The behaviour of large air cushion supported structures in waves

The behaviour of large air cushion

supported structures in waves

J.A. Pinkster & A. Fauzi

Deift University of Technology

Y. moue & S. Tabeta

Yokohama National University

Report 1183-P

December 1998

Proceedings of the Second International

Confe-rence on Hydroelasticity in Marine Technology,

Fukuoka, Japan, 1-3 december 1998.

TU Deift

Faculty of Design. Engineering and Productiony Department of Marine Technology

PROCEEDINGS OF THE SECOND INThRNATIONAL CONFERENCE

ON HYDROELASTICITY IN MARINE TECHNOLOGY

FUKUOKA/ JAPAN! 1-3 DECEMBER 1998

Hydroelasticity in Marine

Technology

Edited by

M. Kashiwagi

RIAM, Kyushu University

W. Koterayama

RIAM, Kyushu University

M. Ohkusu

RIAM, K ushu University

Research Institute for Applied Mechanics (RIAM),

Kyushu University! FUKUOKAIJAPAN/1998

The texts of the various papers in this volume were set individual/v bi typists under the supervision of each of

the authors concerned.

Published bi

Yomei Printing Cooperative Society, Fukuoka, JAPAN

(Tel: +81-92-582-5229, Fax: +81-92-582-5281)

ISBN 4-87780-001-8

Copyright©l 998, RIAM. Kyushu University

All Rights Reserved

Hydroelasticity in Marine Technology, Kashiwagl et al. (eds)© 1998 R!AM, Kyushu University

Contents

Preface

xi

Keynote lectures

Hydroelasticity of high-speed vessels

O. M. Fa/tinsen

Optimal structure for large-scale floating runways

15

W C. Webster

VLFS- I

An assessment of hydroelasticity for very large hinged vessels

27

C.-H. Lee & J. N. Newman

3D hydroelastic analysis of module linked large floating structures using quadratic

37

BE-FE hybrid model

T Hamamoto

An eigenfunction-expansion method for predicting hydroelastic behavior of

47

a shallow-draft VLFS

J. W. Kim & R. C. Errekin

Hydroelasticity of curved flexible slender structures

61

A. R. Ga/per & T Mi/oh

VLFS-II

A boundary element method to describe the excitation of waves in a very large

69

floating flexible platform

A. J. Hermans

Structural drag and deformation of a moving load on a floating plate

77

W Yeung & J. W. Kim

Numerical calculation of hydroelastic behavior of VLFS in time domain

89

Ohmatsu

Effect of short-crested irregular waves on response of a very large floating structure

99

N. Ma & T Hiravama

Impact problems

Unsteady hydroelasticity of floating plates

₁₀₉

A. Korobkin

A time-domain simulation method for hydroelastic impact problem

119

K. Tanizawa

Impact of rigid and elastic systems over the water surface

₁₂₉

A. lafrati, A. Carcaterra, E. Ciappi & E. E Campana

On the impact phenomenon of an elastic plate on a water surface in small attack angles

₁₃₉

S. Okada & Y. Sumi

Array of cylinders

Numerical study of complex flow around an array of circular cylinders in various

₁₄₅

arrangements

S. Etienne, Y M. Scolan & B. Mo/in

Resonances in wave diffraction/radiation for arrays of elastically-connected cylinders

155

T Utsunorniva & R. Eatock Tar/or

Hydrodynamic interactions among a great number of columns supporting a very large

165

flexible structure

M. Kashiwagi

VLFS -Ill

Hydroelastic behavior of a large floating platform of elongated form on head waves

177

in shallow water

M. Ohkusu & Y. Na,nha

A hybrid element approach to hydroelastic behavior of a very large floating structure

₁₈₅

in regular waves Seto & M. Ochi

An analysis of the hydroelastic behavior of large floating structures in oblique

waves 95

H. Sim & H. S. Choi

VLFS -IV

Some efficient calculation techniques for hydroelastic analyses of a very large floating

₂₀₁

structure in waves

H.

_{Kagemoto, M. Fujino, M. Murai & T Zhu}

Hydroelastic analysis of semi-submersible type VLFS capable of detailed structural analysis

₂₁₁

K. lili/na, K. Yoshida & H. Suzuki

On B-spline element methods for predicting the hydroelastic responses of

a very large ₂₁₉

floating structure in

waves X. Lin & M. Takaki

Reduction of the motion of an elastic floating plate in waves by breakwaters

₂₂₉

S. Nagata, H. Yoshida, T Fujita & H. lsshiki

VLFS- V

The oblique incidence of surface waves onto the leastic band

₂₃₉

I. V Sturova

The response of VLFS with anisotropic rigidity in waves

₂₄₇

T. Mikami, M. Kobayashi. K. Shimada, S. Miyajima, M. Kashiwagi & M. Ohkusu

Timoshenko equation of vibration for plate-like floating structures

255 H. Endo & K. Yoshida

Modal analysis experiment of clamped-free elastic cylindrical shell surrounded by

water 265

O. Saijo, A. Nishida & F Mivashita

Cable, towed vehicles

High frequency motion components in a large dynamic duty umbilical

₂₇₃

G. J. Lyons, H. H. Cook & M. Ashworth

A novel computer simulator for cable-towed submerged vehicles

₂₈₃

Z H. Zhu & B. C. Morrow

Prediction of towing cable tension and comparison with field experiment

293 W Kotera rama. N. Yamawaki, T Yokobiki & H. Que

Risers and pipelines

Deep water drilling riser systems

₃₀₁

ishida. M. Ohas/ii, H. Fujira & H. Yasukawa

Damping of VI V-induced axial vibration in deep

sea risers

₃₀₉

E. Huse & G. Kleit'en

Axial stress-control of pipe string for mining manganese nodules in deep

sea 31 7

by using the string with non-uniform cross section

G. Cui, K. Aso & H. Doki

Flow-induced vibration

Vortex induced vibrations of pipe in high waves. Field measurements

₃₂₅

N-E. Ortesen Hansen & B. Pedersen

Added mass and oscillation frequency for a circular cylinder subjected

to vortex 335

induced vibrations and external disturbance

K. Vikestad, C. M. Larsen & J. K. Vandiver

A comparative validation study of Navier-Stokes codes applied on vortex induced vibrations

345

R. Krokstad, F Solaas & J. Da/heim

Third-harmonic diffraction forces on a truncated cylinder

₃₅₇

S. Karo & B. Teng

Sea quake and Tsuna,ni

Wave and earthquake response of large offshore platforms

₃₆₅

K. Kawano & K. Venkararamana

A study on prediction method of time history response of very large floating offshore

375

structure by sea shock force

M. Bessho, H. Maeda, K. Masuda & K. Shimiu

Experimental and Numerical study on the hydroelastic behavior of VLFS

_{under Tsunami 385}

S. Sakai, X. Liu, M. Sasarnoro & T Kagesa

Wave response analysis of an elastic floating plate in a weak current

₃₉₃

E. Watanabe, T Ursunomiva & S. Taenaka

High-speed vessels

Forward speed effect on the structure responses of a ship travelling in waves

401

S. Du, Y Wu & W G. Price

Wave-induced hydroelastic response of fast monohull displacement

_{ships 411}

J. J. Jensen & Z. Wang

A practical method for the hydroelastically pseudo-nonlinear ship structural dynamic

419

response in regular waves

} ¡noue, K. A. Hafez & M. Arai

Ship-related problens

Analysis of hydroelastic effects on propeller performance in steady and unsteady flows

429

D. J. Georgiet' & M. Ikehata

Coupled low frequency responses of vessels and their non-linear moorings in random waves

439

A. Sarkar & R. Eatock Taylor

Numerical solution of the flow past a freely oscillating body in waves and current

₄₄₉

G. Graziani, M. L.andrini & O. M. Fa/tinsen

SES and other vessels

Computation of cobblestone effect with unsteady viscous flow under a stern seal bag

461

of a SES

N. Hirata & O. M. Fa/rinsen

Sea trial test results on the wear characteristics of SES bow seal

fingers 47 1 Yamakita & H. Itoh

Comparison of simulated global stresses with full-scale measurements on an aluminium

477

fast patrol vessel

P Kannari, P Klinge, S. Rintala, T Karppinen, T P.1. Mikkola & A. Ranianen

Time-domain analysis of ship motions and hydrodynamic pressures on a ship hull in waves

485

Elastic structures

The behaviour of large air cushion supported structures in waves 497

J. A. Pinkster A. Fauzi, Y. ¡noue & S. Tabeta

Hydroelastic coupling of beam structural model with 3D hydrodynamic model 507

S.

Malenica

Statistical linearization of cable equations: an application to a guyed tower offshore structure 513

A. Culla & A. Carcaterra

Author index 522

Hydroelasticify in Marine Technology, Kashiwagi et al. (eds)© 1998 RIAM, Kyushu University

Preface

Hydroelasticity is of concern in various areas of marine technology. such as high speed vessels,

floating airports. floating bridges. buoyant tunnels, risers, various cable systems and umbilical for

ROV, and flexible containers for water transport and oil spill recovery.

Analysis for design of such

structures or systems necessitates integration of hydrodynamics and structural mechanics

hydro-elasticity plays the key role.

There has been significant recent progress in research into hydroelastic phenomena and the

topics of hydroelasticity are becoming more of current interest. One reason is that the innovation

in

marine technology requires it.

Another will be that we are now ready for analysis of complex

hydroelastic phenomena which would not be amenable to numerical implementation if it

were not

for high speed computers. Hydroelasticity in marine technology is making

a new frontier discipline.

The purpose of the Second Conference in Fukuoka following the first

one in Trondheim in

1994 is to bring together experts on both hydrodynamics and structural dynamics from various

fields of application and encourage the exchange of ideas and information among various disciplines

in science and engineering. We hope it will promote development of hydroelasticity in marine

technology and serve as a forum for this new discipline of applied mechanics.

The Conference fortunately attracts interest of many scientists and engineers. Topics of the

papers found in these proceedings are of wide spectrum. Two keynote lectures give overviews of

two important areas, hydroelasticity of high-speed vessels and design of large-scale floating

runways. One significant topic of the Conference is hydroelastic behavior of very large floating

platforms of extremely small rigidity relative to their size. lt is studied in various

contexts, in waves.

seaquake and Tsunami. Enipact of a flexible body on the water surface is one example of high speed

hydroelasticity increasingly attracting our attention.

Flow induced vibration is always a topic of

our concern: hydroelasticity of flexible slender structures for various applications in marine

technology,

risers,

pipelines and umbilical

for submerged vehicles are vigorously

studied.

Hydroelastic behavior of flexible membrane structure such as the seal bag of a Surface Effect Ship

(SES) is an important topic in the proceedings.

_{Local and global hydroelastic effects are crucial in}

high-speed vessels and recent progress focusing on nonlinearity are presented.

Makoto Ohkusu

Chairman of the Organizing Committee

Research Institute for Applied Mechanics, Kyushu University

The behaviour of large air cushion supported structures in waves

J. A. Pinkster & A. Fauzi

Ship Hydroniechanics Laboratory, Delft University of Technology, Deift, The Netherlands

Y. moue & S. Tabeta

Department of Naval Architecture ¿ Ocean Engineering, Yokohama National University, Yokohama, Japan

ABSTRACT: A computational method to determine the effects of air cushions on the behaviour of floating structures in waves at zero speed 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 mean wetted surface of the structure are described by panel distributions representing oscillating sources. Model tests were carried out with a captive model to validate computed data with respect to wave forces, added mass and damping and cushion pressures. Model tests were also carried out with a simple, free-floating rectangular barge in regular waves. Results of measurements and computations are compared with respect to cushion pressures, motions and mean second order wave drift forces.

i

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. (1972) These inverted bell-shaped units with open bottom were transported to location l)y 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 n the spaces between the skirts. See Kure et al. (1988). 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 loca-tion. Air cushions had been used previously (1974) to increase the buoyance of condeep structures but in the case of Gullfaks C the relative magnitude of the air cushion buoyancy was much greater.

Berthin ei al. (1985) describe the use of air cush-ions in the floatout operation of the Maureen Gravity Platform from the construction yard. In this case the air was also released on 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

Hydroelasticity in Marine Technology, Kashiwagi el al. (eds)© 1998 RIAM, Kyushu University

497

the air cushions was to reduce the draft of the structure to allow transportation over a shallow water area. Ex-cept for the case of the Khazzan Dubai units, the static characteristics of air cushions were of main importance. In Burns et al. (1972), 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 et al. (1974),Faltinsen et al. (1991) and Nakos ei al. (1991).

In the 70's the Seatek Slo-Rol system was intro-duced Lo reduce the wave-inintro-duced 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

pon-toon of the unit.

As a result of the application of this system, the roIl 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. Iwata et al. (1986) and Iwata et al. (1987) studied

the motions in waves of a catenary moored structure partially supported by an air cushion. Numerical stud-ies were carried out using a two-dimensional seperate region method based on the eigen inction expansion of velocity potentials. A linearized adiabatic law was applied for the air pressure in the cushion. Some model experiments were carried out in a wave tank in order to validate the numerical niethod.

lii recent years, a pneumatically stabilized platform lias been investigated for application as a permanent maritime platform in an open sea environment. See Blood (1996). This concept incorporated 75 indepen-dent air filled cylinders which were open to the sea on the underside. Model tests were carried out to deter-mine the air-pocket factor as defined by Seidel (1980). Results of such investigation indicate that air cush-ions can modify the behaviour of structures in waves considerably and justifies a more detailed investigation into t hie effects of various air cushion configurations

Based on this observation it was decided to modify an existing linear 3-dimensional diffraction code DEL-FRAC of (he Deift lJiiiversit.y of Technology to take into account the effect of one or more air cushions un-der a structure floating ri waves at. zero forward speed.

In tins paper, a review rs given of the main

el-ennents of (lie theory underlying (lie computational method 'The nurrierical modelling of the structure and the ai, cushions is treat.ed and results of comparisons between computations and model tests are given of the

wave-induced air-cushion pressures, wave forces,

hydro-ulyriarnic coefficients, motions and mean second order wave drift forces on a barge-shaped vessel supported

b one or by two air cushions. In a previous paper,

see Pinkster (1997), an outline of the theory has been given along with examples of the effect of air cushions under different structures based on computations only

2 AIR CUSIIION DYNAMICS

The theory is given for an air cushion supported con-struction 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 iìo air leak-age 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. The wave frequency air pressure variations within a cushion are determined by the change in cushion volume through the linearized polytropic gas law:

Lp = vol * n

* (PO + p)/vol (1)

in which:

498

vol = mean air volume in cushion

po = atmospheric pressure

Pc = mean excess of pressure in cushion

vol = wave frequency volume change in cushion = pressure variation relative to mean

cushion pressure = gas law index

For wave frequency pressure variations adiabatic corì-ditions are assumed. In that case = 1.4.

3 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 determined based ori linear 3-dimensional potential

theory.

Use rs made of a right-handed, earth-fixed O

X,

-A2 X3 system of axes with origin in the mean water level and X3-axis vertically upwards. The body axes

G xi 2 x3 of the rigid part of the construction

has its origin iii the center of gravity of the construc-tion, with Xj axis towards the bow, x2 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

el-evationi is as follows:

((X1, X2,t) = (2)

in vhiclì

k = wave number

a

= wave direction, zero for waves from dead astern

w = wave frequency

Co = wave amplitude

ç = wave elevation

The motions of the rigid part of the construction in the j-mode relative to its body axes are given by:

z (t) =

(3)

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 as

follows:

cIi(X1,X2, X3, t) = (Xj, X2, Xa)ewi (4)

The potential ç satisfies Laplace's equation, the lin-earised boundary conditions on the free surface out-side the body, the boundary condition at the sea-floor and, excepting the undisturbed incoming wave poten-tial, 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 poten-tial must satisfy the no-leak condition at the unknown,

moving free-surface and also the requirement of a spa-tially equal but time-dependent pressure in each cush-ion. These requirements _{are not automatically met so} besides the incoming wave potential, additional

poten-tials 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 q then follows from the

super-position of the undisturbed wave potential o, the wave

diffraction potential d, the potentials associated with the 6 d.o.f. motions of the rigid part of the

construc-tion _{and the potentials associated with the vertical}

motions of the free-surface within each cushion, =

fJ

c(cdSc} j=1 c=1

(5)

in which:

= potential of undisturbed incoming wave = diffraction potential

1,c = potential associated with vertical motions of the free-surface in the c-cushion

xi = rigid body motion in the j-mode

(C = vertical motion of free-surface in c-cushion

Sc free-surface area of c-cushion C = total number of independent,

non-connected cushions

lu the above equation the undisturbed _{wave potential} and the diffraction potential d together decribe the flow around the captive structure under the assumption that the free surfaces witlnn each of the air cushions is also rigid and non-moving.

The potentials _{j are a-csociated with the flow around} he structure oscillating in stil water under the

assurnp-tion that the free surface within each air cushion is rigid and fixed.

The potentials q5 are associated with the flow around t.he captive structure as induced by the vertical

mo-tions (. 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)jk(XcO_+Xjfl)

(6)

w2 coshkh

The fluid pressure follows from Bernoulli's law:

p(Xj,X2,V3,t) =

_{p-a-- =p(Xj,X2,X3)e"'}

(7)

p(Xi,X2,X3) =p

(8) and: 6 C p = 2{(

+d)(+&x+

U c(cdSc}

j=I c=1 (9) 499 4 _{NUMERICAL APPROACH}

When considering a conventional rigid body, it is

cus-tomary to determine the wave forces on the captive structure based on the undisturbed wave potential _o, the solution of the diffraction potential

_{'d 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 çb- The motions of the structure are then determined by solv-ing a 6 d.o.f. equation of motion taksolv-ing intoaccount

the wave forces, added mass and damping and

restor-ing terms.

With a construction partially supported byone 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

cou-pling effects. No data is obtained on the wave forces or added mass and damping of the structure including the effects of the air cushions. The second method, in which the wave forces and added mass and damp-ing includdamp-ing effects from the air cushions are deter-mined as the solutions of separate body or multi-d.o.f. problems is also treated in this paper. See also Pinkster (1997) In that case the motions of the

struc-ture in waves are determined as the solution ofa normal 6 d.o.f. equation of motion.

For both methods the rigid part of the _structure is modelled in the usual way by means of panels rep-resenting pulsating sources distributed over the mean

underwater part of the construction.

The free surface within each air cushion _{is aLso} mod-elled 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 restor-ing and aero-static restorrestor-ing 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} cou-pling and damping coucou-pling exists between all free

sur-face panels and between free sursur-face panels and the

rigid part of the structure.

4.1 Direct solution

structure and the vertical motions of free surface panels within the air cushions considers the total system in terms of a multi-body problem with mass, damping and spring coupling. The number of degrees of freed orn amount to:

D.O.E. = G + N, (IO)

in uìich:

N, = number of panels in cushion-c

and the number 6 accounts for the six degrees of free-dom of the rigid part of the structure.

The equations of motion for this case are as follows:

D O.F

w2(M + a) - iwb

+c,}x,

Xr

(Il)

in which:

ri = 1, D.O.F.

= mass coupling coefficient for force in n-mode due to acceleration in the j-mode. Zero for cushion panels

added mass coupling coefficient = damping coupling coefficient. ci,, = spring coupling coefficient

z) = mode of motion

Xr, wave force in the n-mode

In the above equatiomi it is understood that j = 1, 6 and n = 1, G represent motion and force modes respectively

of the rigid part of the structure. The case 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 arid the vertical forces on free surface pan-els in the cushions.

i > 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,,, t.he added mass and damping coupling coefficients a,,j and b,, are determined in the same way as is customary for a multi-body system. The mean underwater 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 dis-crete pulsating source distribution over the structure and the free surfaces of the air cushions is as follows:

N

1,(J)=

(12)

in which:

500

N1 = total number ofpanels on the structure

and the free surfaces of all cushions

= (X1,X2,X3) = location

ofa field point

A = (A1, A2, A3) = location of a source G(X, A) = Green's function of a source in A

relative t.o a field point X

= surface element of the body or the mean free surfaces in the air cushions

= strength of a source on surface element s due to motion mode j

(X) = potential in point due to j-mode of motion

The unknown source strengths c are determined based on boundary conditions placed on the normal

velocity ofthe fluid at the location of the centroids of the panels: N,

-

Urn

?)+

j

l7r an (13) wit h: 771 = 1, N1 (14)

The right-hand-side of the above equation depends ori the case to he solved. If the source strengths for determination of the diffraction potential are required the normal velocity becomes:

an,,, -

anm (15)

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 coeffi-cients are found by applying appropriate normal ve-locity requirements. For the six rigid body motions (j = 1,6) ofthe structure:

an,,,

aj

= (16)

in which the panel index mn covers only the panels on the structure. mj are the generalized directional cosines for the panels on the structure given by:

72m1 = COS(flm,Xi) = COS(flm,X2) = COS(Tlm, z3) m4 = Xm2Tim3 - Xm371m2 m5 = Xm3flmi - Xm171m3

rimS XmiTlm2 - Xm2flmi (17)

501

are related to the change in air pressure in an air cush-ion due to, for instance, unit vertical displacement ofa

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. Con-versely, displacing the structure in any of the three

ver-tical modes of heave, roll or pitch will change the vol-ume of an air cushion thus inducing pressure changes and as a consequence forces on al! free surface panels and on the structure itself.

For the determination of the aerostatic part of the

restoring terms, use is made of the hinearised adiabatic law given in equation (1)

Based on the wave forces and added mass and damping coefficients, the wave frequency motions of the struc-ture and the cushion panels are determined by solving equation (11). From these results other quantitiesmay

be derived such as the air cushion pressure variations and the mean second order wave drift forces.

For the computation of the mean horizontal drift forces we have made use of the far-field formulation as given by Faltinsen ei al. (1974). This method can be easily applied to both the rigid part of the structure and the free-surfaces of the air cushions.

A liniitation of the direct method for deterniining

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 is taken. 4.2 The second approach

Starting point of the second method is again

equa-tion (13) which expresses the normal velocity boundary condition on the structure and the cushion panels. When determining the wave forces including the effect of the air cushions it is necessary to solve the cushion panel motions for the case that the structure is captive. Having derived the pane! motions it is then possible to determine the total forces on the captive structure tak-ing into account the wave forces in the structure, the

added mass and damping coupling effects due _{to the} cushion panel motions and the air _{pressure variations} in the cushions. In order to determine the various ef-fects the following steps are taken:

The source strengths _{are determined by solving} equation (13) for the boundary condition of equation

(15). _{Based on these results the wave forces on the}

structure without the effect of the motions of the free-surfaces in the air cushions are found. From these cal-culation the wave forces on the fixed panel cushionsare

also found.

The added mass and damping coupling coefficients as-sociated with vertical motions of the individual cushion panels are found by solving for the source strengths _Qj Tim = normal of m-pane!

mi = co-ordinates of the centroid of a panel

relative to the body-axes

For this case the normal velocity components on all cushion panels are 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 is zero except for one cushion panel at a time for which the following value holds:

thI)m

(18)

where the 1 follows from the fact that the free surface norma! is pointing in the negative X3-direction. From the solutions of the source strengths for all these cases the wave force vector X and the added mass and damping coupling coefficients can be obtained. The wave force follows from:

N,

xl = -

('Ok + Çlidk)Tl,.,.1Sk (19)

iii which:

diifraction potential at k-panel obtained from equation (12)

X,. = wave force in the n-mode, n = 1,6 for the structure

N,. = number of panels involved in the force

ii the ri- 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 part. = generalised directional cosine of

k-panel related to ii-mode area of k-panel

The added mass and daniping coefficients follow from:

a,.

Re[p

_7iflLS]

k=i

= Irn[pk

hkJ

(20)

k=i

in which:

= motion potential value on k-panel obtained from equation (12).

The restoring coefficients c,. in general consist of two contributions i.e. an aerostatic spring term and_an hy-drostatic spring term.

The hydrostatic restoring term is equal to the product of waterline area , specific mass of water and accelera-tion of gravity. This applies to both the structure and the free surface panels. The aerostatic restoringterms

in equation (13) for the case of vertical oscillations of each cushion panel individually using normal velocity boundary condition of equation (18).

Finally the motions of the cushion panels are de-terinined by solvrng the equations of motion for these panels using the above mentioned wave forces and added mass and damping coupling coefficient as well as the spring coupling coefficients based on the aerostatic restoring coefficient based on equation (1) and the hy-drostatic restoring coefficients of the cushion panels:

DOF

a,1 -

+ c}x3 = X

(21)

t li

n = 7, D.O.E. (22)

In this equation, the added mass and damping co-efficients and the wave forces are the same as applied in equation (Il). Froni tue solution of the equations of motions of the cushion panels the total wave forces on the captive structure can be determined as well as the pressure variations within the cushions.

The next step

is to determine the added mass and

(larliping of the structure including the effect of the free surfaces in the air cushions.

in order to accomplish this it is first necessary to de-termine the solution of the source strengths and fluid pressures for the case that the structure is oscillated while time cushion panels remain fixed. This is accom-plished by solving equat.ion (13) for the case that. the normal velocities on the panels of the structure are in accordance with equation (16) while the normal veloc-ities on the cushion panels are equal to aera.

For each of the six modes of motion of the structure, this results in hydrodynarnic loads on the structure and ou the cushion panels. To these loads we also need to add the aerostatic forces since the oscillations of the structure change the pressure in the cushions.

Based on the total forces on the cushion panels, the added mass and damping coupling coefficients and the aera- and hydrostatic spring coefficients the equations of motion of the cushion panels can again be solved:

DOF 2 2

- iib,, + c)x

X (23) vit.h:

n=7,D.O.F.

(24)

In this equation, the coefficients and wave forces are again in accordance with equation (11). From the solution of the motions of the cushion panels, the addi-tional hydrodynamic and aerodynamic contributions to time forces on the structure can easily be obtained thus

502

yielding the added mass and damping includuig

cush-ion effects. Based on the thus obtained added mass, damping and wave forces the wave frequency motions of the structure can be determined from the normal six degrees of freedom equations of motion:

2

_{+ a) - iwb,, + c1}x = Xn}

(25)

j=i

with:

ri= 1,6

(26)

In tins equation the added mass, damping and

spring coefficients as well as the wave forces apply to the structure only and include the effects of air

cush-i on s.

5 VALIDATION OF COMPUTATIONAL

MODEL

5.1 CeneraI

As the purpose of (lie tests was to valid ate the com-putational method, results of computations and model tesis are given for the actual model and arc not. extrap-olated to any full scale concept. Extrapolation to full scale concepts entails discussion wit.h respect to the in-fluence of the model and full scale elastic properties of the air cushion. This aspect will iiot be adressed in this contribution. Suffice mt to say that the computational

method is capable of taking into account. full scale cases

through equation(i). Model tests aimed at validating the computational method for full scale structures how-ever introduce additional modelling problems with re-spect to the air cushion stiffness. See, for instance, Moulijn (1998). By carrying out the comparison on model scale tins complication is avoided.

5.2 The test facility

Model tests were carried out in No.1 towing tank of the Ship Hydromechanics Laboratory of the Delft University of Technology. This facility measures 140 m x 4.25 m x 2.5 m. It is equiped with a hydraulically operated, flap-type wave maker by means of which

reg-ular or irregreg-ular waves can be generated.

5.3 Captive model tests

For these tests a simple rectangular barge model measuring 2.50 ni x 0.70 m was constructed of wood. The model consisted of a horizontal deck surrounded by vertical side and end walls. The depth of the model (wall depth) amounted to 0.50 m. The draft of the barge measured to the lower edge of the side and end

walls amounted to 0.30 m. For tests including two cush-ions, a vertical bulkhead was added midships. The wall and deck thickness amounted to 2.0 cm. For all tests the static air pressure in the cushion(s) was increased relative to the ambient. pressure to bring the mean wa-ter level inside the cushions to 0.15 w below the mean waterline of the barge. For t.he tests the model was attached to the vertical legs of a vertical motion oscil-lator fixed to the carriage of the basin. A sketch of the model set-up the positions of force transducers and of pressure transducers is shown in figure 1. The dis-placement of these barges amounted to 0.2625 m3.

For the computations of the wave forces, added mass, damping and air pressures, panel models of both barge variants were developed. These are shown in fig-ure 2 for the 1-cushion and 2-cushion cases. In these figures, the inner horizontal panels are at the mean

wa-terlevel of the free-surface in the air cushions.

5.3.1 The model test program, measurements and results

Cap t.ive model tests were carried out in regular head

waves for both cushion configurations. Measurements included the heave force, pitch moment, cushion

pres-sures and the water elevation inside the cushion. Forced heave oscillation tests were carried out with the longitudinal axis of the model at right-angles to the axis of the basin.

A comprehensive set of measurement data from 1)0th the captive model tests in waves and the forced

os-cillation tests mn stil water are given by Tabeta (1998). In this paper the maui results are shown. For the cap-tive tests in waves these consist of R.A.O.s of wave forces and air pressures. For the forced heave oscil-lation tests these consist of values of added mass and damping and the cushion pressure RAUs.

The results for the captive model tests in waves are shown in figure 4 through figure 9. Results of the forced oscillation tests in heave are shown in figure 10 through figure 15. In all figures the corresponding computed data are also show n.

5.3.2 Discussion of results

The results of the heave force measurements shown

in figure 4 and figure 6 show near-zero forces at

frequen-cies corresponding to about 5 r/s and 7 r/s. At these frequencies the wavelength equals 1.0 and 0.5 times the barge length respectively The cushion pressure shown in figure 5 for the one-cushion barge shows a strong resemblance to the heave force in figure 4. This is due

503

to the fact that the barge sides are very slender and that the heave force is almost entirely due to air pres-sure variations. For the two-cushion barge the results shown in figure 8 and figure 9 on the air pressures in the fore- and aft cushions also show similar trends to the heave force in figure 6. The similarity is somewhat less than is the case with the one-cushion barge how-ever. This is due to the fact the cushion pressures are not in phase. This is demonstrated by the results in figure 7 which contains the R.A.O. of the pressure dif-ference between the fore- and aft cushion. In all cases the correlation between the computed and measured heave forces and cushion pressures is good. Computa-tians tend to overestimate the pressure and force peaks at 7.5 r/s shown most clearly in the results of the one-cushion barge.

The heave added mass and damping of both barges are shown in figure 10 through figure 13. For heave motions, both added mass and damping should be the same for both barges. This is a result of the fact that the barges are symmetrical about the midship. The re-suits indicate that results are indeed almost thesame.

Some differences are seen in the computed results for both barges. See for instance the peak in the heave damping for the one-cushion barge in figure 11. This peak is not present in the results given in figure 13 for the two-cushion barge. These differences in the

com-puted results are attributed to numerical effects related to the panel models.

Finally, result on the air cushion pressures during the heave oscillation tests are shown in figure 14 and figure 15. Again, due to symmetry, the results for both barges should be almost the same. As before, this is generally true although the computed results for the one-cushon barge show larger oscillations at

frequen-cies above 7.5 r/s. Correlation between computed and measured data is good.

5.4 Free-floating model tests

For these model tests a simple rectangular barge model was constructed of wood. The dimensions of the barge, specifically the thickness of vertical side and end walls, were selected to obtain sufficient transverse and longitudinal static stability in free-floating condition.

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 in-creased 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 t.he air cushions were tested i.e. a one-cushion arrangement and a two-cushion arrangement. In both cases the air cushions account for about 62% of the total displacement. The main particulars of both arrangements are given in

Ta-ble I.

Table I: Main Particulars of Air-cushion Barges

The data in this table shows that the main difference between the one- and two-cushion barges lies in the lon-gitudinal GM-values which is much higher for the

lat-ter. 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. This does not occur with the one-cushion barge.

Panel niodels of the barges are shown in figure 3.

5.4.1 The model test program, measurements and results

The model tests were carried out in regular head

waves 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 transduc-ers 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 difference between

504

the fore and aft cushions was also measured.

From the tests in regular waves the R.A.O.s of the pressures and motions and the transfer function for the mean second order wave drift force were obtained. The results of computations and model tests are shown in figure 16 through figure 27.

5.4.2 Discussion of results

The surge motions shown in figure 16 and in fig-ure 20 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 17 and in figure 21 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 cor-responding 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 arid aft cushions being compressed compensate each other to produce a mininial heave force. In the two-cushion case we expect a relatively large pitch moment at this frequency since the fore and aft cushion pres-sures will be in counter-phase.

The pitch motions of the barges are shown in figure 18 and figure 22. 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 frequency for pitch is at 4.33 r/s which coin-cides wit.h the peak pitch response. Due to the one-cushion arrangement, the one-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 stabil-ity, 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 con-siderably less. The correllation with measurement is also somewhat better. This is likely to be related to the fact that at the pitch resonance frequency viscous effects play a smaller role in the two-cushion case. The cushion pressures are shown in figure 19 for the one-cushion barge and in figure 23 and figure 24 for the fore and aft cushions respectively of the two-cushion

Quantity Units 1-Cushion

Barge 2-Cushion Barge Length ro 2.50 2.50 Breadth ru 0.78 0.78 Draft ru 0.15 0.15 Depth ru 0.30 0.30 Displacement ru3 0.130 0.130 KG ru 0.30 0.30 GM(transv.) ni 0.11 0.11 GM(long.) ru 1.32 5.95 kxx ru 0.223 0.223 kyy ru 0.751 0.751 kzz ni 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 5.00

barge. The correlation between measurement and com-putation is better for the one-cushion barge. The pres-sure amplitudes for the two-cushion barge are gener-ally 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 pres-sure R.A.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 25 the R.A.O. of the pressure difference be-tween the fore and aft cushions of the two-cushion barge is shown. Comparing this value with the

pres-sure R.A.O.s for the separate cushion in figure 23 and figure 24 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 26 and figure 27 respectively. The cor-relations 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 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.

6 CONCLUSIONS

From the results of the comparisons between the

re-sults of model tests and computations based on

3-dimensional linear potential theory it is concluded that in general hydrodynamic added mass and damping and cushion pressures obtained from oscillation tests and

the wave forces and the cushion pressures obtained from captive model tests in regular wave can be pre-dicted with good accuracy. This is also the case for the cushion pressure, motion and mean drift force char-acteristics of the free-floating, soft moored barges in regular head waves. Differences between the results for both barges are in some cases clearly related to the cushion arrangement.

The result.s 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 investigate these relatively unknown and

un-tried concepts for large floating structures.

505

REFERENCES

Burns,G.E.e al., 1972 "Dynamic Submergence Analysis of the Khazzan Dubai Subsea Oil 'ranks " ,

Pa-per No. OTC 1667, Offshore Technology Conference,

Houston

Kaplan,P.e al., 1974 "A Simplified Representation of the Vertical Plane Dynamics of SES Craft ", AIAA

Paper No. 74-314, American InsituLe of Aerodynamics and Astronautics

Faltinsen,O.M.e al., 1974 "Motions of Large Structures in Waves at Zero Froude Numbers ", ini.

Symp. Dynamics of Marine Vehicles and Structures in

Waves, London

Seidel,L.H., 1980 "Development of an Air Stabi-lized Platform ", University of Hawaii, Dept. of Ocean Engineering, Technical report submitted to U.S. De-partment of Commerce, Maritime Administration

Berthin,J.C.ei al., 1985 : "Installation of Maureen Gravity Platform over a Template ", Paper No. 4876,

Offshore Tech nology Conference, Houston

Iwata,K.ei al., 1986 "Characteristics of Motion Response of a Floating Structure supported by Air-cushion in Waves ", Proceedings of Coastal Engineer-ing, JSCE (Japan Society of Civil Engineering), Vol. 33, (In Japanese)

Iwata,K.ei al., 1987 "Characteristics of Motion Re-sponses of a Moored Floating Structure supported by Air-cushion and deformation of Waves around it", Pro-ceedings of Coastal Engineering, JSCE, Vol. 34, (In Japanese)

Kure,G. et al., 1988 _{"Record-Breaking Air Lifting} Operation on the Gullfaks C Project ", Paper No. OTC 5775, Offshore Technology Conference, houston

Faltinsen,O.M. et cil., 1991 : "Speed Loss and

Oper-ability of Catamarans and SES in a Seasway ", Fast '91

Conference, , Trondheim

Nakos,D.E. et al., 1991 "Seakeeping Analysis of Surface Effect Ships ",Fast'91 Conference, Trondheim

Blood,H., 1996 "Model Tests of a Pneumatically Stabilized Platform ", Ini. Workshop and Very Large

Floating Structures, Hayama

Pinkster,J,A., 1997 "The Effect of Air Cush-ions under Floating Offshore Structures ", Proceedings

Boss'97 Conference, Deift

Moulijn,J., 1998 : "Scaling of Air Cushion Dynam-ics ", Report No. 1151, Laboratory of Ship

Ilydrome-chanics, Delft University of Technology, Deift

Tabeta,S., 1998 _{"Model Experiments on Barge} Type Floating Structures Supported by Air Cushions",

Report 1125, Laboratory of Ship Hydrom.echanics, Delft

aüSm i l =0.5m

AZ

X

- - (aft cushion) - -(fore cushion).

015m

015m E

Figure 1: Set-up for captive- and oscillation tests

1116 panels

Figure 2: Panel models of one- and two cushion barges for captive model and oscillation tests

Figure 3: Panel models of one- and two cushion barges for free-floating tests

T

0.50 m

E 20 15 10 2 o o 2 4 6 wave frequency n r/s

Figure 8: Pressure forward cushion 2-cushion barge

10 8

EG

o-2 o 2 10

E6

V,

z

o o 2 4 6

wave frequency in ris

8

Figure 5: Pressure cushion 1-cushion barge

4 6

wave frequency in r/s

Figure 7: Pressure difference 2-cushion barge

2 4 6

wave frequency in ris

8

Figure 9: Pressure aft cushion 2-cushion barge

10 10 10 Computed Measured

-r

o

n 20 15 E 10 o 10 8

E6

a-4 10

wave frequency in ris

Figure 4: Heave force 1-cushion barge

10

E6

a-10 8

E6

co a-o 2 4 6 8 10 wave frequency in r/s

Figure 6: Heave force 2-cushion barge

4 6 lo

heave frequency in ris

Figure 10: Heave added mass 1-cushion barge

4 6 10

heave frequency in r/s

Figure 11: Heave damping 1-cushion barge

8 lo 2.5 2.0 1.5 1.0 0.5 O

2.5 2.0 1.5 C C 2 1.0 0.5 O 0 2 4 6 8 10

heave frequency in ris

Figure 12: Heave added mass 2-cushion barge

60 40 E û-20 O 1.2 0.9 0.6 0.3 O 0 2 4 6 heave frequency in r/s O 8 10 0 2 4 6 heave frequency in r/s 2 O 60 40 E Q-20 1.0 0.8 E0.6 E 0.4 0.2 1.0 0.8 E 0.6 co û-- 0.4 0.2 O O

- Computed

o Measured I 0 4 6 8 10

heave frequency in ris

Figure 13: Heave damping 2-cushion barge

t 2 4 6 8 1 ₁₀ wave frequency in r/s

-

D - o I

t

-

+ o -!

Figure 14: Pressure cushion 1-cushion barge Figure 15: Pressure cushions of 2-cushion barge

0 2 4 6 8 10 4 6 10

wave frequency in r/s wave frequency in r/s

Figure 16: Surge motion 1-cushion barge Figure 17: Heave motion 1-cushion barge

Figure 18: Pitch motion 1-cushion barge Figure 19: Pressure cushion 1-cushion barge

2 4 6 8 10 wave frequency in r/s 8 10 10 8 400 300 E 200 100

1.2 0.9 0,6 0.3 400 300 E CI) 200 C) Q) -C 100 E co Q-O O 2 4 6 wave frequency in r/s 4 6

wave frequency in ris

Figure 24: Pressure aft cushion 2-cushion barge

1.0 0.8 E0.6 E 0.4 0.2 4 3 E o-E Q--2 o

- Computed

o Measured D O 10 0 2 4 6 wave frequency in r/s 2 wave frequency in r/s

Figure 25: Pressure difference cushions of 2-cushion barge

-8

-6

10

[iaL

Figure 20: Surge motion 2-cushion barge Figure 21: Heave motion 2-cushion barge

Figure 22: Pitch motion 2-cushion barge Figure 23: Pressure forward cushion 2-cushion barge

2 4 6 8 10 0 2 4 6

wave frequency in r/s wave frequency in r/s Figure 26: Drift force 1-cushion barge Figure 27: Drift force 2-cushion barge

2 4 6 8 10 wave frequency in r/s 10 8 O O 2 4 6 8 10 wave frequency in r/s 8 10 -8 -6 E

z

-2 o