Book Ref. T098001
Vth WEGEMT WORKSHOP
fl~aLOGEA& WAVE
STRUCTUlRES AM© SIP3
W (4th September 1998 Venue EGEMT University Toulon-Var France W~EGEMT A European Association of Universities in oU Lo.O VAR Marine Technologies and related Sciences
Vth WEGEMT WORKSHOP
NON-LINEAR WAVE ACTION ON STRUCTURES AND SHIPS
University Toulon-Var France
4th september 1998 * AZEMER
Ville de La Seyne Sur Mer N S I
CQNSEIL A- GENERAL
Information / Workshop Secretary Vincent REY, Wegeit Workshop Secretary. ISIiTV. 1I1 56. 83162 I.i Valcue dui Var (cdex. FRANCI." Fax: +33 (0)4 94 14 24 48. E-mail: rey (0isit,.uii v-t In. f' ABOUT WEGEMT
WEGEMT is a European Association of 42 Universities in 17 countries. It was formed in 1978 with the aim of increasing the knowledge base and updating and extending the skills and competence of engineers and postgraduate students working at an advanced level in marine technology and related sciences.
WEGEMT achieves this aim by encouraging universities to be associated with it, to operate as a network and therefore actively collaborate in initiatives relevant to this aim.
WEGEMT considers collaborative research, education and training at an advanced level, and the exchange and dissemination of information, as activities which further the aim of the Association.
N-B For marine technology and related sciences, WEGEMT includes all aspects of offshore oil and gas exploration and production, shipping and shipbuilding, underwater technologies and other interdisciplinary areas concerned with the oceans and seas.
ABOUT THE PUBLICATION
This publication represents a series of lecturers' papers, which were presented at a one-day Workshop entitled Non-linear Wave action of Structures and Ships presented at the Universit6 Toulon Var on Friday 4 September 1998.
Published by WEGEMT
ISBN Number: 1 900 453 08 8
This volume has been made available so that it contains the original authors' typescripts. The method may from time to time display typographical limitations. It is hoped however, that they do not distract the attentions of the reader. Please note that the expressed views are those of the individual authors and the publishers cannot accept responsibility for any errors or omissions. PROGRAMME
Friday 4 September 1998:
Registration Welcome An overview of flow induced vibrations with emphasis on Vortex Induced Vibrations (VIV) for marine risers Pr. Geir MOE, N7INU, Norway
Slamming Pr. Odd FALTINSEN, Marintek, Trondheim, Norway
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.
Semi-analytical methods for some linear and nonlinear problems of diffraction- radiation by floating bodies Mr. Sime MALENICA, Bureau Vedtas, Paris,France
Station keeping and slowdrift oscillations Pr Joe PINKSTER, Tech. Univ. of Delft, The Netherlands
Design load predictions by non-linear strip theories Dr. Jorgen JUNCHER JENSEN, Tech. Univ. of Denmark., Denmark
Non-linear interactions between waves and a horizontal cylinder Pr. John CHAPLIN, City University, London, U. K Transformation of non linear waves in shallow water and impact on coastal structures Pr. Stephan GRILLI, University of Rhode Island, USA LIST OF PARTICIPANTS
NOM PRENOM UNIVERSITE A.ALBERS Pr. A. Delft University of Technology ALLWOOD Dr. R Cranfield University BERHAULT Christian Principia R&D et ISITV BOOTE Pr. D. UniversitA di Genova
CARDO Pr. A. Universith 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 DE JOUETTE Christine Principia R&D et ISITV DEVENON Jean-Luc Univ. Toulon-Var DEVILLERS Jean-Franqois Ecole Nationale Sup6rieure de ______Techniques Avanc6es
______Cyril Univ. de Provence FLISNPr. Odd MARIN*TEK FEDDr. Graham FUGRO GEOS FRATPierre Sirhena FRAUNIB Philippe Univ. Toulon-Var FUMEY Fr~idrfic Global Maritime GAILLARDE Gujihem MARIN GRANT Jim WEGEMT Secretary GRIIETHUYSEN Pr. J van University College London GRILLI Pr. Stephan University of Rhode Island GUIGNARD Stephan U T V et Principia R&D HENRIKSEN Martin University of Science and ______Technology Norway JUNCHER JENSEN Jorgen Tech. Univ. of Denmark KRUPPA Pr. C. Technische Universitact Berlin LECALVE Olivier Univ. Toulon-Var MALENICA Sime Bureau Veritas MATUZIAK Pr. J. Helsinki University of ______Technology MELING Frode MARINTEK MOE Pr. Geir Norwegian University of Science
______and Technolog NTNU MOLIN B . ESIM NIJNEZ BASANEZ Pr. J. Escuela Tecnica Superior de ______ngeiers N vales PAPANIKOLAOU Pr. A. National Technical University of Athens PELISSLER Pr M.Claude Univ. Toulon-Var PINKSTER Pr. Joe Tech. Univ. of Delft REINI-OLDTSEN Svejn-Axne MARIN4TEK RBSCH Franqois Univ. Toulon Var REY Vincent Univ Toulon Var RIGAUD, Stdphane Principia R&D RUSSO-KRAUSS Pr. G. UniversitA Federico H1 SCOLAN Yves-Marie ESIM SEN Pr. P University of Newcastle Upon
______Tyne TRASSOUDAINE Damuien S BM WIETASCH Pr. K W Gerhard Mercator Universitdt ZALAR Mirela Bureau Veritas Vortex-Induced Vibrations in Water
Pr Geir MOE. NTNU. Norway Vortex-Induced vibrations in Water A review paper by professor Geir Moe July 1998
Abstract This paper gives an overview on transverse lock-in on sectional models subjected to a steady, uniform water current. Implicationsfor VIV motions of line-like structures are considered
Introduction Structures subjected to a steady or oscillatory flow may experience flow-induced vibrations that are caused by several mechahisms, for instance
* Turbulence induced vibrations ("Buffeting") " Galloping and Flutter * Static divergence * Flow interference, e. g. between members of a group * 'Jumps' from a position on the seafloor or a wall * Drag-crisis induced vibrations * Vortex-Induced Vibrations (VIV)
In this review due to time and space limitations only the last of these phenomena will be dealt with, and the flow will be assumed to be steady. Furthermore situations with purely in-line VIV motions will not be considered.
Vortex-induced vibrations (VIV) have fascinated people from the dawn of history. Thus the tones produced when wind created VIV on a collection of cylindrical objects, a so-called 'Aeolian harp' or 'Wind harp', was reported by Aristotle to be 'the music of the spheres'. It was also said "through it the spirit of the wind carries the Muses across the earth's surface", and "the music of its silvery strings gives voice to the Goddess Earth". Later one of the first scientific studies in the vast arena of hydrodynamics was conducted by Leonardo da Vinci who when studying the flow behind a circular pile in water drew the famous pen drawings of vortex shedding. His attitude was experimental, as this quote shows: "Remember, when discoursing about water, to induce first experience, then reason ". This was a bold departure from the 1000 year long 'scholastic' tradition that up to his time had dominated science. It had relied on 'pure thinking' and on interpretation of the text of earlier 'sacrosanct' works. Also the quote might suggest that Leonardo sometimes found the behaviour of water to be surprising, or even confusing. (Also today many of us may have the same feeling, for instance when working with VIV problems!)
In 1878 Strouhal on the basis of experiments showed how the vortex shedding frequency could be predicted for a rigidly held cylinder of known diameter in a flow of known velocity. Since then a vast number of studies have been conducted on VIV both in air and water. Presently there is an increased interest in VIV of long cylindrical members in water, because of the development of hydrocarbon resources in depths of 1000 m or more. There turns out to be a significant difference between VIV in water and VIV in air, and the present review will concentrate on VIV in water. 2
VIV on a section model The simplest case of VIV considers a section model in which a short, rigid circular cylinder is situated in a steady fluid flow perpendicular to the cylinder axis, supported on springs transversely to the flow, while being either restrained or spring-supported in the direction of the flow. The situation is intended to be two-dimensional, and as a consequence the cylinder length L must either be roughly 40 times its diameter D, or some special means must be employed to prevent three-dimensional flow around the ends, so-called "end effects". For this purpose end plates have often been used, even if these may affect the motions in other ways than desired, e.g. by increasing the drag force in both directions.
If the cylinder is rigidly held then the Strouhal number S (to be defined presently) is a constant close to 0.2 over a wide range of Reynolds numbers, so that the vortex shedding frequencyf, can be predicted as follows
S- '= U /(fD) (1) f. =SU/D (1')
Here U is the undisturbed flow velocity and D is the cylinder diameter.
Fig 1.Section model: A rigid cylinder on transverse springs in a steady, uniform flow
Experiments conducted in the situation shown in figure 1 may be denoted self-excited tests, because the cylinder in a certain velocity range is excited by the vortices it creates itself An oscillating cylinder will have a well-defined natural frequencyf, and it appears reasonable to expect vortex induced vibrations when the flow velocity is such thatf, as determined from (P'), equalsf,. Indeed vibrations will occur for this velocity, but when the flow velocity is gradually increased beyond that value, it turns out that vibrations of similar or larger magnitude will occur over a range of velocities. This is associated with a delay of the shedding frequencyf, compared to the value predicted from (1'), so that it matches the frequency of the cylinder motions. This phenomenon is commonly denoted 'lock-in', 'lock-on' or 'synchronization'. It is then customary to form a quantity similar to the inverse Strouhal number S-, but based not onf,, but either on some reference natural system frequency,f , or on the frequency at which the lock-in vibrations occur, f, viz.
U' = U/ (fD) (2) U,.,,, = U / (fD) 3
Lock-in vibrations in air will always occur at the natural frequency, so thatf=fý, and from (2) Ured and U,.t,, will be identical. In water, however, the frequency of the oscillations may vary considerably through the lock-in range. This effect is attributed to variations in the added fluid mass, which in water is large relative to the structural mass. Self-excited vibrations in water are normally described on the basis of U, d, i.e. the water velocity for a given cylinder is normalized by division with a constant quantity. Thus the velocity range for which VIV will occur will be simple to envision. The constant value to be used forfý is usually taken to be the system natural frequency in air, but especially in the past, the natural frequency in still water was often used for fý. In forced vibration tests or when measuring VIV on prototypes the known quantity is the frequency at which the vibrations actually occurs, i.e.f will be the obvious normalizing frequency, so that U,.t,, should be used. In Wu (1989) and Moe et Wu (1989) Ured was called "nominal reduced velocity" while U,,,, was called "true reduced velocity", but as of today these names are not in widespread usage.
The motion amplitude and the width of the lock-in range depend, besides U,,d, on other parameters. E.g. for the amplitude
AYv / D = f (U ,,,,m, / maispt,4 s,Re,cru /1U ) (3) in which m, is the structural mass per unit length, mai.,pt is the displaced fluid mass (each per unit length of the cylinder), and 5, is the structural damping given as the so- called "logarithmic decrement". Turbulence is here described, rather crudely, by the turbulence intensity, which is the standard deviation of the flow velocity relative to the mean velocity.
Traditionally the motion amplitude is predicted on the basis of a combined stability parameter Ks, also, especially in wind engineering called 'the Scruton number'. Alternatively a so-called response parameter Ar, also called 'the reduced damping parameter' is used. Their definitions are
K, 2m, , 'plD (4) A, = 2r U:,,2, K,
Here p is the fluid density, so that m,/pD2 represents the ratio between the structural mass and the displaced fluid mass. (Apart from the constant 4/n.) Thus the mass ratio and the structural damping have been combined into one parameter. In the literature in the above formula for A,, S2 is usually written instead of (U,,,, - , i.e. the concept of a vibrating cylinder Strouhal number is employed. S will then have a value smaller than 0.2, say about 0.15 for subcritical flows.
The dependence of the motion amplitude on ., can be developed from the dynamic equation for a cylinder on transverse springs, assuming also that the transverse motions are harmonic. This will depend on assumptions that are not satisfied for vibrations in water, but may be acceptable if air is the surrounding medium. 4
y = AY sin wt mr +5+cP+ky=O.5pU 2DC,.sin(wt-9) (5) = 05 pU 2D(CQ sin weo- CQ coswot)
The sine term on the right side of the latter equation above may be interpreted as the transverse added mass term and the cosine term as the transverse fluid damping term which when negative, may drive the vibrations. Normalizing by the total transverse
rn,+,=rn+0± pU'DC., .2 (6) o., = k /rn,0, 2gi,. = c rn,., 5 + [2qm6p- 0.5pU2DC, coswt / m,, ] + c2y=0 (7)
The first and last terms on the left side of (7) now balances, determining the frequency of the oscillations equal to o'.,=w =24-f since this is the frequency at which the lock- in vibrations will occur, i.e. the frequency associated with U,.,,e.. From a balance between the damping and the load component in phase with velocity, i.e. the expression in the square bracket in (7), the amplitude may be expressed as
2A'q'o&ýw = -pU2 DC,, /(2rm,,,) 2 . A_.._ pU C4 •._ I pD' U Cd, (8)
D 4qOrnm,,, 4 rn,,0 W 2D,
In the last expression in the bottom equation is recognized first the inverse of the mass ratio, then a constant times u r.,-,e and finally the inverse of the damping ratio that for small values is a constant times the logarithmic decrement, i.e. one obtains
.4, _Cj C, D KU.. A (9)
The stars on K, and Ar are introduced to indicate that these quantities are based on the total mass as given by (6). In air m,,, is practically speaking equal to the structural mass mi, thus the original definitions of K, and J, in (4) are recovered. Provided the component of the lift force in phase with the transverse velocity, Cad, is a constant and thus independent of motion amplitude and reduced velocity, then the result will depend on A, only. This may be an acceptable approximation for cylinders in air, and then the vibration amplitude may be predicted on the basis of A, and if one further assumes that U,.i,,e does not vary much from a value of 6 to 7, then K, may equivalently be used. One of the most updated plots made on the basis is of A, is shown below, from Griffin (1985). It is seen that the general trend of the results are captured, but the outcomes may differ by a factor frequently as large as 2. 5
10.0 MARINE STRUCTURES
6.0. MARINE CABLES 2AID X. 2.0 RANGE OF -44 1.0 LIMITING DISPLACEMENT
0.6 0.4 0 0
0.2 4
0.I--
0.005 0.01 0.02 o.e5 0.1 0.2 0.5 1.0 2.0 5.0 10.0 Response parameter, A,
Fig 2. Lock-in crossflow double amplitude (2A) as afunction of A,. Maximum value in lock-in rangefor equivalent section model, (based on (10)). From Griffin (1985)
Forced vibration tests for a section model The important Cs, coefficient is presented in figure 3 for the in-linefixed case and in figure 4 for the in-line spring-supportedcase. The cylinder is in both situations forced to oscillate in the transverse direction at a frequency equal to that at which the self- excited vibrations occur. In the in-line fixed case the cylinder is guided by the geometry of the test rig to move in the transverse direction only, while for the in-line spring-supported case the cylinder is supported at springs with a stiffness that produces roughly the double in-line natural frequency compared to the transverse oscillation frequency. This is done to trigger in-line vibration response to each shed vortex, and thus facilitate the figure eight cylinder motions that usually occurs under prototype conditions, see e.g. Vandiver (1983). The tests are run at a number of prescribed reduced velocities and amplitudes. From the figures is seen that for motion amplitudes of roughly one diameter Cd+ changes sign, which means that when increasing beyond this value, instead of receiving energy from the flow, energy will be extracted. According to this, the maximum transverse amplitude Ay, can in this case be no larger than one diameter. This agrees well with the observed amplitudes from the self-excited tests, see the points marked with the letters a to e in figure 4.
In water C.,,.and hence the added fluid mass will depend on Ud, see figure 5 and 6. It is seen that under lock-in conditions the added mass is usually negative, and the more so the larger U,,d is. Therefore the oscillating frequency will vary through the lock-in range, more pronounced for lighter cylinders. In the lock-in range Cd, and C., are also strongly dependent on the amplitude, see figure 3 to 6 below. Therefore there is little justification for the use of some fixed values for the damping and mass ratios to be used in Ar orK,. In the defense of figure 2 it may be argued that the data points fit the curve reasonably well, also for the portion that represents bodies in water. However in this situation the vibration amplitude will be determined mostly by where CQ, crosses 6 over from positive to negative, and this is independent of damping or mass ratios. But since experiments with cylinders in water tend to result in vibration amplitudes around one diameter, data from experiments can be plotted against just about any parameter and a respectable fit (to A/D=1) will still be obtained.
f-' t'I 1.5 .5 rorcsa salt-.xclttdl sul-f--lClt~ad
s: Vr-.0004 a: Vr-. 1183 f: Vr-6.6923 A
*: Vr-G.4036 t: Vr-6.2391 q: Vrý.8526
w: Vr-6.7985 C: Vr-6. 1936 N.: Vr-7. 1055
0: Vr-7.2040 a: Vr-e.2502 1: Yr-7.2148 11 . 7 z: Vr-7.4003 0: Vr-6.535 "-
0.5 ---
- - -; 4
-1 ii 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Ay/O
Fig 3. Cd, versus A/D for some values of Ure. In-line fixed condition, a to e represent self-excited cases. Smooth cylinders
1.5 forced 3.If-exCiteO
u:jr-6.BO4 r-6.5774A
- ur-6.6652 0- .,-6.6590
0: Ur-6,9481 4: ur-6.9328"" "
ur-7.2909 e: .,-7, WI•,
--- 3 ' K,
a '. N ,:;,
-t *' , -,. r" I / "j
0 0.2 0.4 0.6 0.8 1 1.2
Fig 4. Cav versus AlD for some values of U,.,,. In-line spring-supported a to e represent self-excited cases. Smooth cylinders 7
1.5
0.5 ,k
0I -
-0.5 rorced selt--,xted *elf-xcited-
0: Vr-6.000A *: Yr-6. 1183 f: Yr-8.6923
*: Y--,4036 b: Yr-6.2391 g: VP-6.S525
K: Yr-6.7965 C: v-I. 1936 M VP-7.1055
0: Vr-7.2040 0: vr-6.2502 1: Vr-7.2146
X: Vr-7.4003 a: Yr-6.5357 -1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Ay/D
Fig 5. CG, versus A/D for some values of U,,,,. In-line fixed a to e representself- excited cases. Smooth cylinders
3 II
...... - ,, ..
, J . .-- -.. 'I
I ,' ,,
forcea self-excitea '
r-:6'r eo80i a: vr-6.5774•V I/r-6.6652 D: Vr-6.6590
,v-6.7203 c: Vr-6.7115 *
o: vr-68404 a: Vr-6,8341
4 I : Vr-6+945I a: r'-6.9382
0 0.2 0.4 0.6 0.8 1 ,2 Fig 6 Cm versus A/D for some values of Ur,.,.e. In-line spring-supported a to e represent self-excited cases. Smooth cylinders
Sarpkaya (1995) presented a detailed comparison between his own, Gopalkrishnan's and Wu's experimental results for Cd, and Cmv for the inline-fixed cases. There was relatively little scatter. 8
Self-excited Vortex-Induced Vibrations (VIV) for section models In figure 7 and 8 from Moe et al (1994) are shown some experimental results for the VIV motion amplitudes, A,. In figure 7 tests conducted at subcritical Reynolds numbers are shown as a function of the reduced velocity Ured for several values of the ratio between the in-line and the transverse natural frequencies,fý andfy respectively. Under transverse lock-in one full vibration cycle in the transverse direction will occur when two alternating vortices are shed, while the in-line force during the same time segment has undergoes two full cycles, one for each shed vortex. One might therefore expect that a frequency ratio offjlfy =2 would represent the situation in which lock-in vibrations in both directions are excited to their full potential, even though the complex variations in transverse added mass might blur the results to some extent. However from figure 7 is seen that while a frequency ratio of 2 exhibits large motions over a very wide velocity range, the largest maximum motion amplitudes at say 1.2 diameters, occur for frequency ratios of 1.2 and 1.7. Also it can be seen that the in- line fixed case is the most benign among those investigated.
1.5
A/D ./f = 1.2 1.7 • '•,f• 2.25
N. "' -2.0 2'i ...... , \ ' I .. " ----=
3 4 5 6 7 8 9 i0 it 12 Ur Fig 7. Transverse motion amplitude versus Ut red. SubcriticalReynolds numbers. Smooth cylinders
In figure 8 the subcritical VIV results for the frequency ratio of 2 is replotted, together with the results from tests in the Reynolds number range 40000 to 400000, i.e. well into the critical Reynolds number range. Ure,=6 corresponds namely to Re=200000, .which for this case of a smooth cylinder in low turbulence flows may be taken as the limit between subcritical and critical flow. The figure indicates that the motions are at least as large in the critical range as for the corresponding subcritical case. In Moe et al (1994) is also presented some experimental results for rough cylinders in critical flow, indicating that the motion amplitudes are reduced to roughly one half diameter. 9
AyID
1.02.0
3 4 5 6 7 5 9 10 11 12 Ur Fig 8. Transverse motion amplitude versus U,,d. Critical (full line) and subcritical Reynolds number results. Frequency ratio:ffA=2. Smooth cylinders
Experiments with fixed cylinders in critical Reynolds number flows show a narrow, disorganized, and turbulent wake which gives rise to small lift forces, and it would therefore be reasonable to expect that VIV motions in this Reynolds number range would be small. Since figure 8 suggests that the motions are large, one is led to postulate that when the cylinder is moving its motions organize the flow in the wake into well-defined vortices. Figure 9 from Sarpkaya and Isaacson (1981) shows the Strouhal number as a function of the Reynolds number. In the critical Reynolds number range most researchers report a Strouhal number that increases to a maximum of 0.45. This, however, is based on the zero-upcrossing frequency of the velocity process and may be caused by the wake turbulence. It does therefore not necessarily imply organized and stable vortices. In figure 9 a curve is plotted that indicates that the cylinders will oscillate at frequencies consistent with a Strouhal number of about 0.2, i.e. that lock-in will start at roughly the same U~d value as in subcritical flows.
0.5
Z;; 0.4 Z;- - > 0 -9# 0.3 - •o • o•-- • ,• REGION OF TURBULENTCLIDE VORTEX TRAIL c " 0.5 '
r 0.1 a CYLINDERA TH SR L/ CYLINDER OSCILLATING. AT J •REGIONVORTEX WHERESHEDDING TUE , oJITS NATURAL FREGUENY I PL) FRTEQUENCYCANEBE
C DOMINANTIN A SPECTRUM FREaUEN CYI%h
REYNOLDS NUMBER, Re Fig 9. The Strouhal-Reynolds number relationshipfor circularcylinders. (From Sarpkqya and Isaacson, 1981, after Lienhard, 1966) 10
The average drag on cylinders undergoing VIV motions is considerably larger than on fixed cylinders. In figure 10 from Moe et al (1994) is shown the coefficient for the average drag force CD versus Ued for some values of the frequency ratiof,/fy. It is seen that large drag amplifications are the rule in the entire lock-in range, and that values as large as Cr-4 occur. 2o
3.0 2 -- f.0... \ \ -""-. 2.0 \ 2 . 25 -
1.0 " fixed
(10 1 3 A 5 6 7 8 9 10 ii 12 Ur
Fig 10. Coefficient of the mean dragforce CL versus reduced velocity Ured. Subcriticalflow. Smooth cylinders
In figure II is shown the coefficient of the mean drag force from the previously cited tests by Moe et al (1994) in the Reynolds number range 40000 to 400000 in which for Urea>6 the flow is in the critical Reynolds number range. The curves marked "free to move" represent drag on a cylinder that is vibrating under lock-in at af,/fý=2. The curves marked "fixed" represent situations where the rig motions were restrained, but where small vibrations still occurred. The results are therefore not quite representative for fixed cylinders. Results from "(almost) fixed" and "self-excited" cases at the same, high Reynolds number are plotted at a the same U,,d value, to give an idea of the change in the average drag force due to lock-in vibrations. It cai be seen that the drag coefficient on vibrating, smooth cylinders in the critical Reynolds number range is roughly equal to 0.5, i.e. significantly larger than for a fixed cylinder, but very much lower than for the corresponding subcritical VIV case. C0 for rough cylinders in the critical range (which now starts at a lower Reynolds number) appears to be about 1.0, regardless of whether vibrations occur. CD
free to oscillate
0 2 4 6 8 10 '12 141. 6 18Ur
Fig IL. Coefficient of the mean dragforce C& versus reduced v'elocity Ured. Critical Reynolds numbers above about Ured 6 , smooth and rough cylinders
Consistency of the shedding process In order to predict the behaviour of line-like structures, or structures made up of slender cylinders, the consistency (regularity) of the vortex-induced forces appears to be of paramount importance. To fix ideas consider a vertical marine riser with diameter D-0.50 m and length L=1000 m subjected to a steady current that for simplicity now may be considered to be unidirectional and nearly uniform. The question is whether the lift forces associated with vortex shedding will trigger vibrations in a given vibration mode. Initially it may be useful to consider how the regularity and consistency of the loading should be assessed. The spanwise correlation coefficient p(211) of the forces at two points Al apart is a widely used measure in this context, but interpretation of such correlation data may not be straightforward. One problem is associated with the phase of the motion. Very long structures such as risers may vibrate at modes where there are phase delays along the riser axis, as is most apparent in the limiting case of an infinitely long riser for which a typical vibration case might be a travelling wave with diminishing envelope. Consider for simplicity the situation in which the riser velocities at two points are harmonic, but 90 degrees out of phase with each other. Then the correlation coefficient between these two velocities is zero, and so is the correlation coefficient betweten two loadings which each is perfectly in phase with the above mentioned velocities. However, these loads will be perfectly phased to put energy into the system, since the work at any location is the integral over time, of force times velocity. Therefore it is the phase between local force and local velocity that matters, not the phase between the forces at different locations.
Another problem is associated with irregularity of the forces. From figure 12 can be seen that the force time history varies much more pronounced than the displacement history. It is then tempting to model the force as the sum of a deterministic part that drives the motions of the cylinder, plus a part that may be denoted "irregular", "spurious" or "random". The deterministic component can be described by the lift force component with coefficient C., in (5) which by definition is perfectly correlated with the cylinder harmonic acceleration Y9,plus the lift force component associated with CA. which is similarly correlated with the cylinder velocity. These coefficients 12
can, at least in principle, be used iteratively in (5) to predict the motions caused by the deterministic part of the VIV lift forces. The irregular part of the force is local in time and space, and may have little influence on the motions. This will be truer the smaller the spanwise and temporal correlations of the irregular force components are)
vf& " I ~ i[|...'q~ A* ... ..flA
i \] *I .1 -'V i
SI- ,.1 ,
a) self-excited, in-line fixed, Vr=5.51 b) forced, in-line fixed. Vr=5.51
:mp • 11•1 A/k iJ?½ !l~
c) self-excited, in-line spring supported, d) foced, in-line spring supported V =5.93 V =5.93 I r
Fig 12. Displacement time series (broken line) andforce time series (full line)
In figure 12 are shown time histories of cylinder displacement and the accompanying force on a sleeve 1 diameter wide. Case a) and c) represent self-excited vibrations which are seen to be nearly harmonic with an average amplitude of about 0.8 diameters and variations between individual amplitude peaks of about ± 10 %. The associated force is only approximately harmonic, and exhibits disturbingly large variations between the peaks from different cycles, especially for case a). In b) and d) are shown the results for forced vibrations at the same frequency and average amplitude as for the corresponding self-excited situation. It may be seen that the resulting force time histories are more nearly periodic, but not very close to harmonic. It is seen that the forced case fails to mimic the detailed behaviour of the self-excited case, but the averaged forcing may still be the same for the two situations. 13
The correlation of the lift forces onfixed cylinders probably depends strongly on Reynolds number and turbulence intensity. In a series of experiments by Toebes (1969) the spax~wise correlation coefficient of the fixed cylinder forces at two points that were 7 diameters apart was found to be practically zero, while it was 50% for a spacing of 2 diameters. Figure 13 shows variance spectra for 256 second long records from which the 10 s segments in figure 12 originated. The spectra for the accompanying displacement time history result in a very narrow spectrum at a single frequency. The spectra for the measured force time histories are shown in figure 13. They resemble line spectra, since for both of the in-line spring-supported cases there were quite large third harmonic components, while each "line" represents a narrow spectrum with about 1% bandwidth. A recent numerical study of the same problem by Meling (1998) also showed significant third harmonics in the force spectra.
2 2 2 S, ) (NH - s/rad) S, Ccl (N " s/tad) S,(-) (N • s/rad) Sf(-) (H2 . /rad) .,a) 1.5 b) .15 20d)
.3- 15 1.0 .10
.2, 10
.5 .05 .1 5
.0 .0 .00 •00 0 20 40 0 20 1.0 0 20 /0 0 20 40 .(rad/s) w(rad/s) w(rad/s) mtrad/s) Fig 13. Variance spectra of the lift force of which 10 s segments are shown in figure 12. (a) Self-excited, in-line fixed, (b) Forced, in-line fixed, (c) Self-excited, in-line spring supported (d) Forced, in-line spring supported
*O lf-lxcltod I
' a:Vr-5. 4444 - p :Vr-5.4164
c: r-,5.4986 . -
a: vr-5.5179 t',
,6 vr-5.7383 0 : Vr-5.9365 /" •..._-.- 2.-/', "-.,• -..
I. IC ' C "
o , " "forced 0.2 / / A /, , *: Vr-5.4028
- 0: vr-5.8016
x: vr-6.0004
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 . Ay/D Fig 14. Spanwise correlation of lit forces on two force sleeves 6 diameters apart, versus AY ID. Middle to high parts of the lock-in range. In-line fixed, forced and self- excited cases. From Wu (1989) 14
0.6
0.4 • -- . ------.5-
0.2 - -
c a - . ',,. ,
fo.cd s.1f-clitvd s5elf-excited 0 -0.2 P*: Yr-6.0004 a: Vr-6.11a3 f: Vr-e.6923 +: Vr-6.4035 b: Vr-6.2391 g: Vr-6.B526
0: VP-6.7985 C: Yr-. 1936 h: Vr-7.1055
-0.4 0: Vr-7.2040 W Vr-f.2502 1: Vr-7.2148
PX: Vr-7.4003 0: Vr-. 5357
-0.61 , ,•J 0 0.1 0.2 0.3 .0.4 0.5 0.6 0.7 0.8 u.s Ay/'0 Fig 15. Spanwise correlationof lift forces on two force sleeves 6 diameters apart, versus Ay/D. Upper part of the lock-in range. In-line fixed: forced and self-excited cases. From Wu (1989)
For the in-line fixed case the lift correlation coefficient on two force sleeves 6 diameters apart on a section model is shown in 14 and 15, from Wu (1989). It may be is seen that the correlation is quite high for the three self-excited cases below U,,,,=5.5, then falls off and becomes negative in the upper part of the lock-in range. Anand (1985) reported similar results. The correlation coefficient from most of the forced vibration tests was relatively low, usually between 0 and 0.5. However for Ur.,tt equal or below 6.0, the correlation coefficient grew from a value of 0.5 to about 0.8 as Ay/D grew from 0.8 to 1.2. Thus the difference between results from forced and free vibration tests that presumably covered the same situation, was rather significant, and in general the correlation between the lift forces was rather low. Especially the negative values for the correlation coefficient that occurred at the upper part of the lock-in range for the in-line fixed situation is hard to accept. Measurements indicate that the transverse forces in these situations are dominated by the transverse added mass which is large and negative, but varies strongly in magnitude for a given time history. One might further speculate that the low forces are associated with positive added mass situations. It isalso know that a large force on one force sleeve is usually associated with a small force on the other, and then negative correlation might occur.
In figure 16 and 17 similar results for the in-line spring-supported case are given. It can be seen that in this situation the correlation coefficient is very much higher. Excepting small amplitudes below say 0.3 diameters, it is only for forced vibrations above Ur.,,,=6.5 that the correlation coefficient drops below about 0.8, but it still stays above 0.6 for all cases. All self-excited cases except the lowest one, have a correlation coefficient in the 0.9 to 0.95 range. These in-line spring-supported results supports the assumption that the cylinder motion governs vortex shedding, so that lock-in vibrations can be calculated from the deterministic part of the force history. 15
0.6 - -. j::. -forced
-: 0.5 •--- Vr-6.2158 . 4: Vr-6.3015 0:o Vr-4. 3667 0.2 x: Vr-4. 4702
selt -eIxcited( - 0 : VP-0. I.20
b: Vr-6.2110 -0.2 - c: Vr-6.2975
S€a: Vr-6.3860
-0.4 , * Vr-4. 4669
-0.6 0 0.2 0.4 0.6 0.6 1 ky/fl 1.2
Fig 16. Spanwise correlationof lift forces on two force sleeves 6 diameters apart, versusAy ID. Middle to high parts of the lock-in range. In-line spring supported, forced and self-excited cases. From Wu (1989)
f orced w: Vr-6.5804 ,". p•
: Vr-6.5652 • - 0.9 -.+:I I IN .*: Vr-E.7203 , , , - ..-
o: Vr-6.8404 ,t = -~V it 'I • .-- 0. Vr6.948I U ;
0.7 2,'
c , - Lo .6 * I......
' salt-excited a:rVr.. 5774
0.5 : Vr-6.7116
0.4 e:Vr -6.93112 o0.2 0.4 0.6 0. 1 1.2 AY/D
Fig 1 7. Spanwise correlation of lift forces on two force sleeves 6 diameters apart, versusAY/D. Upper part of the lock- in range- In-line spring-su-pported, oced and self-excited cases. From (1989) SU 16
VIV for long, line-like structures Iwan (1975) has suggested a model for the estimation of the amplitudes of VIV motions for long, line-like structures, on the basis of the results from self-excited tests on section models. Introducing the i h modal shape function y'i he expressed the modal amplitude at its maximum value along the span, (Ay/D),, by multiplication of the vibration amplitude determined for a section model, (Ay/D),,, by a factor y, and found the following expressions
[ A Y/ D] , _= y [ AY / D j , .
y =,J:,)/,(z)dzj/bvdzdzj (10) 0 a J L LJm
For sinusoidal shape functions this yields y=1.115, results for other geometries may e.g. be found in Blevins (1990).
The development that resulted in (10) rests on a quite sophisticated theoretical model, but some of the assumptions that are made, appear to be questionable for VIV in water. Moe (1991) pointed out that the model uses a constant added mass equal to the still water displaced mass, which according to figure 5 and 6 is highly inaccurate. This choice leads to the assumption of a constant lock-in vibration frequency which in its turn results in a much too narrow lock-in range. Furthermore, in this development there is no provision to specify whether the structure is fixed or spring supported the in-line direction. The lift force is modeled by a van der Pol nonlinear differential equation whose validity it is difficult to assess. All the same it can not be ruled out that the calculated y factors may happen to give reasonable values in engineering practice. This author has seen no comparisons between predictions based on (10) and well documented prototype measurements. However, the results may not be very much off the mark, since the results based on (10) and on iterative use of(5) both will adjust the sectional model results by rather small amounts. The section model (A)/D) value will usually be slightly below the value where Cd&=0. Beyond this amplitude Cd, rapidly becomes quite large, so that significant larger vibration amplitudes can not occur, since then large amounts of energy would be removed. And the above procedure, as quoted, predicts that the amplitude for a sinusoidal mode shape will increase by only 11.55%, relative to the section model results.
At this time it appears most rational to base the predictions of VIV motions for line- like structures on modal analyses using formula (5) with experimental data such as those presented in figures 3 to 6 iteratively to determine the vibration frequency and the driving force for transverse VIV motions, until a energy balance is obtained. From what has been said before there are at least two problems that need consideration.
The first problem is whether the irregular (spurious) part of the loading will affect the motions of the structure. This can not be ruled out, but is more likely to be significant on sectional models, than on long structures.
The second problem is whether a modal vibration can develop from a static situation. since for infinitesimal motions the forces are quite unsynchronized. This must be 17
answered largely by observations of behaviour in the field, or from experiment on long models. The evidence so far suggests that such motion may occur if the flow is sufficiently uniform. Thus as early as 15 years ago Vandiver (1983) reported of transverse VIV motions of about one diameter for a cable with L/D=720, vibrating in its 3rd mode. He also measured magnification by a factor of up to 2.5 of the average drag force. Later experiences confirm that VIV motions can occur even for very long risers, provided the conditions are right. There are some indications that the vibrations on long risers in some cases may be harmonic in time, but that there may be phase differences between points along the riser axis. Vibrations usually occur mainly in one mode, but may suddenly switch to another mode. Also in some other cases non- lock-in vibrations are observed, and there is as of today difficult to define when lock- in vibrations will no longer occur. Some computer programs for prediction of the motions of line-like structures under the influence of a unidirectional shear current exist, e.g. SHEAR-7 by Vandiver (1998). The predictions made will depend critically on the input supplied by the user. Informations on such tools and on most of the more ambitious measurement programs on prototype risers are confidential.
There is also currently a quite rapid development of numerical tools to solve VIV problems. Some recent Norwegian work in this area is by Dalheim (1996), Halse (1997), and Meling (1998).
Conclusions The present paper attempts to give an overview of the findings concerning transverse lock-in vibrations of sectional models subjected to a steady, uniform water current. Some points that have been stressed are:
* There is a need to distinguish between two versions of the reduced velocity herein called nominal and true reduced velocity, Ured and Urr.,e respectively. * It is claimed that it is desirable to split the lift coefficients in two components, one affecting the vibrating frequency, the other the system energy balance. Both are nonlinear functions of the amplitude, and are highly dependent on other parameters. * Force coefficients from forced vibrations tests are shown. * Observed VIV behaviour of self-excited section models is presented. * The results from the spanwise force correlation coefficient must be interpreted with care. * It-is claimed the section model results may be used to predict the motions of line- like structures, such as risers, provided the flow is reasonably uniform.
References Anand, N. M.: "Free span vibrations of submarine pipelines in steady and wave flows", Dr. ing. thesis, Civil Engineering department, Norwegian Institute of Technology (NTH-). 1985
" Blevins, R. D.: Flow-Induced Vibrations, 2 d ed., Van Nostrand-Reinhold Co, New York, 1990 18
Dalheim, J.: "Numerical prediction of vortex-induced vibration by the finite element method", Dr.ing. thesis, Civil Engineering department, Norwegian University of Science and Technology (NTNU), 1996
Griffin, 0. M.: "Vortex-induced vibrations of marine cables and structures", NRL Memorandum Report 5600, Naval Research Laboratory, Washington, D. C., June 1985
Halse, K.H.: "On vortex-shedding and predictions of Vortex-induced vibrations of circular cylinders", Dr.ing. thesis, Marine technology department, Norwegian University of Science and Technology (NTNU), 1996 lwan, W. D.: "The vortex-induced oscillation of elastic structural elements", J of Engineering for Industry, ASME, Vol 97, 1378-1382, 1975
Meling, T. S.: "A numerical study of flow about fixed and flexibly mounted circular cylinders", Dr.ing. thesis, Stavanger College/NTNU, 1998
Moe G., Holden K, Yttervoll P.O.: ISOPE-94-HM-2, Motions of Spring-Supported Cylinders in Subcritical and Critical Water Flow, Proceedings ISOPE-94, Osaka, Japan.
Moe G., Wu Z.J.: The Lift Force on a Vibrating Cylinder in a Current, Proceedings OMAE Europe 1989 Conference, Vol III, pp 259-268, Hague, Netherlands, Mar 1989. Also in J of OMAE, Vol 112, pp 297-303, 1990.
Moe G.: "An experimentally based model for prediction of lock-in riser motions", Offshore Mechanics and Arctic Engineering Conference 1991, Vol 1-B, pp 497-506
Sarpkaya, T.: "Hydrodynamic damping, flow-induced oscillations, and biharmonic response", J of Offshore Mechanics and Arctic Engineering, ASME, Vol.117, pp: 232-238, 1995
Sarpkaya, T. & Isaacson, M. (1981): Mechanics of wave forces on offshore structures, Van Nostrand Reinhold Company.
Toebes, G.H.: "The unsteady flow and wake near an oscillating cylinder", ASME Journal of Basic Engineering, Vol 91, pp. 493-502
Vandiver, J. Kim: "Drag coefficients of long, flexible cylinders", Offshore technology conference, Paper OTC 4490, 1983
Vandiver, J. Kim: Personal communications, 1998
Wu, Z.-J: "Current induced vibrations of a flexible cylinder", Dr.ing. thesis. Civil Engineering department, Norwegian Institute of Technology (NTH), 1989 Slamming
Pr Odd FALTINSEN, Marintek, Norway SLAMMING
by
Odd M. Faltinsen Department of Marine Hydrodynamics Norwegian University of Science and Technology N-7034 Trondheim, Norway
SUMMARY
Theoretical methods for slamming on ships are presented. The topics are based on the author's research background. Two 2-D numerical methods applicable for rigid structures are discussed. One of the methods simplifies the dynamic free surface condition and details of the jet flow. Flow separation from sharp comers are incorporated. It is demonstrated that satisfaction of exact body boundary condition is more important than satisfaction of exact free surface conditions. The effect of local rise up of the water is significant. Water entry of an elastic hull with wedge-shaped cross-sections is analyzed. The effect of local structural vibrations on the flow is incorporated. The theory has been validated by comparing with full scale experiments of wetdeck slamming. Hydrodynamic and structural error sources are discussed. The importance of hydroelasticity as a function of deadrise angle and impact velocity are discussed.
1. INTRODUCTION
Impact between the water and a ship, i.e. slamming, can cause important local and global loads on a vessel. Different physical effects may have an influence. When the local angle between the water surface and the body surface is very small at the impact position, an air cushion may be formed between the body surface and the water surface. Compressibility of the air influences the air flow. The air flow interacts with the water flow, which is influenced by the compressibility of the water. When the air cushion collapses, air bubbles are formed. The large loads that can occur during impact between a nearly horizontal body and a water surface, can cause important local dynamic hydroelastic effects. This can lead to subsequent cavitation and ventilation. These physical effects have different time scales. The important time scale from a structural point of view is when maximum stresses occur. This scale is the highest wet natural period for the local structure. The effect of compressibility and the formation and collapse of an air cushion are.significant initially and normally in a time scale much smaller than the time scale of when maximum stresses occur. This may occur for wetdeck slamming. By wetdeck is meant the structural part connecting the sidehulls of a multihull vessel. Compressibility and air cushion formation will then have a smaller effect on maximum local stresses relative to dynamic hydroelastic effects. The largest impact pressures occur initially and will have a minor effect on the maximum stresses. It is an initial force impulse that matters. The reasons are that the largest pressures have a short duration relative to the highest natural period for the local structure and have a small spacial extent at a given time instant. Theoretical and experimental studies of wave impact on horizontal or nearly horizontal elastic plates of steel and aluminium are presented by Kvilsvold and Faltinsen (1995), KvAlsvold (1994), Kv~lsvold et al. (1995), Faltinsen (1997), Faltinsen et al. (1997), Haugen et al. (1997). It was demonstrated that dynamic hydroelastic effects are significant. Faltinsen (1998) studied slamming against an elastic hull with wedge-shaped cross-sections penetrating an initially calm water surface. This is discussed in more details in the main text.
When the local angle between the water surface and the body surface is not small at the impact position, local hydroelastic effects are not important for slamming on ship cross-sections. The slamming pressures can then be used in a static structural response analysis to find local slamming induced stresses. The air flow is unimportant and the water can be assumed incompressible and the flow irrotational. Since water impact is associated with high fluid accelerations, gravity is less important. The local rise-up of the water has a significant effect. The spray by itself is not important for slamming. The pressure inside the s•ray is close to atmospheric. But the generation of spray is associated with high pressure gradients on the hull. This is more dominant, the smaller the local angle between the water surface and the body surface is at the impact position. The water entry loads on a ship cross-section with bowflare will introduce global hydroelastic effects (whipping) of the ship. Flow separation from knuckles should then be accounted for. Two numerical methods for water entry of ship cross-sections will be reported. They have been validated by comparisons with model tests. One of the methods is exact within potential theory. The other represents a simplification and is more robust for engineering use. The theoretical and experimental studies show that common engineering methods to predict water entry loads on ship cross sections give too low maximum force and wrong time history of the force.
2. SLAMMING LOADS ON RIGID SHIP CROSS-SECTIONS
Slamming on ship hulls is often categorized as bottom slamming and bow flare slamming. The physics of bottom slamming has similarities with wetdeck slamming. When a bow flare section of a ship enters the water, the local loads around the flare are not significantly influenced by hydroelasticity. The pressure distribution can for instance be estimated by the nonlinear boundary element method developed by Zhao and Faltinsen (1993). This method accounts for the local rise-up of the water and the spray generation during entry. The fine details of the spray are neglected, but this is believed to be unimportant for slamming pressures and integrated water entry loads. A reason is that the pressure is close to atmospheric pressure in the spray. Gravity is neglected since fluid accelerations are initially dominating relative to gravitational acceleration. Gravity may play a role at a later stage of the water entry. However, it is a priori believed that introduction of gravity in the numerical model will not cause any problems. The numerical method solves the two-dimensional Laplace equation for the velocity potential as an initial value problem. The exact free surface conditions without gravity and the exact body boundary condition are satisfied at each time instant. An important feature is how the intersection between the water surface and the body surface is handled. Since the fine details of the spray are not studied and the pressure can be approximated as atmospheric in the spray, the spray is excluded in the boundary element formulation. Control surfaces normal to the body surface is drawn through the spray root(see lines AB and CD in Fig. 1). These surfaces can be handled in a similar way as a free surface.
SF \ ..... _-., ," ... SZ. ..
Fig. I Control surfaces used in nonlinear boundary element method by Zhao and Faltinsen (1993) describing water entry of two-dimensional rigid structures. The advantage of doing this is that the intersection between the water and the body at the free surface can be handled in a more robust way. Following the details of the jet flow associated with the spray could cause a small intersection angle between the free surface and the body surface. Small numerical errors would cause large errors in the prediction of the intersection.points. This can then destroy the numerical solution. This is a more severe problem the more blunt the body is. Since the free surface.will have a high curvature close to the body surface during water entry of a blunt body, the free surface shape was described in Zhao and Faltinsen (1993) by a higher order description. If straight line elements are used, artificial mass may be generated and destroy the accuracy.
Very good agreement with similarity solutions for wedges with deadrise angles between 40 and 810 and with asymptotic solutions for small deadrise angles were documented by Zhao and Faltinsen (1993). The water entry velocity is constant. There is a peak in the pressure distribution at the spray root close to the free surface when the deadrise angle is less than approximately30" (see Fig. 2). This pressure peak is what is often referred to as the slamming pressure. When the deadrise angle is less than 20° the pressure distribution becomes more and more peaked and concentrated close to the spray root and sensitive to the deadrise angle. A consequence of this is that rolling could have an important effect on the slamming loads. Faltinsen (1993) validated the method by comparing with drop tests of a bowflare section with a constant heel angle of 22.50.
a- 200 ------a = 250 .-- ..160 -- 16.0 P-Po aa- 400300 2 0.5p V a - 450
a.- 500 12.0 a = 600
.... a - 700 ...... a = 810 8.0------//-
4.0 ,,,O!
0.0 ...... -... "z -1.0 -0.5 o.0 0.5 1.0 Vt
Fig. 2 Predictions of pressure (p) distribution during water entry of a rigid wedge with constant vertical velocity V. p0 = atmospheric pressure (Zhao and Faltinsen, (1993)).
The original method does not include flow separation from knuckles or convex surfaces. Zhao et al. (1996).have extended the original method to include separation from knuckles. A Kutta condition implying tangential velocity and continuity in the pressure at a separation point is satisfied. The pressure on the body surface becomes more uniformly distributed in space after flow separation. The magnitude of the pressure is still significant. Predicting flow separation from convex surfaces have not been studied. This is a harder problem because the separation point is a w-iori unknown. An itcration procedure is needed. The flow separation cannot be determined by a viscous flow analysis. The duration of the water entry is too short to develop velocity profiles with zero shear stress at a point on the surface. The latter is the normal criterion for flow separation due to viscous effects. The situation is believed to be more similar to cavity flow past a blunt body like a circle in cross-flow.
Even if hydroelasticity does not affect the local loads during water entry of a ship cross section, it may play an important role in a global analysis. By considering the ship hull as an elastic beam, the integrated water-entry force on for instance a bow flare section causes transient hydroelastic response (whipping) of the ship. The commonly used methods to predict whipping due to water entry of ship cross-sections do not account for the local rise-up of the water. Zhao et al. (1996) demonstrated that local rise-up of the water is important for water entry forces on bow flare sections. Its importance will increase with increasing relative vertical velocities between a ship cross-section and the water. The smaller this relative velocity is, the more important is Froude-Kriloff and hydrostatic forces. By Froude-Kriloff forces is meant the pressure loads due to the incident waves only. Zhao et al. (1996) also showed that the hydrodynamic water entry force is not negfigible when the flow has separated from the knuckles. The peak in the vertical force occurs when the spray root is at the knuckle.
If the hydrodynamic vertical water entry forces are expressed in terms of the time derivative of infinite frequency added mass as a function of submergence relative to undisturbed free surface, the force part after flow separation from the knuckles will be negligible. This is common to do. An approach like this will also give too low maximum force and a wrong time history of the force. The reason is that an important part of the force is associated with the rate of change with time of the wetted area. The local rise up of the water implies a larger rate of change of the wetted area.
At a late stage of the water entry, it is of interest to compare the numerical results of vertical force with theoretical drag coefficients for steady symmetric cavity flow past a blunt body. These values are a function of the cavitation number. Knapp et 2 al. (1979) defines the cavitation number as K=(p 0 -p8)I0.5pV for water entry. Here V is the velocity of the body, p0 is pressure in undisturbed fluid at the depth of the nose of the entering body , p2 is the cavity pressure. The cavity pressure is the same as atmospheric pressure in our case. Further p is the mass density of water. According to Knapp et al. (1970) CD is 0.745 for two-dimensional symmetric steady supercavitating flow (K=0) past a wedge with interior angle 120 2 at the nose. CD is defined as C0 =F/(O.5pV B). Here F is the total force and B is the maximum breadth of the section. This body shape was studied by Zhao et al. (1996). Since gravity is neglected in the numerical computations, the water entry force on the wedge should approach the results for supercavitating flow when the submergence goes to infinity and the drop velocity is constant. The results show that the unsteady force part reduces.slowly after a rapid decrease just after flow separation from the knuckles. The results were plotted as a function of the inverse of the submergence of the wedge. This makes it easier to estimate asymptotic values when Vt goes to infinity. Here tis the time variable. The results indicated that the computed CD-value for a wedge with deadrise angle 30" approaches an asymptotic value close to 0.745 when the submergence goes to infinity. The numerical simulations should ideally have been continued for larger submergences, but numerical difficulties were encountered. The computations indicated that the cavity becomes infinitely long when the submergence goes to infinity. It is expected that the cavity will be finite if gravity is included. The deceleration of the section will also affect the solution.
The presented method does not solve secondary impact problem. An example on this is water entry of a cross-section with a sonar dome. The jet flow separating from the sonar dome can cause secondary impact on the hull. The same may occur due to flow separation from the bottom of a heeled structure entering the water.
A simplified solution for water entry ofa two-dimensional body was presentejby Zhao et al. (1996). The solution is more numerically robust and faster than the original method by Zhao and Faltinsen (1993). The quality of the predictions is believed to be satisfactory for engineering applications. The method is a generalization of Wagner's method. Wagner (1932) developed an asymptotic solution for water entry of two-dimensional bodies with small local deadrise angles. The flow was studied in two fluid domains. The inner flow domain contains a jet flow at the intersection between the body andtthe free surface. In the outer flow domain the body boundary condition and the dynamic free surace condition "---were transformed to a horizontal line. The kinematic free-surface condition was used to determine the intersection between the free surface and the body in the outer flow domain. Satisfaction of the kinematic free surface condition implies that the displaced fluid mass by the body is properly accounted for as rise up of the water. This is not true for a von Karman approach that does not account for the local rise up of the water.
In the generalization of Wagner's solution to larger local deadrise angles only the outer flow domain solution is analyzed. A main difference from the Wagner theory is that the exact body boundary condition is satisfied at each time instant. The wetted body surface is found by integrating in time the vertical velocities of fluid particles on the free surface and finding when the particles intersect the body surface. Wagner did also that, but he could use analytical solutions due to the simplified boundary conditions. The dynamic free-surface condition is the same as Wagner used. This is a simplification relative to the more complete method presented by Thao and Faltinsen (1993). The pressure is calculated by the complete Bernoulli's equation without gravity. It has not been possible to find an inner flow solution near a spray root that matches the outer flow solution for finite deadrise angles. This would have made it possible to exclude in a rational way the large negative pressures that occur at the intersection points in the outer flow solution. The procedure is simply to neglect the negative pressures. The theory was verified by comparing with the fully nonlinear solution and validated by comparisons with model tests. The simplified theory shows the importance to satisfy the exact body boundary condition and include the local water elevation at the hull. The exact dynamic free-surface condition is less important.
The simplified nonlinear analysisby Zhao et al. (1996) was extended by Zhao et al. (1997) to include flow separation from fixed separation points. This is done by an iterative process. A shape of the separated free-surface is assumed and the kinematic free-surface condition is satisfied on the assumed free surface. The functional form of the separated free-surface is found by an analytical solution close to the separation point. The shape of the separated free surface is iterated until the difference between the pressure on the separated free-surface and atmospheric pressure is minimized. A Kutta condition is satisfied at the separation points. The solution was verified by comparing with the fully nonlinear solution by Zhao et al. (1996) for water entry of wedges with knuckles. The method was also validated by comparing with experiments from drop tests of a wedge and a bow flare section with knuckles.
Generalizations of the two-dimensional methods to three-dimensional flow are needed. An asymptotic theory for water entry of an axisymnmetric body with small local deadrise angles is presented by Faltinsen and Zhao (1998). This represents a valuable tool for verification of three-dimensional methods. It is demonstrated that local rise up of the water at the impacting body is also significant for three- dimensional flow. The simplified numerical method derived for 2-D flow was generalized by Faltinsen and Zhao (1998) to water entry of axisymmetric bodies. The method was verified by comparing with the asymptotic method and validated by comparing with experiments for spheres and cones. The more exact 2-D method by Zhao and Faltinsen (1993) have been generalized to axisymmetric water entry by Zhao and Faltinsen (1998). It is demonstrated that satisfaction of exact body boundary conditions is more important than satisfaction of exact free surface conditions. 3 LOCAL HYDROELASTIC SLAMMING LOADS
The importance of hydroelasticity on local slamming loads is studied by Faltinsen (1998). Water entry of a slender hull with a wedge-shaped cross-sections is analyzed. By slender is meant that the length of the hull is large relative to the cross-dimensions. The deadrise angle P of the cross-sectional shape is assumed small (see Fig. 3). Hydroelasticity is accounted for. The stiffened platings between two transverse girders on each side of the keel are separately modelled. The stiffened plating consists of a
VZl
Fig. 3 Water entry of wedge-shaped elastic cross-section plate and longitudinal stiffeners (see Fig. 4). Since syrnuntry about the centre plane is assumed, only one of the stiffened platings is analyzed. The main emphasize is on strains in the longitudinal stiffeners. Orthotropic plate theory and Vlasov/Galerkin method is used (see Szilard (1974)). The Vlasov approach expresses the lateral deflections in terms of eigenfunctions of vibrating beams. The hydrodynamic problem is solved by a generalization of Wagner's method (1932) by including the effect of elastic vibrations. The details of the flow at the spray roots are not included. The method is similar as used by
y y=b fasvenet
xr--L , -Transvers
Fig. 4 Stiffened plating consisting of plate and longitudinal stiffeners.
Kv3,1svold and Faltinsen (1995). They considered hydroelastic impact of a beam on waves. Faltinsen (1998) assumes initially flat free surface and that the heel angle of the cross-section is zero. The hydrodynamic problem is solved for different cross-sections between two transverse stiffeners. Two- dimensional Laplace equation is satisfied for each cross-section. Unknowns in the hydroelastic problem are the mode amplitudes and the wetted surface along the hull. These are expressed in terms of coupled nonlinear ordinary differential equations in time. The coefficients in the differential equations can to a large extent be analytically expressed. This has a clear advantage in the numerical time integration which is performed by a Runge-Kutta 4th order scheme. The theoretical method was validated by comparing with full scale test of wetdeek slamming that have been carried out with the Ulstein Test Vessel. This is a high speed catamaran. Strains were mcasured in two areas in the forward part of the wetdeck. The main part of the strain gauges were mounted on the top of longitudinal stiffeners. Representative cross- sections of the catamaran at impact positions I and II are shown in Fig. 5. Impact position I is forward of position H. The wetdeck has wedge-shaped cross-sections. The deadrise angles are between 50 and 150 in the impact areas. Details of the measurement program are reported by Aarsnes and Hoff (1998).
•~~W ;110 ,1#
V•lI trflI(
Fig. 5 Representative cross-section of Ulstein Test Vessel at impact positions I and Il.
The water hits impact position II first in the comparative studies. The stiffened plate model contains three longitudinal stiffeners at position 11. The numerical and experimental results are in qualitative agreement. After the maximum strains have occurred at the middle longitudinal stiffener at impact position II, the theoretical values drop faster to small values than the experimental values. One reason to the experimental behaviour is probably that the interaction with the sidehulls becomes stronger than the theory assumes. The theoretical values become negative at the end of the time simulation. The reason is that the negative "added mass" pressure component associated with global shipaccelerations dominates. The experimental values also become negative. The presence of the large ship accelerations has also an important effect on the maximum strain. This is due to the reduction of the water entry velocity with time. It is demonstrated that the results are sensitive to the impact velocity. High frequency oscillations are present in some of the strain results. The oscillation amplitudes increase with time until they seem to reach a constant value. The high frequency oscillations are associated with the second mode which has a wet natural period of 0.0093 s for a fully wetted plate. The wet natural period of the first mode is 0.057 s. The behaviour is explained as a physical instability. The hydroelastic orthotropic plate theory is approximate. The error sources are both of structural and hydrodynamic nature. The structural model has been verified by comparing with finite element calculations. A quasi-steady model was used. This means that the instantaneous pressure distribution on the rigid structure was used in a steady structural analysis. When the water has passed the position of the middle longitudinal stiffener and the strain in this stiffener has reached its maximum value, the agreement between orthotropic plate theory and the finite element method is good. When the strain at the longitudinal stiffener closest to the "keel" of the wetdeck has its maximum value, the agreement is less good. This occurs when a small part of the plate is wetted. The results by the orthotropic plate theory are not sensitive to number of modes if at least two modes are used in the y-direction (see Fig. 3). Hydrodynamic errors occur because the body boundary and free surface conditions are approximated. Further the quadratic velocity term in Bernoulli's equation is neglected in calculating the pressure. These effects can be assessed by comparing with the results by Zhao and Faltinsen (1993). They presented numerical results for water entry of rigid wedges with constant velocity. Gravity was neglected and potential flow was assumed. When the deadrise angle is 15", Zhao and Fahinsen (1993) predicts a vertical water entry force f-', 5.5 p V't. Our theoretical approach gives Ios p vt. This gives an indication of errors in strains due to approximation of Bernoulli's equation, body boundary and free surface conditions. Since the differences in the loading by the methods are clearly largest at the spray roots, the difference in the two force results is not a conservative estimate of error. The error will decrease with decreasing deadrise angle. The effect of gravity, compressibility and air cushions are believed to be small. Hydrodynamic interaction effects with the sidehulls will be present. This have been discussed earlier in the chapter. The hydroelastic boundary value problem has been approximated. A mean deflection velocity has been used in formulating the body boundary condition. A more complete formulation is needed to assess the error. But the studies by KvAlsvold and Faltinsen (1995) indicate that this is not a major error source. Hydroelasticity mattered but did not have a dominant role in the comparative study with the full scale measurements. Its importance will increase with decreasing deadrise angle and increasing impact velocity. This was examined by a parametric study where non-dimensionalized parameters are introduced. Even if P3is assumed small in the analysis, results were presented for quite large P3. This is needed to clearly illustrate the dependence on hydroelasticity. The non-dimensional parameters can be found by non-dimensionalizing the equations in a similar way that Kv.lsvold and Faltinsen (1995) did in their study of impact of a horizontal elastic beam on waves of different steepnesses.
4 CONCLUSIONS
Theoretical methods for water entry of two-dimensional bodies are discussed. When the local angle between the water surface and the body surface is small, hydroelasticity should be considered. In the first part it is assumed that the structure is rigid and compressibility effects can be neglected. Two 2-D numerical methods are discussed. Both methods satisfy the exact body boundary condition and account for the local rise up of the water. One of the methods satisfies the exact free surface condition without gravity. The studies show that it is more appropriate to approximate the free surface conditions than the body boundary condition. Local rise up of the water must be included. Flnw separation from sharp corners has been incorporated. Gravity has been neglected, but should be incorporated after flow separation occurs. The loads after flow separation have been found to be non-negligible.
Water entry of an elastic hull with wedge-shaped cross-sections is discussed theoretically. The effect of local structural vibrations on the fluid flow and rise up of the water is incorporated. The theory has been validated by comparing with fullscale experiments on wetdeck slamming. The results are strongly affected by the global accelerations of the vessel. When the deadrise angle goes to zero, the results agree reasonably with asymptotic hydroelastic impact theory (Faltinsen (1997)). Hydrodynamic and structural error sources are discussed. Systematic studies on the importance of hydroelasticity as a function of deadrise angle and impact velocity are reported by Faltinsen (1998). The smaller the deadrise angle is and the larger the impact velocity is, the larger the influence of hydroelasticity is. This is quantified by Faltinsen (1998) in terms of non-dirmensionalized parameters. When the effect of hydroelasticity is large, it implies lower maximum stresses than predicted by a quasi-steady theory.
REFERENCES
Aarsnes, J.V. and Hoff, J.R., 1998, "Full Scale Test with Ulstein Test Vessel", Marintek Report, Trondheim, Norway. Faltinsen, 0., 1990, "Sea Loads on Ships and Offshore Structures", Cambridge University Press. Faltinsen, 0., 1993, "On Seakeeping of Conventional and High-Speed Vessels", 15th Georg Weinblum Memorial Lecture, J. Ship Res., Vol. 37, No2, June, pp. 87-101. Faltinsen, O.M., 1997, "The Effect of Hydroelasticity on Slamming", Phil Trans. R. Soc. Lond. A, 355, pp. 575-591. Faltinsen, O.M., 1998, "Hydroelasticity of High-Speed Vessels", 2nd Int. Conf. on Hydroelasticity in Marine Technology, Fukuoka, Japan. Faltinsen, O.M., Kv.lsvold, J. and Aarsnes, J.V., 1997 "Wave Impact on a Horizontal Elastic Plate", Journal of Marine Science and Technology, Vol. 2, No. 2, pp. 87-100. Faltinsen, O.M., Zhao, R., 1998, "Water Entry of Ship Sections and Axisymmetric Bodiss", AGARD Report 827, High Speed Body Motion in Water. Haugen, E.M., Faltinsen, 0., Aarsnes, J.V., 1997, "Application of Theoretical and Experimental Studies of Wave Impact Wetdeck Slamming", Proc. FAST'97, Sydney, Australia. Knapp, R., Daily, J.W., 1970, Hammitt, H.F.G., "Cavitation", Mc-Graw-Hill Book Company. Kv~lsvold, J., 1994, "Hydroelastic Modelling of Wetdeck Slamming on Multihull Vessels", Dr.ing. thesis, Department of Marine Hydrodynamics, Norwegian Institute of Technology, MTA-Report 1994:100. KvAlsvold, J. and Faltinsen, O.M., 1995, "Hydroelastic Modeling of Wetdeck Slamming on Multihull Vessels", Journal of Ship Research, Vol. 39, No. 3, Sept., pp. 225-239. Kv.lsvold, J., Faltinsen, O.M. and Aarsnes, J.V., 1995, "Effect of Structural Elasticity on Slamming Against Wetdecks of Multihull Vessels", Proc. PRADS'95, Korea, The Society of Naval Architects of Korea, pp. 1.684-1.699. Szilard, R., 1974, "Theory and Analysis of Plates. Classical and Numerical Methods", Prentice- Hall, Inc. Englewood Cliffs, New Jersey. Wagner, H., 1932, "Uber Stoss- und Gleitvorgange and der Oberflache von Fldssigkeiten", Zeitschr. f. Angew. Math. und Mech., Vol. 12, No. 4, pp. 193-235. Zhao, R. and Faltinsen, O.M., 1993, "Water Entry of Two-Dimensional Bodies", J. Fluid Mech., Vol. 246, pp. 593-612. Zhao, R., Faltinsen, 0., Aarsnes, J.V., 1996, "Water Entry of Arbitrary Two-Dimensional Sections With and Without Flow Separation", Proc. 21st Symp. on Naval Hydrodynamics, Trondheim, Norway, National Academy Press, Washington DC. Zhao, R., Faltinsen, 0., Haslum, H., 1997, "A Simplified Nonlinear Analysis of a High-Speed Planing Craft in Calm Water", Proc. FAST'97, Sydney, Australia. Zhao, R., Faltinsen, O.M., 1998, 'Water Entry of Arbitrary Axisyrnmetric Bodies with and without Flow Separation", Proc. 22nd Symp. on Naval Hydrodynamics, Washington DC. loads on barges and A quasi-3D method for estimating non-linear wave FPSO's
Pr Philip CLARK, Heriot-WaIt University, U.K action on offshore Non-linear time domain solutions for wave and current structures
Mr Pierre FERRANT, Sirhena, France h V WEGEMT Workshop on Nn-Linear Wave Action o Structuires antd Shi;,s. UTV-ISITV, Sepieniber 998
NON-LINEAR TIME DOMAIN SOLUTIONS FOR WAVE AND CURRENT ACTION ON OFFSHORE STRUCTURES
Pierre FERRANT SIREHNA NANTES - FRANCE
ABSTRACT
We present significant and validating results of a fully non linear time domain diffraction model for free surface potential flows. The theoretical basis as well as its numerical implementation are first recalled. Then a panel of recent results from the model, including strong non linear effects, is presented. Most of these applications include comparisons of simulated loads and runup with experiments or higher order diffraction theories, and assess the accuracy and efficiency of the numerical scheme, at least on simple geometries submitted to regular waves of finite amplitude with or without superimposed current. Finally, the challenge of extending the formulation to more general geometries and sea states is discussed.
INTRODUCTION
In the design process of offshore structures, the estimation of wave action is of primary importance. Wave loading condition the design of positioning systems and of the structure itself, while the wave runup estimation affects the vertical position of the deck, with constraints linked to the operability of the structure and the limitation of slamming loads. On both wave loads and wave runup, nonlinear effects are very sensible, especially in design conditions. For example, the so-called 'ringing' phenomenon discovered some years ago on offshore structures can be shown to be connected to third or higher order effects. The wave runup itself is driven by local wave-body interactions which are generally maximum at the waterline (except in some cases for multi-column structures, due to resonant interaction effects), and is thus naturally subject to strong nonlinear effects. In these conditions, it would seem wise to rely on fully nonlinear simulation methods for estimating wave actions on offshore structures. However, despite continuing advances both in hardware and numerics, the applications of direct three dimensional fully non linear models for wave-body interaction problems remain limited and require considerable computing resources for reasonably accurate results (Celebi et at 1998). At an intermediate level of complexity and computing demand, the time domain second order approximation of the fully non linear problem may represent a valuable alternative in moderately steep regular waves, see e.g. Isaacson & Cheung (1992), Skourup et at (1997). or Kim et al (1998), but misses significant nonlinear features of the diffraction phenomenon when applied to steep transient wave packets (Ferrant & Guillerm 1998). The present fully non linear model is based on an original approach in which an explicit description of the non linear incident wave is exploited. The problem is then formulated for the diffracted wave elevation and potential. The resulting scheme appears to be accurate and computationally more efficient than usual direct approaches, while relying on a consistent and fully nonlinear treatment of boundary value problem. The present version of the numerical model offers the choice of regular incoming waves of finite amplitude modeled by a stream function theory (Rienecker & Fenton 1981), with a uniform current possibly superimposed to the waves. The extension to arbitrary nonlinear long-crested wave packets is under implementation (Ferrant 1998b). In the case of regular waves without current, significant results were initially published in 1996 (see e.g. Ferrant 1996). In Ferrant (1998b), higher order wave loads on a thin vertical cylinder in long waves are successfully compared to experimental results from the University of Oslo (Huseby & Grue 1998, Grue & Huseby 1998), while extensive comparisons with the third order theory of Malenica & Molin (1995) may be found in Ferrant et al (1998). Preliminary results comparing simulated first, second and third harmonics of the inline force on a truncated surface-piercing cylinder to results derived from experiments performed by Marintek (Krokstad &
Now wit.h I.AIF-11IN, Ecole Centrale deI Nantes. Farcn Stansbcrg 1995) have also recently been obtained and will be given in the present paper. In regular waves plus current, extensive results may be found in Ferrant (1998a), including a parametric study on the combined non linear influence of wave steepness and current strength on the wave runup about a vertical cylinder. In Bijchmann et al (1998) runups computed with the present fully nonlinear model are compared to results from the .i finite order model of BUchmann et al (1997), validating both models in the low steepness-low Froude regime, while the importance of nonlinear (higher than second order) effects in more severe conditions is highlighted. A preliminary comparison with experimental runup data from Ifremer (Le Noac'h et al 1997), both with and without current, may also be found in Ferrant (1998b).
A review of these recent results is presented in this paper, attesting the accuracy and efficiency of the present nonlinear diffraction formulation, in the case of nonlinear regular waves with or without current. At last, the extension to more general geometries and sea states is discussed.
THEORETICAL FORMULATION
Semi-Lagrangian formulation
We consider a three-dimensional fluid domain (D), bounded by a free surface F and a set of N solid boundaries Si. These boundaries include the surface of a fixed offshore structure, as well as the sea floor at finite distance. The domain is of infinite extent in the horizontal directions. The fluid flow problem is formulated in the frame of potential flow theory. The fluid velocity thus derives from a scalar potential satisfying Laplace's equation at any point of the fluid domain:
V(M.t) = V4,(M,t) for M e•(D) (I)
A4'(M,t) = 0 for M r (D) (2)
On the free surface, both kinematic and dynamic conditions must be satisfied. The kinematic condition states that the mass flux through the free surface is zero, and writes, in Lagrangian form:
DM = V4(M)] forMe F (3) Lit
If surface tension is ignored, the dynamic condition expresses the continuity of the pressure across the free surface, and derives from Bernoulli's equation:
D =(M,)-z +/Ivrll for MeF (4) Di where D/Dt stands for the material derivative. Equations (3) and (4) suppose a fully Lagrangian description of the free surface, with markers identified as material points. In the present paper, the formulation will be modified by inhibiting the horizontal motions of free surface markers, leading to a semi-Lagrangian description. In such a formulation, the free surface vertical co-ordinate becomes implicitly single-valued, and may be expressed as:
z =l(x.y.t) forM(x.yz.t)e F (5)
Plugging this notation into (3) and (4), and after some manipulations, we obtain new forms of the non- linear kinematic and dynamic boundary conditions, in which a fixed projection of free surface markers in the x-y plane is implied:
= D (M) arlD for z=rt(x.y,) (6) di ra)xax 5hY(y +IIV1n2+ Ila)frz jx .t 'it 2 8at f1Z (7) where TI is the free surface elevation, z is positive upwards with its origin on the mean position of the free surface, and non dimensional quantities are assumed, based on a reference length L and with the acceleration of gravity g as the acceleration of reference. In the present paper, L is the water depth h.
On fixed material boundaries, no-flux Neumann conditions are applied:
=0 for ME Si, i=I.N 5T (8)
Separation of an explicit part of the solution
In some situations, it is practical to re-formulate the general problem described in the preceding section, by substracting from the total flow a contribution which may be explicitly described. The aim of such a procedure is to be left with a modified problem for the remaining part of the flow which will be easier to solve. This is what is usually achieved in linearized radiation-diffraction theory when prescribing the incident wave potential and solving the problem for the diffracted flow alone. Here the situation is sensibly more complicated, since the boundary-value problem is nonlinear. However, the field equation itself, namely Laplace's equation, remains linear so that we may write:
4) = (D,+ 4)d for M E (D) (9) where 'D, is a scalar potential satisfying Laplace's equation in the whole fluid domain. In the same manner, in a single-valued description of the free surface, we may write:
(x,3% ) = 77,x.Y,yt) + 71d(X, Y, t) (10) where r1,is some explicit time dependent function of the horizontal co-ordinates. Plugging (9) and (10) into (6) and (7), we are left with free surface conditions for 45 and 17,, in which terms depending on 0, and r7,are supposed to be evaluated explicitly.
dns =ac• aeD an 8kc' ar7, for z=Tn(x,yrt) (11) di az xax ay Eyy at =_+_% IV(D. -0%,at)a ýdt "1 I- for z=r7(x,y.t) (12) dt 2 T~ at 7Z
Note that at this point, 0, and 174are totally independent of each other, the only constraint is on P,•which must satisfy Laplace's equation throughout the fluid domain.
Specification of the incoming flow
In the present study, the incoming flow is composed either of a uniform current superimposed to non-linear regular waves, or of short duration wave packets. In the former case, the incident wave is modelized by a stream function model (Rienecker & Fenton, 1981). The model for wave packets is discussed in the sequel of this paper. In the stream function model, which is valid for both deep and shallow water, the velocity potential and elevation of a periodic wave traveling over a horizontal bottom are expressed as Fourier series. Nonlinear free surface conditions are satisfied exactly at some equidistant points distributed over one wave length, so that the error in the solution is driven by the truncation of the Fourier series only. The incident wave elevation is given by:
%(, Y, = + cost jk(- -,c)) (13) 2i and the potential by:
N. cosh(jk (z +h)) .(14) 4) i(x, y, t)= U (x - x, ) + cBs sin(jk (x.- x,, - ct)) cosh(jkh) where U is the current speed, c the wave celerity, k the wave number and N. the order of truncation of the Fourier series. The incoming flow being specified, conditions at infinity simply write:
D(M.) Mt---3- i(M t) (15)
Formulation for the perturbation flow
The most evident choice for 4. is Pi, for example by examining the simplification induced on condition at infinity, and the fact that after this choice, a major contribution to the total flow will be represented analytically, with advantages in terms of global accuracy. For identical reasons, we shall also apply r, = 77iThe resulting boundary value problem to be solved for (0d. 77d) is thus:
A'a(M .) = 0 formM (D) (16)
On the free surface (z=fl(x,y~t)):
d-'-=-(/ d) + - I -ar-. a _. (17) _i - '(7 1d F +V(~ (1(1+7)
dt Iz +a ac at 21 at az at
In these equations, .'stands for T .alax+j.alay, and terms involving Oi and r/iwill be evaluated exactly from the stream function model. Damping terms involving functions v(R) have also been included. The role of these terms is to absorb the diffracted waves on the outer part of the free surface mesh. More details on the implementation of this absorbing zone technique are given in a following section.
Initial conditions in the domain correspond to the incident wave without perturbation:
aDd(M- =0) =0 formM (D) (19)
On the body, the no-flux conditions will be written:
-•n for ME S1 (20) an an where og(t)is a smooth ramp function varying from 0 to I during the beginning of the simulation (typically I to 2 wave periods). It must be emphasized that the boundary-value problem being non-linear, equations (17) and (18) will be satisfied on the instantaneous free surface position, so that the incident potential 0, may possibly be evaluated above the undisturbed incident wave z=rlx,yvt ) . This is possible here because the stream function potential has a continuous prolongation above the incident wave. One of the advantages of this formulation is that the incident wave is treated explicitly, so that it is not altered during its propagation from the outer surface of the computational domain to the body. In addition, the perturbations represented by (0adrTq)are composed of waves propagating in radial directions, allowing relatively coarse meshes to be adopted far from the body, in tile absorbing zone. This results in considerable savings of memory and CPU time, without loss of accuracy. On the other hand, this formulation is not universal and rely on the availability o)f an explicit model for the incident wave. with appropriate continuity properties across the incident wave The initial boundary problem being properly posed, a solution procedure based on a boundary integral equation method is set-up. Applying Green's identity to the velocity potential and to the free space Green function (Rankine source) G(M,M')=-1J/4rMM, we obtain an integral representation of the velocity potential (P,:
nm(M) •(M~t0 47f
all, ( 21 where Q2(M) is the internal solid angle at point M. fl(M)=2,r for M on the fluid boundary surface Y, and Q(M) = 4% in the fluid domain. Applying a Dirichlet condition on the free surface and Neumann conditions on the rest of the fluid boundary, we are left with a mixed Fredholm integral equation of the first kind on the free surface, and of the second kind elsewhere. This equation will be solved at fixed time t for Clo Ian. The solution procedure adopted to solve the initial boundary value problem pertains to the family of mixed Euler-Lagrange methods and involve a combination of numerical schemes described in the following section.
NUMERICAL METHODS
Boundary element method
A boundary element method is used for the solution of the boundary integral equation formulation of the problem. The method is based on iso-parametric triangular elements distributed over the different boundaries. A piecewise linear, continuous variation of the solution over the boundary is thus assumed, and collocation points are placed at panel vertices. Meshes are made of an assembly of different patches, with the assumption of continuous normal on each of them. On intersection lines between two patches, two collocation points are considered for one single geometrical location, and the boundary conditions corresponding to the two surfaces are both satisfied. This discretization scheme reduces the integral formulation to a linear algebraic system to be solved for the normal velocity on Dirichlet boundaries (free surface) and the potential on Neumann boundaries. This system is assembled from the influence coefficients of linearly varying distributions of sources and dipoles on boundary elements. Analytical formulas for the near field, and different approximate formulas for the intermediate and far field are implemented. These coefficients are factorized with respect to sources or dipoles density at panel vertices, which are selected as control points. This scheme results in full, non-symmetric square systems of equations to be solved for the singularity distributions at the boundaries of the computational domain.
Solution of linear systems of equations
In 3D non linear applications large systems of equations have to be solved at each time step, and any O(N3) solution algorithm is absolutely unsuitable. In the present formulation, the linear systems to be solved are full and non-symmetric. The GMRES scheme of Saad & Schultz(1983) is applied to the solution of these systems. The method is used with diagonal pre-conditioning, which reduces the necessary number of iterations by a factor close to 2, at no additional cost. Further reductions of the number of iterations may be obtained with more elaborate pre-conditioning techniques, but the cost of such pre-conditionings applied to full matrices is not negligible and may annihilate the advantage of the reduction of the number of iterations.
Interpolations and smoothing at the free surface
In the present application, the projection of free surface nodes on the x-y plane is time-invariant, and the mesh is structured in the azimutal direction, that is nodes are distributed on circles surrounding the body, with a regular spacing on each circle. This situation allows us to apply a bi-cubic interpolation scheme for both the potential and the vertical co-ordinate at the free surface. These quantities are first interpolated at each radial station as functions of the azimutal angle. Then: interpolating splines are computed in the radial direction, at each free surface node. We are thus left with C, representations of 0 and rl at the free surface, which allow accurate and straightforward evaluations of the velocity components and oi" the normal vector, by direct differentiation of the interpolating splines. This procedure has proved to be sensibly more accurate than the local polynomial fitting used in the general case, but cannot directly be applied on arbitrary geometries. The structure of the free surface mesh is also exploited for the application of smoothing procedures at the free surface. This smoothing may become necessary for removing saw-tooth instabilities appearing during the simulation of strongly non-linear flows. Here we use'five points formulas based on Chebyshev polynomials, applied sequentially in each parametric direction. For simulations presented in this paper, the smoothing proved necessary in the strongest cases, and was applied typically every five time steps.
Time marching scheme
After the solution of the boundary value problem and the computation of the fluid velocities and normal vector at the free surface, free surface conditions (17) and (18) considered as ODE for C1and r) are advanced in time. A fourth order Runge-Kutta scheme is used for that purpose, requiring four solutions of the boundary value problem per time step. In order to reduce CPU times, results presented in this paper were obtained using the "frozen coefficient" approximation, in which influence coefficients are updated only one time per time step, while four solutions of the boundary value problem are performed.
APPLICATIONS AND VALIDATION IN REGULAR WAVES AND CURRENT
The table below gives an overview of results and comparisons recently performed with our fully nonlinear diffraction model. All cases concern vertical surface-piercing circular cylinders of radius R submitted to regular waves with amplitude A and wave number k in water of depth h, with possibly a uniform current of velocity U superimposed to the waves. These cylinders are bottom-mounted, except in cases corresponding to experiments from Marintek and Ifremer. Values given in the table refer to cases for which the present numerical scheme has been run. Other cases are available in the experimental data bases from UiO, Marintek and Ifrerner are presently being or will be investigated shortly.
Case Answave compared to: Loads Runup kh kR kA U/(gR) A 3rd Order theory 8. 0.05 to 0.30 0.06 0. Malenica & Molin X (1995) 8. 0.20 0.04 to 0.12 0. B Exp. UiO 4.9 0.245 0.05 to 0.145 0. Huseby & Grue X (1998) C Exp. Marintek 12.4 0.39 0.145 0. Krokstad & Stansberg X 0.215 (1995) 111 D Exp. Ifremer 9.43 0.235 0.141 0. A. Le Noac'h et al X (1997) 6.04 0.151 0.09 0., +0.16 E 2" Order Wave + 1.0 1.0 0.0025 to 0.15 0. to +I/-0.I I" Order Current X Skourup et al (1997)
Except for case E which is in the diffraction regime of intermediate depth waves, all other cases correspond to the deep water, long wave regime, corresponding to the possible occurrence of ringing which motivated most of the experimental and theoretical studies reported in the table. In the following sections, brief overviews of each of the comparisons referred to as cases A, B. C and E will be given, with key results and comments. For case D, limited comparisons described in Ferrant (1998b) will not be repeated, a more comprehensive comparative study being planned in a near future.
Case A
In this case, fully nonlinear simulations were run first with the objective of recovering the 3'd order results of Malenica & Molin (1995). Fully nonlinear simulations were thus run for small steepness, kA=0.16. for a fixed wave number kh=8., and for different cylinder radii such that kR varies between 0.05 and 0.30. Time series of the horizontal forces on the cylinder were analysed by a moving window Fourier scheme, and results for the third harmonic compared to Malenica & Molin's values. The comparison is given in figure I. were real and imaginary parts of the normalized third harmonic force given by both approaches are plotted. The agreement is extremely good over the whole range of kR. Mean force, first and second harmonic were also compared, with a similar level of agreement, see Ferrant et al (1998). A second investigation was devoted to the influence of wave amplitude on third harmonic force components. Simulations were run for kR=0.20, and for increasing wave amplitudes, kA=0.04 to 0.12. Figure 2 shows that nonlinear results tend pretty well towards third order analysis results when the amplitude is reduced. This is a non trivial result, considering the very low absolute value of the third harmonic which has to be extractee from the total time series. For example, for kA--0.04, F..3 F' is less than 104 . For larger amplitudes, an appreciable variation of the normalized third harmonic components is observed.
F,(3) 0.5 pgA3 .. 0 0.3
0.1 o
-0... lOm(F3)'- rd order model
-0.1 - RfFJJ- 3rdorderntmdel
-0.3 0 Re(F3). Fully .n linea,
-0.5 , I _I j 0.00 0.10 0.20 0.30
Figure I: Third Harmonic force components, kh=8., kA=0.06
F(3)1 pgA 3 3rd Order 0.20
O-- Re(F3). Fully mm linear 0.10 - hm(F3) - Fully non linear 3rd Order
0.00
-0.10 0.000 0.005 0.010 0.015 0.020 A/h Figure2: Influence of wave amplitude on third harmonic force components, kh=8.. kR=0.20
Case It
In Huseby & Grue (1998), experimental data on wave loads about a thin vertical in long regular waves were presented, in the form of the harmonic coefficients of the steady-state horizontal force on the body. Harmonics up to the 3wo lorce were given, for a wave number kR=0.245, and for different wave amplitudes. The present nonlinear diffraction model was run in similar conditions, and a very good agreement was observed on the amplitude and phases of experimental and numerical results, including the variation of normalized lorce coefficients with the wave amplitude, see figures 3 and 4 . This motivated a deeper comparison, in which measured and simulated wave loads are compared up to the seventh harmonic (Grue & Husehy. private 0.50 ------.
3.0 0-0 Experiments U. Oslo 2.5 0-0G-O-( Simulation (Answave) 2-0O - Experiments U. Oslo 0.402.0 G---O Simulation (Answave) 1.5
* 2 1.0 * 0.30 0'- 0.0 S-0.5 >0.20 cn -~-1.0 -1.5 0.10 -2.0 -2.5 -3.0 0.00 -3.5 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 Ak Ak Figure 3: 3rd Harmonic Force Amplitudes Figure 4: 3rd Harmonic Force Phases kR=O.245, kh=4.9 kR=0.245, kh=4.9 0.40 I3.5 -cExperiments U. Oslo 3.0 G-O- Simulation (Answave) 2.5 0.30 2.0 'C) 1.5 < ~1.0 * C1 0.2 ~ 020'- 0.0 * < -1.0 -1.5 0.10 -2.0 -G--E] Experiments U. Oslo -2.5 -G--O Simulation (Answave) -3.0 0.00 -3.5I I 0.00 0f05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 Ak Ak Figure 5: 5Il' Harmonic Force Amplitudes Figure 6: 511'Harmonic Force Phases kR=0.245. kh=4.9 kR=(X245. kli=4.9 communication. 1998). Full results of this extended comparison may be found in Ferrant (1998b), showing a strikingly good agreement in every respect, except may be for the second harmonic, exhibiting a sensible and constant difference between experiments and simulations. We simply give here in figures 5 and 6 the comparison of the fifth harmonics- A monotonic decay of the normalized force coefficients with the wave amplitude is t observed, a trend appearing also on the 6th and 7" harmonics. The global level of agreement between experiments and nonlinear simulations is extremely satisfactory and is an indication of the reliability of both experimental and numerical models. Considering that the free surface mesh used in the present simulations is not supposed to correctly resolve waves shorter than 3o free waves, the observed level of agreement up to the 6Oh or 7th force seems to indicate that higher harmonic forces are not linked to short diffracted free waves. This should be investigated in a forthcoming study. Case C Simulations for this case were undertaken very recently. The conditions correspond to two experimental tests on a truncated vertical cylinder in regular waves. The experiments were performed in Marintek, and are reported in Krokstad & Stansberg (1995), where first, second and third harmonic forces derived from the experiments were given. The water depth is 10m, the wave period is l.8s. and two wave amplitudes, A=0.l 184m and A=O.1715m were considered. The cylinder radius is 0.3125m and the draft is 0.938m. The first, second and third harmonic forces derived from the present fully nonlinear simulation method are compared to experimental values in table 2 below. T=1.8 s kR=0.39 A F1 Exp. F1 Num. F2 Exp. F2 Num. F3 Exp. F3 Num. 0.1184m 487N 485N 19.8N 172N 8.2N I1.0N (kA=O. 145) 0.1715m 709N 692N 24.3 N 24.4 N 24.2 N 25.7 N (kA=0.2 15) _ The agreement is again extremely satisfactory, up to the third harmonic amplitude, especially for the largest " amplitude. Note the better agreement on second harmonics than for case B, possibly linked to the different scales of the experimental facilities. Unfortunately, phases were not reported in Krokstad & Stansberg (1998) an thus cannot be compared. Case E In this case, wave and geometry conditions are sensibly different, with kh=l.0 and kR=l.0, with a fat bottom-mounted cylinder (R/h=l.0). A parametric study on the non-linear influence of both the wave amplitude and the strength of a uniform current superimposed to the wave on the wave runup about the cylinder was undertaken. First results were compared with those of a finite order time domain model and reported in B1chmann et al (1998). This comparison essentially revealed the convergence of both models in the low Froude-low steepness regime, while sensible differences due to the growing importance of nonlinear effects were highlighted in more severe conditions. The nonlinear simulations were further analyzed in Ferrant (1998a), where profiles of the harmonic components of the wave elevation at the cylinder waterline were discussed. Figures 7 to 10, give the harmonic components of the wave profile at the cylinder waterline for the largest wave height, Hfh=0.30, and for five values of the Froude number, Fr=0.O, +/-0.05 and +/-0.10. The fundamental harmonic presents sensibly linear variations with respect to the Froude number at the upwave and downwave locations, while the behavior is not so clear around =---./2. The second harmonic is significant all around the cylinder, with a marked maximum close to P-=0.9n. The influence of the Froude humber on the second harmonic is relatively smooth. The third harmonic shows moderate variations around the cylinder for positive values of Fr. while large amplifications between 13--0.4nt and ---0.8"t are observed for negative Froude numbers. 1 Fr=--lOO 2.5 , 1 0.20 Fr=.05 Fr=-O. I0 0 0----.2.-o Fr=-O.050 - Fr=+O.O05 1.5 - Fr=-O.OO 0.00 Fr=-+O.OD1 Fr=-O.05 -00 . 1.0 - ".1 -0.20 0.5 -0.30 0.0 I I I I 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 7: Mean value of the wave profile.H/h=0.30 Figure 8: 1"harmonic of the wave profile.H/h--0.30 ' I I I I I 0 .5 0 -F -0r ,F.0.40 -- -o 0.20 . -,o 0 Z0.30 -' a Fr-O 05 Sý00 --. 0 ýFrre.OI 0.20 0.10 0.10 0.00 0.00 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 9: 2nd harmonic of the wave profile Hlh=0.30 Figure 10: 3'd harmonic of the wave profile H/h=0.30 EXTENSIONS OF THE MODEL Nonlinear wave packets The key of the separation of incident and diffracted flows in the fully nonlinear diffraction model described above is the availability of a prolongation of the incident wave potential above the incident wave. In regular waves, the stream function allows the estimation of the incident wave potential and velocities below or above the incident- wave without difficulty. When accounting for transient waves, things are sensibly more complex. One possible solution for representing the incident flow would be to run a two dimensional nonlinear time domain free surface model to obtain a transient nonlinear incident wave from a given wave maker motion time series. However, potential and velocities would have to be evaluated by some kind of extrapolation, which is not recommended when accurate results are expected. Another solution is being developed. The entry of the process is the record of the incident wave packet at a specific location Xo on the free surface. The signal being bounded in time, the corresponding nonlinear incident flow can be reconstructed using a Fourier series expansion in space and time (Baldock & Swan 1994, Chaplin 1996). The coefficients of this double series are determined through the minimisation of an objective function based on the errors on kinematic and dynamic conditions in the vicinity of Xo. The resulting representation of the incident wave packet is continuous across the incident free surface. The nonlinear diffraction model with separation of the incident potential and velocities may thus be applied, just as with the stream function model. Extension io long duration irregular incoming waves, possibly based on a local Fourier approximation for nonlinear irregular waves, along similar lines as described in Sobey (1992). may also be planned. More general geometries. One of the keys of the accuracy of fully nonlinear free surface potential flow simulations lies in the calculations of free surface slope and fluid velocities at free surface nodes. In the applications presented in this paper, bi-cubic spline interpolations in radial and circumferential parametric directions were applied. With more complicated geometries of the free surface mesh, for example around a multi-column structure such as a TLP, this schem cannot be applied any more. Instead, local polynomial fittings and finite difference formulas may be applied, as in the time-domain second order solution of the interaction of wave packets with a cylinder in a wave tank (Ferrant & Guillerm 1998). This alternative is currently being tested in the fully nonlinear version of the code, and will aloow the simulation of fully nonlinear diffraction problems around realistic offshore structures. CONCLUSION A review of recent results obtained with the software ANSWAVE have been given. Most of these results have been compared with available results from experiments, or finite order theories. The overall agreement is very satisfactory and gives a high level of confidence in the model.. Future work include the extension of the model to more general sea states and body geometries. The possible limitations of the model when compared to experiments or real scale measurements, due to flow separation in large current values or for large A/R ratios should also be investigated. Acknowledgments: The development of the nonlinear free surface flow simulation model Answave has been supported by the French Ministry of Defense, through various research contracts. Recent developments were achieved in the frame of french CLAROM projects. funded by CEP&M. The author would like to thank Professor John Give for valuable discussions and exchange of results. The help of IFREMER/DITI/GO/HA who supplied experimental runup values in waves and current is also greatly appreciated. REFERENCES T.E. Baldock & C. Swan, 'Numerical calculations of large transient water waves', Applied Ocean Research, vol. 16, pp 101-102, 1994 B. Biichmann, J. Skourup, K.F. Cheung, 'Runup on a structure due to waves and current', ISOPE'97, Honolulu, 1997 B. Wiichmann, P. Ferrant, J. Skourup. 'Runup on a cylinder due to waves and current: fully nonlinear and finite order calculations', Proc. 131hW.W.W.F.B., Delft, 1998. M.S. Celebi, M.H. Kim, R.F. Beck, 'Fully nonlinear 3D numerical wave tank simulation', J.S.R., Vol. 42, N0 l, pp 33-45, 1998. J.R. Chaplin, 'On frequency-focusing Unidirectional waves', Int. J. Offshore and Polar Engg., Vol. 6, n02, 1996. P. Ferrant, 'Simulation of strongly nonlinear wave generation and wave-body interactions using a 3D MEL model', 21" ONR Symposium on Naval Hydrodynamics, Trondheim, 1996. P. Ferrant, 'Runup on a cylinder due to waves and current: potential flow solution with fully nonlinear boundary conditions', Proc. 8th Int. Conf. Offshore and Polar Engg, ISOPE'98, 1998a. P. Ferrant & P.E. Guillerm, 'Interaction of second order wave packets with a vertical cylinder', Proc. 8"' Int. Conf. Offshore and Polar Engg, ISOPE'98, 1998. P. Ferrant, 'Fully nonlinear interactions of long-crested wave packets with a three dimensional body'. 22"m ONR Symposium on Naval Hydrodynamics, Washington, 1998b. P. Ferrant, S. Malenica, B. Molin, 'Nonlinear wave loads and runup on a vertical cylinder', Chapter in: Nonlinear Water Wave Interaction, Advances in Fluid Mechanics. Eds M. Markiewicz & 0. Mahrenholhz, Computational Mechanics Publications, 1998. .1.Grue & M. Huseby, 'Higher harmonic forces on a vertical cylinder', Private commnunication 1998. NI. Iluseby, J. Grue, 'An experimental investigation of higher harmonic forces on a vertical cylinder in long waves', Proc 13"' Int. Workshop on Water Waves and Floating Bodies, Aalphen an den Rijn, 1998. Isaacson, M, and Cheung. KF. 'Time Domain Second Order Wave Diffraction in Three Dimensions'. .. Watcrways. Port, Coast. and Ocean Engg., ASCE. Vol. 118, Part 5.496-515., 1992 Y. Kim, D.C. Kring, P.D. Sclavounos. 'Linear and nonlinear interactions of surface waves with bodies by a three dimensional Rankine panel method' Appliead Ocean Research. Vol. 19. pp 233-249, 1998. J.R. Krokstad, C.T. Stansberg, 'Ringing load models verified against experiments'. Proc. OMAE'95 Conference, 1995. A. Le Noac'h, D. Buisine, M. Le Boulluec, 'Surklfvations de la houle autour d'un cylindre', Ifremer Report DITI/GO/HA RI IHA97, 1997. S. Malenica, B. Molin, Third harmonic wave diffraction by a vertical cylinder', J. Fluid Mech ., Vol. 302, pp203 - 229, 1995. J.N. Newman, 'Second_order diffraction loads upon three dimensional bodies'. J. Fluid Mech., Vol. 320, pp 417- 443, 1996 M.M. Rienecker, J.D. Fenton. 'A Fourier approximation method for steady water waves', J. Fluid Mech, Vol. 104, pp 119-137, 1981. Y. Saad, M.H Schultz, 'A generalized minimal residual algorithm for solving non-symmetric linear systems', Res. Report Yale University RR-254, 1983. Scolan, YM, Le Boulluec, M, Chen, XB, Deleuil, G, Ferrant, P, Malenica, S, and Molin, B (1997), "Some Results from Numerical and Experimental Investigations on the High Frequency Responses of Offshore Structures"; Proc BOSS'97 Conference, Delft, The Netherlands. Skourup, J, Biichmann, B, and Bingham, HB (1997). "A Second Order 3D BEM for Wave-Structure Interactions", Proc. 12"' Int. Workshop on Water Waves and Floating Bodies, Carry-le-Rouet, France. R.J. Sobey, 'A local Fourier approximation method for irregular wave kinematics', Applied Ocean Research, vol. 14, pp 93-105, 1992. by Semi-analytical methods for different problems of diffraction-radiation vertical circular cylinders Mr Sime MALENICA, Bureau Veritas, France SEMI-ANALYTICAL DIFFERENTS METHODS FOR PROBLEMS OF DIFFRACTION-RADIATION BY VERTICAL CIRCULAR CYLINDERS time MALENICA Presented at the 5th WEGEMT Workshop Toulon, 4th September 1998. SEMI-ANALYTICAL METHODS FOR DIFFERENT PROBLEMS OF DIFFRACTION-RADIATION BY VERTICAL CIRCULAR CYLINDERS Sime MALENICA Presented at the 5th WEGEMT Workshop Toulon, 4th September 1998. 1 Introduction As in the other fields of mechanics, the analytical methods represent an important analysing tool in marine hydrodynamics. The analytical approach is interesting for different reasons : it gives the reference results for numerical codes verification, it gives the physical insight to some complicated problems, it can be used as a simplified predesign tool, etc. This approach is of course limited to some simplified geometries (cylinders, spheres, ... ), and the case of one or more cylinders, truncated or not, will be considered only. Presented methods are basically eigenfunction expansions whose complexity depends on the boundary conditions. The hydrodynamic boundary value problem (BVP) is formulated within the usual assumptions of potential flow and is additionally simplified by the perturbation method. By using this approach, the highly nonlinear problem decomposes into its linear part and the higher order (second, third, ...) corrections. Also, the periodicity is assumed so that the time dependence can be factorized i.e. the frequency domain formulation is adopted. As far as the free surface flow is concerned, only the cases without or with small forward speed are sufficiently simple to be solved semi-analytically. The problem of the floating body advancing in waves with arbitrary forward speed is far more complicated. These remarks are also valid for the general numerical methods where the case of arbitrary forward speed, even linearized, is still too difficult from numerical point of view, and "it is fair to say that there exists at present no general practicalnumerical method for the wave resistanceproblem" [9], and even less for the general seakeeping problem. 2 General theory Classical assumptions of perfect fluid and irrotational flow inside the fluid domain fl, are adopted. We define the right-handed coordinate system (z, y, z) fixed to the body, with z = 0 the undisturbed free surface, the axis z pointing upward. The sea bottom SBT is assumed to be horizontal plane placed at z = -H. With these assumptions, a nonlinear boundary value problem for the velocity potential t(x, y, z) can be formulated At= 0 xefl (1) (V$ - v)n= 0 xE SBUSBT (2) oT•t2_ + 9:-Z-z°+ 2VtV 5-t- + vtV(V4DVt) =0 , S- (3) where E is the exact position of the free surface defined by . 18$9 1- -- ;t• = -(4) +z and v is the velocity of the rigid boundaries. The above BVP should be completed by an appropriate radiation condition at infinity. The convention used throughout this paper is that the normal vector n is pointing out of the fluid domain. In order to simplify the BVP (especially the free surface condition) we proceed in two steps. First we assume the displacements to be small and we express the quantities at the instantaneous position by Taylor series developments with respect to their mean position, and after that we introduce the perturbation series with respect to the wave steepness ( e = k0 A with c0 - wave number, A - wave amplitude ) which allows the decomposition of the nonlinear problem into the less complicated first, second, third, ... order approximations. Even if the methodology remains the same for U = 0 and U $ 0 there are some differences in the application, so we consider two cases separately. 2.1 The zero forward speed case In the case of no forward speed, the Taylor series expansion gives for the free surface elevation I118$ 1 1 at 42$ -- = -(--+ - )l +o( ) (5) 2 and for the free surface condition Ot2 + g oO+ V4V(O) -- + -24) 524 + 2 15, 24), 03,41 g a~t Saz 2t Tt 8z 1(1 &D5a24p 14V.) ",D+ 092D 9-(g9 -D -t2WVt-a0 z +gba-jZ-) + -• ( )2( 54tO g8 ) + O(4)(6 -W In this at ata2 Z case the perturbation series for (6) the potential has the following form 4, = eC(O) + 620(2) + E3(3) + 0(, 4 ) (7) Also we assume time periodicity at frequency w for the flow at first order from i w{e(1)e-iet} from which we easily deduce (8) the form of the higher order potentials S20(2) = @(2) + ({•,(2)e_2i•t} (9) C3q5(3)= R1-{(3)e-l") +R (3)e-3iwt} (10) The similar perturbation series is assumed for the free surface elevation interest): = (and all other quantities of 2 = +E e E(2) + E3=±(3-O ) (11) - 4{7Me't} + q(2) + R{r,(2)e-2iwt} I+ R{1/(3)e-i't} 3)e - 3 After introduction + R{1n( it} + 0(e 4 ) of the perturbation series (7,11) in (6) and (5) we obtain the following conditions and free surface free surface elevations at the corresponding orders o(e) + - 0 (12) Or(1) = -)WMI. W (13) 0(0 2 )9 -41lW- (1) + M A5'2 ()1 (14) 17(2) _ 2 2iw W(2) 1 V•(1)7Vol) 1 W()WM15 g 4g 215 0(0~) -9vwts) + 3i=w2g 2 "O v z -VW (2)02 ( 1) [ ¾(2)) 4'O)( 2 5Z Sz +- v(9v (1 VWo(I)) -W') 2 -•- "Oz--'- T (16) where Of course,V =w2y. all these potentials must satisfy Laplace the fixed equation in the fluid domain, zero boundaries and corresponding radiation normal velocity on We conditions which will be discussed precise here that only the first order later. 2 quantities at frequency w, the second w and the third order quantities order quantities at frequency at frequency 3w will be considered can I)e performed for the steady here, although the similar analysis second order quantities and the w component of the third order quantities. 3 2.1.1 Wave loads They are obtained by integrating pressure over the wetted surface of the body F = Lsa pndS (17) where the pressure is calculated from the Bernoulli equation = - -e(Vf)? (18) Careful analysis should be performed to collect the terms of different order. The problem is simplified for fixed bodies and following expressions can be obtained O(E) 0 ) F = L8.iwoppO'ndS (19) 0(02) F LSo(2iw9Ow2-0= oV•7'lVp(1')ndS + V9 fC0o 7 77ndC (20) O(W) F(3) (&W"(3 )3- !vVq°(1VW(2))ndS + !1 9 r7[(1)l(r(2) - 4('r7,(1))ndC (21) i5 80 2 LBOn 2.2 The small forward speed case In this case only the linear problem will be considered. Due to the presence of small forward speed U in the positive x direction (or the current in the negative x direction), the perturbation series has the following form : 2 4= U(0) + eu + 0(e ) , =U() + =u + O(e 2 ) (22) where 0(o) is the steady potential due to the presence of current, and can be further decomposed to the uniform current and the body perturbation €, 00) = ý - X. On the other hand, we have chosen to describe the problem in the coordinate system (z, y, z) fixed to the body so that the incident potential can be written as : 600l)(xt)= R{R?{p1e)e-it} igAcoshko(z_+ H).iko(zcos0+ysin3)ei(wkoUcosI3)t(= (23) w cosh/ 0 H where We denotes the well known encounter frequency (w, = w - k0 U cos 0), / is the incidence angle i.e. angle between the direction of wave propagation and the positive x axis, g is the gravity acceleration, A is the amplitude of the wave, cwis its frequency in the earth fixed corrdinate system and ko is its wavenumber v = ko tanh koH. This allows us to write the following expressions for the potential and free surface elevation t - E0 (x, t) = )e .t} , s--"C-{S(p = 1 (24) After perturbing the original BVP (3) we obtain at different orders 0(1) --Z = 0 (25) =(O) =0 (26) O(E) 8 2u - (O)VoWU g s . 22iwUV= (27) 0 ,' = -I [-iw'"O + UVoe VoWU1 (28) 9 4 where Vo denotes horizontal gradient Vo= (7,t&,0). In this section we will also consider the case of a freely floating body so that the radiation problem should also be formulated. We write first the general decomposition of the total potential W" into the incident W1', diffraited W'D and the radiated Wu parts 6 W = 1)+ iwe qo • (29) j=I where Cu stands for the body displacements. According to this decomposition, the following body boundary conditions for different potentials can be deduced 8oY,_ aA' • 8o•R iu On On 8 a + m j n' = w" (30) where (nl,n 2 ,n 3 ) = n , (n 4,ns,ns) = ^An (31) (°) (0° (mj,m 2 ,m3 ) = -(n. V)V¢ (m4 ,ims,iMs) = -(n. V)(x A Vý ) (32) On the free surface we have 0 -wsc42iw ~UV4 ~)V cp4~ ~+ p'R,(. + _ ~ + 2 (°) _weoaW _ 2iwUVo+(O)Vo OUj + gs8Z j z€ = 0 (33) 2 • Os o Y 89 '( 1 1 -WewD + g D + 2iwU2 - 2iweUVoOVo(W + W)) + iW0 U(soý + so, ) -= 0 (34) Oz axIz 8z 2 The small forward speed problem defined above can be solved directly in that form but usually new perturbation series with respect to forward speed parameter r = Uw/g is introduced (incident potential is not affected by this perturbation) W," = W-() + rso(1) + O(r 2 ) (35) Due to above, the free surface condition decompose to 0(I) 01)1)9 -vo(W) + ý = 0 (36) O(r) 1 +W - -2i 2ko cos/0opM + 2i Vo ýVosWM - iW (37) and the free surf-ace elevation 7fA = 7(l) + ri70l) to : /(i) = ( ) (38) 9 7( - iak[o(1I) - k°cos/34l) + !Vo$(O)VpO(1)J (39) 9 Vu, .The inconvenience of the approach above is that the additional perturbation series (35) cause the secu- larity of the solution, i.e. the unphysical growth for R -+oo. It has been shown in [181 that this problem can be solved using the multiple scale analysis. ( We note here that the potential 9p is exactly the linear potential from the zero forward speed case, and the potential Wo(1) is its first order correction with respect to the forward speed. FLurther onward, this potential will be decomposed in two parts. In fact, an explicit particular solution satisfying the free surface condition _,,1,)a•o- 1) ( + -2i-- - 2ko cosi3so 'M (40) 89z r and Laplace equation in the fluid, can be found [8, 16]. One of the possible expressions is 3 W01) -2(iy+ kocos/ ) aWr (41) An homogeneous solution should be added to the particular solution above, in order to annul the normal velocity induced by 9pV'). This homogeneous part of the solution is of the same type as WM' and the same method can be used for its evaluation. The remaining part of the treated~intesaewy sW(2) (3) free surface condition (37) will be treated in the same way as 'v and ý4}D because no particular solution equivalent to (41) can be found. However, due to the rapid decay of the forcing function (Q = 2iVo0Vop,() - iwo) ) the radiation condition for this part (denoted 411)) of the potential W(") will be of the Sommerfield type (69). 2.2.1 Wave loads In the case of small forward speed the situation is complicated by the presence of the steady potential but is simplified because we limited ourselves to the linear and steady second order loads, both of which can be obtained by knowing only the first order potential w' and the steady potential 4. " First order loads The following expression for the first order force can be obtained F" = - IJ fE(-iwwu + UV•$•uo~V )ndS (42) The global forces are usually decomposed into the so called excitation force associated with the diffracted and incident potential, and the radiation force which is written in the form of added mass and damping coefficients : = oJLfi II +ewo)+ UVtsW)V(,p()+ +o)1nidS (43) w,20a += ~j o !0 (-,2pR, + iwUVý( 0)Vwu.j)njdS (44) * Steady second order loads The steady second order loads are important for the analysis of the moored floating structures slow drift motions. The case of small forward speed is particularly important because it permits the evaluation of one component of the slow drift damping, namely the wave drift damping. In fact this damping is defined as a derixative of the steady second order force with respect to the forward speed, and it can be calculated only after the resolution of the small forward speed diffraction-radiation problem. Only horizontal components of the steady second order forces will be considered here and the so called far-field expression for these forces is 9Rer t [VW'VU'*no - 2Vomp" ]dS- .. 0 W o½nodC} (45) The wave drift damping coefficient B = -O8t/8U is usually obtained by numerical differentiation after calculating ],u for two small forward speed. Compared to the case without forward speed the evaluation of the steady second order forces is much more complicated and that is why some authors tried to propose some simplified methods to quantify these loads. An extremely simple 3D formula for wave drift damping coefficient was proposed in [7]. It turned out that, in the case of the fixed single or group of bottom mounted cylinders, this formula gives exactly the same results as the complicated theory presented here, and this up to four significant digits. An theoretical proof of the formula was proposed in a controversy paper [2] but it is not clear yet if the formula effectively should work for the general case, and one of the conclusions of the present paper is that it shouldn't. Anyway, we recall here the expression for the x component of the wave drift damping coefficient B11 for finite water depth [16] u311 [("7W) M) +2 - M I8(w) sin ) (46) where F1 is the zero forward speed steady second order force (so called drift force), 3 is the wave incidence and a is the ratio between the group velocity and phase velocity of waves a = 1/2 + koH/ sinh 2koH. The utility of the formula is evident because it needs only the zero forward speed case to be solved which- becomes nowdays quite trivial task. 2.3 Hydroelasticity Hydroelasticity contains another type of problems where the presented methods can be used. It concerns the bending of the vertical column under the action of waves. In fact, hydroelasticity is an important problem for large floating structures which have the natural frequencies sufficiently small to be excited by the common wave spectrum. The elastic displacements of the body become of the same order as the rigid body displacements and should be treated together. The method we use to treat this kind of problems is explained more detailed in [20] and consist of coupling the beam finite element structural model with the 3D hydrodynamic model. Here we recall just the basic steps. 2.3.1 Theoretical assumptions The cylinder is assumed to be a slender beam with the horizontal displacement W(z, t) = R{w(z)e - w't along the height. Differential equation for the structural deflection of the beam, in the simpliest case, can be written in the form [27] : -W2 + &d2(EI•dW) (47) where m(z) is the mass distribution, E(z) is the module of elasticity, I(z) is cross-sectional moment of inertia and q(z) is the local pressure force acting on a horizontal section of the cylinder. The numerical method used here is the well known finite element method based on the so called "dis- placement" formulation. 2.3.2 Beam finite element Figure 1: Finite element model. As far as the structural problem is concerned, the theory we use is the classical one [27] and is repeated here only for completeness. We define an isolated finite element k of length I and we assume the displacement k w (z) (in the local coordinate system) within the element in the following form 4 wk(z) = Nik(z)6d = {N kT (6}k (48) 7 where T signify the transpose operation, {d}Y is the vector of nodal displacements Wk and nodal slopes (8w/8z)', and (N}" is the vector of the shape functions : w•k 3(a 2 3 2 =(f) + (2P }(f) 3k - 2(1 3 1 () 2_1)3+((49) After the discretisation of the equation (47) using the method of the beam potential energy minimisation, we obtain the well known matrix formulation of the problem : (-W2[Mjl + [K]k){6}) = (f{/k + {F)k (50) The matrix [M] k and [K) k denote respectively mass and stiffness matrix of the beam, with the following generic elements : ' Smt= INj NPkPdz k, j = EkI k f I d2Ni d&N( c- --- -dz (51) whith Ek l denoting the averaged value of the stiffness factor of the element and mA' is the averaged mass per unit length. After performing the integration in (51) the following expressions are obtained [6 31 -6 31 1 [156 221 54 -1311 k [K]k 2EkI 31 212 -31 12 3 " -6 -31 [M~k =.mkl 221 412 131 -312 2 (2 6 -31 ' f'2-0 54 131 156 -221 (52) 31 12 -31 212 -131 -312 -221 412 The vector {F) k is the vector of the concentrated forces and moments at the ends of the element MIk {Fz; 1k (53) M2 Finally the vector {f}k represents the action of the distributed external forces which are in our case due to the water pressure. The elements fi" of this vector are obtained by the following expression : Ji = qkNkdzq (54) As it will be shown later, the external forces distribution qk(z) is obtained by solving the hydrodynamic problem. 2.3.3 Assembly Once the characteristic matrix and vectors of each element is calculated, the assembling is performed and the following linear system of equations is obtained (-w 2 [M] + [K]){6) =-f1 + (F) (55) The assembly procedure is the classical one [27] and will not be repeated here. We note just that the dimension of the linear system of equations is 2(N, + 1) and that the global displacement vector is = {WIJ,0,,w2,0 ..,.... wei, Vi+, Oi+l,.. W., ON.,WN.+,ON.+ (56) In the case of "dry" structures the resulting matrix [M] and [K] are banded around the diagonal with the half-band width equal to 4. The vector IF) is the vector of the concentrated external loads, if any. 8 2.3.4 Hydrodynamic problem The hydrodynamic problem is considered within the usual assumptions of the potential flow and the linear case is considered only. The total potential WO) is decomposed into incident , diffracted W(D() part, and a part due to the ship displacements v). We write: N. 4 'P(,) (1) + ( -i) (w) (Z1) - (() k=i j=1 The boundary condition on the body surface asoI()/8n - -iww(z)n. gives the following conditions for different potentials =n MI) )(z)cos6 ýn On ' On =-N, (58) The BVP's are completed by the Laplace equation in the fluid domain, radiation condition for diffracted and radiated parts and the following free surface condition W + g = 0 (59) We note that the diffraction potential is exactly the first order diffraction potential without forward speed and that the radiation potentials satisfy similar kind of the BVP as those for the rigid body radiation, so that the same method can be used for their resolution. The pressure is obtained from the Bernoulli's equation : p = -egz +- iwo4 ( ) (60) and is integrated over the body surface to give the forces which compose the right hand side vector If). First we consider the hydrostatic part -pgz. It can be shown that, in the case of the vertical cylinder, this component can be written in the following form : N 4 -HIL 21311ndSbjkp (61) .• f 6ffk zP=l j=l where < is the hydrostatic stiffness matrix coefficient defined by kfk = -ega2 1 N, d-2 -- z (62) with Sk denoting the part of the column which corresponds to the k-th finite element. We consider now the dynamic part of the pressure iwgp('), which is first divided into the part associated 1 *with the potentials WOp ) and W(p) independent of the body displacements and the part associated with W.) dependant of the displacements. We write for W, + ý) fDk = iWO (w, + 0))NndS (63) and for 4(l) 2 Nnd=2 N. 4 ~ B : i," k~ = OW• [ w4 ) Nn.dS = O2 bp w(') (64) p=1 j=i In order to write the matrix equation in the convenient form, the above expression is rewritten as N. 4 = + (65) 1 P= j=1 9 with the added mass apk and damping bP coefficients, defined as follows ap={fffR (l)= n. , jpk b.... J, ff . ()Pn dk A, I- II j -. fý R ~ (66) We note that the added mass and damping coefficients (66) have finite value for all p, what means that the matrix of the resulting system of equations will be full contrary to the case of the "dry" problem, as mentioned before (sect. 2.3.3). The assembly of the resulting coupled system of equations is now slightly more complicated but still straightforward so we can write : (-w2([M] + [A]) - iw[B] + [K] + [K]H) {6) = {f}D + {F) (67) The solution of this linear system of equations gives the global vector of nodal displacements which complete the solution for the potential (57). 3 Solution for the potentials We can see that, except for the stationary potential ý, all the potentials w('), W(2), •(3), (11)1 9 RjMP involved in the analysis satisfy the similar kind of the boundary value problem. The free surface con- dition, which is the most difficult task to satisfy, is of the same type for all cases, except in the linear case @W), Al)) it has homogeneous form (12,59), and in the nonlinear (W(),Ps)) or weakly nonlinear cases it is nonhomogeneous one (14,16,37). The same procedure will be used for each of these potentials. Thus, we define the following boundary value problem for the generic potential 4 AO=0 00>z>-H -a + •z=Qr,) z = 0 az= n - v(z, 6) on the cylinders (68) S0 z = -H radiation condition r -- oo For the sake of clarity, the potential 4P is further subdivided in two parts 4' - 4 B + 4 Q with the following BVP-s for each of them: A4O =0 0> z> -H -aOB + ' =O z=O0 OOCB -n' = (z,O) on the cylinders a n o(69) -90= 0 z=-H Oz- I k\im[ Vor'B )] =Q0 r -4+o 10 AQ=Q0 0>z>-H • 0+ --- Q(r, 0) z = 0 Oz radiation condition r 0-0o The potential ObB is the part of the potential caused by the imposed normal velocity on the body boundary, while the OQ is the part caused by the forcing on the free surface. Two parts are fundamentally different and will be treated separately. In the case of small forward speed another kind of the BVP should also be considered. In fact the potential 4 can not be treated as a special case of the potential 0, and an independent BVP should be solved: AO=0 0>z> -H -= 0 z=0 =0 on the cylinders (71) -z=0 z=-H •0 r -400 This potential is of the local type and does not generate the waves. On the other hand, the cylinder configurations which will be treated here are " bottom mounted single cylinder * single truncated cylinder " array of bottom mounted cylinders We note here that the case of the truncated cylinders array will not be treated explicitly but it can be constructed as a combination of the single truncated cylinder case and the case of bottom mounted cylinders array. 3.1 Bottom mounted single cylinder 3.1.1 Potential ObB The general methodology for this potential was first given in [12] for finite or infinite water depth and for two or three dimensional case. The method is basically eigenfunction expansion which is possible due to the relatively simple geometry. We briefly resume the basic steps. The solution is assumed in the following eigenfunction expansion form O= 1 [fo(z)[%noHm(kor) + > f.(z),3m.Km(k.r)]eimo (72) n=I where Hm are Hankel functions of the first kind Hm = Jn + iYm and Km are the modified Bessel functions. The functions f,(z) are defined by fo(z) = cosh ko(z+H) f(z) = cos k.(z + H) coshkeH cosk.H (73) with a = ko tanh koH = -k. tan kH. Application of the boundary condition on the cylinder (69) and the use of orthogonality of eigenfunctions, gives the expressions for the unknown coefficients fmn : 2C0 P oZ..zd i3,o = - koH,(ko 0 fo(z)v(z)dz (74) _ 2Cn to mn = - k2K'ka) ]0 fn(z)vn(z)dz (75) where Co and C. are defined by : 0 0- 2 --1dz Co = [2] Ic C. =_ 121 ,(2Z)dzJf(z] (76) and vm(z) follows from the Fourier series expansion v(z) = > vm(z)etM8 (77) 3.1.2 Potential 'Q The situation is now more complicated and eigenfunction expanmion method alone is not sufficient.The integral equation method should be introduced. The method presented here is based on using the integral equation technique with the classical Green function expressed in its eigenfunction expansion form [13]. This method is inspired from the method presented in a relatively unnoticed (publication in Japanese) paper [21] where the interaction of vertical circular cylinder with water waves and small current was treated. We note here that two alternative methods are also possible [3, 24]. The first one uses the special kind of the Green function which satisfy the condition of zero normal velocity on the cylinder, while the second one uses the Weber transform technique. The final expressions are essentially the same. We describe now briefly the general method presented in [17]. First we write the Green function for two points, x = (x, y, z) = (r, 0, z) and ((p,,)71, = (pt ), in its eigenfunction expansion form G(x,C)= > GC(r,z;p,0)e 0(e-0) (78) with GC(r,z;p,() = 2' Coo(j.(kor)H.(kop)J~l)Io (Hm(kor)J(kop) fo(z..o(C 1 (.ImK.(knr)IK(knp)) .z) -7r . v"krK~kp)(r <> p)p (79) The integral equation for the BVP (70) can be written in the form __O0(X))+fo - rOG(Xe)nao (40(.) k p = -sG(x,C)QD(p,t9)dSs, (80) It can be shown that the integral over the surface at infinity S. disappears in all cases considered here [17], which represents, in some way, tlh, radiation condition for 4 Q. 12 The next step is to write the solution for 'PQ on the cylinder, in the eigenfunction expansion form OQq(a, 0,z) = E OQqn(a, z)eimO = Z[fo(z)Amo + Z fn(z)Amn.Jeim (81) -- -W n=1 If we write now the integral equation for one point inside the cylinder, r = a - 6, (a > 5 > 0) we can deduce the value of the Am,- coefficients by using the orthogonal property of eigenfunctions : Amo = Co2 f Hm(kop)QDm(p)pdp = 2Cnfýo Km(knP)QDm(p)pdp (82) koaHg,(koa) ' knaKgý(k.a) By returning this solution in the integral equation we obtain the expression for each Fourier mode of the potential at any point in the fluid : 4 Qm(r, z) = riCofo(z)H,(kor) [Jm(koP) - ZmoHm(kop)]QDm(p)pdp +2 E Cnfn(z)Km(knr) jr[l(k.p)- Km(kp)]Q (P)PdP n=1 +fC~O()~m(kr -Zmfm~orJ m~opQinpad + 1iCofo(z)[Jm(kor) - Zmogm(kor)] HD +2 C~f.(z)[I.(knr)+2 -Z.nK.K(knr)] jo• K.(kPQ.PKmn(knP)QDin(p)pdp P (3(83) where ko a ) 'JI J~o) m nmn~ Zro (84) = H.(koa) Z " K=I(k.a) As we can see, we represented the solution, of the BVP with nonhomogeneous free surface condition, as a series of eigenfunctions which individually satisfies the homogeneous free surface condition. However it can be shown [3] that this series satisfies an nonhomogeneous condition in the limiting sense z - 0-. This will cause the appearance of the logharitmic type of singularity when calculating the potential on the free surface. Another problem arises in the calculation of the infinite integrals in (83) which generally include slowly decaying functions so that the asymptotic expressions should be exhibited. In summary, we should be very careful (3, 17J when evaluating the different integrals and infinite sums involved in the above expressions. 3.1.3 Potential In the case of the bottom mounted vertical circular cylinder, the expression for the double body potential is very simple, and is represented by the dipole potential -2-cos 0 (85) r .3.2 Single truncated cylinder The configuration and the notations for this case are presented on the figure 2. 3.2.1 Potential OB First we consider the diffraction case which means that v(z) becomes the radial derivative of the incident potential denoted here by 0t. The method used here is inspired from [10], and is also based on the use of eigenfunction expansions in a respective domains. The eigenfunction expansion for the interior region can be written in the form Com = E, tbI' (r)l"l + E b,.cosA,(z + H)I.(AXr)]eime (86) 13 ab , q6 + y' Figure 2: Different fluid domains. with An = nir/(H - D), and I. denoting the modified Bessel functions. In the exterior domain we write the eigenfunction expansion in the form similar to (72) =C > [cMofo(z)Hm(kor) + S cýnfn(z)Km(knr)]eime (87) n=1 Orthogonality of the eigenfunctions in z direction is now used, and the boundary conditions on the cylinder (r = a, 0 > z < -D) and continuity conditions on the intersection surface (r = a, -D > z < -H) are written in the form j (?Cz + 0b)dz = ndz " (88) JHP ) cos CO (z+H)dz = CI Ancos(z + H)dz n =1, oo (89) 0 Lu (Vt+tfffldz = f '-rf°(z)dz (90) f 0 a5 HTr (V + 4" [-)fO(z)dzj TH nfz)dz n = 1,00 (91) We can now write the linear system of equations for each Fourier mode m = -00, 00 bMo M 00 0. mal = C~t m4,t btmD"0 + nD-?+ Foit 1 1, (93) tm~o E CM nnn 00 n=1 + cm° = bI°E• +EbmnEm Z Go (94) 0 cm - b' "t't +rLmE~ G =1oo(5 n=I IIIb, = EG + b + G = 1,00 (95) 14 which can be rewritten in the matrix form : I) c•') = (Gcm) (96) with ...... E m'p ... (97) \Dmo¢ ... Dm, Eo ¢ ... Ej-, ' to In§7 B 'M= [b'M'o, b" 1,... b'ý n] T C O = [C¢O. , cO ,-, 0¢ ] (98) , 1 F` = [F0o F 70 ,' ,F,Fm] T Gmo = [C"" , Gm t.., crnflr (99) and I denoting the unity matrix. All the coefficients in the matrix equation above are easily deduced from (88) to (91). The solution of the system gives the unknown coefficients b~n and ctn which terminate the calculation of the potential. In the case of radiation the procedure is similar except that an particular solution should be added to the expression for the potential in the interior region. In fact the eigenfunction expansion (86) satisfies the condition of zero normal velocity on the bottom of the cylinder which is not the case for all radiation problems. Only three independent radiation problems need to be considered. a Surge i = 1 The boundary conditions for surge are 0 In/oza = z= -D,a > r > 0 (100) RI /8r = cos6 I r =a,0 > z > -D (101) so that there is no need for additional particular solution but only the modes m = 1 and m = -1 in the expansions (86,72) should be included. * Heave j = 3 The boundary conditions for heave are "aZ -3= 1 z = -D, a > r > 0 (102) pR3 9/8r = , r=aO>z>-D (103) so that the particular solution can be chosen as (i)in 1 2 r2 R3 --- (z - -) (104) and only the zeroth Fourier mode is included in the expansions (86,72). * Pitch j = 5 The boundary conditions for pitch are /iz=-rcos=osR z = -D,a > r > 0 (105) (1)i.. OR3 /Lr = 0 r=a,O0>z>-D (106) so that the particular solution becomes (I)in r 2 - 2 S= -)cos0C-( (107) and only the modes in = 1 an(ln = -1 in the expansions (86,72) should be included. Beside that, the procedure is comphltely similar and will not be detailed here. 15 3.2.2 Potential OQ The method remains very similar. In fact, the solution OQ for the complete cylinder (81,83), plays here the role of the incident potential 4B'. All expressions and general methodology leading to the linear system of equations are exactly the same. However, the case of o(11) is a little bit different. In simple since the particular fact, it is more solution is explicitly known and can be used directly as the incident t potential 0 for the homogeneous part [15J. 3.2.3 Potential Even if the methodology remains the same for this case, the eigenfunctions expansion in the exterior region changes due to the different free surface condition -o •,z= Z [cý. 0 (a)I + Ec 4. cospju(z + H)Kg(p.r)leim (108) with p. = nmr/H. The rest of the procedure is the same as in the case of OB. 3.3 Array of bottom mounted cylinders The geometry and basic definitions are shown on the figure 3. y Figure 3: Basic configuration. 3.3.1 Potential tB The method which will be explained here is essentially based on the work [14] where the linear diffraction by an array of bottom mounted cylinders was presented. First we assume the following eigenfunction expansion for 0B1 N. m 0 fo(z)mo= oH.(kork) + Zh(z)D.uiZkLKm(klrk)]eim9h (109) k=i M=-1 1=1 16 After applying the boundary condition on each cylinder using the Graft's addition theorem for Bessel functions, we obtain the following systems of equations for the unknown coefficients D'i: Dk.s + Z I D eD(nm)dihHnm(koRj&) = (110) j$Ek n=-oo D k + (-1)m K DiZ le'(n-m)ojIKn.-m(kiRjk) = l-ta (111) j;Ak n=-wo m = -oo, oo k = 1,NI , =1,oo where 7 2Co0 0 fo(z)V (z)dz 2C0 fto koJg(koak) f H kIo)(k a ) - fH(Z)Vm(Z)dz (112) Near the cylinder k, rk < Rjk, Vj $ k, this expression can be simplified using the Graff's addition theorem together with (110,111). The simple expression in terms of the k-th cylinder local coordinates is: IPB(rk,Gk,z) = • { fo(z)[DmnoZnoHm(kork:) + (Yno - D~o)Sm(kork)] a ° +KfL(z)[D~,,Zkm 1Km(ktrk*) + (yk,~, - Dk,,l)I,.(ktrk)]l}etin (113) l=1 3.3.2 Potential 1OQ As in the single cylinder case we start by writing the Green's theorem for one point outside of the fluid domain: d a8 0 dS G QdS (114) 1 8sn (114) The potential on the cylinder k is then assumed in the form 10k(ak,Ok,Z) = (Z) + Bmuf(z)]ei (115) In=--o 1=1 After writing the Green's theorem (114) for one point inside the cylinder k -- (rk = ak - 6, 0 < 6 < ak), carrying out the integration by C, using the orthogonality of the functions fý(z) and the Grafi's addition theorem for Bessel functions, exploiting the orthogonality of the functions ei ms and rearranging the different terms we obtain the following systems of equations for the unknown coefficients Bkm: j =- 0 = ) (116) Zj3kak n (koa) 7rakkoHrn(koak Hm(kopk)e-CiOQ(Pk, Ok)dS 3 BI + (-i)M' _ BD ,•, i Knmrn.(kIRjk)ein)ai& j ak n=->• " (ki~aj )" . (117) c, f[s Km(kjPk)e-sr'kQ(Pk't9k)dS lrakkIKrn(kjak) ISr mn= 0, 0 , k=1,N , I=1,o 17 This completes the solution for the potential Ob on the cylinders. Once the potential on the cylinders is found, the potential at any point in the fluid can be calculated using the Green's theorem. The resulting expression is N . M i I = ZE27raj • [B.--0 J, (koaj)fo(z)Hr(korj) j=1 + Z i e - S•"7l m(knaj)f(z)K'(kj] 11S, GQdS (118) As in the case 4B of this expression can be simplified near the cylinders (rk < Rik,Vj) 0oMoz ak =Q z f -2rkfo(z)B~o[J,,(koak)Hm(kork) - H.(koak)Jm(kOrk)J M-00 00 +I: akkfjf(z)Bnj [I. (kLak)KmQVjrk) - K.(klak)Im(klrk)] 1=1 + {inrCofo(z)[Hm(kork)Jm(kopk) - Jm(kork)Hrn(kopk)] 00 +2 E C1.f(z)[Km(krk)I.(klPk) - Im(klrk)Km(kLpk)])Qm(pk)Pkdpk }eime& (119) l=1 For rh = ak expression reduces to (115). The numerical implementation of the method should be performed very carefully because of many conver- gence problems associated with the eigenfunction expansions, Graff's theorem and especially numerical integration over the free surface. For details we refer to [19]. 4 Results and discussions We start by presenting the results for the exciting forces for the fixed bottom mounted cylinder, up to the third order. On the figures-4,5 and 6 first, second and third order exciting surge forces are presented for 2.0 0.0' -1.0- -2.0- -3.0-' -4.0- -so- / -6.0- 0.00 0.50 1.00 1.50 2.00 k~a 2.50 Figure 4: Real and imaginary part of the- first order force on the bottom mounted cylinder of radius a in the water depth H = 10a. the cylinder of the radius a standitig in i1he water of the depth H = 10a. As usually, the total second and third order forces are decolnposed inil dilrerent components associated with the different contributions 18 6.0- 3.0- 4.0- 0.0--t *1.0 -3.0 - 2.00- 1.0- 410- 0-3. 4 -4.01.-. p 2) T 0.00 0.50 1.00 1.50 2.00 A:a2,50 0.00 0.50 1.00 1.50 2.00 ka2.50 Figure 6: Real (left) and imaginary (right) part of thethirnd order force on the bottom mounted cylinder rduof raisa thein water depth H 10a.n 190 2 ) * F, - part of the second order force induced by the quadratic products of the first order quantities * - part of the second order force induced by the second order potential (2 ) F(2 ) (22 ) " F 2 1 F * F~ - + F2) F(3)( part of the third order force induced by the triple products of the first order quantities FM2 - part of the third order force induced by the products of the first and second order quantities 3 * F4 )- part of the third order force induced by the third order potential 3 " F(3) F + F43) + F( ) FRom the practical point of view, a disappointing result shown by these figures is that all contributions to the forces are important in whole practical range of frequencies and none of them can be neglected. Thus, some approximation methods based on the long wave theories have very restricted range of application and often have no practical utility [17]. We turn now to the problem with small forward speed and first we show the results for the first order surge exciting forces on the bottom mounted cylinder. On the figure 7 this force is presented for three different 4.00 -Fa, - -OMs gaga2A -- Fjp --0. 2.00.1.50. 1.00 . 0.00 0.50 1.00 1.50 ka 2.00 Figure 7: First order surge exciting force. on the bottom mounted cylinder for three different values of Froude number. Cylinder has radius a, water depth is H = a and wave incidence is = 0. values of the Froude number F, = U/l/d. The added mass and damping coefficients for the same case are presented on the figure 8. Interesting results concerning the wave drift damping coefficients are shown on the figures 9 and 10. The figure 9 concerns the case of the fixed cylinder, and in this case the results obtained by the simple formula (46) are indistinguishable from the semi-analytical calculations. Contrary, the case of the freely floating cylinder presented on the figure 10 indicates different conclusion because there are important differences between two classes of results. Surprisingly the case of / = 7r/4 shows again complete agreement between the simple formula and the analytical calculations!? We note that the calculations of the wave drift damping coefficient were performed both by the far-field method (45) and the near-field method (not presented here) and that two classes of results are identical. Additionaly, the completely numerical results obtained by Grue [11] also agree with the analytical results. On the figure 11, we present now one result useful for more detailed benchmarking purposes. It concerns the free surface wave elevation around the cylinder. The cylinder is from the previous case and wave number is k0 = 1.2. We note important. modifications of the free surface influenced by the presence of 20 4.00- 2.50-- al F.3 - -. 03 ...... Pa23P...... Fs0.0 too- OG 1.00 0.0 0.50 1.00 ISO tOO 0LO 0OM 1.M ISO zoo0 Figure 8: Added mass (left) and damping (right) coefficients for the bottom mounted cylinder and for three different values of FrYoude number. Cylinder has radius a, water depth is H =a and wave incidence is 0 0. B13 , pgA~a 0.80 0.40 / 0.r/4 0.20 0D 0. Gr. 0.00 0.50 too00 ISo 2M0 Figure 9: Wave drift damping coefficient B13 for a fixed bottom munted cylinder. Cylinder has radius a and water depth is H =a. 21 1 .8 - - *-20 o * BgA a -o, -- /"'m't.,,,' (10), *a .- ,)r -- I .*'(1o) 1.20- +Aon 0.80- D 0.0 0.80- 0.0 0.0- 0.440 0.20 0.20-0•0- 0. 00- 0 O.C . 0.00 0.50 1.30 IM 2.o oW o.5o0S0 i0I= I-o ZOO Figure 10: Wave drift damping coefficient B,1 for freely floating complete cylinder and for incidences )3 = 0 (left) and fi = ir/4 (right). Cylinder has radius a and water depth is H = a. the current, which has a direct consequence on severe modifications of the steady second order loads. Another interresting phenomenon concerning the wave-current-floating body interactions is the secularity 1.20- 1.60- t.00- 0•..• ...... -- 7n-0S .... 0400 , , I ...... i +.. . 0.00 0.50 1.00 1.50 2Oo Figure 11: Free surface wave elevation for the fixed bottom mounted cylinder. Cylinder has radius a, water depth is H = a, wave incidence is 0 = vr/4 and wave number koa = 1.2. (unphysical growth of the solution far from the body) of the additional perturbation by r (35). In spite of the recognised secularity of the solution the perturbation by r is common approach to treat this problem [26]. Only recently [25, 5] it was numerically shown that two solutions are the same as far as the forces are concerned. This was further confirmed by the multiple scale approach [18] and theoretically shown that two solutions should be the same on the body. However, two solutions differ very quickly when we leave the body so that the secular approach should not be used in the calculation of the wave elevation near the body (between the column of the multi column structures for example) because the elevation can be seriously overestimated. Figure 12 which represent the view of the potential (proportional to the wave elevation) on the free surface around the fixed cylinder confirms the above constatations. We turn now to the case of the truncated cylinder and on the figure 13 we first present the values of the first order surge and heave exciting forces for different draught of the cylinder with the radius a in the 22 Figure 12: Secular (left) and non-secular (right) solution for the imaginary part of the diffracted potential on the free surface. Cylinder radius is a = 1., the water depth is H = a, wave incidence is )3 = 0 and wavenumber is ko = 0.7. 6.0 3.0" IP IIF(1)I 5 D 0 - ---. 3.0- U- 2.0- 1.0. 1.0 ,o o . . 0.0 0.2 04 06 08t 1.0 .!2 .4 1'6 I'S 0o 0.0 0.2 0A 0.6 0.8 1.0 1.2 1.4 L6 11 2.0 k 0a koa Figure 13: First order surge (left) and heave (right) exciting forces on bottom mounted vertical cylinder. The cylinder radius is a, water depth is H = 5a and wave incidence is = 03. Dott dashed line: D = 4a; dashed line: D = 3a; dotted line D = 2a; full line D = a. 23 water of depth 5a. More interresting results are shown on the figure 14. It concerns the comparisons of 14 -. ,A2 09- . I.0L 01- I,0- 0.7- 0.6 - 04A 04-- 01-1 0.0 - 0 _ 0.0 U. 0,1 06 04 1.0 1.2 .A 1.6 12 LO0 0.0 02 0.4 0.6 0.1 10D .2 1.4 1A IiA 20 koa koa Figure 14: Wave drift damping coefficient B1, for a fixed truncated cylinder. The cylinder radius is a, water depth is H = 5a and wave incidences are 0 = 0 (left) and ,6 = ir/4. Dotted line: semi-analytic for D = 4a ; full line: semi-analytic for D = a; squares: formula (46) for D = 4a; circles: formula (46) for D = a. the wave drift damping coefficient calculated by the simple formula with one obtained according to the semi-analytical method described here. It can be seen that the formula works quite well for deep-draught cylinder but not for the "short" one. Contrary to the case of the freely floating bottom mounted cylinder, the noticeable differences exist between the formula and semi-analytical calculations also for ,6 = 7r/4. We consider now the case of an array of bottom mounted circular cylinders. On the figure 15 we show again the results for the wave drift damping coefficient. In this case, the theory and the formula give exactly the same results. The most complicated results obtained by the semi-analytical approach described 15.0" B, pgAa 100- 05.0 0.0 0'2 04 0A 0'. 1.0 12 1.4 16 I k0a Figure 15: Wave drift damnping coeflicient BI, for an array of four bottom mounted cylinders of the radius a placed at the corners of the square of side length equal to 7a. Dotted line: /3 = 0.,H = 1.; full line:/3 = 0.,H = co; dot-dashed line: P3 = sr/6,H = 1.; dashed line: 63= sr/6,H = oo. here, are presented on the last figures 16,17. They concern the values of the free surface elevation up to the second order for an array of cylinders. As for the forces on the figure 5, we use the equivalent notations for the free surface elevai i,t.s. It is believed that the second order theory can explain the wave 24 43 - 1 -6 Iq(21)1 2 A TkoA 4,0- 14.0- 3j- 2.0- 35.0 2IDA -A- --- IS ----...... =- 6.0 " ...... 5.0 -...- 46b .,- ,.- .. 03 - LO-0 -5.0 .40 -3.0 -2.0 -1.0 0.0 5D .0 O 0 3 4A 5.0 -5.0 -. 0 -3.0 -to -1.0 0.0 1.0 2L0 3.0 40 5.0 19.0- - t _D____ In(22)1 1,7(2) 1 6 0 56.0WA- oA- 14.0 14.0- 0 .0 -" 1 0 -0 IoAo- 10.0 = 50- 5.0- 6.0 60 2.0 - 2 ". ""-"0- 0.0 - T -r- ir- 0 - 0.0 ,I r - -50 4.0 -30 -20 -1.0 00 5.0 10 11 40 50 -5.0 -4.0 -30 -2.0 -5.O 00 5.0 2.0 3.0 4.0 5,0 Figure 16: Components of non-dimensional first and second order wave elevation amplitude along x = y for an array of four bottom mounted cylinders of radius a placed at the corners of the square of side length 4a. Wave incidence is 3 = sr/4. Solid line koa = 166; dashed line: koa = 0.468; dotted line: k 0a = 0.754. 25 14.0' 16.0- A(koA)"' / 12.0- A(kA)"-' 126.0. /4- // 4A0 0 . ------. 0.0 0.4 oJI I.2 8.6 2.0 0.0 0.4• Oi ( --r)/- 1.21 1,6 2.0 ( - )l --- - 0.0 0 ---- A(ko A)"-' 10.0 8.0- 6,0. 4.0 - 2,0 0.0 04 0.1 .2 1.6 2.0 (0,+-- 0)/', Figure 17: Components of non-dimensional first and second order wave run-up amplitude an array of four bottom around cylinders for mounted c 'hl/ers of radius a placed at the corners Wave incidence of the square of side length 4a. is j = r/4 and koa = 1.66. Dashed 1 line: IuI7)I/A; dot-dashed line: 1(2 )Il/koA2; dotted 2 2 line: 17( )I/koA2; full line: 1,7( 2)I/koA2. 26 amplifications around the off-shore structures which are not captured by the linear theory. In some extent this is confirmed by these figures where we can see that in some cases, the second order component of the free surface elevation can dominate the first order one. The numerical complexities in the evaluation of these quantities are numerous, however the semi-analytical method remains superior'to the complete numerical models because of its rapidity and precision. Finally we consider the problem of hydroelasticity of the vertical column. The case chosen to show the efficiency of the method is that from [23]. The cylinder has a radius of 10m, in water of depth 200m. The distributed mass along the length of the cylinder is assumed to be half of its displaced mass. A concentrated mass mo equal to the total displaced mass (twice the distributed mass) is located at the free 3 - 2 surface. The stiffness factor of the cylinder is chosen such that EI/H = 0.41m 0s . The amplitude of the displacement at the top of the column for different wave periods is shown on the figure 18a. We can observe the highly tuned resonance at approximately T = 6.5s. Finally, on the figure 18b we present the deformation of the column for T 6.5s, in an instant when the top of the column reaches its maximal displacement. W/A W/A 8.0 1.0- 0.6 -0.6" 0.4- -0.4- 0.2 0.2 00" 0.0 350 55 6.0 62 7.0 7 8,0 SS 9.0 9U 0o0 -Mb0 -1;0.0 -1.0 -50.0 &0 T z Figure 18: Amplitude of the displacement at the top of the column for different wave periods (left) and elastic deformation of the column at T = 6.5s. The circles on the left figure are the results from Newman (1994). 5 Conclusion The methods to treat the various kinds of the hydrodynamic problems for different cylinder configura- tions are presented here. Higher order diffraction at zero forward speed, diffraction-radiation at small forward speed and hydroelasticity were considered. The boundary value problems emerging from the different cases are very similar and all of them can be treated by the same methods. The efficiency of these methods is demonstrated by various examples. The analytical methods have their place in the marine hydrodynamics because of their relative simplicity. Due to their high precision, only this kind of method allows the highly nonlinear developments with sufficient confidence. These results are essential for validation of more general numerical codes, which become more and more ambitious due to the rapid computer developments. References I1] ABRAMOWITZ M. & STEGUN I., 1970. : "Handbook of mathematical functions.", Dover. [2] ARANHA J.A.P., 1996. : "'Second order horizontal steady forces and moment on a floating body with small forward speed.". 1. Fluid Mech., Vol. 313, pp. 39-54. 27 [3] CHAU F.P. & EATOCK TAYLOR R., 1992. : "Second order wave diffraction by a vertical cylinder", J.Fluid Mech., Vol.240, pp. 571-599. [4] CHEN X.B., 1991. : "Second order high frequency loads on tension leg platform columns.", Institut Francais du Petrole, Report No. 38741. [5) CHEN X.B. & MALENICA S., 1998. : "Interaction effects of local steady flow on wave diffraction- radiation at low forward speed.", Int. Journal of Offshore and Polar Engg., Vol. 8/2, pp. 102-109. [6) NOBLESSE F. & CHEN X.B., 1995. :"Decomposition of free surface effects into wave and near-field components", Ship Technology Res., Vol. 42/4. [7] CLARK P.J., S.MALENICA & B.MOLIN, 1993. :"An heuristic approach to wave drift damping", Appl. Ocean Res., Vol.15. [8] EMMERHOFF O.J. & P.D.SCLAVOUNOS, 1992. : "The slow drift motion of arrays of vertical cylin- dres.",J. Fluid Mech., Vol.242, pp. 31-50. [9] FALTINSEN O.M., 1990. : "Sea loads on Ships and Offshore Structures.", Cambridge University Press. [10] GARRETT C.J.R., 1971. :"Waves forces on a circular dock", J. Fluid Mech. Vol. 46, pp. 129-139. [11] GRUE J., 1993. : Personal communication. [12] HAVELOCK T.H., 1929. : "Forced surface waves on water.", Philosophical Magazine 8, pp. 304-311. [13] JOHN F., 1950. : "On the motion of floating bodies II", Comm. Pure Appl. Math., Vol.3, pp. 45-101. [14] LINTON C.M., EVANS D.V., 1990. : "The interaction of waves with arrays of vertical circular cylinders", J.Fluid Mech., Vol.215, pp. 549-569. [15] MALENICA S. & S.ETIENNEI 1995. : "A propos des mdthodes semi-analytiques pour les diffdrents probl~mes de diffraction-radiation par un cylindre tronqu6.', Proc. of 5ieme Journ6es de l'Hydrodynamique, Rouen, France. [16] MALENICA S., P.J.CLARK & B.MOLIN, 1994. :"Wave and current forces on a vertical cylinder free to surge and sway", Appl. Ocean Res., Vol.17. 1995. [17] MALENICA S., MOLIN B., 1995. : "Third harmonic wave diffraction by a vertical cylinder", J.Fluid Mech., Vol. 302, pp. 203-229. [18] MALENICA S., 1997. : "Some aspects of water wave diffraction-radiation at small forward speed.", Brodogradnja, Vol. 45, pp.35-43. [19] MALENICA S., R.EATOCK TAYLOR & J.B.HUANG, 1998. :"Semi-analytical solution for second order water wave diffraction by an array of vertical cylinders.", Submitted to J.Fluid Mechanics. [20] MALENICA S., 1998. : "Hydroelastic coupling of beam structural model with 3D hydrodynamic model.", 2nd Int. Conf. on Hydroelasticity, Fukuoka, Japan. [21] MATSUI T., YEOB L.S. & lIrIIjtOSHI S., 1991. :"Hydrodynamic forces on a vertical cylinder in current and waves", J. of the Soc. of Naval Arch. of Japan, Vol.170, pp. 277-287 (in Japanese). [22] MOLIN B. 1994.: "Second order hydrodynamics applied to moored structures.", Ship Tech. Research, Vol. 41/2. [23] NEWMAN J.N. 1994. : "Wave effects on deformable bodies.", Applied Ocean Research, Vol. 16, pp. 47-59. [24] NEWMAN J.N. 1996.: "The second oider wave force on a vertical cylinder" J.Fluid Mech. [25] NOBLESSE F. & X.B.CIIEN. 1995. :"De'romposition of free surface effects into wave and near-field compl)onentst", Ship Tlchnioh,'gy Ics., Vol. -12/4. 28 [261 NOSSEN J., J.GRuE & EPALM, 1991. : "Wave forces on three-dimensional floating bodies with small forward speed",J. Fluid Mech., Vol.227, pp. 135-160. (27] SENJANOVIC I., 1986. : "Finite element method in the analysis of ship structures", University Zagreb, Faculty of Mechanical Engg. and Naval Arch. 29 Station keeping and slowdrift oscillations Pr Joe PINKSTER, Tech. Univ. Of Delft, The Nederlands Motions and Drift Forces of Air-supported Structures in Waves J.A. Pinkster and A. Fauzi Ship Hydromechanics Laboratory Delft University of Technology Mekelweg 2, 2628 CD Delft, 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 mean 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.[1]. 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 Oullfaks C the relative magnitude of the air cushion buoyancy was much gTeater. 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 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 fumction 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.[l], 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. [8]. 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. Model 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- ttures 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 Delft Uniiversity 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- plttational 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[1O], an outline of the theory has been given along with examples of the effect of air cushions under different structiucs b;asal on computations only. 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. The wave frequency air pressure variations within a cushion are determined by the change in cushion volume through the linearized polytropic gas law: Ap = -Avol * ng * (po + p)/voo (l) in which: vol = mean air volume in cushion P0 = atmospheric pressure Pc = mean excess of pressure in cushion Avol = wave frequency volume change in cushion Ap = pressure variation relative to mean cushion pressure 779 = gas law index For wave frequency pressure variations adiabatic conditions are assumed. In that case fg = 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 0 - X, - X2 - X3 system of axes with origin in the mean water level and X 3-axis vertically upwards. The body axes G - X1 - X2 - X3 of the rigid part of the construction has its origin in the center of gravity of the construc- tion, with x, axis towards the bow, X 2 axis to port and x3 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 is as follows: ((N,, • X 2 , t) = (oei(lcoso+x-sn>(2) in which: k = wave number 0 = wave direction, zero for waves from dead astern w = wave frequency = wave amplitude = wave elevation The motions of the rigid part of the construction in the j-mode relative to its body Iaxes are given by: :,j(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 4' as follows: - t 4'(X 1 ,X 2 , X 3 , t) = 4¢(X,, X 2, X 3)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 00, the wave diffraction potential Od, the potentials associated with the 6 d.o.f. motions of the rigid part of the construction Oj and the potentials associated with the vertical motions of the free-surface within each cushion, ¢, 6 C 0 = i40 O)( E 3i+ -,Js ý.dSc (5) j=l C=l S c in which: 0o = potential of undisturbed incoming wave Od = diffraction potential &k = potential associated with vertical motions of the free-surface in the c-cushion X3 -= rigid body motion in the j-mode C, = vertical motion of free-surface in c-cushion S, = free-surface area of c-cushion C = total number of independent, non-connected cushions In the above equation the undisturbed wave potential 00 and the diffraction potential od together decribe the flow aroumd the captive structure under the assumption that the free surfaces within each of the air cushions is also rigid and non-moving. The potentials Oj 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 fixed. The potentials 0, 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 lohg-crested regular wave in water of constant depth h is given by: g0g., = coshk (X 3 + h) eik(X C0S4+X 2 Sin,) (6) L2 coshkh The fluid lressure follows from Bernoulli's law: p(X ,,X, . X3, t)= = X 2,X3)e (7) with: 6 C p(X 1 ,X 2 ,X 3) = P 2 + Od)(0 + E ¢+-odso} (8) j=l 5,-- 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 0o, the solution of the 4 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 4y. 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 mulii-d.o.f. problems is treated in Pinkster [10]. 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 acro- static restoring characteristics. Each free surface panel (body) has one degree of freedom being the vertical motions of panel a 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 str uctuire. The direct method of solving for the motions of the structure and the vertical motions of fr'ce surface panels within the air cizshiouys considers the total system inlterms of :i miilti-body pIoblcni with inass. daalnq g and slpring roupling. The number of degrees of freedom amount to: C D.O.F. = 6 + N (9) c=1 in which: A = 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.E. 2 Y. {-w (M,.j + a~j) - iwbnj.+ cj}xj = Xn...... = 1, D.O.F. (10) j=1 in which: Mý -= mass coupling coefficient for force in n-mode due to acceleration in the j-mode. Zero for cushion panels a~j = added mass coupling coefficient b~j = damping coupling coefficient c%, = spring coupling coefficient xj = mode of motion Xý = 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. The case of j > 6 and n > 6 represent the coupling between the panels of the free surfacesof the air cushions. The case of j = 1, 6 and n > 6 represent the coupling between the six motion mnodes of the rigid part of the construction and the vertical forces on free surface panels in the cushions. j > 6 and n7 = 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,0j and bj 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: 47rE rj()(,,)S (11) in which: N, = total number of panels used to describe structure and free surfaces of all cushions X = X1,X 2,X 3 = a field point A = A,, A 2, A 3 = location of a source ,'(X. .4) = Green's function of a source in A relative to a field point . AS, = surface element of the body or the mean free surfaces in the air cushions aj - = strength of a source on surface element s due to motion mode j Oj(X)= potential in point X due to j-mode of motion The unknown source strengths aj are determined based on boundary conditions placed on the normal velocity of the fluid at the location of the centroids of the panels: - -0amj(X) + Nt -a - - La 1 )c(xA)As, 0-t,,..mr = 1, Nt (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: a(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: ---= n,_7 ...j = 1,6 (14) in which the panel index m covers only the panels on the structure. nrj are the generalized directional cosines for the panels on the structure given by: n., = cos(n.,zX) n.2 = cos(n.,,x 2 ) nfn 3 = COS(n.n, x 3 ) T7=4 = Xm277n 3 - Xm3f77n2 nlns X = X,,r 3 71, I - mIrn3 '? = x.f7m2Z, - Xm2n7,, (15) in which: Xmn = 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: aOne, 1 (16) where the -I follows ft-out the fact that the free surface no-nlal is pointing in the negative .V:ý-directiot. From the solutions of the source strengths for all these cases the wave force vector X, and the added mass aj and damping coupling coefficients bj can be obtained. The wave force follows from: x = P2 Z k + 4hjnlkASnk (17) k=1 in which: Od, = diffraction potential at k-panel obtained from equation (11) Xý = 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 part. n,, = generalised directional cosine of k-panel related to n-mode ASý' = area of k-panel related to the force in the n-mode The added mass and damping coefficients follow from: N, atj = -Rep)t k~ k k=I N,, bj= -JmirpwEkjknflASflk] (18) k=1I in which: Ojk = motion potential value on k-panel obtained from equation (11). The restoring coefficients cj 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 waterline area , 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 lincarised adiabatic law given in equation (1) Bascd 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 (10). 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 forces we 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 structure and the fice-surfaces 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[10). In the present contribution only the cushion pressures, 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 mn 0.05 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 arrangement. 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 0U02 m was added at the midship location which separated the fore -and aft cushions. In both cases the air cushions account for about 62,7 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. This does not occur with the one-cushion barge. Table I: Main Particulars of Air-cushion Barges Quantity Units One-Cushion Two-Cushion Barge 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 m 3 0.130 0.130 KG m 0.30 0.30 GM(transv.) m 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 5.00 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(l). Model tests aimed at validating the computational method for full scale structures however introduce additional modelling problems with respect to the air cushion stiffness. See, for instance, Moulijn[1 1]. 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 sirge 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 pressire differential between the fore and aft cushions was also measured. From the tests in regular waves the R.A.O.s of the pressures and motions were obtained by Fourier analysis. The quadratic transfer fimction of the mean second order surge drift force was obtained by dividing the mean sufrge force by the square of the amplitude of the imdisturbed 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 2.0r/s to 1O.Or/s. The results of computations and model tests are shown in figure 3 through figure 14. Re- stilts for the one-cushion arrangement are given in figure 3 through figure 7. The results 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 in 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 pitch resonance ftequeiicv viscous effects ])la . a sijaller role in the two-cushion case. 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-cushiion barge spatial equalisation of pressure takes place over a larger cushion area. The one-cushion pressure 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 1 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 useflfill 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 [2] 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 [3] Faltinsen,O.M. and Michelsen,F.:'Motions of Large Structures in Waves at Zero Froude Numbers', Int. Syrup. Dynamics of Marine Vehicles and Structures in Waves, London, 1974 [4] Seidel,L.H.:'Development of an Air Stabilized Platform', University of Hawaii, of Ocean Engineering, Dept. Technical report submitted to U.S. Department of Commerce, Maritime Administration, 1980 [5] 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 [6] Kure,G. and Lindaas,O.J.:'Record-Breaking Air Lifting Operation on the Gullfaks C Project', Paper No. OTC 5775, Offshore Technology Conference, Houston, 1988 [7] Faltinsen,O.M. et al. :'Speed Loss and Operability of Catamarans and SES in a Seasway, Fast'91 Conference, Trondheim, 1991 [8] Nakos,D.E. et al.:'Seakeeping Analysis of Surface Effect Ships', Fast'91 Conference, Trondheim, 1991 [9] Blood,H.:'Model Tests of a Pneumatically Stabilized Platform', Int. Workshop and Very Large Floating Structures, Hayama, Japan, 1996 [101 Pinkster,J.A.:'The Effect. of Air Cushions ruder Floating Offshore Structures', Pro- ceeding Boss'97 Conference, Delft., 1997 [11] MoulijnJ.:'Scaling of Air Cushion Dynamics', Report No. 1151, Laboratory of Ship Hydromechanics, Delft. University of Technology, Delft, 1998 Figure 1: Panel model of the one-cushion barge. Total No. of panels - 444. Figure 2: Panel model of the two-cushion barge. Total No. of panlels = 474. 1,2 1.0 -CAL 0 EXP 08 0 E 0 0 0.6 E 0.4 0 1 20 3 0 5. 0.0 ...... 5...... ? , 0123 4 5 6 78910 frequency [rad/sj Figuire 3: Singe motions of one-cl-shion barge. 1.0 0 CAL 0.8 0 EXP 0,6 0.20 00.00 0.0 0" 0 * 0 .0 ...... 0, ... , , , , , . . 0 1 2 3 4 5 6 7 8 9 10 frequency [rad/sl Figuire -1:Heave motions .()f le-clusllionl IJA.r(e 300 250 -CAI 0 EXP ,200.80 T S150 E 0 m 100 50 o...... Jll...... 0 0 1 2 3 4 5 6 7 8 9 10 frequency Irad/si Figure 5: Pitch motions Of one-cuIshion barge. 0.5 CAI 0.4 0 EXP 0 ýa 0.3 0 Ea 0 2 -o . * / 0\ 0 0.1 0 00 0.0 ...... 0 1 2 3 4 5 6 7 8 9 10 frequency jrad/sj Figi n 6: Cui shiion pienssimir of orico-rilsislhhIniu o * 0_O *_. _ -10 9 Q/ \ / -2 /, O p0\ 00 S -3 V/.. -4 & - -5 0 -8-6 -.- CAL -7 O EXP 0 1 2 4 5 6 8 9 10 frequency [radls] F'ig'ure 7: Mlean surge drift, force of one-cushion barge. 1.2~ -7. 0 // "/ A 8 0 I I A. A J ' l l • . I * i 1.0'"\0 • • •Ai i A A i*I A.. 1.I .l.• . . . . . l.l...... * I. i I l EXP ...... 0 1 2 3 4 5 6 7 8 9 10 0)0 frequency [rad/l] Figi~Llle 7: Mean - 0.6 tllrg drlift.i~l forc OfW n-mishiO baruge. Eo 00 0.4 0 0 0/ 0010 0 12 3 4 5 6 7 8 9 1 frequency [rad/sJ 1 SiginoQ 8: St li-go motinn s ofI two-ct shiti baiigu - 1.0 0D.0 .- CAL 0.8 *8 - O 0 E E0.4 0.0 " 0.2 0 0 00 0.0 0 0 r-xo 0 1 2 3 4 5 6 7 8 9 10 frequency [rad/sj Figure 9: Heave motions of two-cushion barge. 120 , CAL 100 . 0 EXP 6 0 \ 0~0 0 60 00 = .8 E. 40 Cý /0 20 ...... 0 1 2 3 4 5 6 7 8 9 10 frequency [radfs) Figil'e 10: Pitch motiols. of two-crhisioii barge. 2.0 1,6 0 EXP"•I o 7i6 1.2 1A0 0. 0 0o/ .0 E//00 0 0.6 0 o4 . o , . '0 , , \' 0.2 \ 0.0 0 1 2 3 4 5 6 7 8 9 10 frequency [radls] Figiure 11: Cilshion pressuLre forward cushion of two-cushion barge. -2ý8 2 6. 2.4 .- CAL 2.2 0 EXP 2.0 ' 1.6 1.4 0.---1,2 / k E1.02 0,8 .0000. 0 (1 . 0.2 0o 0.6 .."8 .0 O0. 0 1 2 3 4 5 6 7 8 9 10 " frequency (radis) Figure 12: Ciushion pressuire aft. (iushion of two-c:iishion barge. 4.5 4.0 CAL 3.5 0 EXP ao. • 2.0. 0)0 1.5 / o 0 1 .00 6, tO0 . oO , ,.•-. , .0 0.5 0.0 ...... 0 1 2 3 4 5 6 7 8 9 10 frequency [radlsJ Figure 13: PressuLre difference between fore and aft cushions of two-cushion barge. 0.0 0 odD -0.5 0 0 0 -1.0 -1.50 0 0 2L -.2. . 0 T -2.0 oo 0 -2.5 3' 5 -3.0 o/ -a CAL i -40 EXP "8' 1-1Sig drffreo'i~eMea ~\ 0 1 2 3 4 5 6 7 8 9 10 frequency (radlsj F'igi re iI-: Nicai .i'-ge(1111, t'r)1Idrift f t~wo-etisihii I.,;u't,._, Design load predictions by non-linear strip theories Dr Jorgen JUNCHER JENSEN, Tech. Univ. Of Denmark, Denmark Vt" WEGEMT WORKSHOP Non-Linear Wave Action on Structures and Ships University Toulon-Var 4t1h September 1998 DESIGN LOAD PREDICTION BY NON-LINEAR STRIP THEORIES by Jergen Juncher Jensen Technical University of Denmark Department of Naval Architecture and Offshore Engineering Building 101 E DK-2800 Lyngby ABSTRACT: Some methods for predicting global stochastic wave load presented. The methods take into account responses in ships are the elastic behaviour of the ship and at least some of the non- lineartities in the wave-induced loadings. Numerical results obtained for actual ships are reviewed with special design procedures emphasis on their usefulness in covering both extreme responses and fatigue damage predictions. 1. Introduction In the design of marine structures, the wave loads play an important role,' both because the forces are of equal magnitude to those arising from gravity forces and because of the stochastic nature of the ocean waves. To obtain rational design procedures for marine structures it is therefore necessary to derive expressions for these wave forces in terms of relevant quantities characterising the ocean waves and the structure itself. Not only the mean values of the loads are required but also stochastic moments or the full probability distribution functions must.be estimated in order to perform a reliability analysis of the structure. The accuracy to which these probability distributions are estimated is crucial for the reliability analysis, as inherent stochastic uncertainties the in the wave loads are much larger than the strength uncertainties of marine structures. The wave-loading analysis can generally be divided into the following steps: - a stochastic characterisation of the wave elevation and wave particle kinematics; - a derivation of equations relating the required responses with characteristic wave parameters; - an application of suitable stochastic methods for extreme value statistics and fatigue damage estimates. In a purely linear analysis, the stochastic sea is modelled by a linear superposition of small amplitude sinusoidal waves, and linear deterministic response relations, so- called transfer functions, are applied for each sinusoidal wave component. Consequently, both the sea and the response become Gaussian distributed. Extreme value statistics in stationary sea states characterised by a time-independent wave spectrum can be obtained analytically (Cartwright and Longuet-Higgins, 1956), and, by semi-empirical methods based on independent peak statistics, extended to cover the design life of the structure. This type of analysis was first applied to ships by St. Denis and Pierson (1953) using transfer functions based on a two-dimensional hydro- dynamic velocity potential description of the interaction between the ship and the waves. Since then many improvements have been made based on both two- dimensional and three-dimensional hydrodynamic formulations of the velocity potential. The two-dimensional theories are usually named strip theories. Generally, the motion characteristics of ships are predicted quite well from these linear theories, but significant non-linearties are observed in measurements of the sectional forces in the hull beam. Two such examples are shown in Figure 1 and Figure 2. Both of them are concerned with the vertical bending moment amidships which is the most important wave load parameter in the design of ships larger than 100 m (i.e. dominated by beam bending rather than local pressure on the hull platings). Figure 1 shows the probability distribution functions for the peak (sag) and trough (hog) values of the bending strain obtained form measurements in a container vessel during an operational period of about 20 minutes. A sea state can normally be assumed stationary within such a short period and the theoretical linear distribution function will be the same for both the peak and trough values and close to a Rayleigh distribution. As seen from the figure, non-linearities tend to increase the sagging bending moment significantly and slightly decrease the hogging bending moment. 5 I I Tz:= 6.5s 4 RMS(E)= 6.5 3 10 sag 3 - oV • 0 payteigh V• VV hog V I - V Vho 0 0.5 0.9 0.95 0o.9} Figure 1. Short term statistical representation of the peaks &. in the wave induced bending strain E derived .from Northern Atlantic measurements on CTS TOKYO EXPRESS (1018 GMT Dec 27, 1973). A low pass filter was applied to remove contributions from the 2-node vibration taking place around 5 rad/s.After Hackmann (1979). The same trend is noted in Figure 2, showing bending stresses obtained from measurements in a specific class of frigates. Altogether measurements covering an operational period of 9 years are included. The measured results have been obtained in different sea states, yielding a distribution function far from a Raylcigh distribution, 2 and more resembling extreme value distributions of the Gumbel or Weibull type. However, also in this case a linear analysis will lead to exactly the same probability distribution for the sagging and hogging stresses. In design procedures the significant difference observed between the magnitude of the sagging and hogging bending moments at the same probability level must be accounted for and in the next chapter - an outline is given of a quadratic (second order) strip theory, capable of predicting at least some of these non-linearities. The methods is illustrated by some recent calculations for the wave bending moment in different vessels both in severe stationary seaways and covering the whole operational life of the ship. 100 Stress (MN/m 2 ) sag 50 V VV V V • •VV v hog V. 0 I I I I 0.99 0.9 0.5 0.2 10-1 10-, 10-3 10-4 10-S Probability of exceedance in a 4 hour period Figure 2. Gumbel plot of long-term deck stress measurements in a narrow beam LEANDER Class frigate. A probability of exceedance of 2 - 10-5 corresponds to a 20 years' return period After Clarke (1986). 2. A quadratic strip theory The non-linearities responsible for the difference between the hogging and sagging responses shown in Figure 1 and 2 are mainly the variations in added mass, hydrodynamic damping and water line breadth with the relative immersion of the hull. In addition, non-linearities in the wave profile and associated wave kinematics can be important, too. All these non-linearities have asymmetric terms and thus a second order model should be appropriate for moderate non-linearities. It should be mentioned, that a similar analysis involving roll motions of the vessel would need a cubic modelling, as the roll motion is symmetric around the mean position. A simplified one degree-of-freedom analysis dealing with this problem was presented by Ness, McHency, Mathisen and Winterstein (1989). Here only the vertical wave-induced loads are considered. As discussed in the introduction, the two-dimensional strip theory usually yields predictions close to measurement in small amplitude waves. A non-linear generalisation of the linear strip theory therefore seems to be an appropriate starting point for the derivation of a non- linear wave loads analysis. Jensen and Pedersen (1979) proposed the following expression for the vertical force q(x,t) per unit length acting on the hull at the longitudinal co-ordinate x: 3 q(x,i)=- [.tm('x)--- "t 1 [Di -r •zp 1 Im D+Q N(zx)--t+ JB(zx)y l:+,dz (2.1) Here the relative displacement 2(xt) is the difference between the absolute vertical motion w(x,r) of the ship and the surface elevation h(x,t) corrected for the local wave particle accelerations (the Smith effect). The two-dimensional added mass mi, hydrodynamic damping N and water line breadth B are all taken as functions of the instantaneous relative displacement F and, as usual, of the longitudinal co-ordinate x. The last term in equation (2. 1) is the instantaneous Froude-Krilov force on the section due to the undisturbed pressure p in the waves. Finally the time is denoted by t and DIDt is the total derivative taking into account the forward speed of the vessel. For small wave amplitudes and correspondingly small displacements F, equation (2.1) reduces to the linear relative motion strip theory proposed by Gerritsma and Beukelman (1964). The unknown is the absolute displacement w(x,t). This function is determined by a Timoshenkco beam modelling of the hull, with the external forces given by equation (2.1) in addition to the inertia forces from the mass distribution of the ship. Petersen (1992) and Xia et al (1998) among others solved the governing equations by direct time simulation procedures. Although such analysis is straightforward, it is very time consuming to calculate sufficiently long time series for reliable extreme value predictions. An alternative when the non-linearities are moderate is a formal second- order expansion of the response around its mean value. Thereby, analytical and asymptotically correct statistical predictions are readily obtained. To account for the frequency dependent hydrodynamic forces, this perturbation scheme should be performed in the frequency domain, yielding linear and quadratic frequency- dependent transfer functions. A full description of this perturbational approach applied. to equation (2.1) and the governing equilibrium equation for a free-free non- prismatic Timoshenko beam is given by Jensen and Pedersen (1979) with some modification to account for oblique waves and combinations with torsional and horizontal wave bending loads added later (Jensen, Pedersen and Petersen, 1992). The final result for the vertical bending moments M(x,t) can be given as 2 M(x,t) = MII(x, t) + MI )(x,t) (2.2) where = - ,(x) M, (x)]costkM-, + xWM in,} MI'(XI)I+ Corr X + ,=I M~~ (2.3) and 4 M(2(xt)= • : {_,[(4.,4,- 4,..,..)M•'(x)- (s,4,+. +4.÷.s,)M,;(x)lcos(o +wg,)t + + )A-(x) + r) (x)]cos (t, -ltD )t + - 4.÷n4,÷n )A, (x) + (4,4,,. + 4,..4, )M (x)lsinw (1n, + t, )t t + + 4,,,, )M- (x)c- (4,,o - s,.s, ),- (x)]sin (q, - ti, )t} (2.4) Here the variables Fi are defined in terms of the amplitudes ai and phase lags 6i of the n linear individual wave components used to define the wave elevation and wave kinematics in the second order wave theory applied: 4, = a, cos6, and + = ai sin 6, (2.5) Analytical expressions for the coefficients M', M'•+,a=c,s are given in the previously cited references. These coefficients can be directly related to the linear and quadratic, complex-valued transfer functions which follow from a Volterra series formulation of the complex time-invariant dynamic system (Jensen, Petersen and Pedersen, 1990). The frequency of encounter W,is given by a?2 51, =w,--'-VcosO (2.6) g where coi is the wave frequency, V the forward speed of the ship. g the acceleration of gravity and ý the angle between the ship heading and the wave direction for the uni- directional (long-crested) wave system assumed. Generalizations to short-crested waves are possible (Longuet-Higgings, 1963), but will greatly increase the computational task involved, as also noted by Langley and McWilliam (1993) in a study of offshore structures. To model a stationary stochastic seaway, the wave amplitudes ai and the phase lags Oi are chosen so that Fi and 4i,+ are jointly Gaussian distributed with ji uniformly and independently 2 distributed and with 1/2ai proportional to the wave energy in the corresponding frequency range. From equations (2.2)-(2.4) the time derivatives to any order can be obtained analytically. Usually thespectral bandwidth of wave responses like the vertical bending moment is rather narrow and thus only the joint probability distribution of the response and its time derivative is needed in the peak value analysis. For more broad-banded responses, also the second derivative must be included, resulting in a quite substantial increase in complexity in the statistical analysis (Longuet-Higgins, 1964). The probability density function of the response in the stationary seaway is obtained conveniently (Vinje, 1974), by introducing the standard normal distributed variable -, 4(2.7) 5 where Vi is the variance of ýi and applying a suitable normalization of the response. At a given longitudinal coordinate x the normalized response • and its time derivative 4can be written as 2n 2n 2n X2x, +eXXAozz , (2.8) 2n 2n 2n I= YX ,+--IIrF;,z, (2.9) ,I5 t=Ij=1 where the deterministic coefficients Xi. e, Aij. y and rij follow from equations (2.3) and (2.4), evaluated at an arbitrary time t, say t = 0. The normalisation of the response and its time derivative are made such that two-norm of the matrices X1, yi, and Aij, are equal to unity. Thereby, the coefficient e becomes a measure of the degree of non- linearity. Using cumulant generating functions, Longuet-Higgins (1963) constructed a formal procedure for calculating the joint probability density function fa to any order in c. The result can be written (Vinje, 1974) fI4=--x (_,lA •i-'2 -2)1+er qna(p,O) fc(,¢ = 2z"exp ... --V/2" ll+',/ I p---oq--,• f ,-p.O) H,rHep(o)Heq}(2.10) where Hep() is the Hermite polynomial of orderp. The constants H' are determined from the cumulants of the joint distribution as obtained directly from equation (2.8) and (2.9). An efficient numerical procedure to calculate H7' is given by Jensen and Pedersen, (1978). Finally, because of the narrow-band assumption, the distribution function Fy(y) of the individual peaks y can be approximated by FY(y) = I -v(y) v4.(y) (2.11) where vg'y) is the mean upcrossing rate of the leyel y: v;(y) = J fe (y,0)d4 (2.12) 0 and where 5 is the value of y maximazing vý. The only drawback in this procedure is the slow convergence of the summation over r in equation (2.10). Actually, even for moderate values of c the summation can consist of alternating positive and negative terms increasing with r in magnitude. Thereby, negative values for the probability density function can appear yielding useless results. As an alternative, albeit approximative, stochastic analysis, equation (2.8) can be used to derive a monotonic relation between the response ý and a standard normal 6 distributed variable U. For responses showing an increasing degree of non-linearity with excitation, Winterstein (1985) suggested a cubic Hermite (or polynomial) series expansion 3 Lb, U' ,---0 (2.13) in which the coefficients bi are determined from the four lowest statistical moments (mean value, variance, skewness and kurtosis) of the response as given by equation (2.8). To obtain the peak value distribution Fy(y), equation (2.11), the following approximation is taken vC(y) = exp (-½/2U2(Y)) (2.14) v(P) in which u(y) is the real solution to 3 y= b,u' (2.15) i__O Contrary to the solution based on the full joint probability density function, equation (2.10), the polynomial expansion behaves well for all magnitudes of the non- linearities as long as the transformation (2.13) is monotonic (i.e. b2 < 3b~b3). However, even if some correlation between ý and 4implicity is imbedded in equation (2.14), this correlations not based on any specific calculation, like equation (2.9), of 4 . Despite this, the method works well for many different problems where exact solutions are available (Winterstein, 1985). In an analysis of the responses shown in Figure 1, both statistical approaches yield quite similar results (Jensen, Petersen and Pedersen, 1990) and, furthermore, these results compare well with the measurements. For design purposes the probability Q (Y(T)) that the maximum peak during a given time period T exceeds the level y is needed. This probability can be obtained either from a conventional Poisson upcrossing model or from order statistics, yielding asymptotically equal results for small values of Q. Long-term predictions are obtained by a proper weighing of the stationary results making use of scatter diagrams and speed reduction in heavy sea, see e.g. Jensen and Dogliani (1996). Finally, fatigue damage estimations should be mentioned. Due to the large skewness of the response (stress) process, the usual assumption that the probability distribution of the stress range Aa, can be obtained by replacing the stress a by Aa/2 in the probability distribution of a is rather dubious. Instead it has been suggested by Winterstein and Manuel (1989) to determine the stress range distribution from the assumption that a peak is followed by a trough with the same upcrossing rate. Thereby, the expected value E[Aam] of the stress range raised to the power m. given by the S-N curve for the structural detail considered, becomes (Nielsen et al., 1994) 7 EiAa-]=JIAar-(u)ue-"du=IJ[a(u) -ar(-u)] ueU'dA 0 0 (2.16) f 2m[c,u + C3u3 1'ue-"'udu 0 assuming the stress a is given in a form similar to Eq. (2.15): 3 a = lc,u' (2.17) ,=O A typical value for m is 3, yielding E[Aar]= 24 [c + 15c c, + 315c'] (2.18) This result can also be given in terms of the kurtosis of the stress distribution, Winterstein (1985). 3. Numerical Results Verification of the results predicted from the strip theory formulation, Eq. (2. 1), has been made for several ships. Usually, the vertical bending moment distribution agrees quite well with results obtained in model tests. Figure 3 shows a comparison between the second-order predictions, a full non-linear time-domain solution by Xia et al. (1998) and model tests by Watanabe et al. (1989). The tests were carried out for the S 175 container ship sailing in regular waves with a wave length equal to 1.2 tinies the ship length and a wave amplitude to ship length ratio of I to 60. Head sea condition with a Froude number of 0.25 was considered. From the figure significant non- linearities are observed with the sagging bending moment being more than twice the hogging bending moment in this test conditions. It is also seen that the agreement between the measurements and both the calculation procedures are very good. Hence, the formulation Eq. (2.1) seems to be able to predict the wave-induced sectional loads with sufficient accuracy for design purposes. The quadratic theory has been used to obtain design values for the midship bending moments in various types of vessels. Two different approaches are used. In the first, Mansour and Jensen (1995), the design value is taken as the most probable largest value during a 3 hours operation in a short-term severe sea state characterised by a Pierson-Moskowitz wave spectrum with a significant wave height H, = 15 m and a zero crossing period T, = 12 s. This rather unrealistic sea state is often used as an equivalent to a complete long-term analysis, c.f. Bach-Gansmo and Lotsberg (1989). A low forward speed and a heading angle equal to 135 degrees are assumed. 0.06 0Steady - state Time-domain -.Quadratic 0o0.03 Experiments 0 -0.03 AP FP Figure 3. Non-linear sagging (positive) and hogging (negative) bending moments the Si 75 Containership of sailing in the regular wave, A = 1.2L, a = L160,; Fn = 0. 25. The comparison is made for the experiments (Watanabe et. al., 1989), the time- domain method (Xia al., et 1998) and the quadratic strip theory (Jensen and Pedersen, 1979). K3 C 2 ''4 * :1K 3 = skewness 0.4 x :K = kurtosis C3 4 0 . 4 - X 3 .4 C2 0.3 P cl 3.3 0 0.2- 3.2 F X a .0 x P 0.1- T x -3.1 X F 01 3 0 T 0.1 0.2 CF Figure 4. Variation 0.3 of skewness and kurtosis with the bow flare coefficient C4."for six ships. Results are for the midship sagging bending moment. For hog the sign of the skewness should be changed The vessels considered are a tanker (T), a frigate (F) an oil-bulk-ore carrier (0), a floating oil production vessel (P) and two containerships (Cl, C2). The degree of non- linearity in the bending moment is depicted in figure 4, showing the skewness M3 and kurtosis x4 of the response. For comparison, a linear response to a Gaussian sea will K have 3 = 0 and K4 = 3. The different ships are here characterized by one single parameter, the bow flare coefficient CF, measuring the flare of the forward 20 per cent of the hull. Of course, this is a rather crude measure, which cannot possibly completely describe the difference in hull forms seen among these ships. However, a clear trend towards increasing skewness and kurtosis with flare coefficient is seen. The skewness is responsible for the difference between the sagging and hogging moments, whereas a kurtosis greater than 3 implies a larger probability of obtaining larger peak and trough values than in the Gaussian case. From the polynomial (or Hermite) series approximation, equation (2.13), the ratio between non-linear and linear design values of the bending is obtained. The results are shown in Figure 5. Again some spreading in the results is seen, but the trend is an increase in the sagging and a decrease in the hogging bending moment with bow flare. A drawback in this analysis is the rather arbitrary choice of equivalent sea state and that neither the ship service speed nor different heading angles are taken into account. This can only be done through a full long-term analysis in which the wave statistics in the different ocean areas covered by the ship's operational profile is weighted and combined with a speed reduction procedure in severe sea states. Such an analysis has been performed by Jensen, Banke and Dogliani, (1994) for 7 ships, some of them identical to those used in the above-mentioned short-term analysis. The main results are given in Figure 6. The abscissa is now a combined bow flare and forward speed coefficient Ciw: CFV= CF+0.5 Fn (3.1) Clearly the same trend and the same range for the ratio between the non-linear and linear predictions for the design values of the wave-bending moments are found in long-term analysis as in the stationary, short-term case. It should be noted, that because of the very large extreme bow motions yielding both bow emergence and deck wetness, a proper definition of the flare must be used, see the last cited reference. 1.5 M 0M=Msag Mlin x : M =Mhog 0 P F S T T X 0.1 0.2 CF 1 I F x x 0 cX P 0.7 C2 Figure 5. Variation of the ratio of hogging and sagging moments to the linear inomient with the how flare coeficient ('..fior six ships. IM 1.5 M~ M = Msog7 M6 7 Mlin x M= Mhog 5 o2 3 4 00O p1 1.0 0.1 x 0.2 0.3 CFV 0.4 X 3 x X X 4 5 6 7 0.7 Figure 6 Ratio between the median values of the long-term extreme values of the wave-induced sagging and hogging bending moments amidship and the corresponding results from linear strip theory for 7 ships (1: tanker, 2: dry cargo ship, 3+5:frigates,4: floating production vessel 6+7: containerships). Finally, a study in a moderate seaway will be mentioned, Jensen (1996). The aim of this analysis was to investigate the effect of hull flexibility on the wave-induced loading on high-speed mono-hull ships. Whereas the hull flexibility normally can be ignored for ships built of steel the use of aluminium or glass-fibre-reinforced plastic (GRP) will lower the hull bending stiffness. Thereby dynamic amplification of the two-noded vibration mode can be significant mainly due to the second order sum terms. 4. Conclusions Ships and offshore structures are among the most complicated and expensive man- made structures. Any failure of such structure can result in loss of many human lives and immense environmental damages. Therefore, detailed engineering calculations must be performed in the design stage. A significant loading on these structures are imposed by ocean waves. Due to the stochastic nature of these waves, the structural response must also be treated in stochastic terms. For design calculations, the extreme loadings expected during the operational life of the structure are required. These loadings will normally be associated with rare events with extremely high waves. A non-linear description of both the waves and the wave load responses is therefore often needed. The present paper is concerned with the determination of the sectional forces in ship hulls. Here a second order, frequency domain method is reviewed and used to obtain design values for several specific ships of different types. II References * Bach-Gansmo, 0. and Lotsberg, I. (1989) "Structural Design Criterias for a Ship type floating production Vessel" PRADS, Varna, Bulgaiia, Paper No. 87. Cartwright, D.E. and Longuet-Higgins, M.s. (1956) "The Statistical Distribution of the Maxima of a Random Function". Proc. Royal Soc. London, Serie A, vol. 237, p. 212-232. Clarke, J.D. (1986) "Wave Loading in Warships". Proc. Advances in Marine Structures, eds. Clarke and Smith, Elsevier Applied Science, London. Denis, M.St. and Pierson Jr., W.J. (1953) "On the Motions of Ships in Confused Seas", Trans. SNAME, Vol. 61, pp. 280-357. Gerritsma, J. and Beukelman, W. (1964) "The Distribution of the Hydrodynamic Forces on a Heaving and Pitching Ship Model in Still Water", Fifth Symposium on Naval Hydrodynamics, ACR-1 12, Us Office of Naval Research, Washington, D.C. pp. 219-251. Hackmann, D. (1979) "Written discussion to Jensen, J. Juncher and Petersen, P.Terndrup (1979)". Jensen, J. Juncher and Pedersen, P. Temdrup (1978) "On the Calculation of the Joint Probability Density of Slightly Non-linear Stochastic Processes", GAMM-Tagung, Lyngby 1977, ZAMM, Vol. 58, T 481-T 484. Jensen, J. Juncher and Pedersen, P. Temdrup (1979) "Wave-induced Bending Moments in Ships - A Quadratic Theory", Trans. RINA, Vol. 121, pp. 151-165. Jensen, J. Juncher, Petersen, J.B. and Pedersen, P. Temdrup (1990) "Prediction of Non-linear Wave-Induced Loads on Ships", Proc. IUTAM Symposium on the Dynamics of Marine Vehicles and Structures in Waves, Brunel University, London, June 24-27. Jensen, J.Juncher, Pedersen, P. Terndrup and Petersen, J.B. (1992) "Stresses in Containerships", Jahrbuch der Schiffbautechnischen Gesellschaft, Vol. 86, paper St5. Jensen, J. Juncher, Banke. L. and Dogliani, M. (1994) "Long-term Predictions of Wave-induced Loads Using a Quadratic Strip Theory", Proc. NAV'94, Rome, Italy, October 5-7. Jensen, J. Juncher and Dogliani, M. (1996) "Wave-induced Ship Hull Vibration in Stochastic Seaways", Marine Structures, Vol. 9, pp. 353-387. Jensen, J. Juncher (1996) "Wave-induced Hydroelastic Response of Fast Monohull Ships", Proc. CETENA Seminar on Hydroelasticity for Ship Structural Design, Geneva. 12 Langley, R.S. and McWilliam, S. (1993). "A Statistical Analysis of First and Second Order Vessel Motions Induced by Waves and Wind Gusts", Applied Ocean Research, Vol. 15, pp. 13-23. Longuet-Higgins, M.S. (1963) "The Effect of Non-linearities on Statistical - Distributions in the Theory of Sea Waves", Journal of Fluid Mechanics, Vol. 17, pp. 459480. Longuet-Higgins, M.S. (1964), "Modified Gaussian Distribution for Slightly Non- Linear Variables", Radio Science, National Bureau of Standards, Vol. 68D, No. 9, pp. 1049-1062. Mansour, A.E. and Jensen, J. Juncher (1995) "Slightly Non-linear Extreme Loads and Load Combinations", J. Ship Structures, Vol. 39, No. 2, pp 139-149. Ness, O.B. McHenry, G. Mathisen, J. and Winterstein, S.R. (1989) "Non-linear Analysis of Ship Rolling in Random Beam Waves", Proc. STAR Symposium on 21st Century-Ship and Offshore Vessel Design, Production and Operation, April 12-15, SNAME. Nielsen, L.P. Hansen, P.Friis, Jensen, J. Juncher, Baatrup, J. and Pedersen, P. Temdrup (1994). "Reliability Based Inspection Planning of a Jack-up Rig", Proc. BOSS'94, Boston USA, July 12-15. Petersen, J.B. (1992) "Non-linear Strip Theories for Ship Responses in Waves", Report Dept. of Ocean Engineering, Technical University of Denmark. Vinje, T. (1974) "On the Statistical Distribution of Maxima of Slightly Non-linear Variables", Rep. SK/M 27, Trondheim, Div. of Ship Structures, the Univ. of Trondheim. Winterstein, S.R. (1985) "Non-Normal Responses and Fatigue Damage", J. Engineering Mechanics, ASCE, Vol. 111, No. 10, pp. 1291-1295. Winterstein, S.R. and Manuel L. (1989) "Non-Gaussian Response of Offshore Platforms Fatigue", J. Structural Engineering, ASCE, Vol. 115, No. 3, pp. 749-752. Xia, J., Wang, Z. and Jensen, J. Juncher (1998) "Non-Linear Wave Loads and Ship Responses by a Time-Domain Strip Theory", DCAMM Report No. 569. Technical University of Denmark, Lyngby, accepted for publication in Marine Structures. 13 cylinder Non-linear wave interactions between waves and a horizontal Pr John R. CHAPLIN, City University, UK WEGEMT WORKSHOP NON-LINEAR WAVE ACTION ON STRUCTURES AND SHIPS University Toulon-Var, France, 4 September 1998 NON-LINEAR WAVE INTERACTIONS BETWEEN WAVES AND A HORIZONTAL CYLINDER John R Chaplin Department of Civil Engineering, City University, London ECIV OHB [email protected] 1. INTRODUCTION The flow around a horizontal cylinder beneath waves in deep water, where the wave crests are parallel with the cylinder's axis, reveals some unexpectedly rich physics. This flow, and the related case of a cylinder moving around a circular path relative to the surrounding fluid, may be important in a variety of applications related to marine structures and ships, including Tension Leg Platforms, submarines, risers, and wave-power devices. Figure 1 shows an outline sketch for the problem, which is defined by four independent dimensional parameters: the cylinder's radius c, the submergence of its axis below still water level h, the wave frequency w, and wave amplitude A. The key independent dimensionless parameters are kc, h/c, and kA, where k is the wavenumber. These notes summarise some of the most important non-linear features of the problem in the regime kc<4, h/c<4, and at wave amplitudes less than those corresponding to separation. Some unpublished results are included (from Chaplin & Retzler, 1997, and Chaplin, 1998). 2nr/k (ok h 2c Figure 1. Definition sketch 2. LINEAR SOLUTION FOR TIlE PIOTEN'l'IAL FLOW According to the linear solution (forkA< The loading on the stationary cylinder beneath waves can be separated into an oscillatory part and a steady part. According to the linear theory, the direction of the former rotates once in every wave period (lagging the acceleration vector of the undisturbed incident flow by Yp), and its magnitude can be expressed in terms of an inertia coefficient as C. pnc2Acw2e-". (1) The inertia coefficient found from Ogilvie's solution is denoted C,, . The corresponding steady part of the force is directed upwards, and its magnitude can be written 2 2 2 - 2 Cy, pIE cA () e k. (2) The coefficients Cm. and CY, and the phase angle y, are in general functions of kc and h/c and are shown in contour plots in figure 2. For the case of a deeply submerged cylinder (hfc>>»), Ci approaches 2. As the relative submergence is reduced, Cm. increases if kc is less than about 0.5, but decreases if kc is larger than this. Another interesting feature shown in figure 2 is that for a given kc, C9 is almost independent of the relative submergence for hlc > 1.5. This force can be attributed to the steady part of the convective component of the particle acceleration that is directed vertically upwards. 3. NON-LINEAR SOLUTIONS FOR THE POTENTIAL FLOW Higher order analytical or numerical solutions of the potential flow problem have more recently identified the most important features of the non-linear potential flow problem. Mclver & Mclver (1990) showed that there is no reflection at the fundamental wave frequency also at second order, and Palm (1991) proved that there is no reflection in the leading order component at any harmonic frequency. These conclusions are supported by numerical computations by Liu, Dommermuth & Yue (1992), which also provide predictions for the attenuation of the transmitted wave at the fundamental frequency, and, like Vada (1987), Wu (1991), and Riley & Yan (1996), for the amplitude of the second order transmitted wave at the second harmonic. Riley & Yan also showed that there is a. lowering (or 'set-down') of the mean water surface over the cylinder. As a result of this more recent work, it seems that the non-linear potential flow problem can be said largely to have been solved - either by perturbation expansions at low orders for waves of moderate steepness, or by fully non-linear numerical solutions for more severe cases. -1.-50' ' .. .. - " .- . io'. -- - - 2. -2.0 ---- ...... ' iI -2.5 -3.0a) - -- -1.2; ~ 201 ~ (b) ri r -2.5 -o3. 5 0.5 1.0 1.5 20 2.5 3.0 !:3 kc Figure 2. Contours of (ac) C.,, (b kf1 , (c C,, computed froin Ogilvie (1963). 4. VISCOSITY-INDUCED CIRCULATION An interesting and important feature of the problemn is that though inviscid predictions of sonic characteristics of the flow and the forces are in excellent agreement with measurements (see below), viscosity has a major effect on others, even when the amplitude of the motion is small and there is no gross separation, that is when kAei1h<> 1 the steady streaming velocity at the outer edge of the boundary layer can be shown to be 3A2(0e/-2kc. This is equal to the speed of the undisturbed incident flow when K = T13, where K is the Keulegan Carpenter number 7tAe-k/c. 5. MAGNUS LIFT Circulation around a body in a cross-flow produces a Magnus lift force, and in the present case this acts in the direction opposite to that of the inertia force mentioned above. If the steady circulation around the cylinder were to spread uniformly into the ambient flow (in the manner of a potential vortex), it would generate a lift force of 6pc 3r'2K3/3t2per unit length, negating a substantial part of the potential flow inertia force. The effective inertia coefficient would then be C,, = Cmj - 6 K'1nr . (3) The elements of this mechanism are sketched in figure 3 (where the cylinder is shown with a rounded-square pontoon section, for which the same process occurs). Force measurements seem to confirm the existence of this mechanism, and reveal substantial reductions in the inertia coefficient from its potential flow value (e.g. Chaplin, 1984, Otsuka & Ikeda, 1996); in some cases (particularly for small values of h/c) they are even larger than those mentioned above. Some examples are shown in figure 4, where observed values of C,, are plotted against K for various values of h/c. A horizontal line corresponding to each set of data represents the linear potential flow solution C., (figure 2), and shows that this is valid only at very small values of K. For moderate Keulegan Carpenter numbers and larger submergences, the measured inertia coefficients are initially quite close to a relationship of the form 2 C,, = C,, -R K . (4) but with R taking values different from the 6/it' of (3). For smaller h/c the behaviour of C,, with increasing K is rather different, since then it appears to match C,,, to start with, before dropping at a gradually reducing rate. Nevertheless it is useful to fit the data for small K to equation (4), and the results - and the corresponding ranges - are shown in figure 5. 4I Beneath a crest Beneath a down-crossing velocity acceleration l velocity , acceleration (a) Velocity and acceleration vectors Iii F -. -F IF (b) Forces predicted by ideal fluid flow computations (c) Circulation generated by steady streaming incident flow Lift , circulation -t circulation I H4 1 "Lift incident flow (d) Lift due to the combination of circulation and incident flow F-Lift * F-Lift (e) Total forces from (b) and (d) * Figure 3. The mnechanismn of viscosity-induced circulation and lift 3.0. .. Sh/c=1.22 2.5 . 1.34 - K~A~1.76 2.0~t .2.26 1.5 - rl•• - 1.0 -: -- • . . . • , 0.0 0.5 1.0 K Figure 4. Measured inertia coefficients as functions of Keulegan Carpenternumber at various submergences (Chaplin & Retzler, 1997) R in equation (4) for kc =0.201 10 Mdn Kat wich the loading fits equation (4) 0 Chaplin (1984) kc =0.206 0 2 3 4 5 6 h/Ic Figure 5. Non-linear behaviour of the coefficient at kc = 0.2 and various submnergences (Chaplin & Retzler, 1997). 6. THE DOUBLE BOUNDARY LAYER These and other experiments suggest also that the flow is considerably more complicated than that briefly described above. First. the steady streaming is not the same at all points around the cylinder (since the velocities are greatest near the top). In these conditions analytical arid numerical solutions point to the existence of an outer boundary layer that prowides a Iransilon 6 to a potential vortex motion around the cylinder that is uniform (Yan & Riley 1996). This means that the outer circulation is not related in any simple way to the steady streaming in the inner boundary layer, that close to the cylinder the circulation is a function of radius, and therefore that the Magnus lift formula is not a good model for the non-linear force. Yan & Riley showed that the outer circulation increases as the cylinder's submergence is reduced, in- a way that is qualitatively similar to what can be inferred from the force measurements, but numerical agreement with laboratory data is not good. 7. THE INSTABILITY In reality the circulating flow is unstable, and it develops a secondary motion with a three- dimensional structure (Chaplin, 1993) in which hoop vortices encircle the cylinder. This is clearly related to the Taylor vortices that occur in the annular space between two concentric rotating cylinders, but in the present case the circulation is self-induced, and the outer cylinder is absent. Visualisations of the out-of-plane secondary flow are shown in figure 6. These were obtained by viewing the cylinder from below (through the floor of the tank) after coating it with neutrally buoyant dye. As waves passed over the cylinder, the dye was drawn along the cylinder's axis to form periodic regions of higher concentration, as shown in figure 6(a). Subsequently, the dye was drawn away from these regions into the surrounding flow in the form of radial plane jets which were highly unstable to disturbances in the axial direction. Figure 6(b) to (d) show the jets rolling up to form rather disorganised vortices wrapped around the cylinder. This secondary motion promotes mixing in the radial direction, but non-uniformity in the axial direction. This is likely to have a significant effect on the effective circulation, and therefore on the non-linear loading. 8. FREE SURFACE NON-LINEARITIES: (a) Limiting Wave Height The first sign of wave breaking over the cylinder appears as a thin layer at the free surface that is briefly accelerated to travel faster than the wave profile and is preceded by capillary waves, but it seems not to be associated with much loss of energy. Nevertheless, it probably represents the conditions at which analytical and numerical solutions of the flow can be expected to break down. The amplitude A of the fundamental component of the limiting incident wave defined in this way is plotted in figure 7. as a function of kc and h/c. Also shown is the limiting amplitude for undisturbed waves, derived from Schwartz's (1974) result of 0.2917 for the limiting value of k'A in deep water, where k' is the corresponding finite amplitude wavenumber. In these conditions, k/k' = 1.1928 tCokelet, 1977). It seems that for kc>0.4the limiting wave amplitude is independent of relative submergence hlc. 9. FREE SURFACE NON-LINEARITIES: (b) Instantaneous Free Surface Profiles Figure 8 shows measured and computed instantaneous free surface profiles for kc = 0.56, h/c = 1.5. In figure 8(a), A/c = 0.16. giving conditions identical to those of a solution of the potential flow problem to second order hy Vada (1987), whose results are indistinguishable from those made to the same order but by a different approach by Riley & Yan (1996). Ilowever. as can be seen from figure 7,this wave breaks at the cylinder. .7 (a) (b) (d) Figitre 6. Visualisations of the secondary flow around a horizzontal cylinder ben eath wavYes; kc =0.1, h/c =5, K = 0.25. h/c= 1.5 irrsiting height 1.75 (uvu 4 waves) 32 : 1.0- *** ÷ A/c -' Ul ,1J , l 0.1 0.0 0.5 1.0 1.5 2.0 k-c Figure 7. Conditions in which wave breaking first occurs over the cylinder. The line is computed from the limiting amplitude in deep water given by Schwartz (1974). It is therefore not surprising that agreement between the measurement and the potential flow solution is poor, particularly at the second harmonic. Figure 8(a) also shows the result of the linear potential flow solution, in which for this case it is found that the fundamental wave component undergoes a phase shift 2y, = 53.20, as it passes over the cylinder. This can be seen, relative to the profile of-the undisturbed wave extended to the right of the cylinder. The phase shift at the fundamental frequency in Riley & Yan's data is also found to be 53.20; in the measured profile it is 42.80. In a second case with the same wave frequency and cylinder submergence, the amplitude was reduced to A/c = 0.107, just less than that necessary to cause breaking The measured instantaneous wave profile for this case (for which the measured phase shift was 50.50) is shown in figure 8(b). It is seen to be in very good agreement with the results of a second order solution provided by Riley & Yan (1998). 10. FREE SURFACE NON-LINEARITIES: (c) Effect of viscosity on the phase shift It is reasonable to assume that circulation around the cylinder may have some effect on the phase lag that the waves experience as they pass over the cylinder. As a model of this process it is helpful to take the case of a stationary line vortex (of clockwise rotation) beneath waves (travelling from left to right). It can be shown that the effect of the vortex in these conditions is to generate a phase lead in the waves. Accordingly, the self-induced circulation around a cylinder can be expected to have a similar effect, leading to observed phase shifts less than 2qyI. Results shown in figure 9 suggest that this is indeed the case, though at small submergences the effect is weak compared with non-linear contributions inherent in the potential flow. 9 0.3 (a) 0.2 0.1. nlc&oo",.0I:0 / ,' -0.1 \.. . -- -0.2 ..... measurement ",k . ; -- -- undisturbed incident wave -0.3 linear solution (Ogilvie 1963) -- 2nd order solulions: Vada (1987), Riley & Yan (1996) 0.2 ) ...... Yan (998) 0.1 IRiley& -ow0.0-.N / -... " -0.1 -0.2 -I0 0 10 20 x/c Figure 8. Instantaneous water surface profiles at the instant at which the undisturbed wave would have a zero down-crossing at the location of the cylinder (x = 0): kc = 0.56, h/c = 1.5. In (a), for an incident wave amplitude A/c of 0.16 (at which the waves break over the cylinder), the measured water surface elevation is compared with linear and second order solutions. In (b) correspondingresults are shown for A/c = 0.107 (at which no breaking occurs), and are compared with Riley & Yan's second order solution. Measurements are from Chaplin (1998). 4 kc- 0.1, h/c 4 '• . ' , 2y i y 01) oi 0C 0 2 A/c Figure 9. Phase lag at the fundamentalfrequency; the contribution due to circulation around tile cylinder is denoted y, and is proportionalto (A/c)2. Measurementsfrom Chaplin (1998). 10 11. CONCLUSIONS These notes summarise some of the-elements of the rather complicated flow around a horizontal cylinder beneath waves aligned with its axis parallel to the wave crests. It is seen that in the regime of concern here (kc<4, h/c<4, K< 1) no reliable predictions have yet- been made of the loading on the cylinder. In conditions that are traditionally associated with the diffraction regime, viscosity is found to be very important, and the flow is inherently three-dimensional. REFERENCES Batchelor G K, 1967 'An introduction to fluid dynamics', Cambridge University Press. Chaplin J R (1984) 'Non-linear forces on a horizontal cylinder beneath waves' J. Fluid Mech. 147, 449-464 Chaplin I R (1993) 'Orbital flow around a circular cylinder: part 2 - attached flow at larger amplitudes' I Fluid Mech., 246, 397-418. Chaplin I R, 1998, 'Non-linear wave interactions with a submerged horizontal cylinder', paper submitted for publication. Chaplin J R and Retzler C H, 1997, 'Orbital flow about a submerged cylinder', Final report on EPSRC Research Grant GR/J54031: OFF 135. Cokelet E D, 1977 'Steep gravity waves in water of arbitrary uniform depth' Phil. Trans. Roy. Soc. Lond. A286 183-230 Y, Liu Dommermuth D G and Yue D K P, 1992 'A high-order spectral method for nonlinear wave-body interactions' I Fluid Mech 245 115-136. Mclver M and Mclver P, 1990, 'Second-order wave diffraction by a submerged circular cylinder' J.Fluid Mech 219 519-529. T Ogilvie F (1963) 'First- and second-order forces on a cylinder submerged under a free surface', J.Fluid Mech., 16, 45 1-472. Otsuka K and Ikeda Y (1996) 'Estimation of inertia forces on a horizontal circular cylinder in regular and irregular waves at low Keulegan Carpenter numbers', Applied Ocean Research, 18, 145-156. Palm E, 1991 'Nonlinear wave reflection from a submerged circular cylinder. J Fluid Mech 233 49-63 Riley N and Yan B (1996) 'Inviscid flow around a submerged circular cylinder induced by free-surface travelling waves' J Engrg Math, 30, 587-601. Riley N and Yan B (1998) Private communication. Schwartz L W, 1974 'Computer extension and analytic continuation of Stokes' expansion for gravity waves' J Fluid Mech 62 553-578 T, Vada 1987 'A numerical solution of the second-order wave-diffraction problem for a submerged cylinder of arbitrary shape' J Fluid Mech 174 23-37 Wu G X, 1991 'On tie second order reflection and transmission by a horizontal cylinder' Applied Ocean Research 13(2) 58-62. Yan B and Riley N (1996) 'Boundary-layer flow around a submerged circular cylinder induced by free-surface travelling waves' J Fluid Mech., 316, 24 1-257. II Transformation of non-linear waves in shallow water and impact on coastal structures Pr Stephan GRILLI, University of Rhode Island, USA Depth inversion in shallow water based on properties of nonlinear shoaling waves Stephan T. Grilli* (*) Dept. of Ocean Engng., University of Rhode Island, Narragansett, RI 02882 (http://www.oce.uri.edu/,-'grilli). ; i" " ...... : ...... i...... .~~~....:.. .'. . ,..::.••...... 4 0.1- Sponsor (1994-97) US Naval Research Laboratory (SSC), remote sensing division (Dr. Peter Smith) Goals of the project * Calculate fully nonlinearproperties of (2D) shoaling waves on cylindrical beaches as a function of depth h(x) and incident wave characteristics (Ho, T), up to the BP : e.g., H(h), c(h), sk(h),... * Compare results with predictions of approximate wave theories (Stokes lst,3rd; Fourier Steady Wave theory; Boussinesq...). * Identify wave properties useful for depth inversion and develop Depth Inversion Algorithms (DIAs) based on these. " Calculate small scale surfacefeatures of highly nonlinear waves (not yet addressed). Methodology e Simulations are made using a Numerical Wave Tank (NWT) solv- ing Fully Nonlinear Potential Flow (FNPF) eqs., and equipped with wave generation and absorption (9 years of successive de- velopments/refinements). " Present studies only address periodic waves on monotonously decreasing depth or over barred-beaches. The method, however, is general and wave groups or irregular waves will be studied later, as well as more irregular topographies. * Extension to 3D is a matter of computational efficiency (LONG- TANK, LPA, hybrid,...). Components of NWT (Grilli et al., 1989-1997) z/h (a) 0.2 0 -0 .2 . .... H...... ---- I.... -0.4 h -0.6 . -0.8 ... .-...... _ - I•----i" •...... i...... 0 10 20 30 40 50 60 70 80 90 0 z/h zo (b) ...... •"t ...... • ..t ...... -0.20 ••...... ' I -0.1 iI H ...... - ' E,rT. . . -0.2I•' ...... "' " " :'-" ' '' ...... -03 ...... -...... -0.43 62 63 64 65 66 67 68 69 70 o * Solution of.continuity equation in Q(t) by a higher-order BEM; nonlinearfree surface boundary conditions are time updated using an explicit 3rd-order Eulerian-Lagrangian scheme. Adaptive regriddingon Pf (Grilli and Subramanya, 1994,1996). * Wave generation : Streamfunction Waves (SFW) with zero-mass- flux, at ~rl (Grilli and Horrillo, 1996,1997). " Wave absorption: in an Absorbing Beach (AB) with : (i) free surface absorption (Cointe, 1990); and active absorption at 17r2 (Clement, 1996). Adaptive calibration of AB to absorb incident wave energy at x = x1 (Grilli and Horrillo, 1996,1997). Principle of Depth Inversion Algorithms (DIAs) z/h o~ (a) -0. 2 - ...... L...... I ...... q... -0.2 -0. -0.6 -0.8 . -1~ 0 10 20 30 40 50 60 70 80 90 z/h z o (b) 0 .1 -• ' -0.2 -0.3-0.4 h ý.._ ...... 62 63 64 65 66 6'7 68 6'9 70 * Principle of depth inversion "use measured wave celerity c and a wave theory (dispersion relationship) to calculate Pt * Most state-of-the-art DIAs use linear wave theory (LWT) or first- order NSW theory to process phase information calculated using a MTF (modulation of the slope of scatterers) from remote sensing data (e.g., Duncan et a]., 1997) or video imaging methods (e.g., Lippmann and Holman, 1989). * Wave nonlinearity, however, increases c (amplitude dispersion function of H) -* errors on depth prediction with LWT may readh 80%. For more accuracy, wave height information is needed (difficult to measure; SAR). Validation of NWT Earlier work shows the NWT accurately models propagation/shoaling of highly nonlinear waves over arbitrary topography : * Solitary wave shoaling and breaking over slopes (Grilli et al., 1994,1997) : computations are within 2% of experimental mea- surements up to breaking (cf. Dommermuth et al., 1988, for deep water waves). e SFWpropagation over constant depth : many cases with long or short waves (Grilli and Horrillo, 1997). * Shoaling over a bar : Experimental set-up of Ohyama et al. (1994), with T' = Tjg/-/h = 9.9 and H, = Ho/h = 0.05. z/h 0.5 ...... 0 .52 . .. 0 ...... ,...... * xi - o.5 100 0 10 20 30 40 5 0' 60 70 o Solitary waves (Grilli et al., 1994,1997.) T11 h(a 05 -0 .5 ...... -...... 3 .5...... -1 - ~ -x/h 29 30 3'1 32 33 34 35 36 37 38 39 0. 36 36.5 37 ~ 37.5 3*8 0 0.4 If-n 0.3l 0.2 -0.1-- 37 38 39 40 41 42 43 44 45 46 47 Deep water periodic wave over constant depth (Grilli and Horrillo,1997) 0.21 -0 .1 ...... 05 1 0 15 20 2 0.04 -0.04 0 20 40 60 80 100 120 140 160 R(%C) 4 3 ...... -...... 2b 60 70 80 90 -100 110 120 130 140 150 Intermediate water periodic wave over constant depth (Grilli and Horrillo,1997) T/ h 0.4 ( 0 .2..0 ...... M ...... -0...... i...... • ...... •...... -0.2 -1 0 10 20 30 40 50 60 70 80 90 400 110 120 130 17/ h (e) 0.2 ...... 0.. - ...... i...... i...... 0.1 0 -0 .1 ...... ' ' ' '' ' t 50 100 150 200 250 300 350 400 R(%) (f) 6 10 .... 0. 200...... 250...... 51-1 30 3 ...... 3 0 ...... 4 00...... Periodic wave over a bar: (Grilli and Horrillo,1998) z/h 0. 0 0 l~x/h o 10 20 30 40O O 1.5 -j 0 0.5 1 1.5 2 Periodic wave over a bar: (Grilli and Hor~rillo,1998) z/h 0.5 - 0 0 10 2'0 30 40 50 60 70 71 0.5 - -0.5 L tt ~ 0 0.511. 77 (d) 1.5 j 0 00.5 1 1.5 2 Calibration of NWT Results sensitivity to beach location z/h 0 0 T r be a ch ga g e s H , T' . 5 a d t h 0 ' ...... -0 .2 .-...... _...... V...... i...... _ ,...... A P...... -.0 ...... L.... ...... - .,.""...... -. ,...... 0 5 10 15 20 25 30 35 40 45 * Three beach locations for H" = 0.04, T' =6.5, at depth hiI 0.07, h2 = 0.10, and h3 = 0.15. * Surface elevation as function of x' and t' (at 10-12 gages). * Wave by wave envelope analysis, local properties as function of x•': Ks,c/co, S2/Si. Surface elevation at gages 1, 5, 7, 9, 10 0.06 0.04 0.002 ,0 -0.02 60 70 80 go 160 1 0 120 0.08- 0.,02 - 0 . . -0.02 ...... A...... -0.04 60 7'0 80 90 100 110 120 0.06- 0 .04 ...... 0.,02 --. . 0 - 0 .0 2 ...... -0.04...... 60 70 80 90 100 110 120 Wave by wave envelope analysis, local properties KK a (a) 2 .2 ...... t ...... 2 .0...... 2. 7 ...... '•...... 1.81.6 Fi...... +...... -...... 1.6•...... ! ...... +...... +...... - 1.6 1. 2 ...... +...... + ...... 1.21. 0 ...... •••...... -...... +...... =...... 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 C/C (b) o.7 .o. .I...... I.... [. . . .I. . 0.605 : i i 0.5 0 ...... " 0.55 0.350 03 0 . .~ '~...... '....M- ...... I , , , l , , 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 2 S (c) , , I...... I ...... I..... 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Calibration of NWT Result sensitivity to discretization e Spatial and temporal discretizations (Ax, At) are selected to en- sure global accuracy in NWT. Convergence ; refinement vs. small scale features, for highly nonlinear waves. * Two discretizations for waves shoaling over a slope, with mini- mum L/Ax = (- ) 40;( )16. e Two cases with: (H'o = 0.04, T = 6.5, s =1:35) or (H o = 0.06, T' = 5.5, s =1:50). * Surface elevation as function of x' at T'/4 intervals. Case 1: H,,=0O.04,T' =6..5,s =1:35 '7 (a) 0 .1 ...... 0 0 51. ,8 1,9 20 211 - - 212 017 0.05 1,8 19 20 21 2 2 0.1 -0.105__ _ _ 18 19 20 21 22 18 1,9 2'0 ý 21 2 2 Case 2: H, z=0.06, T' 5.5. s =1:50 0.1 7 -0.05 0.1 252 7829 30 0.1 A____ -0.05- 25i6i282 9 30 0.05 -0.05- 22622829 30 Shoaling of periodic waves over a 1:50 slope z/h 0 0 .2 -: ...... c).4 .... ý ý :- .. ' -I 0 AP'- .6 ...... 0. 68 ...... 0...... b...... - ...... •< -...... A,...... ' i - -0 .8 ...... A B..>... -1 - I''' 'II II'' ' ' ' ' I' I ' 'I I '' I - / 0 10 20 30 40 50 60 o * Local wave properties for H" =0.04, 0.06, and 0.08, and T' = 5.5, 6.5, 7.5; effect of increased nonlinearity. 9 Wave by wave envelope analysis of local properties as function of': K 5 1,C/C o ,S 2 /SI. * Comparison with wave theories linear, cubic, FSWT (Sobey and Bando, 1991). Local shoaling wave -properties on plane 1:50 slope C/C 0.7- 0.2kh 0.1 0.2 03.40.5 K= HI-H S 0 2.4- 22 9 3 0.8 1 L - I- - kh 0.1 0.2 0.3 0.4 0.5 S/S 2 1 7,,.I F 02 0 . 0.5 .. 0 .6 .. 0.7. Effect of wave height for a 1:50 slope (T' =5.5). Theories Ks (a) 180- 1 ...... 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 C/C0 (b) 0.60 0 .4 0 ...... 0.35 - 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 S2 /SI(C) 5 - 2~ h. 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Depth Inversion Algorithms Monotonous slopes * Assume c(x) = Axp/At and L(x) = (L 1 + L 2 )/2 are available from phase measurements for N points. Wave period T is the mean of L/c values (and k. = (27r) 2/gT 2). * Use empirical curve fit to fully nonlinear computations for 9 cases of wave shoaling over plane 1:50 slope (exact slope does not matter for a mild monotonous slope between 1:35 and 1:100; Grilli and Horrillo, 1997) : C co f(koh, koHo) c/C 0 0 .7 .. 0.6 ...... -...... -...... - ...... 00.5-...... -...... 0 .3 ;• ...... -...... -..._...... _...... • 0 .2 "...... 0.1 kh 0 0.1 0.2 0.3 0.4 0.5 " Estimate koHo from empirical fit to computations, using : (i) DIAl : H(x) information; (ii) DIA2: wave asymmetry informa- tion s2/sI(x) •_L 1/L 2. * Invert c relationship to get h for cases with more complex waves and topography than initial computations. Depth Inversion Algorithms (contd.) * DIAl : Uses wave height variations in numerical wave tank (g, is linear wave steepness; dashed line in figure) H = Ho(c/c 0)gi(koh, koHo) H1c/H c o 0 8 . . . I...... I..... 7 - 6 ...... -_ 5 ...... M...... 66 2~ ...... -...... 3 3 ...... 2 --...... ,...... 1 - k 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 * Assuming wave height data H(x) is available, T . L(x)/€c(x)is first calculated then (ko, co), and Ho is iteratively calculated using both c(x), with f(koh, koHo),as Ho _ H(x)/(c(x)gl (koh, koHo)/co). Depth Inversion Algorithms (contd.) * DIA2 : Uses spatial wave asymmetry variations in numerical wave tank : S2/SiI Li/L 2 g2(kh, koHo) S/s 2 I 7 . 6 65...... i ...... - . - ...... 5 ...... -...... 7 3 . . . I...... -...... 0.2 0.3 0.4 0.5 0.6 0.7 S/S 2 1 7 - ...... IrI...... I ...... 4 : ...... 5...d- ...... :...... •...... - __ 5 ...... 2-...... i...... i...... 3 1- • kh 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 * Assuming locations of phase lines for successive crests and troughs are available (as in DIAl), T -- L(x)/c(x) is first calculated then (ko, co), and Ho is iteratively calculated using both c(x), with f(koh, hoHo) and 92(kh, k-Ho). Validation of DIAs : Natural slope cases * Depth inversion for a Dean's equilibrium beach with average slope 1:50, H'o = 0.06 and T' = (a) ni 5.5; (b) n2 6.5; (c) n3 7.5. * ( ) True bottom topography h(x); (------) estimated topography with DIA 1; (- - -) estimated topography with linear dispersion relation. ho is a reference depth scale. h/hoh/ (a) -0.1 - 0. 1 ...... 0 .5 ...... i!...... - .. .. h/h 10 15: 20 25... 30... 10 15 20 25 30 h/h * (b) -0 .1 , I . . . I. . 0.1 0.3 1 0 0.6 ,i x/h 1 0 is 20 25 30 0 h/h, -0.2 - ,J . . -0.1 ! 0...... ? ...... :...... , . . .. 0.1 0.6 ix/h 10 is 20 25 30 (a) 0.8 0.6 ...... 0.4...... •...... •...... 0 4....4 ...... : ...... 0...... T...... H -.-...... i ...... ! ...... ' -0-.4 .2...... i...... hlh...... i ...... -0.6 ~ ~~~ . , , , :,• , ,,, , ...... _.... -0.4 kh 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 -0.2 iii_ , 0.8 I 0.6 _•• ~~~...... C/ C ,...... i ...... i...... 0.2 - ...... _: ....-...... - .'-...... 0...... 7-...... -...... • ...... ! 0.2 ...... -...... (c) .....- . ...------...... -0 .2 ...... 0. 6 kh 0.8 0.7 0.6 0.5 0.4 0 .3 0.2 0 (C) 0.8 , . . . I 0~ ...... ~6 ...... -- C ...... 0 .46 ...... " ...... -0.6 0.Depth ( ...... -0. .a...... vh ...... i ...... - -0.6 H/0.cc;2--- esiae ,, value wit DI~l ' -- kh 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Depth inversion cases :.(a) nl; (b) n2; (c) n3. ( - ) True h, H/Ho, c/co; ( -----..- ) estimated values with DIAl; (--- -- ) estimated values with linear dispersion relation. Validation of DIAs : Broken slope cases * Depth inversion cases (H', = 0.035;T' = 11.47): (a) bi (1:50, 1:70, 1:35) (-); (b) b2 (1:50, 1:100, 1:35) (-). S(- ) True bottom topography h(x); (-- ---) estimated topography with DIA 1; (- - -) estimated topography with linear dispersion relation. ho is a reference depth scale. h/h o (a) -0.1 0 . 0...... 0.2 ...... : ...... 7 ...... 7 0.4 0.3 ...... •...... • , ...... ,...... --....--- ' .' • -...... - 0 .5 - ...... i...... 0.30.1. .. . 0.2 10 15i" 20 25 30 0 h/h o ~(b.) -0 .2 - ,! , . -0 . 1 ...... 0.4 ...... _...... i...... i...... 0 .5 ...... 10 15 20 25 30° (a) 0.4 0 .2 ...... - ...... o -0.2 ...... i...... •...... , ' ...... i i...... - . ".. -0 .4 -0.4...... ''..... r ...... -0.6 kh 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 (b) 0.2 .. . - /c.,c! ...... 0 .2 0...... 3 5 0. 0. 0 2 -0.2 -- -) e I e v a l u e -0.... .2 ...... w .. disp rsio rel tio . . -0. 4 ...... -h•;,...... i...... -0.6 ' , . .' 0.45 0 --r "'----'L'--- . .. . k h 0.4 0.35 0.3 0.25 0.2 0.15 0.1 Depth inversion cases : (a) bl; (b) b2. (- ) True h, H/Ho,,,C/Co; ------.) estimated values with DIMl; ( - -) estimated values with linear dispersion relation. Validation of DIAs : Error analysis Data for validation cases. Errors for DIAl Case H,1: T' I."IT, . IrI't, Em ir'1SJ"I nl 0.0653 5.5 0.06 5.52 0.0646 2.1 8.0 14.6 50.3 0.6 2.7 n2 0.0636 6.5 0.06 6.52 0.0625 1.7 7.1 17.0 57.6 0.5 2.0 n3 0.0614 7.5 0.06 7.51 0.0602 2.0 13.9 20.5 67.9 0.4 2.0 bI 0.0352 11.47 0.04 11.55 0.0349 3.6 17.2 15.6 47.6 1.2 2.8 b2 0.0352 11.47 0.04 11.50 0.0350 2.7 8.6 16.2 50.8 1.1 3.8 Errors for DIA2, rms : 2.1, 3.4, 3.2% (n1, n2, n3); 5.7, 3.9% (bl, b2) Relative error on depth with: DIAl ( . .); (- - -) LWT E (a) -20 - I I I . 0 ... - .. - 4 0 -...... " - a 60 b, 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 E (b) -20- 1 0i : ------• a" - b 2 0 ...... - ...... i ...... ! ~~~~~~~...... ' . ...." . ':...... " " ' ' ...... % b " 4 0 ...... !...... -...... 60 ...... - ...... -...... c 80 'kh 0.8 0.6 0.4 0.2 Wave shoaling over barred-beaches z/h 0. 2T -0. -0.5 -I ...... X _. 20...... 21.2.2.2.25 2 ..7.8.2.3 0.0215 2. 5 0 35 4 0. 00 2021 ~ 2.33405~ 5 0. 79 528 9 3 0.162 65 % W 05 0.08~jj .....~ 1 ....7- ...... 7 0...... 0.... 4 ...... 1.. ... Case 1 : H,' =.06, T' =5.5 z/h 0.1 0 i/h ...... o.0 .... (b)...... z/h (b) 0.1 0 . 26 2 ...... 4 0 0- ...... 14 16 18 20 22 24 26 28 30 Case 2: Ho =0.061 T' 6.5 znh 0.08 0 .0 6 ...... 0.04 ...... -0.02 ... .. -0.04 16 18 2i0 22 1ý 26 ~ zIP, 0.06 0.02.... -0 .0 2 E I I ~ zib o. (c) 0.06 0.04 0.02 -0.02 2a/H 0 .8 ...... 0.6 ...... I -..... 0.4 Case 3: H' =0.061 T' = 7.5 00 z/h 0 (a) 0.1 1 0 28 /.H ...... 0. . .0.. 6 ...... * Wave height variation H/H 12 0.8 x/h 1 4 16 18 20 22 2'4 26 28 H/H 16 1 146 H/H (C) 2 0.8JI h 14 015 028 025 0. 0.3 04 045 058 Wave asymmetry variation 2sf/a) 100 2 4 ...... (.... 0 0~ 0' 1'6 1 8 210 2i2 2 4 2 6 x/h0 2st5 10- 8 ...... h7 6 ...... x/h 14 1 6 1 8 2 0 22 24 26 .28 0 Wave celerity variation C/ (a) 0.65 1IH 0.55 ...... 0 .5 ...... 0 .4 5...... I...... 0 .4 ...... 0.35 / 0.65 0.45 ...... 0.55k 0.5 01 0165 02 0.2 0.3 0.3 0. 0.4 30.5 Wave nonlinearity parameters H/h (a) 1.2- .T I 1 L 2?- 0 .8...... i...... T ...... I ...... 0 .6 ...... -...... - :' : • ' ...... - 0 .204 . '...... -...... I.. - ...... 14 16 18 20 22 24 26 28 0 0 0.18 0 kH/2 (b) 0 .14 --...... ! ...... i...... 0 .1 2 ...... 0. 12 _- ...... i...... i...... i...... "...... --.. ,. -i...... :.. - 0-'-...... Y...... i...... "...... ''...... '.".7" ' 0.02-0.60 _ = -...... 0 "-...... I ...... ;...... •., ...... / 0.06 - 0 .04 I ' '. I ' ...... ''...... 14 16 "18 20 22 24 26 28 30 0 kh 12 ...... 0.8 -...... i...... i...... 0 .6 ...... :1-2h...... 0 .4 - -: ...... "" ...... 0.2 ...... '...... - ...... I... .. '.... '.. .. 0 1820 14 16 18 20 22 24 26 28 30 Application of DIAl to depth prediction (cases 1,2,3) h/h hlhý (a) 0.1 0.2. ... C -.. 0.3 " . 0.4 1 0.5 -- 0.6...... ' . x/x 14 16 18 20 22 24 26 28 30 h/'h o(b)...... !...... 0.1 . . 0.2 . 0.3 0.4 0.5 0.6 ' "Zt Xix 14 16 1 8 20 22 24 26 28 30 "-(1 2 3 30 Ih Depth Inversion Algorithms: conclusions * In strongly nonlinear situations (e.g., close to breaking), the FNPF-NWT provides a numerically exact solution to the wave transformation problem in arbitrary bottom geometry. Zero-mass-flux SFW's provide clean incident waves in deep and intermediate water, for arbitrary nonlinearity. The absorbing beach (AB/AP) leads to small reflection. Reflec- tion from the AB, however, could be specified to an arbitrary value. e Nonlinear shoaling wave properties are obtained in the NWT. Their comparison with results from linear,weakly-nonlinear, and Fouriertheories allows to define ranges of applicabilityand lim- itations of these theories. For monotonous mild slopes (1:35 - 1:100), detailed bottom ge- ometry is not a major factor if k~h is used -- only h(x) matters. For barred-beaches,however, local wave properties significantly change over and beyond the bar berm (modulation region). Parameters useful for depth inversion are calculated : celerity c, asymmetry s 2/s 1 , steepness kH -* a combination of crest height and celerity measurements plus relative locations of next and pre- vious troughs (s1 , s2) -- good prediction of h(x) for monotonous slopes with DIAl, DIA2. * For barred-beaches, wave properties become multiple-valued in the modulation region. Hence, DIA l does not work well in the modulation region. New methods are needed. Depth Inversion Algorithms : future work * The data base of computed cases in the FNPF-NWT should be enriched and the empirical relationships improved. Parametric results for barred-beaches should be obtained and analyzed to derive specific DIAs for the modulation region. * Cases with mildly irregularincident waves such as narrow-banded groups should be addressed. A localized spilling breaker model should be added. 9 High-resolutioncomputations should be made to provide detailed wave shape close to breaking to remote sensing modelers. No other method using approximate wave theories can provide such results (e.g., Boussinesq equations). * Three-dimensional effects should be introduced. Components of NWT (Grilli et al., 1989-1996) z/h -0 .020 h...... ' ...... :L -0.4 ...•. .r ...... i...... i...... •...... ' -F -0.86-0.4 J ...... -* -.0 ...... ,,...... zAB .L .+...... - - 1 '...... -1.2 ,-,-I ' ' ' I ' x/h 0 10 20 30 40 50 60 0 * Solution-of continuity equation in Q(t) by a higher-order BEM; nonlinearfree surface boundary conditions are time updated using an explicit Eulerian-Lagrangian scheme. Adaptive regridding on 1f (Grilli and Subramanya, 1994,1996). * Wave generation: Streamfunction Waves (SFW) with zero-mass- flux, at rli (Grilli and Horrillo, 1996,1997). * Wave absorption : in an Absorbing Beach (AB) with : (i) free surface absorption (Cointe, 1990); and active absorption at Q7,2 (Clement, 1996). Adaptive calibration of AB to absorb incident wave energy at x = xI (Grilli and Horrillo, 1996,1997). Conclusions e In strongly nonlinear situations (e.g., close to breaking), the FNPF-NWT provides a numerically exact solution to the wave transformation problem in arbitrary bottom geometry. Zero-mass-flux SFW's provide clean incident waves in deep and intermediate water, for arbitrary nonlinearity. The absorbing beach (AB/AP) leads to small reflection. Reflec- tion from the AB, however, could be specified to an arbitrary value. * Shoaling wave properties (local and integral) are obtained in the NWT. Their comparison with results from linear, weakly- nonlinear,and Fouriertheories allows to define ranges of appli- cability and limitations of these theories. Detailed bottom geometry is not a major factor within a range of mild slopes (1:35 - 1:70) ===> only h(x) matters. Parameters useful for depth inversion are calculated : celerity c, asymmetry s2/s1 , steepness kH - - - > a combination of crest height and celerity measurements plus relative locations of next and previous troughs (s1 , s2) --- > prediction of h(x). * Other types of incident waves and more complex bottom topogra- phy can be used. Small scale geometric features can be calculated. WEGEMT Supported by the European Commission through the c/o The Royal Institution of Naval Architects Training and Mobility of Researches (TMR) Programme 10 Upper Belgrave Street LONDON SW1X 8BQ UK Tel: +44 (0) 20 7838 91497'M Tax: +44 (0) 20 7838 9147 http://www.wegemt~org.uk T~a9ing/- r- and Mobility of Researchers