Motions and Drift Forces of Air-supported
Structures in Waves
J.A. Pinkster
Report 1156-P Project Code: 962
4th September 1998 Presented on the Fifth WEGEMT Workshop "Non-Linear Wave Action on Structures and Ships ", University Toulon-Var, France, An European Association of Universities in Marine Technologies and related Sciences
Edited by Vincent Rey, Institut des Sciences de L 'Ingenieur de Toulon et Du Var
TU Deift
Faculty of Mechanical Engineering and Marine Technology Ship Hydromechanics LaboratoryUNIERSITE
IOU IONVAR
Ville de La Seyne Sur Mer
ii
î
EGEMT
rsrv
A European Association of Universities in
Marine Technologies and related Sciences
Vth WEGEMT WORKSHOP
NON-LINEAR WAVE
ACTION ON
STRUCTURES AND SHIPS
University Toulon-Var
France
4th september 1998ui6w
Ci
NSEI
L
GENERAL
Information / Workshop Secretary
Vincent REY, Wegemt Workshop Secretary, ISITV, BP 56, 83162 La Valette du Var Cedex, FRANCE
Fax: +33 (0)4 94 14 24 48. E-mail: rey@isitv.univ-tin.fr
PROGRAMME
Thursday 3 september 1998:
For those delegates who intend to travel on the day before the Workshop. 17h00: Visit of the Ocean Basin FIRST, La SeyneíMer.
20h00: Reception at Maison Pacha-Tamaris, La Seyne/Mer.
Friday 4 september 1998:
08:00 - 09:00: Registration
09:00 - 09:10: Welcome
09:10 - 09:50: An overview of flow induced vibrations with emphasis on Vortex Induced Vibrations (VIV) for marine risers
Pr. Geir MOE, NTNU, Norway
09:50- 10:30: Slamming
Pr. Odd FALTINSEN, Marintek, Trondheim, Norway
10:30- 11:00: Coffee
11:00 - 11:30: A quasi-3-D method for estimating non-linear wave loads on
barges and FPSO's
Pr. Philip CLARK, Heriot - Watt University, Edinburgh, U. K.
11:30 - 12:00: Non-linear time domain solutions for
wave and current
action on offshore structures
Mr. Pierre FERRANT, Sirhena, Nantes, France
12:00- 12:30: Semi-analytical methods for some linear and nonlinear
problems of diffraction-radiation by floating bodies
Mr. Sime MALENICA, Bureau Ventas, Parish France
12:30 - 13:50: Lunch
13:50- 14:30:
Station keeping and slowdrift oscillations
Pr Joe PINKSTER, Tech. Univ. of Deift, The Nedenlands
14:30 - 15:10: Design load predictions by non-linear strip theories
Dr. Jorgen JUNCHER JENSEN, Tech. Univ. of Denmark, Denmark
15:10 - 15:30: Coffee
15:30 - 16:00: Non-linear interactions between waves and a horizontal
cylinder
Pr. John CHAPLIN, City University, London, U. K.
16:00 - 16:30: Transformation of non linear waves in shallow water and
impact on coastal structures
Pr. Stephan GRILLI, University of Rhode Is1and USA
16:30 - 17:00: Wave-breaking impact on coastal structures : VOF modeling
Mr. Stephan GIJIGNARD, UTV and Principia S. A.
Saturday 5 september 1998:
LIST.O'F PARTICIPANTS
PRENÒM UÑIVERSITE
AAL BERS Pr. A. Deift University of Tèchnokgy
A' WOOD Dr. R Cranfield University
BERHAULT Christian Principia R&D et IS1T
'BOOTE Pr., D. Università di Genova
CARDO Pr.. A. Università degli Studi di Trieste
CHAPLIN Pr. John City University London
CHUDLEY Dr., J. University of Plymouth
CLAMOND Didier Univ. Toulon-Var
CLARK Pr. Philip Heriot - Watt University UK.
CORDONNIER J. P Ecole Centrale de Nantes
CORRIGNAN Philippe Sirehna
Christine Principia R&D et IS1TV DE JOUETFE
DEVENON Jean-Lue Univ. Toulon-Var
DEVILLERS Jean-François 'Ecole Nationale Súpérieure de
Techniques Avancées
DULOU yrilP Univ. de Provence
FALTINSEN Pr. Odd MAR]NTEK
FELD Dr. Graham FUGRO GEOS
FERRANT Pierre Sirhena
FRAUNIE Philippe Univ. Toulon-Var
FUIvIEY Frédéric Global Maritime
GAILLARDE Guilhem MARIN
GRANT .Jim WEGEMT Secretary
GRIETHUYSEN Pr. J van University College 'London
GRILLI Pr. Stephan University of Rhode Island
GUIGNARD Stephan U T V et Principia R&D
HENRIKS EN Martin University of Science and
Technology Norway
lUNCHER JENSEN Jorgen Tech. Univ. of'Denmark
KRUPPA Pr. C. Technische Universitaet Beriiñ
LECALVE Olivier Univ. Toulôn-Var
MALENICA Sime Bureau Ventas
MATUZIAK Pr. J. Helsinki University of
Technology
MOE - Pr. Geir Norwegian University of Science and Technology NTNU
MQLII ESIM
NUNEZBASAN2
-
-1r J.
Escuela Tnica.SùperiôrdeiIngeiiàros Nava.les
PAPANIKOLAOU.
Pr. L
NatiOnal TèchnicalUniversityofAthens
PELISSIER Pr M.Claude Univ.TOuIon-Var
PINKSTER Pr. Joe Tech. Uúiv. ofDelft
REINHOLDTSEN Svein-Ame
M '
KRESCH François Univ. Toulon -Var
REY Vincent Univ Toulon Var
RIGAUD
j Stéphane Principia R&D
RUSSO-KRAUSS Pr. G. Università.Federico IT
SCOLAN Yves-Marie ESIM
SEN Pr. P University of Newcastle Upon
Tyne
TRASSOUDAINE Damien S B M
WIETASCH Pr. K W Gerhard Mercator Universität
About WEGEMT
WEGEM'T'isaEuropean associationof37 universities in 17countries. Itwas formecLby IS universities from '10;Eüropean'cóúntiies in. i975wththe aim of
increasing the knowledge base, and updating and extendmg the skills and
competence of engineers and postgraduate students workingat an advanced frvel in marine technok)gy and related sciences.
WEGEMT achievesits aim ityencouraging member universities to:
- Operate as a network and therefore actively collaborate in initiatives reláted' to this purpose
- Cònsider collaborative research,education and trainingactivitiesat an advanced! level
- Exchange and disseminate information in marine technology at an advanced
level to further its aims.
WEGEMT has organised 30 schools, three còlloquia and fOur workshops since 1978 andl further events are planned beyond 1999. Informationon the price and availability of the proceedings of past.events can be obtained from the WEGEMT
Secretariat in London (Fax: +44 (0) 171 838 9147): or via the web site at
http://wwwwegemt.org
About ISITV
ISITV is a division of the 'University Tòuion-Var, it. is divided into four
departments:
* Ingénierie Marine: Sea Technology
* Ingénierie Télécommunication: Telecommuniôations
* Ingénierie Mathématique, Modélisation et Informatique Scientifique: Applièd'
Mathematics
* Ingénierie Matériaux: Matcùiai
The educational: program of the department "Ingénierie marinefl covers the
fundämental processes of the seas and associated engineering. One can distinguish three distinct application areas: general oceanography, coastal
processes and associated engineering, offshore and marine technology
Laboratory equipment (wave taiík, towing tank with irregulär wave abiit) iS!
'used to fàmiliarise the students with the fundamental processes and their
ISITV
INSTITUT DES SCIENCES DE L'INGENIEUR
DE TOULON ET DU VAR
ISITV is a division of the Université dc Toulon et du Var.
The Wave Tank
- coastal processes and associated engineering. This area is largely
covered by a combination uf Physical Oceanography and Marine Technology. This combination of courses leads to a thorough
understanding of coastal wave diffraction, littoral currents and wave
drift. manne sediment transport, harbour oscillations, pressure and
forces on sea defence structures, behaviour of lides and inllucncc of
extreme waves. This is truc for estuarine arcas, beaches, and artificial fono of wave attenuation. Laboratory equipment is used to familiarise the students with the fundamental processes.
ISITV is divided into four deparunenis:
Ingénierie Marine is the department of Sea Technology
lngéniéric Telécommunication is the département of télélécommunications Ingénierie Mathématique. Modélisation et Informatique Scientiliquc is the department uf Applied Mathematics
Ingénierie Matériaux is the dcpanment of Materials
The e(lucational program uf the depanment "Ingénierie narine' covers the fundamental processes of the seas and associated engineering. One can distitiguish three distinct application areas.
I
- offshore and manne technology cover offshore structures and marine foundations (including offshore pipeline design. installation and stabilization and remotely-operated vehiclesl. The wind, wave and current forces on structures arc dcnved and titen used to yield the motions of the structures. Naval Architecture in the true sense of the word is not given. although naval hydrodynamics form part of tIme overall curriculum. The laboratory consists, in addition to the above, of a towing lank with a irregular wave ability. TIte carriage is sufficiently fast to study steady and tinsmcadv airliow theory. impact. ship resistance and motions.
Field Practical Work
The overall cumculum is extremely dense with scientific courses, with advanced mathematics, probability theory, numerical solution of partial differential equations, in addition mo courses on data processing, numerical filtering, structural engineering, thermodynamics, artificial intelligence. to give a few examples. Most graduates master ''eral programming languages in addition to such standard programs as ANSYS, MATLAB, PHOENICS and CADDS programs.
The Towing Tank
general oceanography and current behaviour. Students arc nade familiar with the generation process of wind waves, the generation of swell. tsunamis, the energy tranler. wind driven curreiits and geiicral circulation currents, Coriolis forces and their effects. Included arc courses on gcodynamics and meteorology. Laboratory experiments arc available to show the fundamental processes such as the venturi effect. the vortex behaviour duc to rotation. Field trips are included to show the
use of oceanographic sampling at sea. including turbidity. sonar.
suspended sediments, dynamic positioning.
UNIVERSITÉ
TO V LO N VAR
-..",) - --L ' :LY
//
V
V
II II IILongueur totale Longueur utile Largeur totale Largeur utile Profondeur Profôndeur de la fosse centrale Plage amortisseuse Atténuateurs :transver Générateurs de hoúle Type de houle Générateurs de couran Vitesse dù courant Plancher mobile Chariot
BGOFISTflCØE TECHN QUE
65m, 24m 26m 16m 5m 10m parabolique structure anéchoïque 12 batteurs "canard" à transiation horizontale régulière, irrégulière, aléatoire Hauteur maxi 0.8 m Période maxi 3 secondes Périodemini 0.3 seconde
12 pompes électriques à hélice, installée du côté de la plage parabolique dépend de la position du fond mobileA 3 mètres: Courant avant 0.5 rn/s Courant arrière 03 zn/s tOUte la surfacede O à5m inclinaison de 0° à 70
xY
Vitesse selon X 1.5m/sVitesse selon Y O8flaJs j
Les canaux de recirculation et la Chambre de tranquillité1
rendent le bassin particulièrement performant dans une
configuration compacte
Les atténuateurs transversaux: 'rendent le bassin
particulièrement performant
6 m le plancher mobile occupant 1m
pourvue de lattes horizontales
évitent la génération de houles stationnairestransverses moteurs électriques continus avec asservissement pilotage ARMA
possibilité ultérieure de houle croisée
Soit au 1150e des vagues de plus de 35 mdc haut, et une
gamme de période de 4 à 21 secondes
réversible- débit 25 m3 par seconde
- le planchermobile permet de fafrevarierla vitesse
- permet inversiön du 'courant et de la houle
permet de jouer sur la vitesse du courant, et diñterven
sur 'lés maquettes rapidement, sans vider lè bassin, Contrôle : calculateur
trajectoire parfaitement icontrôléó, arbitraire, r4pitiòns
Document 1300.94 BC/ma 19décembre1994
Station keeping and slowdrift oscillations
Motions and Drift Forces of Air-supported
Structures in Waves
J.A. Pinkster and A. FauziShip Hydromechanics Laboratory Deift University of Technology
Mekeiweg 2, 2628 CD Deift, The Netherlands
Abstract
A computational method to determine the effects of air cushions on the behaviour of floating structures in waves is described and validated through comparison with results
of model tests. The computational method is based on linear three-dimensional potential theory using linearised adiabatic law for the air pressures in the cushions. The water
surfaces within the air cushions and the meai wetted surface of the structure are described by panel distributions representing oscillating sources. Results of computations and model
tests with a simple rectangular barge in regular waves are compared with respect to
cushion pressures, motions and mean second order wave drift forces.
Keywords
Air cushions, motions of floating structures, moored vessels
Introduction
The use of air cushions to support floating structures has been known for a long time in
the offshore industry. Among the first large structures which were partially supported by
air were the Khazzan Dubai concrete oil storage units installed in the Persian Gulf in the early 70's. See Burns et al.[1J. These inverted bell-shaped units with open bottom were transported to location by sea using air to supply the buoyancy. On location the air was released and the structure was lowered to the sea-floor.
The Gullfaks C Condeep structure was lifted to a buoyant condition from its position in the construction yard by pumping compressed air in the spaces between the skirts. See Kure and Lindaas [6]. In this operation 96% of the buoyancy was provided by the air
cushions. After floatout from the construction yard the air was released and the unit
completed in a deep water location. Air cushions had been used previously (1974) to increase the buoyance of condeep structures but in the case of Gullfaks C the relative
Berthin et al [5] describe the use of air cushions in the floatout operation of the Maureen
Gravity Platform from the construction yard. In this case the air was also releasedon
reaching a deeper location. In the above cases the use of air cushions to support large floating structures was temporary and in most cases took place in calm conditions. The main function of the air cushions was to reduce the draft of the structure to allow trans-portation over a shallow water area. Except for the case of the Khazzan Dubai units,
the static characteristics of air cushions were of main importance. In ref. [1], the dynamic
behaviour of the Khazzan Dubai units during the lowering phase at the final location is described. No reference is made to the behaviour of large cushion-supported units in waves.
For many years, much attention has been paid to the development of fast waterborne sea
transport based on air cushion technology as applied to ACV and SES craft. Numerical methods have been developed for the prediction of the behaviour of such craft in the
design stage. See, for instance, Kaplan [2],Faltinsen [7] and Nakos et al. [81.
In the 70's the Seatek Slo-Rol system was introduced to reduce the wave-induced motions
of jack-up platforms in the floating mode. In this system the weight of the structure is partly supported by an air-filled chamber located around the perimeter of the pontoon of the unit.
As a result of the application of this system, the roll and pitch motions of jack-ups in
waves are reduced thereby reducing the dynamic loads in the jack-up legs in the wet tow
mode. According to the developers, motion reduction is partly due to the lowering of the effective metacentric height and partly due to a reduction in the rolling and pitching moments on the structure.
In recent years, a pneumatically stabilized platform has been investigated for application
as a permanent maritime platform in an open sea environment. See Blood[9]. This con-cept incorporated 75 independent air filled cylinders which were open to the sea on the
underside. lVlodel tests were carried out to determine the 'air-pocket factor' as defined by
Seidel [4].
Results of such investigation indicate that air cushions can modify the behaviour of
struc-tures in waves considerably and justifies a more detailed investigation into the effects of
various air cushion configurations.
Based on this observation it was decided to modify an existing linear 3-dimensional diffrac-tion code DELFRAC of the Deift University of Technology to take into account the effect
of one or more air cushions under a structure floating in waves at zero forward speed.
In this paper, a review is given of the main elements of the theory underlying the
com-putational method. The numerical modelling of the structure and the air cushions is treated and results of comparisons between computations and model tests are given of the wave-induced air-cushion pressures, motions and mean second order wave drift forces
on a barge-shaped vessel supported by one or by two air cushions. In a previous paper,
see Pinkster[10], an outline of the theory has been given along with examples of the effect
Air cushion dynamics
The theory is given for an air cushion supported construction consisting of one rigid body
and one or more air cushions which may or may not be interconnected. The air cushions are passive and there is no air leakage or induction. The air cushions are bounded by
the rigid part of the construction which extends sufficiently far below the mean waterlevel
within an air cushion in order to ensure that no air leakage will occur. Thewave frequency
air pressure variations within a cushion are determined by the change in cushion volume through the linearized polytropic gas law:
= vol * flg * (PO + p)/vol (1)
in which:
mean air volume in cushion atmospheric pressure
mean excess of pressure in cushion
wave frequency volume change in cushion
pressure variation relative to mean cushion pressure
flg
= gas law index
For wave frequency pressure variations adiabatic conditions are assumed. In that case
ri9 = 1.4.
Fluid dynamics; Potential Theory
The air cushions and the rigid part of the structure are partly bounded by water. The interaction between the air cushions, the structure and the surrounding water are
deter-mined based on linear 3-dimensional potential theory.
Use is made of a right-handed, earth-fixed O - X1 - X2 - X3 system of axes with origin
in the mean water level and X3-axis vertically upwards. The bodyaxes C -
-
-of the rigid part -of the construction has its origin in the center -of gravity -of theconstruc-tion, with x1 axis towards the bow, r2 axis to port and r3 axis vertically upwards. The wave elevation and all potentials are referenced to the fixed system of axes. In regular
long-crested waves the undisturbed wave elevation isas follows:
(X1, X2, t) = (2)
in which:
wave number
wave direction, zero for waves from dead astern wave frequency
wave amplitude
(
= wave elevationThe motions of the rigid part of the construction in the j-mode relative to its body axes
are given by:
r (t)
(3) k=
a
=
w=
(o=
vol -Po=
Pc =¿vol =
Lp =in which the overline indicates the complex amplitude of the motion. In the following the overline which also applies to the complex potentials etc. is neglected. The fluid motions
are described by the total potential 'I as follows:
F(X1,X2,X3,t) = (Xi,X2,X3)e"
(4)The potential satisfies Laplace's equation, the linearised boundary conditions on the free surface outside the body, the boundary condition at the sea-floor and, excepting
the undisturbed incoming wave potential, the radiation condition. On the rigid part of the body surface a no-leak condition has to be satisfied while at the free-surfaces of the air cushions the potential must satisfy the no-leak condition at the unknown,
moving free-surface and also the requirement of a spatially equal but time-dependent pressure in each cushion. These requirements are not automatically met so besides the incoming wave potential, additional potentials are introduced which represent pulsating source distributions over the mean wetted surface of the rigid part of the structure and
over the mean free-surface of the cushions.
The complex potential then follows from the superposition of the undisturbed wave potential , the wave diffraction potential 'd, the potentials associated with the 6 d.o.f.
motions of the rigid part of the construction j and the potentials associated with the
vertical motions of the free-surface within each cushion, ç
= ic{(
+ d)(O + 2ixi + i2 -fi
&CdS
6 C
(5)
j=1 c=1
in which:
= potential of undisturbed incoming wave çbd = diffraction potential
& = potential associated with vertical motions of the free-surface in the c-cushion rigid body motion in the j-mode
vertical motion of free-surface in c-cushion free-surface area of c-cushion
total number of independent, non-connected cushions
In the above equation the undisturbed wave potential qo and the diffraction potential çbd
together decribe the flow around the captive structure under the assumption that the free
surfaces within each of the air cushions is also rigid and non-moving.
The potentials / are associated with the flow around the structure oscillating in stil water
under the assumption that the free surface within each air cushion is rigid and ftxed.
The potentials qc are associated with the flow around the captive structure as induced by
the vertical motions ( of the free surface within each cushion.
The velocity potential associated with the undisturbed long-crested regular wave in water
of constant depth h is given by:
g coshk(X3 + h) 6ik(XIcostY+X2sinc()
- w2 coshkh
The fluid pressure follows from Bernoulli's law:
p(X1,X2,X3,t)
= p--
=p(X1,X2,X3)e-u',txi
=
(C =
Sc
=
with:
6 C
p(X1,X2, X3) = p = pw2{(çbo + qd)Co + >çbjxj + -
1f
qdS}
(8)c=1
Numerical approach
When considering a conventional rigid body, it is customary to determine the wave forces
on the captive structure based on the undisturbed wave potential q5c, the solution of the diffraction potential and the added mass and damping of the structure oscillating in
any one of the six modes of motion in stil water based on the solution of the motion
potentials . The motions of the structure are then determined by solving a 6 d.o.f.
equation of motion taking into account the wave forces, added mass and damping and restoring terms.
With a construction partially supported by one or more air cushions different approaches
may be followed in order to determine the motions of the structure, the pressures in the cushions and other relevant quantities such as the water motions within an air cushion. In the following a direct method is treated which solves the motions of the structure, the
free-surface behaviour within the air cushions and the cushion pressures as the solution of
a multi-body or multi-degree-of-freedom problem with added mass, damping and spring coupling effects. No data is obtained on the wave forces or added mass and damping of the structure including the effects of the air cushions. A second method, in which
the wave forces and added mass and damping including effects from the air cushions are
determined as the solutions of separate multi-body or multi-d.o.f. problems is treated in Pinkster [lo]. In that case the motions of the structure in waves are determined as the solution of a normal 6 d.o.f. equation of motion.
Direct Solution
For both methods the rigid part of the structure is modelled in the usual way by means
of panels representing pulsating sources distributed over the mean underwater part of the construction.
The free surface within each air cushion is also modelled by panels representing source
distributions lying in the mean free surface of each cushion. This level of the mean free
surface may be substantially different to the mean waterlevel outside the structure and
also different for each cushion.
Each panel of the free surface within an air cushion is assumed to represents a body
without material mass but having added mass, damping, hydrostatic restoring and
aero-static restoring characteristics. Each free surface panel (body) has one degree of freedom
being the vertical motions of panel n within cushion c.
-It will be clear that properties such as added mass coupling and damping coupling exists between all free surface panels and between free surface panels and the rigid part of the structure.
The direct method of solving for the motions of the structure and the vertical motions of free surface panels within the air cushions considers the total system in terms of a
freedom amount to:
=
mass coupling coefficient for force in n-mode due toacceleration in the j-mode. Zero for cushion panels
added mass coupling coefficient damping coupling coefficient spring coupling coefficient mode of motion
wave force in the n-mode
In the above equation it is understood that j = 1, 6 and n = 1, 6 represent motion and force modes respectively of the rigid part of the structure. Thecase of j > 6 and n> 6
represent the coupling between the panels of the free surfaces of the air cushions.
The case of j = 1, 6 and n > 6 represent the coupling between the six motion modes
of the rigid part of the construction and the vertical forces on free surface panels in the
cushions.
j > 6 and n = 1, 6 represent the coupling between the vertical motions of the free surface panels in the air cushions and the six force modes on the rigid part of the structure. The wave forces X, the added mass and damping coupling coefficients a and b3 are determined in the same way as is customary for a multi-body system. The mean under-water part of the structure is discretised into a number of panels representing pulsating sources as is the case with each free surface panel within an air cushion.
The contribution to the total potential due to the discrete pulsating source distribution over the structure and the free surfaces of the air cushions is as follows:
=
3(A)G(X,A)zS3 (11)in which:
N = total number of panels used to describe structure and free surfaces of all cushions
X
= X1,X2,X3 = afield point
A= A1, A2, A3 = location of a source
G(X,A)= Green's function of a source in A relative to a field point
anj
=
=
cfi=
xi=
xn
=
D.Q.F. =6+N
(9) in which:N= number of panels in cushion-c
and the number 6 accounts for the six degrees of freedom of the rigid part of the structure. The equations of motion for this case are as follows:
D.O.F.
i
+c}x =X
n= 1,D.O.F.
(10)i=l in which:
LS8 = surface element of the body or the mean free surfaces in the air cushions o = strength of a source on surface element s due to motion mode j
ç5j () = potential in point )? due to j-mode of motion
The unknown source strengths i are determined based on boundary conditions placed on the normal velocity of the fluid at the location of the centroids of the panels:
Nt
-
Tmj(X) +I:
a3(A)-G(, A)I.S3
=
.m= 1,N
(12)The right-hand-side of the above equation depends on the case to be solved. If the source
strengths for determination of the diffraction potential are required the normal velocity
vector becomes:
anm -
anm (13)It should be remembered that in this case the wave loads due to incoming waves and
diffraction effects are defined as being the loads on the structure and on the individual
free surface panels in the cushions, all being fixed. The added mass and damping coupling
coefficients are found by applying normal velocity requirements. For the six rigid body motions (j = 1, 6) of the structure:
=nmj...j=1,6
(14)in which the panel index in covers only the panels on the structure. n,, are the generalized
directional cosines for the panels on the structure given by:
m1 =
cos(nm,xi) m2 = cos(nm,x2) = cos(rtm,xa) = Xm2flm3 - Xm3Thm2 m5 = Xm3flml - Xm[nm3 m6 = XmlTim2 - Xm2flmI (15) in which:Xmj = co-ordinates of the centroid of a panel relative
to the body-axes
For this case the normal velocity components on all cushion panelsare equal to zero. For the determination of the added mass and damping coupling arising from the normal
motions of individual cushion panels the normal velocity boundary condition iszero except
for one cushion panel at a time for which the following value holds:
anm i (16)
where the 1 follows from the fact that the free surface normal is pointing in the negative
From the solutions of the source strengths for all these cases the'cVae force yetor .X
and the. added' mass and dmping,coupling coefficients ¡i cn 'be obtained. Thewave
force follows fiom:
± dk)nkASnk
in which:
çbdk = diffraction potential at k-panel obtained from equation (11)
X = wave force in the n-mode, n = 1, 6 for the structure
N '= number of panels involved in the force in the n-mode. for the force on a cushion panel N = 1. For the force on the rigid part of the structure N equals the total number of panels on that parti = generalised directional' cosine of k-panel related to n-mode = area of k-panel related to the force in the n-mode
The addedi mass and damping coefficients follow from:
= Re[P>çbjknflk/Sflk]
= Im[pw
ikThnkSflk}k=1
in which:
3k = motion potential value' on k-panel obtained from equation (il).
The restoring coefficients in general consist of two contributions i.e. an aerostatic spring term and an hydrostatic spring term.
The hydrostatic restoring term is equal to the product of waterlinearea , specific mass of water and acceleration of gravity. This applies to both the structure and the free surface panels. The aerostatic restoring terms are related to the change in air pressure in an air cushion due to, for instance, unit vertical displacement of a free surface panel and the corresponding forces applied to the particular panel, all other panels belonging to the same cushion and the force on the structure. Conversely, displacing the structure in any
of the three vertical modes of heave, roll or pitch will change the volume of an air cushion' thus inducing pressure changes and as a consequence forces on all free surface panels and
on the structure itself.
For the determination of the aerostatic part of the restoring terms, use is made of the linearised adiabatic law given in equation (i)
Based on the wave forces and added mass and damping coefficients, the wave frequency motions of the structure and the cushion panels are determined by solving equation (ho)'.
From these results other quantities may be derived such'as the air cushion pressure vari-ations and the mean second order wave drift forces.,
For the computation of the mean horizontal drift forceawe 'have made use of the far-field formulation as given by Faltinsen and Michelsen [3]:. This method' can be easily applied
to both. the rigid part of the çi.ir' and'. the 'free-surfacea of the air cushions
A limitation of the direct method for determining the behaviour of the structure is the fact that wave forces on the captive structure including the effect of the air cushions or added
mass and damping data for the case of the structure with air cushions oscillating in stil water are not obtained. In order to obtain this data also, a slightly different approach can be taken. See Pinkster[1O]. In the present contribution only the cushionpressures, wave
frequency motions and mean second order wave drift forces will be adressed. Therefore the method to obtain added mass, damping and wave forces including effects of the air cushions will not be treated here.
Validation of computational method by model tests
The test facility and the model
Model tests were carried out in No.1 towing tank of the Ship Hydromechanics Laboratory.
This facility measures 140 m x 4.25 m x 2.5 m. It is equiped with a hydraulically
oper-ated, flap-type wave maker by means of which regular or irregular waves can be generated.
For the model tests a simple rectangular barge model was constructed of wood.
The model consisted of vertical rigid side and end walls extending into the water to a
draft of 0.15 m. The thickness of the side walls amounted to 0.06 m and of the vertical
end walls amounted to 0.02 m.
The rigid horizontal deck of the barge which closed the air cushion(s) was situated 0.15 m above the stil waterline. The depth the barge measured from the lower end of the side
walls to the deck amounted to 0.30 m. The air pressure in the air cushion(s) was increased
relative to the ambient air pressure to obtain a mean waterlevel in the cushion(s) which was 0.05 m below the stil water level in the basin. The air cushion height between the free-surface in the air cushion and the horizontal deck amounted to 0.18 m.
Two arrangements with respect to the air cushions were tested i.e. a one-cushion
ar-rangement and a two-cushion arar-rangement. In both cases the rigid part of the model
consisting of rigid side and end walls and the deck were the same except that for the
two-cushion arrangement a rigid transverse bulkhead with a thickness of 0.02 m was added
at the midship location which separated the fore and aft cushions. In both cases the air cushions account for about 62% of the total displacement. The main particulars of both arrangements are give in Table I.
The data in this table shows that the main difference between the one- and two-cushion
barges lies in the longitudinal GM-values which is much higher for the latter. This is due
to the fact that when the two-cushion barge is trimmed , pressure differences are caused in the fore and aft cushions which contribute significantly to the pitch restoring moment.
Table I: Main Particulars of Air-cushion Barges
As the purpose of the tests was to validate the computational method, results of
com-putations and model tests are given for the actual model and are not extrapolated to
any full scale concept. Extrapolation to full scale concepts entails discussion with respect
to the influence of the model and full scale elastic properties of the air cushion. This aspect will not be adressed in this contribution. Suffice it to say that the computational
method is capable of taking into account full scale cases through equation(i). Modeltests
aimed at validating the computational method for fall scale structures however introduce additional modelling problems with respect to the air cushion stiffness. See, for instance, Moulijn[11]. By carrying out the comparison on model scale this complication is avoided.
For the computations, panel models of the rigid part of the barge and the free-surface of the cushions were constructed. The panel model of the single-cushion barge is shown in
figure 1 and the two-cushion barge is shown in figure 2.
The model test program, measurements and results
The model tests were carried out in regular waves in head seas only. The model was moored by means of a linear soft spring system. The fore and aft mooring springs were
connected at deck-level to force transducers measuring the surge mooring force. The mean
surge drift force was obtained by adding the mean values of the fore and aft surge force
transducers.
The surge, heave and pitch motions were measured using a simple wire/potentiometer
set-up. Cushion pressures were measured and in the case of the two-cushion arrangement, the pressure differential between the fore and aft cushionswas also measured.
From the tests in regular waves the R.A.O.s of the pressures and motions were obtained Quantity Units One-Cushion
Barge Two-Cushion Barge Length m 2.50 2.50 Breadth m 0.78 0.78 Draft m 0.15 0.15 Depth m 0.30 0.30 Displacement m3 0.130 0.130 KG m 0.30 0.30 CM(transv.) in 0.11 0.11 GM(long.) m 1.32 5.95 kxx m 0.223 0.223 kyy m 0.751 0.751 kzz m 0.727 0.727 Roll freq.
r/s
2.96 2.97 Pitch freq.r/s
4.33 4.97 Heave freq.r/s
5.00 5.00by Fourier analysis. The quadratic transfer function of the mean second order surge drift
force was obtained by dividing the mean surge force by the square of the amplitude of the undisturbed incident wave.
The model tests were carried out for wave frequencies ranging from approximately 2.5r/s to 8.Or/s. Computations were carried for frequencies ranging from 2.Or/s to 1O.Or/s. The results of computations and model tests are shown in figure 3 through figure 14.
Re-sults for the one-cushion arrangement are given in figure 3 through figure 7. The reRe-sults
for the two-cushion arrangement are given in figure 8 through figure 14.
Discussion of results
In general it can be seen that the main features of the measured data are well predicted by the computations. Significant differences can be seen in some specific cases but generally
quantitative agreement is good. The repeatability of the model tests was also good as
can be seen from the measured data.
The surge motions shown in figure 3 and in figure 8 indicate a somewhat better correlation between measurements and computations for the two-cushion barge. Near-zero values in the R.A.O.s related to the length of the cushions can be seen to be at different frequencies
for both cushion arrangements.
Heave motions shown in figure 4 and in figure 9 agree well with computations. The
R.A.O. values differ little for both cushion arrangements. The first zero in the heave motions occurs at approximately 5.0 r/s corresponding to a wave length of 2.5 m being
the length of the barge. In the case of the one-cushion barge this equals the length of the cushion. For the two-cushion barge the cushion length is half the wave length. In this case it appears that the vertical forces due to the fore and aft cushions being compressed compensate each other to produce a minimal heave force. In the two-cushion case we expect a relatively large pitch moment at this frequency since the fore and aft cushion
pressures will be in counter-phase.
The pitch motions of the barges are shown in figure 5 and figure 10. It can be seen that the peak of the R.A.O.s occur at about the same frequency. However, the peak value of the two-cushion barge is less than half the value for the one-cushion barge even though
from the aforegoing the pitch moment on the two-cushion barge will be much larger. This
can be explained by taking into account that for the one-cushion barge, the natural fre-quency for pitch is at 4.33 r/s which coincides with the peak pitch response. Due to the one-cushion arrangement, the cushion does not contribute to the pitch damping which,
as a consequence will be low and high pitch motion values will occur. In the two-cushion
case the natural frequency for pitch is at 4.97 r/s. Due to the two-cushion arrangement, the fore and aft cushions contribute to the pitch stability, added mass moment of inertia
and pitch damping. Even though the pitch moment will be larger than in the one-cushion
case, the pitch motions are stil considerably less. The correllation with measurement is also somewhat better. This is likely to be related to the fact that at the pitchresonance
The cushion pressures are shown in figure 6 for the one-cushion barge and in figure 11 and figure 12 for the fore and aft cushions respectively of the two-cushion barge. It can
be seen that the pressure amplitudes for the two-cushion barge are generally larger. This is certainly true for the peaks at the lower frequencies. This is due to the fact that for the one-cushion barge spatial equalisation of pressure takes place over a larger cushion area. The one-cushion pressure RA.O. shows clearly the zeros associated with the ratio between the wave length and the length of the cushion. The zeros in the pressure R.A.O.
at 5 r/s and 7 r/s correspond to a bargelength/wavelength ratio of i and 2 respectively. The situation for the two-cushion barge is less clear. This is related to the fact that the cushion pressures in the free-floating condition of this barge are more dependent on the
pitch motion.
In figure 13 the R.A.O. of the pressure difference between the fore and aft cushions of the two-cushion barge is shown. Comparing this value with the pressure R.A.O.s for the separate cushion in figure 11 and figure 12 shows that, certainly at the lower frequencies, the pressures in the cushions are out-of-phase.
The mean second order surge wave drift forces are shown in figure 7 and figure 14
re-spectively. The correlations between measurements and computations are of more or less
the same quality as is found for other, conventional, floating structures. The drift force values seem to be slightly smaller for the two-cushion barge. For the one-cushion barge a large peak value at 4.5 r/s is followed by a near-zero drift force at about 5.3 r/s. This is close to the frequency of minimum cushion pressure for this barge. The two-cushion
barge does not appear to show such near-zero values at higher frequencies.
Conclusions
From the results of the comparisons between the results of model tests and computations based on 3-dimensional linear potential theory it is concluded that in general the cushion
pressure, motion and mean drift force characteristics of a one-cushion barge and a
two-cushion barge can be predicted with reasonable accuracy. Differences between the results for both barges are in some cases clearly related to the cushion arrangement. Some differ-ences are less easily explained based on the presented data. For a more thorough analysis
data on the wave forces, added mass and damping of such constructions , including the
effect of the air cushions, need to be taken into account.
-The results presented indicate that the computational tool can be usefull to investigate the
merits of air-cushion supported structures in waves and as such can be used to optimise these relatively unknown and untried concepts for floating structures.
References
[1] Burns,G.E.,Holtze,G.C.:'Dynamic Submergence Analysis of the Khazzan Dubai
Sub-sea Oil Tanks', Paper No. OTC 1667, Offshore Technology Conference, Houston, 1972
[21 Kaplan,P. and Davis,S.: 'A Simplified Representation of the Vertical Plane Dynamics
of SES Craft', AIAA Paper No. 74-314, American Institute of Aerodynamics and
Astotronautics, 1974
Faltinsen,O.M. and Michelsen,F.: 'Motions of Large Structures in Waves at Zero
Froude Numbers', Tnt. Symp. Dynamics of Marine Vehicles and Structures in Waves, London, 1974
Seidel,L.H.: 'Development of an Air Stabilized Platform', University of Hawaii, Dept. of Ocean Engineering, Technical report submitted to U.S. Department of Commerce, Maritime Administration, 1980
Berthin,J. C. ,Hudson,W.L. ,Myrabo,D.O.: 'Installation of Maureen Gravity Platform over a Template', Paper No. 4876, Offshore Technology Conference, Houston, 1985 Kure,G. and Lindaas,O.J.: 'Record-Breaking Air Lifting Operation on the Guilfaks C Project', Paper No. OTC 5775, Offshore Technology Conference, Houston, 1988
Faltinsen,O.M. et al.: 'Speed Loss and Operability of Catamarans and SES in a
Seasway, Fast'91 Conference, Trondheim, 1991
Nakos,D.E. et al.: 'Seakeeping Analysis of Surface Effect Ships', Fast'91 Conference, Trondheim, 1991
Blood,H.:'Model Tests of a Pneumatically Stabilized Platform', Tnt. Workshop and Very Large Floating Structures, Hayama, Japan, 1996
Pinkster,J.A. : 'The Effect of Air Cushions under Floating Offshore Structures', Pro-ceeding Boss'97 Conference, Delft, 1997
Moulijn,J.:'Scaling of Air Cushion Dynamics', Report No. 1151, Laboratory of Ship
Figure 1: Panel model of the one-cushion barge. Total No. of panels = 444.
1.2 1.0 0.8 E a) 0.6 Q-E (D 0.4 0.2 0.0 0 2 3 4 5 6 7 8 9 10 frequency [radis]
Figure 3: Surge motions of one-cushion barge.
o 2 3 4 5 6 7 8 9 10
frequency [radis)
Figure 4: Heave motions of one-cushion barge.
1.0 0.8
io.6
E 0.4 (u 0.2 0.0300 250 0.1 50 0.5 0.4 E 1 0.3 Q-Q) D 0.2 0.0 O O 2 3 4 5 6 7 8 9 10 frequency Erad/SI
Figure 5: Pitch motions of one-cushion barge.
o 2 3 4 5 6 7 8 9 10
frequency [radis]
Figure 6: Cushion pressure of one-cushion barge.
100 200 E Q) a) D 150 Q. E w
1.2 1.0 0.8 0.4 0.2 0.0 o o CAL O EXP 3 4 5 6 7 8 9 10 frequency [rad/s]
Figure 7: Mean surge drift force of one-cushion barge.
0 1 2 3 4 5 6 7 8 g 10
frequency [radis]
120 100 80 60 40 20 CAL O EXP O 2 3 4 5 6 7 8 9 io frequency frad/sJ
Figure 10: Pitch motions of two-cushion barge.
O 2 3 4 5 6 7 8 9 io
frequency Erad/sl
Figure 9: Heave motions of two-cushion barge.
1.0 0.8 . 0.6 E a) J 0.4 0.2 0.0
2.0 1.8 1.6 1.4 E l 1.2 o-. 1.0
¡
0.8 0.6 0.4 0.2 0.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 E e 1.0 0.8 0.6 0.4 0.2 0.0 O CAL O EXPFigure 12: Cushion pressure aft cushion of two-cushion barge.
1 2 3 4 5 6 7 8 9 10
frequency [radis]
0 1 2 3 4 5 6 7 8 9 10
frequency [radis]
q) J E z . -2.0 -2.5 -3.0 -3.5 -4.0 0 1 2 3 4 5 6 7 8 9 10 frequency Erad/sl
Figure 14: Mean surge drift force of two-cushion barge.
O 2 3 4 5 6 7 8 9 10
frequency Erad/sl
Figure 13: Pressure difference between fore and aft cushions of two-cushion barge.
4.5 4.0 3.5 E 1i 3.0 Q-C q) 2.0 1.5 o-1.0 0.5 0.0 0.0 -0.5 -1.0
