Newman (1996) and the
Rainey (1995b) force models. In applying the Newman Model, Equations (2.10)-(2.13), all
calculations must be based upon a linear representation of the freely propagating wave
components. In the first case, denoted by Newmaninput, this is straightforward and simply based
on the original JONSWAP spectrum. In the second case, denoted by Newmanmodified, this must
include the local nonlinear evolution of the free wave spectrum which can either be defined
using the Zakharov (1968) Equation or can be approximated by considering the difference in the
spectra corresponding to the focussing of wave crests and wave troughs. This latter approach
isolates the odd-order wave interactions (see Equation (2.19)) which, given the magnitude of the
third-order bound wave components, provides a very good approximation of the modified freewave spectrum4. In contrast, the application of the Rainey (1995b) force model, Equations
(2.14)-(2.18), simply requires the output from the BST model in terms of the water surface
elevation and the underlying water particle kinematics associated with the extreme wave event.
However, it is perhaps important to distinguish between the two third-order forces, FSI in
Equation (2.16) and FSD in Equation (2.18), since the latter is no longer based on a slender body
approximation and may, given the wave conditions employed, suffer from convergence
difficulties. Accordingly, this force component is included within the total Rainey solution,
denoted by Raineytotal, but neglected in the slender body approximation, denoted by Raineyslender.
Figure 3.14(a) concerns the predicted horizontal forces, F(t). Comparisons between the two
Newman solutions (input and modified) suggests that although the modification of the spectrum
does not produce significant changes in the maximum and minimum loads, the transfer of
energy to the higher frequency components ensures that the reversal in the applied loads occurs
much more rapidly. Furthermore, comparisons with the Rainey solution suggests that the
additional nonlinear forces arising due to an improved description of the wave motion account
for a significant increase in the peak loading, with a further increase in the rate of load reversal.
Indeed, even if the surface distortion force, FSD in Equation (2.18), is removed (as is the case in
Raineyslender), the maximum applied load remains higher than that predicted by Newman; this is
despite the fact that the latter solution retains a term equivalent to a linear approximation of FSD.
Data describing the maximum forces provided by each of these models is predicted in Table 3.3.
Figure 3.14(b) concerns the same wave cases and contrasts the time-history of the applied
moment, M(t), predicted by each of the four solutions. Comparisons between these results
follow similar trends to those observed in the horizontal forces (Figure 3.14(a)). However, the
relative magnitude of the changes are somewhat larger, as indicated on Table 3.3. This reflects
the fact that due to moment-arm effects the applied moment is critically dependent upon forces
arising close to the instantaneous water surface. Unfortunately, the additional forces arising due
to both the high-frequency wave components (following the evolution of the underlying wave
4
Full details of this approach are given in Johannessen & Swan (2003) with quantitative comparisons to the
Zakharov Equation provided by Gibson & Swan (2005).
34
spectrum) and the uncertainty in the nonlinear water particle kinematics are a maximum at this
location. Accordingly, the maximum applied moment predicted by the Rainey model
(Raineytotal) is more than twice as large as that predicted by the original Newman solution
(Newmaninput). Furthermore, even if the FSD force is entirely discounted, the Rainey slender
body solution still predicts a 43% increase in the over-turning moment.
These changes are reflected in Figure 3.14(c) which describes the corresponding dynamic
response, RM(t). Within this figure two points merit particular attention. First, comparisons
between the two Newman solutions suggest that the nonlinear changes in the wave spectrum
lead to large excitations at the natural frequency, n. This is entirely consistent with the effect of
the dynamic amplification factor noted above. Second, the response predicted by the Rainey
model, both with and without the surface distortion force, FSD, is significantly larger (see Table
3.3) and is characterised by a typical bursting-type behaviour. Taken together these results
clearly emphasise the importance of the nonlinear contributions arising from both the local
evolution of the wave spectrum and an accurate representation of the water surface elevation
and the underlying water particle kinematics.
35
25
(a)
20
15
10
d (t)
5
0
ï5
ï10
ï15
ï20
ï20
ï15
ï10
ï5
0
t
5
10
15
20
3
(b)
2.5
a(k)
2
1.5
1
0.5
0
0
0.01
0.02
0.03
0.04
0.05
k
0.06
0.07
0.08
0.09
0.1
Figure 3.13 Alternative descriptions of extreme wave events arising in the 100-year
storm. (a) wave profiles, (t), based upon: _____ (d/dx)max linear theory; - - - (d/dx)max
BST theory,
max BST theory. (b) Wave spectra: _____ underlying linear
(JONSWAP) spectrum and - - - modified spectrum based on BST
36
80
(a)
60
F (t)
40
20
0
ï20
ï40
ï10
ï5
0
5
10
15
20
ï5
0
5
10
15
20
ï5
0
5
t
10
15
20
10000
(b)
M (t)
5000
0
ï5000
ï10
4
x 10
1 (c)
RM (t)
0.5
0
ï0.5
ï1
ï1.5
ï10
iF
g
re
u3
.1
4
Loads and responses arising in the two steepest wave cases, (d/dx)max
linear theory and (d/dx)max BST (Figure 3.12). (a) F(t), (b) M(t), and (c) RM(t).
_
Newmaninput; - - - Newmanmodified; _
Raineytotal;
Raineyslender
37
4. LABORATORY DATA: COMPARISONS WITH EXISTING
MODELS
The purpose of the present tests was to mount a scaled model structure in a three-dimensional
wave basin capable of simulating extreme wave events arising in realistic sea states. To measure
the applied loads and the dynamic response of the structure, and to compare this data with the
best-available force models based upon state-of-the-art descriptions of the underlying water
particle kinematics. To achieve the required data, the measuring programme was undertaken in
three stages:
1. The first stage concerned the generation of the desired wave conditions in the absence
of the structure. In these tests detailed measurements of the water surface elevation,
(t), combined with state-of-the-art wave modelling allowed the full characteristics of
the incident wavefield, including the underlying water particle kinematics, to be
specified.
2. In the second stage the model structure was located within the wavefield. Two structural
configurations were considered: a single vertical column and a multiple-column
arrangement, the latter representative of a typical GBS. In both cases the column, or
columns, were held rigid between two load cells. This arrangement allowed both the
total horizontal force, F(t), and the over-turning moment, M(t), to be measured in a wide
range of wave conditions.
3. In the third stage the column, or columns, were allowed to oscillate in the mean wave
direction about their lowermost point, thereby creating a single degree of freedom
system akin to that outlined in Section 2.3. With a laser displacement transducer located
above the water surface, the dynamic response of the structure could be measured in the
various wave conditions and related to the applied loading.
The majority of the data presented was recorded in a new three-dimensional wave basin located
in the Hydrodynamics Laboratory within the Department of Civil and Environmental
Engineering at Imperial College London. This facility has a plan area of 20m x 12m and
operates with a working depth of 1.5m. It is equipped with 56 individually controlled wave
paddles, each 0.35m wide and 0.7m deep. The paddles are bottom hinged with the hydrostatic
load on individual paddles being supported by a system of springs and pullies. The facility has
been specifically designed to produce a wide range of very high quality waves with excellent
directional control. Preliminary tests have confirmed that it is indeed capable of generating
frequency components within the range 0.3Hz f 3Hz, with directional spreads up to and
including ±60° (either side of the mean wave direction). On the opposite side to the wave
paddles, the wave energy is dissipated on a continuous beach with a parabolic shape which
extends to a depth of 0.5m beneath the still water level. Preliminary studies have confirmed that
the reflection coefficient is typically less than 5%.
Within this series of tests two model configurations were considered. The first concerned a
single surface-piercing column and the second a multiple column arrangement representative of
a typical GBS. In both cases the columns were of uniform diameter and were mounted on top of
a rigid box; the height and plan area of the latter being representative of a typical arrangement
of storage caissons. However, both earlier experimental observations (Swan et al., 1997) and the
nonlinear force models outlined in Section 2 suggest that both the nonlinear wave-structure
interactions and the nonlinear or high-frequency forcing is predominantly dependent upon
38
conditions at the water surface; particularly, the steepness of the incident waves, the column
diameter and, in the case of the multiple-column structure, the leg spacing. As a result, the fact
that we have adopted columns of constant cross-section and mounted them on top of some
scaled storage caissons is not of primary importance, provided the latter does not cause a
steepening of the incident wave profiles due to a local reduction in the water depth. On the
assumption that all of the present tests are conducted in deep-water, the storage caissons will not
influence the incident wave characteristics (see Swan et al., 1997) and the model scale is solely
determined by length-scales arising at the water surface. With a nominal length-scale of
ls=1:100, the diameter of the columns is 11.5m at full-scale (or 0.115m at model-scale) and the
corresponding time-scale is defined by the linear dispersion equation as ts=1:10.
Figure 4.1 provides a schematic representation of the model structure; part (a) describing the
single-column and part (b) the multiple-column arrangement. In both cases the horizontal forces
were recorded via two load cells. These were located at either end of the columns and connected
to rigid supports via a rod-end bearing at the bottom and a universal joint at the top. These
released the applied moment, ensuring that the load cells were subject to pure horizontal shear.
Under these conditions the horizontal forces could be measured with an accuracy of ± 0.05%.
In undertaking the dynamic tests, the bottom load cell was replaced by a rigid connection whilst
the top load cell and universal joint were replaced by a guide rail and springs allowing the
column (or columns) to rotate about the bottom bearing in the vertical plane defining the mean
wave direction. A laser displacement transducer, located above the water surface, allowed the
motion of the structure to be determined with an accuracy of ± 0.2%.
In addition to the load cells and displacement transducers, described above, detailed
observations of the water surface elevations were required both with and without the structure in
place. The majority of this data was gathered using resistance-type wave gauges having an
accuracy of ±0.5mm. However, at locations close to the model structure, the extreme water
surface elevations become highly localised and may also involve some degree of over-turning
and/or air entrainment. In these cases the wave gauge data was supplemented by a system of
quantitative flow visualisation, based on an image by image analysis from high definition video.
In the remainder of this Section the laboratory data is presented and comparisons made with the
existing modelling procedures. This is sub-divided into two parts: the first concerning the
single-column case and the second the multi-column case. In each sub-section the flow field
becomes progressively more complex dealing with regular waves, irregular or focussed waves,
the effects of directionality and the onset of dynamic response. The purpose being to work
towards realistic wave conditions; highlighting the importance of the nonlinear processes, the
extent to which they can be modelled, and their significance in terms of dynamic excitation.
4.1 SINGLE-COLUMN DATA
4.1.1 Forcing in uni-directional regular waves
This corresponds to the simplest possible flow conditions. With the column diameter fixed at
D=2a=0.115m and with incident wave periods in the range 0.8T1.4s, this corresponds
exactly to the flow conditions in which the FNV theory should be valid (A/<<1, a/<<1 and
A/a=0(1)). Furthermore, with regular incident waves the harmonic content of the applied
forcing can be readily identified via simple Fourier analysis. Within this flow regime a large
number of cases have been considered, with incident wave steepness varying from linear
(Ak<0.1) to near-breaking (Ak0.4). Given the importance of steep waves, three example cases
are presented herein, full details of which are given in Table 4.1
39
Table 4.1: Regular incident waves: Cases 1, 2 and 3 (T is the wave period, H is the
wave height, A is the wave amplitude or H/2, a is the column radius or D/2, is the
wavelength and k is the wavenumber or 2/).
Case 1
Case 2
Case 3
T
1.00
1.21
1.42
H
0.144
0.124
0.165
Ak
0.273
0.278
0.252
A/
0.044
0.044
0.040
a/
0.034
0.024
0.017
A/a
1.28
1.87
2.30
Figure 4.1: Layout of experimental apparatus.
Figures 4.2 and 4.3 concern wave case 1, 2 and 3 (Table 4.1) describing time-histories of the
horizontal force, F(t), and the over-turning moment, M(t). In each case these loads are
dominated by the linear contribution, but the occurrence of additional high-frequency loads is
also clear. In particular, there is a notable force component occurring soon after the passage of a
wave crest. This is also present in the applied moment, but its effect is amplified suggesting that
the forces responsible occur close to the water surface. This corresponds to what has previously
been referred to as a secondary loading cycle (Grue et al. (1993) and Chaplin et al. (1997)).
However, comparisons between these wave cases, suggest that the nonlinear contributions to
both F(t) and M(t), are surprisingly frequency dependent. Evidence of this is given by the fact
that although all three wave cases are steep, the nonlinear forces and moments arising in wave
case 2 are proportionately larger. This suggests that the relative size of the column, a/, is
critically important in determining the significance of these nonlinear loads. At this point it is
interesting to note that similar comments were expressed in relation to the magnitude of the
nonlinear wave scattering described by Sheikh et al. (2005).
Figure 4.4 concerns wave case 1 on Table 4.1 and contrasts both the total horizontal force and
the horizontal force components arising at increasing harmonics of the incident wave motion
(first, second and third) with the predictions of the theoretical models outlined in Section 2.1.
Several points are immediately apparent:
(i) Neither of the force models, FNV or Rainey (1995b), provide a good description of the
total horizontal forces acting on the column (Figure 4.4(a)).
40
20
(a)
F(N)
10
0
ï10
ï20
0
1
1.5
2
2.5
(b)
20
F(N)
0.5
0
ï20
0
1
1.5
2
2.5
3
3
3.5
3.5
(c)
20
F(N)
0.5
0
ï20
0
0.5
1
1.5
2
2.5
4
t(s)
Figure 4.2 Horizontal force, F(t), due to regular waves interacting with a single column:
(a) Wave case 1; (b) Wave case 2; (c) Wave case 3. _____ F(t) measured, - - - 1st
harmonic
0.5
1
1.5
2
(c)
M(Nm)
5
0
ï5
ï10
0
0.5
1
1.5
2
t(s)
Figure 4.3 Overturning moments, M(t), due to regular waves interacting with a single
column: (a) Wave case 1; (b) Wave case 2; (c) Wave case 3. _____ M(t) measured, - - 1st harmonic
41
20
(a)
F(N)
10
0
ï10
ï20
0
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
20
(b)
F1(N)
10
0
ï10
ï20
0
4
(c)
F2(N)
2
0
ï2
ï4
0
2
(d)
F3(N)
1
0
ï1
ï2
0
t(s)
Figure 4.4 Horizontal forces recorded on a single column, wave case
1. (a) Total force, F(t); (b) 1st-harmonic force, F1(t); (c) 2nd-harmonic force, F2(t); and
(d) 3rd-harmonic force, F3(t). _____ measured data, _____ FNV, _____ Rainey,
- - - Rainey-FSD, _____ Eatock Taylor & Hung
42
10
(a)
M(Nm)
5
0
ï5
ï10
0
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
0.5
1
1.5
2
2.5
10
(b)
M (Nm)
5
1
0
ï5
ï10
0
4
(c)
0
2
M (Nm)
2
ï2
ï4
0
(d)
0.5
0
3
M (Nm)
1
ï0.5
ï1
0
Time(s)
Figure 4.5 Over-turning moments recorded on a single column, wave case 1. (a) Total
moment, M(t); (b) 1st-harmonic moment, M1(t); (c) 2nd-harmonic moment, M2(t); (d)
3rd-harmonic moment, M3(t). _____ measured data, _____ FNV, _____ Rainey (1995),
- - - Rainey-FSD.
43
2
(a)
F2
F3
F
4
F
F(N)
1
0
5
ï1
ï2
0
0.5
1
1.5
2
2.5
3
3.5
4
1.5
(b)
M(Nm)
1
M
2
M
3
M
4
M
0.5
0
5
ï0.5
ï1
ï1.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Time(s)
Figure 4.6 Harmonic analysis of loads recorded on a single column, wave case 3. (a)
Horizontal forces, F2, F3, F4 and F5; (b) Over-turning moments, M2, M3, M4 and M5.
(ii) The first-harmonic loading, denoted by F1 on Figure 4.4(b), is dominant and in very good
agreement with the Morrison's inertia load.
(iii) The second-harmonic loading, denoted by F2 in Figure 4.4(c), is significantly smaller in
magnitude when compared with the prediction of either FNV or Rainey (1995b), where
the differences between the latter solutions merely reflect the order of the wave theory
applied; FNV being based on linear theory and Rainey (1995b) on a nonlinear wave
theory. In this case the significant over-prediction of the measured data is consistent with
the findings of Stansberg et al. (1995), and others, and is believed, in part, to be due to the
neglect of non-slender far-field diffraction effects. These additional terms can be
conveniently calculated, at least for a bottom mounted column, using the analytic model
proposed by Eatock Taylor & Hung (1987). This solution evaluates the force arising from
the quadratic potential, q, in terms of a surface integral; where some care is required to
ensure the proper convergence of the latter. The physical origins of this force are identical
to those discussed by Lighthill (1979), Equation (2.4) in Section 2.1, but its magnitude is
rather different except in cases involving very large water depths and incident waves of
very low frequency. Indeed, Eatock Taylor & Hung (1987) conclude that the
approximations outlined by Lighthill (1979) are unexpectedly restricted in their range of
applicability. However, even with the inclusion of this term, the second-order forces are
poorly predicted, both in terms of their magnitude and their phase.
(iv) The third-harmonic loading, denoted by F3 in Figure 4.4(d), is better predicted, but still
over-estimated by some 60%. In this figure the Rainey (1995b) predictions incorporate
both the surface intersection force, Equation (2.16), and the surface distortion force,
Equation (2.18). If the latter force is neglected on the grounds that it is based upon a
44
perturbation expansion which may become divergent in very steep waves, an argument
adopted by Chaplin et al. (1997) but not fully explained, the resulting solution denoted by
Rainey-FSD is in significantly better agreement with the measured forces. However, the
phasing of the measured third-order forces differs significantly from all of the predicted
solutions.
Overall, there is poor agreement between the measured and predicted forces arising at the
second- and third-harmonic of the incident wave. This view is reinforced by the fact that the
best available solutions presented on Figures 4.4(c) and 4.4(d) appear inconsistent. At a secondorder of wave steepness the inclusion of the second-harmonic forces arising due to the quadratic
potential, Fq, leads to some improvement in the description of the measured forces. However, at
third-order the neglect of the surface distortion force, FSD, also leads to an improved
representation of the third-harmonic forces. Unfortunately, the origins of these two force terms
are intrinsically linked; making it very difficult to justify the inclusion of one but not the other.
The loads arising in this wave case are also considered in Figure 4.5. This concerns the total
over-turning moment, M(t), and it's harmonic components (M1, M2 & M3) and, once again,
contrasts the laboratory data with the various model predictions. These comparisons confirm
that the comments (i)-(iv) noted above are equally applicable to the description of the applied
moment. Indeed, careful comparisons suggest that the predicted moments show even larger
departures from the measured data. This is consistent with the notion that the largest
discrepancies in the description of the higher-harmonic forces arise close to the water surface,
where both the wave motion and the wave-structure interaction produces its most nonlinear
contributions. In calculating the applied moment these discrepancies will be amplified due to
the moment-arm effects, leading to poorer overall predictions.
A partial explanation for the poor agreement between the measured and predicted loads lies in
two factors: first, the flow regime (expressed in terms of D/) in which we have chosen to work
and, second, the steepness of the incident waves. However, previous work (reviewed in detail in
the full report) suggests that it is in exactly these conditions that ringing is likely to occur. In
terms of the flow regime, the forces acting on a column of larger relative diameter are more
likely to be dominated by the second-order distributed forces, particularly those associated with
the quadratic potential, and hence the nonlinear forcing will be closer to the results of a secondorder diffraction solution. Likewise, since most diffraction solutions are based upon some form
of series solution, a reduction in wave steepness might (reasonably) be expected to produce
improved convergence. Whilst this is again valid for large column diameters, it does not appear
to be true for the present flow regime. Indeed, the present tests suggest that although a reduction
in the wave steepness leads to smaller nonlinear loading, the second- and third-harmonic
components show no tendency to converge to the theoretical predictions. This result, together
with the data presented on Figures 4.2 – 4.5, suggest that within the present flow regime the
physical mechanisms responsible for the generation of the nonlinear or high-frequency loading
are not well understood and not fully incorporated within the existing force models.
Further evidence of this is presented in Figure 4.6 This concerns wave case 1, superimposing
the second-, third-, fourth- and fifth-harmonic loads derived from a Fourier decomposition of
the measured data: part (a) describing the horizontal forcing, F(t), and part (b) the over-turning
moments, M(t). These results suggest that in terms of their magnitude, the third- and fourthharmonic forces are larger than the second; whilst the fifth- and second-harmonic forces are of
comparable size. Given that all the force models, with the notable exception of Rainey
(1995a,b), are based upon a formal perturbation expansion in which the magnitude of the
higher-harmonic forces are expected to reduce monotonically, this clearly raises concerns
regarding the convergence of the series expansions. However, in wave cases 2 and 3 similar
plots suggest that the ordering of the terms is more appropriately preserved. Nevertheless, the
45
third-, fourth- and fifth-harmonic loads still appear to be of comparable size. Overall, this
suggests that the high-frequency loading, particularly the applied moments, may be more
significant and more nonlinear than previously expected.
Table 4.2: Incident focussed wave groups (Note: Tp is the peak period, is the peak
enhancement factor and 1An is the linear amplitude sum which corresponds to the
linearly predicted crest elevation).
Case 4
Case 5
Case 6
Spectral
Shape
JONSWAP
JONSWAP
JONSWAP
Tp
An
1.2s
1.4s
1.4s
2.5
2.5
5.5
58mm
76mm
90mm
4.1.2 Forcing in uni-directional focussed wave groups
This section considers the forces acting on a single-column structure (D=0.115m) due to a
number of extreme wave events arising in uni-directional JONSWAP spectra. In terms of the
incident wave conditions, the only simplification concerns the absence of directionality. In all
other respects, these isolated wave events provide an accurate scaled representation of the
extreme waves most likely to provoke a ringing response (see Section 3). Table 4.2 describes
three extreme wave events (wave cases 4, 5 and 6) that will be considered in detail. However,
the results are representative of a much wider range of wave conditions that have been
considered.
Figure 4.7 concerns wave case 5 (Table 4.2). Although this is not a particularly large wave, it
corresponds to a steep wave event arising in a sea state in which significant wave scattering has
previously been observed; the importance of this latter effect being further considered in Section
5.0. Figure 4.7(a) describes the water surface elevation, (t), recorded at the focal position (x=0)
in the absence of the structure. This event is produced by the focussing of wave crests and
therefore represents the highest wave arising in this particular sea state. In effect, it corresponds
to a fully nonlinear simulation of a NewWave event (Tromans et al., 1991). More importantly, it
is representative of the optimised wave profiles, identified using the SRS-method (Section 3),
and shown to be responsible for the largest nonlinear loading and the maximum dynamic
response (Figures 3.3 and 3.4 respectively). Within this figure, the measured water surface
elevations are compared to a number of wave solutions. The first, corresponding to the smallest
wave profile, represents a linear prediction based upon the initial, or input, JONSWAP
spectrum. This solution, denoted by linearinput, only includes the f(Ai) terms identified in
Equation (2.19). As a result, it provides a poor description of the measured data.
Similarly, the second-order model based upon this linear input (0(a2k2)input) includes the secondorder bound waves, represented by g(AiAi) in Equation (2.19), but still under-estimates the
maximum crest elevation by some 30%. In contrast, the linear model based on the modified
spectrum denoted by linearmodified, includes both the underlying linear components, f(Ai), and the
third-order terms, hb(AiAjAk)+hr(AiAjAk). In the absence of additional information, but with
reliable evidence that the third-order bound waves (hb) are small (Gibson & Swan, 2005), all the
third-order terms are treated as freely propagating and the linear spectrum modified accordingly.
This leads to substantially better agreement, which is further improved by the inclusion of the
second-order bound waves 0(a2k2)modified.
In attempting to predict the applied loads, the two linear representations of the sea state, one
based upon the input spectrum and the other based upon the locally modified or evolved
spectrum, are particularly important. Unlike the earlier regular wave cases (Section 4.1.1),
46
Fourier analysis cannot be applied to determine the forces and moments arising at the various
harmonics of the incident wave motion. However, by contrasting the measured loads with the
linearly predicted loads, the latter being based on the integral of Morrison inertia term to the still
water level (SWL), it is possible to identify the total nonlinear contributions to the applied
loading. Although these may be relatively small in comparison to the total applied, it has
already been demonstrated in Section 3 that they are critically important to the onset of dynamic
response. This approach has been adopted in Figures 4.7(b) and 4.7(c) which respectively
concern the horizontal forces and the over-turning moments. Comparisons between the
measured and linearly predicted loads highlight the following points:
(i) Prior to the arrival of the large wave event, t<-2.5s, there is no evolution of the wave
spectrum and the nonlinear loading remains very small.
(ii) With the arrival of the large wave crest the spectrum rapidly evolves, leading to additional
linear components. Although the inclusion of these terms leads to an improved description
of the measured data, the nonlinear loads are also significant. Evidence of this is provided
by the difference between the measured loads and the linearly predicted values based on
the modified spectrum, denoted by linearmodified. In particular, there is clear evidence of the
secondary loading cycle, discussed earlier, involving a high-frequency load reversal
approximately 0.2s after the passage of the large wave crest. Furthermore, the relative
magnitude of the nonlinear components is larger in the applied moments, suggesting that
the origin of these forces lies high in the water column, close to the instantaneous water
surface.
(iii) After the passage of the wave group, t>2.5s, the small difference in the measured and
linearly predicted loads occurs at the dominant wave frequencies and represents the
arrival of unwanted wave reflections from the downstream boundary or beach. Evidence
to support this is provided in the water surface elevation, (t), presented in Figure 4.7(a).
Having demonstrated that the nonlinear loads are significant in the vicinity of a large wave
event, Figures 4.8 and 4.9 again concern wave case 5 (Table 4.2) and contrast the total
horizontal forces with the predictions of the Newman and Rainey force models respectively. In
accordance with previous discussions (Section 2.1), the Newman force model must be based
upon a linear representation of the freely propagating wave components. As a result, Figure
4.8(a) contrasts the measured forces with two applications of this model: the first based upon
the original input spectrum and the second the locally modified wave spectrum. Although the
latter provides a significantly better description of the measured data, important nonlinear force
components remain unpredicted. To further emphasize the importance of the local changes in
the wave spectrum, Figures 4.8(b), (c) and (d) concern the first-, second- and third-order forces
predicted by the Newman model; F1, F2 and F3 in Equations (2.10) – (2.13). These results are
important since they imply that although the local evolution of the wave spectrum leads to
changes in all of the force components, the largest proportional change occurs in the highfrequency forces, F3 in Figure 4.8(d).
Figure 4.9 provides a similar sequence of plots relating to the Rainey force model. In this case,
the two alternative sets of results are based on second-order wave models generated using the
input spectrum and the locally evolved wave spectrum; the corresponding surface elevations
being given on Figure 4.7(a). Furthermore, given the uncertainty in the convergence of the
solution procedure used to calculate the surface distortion force, FSD in Equation (2.18), the
Rainey calculations presented on Figure 4.9 have been undertaken both with and without FSD
included. Comparisons between the measured and predicted horizontal forces, Figure 4.9(a),
again demonstrate the importance of incorporating the local evolution of the wave spectrum.
However, important nonlinear forces remain unpredicted. Indeed, it is interesting to note that
47
the Rainey solution based on the modified second-order solutions provides a slightly poorer
description of the maximum applied force when compared to the predictions of the Newman
model, Figure 4.8(a). This arises because a second-order wave kinematics model will predict
smaller fluid velocities and accelerations when compared to a corresponding linear model. This
effect was fully investigated by Johanessen & Swan (2001) and the second-order model based
on a modified spectrum shown to be in good agreement with detailed kinematics measurements.
This suggests that the description of the maximum loads provided by the Newman model
(Figure 4.8(a)) may be a little fortuitous.
Figures 4.9(b)-(d) concern the individual Rainey forces; FI, FAD, FSI and FSD in Equations
(2.14)–(2.18). These results suggest that both the evolution of the underlying spectrum and the
accurate predictions of the nonlinear water particle kinematics are very important, particularly
in relation to the point loads acting on the water surface, FSI and FSD in Figure 4.9(d). In
considering these results it is important to stress that the Rainey force components are not
rigorously ordered in the sense that, for example, the inertia force FI in Figure 4.9(b) is
integrated to the instantaneous water surface and so includes both linear and nonlinear force
components. As a result, it is not possible to make direct comparisons between the individual
force components isolated on Figures 4.8(b)-(d) and those presented on Figures 4.9(b)-(d).
Nevertheless, the conclusions arising are entirely consistent.
Figures 4.10 and 4.11 provide a similar sequence of results contrasting the over-turning
moments; M(t), with the predictions of the Newman and Rainey models respectively. In both
cases the comments noted above concerning the horizontal forces, F(t), are equally appropriate;
the only difference being that near-surface effects, particularly the secondary loading cycle, are
relatively more significant due to moment-arm effects.
To further explore the extent of the nonlinear loads and, in particular, those components that
cannot be predicted, Figure 4.12 isolates the nonlinear loads by subtracting the linearly
predicted loads from the measured data, and contrasts these with the predictions of the various
force models. Parts (a) and (b) concern the horizontal forces; while parts (c) and (d) relate to the
over-turning moments. In parts (a) and (c) a single line is presented describing the measured
load (forces and moments) minus the linearly predicted load based upon the input spectrum,
where the latter is defined by the integral of Morrison's inertia term up to the still water level.
These records describe the true extent of the nonlinear loading: this is argued on the basis that
although they may include some linear load components, these arise due to the nonlinear
evolution of the wave spectrum. In effect, these records define the unexplained loads that would
be omitted in a typical design calculation.
In Figure 4.12(b) and (d) the nonlinear loads predicted by the Newman and Rainey models are
compared to the measured data minus the linear loads based on the modified spectrum. This
latter result is notably different, and smaller, than that presented in parts (a) and (c); the
difference corresponding to the linear loads arising due to the evolution of the spectrum.
Although these loads occur at high frequencies relative to the input spectrum5, and are therefore
significant in terms of dynamic response, our observations suggest that they are well modelled
by a linear solution provided the evolution of the spectrum can be predicted. However, even
with these terms removed, it is clear that the predicted nonlinear loads are in poor agreement
with the measured data. In particular, the true extent of the secondary loading cycle becomes
clear; the high-frequency load oscillations accounting for 15% of the maximum linearly
predicted force and 23% of the maximum linearly predicted moment, with no indication that
either of the solutions are able to model it. In both Figures 4.12(b) and (c) there is some
5
In deep-water the evolution of a wave spectrum in the vicinity of a large wave event principally involves the
movement of energy to higher frequencies.
48
evidence to suggest that the maximum and minimum nonlinear loads are closest to the Rainey
predictions excluding the surface distortion force, FSD. Further details concerning the maximum
nonlinear loads for this and several other wave cases are presented on Table 4.3.
It has already been noted that the comparisons provided on Figures 4.7 – 4.12 are representative
of many other wave cases. To demonstrate this Figures 4.13 and 4.14 isolate the nonlinear
loading records and provide a similar sequence of plots relating to waves cases 4 and 6; full
details of which are given in Table 4.2. In both cases the magnitude of the secondary loading
cycle is clearly defined; as is the inability of the existing force models to predict it. In wave case
4, on Figure 4.13, there is again some evidence that both the maximum nonlinear force (Figure
4.13(b)) and moment 4.13(d)) are closest to the solution based on the Rainey model excluding
the surface distortion term, FSD. However, this does not apply in wave case 6 on Figure 4.14. In
this example, which corresponds to the largest wave event investigated in the present tests, both
the maximum nonlinear force (Figure 4.14(b)) and moment (Figure 4.14(d)) are similar in
magnitude to the full Rainey solution (including the surface distortion force, FSD). Indeed, it is
interesting to note that in this case the maximum nonlinear force represents some 15% of the
total; whereas the maximum nonlinear moment accounts for more than 27%. Furthermore,
taking into account the large secondary loading cycle, it becomes clear that the passage of a
large wave event gives rise to significant nonlinear loading, the frequencies of which are
sufficient to raise concerns regarding the possible onset of dynamic response. Figure 4.15
considers exactly this case (wave case 6), describing the water surface elevation, (t), together
with the linear and nonlinear components of the horizontal force, F(t) and the over-turning
moment M(t); the latter records providing a clear indication of the magnitude and frequency of
the nonlinear loading. Similar details of several other cases are provided on Table 4.3.
4.1.3 Effects of directionality
The laboratory data presented thus far all relates to uni-directional incident waves. This is
significant for two reasons: (i) With all the wave components aligned in a single plane, a uni­
directional wave form has the largest inline wave front steepness; (ii) previous work, notably
Johannessen & Swan (2003) and Gibson and Swan (2005), has shown that the local evolution
of a wave spectrum is critically dependent on the underlying directionality (Section 2). To
investigate the importance of these effects, the wave loading was recorded in sea states
involving a range of directional spreads. Without exception these results confirm that the
nonlinear loading components, particularly those relating to the secondary loading cycle, are
strongly dependent upon the directionality of the sea state.
To demonstrate this effect, Figure 4.16 concerns wave case 4, providing a description of the
water surface elevation, (t), measured in the absence of the structure, the total horizontal force,
F(t), and the corresponding over-turning moment, M(t), and contrasts the uni-directional data
with three directionally spread wave cases corresponding to a wrapped-normal distribution with
a standard deviation of =10°, 20° and 30°. Comparisons between these cases confirm two
important effects. First, for a given input spectrum, an increase in the directional spread leads to
lower maximum crest elevations and hence reduced wave slopes. This is clearly demonstrated
on Figure 4.16(a) and is consistent with changes in the evolution of the wave spectrum (point
(ii) above). Second, both the horizontal forces and the over-turning moments, parts (b) and (c)
on Figure 4.16, confirm that an increase in the directional spread leads to reduced nonlinear
loading in the vicinity of the extreme wave event. This is particularly prominent in respect of
the secondary loading cycle which is clearly defined in the uni-directional data, but rapidly
diminishes with increasing directionality.
49
0.1
(a)
d(m)
0.05
0
ï0.05
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
20
(b)
F(N)
10
0
ï10
ï20
ï4
(c)
M(Nm)
5
0
ï5
ï4
t(s)
Figure 4.7 Loading on a single column in a focussed wave group, wave case 5. (a)
Surface elevation, (t); (b) Horizontal force, F(t); (c) Over-turning moment, M(t). _____
measured data, _____ linearinput, _____ linearmodified, _____ second-orderinput, _____ second
ordermodified
50
20
(a)
F(N)
10
0
ï10
ï20
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
20
(b)
F1(N)
10
0
ï10
ï20
ï4
5
F2(N)
(c)
0
ï5
ï4
4
(d)
F3(N)
2
0
ï2
ï4
ï4
t(s)
Figure 4.8 Horizontal forces acting on a single column: comparisons with the Newman
force model, wave case 5. (a) F(t), (b) F1(t), (c) F2(t), (d) F3(t). _____ measured data, _____
Newmaninput, _____ Newmanmodified
51
20
(a)
F(N)
10
0
ï10
ï20
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
20
(b)
F (N)
10
I
0
ï10
ï20
ï4
1
(c)
FAD(N)
0.5
0
ï0.5
ï1
ï1.5
ï4
4
(N)
0
SI
SD
2
F &F
(d)
ï2
ï4
ï4
t(s)
Figure 4.9 Horizontal forces acting on a single column: comparisons with the Rainey
force model, wave case 5. (a) F(t), (b) F1(t), (c) FAD(t), (d) FSI(t) & FSD(t). _____ measured
Raineyinput -FSD,
Raineymodified -FSD
data, _____ Rainey input, _____ Rainey modified,
52
10
(a)
M(Nm)
5
0
ï5
ï10
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
10
(b)
M1(Nm)
5
0
ï5
ï10
ï4
3
M2(Nm)
2
(c)
1
0
ï1
ï2
ï3
ï4
2
(d)
M3(Nm)
1
0
ï1
ï2
ï4
t(s)
Figure 4.10 Over-turning moments acting on a single column: comparisons with the
Newman force model, wave case 5. (a) M(t), (b) M1(t), (c) M2(t) (d) M3(t). _____
measured data, _____ Newmaninput, _____ Newmanmodified
53
10
(a)
M(Nm)
5
0
ï5
ï10
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
10
(b)
MI(Nm)
5
0
ï5
ï10
ï4
0.5
MAD(Nm)
(c)
0
ï0.5
ï1
ï4
MSI & MSD(Nm)
3
2
(d)
1
0
ï1
ï2
ï3
ï4
t(s)
Figure 4.11 Over-turning moments acting on a single column: comparisons with the
Rainey force model, wave case 5. (a) M(t), (b) MI(t), (c) MAD(t), (d) MSI(t) & MSD(t). _____
Rainey input -MSD,
measured data, _____ Raineyinput, _____, Raineymodified,
Raineymodified - MSD
54
4
(a)
F(N)
2
0
ï2
ï4
ï6
ï4
ï3
ï2
ï1
0
1
2
3
4
ï2
ï1
0
1
2
3
4
ï2
ï1
0
1
2
3
4
ï2
ï1
0
1
2
3
4
4
(b)
F(N)
2
0
ï2
ï4
ï6
ï4
ï3
4
(c)
M(Nm)
2
0
ï2
ï4
ï4
ï3
4
(d)
M(Nm)
2
0
ï2
ï4
ï4
ï3
t(s)
Figure 4.12 Nonlinear loads on a single column, wave case 5. (a) F-F1input, (b) Comparisons with the Newman and Rainey forces, (c) M-M1input, (d) Comparisons with the Newman and Rainey moments: _____ F-F1input and M-M1input;
_____
F-F1modified and M-M1modified; _____ Newman’s (F2+F3)modified and (M2 + M3)modified;
_____
Rainey - F1modified and Rainey - M1modified;
Rainey – FSD -F1modified and Rainey MSD - M1modified
55
10
(a)
F(N)
5
0
ï5
ï10
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
10
(b)
F(N)
5
0
ï5
ï10
ï4
M(Nm)
5
(c)
0
ï5
ï4
M(Nm)
5
(d)
0
ï5
ï4
t(s)
Figure 4.13 Nonlinear loads on a single column, wave case 5. (a) F-F1input, (b) Comparisons with the Newman and Rainey forces, (c) M-M1input, (d) Comparisons with the Newman and Rainey moments: _____ F-F1input and M-M1input;
_____
F-F1modified and M-M1modified; _____ Newman’s (F2+F3)modified and (M2 + M3)modified;
_____
Rainey - F1modified and Rainey - M1modified;
Rainey – FSD -F1modified and Rainey MSD - M1modified
56
20
(a)
F(N)
10
0
ï10
ï20
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï2
ï1
0
1
2
3
4
20
(b)
F(N)
10
0
ï10
ï20
ï4
15
M(Nm)
10
(c)
5
0
ï5
ï10
ï4
15
(d)
M(Nm)
10
5
0
ï5
ï10
ï4
ï3
t(s)
Figure 4.14 Nonlinear loads on a single column, wave case 6. (a) F-F1input, (b) Comparisons with the Newman and Rainey forces, (c) M-M1input, (d)
Comparisons with the Newman and Rainey moments: _____ F-F1input and M-M1input;
_____
F-F1modified and M-M1modified; _____ Newman’s (F2+F3)modified and (M2 + M3)modified;
_____
Rainey - F1modified and Rainey - M1modified;
Rainey – FSD -F1modified and Rainey MSD - M1modified
57
0.15
(a)
0.1
d(m)
0.05
0
ï0.05
ï0.1
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
20
(b)
F(N)
10
0
ï10
ï20
ï4
10
(c)
M(Nm)
5
0
ï5
ï10
ï4
t(s)
Figure 4.15 Comparison between linear and nonlinear loads recorded on a single
column, wave case 6. (a) (t), (b) F(t), (c) M(t): _____ linear solution, (t), F1(t) and M1(t);
_____
nonlinear solution, (t),(Fdata(t)-F1(t)) and (Mdata(t)-M1 (t))
58
To further investigate the reduction in the nonlinear loading, Figure 4.17 describes the timehistory of the nonlinear contributions to the total horizontal force. This is defined as the
measured force minus the linearly predicted force; where the latter is based upon the locally
modified spectrum and denoted by F1modified. All three wave cases (4, 5 and 6 on Table 4.2) are
considered and in each case the uni-directional results are compared with the three directionally
spread cases considered previously. To complement these results a similar set of comparisons
describing the nonlinear contributions to the total over-turning moment are provided in Figure
4.18. In both cases it is clear that the nonlinear loads are strongly dependent upon the directional
spread. For example, in wave case 6 the maximum force reversal associated with the nonlinear
loading cycle reduces by 42%; while the over-turning moment reduces by 35%. These
correspond to very large changes which, given the frequency at which they occur, may be
expected to significantly alter the dynamic response.
Table 4.3: Forces and moments on a single column in the focussed wave cases 4, 5
and 6
Fmax
Fmin
F1max
F1min
(F-F1)max
(F-F1)min
Mmax
Mmin
M1max
M1min
(M-M1)max
(M-M1)min
Wave Case 4
11.616
-13.207
10.705
-12.095
0.91129
-1.114
5.4288
-5.8519
4.5722
-5.3389
0.85663
-0.51304
Wave case 5
15.485
-16.288
13.549
-15.866
1.9363
-0.42126
6.8219
-6.69
5.3319
-6.5932
1.4901
0.09681
Wave case 6
18.933
-18.417
16.019
-17.142
2.9138
-1.2747
8.4054
-7.2181
6.1538
-6.7845
2.2517
-0.43354
4.1.4 Dynamic Response
To investigate the response of the single-column structure to the applied wave loading, two
cases were considered: Tn =0.28s and Tn=0.46s, where Tn defines the natural period of the
structure; the former corresponding to the fifth-harmonic of the incident waves (with Tp=1.4s)
and the latter the third-harmonic. In both cases the motion of the top of the structure (above the
water surface) was recorded by a laser displacement transducer, s(t), and the results compared to
calculations based upon the single-degree-of-freedom system outlined in Section 2.3. Figure
4.19 provides a representative set of results describing the response of the single-column
structure (Tn=0.46s) in regular waves (wave cases 1, 2 and 3 in Table 4.1). From these, and
other related cases, it is clear that the displacements predicted on the basis of the measured
moments, M(t), are in very good agreement with the measured data. These results confirm that
provided the applied loads can be accurately described, the dynamics of the structure can be
successfully predicted. In contrast, if either the FNV or the Rainey load models are employed,
their inability to describe the applied forces, particularly the high-frequency forces, leads to a
very poor description of the observed dynamics.
Similar conclusions may be drawn from the irregular wave cases. For example, Figure 4.20
concerns wave case 5 (Table 4.2) and presents data relating to both natural periods (Tn=0.28s
and 0.46s). In addition, the importance of directionality in the incident wave is considered; parts
(a) and (b) concerning uni-directional waves (=0) and parts (c) and (d) directionally spread
(=30°) wavefields. In each case (including many others not presented), calculations based
upon the measured moments, M(t), are in good agreement with the observations; whereas those
59
based upon either of the force models (Newman or Rainey) are not. However, it is interesting to
note that with the introduction of directionality, the success of the calculations based on the
force models is marginally improved. This is consistent with the reduction in the nonlinear or
high-frequency loads, due to lower inline wave front steepness, discussed in Section 4.1.3.
Nevertheless, it is clear from these results that the description of the nonlinear loads remains the
key issue in the successful prediction of the dynamic response. This is entirely consistent with
the findings of Section 3.
4.2 MULTIPLE COLUMN DATA
4.2.1 Forcing in uni-directional regular waves
The data present within this Section concerns the forces acting on the 3-column structure
described earlier. Comparisons between these results and the earlier single-column data (Section
4.1) allow the effects of the multiple-column layout to be addressed. Figure 4.21 begins this
process by comparing the total horizontal force, F(t), recorded on the 3-column structure with
the linear sum of the forces recorded on 3 separate columns, with appropriate phasing
introduced to account for their relative spatial locations. This data relates to wave case 2 (Table
4.1) and concerns the structure orientated with two legs in the upstream position. Part (a)
describes the individual force components from the single-column tests; while part (b) contrasts
their sum with the 3-column data. To complement this data, Figure 4.22 provides a similar
sequence of results relating to the applied moments. In both cases, the first evidence of the
secondary loading cycle generated on the upstream legs is well defined by the sum of the single
column data, suggesting that no significant leg interaction effects occur at this stage. However,
the delayed secondary loading cycle due to the flow about the downstream leg is less well
defined. Furthermore, the maximum applied loads, both forces and moments, appear to be
underestimated by the linear sum. This latter effect is almost certainly evidence of strong leg
interactions. However, before commenting on the practical significance of these results, it is
important to assess whether similar effects occur in more realistic simulations of extreme ocean
waves, or whether they simply arise due to the unrepresentative nature of the regular waves
employed. This point is further discussed in Section 4.2.2.
Setting aside any concerns regarding the validity of regular wave tests, several important points
can be noted. Figure 4.23 also concerns wave case 2 (Table 4.1), with part (a) contrasting the
total horizontal force, acting on all 3 columns with the predictions of the FNV and Rainey
models; where the latter solution is calculated both with and without the surface distortion force,
FSD. Fourier analysis of these records, both measured and predicted, allows the first-, secondand third-harmonic of the applied forces to be presented on parts (b), (c) and (d) respectively. A
similar set of plots concerning the total over-turning moment and a description of its component
parts is provided on Figure 4.24(a)-(d).
Taken together, these results are consistent with the earlier single column results in that they
suggest:
(i)
The main discrepancy in the description of both the total force and the total moment
(Figures 4.23(a) and 4.24(a)) lies in the high-frequency effects occurring after the
passage of the wave crest.
(ii)
The first-harmonic components (F1, and M1 in Figure 4.23(b) and 4.24(b)) provide
the dominant contribution to the total loads and are well described by Morison’s
inertia load.
60
(iii)
The phasing of the second-harmonic components (F2 and M2 in Figures 4.23(c) and
4.24(c)) is well described, but their magnitude is significantly over-predicted. This
is, at least in part, due to the neglect of non-slender body diffraction effects.
(iv)
The third-harmonic components (F3 and M3 in Figures 4.23(d) and 4.24(d)) are
poorly described, particularly in respect of their phasing.
A similar sequence of plots relating to wave cases 1 and 3 confirm the comments noted above;
the only difference being that in wave cases 1 and 3 the magnitude of the third-harmonic loads,
F3 and M3, is also very poorly predicted. Once again, these results are consistent with the single
column results presented in Section 4.1.1.
Table 4.4 Forces and moments: ratio of 3-column data to single column data.
F1
F2
F3
M1
M2
M3
Wave Case 1
2.61
2.97
0.36
2.62
2.54
0.59
Wave Case 2
2.87
2.01
1.67
2.97
2.10
1.54
Wave Case 3
2.91
2.92
1.70
3.04
2.96
1.43
Having compared the relative magnitude of the measured and predicted forces, it is also
instructive to consider the magnitude of the loading components measured on the 3-column
structure with those recorded in the earlier single column tests. This data, expressed as a ratio of
the 3-column data over the single-column data, is presented on Table 4.4. As expected, the
linear force components (F1 and M1) yield a ratio just less than 3. This reflects the fact that the
column spacing is small relative to the incident wavelength, so that the phase changes in the
linear loading applied at individual legs is small. Applying similar arguments it follows that the
relative phase changes associated with the second and, in particular, the third-harmonic loads
will be proportionately larger. As a result, the ratios relating to the higher harmonic loads
progressively reduce. The data presented on Table 4.4 confirms this and are consistent with the
calculations based on the SRS-method presented in Section 3.
The frequency content of the applied forcing, expressed as harmonics of the incident wave
motion, is further considered in Figure 4.25. This concerns wave case 1 (Table 4.1); with part
(a) providing a breakdown of the horizontal forces, F(t), and part (b) the over-turning moments,
M(t). Comparisons between these results and the earlier single-column data (Figures 4.6) again
highlights the reduced importance of the high-frequency loads for the multiple column structure.
In the single-column data (Figure 4.6) the third- and fourth-harmonic forces (F3 and F4) are
larger than the second (F2); while the fifth-harmonic forces (F5) were only marginally smaller
than the second. In contrast, the 3-column data (Figure 4.25) suggests that the second-harmonic
forces (F2) are dominant; with F3 and F5 being comparable in size, but only having an amplitude
23% of that of F2; while F4 is slightly larger having an amplitude 39% of that of F2. In all of the
regular wave cases considered, the 3-column data shows that the second-harmonic load
dominates both the forces and the moments, with the magnitude of the remaining terms being
considerably smaller.
61
0.1
(a)
d(m)
0.05
0
ï0.05
ï2
ï1.5
ï1
ï0.5
0
0.5
1
1.5
2
ï1.5
ï1
ï0.5
0
0.5
1
1.5
2
ï1.5
ï1
ï0.5
0
0.5
1
1.5
2
15
10
(b)
F(N)
5
0
ï5
ï10
ï15
ï2
(c)
M(Nm)
5
0
ï5
ï2
t(s)
Figure 4.16 Load variations due to the directional spread, wave case 4. (a) Surface
elevation, (t), (b) horizontal force, F(t), and (c) over-turning moment, M(t).
_____
uni-directional; _____ =10°, _____ =20°, _____ =30°
62
2
(a)
F(N)
1
0
ï1
ï2
ï3
ï2
ï1.5
ï1
ï0.5
0
0.5
1
1.5
2
ï1.5
ï1
ï0.5
0
0.5
1
1.5
2
ï1.5
ï1
ï0.5
0
0.5
1
1.5
2
4
3
(b)
F(N)
2
1
0
ï1
ï2
ï3
ï2
6
(c)
F(N)
4
2
0
ï2
ï4
ï2
t(s)
Figure 4.17 Nonlinear forces acting on a single column, variations with directional
spread. (a) wave case 4, (b) wave case 5, and (c) wave case 6. _____ Uni-directional,
_____
=10°, _____ =20°, _____ =30°
63
2
1.5
(a)
M(Nm)
1
0.5
0
ï0.5
ï1
ï1.5
ï2
ï1.5
ï1
ï0.5
0
0.5
1
1.5
2
ï1.5
ï1
ï0.5
0
0.5
1
1.5
2
ï1.5
ï1
ï0.5
0
0.5
1
1.5
2
3
(b)
M(Nm)
2
1
0
ï1
ï2
ï2
4
3
(c)
M(Nm)
2
1
0
ï1
ï2
ï2
t(s)
Figure 4.18 Nonlinear moments acting on a single column, variations with directional
spread. (a) wave case 4, (b) wave case 5, and (c) wave case 6. _____ Uni-directional,
_____
=10°, _____ =20°, _____ =30°
64
0.04
(a)
s(m)
0.02
0
ï0.02
ï0.04
ï0.06
0
0.5
1
1.5
2
2.5
0.04
(b)
s(m)
0.02
0
ï0.02
ï0.04
0
0.5
1
1.5
2
2.5
3
3
3.5
3.5
0.1
(c)
s(m)
0.05
0
ï0.05
ï0.1
0
0.5
1
1.5
2
2.5
4
t(s)
Figure 4.19 Dynamic response of a single-column in regular waves, Tn=0.46s. (a)
wave case 1, (b) wave case 2, (c) wave case 3 _____ measured data; - - - predictions
based on measured moment, M(t); _____ predictions based on FNV load model;
_____
predictions based on Rainey load model
65
0.01
s(m)
0.005
0
ï0.005
ï0.01
ï0.015
ï1
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0.04
s(m)
0.02
0
ï0.02
ï0.04
ï1
ï3
5
x 10
s(m)
0
ï5
ï10
ï1
0.02
s(m)
0
ï0.02
ï0.04
ï1
t(s)
Figure 4.20 Dynamic response of a single-column in a focussed wave group, wave
case 5. (a) Tn=0.28s, =0°; (b) Tn=0.46s, =0°; (c) Tn=0.28s, =30°; (d) Tn=0.46s,
=30°. _____ measured data; - - - predictions based on measured moment, M(t);
_____
predictions based on FNV load model; ______ predictions based on Rainey load
model
66
60
(a)
40
F(N)
20
0
ï20
ï40
ï60
0
0.5
1
1.5
0.5
1
1.5
2
2.5
3
3.5
2
2.5
3
3.5
60
(b)
40
F(N)
20
0
ï20
ï40
ï60
0
t(s)
Figure 4.21 Forces acting on the 3-column structure, comparisons with earlier singlecolumn data (wave case 2). (a) Single column data, (b) 3-column data. - - - F(t) on a
F(t) on two upstream columns; _
F(t) on single
single column (x=0); _
_
F(t) measured; _
F(t) based on superposition of 3 singledownstream column;
column records
30
(a)
20
M(Nm)
10
0
ï10
ï20
ï30
0
0.5
1
1.5
0.5
1
1.5
2
2.5
3
3.5
2
2.5
3
3.5
30
(b)
20
M(Nm)
10
0
ï10
ï20
ï30
0
t(s)
Figure 4.22 Moments acting on the 3-column structure, comparisons with earlier
single-column data (wave case 2). (a) Single column data, (b) 3-column data. - - - M(t)
M(t) on two upstream columns; _
M(t) on single
on a single column (x=0); _
_
_
M(t) measured;
M(t) based on superposition of 3 singledownstream column;
column records
67
4.2.2 Forcing in uni-directional focussed wave groups
This section considers the forces acting on a multiple-column structure subject to a number of
extreme wave events arising in a uni-directional JONSWAP spectrum. Much of the data relates
to the three-column structure, orientated with two legs in the upstream direction, subject to
wave case 5 (Table 4.2). However, unless otherwise stated, the conclusions are generally
applicable to the wide range of wave conditions considered. Figure 4.26 concerns the horizontal
forces, F(t), acting on the columns of the structure. Part (a) employs the earlier single-column
data (Section 4.1.2) to predict the forces acting on the two upstream legs and the single
downstream leg; while part (b) contrasts the sum of these forces with the measured data. To
complement these results, Figure 4.27 provides a similar set of comparisons relating to the over­
turning moment. In several respects these results are consistent with Figures 4.21 and 4.22.
Most notably, the first high-frequency loading cycle (occurring on the upstream columns) is
well predicted by the linear sum of two single column results, whilst the second is not.
However, there are also important differences, particularly in the amplification of the maximum
predicted loads. Although there is a small increase in the maximum force, appearing as a
slightly larger increase in the maximum moments, these changes are much smaller than those
observed in the earlier regular wave cases. This occurs because the disturbance of the flow field
between the legs of the structure due to the passage of the previous wave is significantly
reduced in a focussed wave group, principally because this wave is smaller in amplitude.
Indeed, the only significant disturbance occurs due to the passage of the largest wave crest and
this produces a modification of the next local force maxima; evidence of which is provided at
t=1.0s on Figures 4.26(a) and 4.27(b).
Figure 4.28 again considers wave case 5, comparing the measured loads with the linearly
predicted loads based on the integral of Morison's inertia load to the still water level; the latter
taking due account of the spatial layout of the columns. Figure 4.28(a) describes the water
surface elevation, (t), and contrasts the measured data with both linear and second-order
solutions based upon the input spectrum and the locally modified spectrum, details of which
have been discussed previously. These linear models (input and modified) provide the input to
the force and moment predictions presented on Figures 4.28(b) and (c) respectively. These
comparisons, which are similar in form to the single-column results presented on Figure 4.7,
confirm that whilst the local modification of the wave spectrum accounts for an important part
of the applied load, significant nonlinear loads are also present in the vicinity of the large wave
event.
Figure 4.29(a) concerns the same test case and contrasts the measured forces with the
predictions of the Newman model based upon both the linear input spectrum (referred to as
Newmaninput) and the locally modified spectrum (Newmanmodified). These results are interesting in
that the latter predictions are shown to be in good agreement with the measured data, despite the
complexity of the flow about a multiple-column structure. Indeed, these comparisons appear to
be at least as good, and perhaps better, than the equivalent single-column results presented on
Figure 4.8(a). An obvious explanation for this lies in the reduced importance of the nonlinear
loads. The latter parts of this figure ((b), (c), and (d)) concern the individual force components
(F1, F2, and F3 respectively) and confirm the importance of the local modification of the
underlying spectrum.
A similar sequence of results are presented on Figure 4.30 in which the measured data are
compared to predictions based on the Rainey model, with separate calculations undertaken with
and without the surface distortion force, FSD. Once again, the results based on the modified
spectrum are in good agreement with the measured data; the small under prediction of the
maximum force perhaps being due to the applications of a second-order kinematics model as
discussed previously. The remaining parts of the Figure ((b), (c), and (d)) concern the
68
description of the individual force components due to the Morison's inertia term, FI, the axial
divergence term, FAD, the surface intersection force, FSI, and the surface distortion force, FSD. In
contrasting these results with the earlier Newman solution (Figure 4.29) it is important to stress
that the individual force components are not directly comparable. Nevertheless, the earlier
comments concerning the importance of the local evolution of the wave spectrum are
reinforced. Furthermore, detailed consideration of the axial divergence force, FAD on Figure
4.30(c), confirms the importance of small changes in the phasing of the wave components as a
large wave group evolves. This accounts for the large differences immediately after t=0. Further
evidence of the apparent success of these models is provided in Figures 4.31 and 4.32 which
compare the measured moments with the Rainey and Newman models respectively.
The data relating to wave case 5 is further considered in Figure 4.33. This seeks to determine
the nonlinear components of the applied load, both F(t) and M(t), and establish the extent to
which they can be predicted. Figure 4.33(a) describes the difference between the total measured
horizontal force and the linearly predicted load based upon the initial input or underlying
JONSWAP spectrum. This record corresponds to the nonlinear force components that would be
typically omitted in a basic design calculation. In contrast, Figure 4.33(b) describes the
difference between the measured force and the linearly predicted force based on the locally
modified spectrum, and contrasts this record with the Newman and Rainey force models; where
the latter is calculated both with and without the surface distortion force, FSD. These results first
demonstrate that the additional ‘linear’ force components arising due to the modification of the
wave spectrum account for a significant proportion of the unexplained loads. Second, although
neither the Newman or the Rainey solutions account for the form of the high-frequency loads,
particularly that part which is referred to as the secondary loading cycle, the maximum
nonlinear force is well defined by the Rainey model based on slender body theory (excluding
the surface distortion force, FSD). These remarks are equally appropriate to Figures 4.33(c) and
4.33(d) which provide a similar set of comparisons describing the nonlinear contribution to the
over-turning moment, M(t).
Figure 4.34(a)-(d) provides an identical analysis of the nonlinear loads, forces and moments,
arising in wave case 6. This case corresponds to the largest wave group considered (Table 4.2)
and, once again, the local modification of the wave spectrum is shown to be very significant in
terms of the nonlinear load prediction. However, in this case both the maximum nonlinear
forces (Figure 4.34(b)) and moments (Figure 4.34(d)) are in closer agreement with the Newman
model or the total Rainey solution; the latter including the surface distortion force, FSD.
Although there clearly remains some ambiguity concerning which terms should be included
within these solutions, the overall agreement between the measured and predicted nonlinear
loads is surprisingly good. Indeed, it is perhaps better than one would expect on the basis of the
earlier single-column results. In large part this reflects the reduced importance of the secondary
loading cycle which, although clearly still present is much reduced in magnitude.
Figure 4.35 contrasts the linearly predicted loads, based on the locally modified wave spectrum,
with the errors in the nonlinear loads predicted by the various force models. Given that all of the
solutions correctly incorporate the linear loads, provided they are based on the modified
spectrum, the latter records correspond to the difference between the measured loads and the
total predicted. It is interesting to note that a comparison between these errors and the total
nonlinear loads (defined by the difference between the measured loads and the linearly
predicted loads) have maximum values that are surprisingly consistent. This implies that the
errors in the predicted nonlinearly loads are as large as the nonlinear loads themselves. This is
true for both the total base shear, F(t) on Figure 4.35(a) and (c), and the over-turning moment,
M(t) on Figures 4.35(b) and (d). These results suggest three important points:
69
100
(a)
F(N)
50
0
ï50
ï100
0
0.5
1
1.5
2
2.5
3
3.5
0.5
1
1.5
2
2.5
3
3.5
0.5
1
1.5
2
2.5
3
3.5
0.5
1
1.5
2
2.5
3
3.5
100
(b)
F1(N)
50
0
ï50
ï100
0
20
(c)
F (N)
10
2
0
ï10
ï20
0
4
(d)
F3(N)
2
0
ï2
ï4
0
t(s)
Figure 4.23 Horizontal forces recorded on a 3-column structure, wave case 2. (a) Total
force, F(t); (b) 1st-harmonic force, F1(t); (c) 2nd-harmonic force, F2(t); (d) 3rd-harmonic
force, F3(t). _____ measured data, _____ FNV, _____ Rainey (FSI+FSD), - - - Rainey (FSI)
70
40
(a)
M(Nm)
20
0
ï20
ï40
0
0.5
1
1.5
2
2.5
3
3.5
0.5
1
1.5
2
2.5
3
3.5
0.5
1
1.5
2
2.5
3
3.5
0.5
1
1.5
2
2.5
3
3.5
40
(b)
M1(Nm)
20
0
ï20
ï40
0
10
(c)
M2(Nm)
5
0
ï5
ï10
0
4
(d)
M3(Nm)
2
0
ï2
ï4
0
t(s)
Figure 4.24 Over-turning moments recorded on a 3-column structure, wave case 2. (a)
Total moment, M(t); (b) 1st-harmonic moment, M1(t); (c) 2nd-harmonic moment, M2(t);
(d) 3rd-harmonic moment, M3(t). _____ measured data, _____ FNV, _____ Rainey (FSI+FSD),
- - - Rainey (FSI)
71
1.5
1
F2
F
3
F4
F
(a)
F(N)
0.5
5
0
ï0.5
ï1
ï1.5
0
0.5
1
1.5
2
2.5
1.5
M(Nm)
1
M
2
M
3
M
4
M5
(b)
0.5
0
ï0.5
ï1
ï1.5
0
0.5
1
1.5
2
2.5
t(s)
Figure 4.25 Harmonic analysis of loads recorded on a 3-column structure, wave case
1. (a) Horizontal forces, F2, F3, F4 and F5; (b) Over-turning moments, M2, M3, M4 and M5
(i) In terms of the overall load predictions, both linear and nonlinear, the single most
important step is to account for the local evolution of the wave spectrum. Comparisons
with Figure 4.34 (parts (a) and (c)) confirm this.
(ii) In attempting to predict the detail of the nonlinear loads, none of the force models provide
convincing results. Indeed, one could question the validity of including them at all.
However, it does not necessarily follow that similar arguments apply to the description of
the dynamic response.
(iii) Overall, the slender body form of the Rainey force model (excluding the surface
distortion force, FSD) produces solutions having the smallest errors, but these still remain
significant.
4.2.3 The Effects of Directionality
Figure 4.36 concerns the horizontal forces, F(t), and the over-turning moments, M(t), recorded
in wave cases 5 and 6 (Table 4.2) and contrasts the earlier uni-directional data with three
directionally spread wave cases corresponding to =10°, 20° and 30°. In both cases it is clear
that the secondary loading cycle progressively reduces in amplitude with the increasing
directional spread. This is consistent with the earlier single-column results (Section 4.1.3)
72
40
(a)
F(N)
20
0
ï20
ï40
ï4
ï3
ï2
ï1
ï3
ï2
ï1
0
1
2
3
4
0
1
2
3
4
50
F(N)
(b)
0
ï50
ï4
t(s)
Figure 4.26 Forces acting on the 3-column structure, comparisons with earlier singlecolumn data (wave case 5). (a) Single column data, (b) 3-column data. _____ F(t) on a
single column (x=0); _____ F(t) on two upstream columns; - - - F(t) on single downstream
column; _____ F(t) measured; _____ F(t) based on superposition of 3 single-column
records.
15
10
(a)
M(Nm)
5
0
ï5
ï10
ï15
ï4
ï3
ï2
ï1
ï3
ï2
ï1
0
1
2
3
4
0
1
2
3
4
30
(b)
M(Nm)
20
10
0
ï10
ï20
ï4
t(s)
Figure 4.27 Moments acting on the 3-column structure, comparisons with earlier
single-column data (wave case 5). (a) Single column data, (b) 3-column data - - - M(t)
on a single column at x=0; _____ M(t) on two upstream columns; _____ M(t) on single
downstream column; _____ M(t) measured; _____ M(t) based on superposition of 3 singlecolumn records
73
d(m)
0.1
(a)
0.05
0
ï0.05
ï4
F(N)
50
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
(b)
0
ï50
ï4
30
(c)
M(Nm)
20
10
0
ï10
ï20
ï4
t(s)
Figure 4.28 Loading on a 3-column structure in a focussed wave group, wave case 5.
(a) Surface elevation, (t); (b) Horizontal force, F(t); (c) Over-turning moment, M(t).
_____
measured data, _____ linearinput, _____ linearmodified, _____ second-orderinput, _____ second
ordermodified
74
50
F(N)
(a)
0
ï50
ï4
40
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
(b)
F1(N)
20
0
ï20
ï40
ï4
10
(c)
F2(N)
5
0
ï5
ï10
ï4
10
(d)
F3(N)
5
0
ï5
ï10
ï4
t(s)
Figure 4.29 Horizontal forces acting on a 3-column structure: comparisons with the
Newman force model, wave case 5. (a) F(t), (b) F1(t), (c) F2(t), (d) F3(t). _____ measured
data, _____ Newmaninput, _____ Newmanmodified
75
F(N)
50
(a)
0
ï50
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
40
(b)
FI(N)
20
0
ï20
ï40
ï4
3
(c)
(N)
0
AD
1
F
2
ï1
ï2
ï4
10
FSI & FSD(N)
(d)
5
0
ï5
ï10
ï4
t(s)
Figure 4.30 Horizontal forces acting on a 3-column structure: comparisons with the
Rainey force model, wave case 5. (a) F(t), (b) FI(t), (c) FAD(t), (d) FSI(t) & FSD(t).
_____
measured data, _____ Raineyinput, _____ Raineymodified,
Raineyinput -FSD,
Raineymodified -FSD
76
30
(a)
M(Nm)
20
10
0
ï10
ï20
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
20
(b)
M1(Nm)
10
0
ï10
ï20
ï4
10
(c)
2
M (Nm)
5
0
ï5
ï4
5
0
3
M (Nm)
(d)
ï5
ï4
t(s)
Figure 4.31 Over-turning moments acting on a 3-column structure: comparisons with
the Newman force model, wave case 5. (a) M(t), (b) M1(t), (c) M2(t) (d) M3(t).
_____
measured data, _____ Newmaninput, _____ Newmanmodified
77
Further evidence of this effect is provided in Figure 4.37 which concerns the same data (F(t)
and M(t)) for wave cases 5 and 6, but contrasts the variation in the nonlinear loads with the
directional spread. These results confirm that the rapid oscillation of the loads immediately
following the passage of the wave crest (t=0) is significantly reduced. Since this is important for
the onset of dynamic response, it is clear that the directionality of the incident wavefield plays a
key role.
4.2.4 Dynamic Response
Figure 4.38 concerns the dynamics of the 3-column structure with natural periods Tn=0.28s and
0.56s in the regular wave cases 2 and 3 described in Table 4.1. In all four examples the
measured dynamics are shown to be in very good agreement with the predictions of the
dynamics model (Section 2) based upon both the moments measured on the 3-column structure
(in locked-down mode) and the moments resulting from the sum of the earlier single-column
data, taking due account of the layout of the structure. In contrast, the dynamics based upon the
moments predicted by the Newman and Rainey models is much less convincing, particularly for
the higher natural periods; Tn =0.56s on Figures 4.38(b) and (d).
Similar results concerning the nonlinear focussed wave group, wave case 5 on Table 4.2, are
presented on Figure 4.39. Two natural periods are considered (Tn =0.28s and 0.46s) and the
importance of directionality is addressed: parts (a) and (b) concerning uni-directional waves
(=0) and parts (c) and (d) directionally spread (=30°) wavefields. Overall, these results are
in broad agreement with the comments noted above. However, it is interesting to note that the
maximum positive displacements are reasonably well predicted by the Newman and Rainey
models, whilst the maximum negative displacements are not. However, with the introduction of
directionality, the latter is reduced and typically in better agreement with the predictions. It is
also interesting to note that whilst the predictions based upon the measured moment are always
in good agreement with the measured displacements, those based on the sum of the singlecolumn moments are not. This is particularly apparent with the larger natural period, Tn =0.46s.
An explanation for this lies in the development of additional inter-column effects. However, it is
significant that these effects do not arise until some time after the passage of the largest wave
crest. As a result, they do not effect either the maximum positive or negative displacements.
78
30
(a)
M(Nm)
20
10
0
ï10
ï20
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
20
(b)
MI(Nm)
10
0
ï10
ï20
ï4
2
MAD(Nm)
(c)
1
0
ï1
ï2
ï4
MSI & MSD(Nm)
6
4
(d)
2
0
ï2
ï4
ï6
ï4
t(s)
Figure 4.32 Over-turning moments acting on a 3-column structure: comparisons with
the Rainey force model, wave case 5. (a) M(t), (b) M1(t), (c) MAD(t), (d) MSI(t) & MSD(t).
_____
measured data, _____ Raineyinput, _____ Raineymodified,
Rainey input -MSD,
Rainey modified -MSD
79
20
(a)
F(N)
10
0
ï10
ï20
ï4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï3
ï2
ï1
0
1
2
3
4
ï2
ï1
0
1
2
3
4
20
(b)
F(N)
10
0
ï10
ï20
ï4
10
(c)
M(Nm)
5
0
ï5
ï10
ï4
15
(d)
M(Nm)
10
5
0
ï5
ï10
ï4
ï3
t(s)
Figure 4.33 Nonlinear loads on a 3-column structure, wave case 5. (a) F-F1input, (b)
Comparisons with the Newman and Rainey forces, (c) M-M1input, (d) Comparisons with
the Newman and Rainey moments. _____ F-F1modified and M-M1modified; _____ Newman’s
(F2+F3) modified and (M2+M3)modified; _____ Rainey-F1modified and Rainey-M1modified,
Rainey- FSD -F1modified and Rainey- MSD -M1modified
80
20
(a)
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(c)
5
0
ï5
ï10
ï4
15
(d)
M(Nm)
10
5
0
ï5
ï10
ï4
ï3
t(s)
Figure 4.34 Nonlinear loads on a 3-column structure, wave case 6. (a) F-F1input, (b)
Comparisons with the Newman and Rainey forces, (c) M-M1input, (d) Comparisons with
the Newman and Rainey moments. _____ F-F1modified, and M-M1modified; _____ (F2+F3) modified
Rainey- FSD-F1modified
and (M2+M3)modified; _____ Rainey- F1modified and Rainey- M1modified,
and Rainey- MSD -M1modified
81
40
(a)
F(N)
20
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ï20
ï40
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(b)
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50
F(N)
(c)
0
ï50
ï2
ï1.5
20
(d)
M(Nm)
10
0
ï10
ï20
ï2
ï1.5
t(s)
Figure 4.35 Comparison between linear and nonlinear loads recorded on a 3-column
structure. (a) F(t), wave case 5; (b) M(t), wave case 5; (c) F(t), wave case 6; (d) M(t),
F(t)wave case 6. _____ F1(t); _____ F(t)-F1(t); _____ F(t)-FNewman(t); _____ F(t)-FRainey;
(FRainey(t)-FSD); with similar descriptions of the moments
82
50
F(N)
(a)
0
ï50
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60
40
(c)
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ï20
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ï60
ï2
30
M(Nm)
20
(d)
10
0
ï10
ï20
ï30
ï2
t(s)
Figure 4.36 Load variations due to directional spread. (a) F(t), wave case 5, (b) M(t),
wave case 5 (c) F(t), wave case 6; (d) M(t), wave case 6. _____ uni-directional, =0°;
_____
=10°; _____ =20°; _____ =30°
83
15
10
(a)
F(N)
5
0
ï5
ï10
ï2
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M(Nm)
(b)
5
0
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20
(c)
F(N)
10
0
ï10
ï2
15
(d)
M(Nm)
10
5
0
ï5
ï2
t(s)
Figure 4.37 Nonlinear load, F(t)-F1(t), variations due to directional spread. (a) F(t),
wave case 5, (b) M(t), wave case 5 (c) F(t), wave case 6; (d) M(t), wave case 6.
_____
uni-directional, =0°; _____ =10°; _____ =20°; _____ =30°
84
0.01
(a)
s(m)
0.005
0
ï0.005
ï0.01
0
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1
1.5
2
2.5
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3.5
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1
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(b)
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0.005
0
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0
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1
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1
1.5
2
0.04
(d)
s(m)
0.02
0
ï0.02
ï0.04
ï0.06
0
t(s)
Figure 4.38 Dynamic response of a 3-column structure in regular waves. (a) Tn=0.28s,
wave case 2; (b) Tn=0.46s, wave case 2; (c) Tn=0.28s, wave case 3; (d) Tn=0.46s,
wave case 3. _____ measured; - - - predicted from M(t); _____ predicted from singlecolumn data; _____ predicted from FNV; _____ predicted from Rainey
85
0.01
(a)
s(m)
0.005
0
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5
x 10
s(m)
(c)
0
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(d)
s(m)
0.01
0
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ï1
t(s)
Fig
re
u4
.3
9
Dynamic response of a 3-column structure in focussed waves, wave case
5. (a) Tn=0.28s, =0°; (b) Tn=0.46s, =0°; (c) Tn=0.28s, =30°; (d) Tn=0.46s, =30°
_
measured; - - - predicted from M(t); _
predicted from single-column data; _
_
predicted from Rainey
predicted from Newman;
86
5.0 NONLINEAR WAVE SCATTERING AND RELATED ISSUES
5.1 BACKGROUND: EARLIER OBSERVATIONS OF WAVE SCATTERING
Recent work undertaken at Imperial College (funded under EPSRC grant number GR/M9807)
involved detailed laboratory observations of the flow about a single surface-piercing column.
For a column of D=11.5cm, very significant high-frequency wave scattering was observed with
an incident regular wave train having a period of T=1.0s. For this case two distinct types of
wave scattering were observed. The first, hereafter referred to as Type 1 waves, is clearly
identified on Figure 5.1. In this example the occurrence of an incident wave crest at the column
leads to large run-up and wash-down on its front face. When the fluid washes down it creates a
disturbance on the water surface which propagates radially outwards, maintaining a
characteristic concentric wave pattern. Half a wave cycle later, an incident wave trough is
located at the column and the negative fluid velocities (opposing the direction of wave
propagation) lead to reduced run-up and wash-down on its back face. At this phase of the
incident wave cycle a second Type 1 wave (of reduced amplitude) is scattered radically
outwards from the downstream face. In a regular wave train this process is repeated with two
Type 1 waves, one from the upstream face and one from the downstream face, being scattered
during each incident wave cycle.
The second type of wave scattering, hereafter referred to as Type 2 waves, is clearly indicated
on Figure 5.2. In this figure the incident waves advance vertically downwards from the top of
each image. Visual images of this kind, together with measured data, have confirmed that the
generation and scattering of Type 2 waves is associated with the motion of fluid around the
surface of the column. As a wave crest moves past, the local disturbance caused by the presence
of the column produces increased water surface elevations on its surface. These advance around
both sides of the column and merge to form a noticeable mound of water at the back face
immediately following the passage of the wave crest. These disturbances then pass through one
another and continue propagating round the surface of the column. At this stage they are now
propagating in a direction opposite to that of the wave motion, and since this occurs close to the
position of a wave crest, the moving disturbance is subject to adverse wave-induced velocities.
This causes the symmetric pattern of disturbances moving in opposite directions around the
surface of the column (one clockwise and the other anti-clockwise) to be significantly
steepened, and for the tail of the disturbance to begin moving away in a spiral fashion. This
process is further enhanced as the incident wave moves on and the counter-rotating disturbances
are subject to favourable wave-induced fluid velocities (negative with respect to the direction of
propagation of the incident waves) as the wave trough rapidly approaches. At some point,
around the position of the wave trough, the disturbances are swept away from the column to
produce a pair of symmetric, but non-concentric, wave fronts which then develop radially and
merge with one another along the centreline of the column.
The first important point to note about these Type 2 waves is that their height is dependent upon
the steepness of the incident waves. Although this is also true of the Type 1 waves, wideranging measurements have confirmed that the individual disturbances associated with each
symmetric Type 2 wave is significantly larger than that associated with each Type 1 wave.
Furthermore, the surface elevation arising immediately upstream of the column becomes even
larger because it involves the merging of the adjacent Type 2 waves. This point is clearly noted
on Figure 5.2.
87
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.1 A perspective view of Type 1 wave-scattering from the upstream side of the
column. The incident wave crest is moving from the bottom to the top of each image;
phases of the incident wave cycle are highlighted by dashed white lines which denote
the position of an incident wave crest. The time increment between images is t=0.08s
88
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.2 A plan view of Type 2 wave-scattering from the upstream side of the
column. The incident wave crest is moving from the top to the bottom of each image;
phases of the incident wave cycle are highlighted by dashed white lines and the time
increment between images is t=0.08s
89
(a)
(b)
Figure 5.3 Nonlinear interaction between an incident wave and Type 2 scattered wave
arising upstream of the column. (a) Video image and (b) Sampled data
A second important point to note is that the time taken for the initial disturbance, to move
around the column is largely dependent upon the size of the column and not the characteristics
of the incident waves. As a result, if the initial disturbance is assumed to be associated with the
passage of a wave crest, the phasing of the incident wave cycle at which the evolving pair of
Type 2 waves has propagated around the surface of the column and begins the process of
scattering in the upstream direction is largely dependent upon the incident wave period. If this
wave period is large, the Type 2 waves will be scattered well before the arrival of the incident
wave trough. As the Type 2 waves move away from the column, the amplitude of the motion (or
the size of the disturbance) reduces rapidly. This is the case with all radially propagating
disturbances. As a result, the opportunity for these waves to interact nonlinearly with the next
incident wave crest is much reduced. Furthermore, even if such an interaction does take place, it
is sufficiently removed from the centreline of the column that it lies outside the foot print of any
over-hanging structure.
90
Figure 5.5 Wave gague layout. (a) Single-column structure and (b) Multiple-column
structure
91
In contrast, if the incident waves are such that the Type 2 waves are scattered in the upstream
direction in the vicinity of a wave trough, they will not have propagated very far nor reduced
significantly in height before they begin to interact nonlinearly with the next incident wave
crest. In this case, significant increases in the maximum water surface elevation will arise close
to the surface of the column. This provides the potential for wave impacts beneath the over­
hanging deck.
This physical explanation provides the first clear understanding as to why the physical model
tests reported by Swan et al. (1997) showed that both the steepness and the period of the
incident waves was critical in determining the extent of the wave-structure interaction, or at
least the subsequent wave-wave interaction. This is particularly important since it is the latter
aspect that provides the increased water surface elevations necessary for wave impacts to occur.
Evidence of this effect is given in Figure 5.3. This provides a video image, Figure 5.3(a), of
two adjacent Type 2 waves, generated from opposite sides of the column, converging on the
upstream radius = and interacting nonlinearly with the next incident wave. The resulting
increase in the water surface elevation, , is presented as a contour plot on Figure 5.3(b),
corresponding to exactly the same time or phase of the wave cycle. Further discussion of these
important events, including the wave conditions in which they are likely to arise is given in
Sheikh et al. (2005).
5.2 HIGH FREQUENCY WAVE LOADING: A PHYSICAL EXPLANATION
Given the poor agreement between the observed and predicted high-frequency loading
described in Section 4, it would seem logical to isolate these loads and determine if they are in
anyway related to the unexpected high-frequency scattering described in Section 5.1. As a first
step in achieving this, it is instructive to consider the measured time-history of the total
horizontal force, F(t), and subtract from it that part of the force that can be confidently
predicted, namely the first-harmonic force corresponding to the integral of the Morrison's inertia
term from the base of the column to the still water level (z=0). Figure 5.4 concerns wave case 1,
defined on Table 4.1, and adopts exactly this approach. In addition to the time-history of the
high-frequency forces defined on Figure 5.4(b), the corresponding water surface elevation, (t),
is given in Figure 5.4(a). Comparisons between these figures allow the phasing of the various
force reversals relative to the incident wave motion to be defined.
Rather than representing the high-frequency forces as a harmonic series based on the frequency
of the incident waves, as was the case in Section 4.1, the present approach is to retain the entire
high-frequency force trace and determine whether any of its key features correlate with the
characteristics of the scattered wavefield. The motivation for adopting this approach arises from
the nature of the Type 2 waves; specifically, the fact that the phasing of these wave components
is dependent upon the time taken for the fluid to move around the circumference of the column.
Since this is not directly related to the period of the incident waves, a harmonic analysis based
solely on the latter will give misleading or, at best, incomplete results.
Points (A)-(F) on Figure 5.4(b) correspond to the local maxima and minima in the measured
high-frequency forcing. Taking each of these points in turn, (A)-(F), it is possible to explain the
physical origins of the high-frequency forcing in terms of the scattered wavefield:
Point (A):
This point corresponds to the flow field noted on Figure 5.2(a). With the run-up and wash-down
of fluid on the front face of the column, the upstream propagation of Type 1 waves leads to a
positive (downstream) force. At this phase of the wave cycle, and several others noted below,
the generation of waves propagating away from the column implies a force applied to the fluid
92
in the direction of wave propagation. In effect, the column is acting as a wave maker, albeit a
static one. As a result, the column experiences an equal and opposite force, in this case in the
downstream direction.
Point (B):
This point corresponds to the flow field noted on Figure 5.2(b). With the passage of a wave
crest fluid driven around the surface of the column, in opposite directions, merges at the
downstream side of the column creating a mound of water. The increased pressures associated
with this build-up leads to a negative (upstream) force.
Point (C):
This point corresponds to the flow field noted on Figure 5.2(d). The tangential movement of
fluid around the circumference of the column, both in the clockwise and anti-clockwise
direction, propagates through the downstream mound (Point (B) above) and continues to move
back towards the upstream face of the column. When these adjacent bodies of fluid collide, at
the upstream face, Type 2 wave scattering is initiated in the upstream direction. For the reasons
noted above, the generation of these waves leads to positive (downstream) forces acting on the
column.
Point (D):
This point occurs half a wave period after point (A) and corresponds to the scattering of Type 1
waves in the downstream direction from the downstream face of the column. This scattering is
produced by the run-up and wash-down on the downstream face of the column due to the
negative fluid velocities arising in the wave trough. Since the magnitude of these velocities is
smaller than those rising in the wave crest, this second Type 1 scattered wave is less pronounced
than that occurring in the opposite direction half a wave cycle earlier. As a result, this wave
scattering leads to negative (upstream) forces that are reduced in magnitude relative to those
occurring at point (A).
Point (E):
This point corresponds to the reversal, or mirror image, of point B: the reduced fluid velocities
arising in the wave trough drive reduced volumes of fluid around the circumference of the
column to form a smaller mound of water on the front face of the cylinder. This leads to positive
(downstream) forces that are reduced in magnitude relative to the negative forces arising at
point (B).
Point (F):
The movement of fluid around the circumference of the column eventually leads to the
scattering of Type 2 waves, again of reduced magnitude, from the back face of the cylinder in
the downstream direction. This corresponds to the mirror image of point (C); leading to negative
(upstream) forces of reduced magnitude.
The correspondence between the applied high-frequency forces, F(t)-F1(t), and the unexpected
nonlinear wave scattering confirms that these are indeed two facits of the same problem. Further
details concerning this important result are given in the full report and in Swan et al. (2004).
5.3 WAVE SLAMMING
The occurrence of wave slamming, or the impulsive-type loading associated with the impact of
the water surface on a structure, is important for a number of reasons:
(i) The nature of the applied loading is such that it is highly localised, often representing the
maximum applied load on an individual member located within the crest-trough region.
93
ï0.1
ï0.1
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ï0.15
ï0.2
ï0.2
ï0.25
28.5
29
0
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30
29
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(d)
ï0.1
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t (s)
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t (s)
0.02
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0
0.5
1
dd/dx
6d (m)
W
Figure 5.6 Contour plots showing the upstream scattering of a Type 2 wave and its
subsequent interaction with the next incident wave crest. Conditions relate to a singlecolumn in regular waves, wave case 1. (a) wo(x,t); (b) w(x,t); (c) (x,t); (d)
(d/dx)w(x,t)
94
(a)
(b)
(c)
(d)
Figure 5.7 Still images showing Type 2 wave-scattering and subsequent wave
slamming. Wave case 1
95
Figure 5.8: Contour plots showing the upstream scattering of a Type 2 wave and its
subsequent interaction with the next incident wave crest. Conditions relate to a
focussed wave case. (a) wo(x,t); (b) w(x,t); (c) (x,t); (d) (d/dx)w(x,t)
96
(ii) The occurrence of wave impacts is often associated with large vertical wave run-up.
Although it is accepted that the deck area may be subject to a certain amount of wave
splashing, significant green-water impacts on the underside of the deck structure are not
typically acceptable in terms of a design condition. Indeed, the design process seeks to
avoid such events through the provision of an effective air-gap. However, recent
experience from several North Sea structures, particularly Gravity Based Structures, has
shown that present design practice in relation to air-gap provision may be insufficient;
with significant wave impacts arising in surprisingly mild storms with return periods of
less than 10 years (Swan et al. 1997).
(iii) With slamming forces typically arising at high elevations above the still water level
(SWL), moment-arm effects ensure they provide a disproportionately large contribution to
the total over-turning moment. This effect, coupled with the impulsive nature of the
applied loading, suggests that wave slamming may contribute significantly to the onset of
structural ringing. Indeed, on the basis of a physical model study of a Tension Legged
Platform (or TLP), Zou et al., (1998) concluded that the occurrence of wave impacts
represents the primary cause of the observed dynamic response.
At first sight this latter conclusion appears to be at odds with the results of the present study
which has clearly shown that the nonlinear potential flow loads, particularly those occurring at
the third- and higher-harmonies of the incident wave, can readily provoke a ringing-response in
the absence of a true slamming event. However, two points are relevant to this discussion. First,
the rapid reversal in the nonlinear potential flow forcing produces an impulsive-type loading
event which may, incorrectly, be interpreted as wave slamming. Second, arguments of this type
imply that a ringing response must be produced by one, and only one, type of loading event.
There is no evidence to support this latter view. Indeed, the possibility that the optimal or most
severe dynamic response arises when two (or more) separate, but related, loading events
combine constructively should not be discounted, particularly in the case of a multiple column
structure.
It is well known that the occurrence of wave slamming is critically dependent on the steepness
of the incident wave profile. In this sense it is clearly related to the nonlinear potential flow
loading. However, if wave slamming is to occur the incident wave profile must be parallel, or
near-parallel, to the surface of the structure on which the impact is about to occur. In the case of
a vertical column this suggests that part of the wave profile must be vertical, or nearly so.
Although large deep water waves are steep, near-vertical wave profiles, whilst not impossible,
are highly unlikely. Indeed, such occurrences imply that the wave has achieved its breaking
limit and is about to undergo large-scale over-turning. Whilst this is a common feature of
shallow water coastal locations, accounting for the occurrence of wave impacts on many coastal
structures, there is no evidence to suggest it occurs in deep water, at least not in the largest
waves. As a storm intensifies, an increase in the wave height is associated with a corresponding
increase in the peak period: the highest waves are therefore longer and not necessarily any
steeper.
These arguments suggest that the largest incident waves are unlikely to produce significant
slamming on a vertical column. However, this takes no account of the unexpected scattering of
nonlinear or high-frequency waves discussed in Section 5.1. Although the amplitudes of these
waves are small, their subsequent interaction with the next steep incident wave takes the form of
a classic long-wave short-wave interaction (Longuet-Higgins & Stewart (1960)) in which both
the amplitude and the steepness of the short-wave (or scattered wave) may be significantly
increased on the crest of the longer incident wave. This will not only produce large increases in
the maximum water surface elevation (relevant to air-gap calculations), but may also produce
significant local steepening of the wave profile relative to that recorded in the absence of the
97
structure. This may, in turn, produce more severe slamming events, with a greater probability of
occurrence.
To quantify this effect a new series of experimental observations were undertaken involving
both a single-column and a multiple-column arrangement. The purpose of these tests was to
investigate the effect of the high-frequency wave-scattering on the incident wave steepness: in
particular, to identify increases in the gradient of the free surface due to the nonlinear interaction
between the high-frequency scattered waves and the subsequent incident waves. In light of the
earlier observations, briefly outlined in Section 5.1, in which the strongest nonlinear wave-wave
interactions arose immediately upstream of the column, observations were undertaken with the
layout of wave gauges depicted on Figure 5.5. In the single-column case, Figure 5.5(a), a total
of 42 wave gauges were equally spaced along the x-axis, with the first gauges located 6mm
from the surface of the column (x=-0.064m) and the spacing between gauges set at x=12mm.
In the multiple-column case, Figure 5.5(b), 20 wave gauges were located within the range 0.531m x -0.151m, with a spacing of x=20mm. In both cases the origin (x=y=0) describes
the geometric center of the arrangement and the incident wavefield advances from the right of
each schematic, propagating in the positive x-direction as indicated by the large arrows.
Given the purpose of the present tests, to identify significant changes in the wave steepness, the
laboratory measurements were undertaken in two stages. The first concerned measurements of
the water surface elevations without the structure in place, wo(t); while the second involved
identical incident wave conditions and provided measurements of the water surface elevations at
the same spatial locations with the structure in place, w(t). Comparisons between these
measurements define the changes in the water surface elevation due to the presence of the
structure. It is these changes, =w-wo , that contain information on the nature of the scattered
waves and their effect on the incident wavefield, particularly any increase in the wave steepness.
Having considered a wide range of incident wave conditions; significant changes in the wave
steepness and hence the occurrence of wave slamming were found to be critically dependant
upon both the period of the incident waves and their initial steepness. These criteria are directly
related to the mechanisms responsible for the wave scattering and the subsequent nonlinear
wave-wave interactions. To highlight the potential importance of these effects, two cases are
considered. The first concerns a regular wave of period T=1.0s incident to a single column
having diameter of D=0.115m, corresponding to D/=0.07. To illustrate the nature of the
interaction arising, the data recorded without the column in place, wo, with the column in place,
w, and the difference between the two, =w-wo, are plotted as contours in time and space,
(x,t), on Figures 5.6(a)-(c) respectively. In addition, Figure 5.6(d) provides a similar
representation of the gradient of the water surface measured with the single-column in place,
(d/dx)w. Taken together these figures provide a condensed form of data representation that
enables the evolution of the wavefield in both space and time to be visualised more clearly. In
each case a horizontal slice across one of the plots defines a time-history of the water surface
elevation, (t), at a given spatial location (x=constant); whilst a vertical slice defines a spatial
record, (x), at a given instant in time (t=constant). In Figures 5.6(a) and 5.6(b) an incident
wave crest is shown propagating in the positive x-direction, towards the column. Evidence of
this is seen in the fact that the dominant gradient of the contours, dx/dt, is positive. However, in
the latter case, Figure 5.6(b), the contours of w(x,t) show significant distortion due to the
presence of the scattered wavefield.
The details of this are clarified in Figure 5.6(c) where the contours of are negative, dx/dt<0,
representing the outward propagation of a Type 2 scattered wavefield. It is evident from this
latter plot that due to the nonlinear wave-wave interaction, between the scattered waves and the
next incident waves, the motion of the scattered wavefield is non-uniform (the contours
98
showing significant curvature), with the interaction being most apparent at those locations in
space and time where the scattered and incident wave crests superimpose. Figures 5.6(c) and
5.6(d) show that at these instances the scattered waves are steepened, with large increases in the
measured free-surface elevations and gradients. Indeed, in this case it is interesting to note that
the maximum free-surface gradient measured in the absence of the column was of the order of
(d/dx)wo =0.5, whereas in the presence of the column this value increased to (d/dx)w=1.2. This
latter value is far inexcess of the limiting steepness of a Stokes wave, corresponding to the onset
of wave breaking for a regular wave train, defined by ak=0.44 or d/dx=0.58.
This data is complemented by Figure 5.7 which provides a series of still images, taken at
t=0.12s intervals, in the vicinity of an extreme slamming event. These images, together with
the data presented on Figure 5.6 illustrate that under certain circumstances Type 2 waves are
first steepened by their interaction with the next incident wave; but that the strength of the latter
in this is such that they are swept back onto the structure. This is highlighted on Figure 5.6(c),
with arrows indicating the direction of motion of a Type 2 wave crest, and by the visual images
on Figures 5.7(b) and (c). Finally, Figure 5.7(d) shows that this occurrence produces a severe
slamming event, with water projected to high elevations on the front face of the column.
The second case concerns a focussed wave group modelled using a JONSWAP spectrum
(Equation (3.6)) with a peak period of Tp =1.4s (or 14s at full-scale) and a peak enhancement
factor of =5.5; where the latter is important to ensure that the sea state is relatively narrowbanded and thus that any extreme wave group consists of more than one large wave event.
Within the present tests the wave groups were formed from 121 wave components, evenly
spaced within the range 30/64H3 f 150/64 and with their relative phasing at the wave paddles
adjusted to produce the required focussed event at a given location relative to the structure. In
addition, the wave groups were directionally spread. For each input frequency, 81 directional
wave components equally spread within the range -40°+40° were adopted, with individual
amplitudes defined by a wrapped-normal spreading function the standard deviation of which
was set at =10° (Equation (3.7)). Within the present tests the location of the extreme or
focussed wave group relative to the structure was varied in order to establish the optimum wave
profile, resulting in the most significant steepening of the incident wavefield.
Observations relating to this focussed wave group are presented in Figure 5.8. This figure
concerns the water surface elevations recorded immediately upstream of a typical 3-legged GBS
(Figure 5.5(b)); the diameter of the individual legs being identical to the single-column
considered previously, D=0.115m or 11.5m at full-scale, and the center-to-center leg spacing
fixed at 0.326m (or 32.6m at full-scale)). The individual plots within this figure are identical in
format to those presented in Figure 5.6; providing contours in the space-time domain describing
wo(x,t), w(x,t), (x,t)=w-wo , and dw/dx(x,t). In this case it is interesting to note that the
maximum wave steepness increases from (d/dx)wo=0.51, measured in the absence of the
structure, to (d/dx)w=1.28 in the presence of the structure; where the latter value is again
significantly higher than the corresponding value for a Stokes wave of limiting height
(d/dx=0.58).
The results presented in Figure 5.8, correspond to a realistic design wave condition and are
entirely consistent with the earlier regular wave tests. Overall, they suggest that the nature of the
wave-structure interaction, particularly the scattering of Type 2 waves, can produce a significant
local increase in the steepness of the wave profile and that, under some circumstances, this may
be swept back into the structure giving rise to the potential for significant slamming events.
99
6. DISCUSSION OF RESULTS: PRACTICAL IMPLICATIONS
The purpose of this study was to quantify the practical importance of ringing type process to
Gravity Based Structures and to highlight the extent to which they can be modelled using
available analysis procedures. In addressing these tasks, three key questions come to the fore:
1. Are significant nonlinear or high-frequency forces generated on common structural
geometrices subject to realistic sea states?
The answer to this is an emphatic yes. These forces, which are readily identifiable up to
the fifth-harmonic of the incident waves, are driven by conditions arising at the water
surface and are critically dependent upon both the steepness of the incident waves and
their directional spread.
2. Can these nonlinear forces be predicted by existing analysis procedures?
In the absence of numerical solutions capable of including the full nonlinearity of both a
steep incident wavefield, including directional effects, and the subsequent wavestructure interaction, involving multiple surface-piercing columns, the answer to this
remains no. Indeed, although the present study has provided new and important insights
into the physical origins of these force components, the mechanisms that cause them are
not fully understood.
3. Are these high-frequency forces and the associated ringing type processes practically
significant?
This is perhaps the hardest question to answer, not least because the importance of the
effects is critically dependent upon the details of the structure involved. Nevertheless,
the present study has shown that if a structure is susceptible to dynamic response,
having a relatively high natural period, Tp=4-5s, the moments applied at the columncaisson connection may be enhanced. However, since the high-frequency forcing is
clearly associated with the steepest waves, it is unlikely to affect the maximum global
loading associated with the largest wave events since these are not, typically, the
steepest waves. In contrast, related effects such as the loss of an air-gap and the
occurrence of wave slamming on the columns may present a significant problem.
Further explanation of these answers, and the reasoning behind them, is provided in the
following sections.
6.1 NONLINEAR FORCING
The laboratory data provides clear evidence of significant nonlinear and high-frequency force
components. These occur in both regular and irregular wavefields, having their largest relative
magnitude in steep uni-directional waves. These forces, and their associated moments, are
driven by conditions close to the water surface. As a result, they are unaffected by conditions
arising at greater depths such as increases in the column diameter or the presence of storage
caissons; the latter being a typical feature of most GBS designs. The only exception of this
occurs in intermediate or shallow water depths where a further local reduction in the water
depth, above the storage caissons, may lead to a significant focussing and steepening of the
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wave profile. This is known to occur at several GBS's and, in such cases, the nonlinear forces
acting on the columns will be much enhanced, as will other related effects (see below).
In a number of regular wave cases direct harmonic analysis allows significant high-frequency
forces up to the fifth-harmonic of the incident waves to be identified. Although, in all cases, the
linear or first-harmonic forces are dominant, the second-, third-, fourth- and fifth-harmonic
forces can be comparable in magnitude. This is unexpected and at odds with the requirements of
a formal perturbation expansion, on which most nonlinear force models are based. One
important and previously unexplained feature of the applied forcing is the secondary loading
cycle, first identified by Grue (1993). For a given column diameter this is shown to be critically
dependent upon the incident wave frequency and rapidly reduces with an increase in the
directional spread; the latter effect being related to a reduction in the in-line wave front
steepness (see Section 6.2 below).
In practice this secondary loading cycle only represents a part, albeit the most visible part, of
the applied nonlinear loads. To identify the true extent of these loads, the linearly predicted
loads have been subtracted from the total measured. The resulting pattern is shown to be closely
correlated with the unexpected high-frequency wave scattering. This process involves the
movement of fluid around the circumference of the column with the scattering of both Type 1
and Type 2 waves. As the incident wave crest approaches an individual column, fluid is driven
up its front face, washes down and produces the first scattered wave form (Type 1). This, in
turn, induces a positive downstream force. A short time later fluid moving around the column
merges at it's back face to produce a negative or upstream force. This fluid continues to
propagate around the column, generating two non-concentric, Type 2 waves from the front face.
This again induces a positive force. This rapid force reversal, together with a similar patter of
reduced magnitude associated with the negative velocities occurring in the wave trough, defines
the nonlinear force components. When considering these forces, two points need to be stressed:
(a) The mechanisms responsible for these forces are completely unrelated to linear
diffraction. Given the size of the column involved, this is not expected to be important.
(b) The rapid force reversals are, in large part, dependent upon the time taken for the fluid
to move around the column. This is independent of the incident wave frequency and,
consequently, the applied forces will not be well modelled by a series expansion based
solely on the harmonics of the incident wave motion. Indeed, such an approach would
be expected to give miss-leading results.
These nonlinear forces will not be significant in all cases. For example, if the column diameter
is too small the wave-induced movement of fluid around the circumference of the column will
induce flow separation and wake formation. In this case, no non-linear wave scattering occurs
and the applied forces are drag dominated. Equally, if the column diameter is too large linear
diffraction will dominate and, even in relatively steep waves, both the water surface elevations
and the applied forces to be well modelled by a second- or third-order diffraction solution.
However, if the flow regime is intermediate to these cases so that the inertial forces or potential
flow forces dominate, (KC5), but linear diffraction effects are not significant, these nonlinear
forces will become significant in steep incident waves. In practical terms, this flow regime
corresponds to column diameters of 10-15m subject to steep waves with peak periods
12sTp14s. This corresponds closely to those conditions in which severe ringing-related
effects (forcing, scattering and dynamic response) are observed.
In all of the cases considered the linear loads, based on the integral of Morrison's inertia term to
the still water level, are dominant. However, within the flow regime noted above, the nonlinear
101
loads may become very significant. For example, in the steepest wave case the applied force
reversals occurring immediately after the passage of a large crest wave may be as large as 40%
of the linear predictions. More importantly, these force reversals occur at frequencies well
above the incident wave frequencies providing the opportunity for the onset of dynamic
response.
At present these nonlinear (or high-frequency) force components cannot be adequately
modelled. The best available solution appears to be based upon the Rainey (1995a) force model
since this provides some nonlinear force components that can legitimately be calculated using a
fully nonlinear wave model. However, even this approach fails to account for the general form
of the rapid force reversals, let alone their magnitude, and there remains uncertainty concerning
the necessity and validity of including the surface distortion force, FSD. Given the nature of
these force components, it is unlikely that they can be predicted by anything other than a full
numerical solution capable of including the nonlinearity associated with both the wave-wave
and the wave-structure interactions. Such solutions are not presently available, but a number of
potential developments are considered in Section 6.5 below.
6.2 WAVE CONDITIONS AND WAVE MODELLING
The study has shown that the generation of significant nonlinear forces is critically dependent
upon both the steepness and the peak period of the incident waves. For columns of diameter 10­
15m, the largest nonlinear loads arise in steep waves having a relatively small peak period
12sTp14s. If the period is larger, as would be the case in a 1 in 100 and (certainly) a 1 in
10,000 year design wave event, the steepness, the nonlinear wave scattering and the nonlinear
forcing will be of reduced magnitude. Clear evidence of the importance of wave steepness is
provided by the rapid reduction in the nonlinear loads with the introduction of directionality in
the incident wavefield. This was also reported in observations of nonlinear wave scattering
(Swan & Sheikh, 2005) and may be explained by a reduction in the inline wave-front steepness.
This appears to be a key parameter in driving the fluid around the circumference of a column.
The present results also confirm that a fully nonlinear description of the incident wavefield is an
important factor in the prediction of the applied loads. This is at odds with the arguments
originally by Newman (1995) who claimed that a linear representation of the wave motion is
adequate. The explanation for this is two-fold:
(i) Although the second and higher-order bound waves may not cause a large change in the
water particle kinematics, the change in the wave shape will involve higher and steeper
wave crests. These changes will effect both the integral of Morison's inertia force up to
the instantaneous water surface and, perhaps more importantly, the evaluation of the
surface point loads which are critically dependent upon both the surface elevation and the
depth-variation in the local kinematic parameters. These effects account for the
differences in the Newman and Rainey force models which are shown to be significant in
respect of the high-frequency forces.
(ii) The importance of the third-order resonant interactions leading to rapid and local changes
in the spectrum defining the freely propagating wave components must not be under­
estimated. In present design practice, a design wave spectrum is specified on the basis of
field data and is assumed to remain constant. However, recent work, (Gibson & Swan,
2005), has shown that important spectral changes will arise in the vicinity of a large wave
event. In deep water these changes involve a transfer of energy to higher frequency
components, coupled with a reduction in the directionality of the wavefield. Although
these changes do not necessarily result in higher crest elevations, they do result in
significantly steeper waves and hence have a large effect on the nonlinear loading.
102
Despite the fact that these latter changes result from nonlinear (third-order) resonant
interactions, they may be interpreted as changes in the freely propagating wave spectrum. As a
result, their effect can be incorporated within the Newman model, in a way not originally
envisaged by its author. This leads to a significant improvement, but the neglect of (i) ensures
that the predictions remain inferior to those of Rainey. However, it is important to stress that
even with the inclusion of the full nonlinearity of the wavefield, involving the utilisation of
essentially exact water particle kinematics (predicted in the absence of the structures), important
aspects of the nonlinear loading cannot be predicted.
Finally, in relation to the wave conditions considered, the majority of the present irregular wave
tests concern focussed wave groups. The details of these groups were chosen on the basis of the
earlier Spectral Response Surface method. These results suggest that, contrary to linear theory,
the optimal dynamic response is generated by a wave group produced by the focussing of wave
crests. The subsequent laboratory tests confirmed that this was indeed the case, and highlighted
the possible excitation of the structure at it's natural frequency.
6.3 DYNAMIC RESPONSE
Having recorded the dynamic response of both single-column and multiple-column structures,
with two natural periods and in a wide variety of wave conditions, it is clear that a
straightforward solution based upon the mass-spring-damper equation will yield satisfactory
results provided the static loads, F(t) and M(t), can be accurately defined. In essence, the
difficulties associated with the prediction of dynamic response boil down to the uncertainty in
the applied loads, particularly the high-frequency loads arising close to the natural frequency of
the structure. Unfortunately, with none of the available force models (FNV, Newman, or
Rainey) providing a good description of these loads, the dynamic response cannot be defined in
terms of predicted quantities alone. If this is to be achieved significant improvements in the
force models are required, taking into account both the nonlinearity associated with the
wavefield and the wave-structure interaction.
In the multiple-column data there is some evidence of additional nonlinear interactions between
the legs of the structure. This occurs in many of the force traces, where the total load cannot be
reproduced by the sum of appropriately phased single-column records, and in the measured
dynamics, where the peak deflection and its subsequent decay is occasionally followed by a
secondary maximum. This latter effect cannot be predicted on the basis of the measured static
loads. However, given that its magnitude is significantly smaller than the initial peak
displacement, it clearly represents a secondary effect which is inconsequential in terms of the
dynamic amplification of the peak loads.
6.4 WAVE SLAMMING
Before discussing the significance of wave slamming, and the wave conditions in which it is
most likely to arise, it is important to stress that both the theoretical calculations based on the
Spectral Response Surface Method (Section 3) and the Laboratory Observations (Section 4)
demonstrate that ringing events can be initiated by the nonlinear potential flow forces without
the occurrence of a specific slamming event. This is consistent with the earlier findings arising
from the Norwegian Joint Industry Project on Ringing and is perhaps at odds with the
conclusions of Zou et al. (1998). However, this does not imply that slamming has no part to
play. Indeed, the occurrence of impulsive loads at the instantaneous water surface represent
exactly the type of conditions in which a transient dynamic response might be expected. In
essence, the present results suggest that these two separate effects may be complementary; the
103
fact that they occur in similar wave conditions, both being related to the process of nonlinear
wave scattering, further strengthens this view.
It has already been observed that the nonlinear or high-frequency potential flow loading is
largest in steep incident waves having relatively small peak periods; with the rapid force
reversals corresponding to the unexpected wave scattering. These effects only require one large
wave event. If the wave group involves more than one large wave the loading pattern is
repeated, but the loads arising in any one wave cycle are largely independent of events arising in
adjacent cycles. The clearest evidence of this is provided by the force data recorded in regular
waves (Sections 4.1.1 & 4.2.1). However, this will not necessarily be the case. If a sea state is
characterised by a high peak enhancement factor (>5.0), indicating a narrow banded spectrum,
an extreme wave group will consist of more than one large isolated wave event. In this case, the
nonlinear scattered waves produced by the first event will have the opportunity to interact
nonlinearly with the next incident waves. This interaction takes the form of a long-wave short­
wave interaction, in which the high-frequency scattered wave will become higher and shorter,
and hence steeper, on the crest of the next incident wave. If the phasing of the scattered wave is
optimal, hence the requirement to have relatively low period incident waves, this interaction
will occur close to the structure and will involve large potential increases in the maximum water
surface elevation. It is believed that this mechanism provides an explanation for the occurrence
of wave impacts on the underside of deck structures.
An alternative but closely related event occurs if the second incident wave is more extreme. In
this case, the high-frequency scattered wave is first steepened (in accordance with the arguments
noted above) but then swept backwards into the structure due to the strength of the next incident
wave. This two-part mechanism, involving nonlinear wave scattering and nonlinear wave-wave
interactions, is important for a number of reasons:
(i) It explains the increased water surface elevations necessary to account for wave impacts
on the underside of the deck structure.
(ii) It explains the large increases in the steepness of the wave profile necessary to account for
the occurrence of wave slamming at high elevations on the column(s) of a structure.
(iii) It is not presently taken into account in design.
Although these events are clearly significant, they do not arise in all sea states: both the
steepness and the period of the incident waves are critical. If the waves are less steep, both the
nonlinear wave scattering and the associated nonlinear forcing will be much reduced in
magnitude. Alternatively, if the period of the waves increases, the scattered waves have more
time to radiate away from the structure before they interact with the next incident wave crest.
This has two implications. First, as the scattered waves move away from the structure they
reduce in size. This is true for all radiating disturbances and means that the subsequent
nonlinear wave-wave interactions will again be of reduced magnitude. Second, the nonlinear
wave-wave interactions will result in increased water surface elevations outside the plan area of
the structure. Furthermore, it will not produce large increases in the steepness of the wave
profile that can be swept back into the structure to create a slamming event.
These restrictions on the relevant wave conditions are important since they imply that incidents
of wave slamming, whether they be on the underside of the deck or on the columns of the
structure, will not occur in conjunction with the typical 1 in 100 or 1 in 10,000 year events.
Although these waves may be very high, they are not necessarily the steepest waves and involve
peak periods that are too large. As a result, these effects should not be added to the ultimate
design loads and do not, therefore, threaten the overall integrity of the structure. Indeed, they
104
should be considered as alternative design states in which highly nonlinear interactions may
result in the local loss of air-gap and/or the occurrence of wave slamming on the columns;
where the latter may contribute to the onset of structural ringing.
6.5 RELATED ISSUES AND MODELLING PROCEDURES
It has already been noted that:
(i) The development of nonlinear or high-frequency loads capable of provoking a dynamic
response,
(ii) The occurrence of vertical jetting leading to a loss of air-gap and wave impacts on the
underside of a structure, and
(iii) The generation of impulsive loads due to wave slamming on the columns,
are all related issues associated with the unexpected scattering of nonlinear or short-wavelength
waves from the legs of a structure. Given that this effect is, at least in large part, driven by the
movement of fluid around the circumference of individual columns close to the water surface, a
series solution based solely on the harmonics of the incident wave motion will not yield
appropriate results. Furthermore, given that (i), (ii) and (iii), are particularly concerned with
steep waves, there remain important concerns regarding the convergence of diffraction solutions
based on a perturbation expansion. Indeed, even if such an approach were to be successful, it
would have to include terms up to at least a fifth harmonic of the incident wave. This precludes
the use of an analytic or semi-analytic model and has led to the conclusion that a fully nonlinear
model must be adopted if the effects noted above are to be described.
At present the authors are unaware of any numerical models that are capable of incorporating
both the full nonlinearity of the wavefield and the wave-structure interaction. For example, the
boundary element model proposed by Ferrant (1998) is very limited in terms of the wave
steepness it can model (ak<0.15); whilst the finite element modelling undertaken by Kim et al.
(2003) is able to reproduce some of the nonlinear wave scattering, but finite element approaches
are notoriously poor at describing highly nonlinear waves. The work of Wu & Eatock Taylor
(2003) is also relevant, particularly in respect of multiple columns, but very little quantitative
data is provided on the nature of the predicted flows and no rigorous validation has yet been
provided.
Looking ahead, on-going developments in boundary element modelling, particularly the
application of multiple flux elements at the fluid-structure boundary and fast multipole methods
to limit computational resources, offer the potential to solve over-turning but non-breaking
wave problems. In the longer term, the occurrence of vertical jetting and perhaps also wave
slamming may be solved by meshless procedures such as Smooth Particle Hydrodynamics
(SPH). However, in the short-term, the absence of available numerical models ensures that
many of these highly nonlinear processes can only be quantified in a laboratory environment.
105
7. CONCLUDING REMARKS
The present study has shown that a typical Gravity Based Structure may be subject to
unexpected high-frequency loading. It occurs in steep waves, with relatively small peak periods,
and is associated with near-surface effects involving the movement of fluid around the legs of
the structure and the scattering of high-frequency waves. These effects are relevant to both
single and multiple-column structures and are not linked to the development of trapped wave
modes between adjacent columns. Given the wave conditions in which this forcing occurs, it is
not expected to add to global design loads based on 100 or 10,000 year events. However, if a
structure is susceptible to dynamic response, with large natural periods (Tn>4s), this forcing
may provoke a transient ringing event.
The high-frequency loads cannot be predicted by existing force models. Contrary to previous
assertions, the nonlinearity of the wavefield plays an important role in determining the nature of
the nonlinear loads. Indeed, in the absence of modelling procedures capable of incorporating
both the nonlinearity of the wavefield and the wave-structure interaction, the single most
important step is to incorporate the best possible description of the incident waves; including
both their directionality and, most importantly, the local evolution of the wave spectrum. With
this achieved, the integral of Morison's inertia term to the instantaneous water surface will fully
resolve the applied linear loading and will provide an initial, albeit incomplete, description of
the nonlinear loads. Further description of the nonlinear loads is difficult, not least because the
introduction of additional nonlinear loading terms can result in increased errors. In such cases,
the applied loading can only be determined by detailed physical model testing, with particular
attention paid to the directionality of the wavefield. However, it is important to stress that these
comments specifically relate to those cases where traditional diffraction solutions are considered
inappropriate, for reasons of column size and wave steepness. For larger column diameters, both
linear and nonlinear diffraction would be expected to become increasingly dominant, with
second- and third-order solutions providing a reasonable estimate of the forces in relatively
steep waves.
The present tests have also shown that the processes of nonlinear wave scattering, responsible
for the generation of the nonlinear loads, may lead to related difficulties concerning the loss of
an air-gap (with impacts occurring on the underside of the deck structure) and the occurrence of
wave slamming at high elevations on the columns. Although these effects tend to be localised,
and will not therefore threaten the overall integrity of a structure, they represent important
design considerations.
Finally, it is important to note that although the present study specifically relates to Gravity
Based Structures, the observed effects are equally appropriate to a much wider class of offshore
structure. With the scattering of high-frequency waves now known to be solely dependent upon
the structural geometry at the water surface, the effects will be equally important to both fixed
and floating structures. As a result, the specification of an effective air-gap or the occurrence of
wave slamming should not be predicted on the basis of crest elevations or wave slopes
measured in the absence of a structure or vessel. Furthermore, with the effects critically
dependent on wave steepness, they will become increasingly important in shallow water.
106
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