E L S E V I E R
P I I ; 8 0 1 4 1 - 1 1 8 7 ( 9 7 ) O O O J O - 2
Applied Ocean Researcii 19 (1997) 35-47 © 1997 Elsevier Science Limited All riglils reserved. Printed in Greal Britain 0141-1187/97/$17.00
Numerical and experimental study of
large amplitude motions of two-dimensional
bodies in waves
N. Fonseca^ C . Guedes Soares'''* & A. Incecik''
''Unit of Marine Technology and Engineering, Technical University of Lisbon, Instititto Superior Tecnico, Av. Rovisco Pais, 1096 Lisboa, Portugal
^Department of Naval Architecture and Ocean Engineering, Hydrodynamic Laboratoiy, University of Glasgow, Acre Road, Glasgow G20 OTL, UK
(Received for publication 22 March 1997)
The hydrodynamic forces and motions of floating prismatic bodies with arbitrary cross section subjected to beam waves are studied in the time domain. The hydrodynamic radiation forces are represented by infinite frequency added masses and convolution integrals of memory functions. The exciting forces are calculated in the frequency domain. The hydrostatic forces are evaluated both around the mean equilibrium position and over the instantaneous wetted surface in order to assess the effect of the non-linearity of these forces. Results of heave and roll motions are presented for a prismatic body with a triangular cross section shape. Comparisons are made between the solutions for linear and non-linear hydrostatic forces, and between the numerical and experimental results. © 1997 Elsevier Science Limited
1 I N T R O D U C T I O N
Tlie problem of floating body motions i n waves have been extensively studied i n the frequency domain. However this approach is meaningful only i f the oscillatory responses are linear. On the other hand, time domain formulations are more general since they are able to deal with arbitrary motions. This is important, f o r example, when studying the non-linear responses o f floating bodies subjected to incoming waves.
Finkelstein [1] and Stoker [2] were the first to discuss the time-domain direct solution o f hydrodynamic problems. These authors applied the f o r m u l a t i o n to the problem o f generated water waves. The use o f time domain analysis to solve the unsteady ship m o t i o n problem was initiated by Cummins [ 3 ] , w h o represented the hydrodynamic forces f o r arbitrary motions i n terms o f convolution inte-grals o f impulse response functions. The f o r m e r impulse response functions are independent of the body m o t i o n . O g i l v i e [4] studied the problem by the same approach and related the time domain m o t i o n equations w i t h the classical frequency domain equations using Fourier analysis.
Indirect solutions o f the two-dimensional unsteady motion problem were obtained by Ursell [5] and Maskell and Ursell
*Author to whom correspondence should be addressed.
[6] f o r a semi-circular floating cylinder, using Fourier analysis o f frequency domain solutions. Two-dimensional direct solu-tions i n time-domain were presented by Ikebuchi [ 7 ] , f o r the hydrodynamic forces o n a body oscillating i n the free-surface, and Yeung [8] f o r the transient heaving motion o f floating cylinders. B y this direct solution the integral equations apphed to the body surface are solved i n time-domain to obtain the velocity potential or source strength, satisfying simulta-neously the body condition and free surface condition.
A different approach was introduced by Longuet-Higgins and Cokelet [9] applied to the wave breaking problem. Here a Lagrangian time-stepping method was used, w h i c h can treat the exact free-surface and body boundary conditions at every instant. Later this has been extended to the wave-body interaction problems by Faltinsen [ 1 0 ] , V i n j e and B r e v i g [ 1 1 ] , V i n j e et al. [12] and by L i n et al. [ 1 3 ] . This method seems to be the most general one, but i t takes very much computation time i n the numerical solution and some numerical difficulties were f o u n d .
Chapman [14] proposed a method where the solution is generated i n terms o f a source distribution representation o f the body wetted surface, and a spectral representation o f the linear free-surface. This method is simpler than the fonner, but, even so, l o n g run times are expected i f one wants to use the exact body wetted surface at every instant.
Herein a twodimensional nonlinear time domain f o r m u -lation is compared w i t h the linear f o r m u l a t i o n and w i t h experimental results. The problem considered is a floating prismatic body w i t h arbitrary cross section shape and oscillating under the action o f i n c o m i n g beam waves. The theory is based on the linear potential flow f o r m u l a t i o n and a linearized free surface and body boundary conditions are used. The radiation forces are formulated i n the time domain and then related to the frequency domain solutions by means of Fourier transforms. The exciting forces are evaluated assuming sinusoidal i n c o m i n g waves acting upon the restrained body. The hydrostatic forces have a non-linear characteristic since they are evaluated over the "exact" wetted surface taking into account the contribution f r o m the linear incident, the d i f f r a c t e d , and the radiated wave.
The present time domain method is not as general as some o f those developed by other authors, but, w h i l e i t can deal w i t h some non-linearities, i t has the advantages o f being relatively simple to implement and the computa-tional e f f o r t is small. These advantages are important when the method is generalised to solve the much more complex problem of ship motions i n waves. W i t h this purpose a strip theory approach can be used as indicated i n Fonseca and Guedes Soares [15].
This w o r k is aimed at a more general research programme intended to study mainly the wave induced vertical motions and loads i n ships. There is the belief, w h i c h is shai-ed by some authors, that the main component o f the non-linearity i n the case o f the loads associated w i t h vertical motions is associated w i t h the hydrostatic forces. Therefore, this w o r k has been developed w i t h this perspective i n mind, being aware that only one aspect o f the non-linearity o f the motions is being accounted f o r by the theory. However, this l i m i t a t i o n has the important consequence o f allowing a computationally simple solution.
The assessment of the accuracy o f the present method can be made i n various ways one o f w h i c h could be to compare w i t h the results o f methods based on more complete formulations. Another approach, w h i c h was the one adopted
i n this w o r k was to compare the results w i t h model experiments.
A detailed presentation o f the f o r m u l a t i o n and some numerical results f o r prismatic bodies with rectangular and triangular cross section shapes were presented i n Fon-seca and Guedes Soares [16] and FonFon-seca et al. [17]. I n this w o r k additional results are presented as well as comparisons against experimentally obtained data. This paper also describes the experimental w o r k program that was carried out at the Hydrodynamic Laboratory o f the University o f Glasgow, where regular incident waves were imposed on a prismatic body o f triangular section f r e e to sway, heave and r o l l . The results are compared w i t h the theoretical predic-tions of both the linear and non-linear models.
The experimental program that was conducted provided information to other motion components i n addition to the ones that were o f interest to compare w i t h the present the-oretical results and thus are o f interest on their o w n since they can be used to check other methods developed to describe the other phenomena also measured.
2 L I N E A R P O T E N T I A L F L O W F O R M U L A T I O N T w o coordinate systems w i l l be defined, the first, X = (y,z), is fixed w i t h respect to the mean position o f the body, w i t h z i n the vertical upward direction and passing through the centre of gravity. The origin is i n the plane o f the undis-turbed free-surface. The other, the body axis system, X' — (y',z'), is fixed w i t h respect to the body.
The prism subjected to the incoming waves w i l l oscillate i n thi'ee degrees o f freedom. Namely it w i l l have two translatory motions i n the y and z directions, respectively, the sway (^2) and heave ( ^ 3 ) , and one angular m o t i o n i n the y-z plane, the roll motion ( ^ 4 ) . The reference systems and the sign conven-tion f o r the displacements are represented i n F i g . 1.
The viscous effects are neglected, i m p l y i n g that the fluid m o t i o n is irrotational and the problem is formulated i n terms of potential flow theory. This means that the fluid velocity vector may be represented b y the gradient o f a velocity
Heave
y Sway
Numerical and experimental study of waves 37
potential:
(1)
I n addition, i n the fluid the Laplace equation must be verified:
= 0 (2)
The velocity potential arises f r o m the contribution o f several phenomena. One results f r o m the incident waves, where the associated potential is described b y the gravity wave theory. The presence o f the body creates a perturba-tion on the incident wave field, w h i c h is accounted by the d i f f r a c t e d potential. Finally, the oscillatory body motions are also responsible f o r motions o f the fluid and these are described by the radiation potential.
I n order to linearize the oscillatoiy potential both the amplitudes o f the incoming waves and unsteady motions must be smafl enough. Thus the fluid velocity potential, caused by the incident, d i f f r a c t e d and radiated wave systems, can be decomposed as:
$ ö ' , z , 0 = * ' + * ° + X ^ f i = 2 , 3 , 4 (3) y = 2
where $ ' and $ ° are, respectively, the incident and d i f -fracted potentials, and is the radiated potential due to an unsteady motion i n the j - m o d e .
T o find the radiated and d i f f r a c t e d potentials satisfying the Laplace equation i n the fluid domain, t w o boundary conditions are used. Based on the assumptions o f small amplitudes o f the incident wave and oscillatory motions, the Idnematic body boundary condition and the free surface boundary condition are linearized. I n this way the radiated and d i f f r a c t e d potentials must satisfy the free surface bound-ary condition: 2.T.R,D dt' + 1 0 * R.D dz = 0
and the kinematic body boundary condition
dn ' dn '' • ^jiij, on So d^ dn' on Sa (4) (5) (6)
where represents the oscillatory m o t i o n i n the y-mode, the dots over the symbols represent d i f f e r e n t i a t i o n w i t h respect to the time variable, g is the gravity acceleration,
Hj are the generalised unit normal vector components
defined as n = (/!2,n3) and >• X n = n4, n being the unit vector normal to the mean wetted surface, SQ, and p o i n t i n g out to the fluid, r is the position vector represented i n the body axis system.
I n addition, at i n f i n i t y the fluid velocities associated w i t h d i f f r a c t i o n and radiation must tend to zero such that;
V^D''^ — 0 as (7)
The incident wave potential, by definition, satisfies the Laplace equation, the free surface condition, and the con-dition at sea bottom.
The hydrodynamic pressure can be evaluated substitut-ing the velocity potential i n the linearized B e r n o u l l i equation:
p dt (8)
where p is the fluid pressure, is the atmospheric pressure, and p is the fluid density.
Integration o f the pressure over the body surface gives the hydrodynamic force:
nt ds
ds (9)
where, w i t h i n the linear approach to the problem, integra-tions o f quantities related w i t h the oscillatoiy potentials are taken over the mean wetted surface {SQ]). The terms on the right side o f eqn (9) represent respectively exciting, radia-tion and hydrostatic forces.
I n order to implement a time domain method all the forces i n the m o t i o n equations must be represented i n the time-domain. T h i s brings no m a j o r difficulties i n the evalua-tion o f the exciting force, since, f o r the present linearized model, these forces do not have the time dependency o f the previous history o f the fluid m o t i o n .
However the radiation forces behave i n a d i f f e r e n t man-ner. The existence o f radiated waves implies a complicated time dependence o f the fluid m o t i o n and hence o f the pressure forces. Waves generated by the body at t i m e t w i l l persist, i n principle, f o r an infinite time thereafter, as w e l l as the associated pressure force on the body surface. This problem can be described mathematically by a con-v o l u t i o n integral, w i t h the fluid motion and pressure force at a given time being dependent on the previous history o f the m o t i o n .
2.1 R a d i a t i o n forces
The f o r m u l a t i o n presented by Cummins [3] w i l l be used to represent the radiation forces i n terms o f u n k n o w n velocity potentials. The basic assumption is the linearity o f the radia-tion forces. That is, i f the body is given an i m p u l s i v e dis-placement o f any k i n d , it w i f l have a certain response lasting much longer than the duration o f the impulse, since the perturbation i n the surrounding water w i l l remain after the impulse. I f the body experiences a succession o f impulses, its response at any t i m e is assumed to be the sum o f its responses to the individual impulses, each response being calculated w i t h an appropriate time lag f r o m the instant o f the corresponding impulse.
due to a m o t i o n i n the /'-mode is represented by:
^ f { t ) ^ l { t ) d j + (10)
where the first term is the potential proportional to the instantaneous velocity o f the body and the second repre-sents the effects of the past history o f the fluid m o t i o n .
Introducing the former expression i n the second term o f eqn (9) results i n the radiation force acting upon the body:
Flit):
•- - hm^j - J __^{Ktj{t-T)kj{t)} dT,
k,j = 2,3A (11)
Indices k,j are associated w i t h forces i n the ^-direction due to an oscillatory motion i n the ;-mode. I t may be shown that the coefficient A^-, w h i c h is dependent only on the body geometry, represents the infinite frequency l i m i t o f the added mass.
The f u n c t i o n K^jit), w h i c h is equivalent to the impulse response f u n c t i o n o f any stable linear dynamic system, is dependent of the time and geometry. K^jit) contains all the memory o f the fluid response. A p p l y i n g Fourier analysis to the equations o f motion the former memory functions may be represented i n terms o f the frequency dependent damping coefficients Btj{co):
% ( 0 = - {Btjiui) cos (o)t)}doi (12)
I t is important to stress that none o f the quantities described above (A^- and Ki,j{t)) is dependent o f the past history o f the unsteady motions. This means that they need only to be calculated once f o r a given cylinder, and then the radiation forces can be evaluated f o r any arbitrary motion using eqn (11).
The relations obtained f o r the radiation forces do not have coefficients dependent on the frequency, thus they are valid to evaluate the radiation forces associated w i t h non-sinusoidal motions, f o r example irregular motions.
The sectional damping coefficients are computed by the Frank's method, while the added masses coiTesponding to the infinite frequency l i m i t are computed by a simplification o f that method which takes into account the zero potential condition on the free surface. Frank [18] represented the velocity potential by a distribution o f sources over the mean submerged cross section. Green functions, satisfying the Laplace equation, the conditions at infinity and the free surface boundary condition, are applied to represent the potential of the unit strength sources. The density o f the sources is an unknown f u n c t i o n , of the position along the contour, to be determined f r o m integral equations derived by applying the kinematics boundary condition on the sub-merged part o f the cylinder (5).
2.2 Exciting forces
F r o u d e - K r i l o v force, represents the effects of the incident wave potential and the second part represents the effects o f the disturbance i n the incident potential caused by the presence o f the cylinder, and is k n o w n as d i f f r a c t i o n force.
I n accordance w i t h the linear wave theory, the incident wave potential corresponding to a wave travelling i n the positive y-direction is given by:
* ' ( y , z , f ) = — ( e ' - ° ^ ) ( e - )(e'-') (13)
where f " is the wave amplitude, and kg = u>llg is the wave number.
The d i f f r a c t i o n part o f the potential is calculated by the Frank close fit method. Finally the time description o f the total exciting forces are given by:
7E/
F f ( r ) = R e { ( f [ + f ° ) e ' " ' } , ^ = 2 , 3 , 4
2.3 Hydrostatic forces
(14)
The hydrostatic forces are the result o f hydrostatic pressure action upon the h u l l . I n the present method these forces are evaluated over the "exact" wetted surface at each time instant taking into account the free surface elevation.
A p p l y i n g Gauss's theorem to the surface integral i n expression (9) representing the hydrostatic term, the hydrostatic forces become:
Fi=Pg (Z/!^) d . F'}{t)^pg^,,.,{t)-pg [ z n f ) As F^=Pg^MB{t)-pg z (15) (16) (17)
where « 2 ' ™ d 7x3'' are the components, respectively, along the y and z axes, o f the two-dimensional unit vector normal to the intersection o f the free surface o f the fluid w i t h the body at each time instant. V„,,(0 is the instantaneous immersed volume under the intersection o f the free surface w i t h the body and ygit) is the y coordinate o f the former volume.
The free-surface elevation on the neighbourhood o f the body is given by the contribution o f the incident wave, diffracted wave, and radiated waves associated w i t h each o f t h e three modes o f motion. Bernoulli's equation is used to evaluate the free surface elevations. The incident and diffracted components are obtained directiy f r o m the frequency domain results. The radiated components of the free surface elevation are obtained f r o m Fourier analysis as f o l l o w s :
1
Hj{t-Tmr)]dT, j = 2,3,4 (18)
According to eqn (9) the linear exciting force is divided i n t w o components, F^ = F^ +F°, where the first part, named
where Hj(t) is the memory f u n c t i o n o f the free surface elevation. The later is calculated f r o m the real (or
-0.04
-0.08
-0.12
Numerical and expenmental study of waves 1.5 1.0 0.5 Ü 0 39 10 2 4 6 Frequency (rad/s) Fig. 2. Coupling coeff. of roll into sway.
imaginary) part o f tlie frequency domain complex a m p l i -tudes o f the free surface elevation associated w i t h motions o f unit amplitude: Hj{t) = f f ^ ( c o ) ^ s i n ( c o O f f ^ M ^ c o s ( a , 0 d w do) (19) 2.4 Equations of motion
N e w t o n ' s second law is applied to obtain the motion equa-tions. The equations respectively f o r sway, heave and r o l l motions are given below, where M is the cylinder's mass and Xq and ya the coordinates o f the centre o f gravity. The term 5 ^ l 4 represents the viscous effects due to the skin f r i c t i o n and eddy components o f the r o l l damping.
{M+A22m+ -pg K24{t-r)Ur)] dT i z n f ) ds = F^{t) (20) {M + A^,)Ut) + \K33ir)Ut-r)] dT -pg i z n f ) ds + p g y „ „ ( 0 -Mg = F f ( 0 (21) -0.5 -1.0
Fig. 4. Memory function for sway.
-t (A24 - Mz'ciUt) + [K42{t - T)UT)] dT + {l44+A44)Ut) + B M t ) K44{t-T)UT)] dT -pg z{yn'^'- zn'^') ds + pgV^Mysit) -yG(t)Mg = F^{t) (22) 3 S I M U L A T E D R E S U L T S
The equations o f m o t i o n are solved b y the fourth-order R u n g e - K u t t a method. The memory functions, Ktj{t), are computed before the routine w h i c h solves the motions equa-tions is called f o r the first time. I n fact these funcequa-tions are computed by a different program since they are independent o f the characteristics o f the m o t i o n . I n this way, several runs can be made w i t h the motions program i n different exciting conditions w i t h o u t having to recalculate those functions.
The sway and heave may be assumed to occur i n an inviscid fluid since the linear damping due to radiation is dominant. However this is not the case o f r o l l m o t i o n and thus the viscous effects due to skin f r i c t i o n and eddy m a k i n g are included i n the r o l l m o t i o n equation. The method adopted to calculate this component was presented by Ikeda et al. [19].
It was f o u n d that i n certain cases the solution o f the r o l l motion is very sensitive to sudden variations o f the exciting moment, thus an exponential ramp f u n c t i o n was adopted to initiate the m o t i o n . This f u n c t i o n increases the incident
Time (sec)
Fig. 6. Numerical results, 3.1 cm and Wg = 3.77 rad/s.
wave amplitude, i n an exponential way, up to the desired amplitude during a time equal to f o u r times the period o f the motion.
Curves o f non-dimensional damping coefficients versus the frequency o f oscillation are represented i n Figs 2 and 3 f o r a two-dimensional body w i t h triangular cross section shape and a beam/immersion ratio o f 2. The non-dimensionalising factors are: pA^^JlglB f o r sway and heave coefficients, pA^B^JlglB f o r the coupling coefficients and pA,B^ \J'2glB f o r the r o l l coefficient. A, and B represent, respectively, the sectional area and the beam.
Examples o f memory functions are shown i n Figs 4 and 5. Results are non-dimensionalised w i t h respect to the same factors as the damping coefficients. These graphs show that the memory effects i n the response vanish after about 10 s. Figures 6, 8 and 9 show simulations o f the sway, heave and r o l l motions. The solid lines represent non-linear solu-fions, where non-hnear hydrostafic forces were used, and the dashed hnes represent hnear solutions, where restoring coefficients were used to represent the hydrostatic forces.
Figure 6 represents a simulation o f the sway m o t i o n con-sidering a wave amplitude o f 3.1 c m and a wave frequency o f 3.77 rad/s. Comparing the linear and non-linear solufions
i t can be observed that the results are similar, except that f o r the second case there is a small steady sway.
The steady sway motion arises f r o m two effects; there is a sway hydrostatic force w h i c h i n general is stronger f o r one o f the directions due to the asymmetry o f the free surface elevation on both sides o f the body, and there is a small i n i t i a l numerical error w h i c h induces a small steady sway velocity, w h i c h is not opposed by a restoring force.
Figure 7 presents a simulation o f the free surface eleva-tion on both sides o f the body, f o r a wave amplitude o f 2.0 c m and wave frequency o f 6.28 rad/s. These free surface elevations are obtained by adding several components, namely, the incident wave elevation, the d i f f r a c t e d wave and the radiated waves corresponding to the three modes o f motion. One can observe that the elevation on the side receiving the incident waves has a higher amplitude than the other side. The reason is because o f the d i f f r a c t i o n , the waves are amplified on incident side and attenuated on the other. These results help to understand the steady d r i f t beha-viour indicated i n F i g . 6.
Figure 8 shows the linear and non-linear heave solutions f o r a regular wave w i t h 2 c m o f amplitude and 5.05 rad/s o f frequency. A small non-linear behaviour can be observed
0 2
Time (sec)
Numerical and expenmental study of waves 41
since the positive amplitudes are higher than the module o f the negative amplitudes. This is due to the coupling through the hydrostatic forces between the heave and r o l l motions. Because o f the V-shape cross section when the body experiences a roll displacement the immersed volume tend to increase, thus an hydrostatic force pointing upwards is induced. Obviously, this tendency is more pronounced f o r wave frequencies close to the r o l l resonance.
Figs 9 and 10 show simulations o f the r o l l motion f o r different regular waves. The first represents the solution f o r a frequency close to the r o l l resonance, thus large
amplitude o f r o l l are obtained. The characteristic w h i c h stands out is the difference between the linear and non-hnear amplitudes (approximately 5 0 - 1 5 ° ) . I n fact, the non-linear restoring moment is much higher than the linear one, since it properly takes into account the V -shape o f the body. On the other hand, the linear metacentric theory is v a l i d f o r small angles o f r o l l and bodies w i t h vertical sides.
The results show another non-linear characteristic, since the r o l l amplitude is greater f o r one o f the sides than f o r the other. The asymmetry i n the response arises because the
Time (sec)
Fig. 8. Numerical results, = 2.0 cm and COQ = 5.03 rad/s.
Time (sec)
Fig. 9. Numerical results, = 3.0 cm and COQ = 5.65 rad/s.
Time (sec)
600 mm
300 mm
• L e n g t h = 150.0 c m
• B e a m at the water Hne (B) = 29.3 c m • I m m e r s i o n ( I ) = 14.65 c m
• Weight of the hull = 11.9 K g • Ballast = 20.25 K g
• T o t a l weight = 32.15 K g
• Zg = - 4 . 0 c m (vertical position of the gravity centre, positive upwards)
• GTt = 8.9 c m (lateral metacentric height) • Iii (roll inertial moment about the gravity centre)
Fig. 11. Test model.
hydrostatic forces are evaluated over the exact wetted sur-face and the free sursur-face elevation on both sides o f the body is asymmetric itself.
4 E X P E R I M E N T S
A series o f experiments were conducted i n the t o w i n g tank of the Hydrodynamic Laboratory at the University o f Glasgow, w h i c h is 77 m long, 4.6 m wide, and 2.7 m
deep, and has a wave maker at one o f the ends and an absorber beach at the other. The wave maker, w h i c h generates regular waves, is driven by an hydraulic pump controlled electronically by a micro-computer.
I n order to obtain non-linear motions, even at moderate motion amplitudes, the model chosen was a prism w i t h triangular cross section. A sketch o f the model as w e l l as the geometric characteristics are presented i n F i g . 11.
The model was ballasted i n such a way that i n the static equilibrium position i t had zero heel and t r i m angles. Then the model was positioned laterally w i t h respect to the length of the tank, at about 1/3 o f the length f r o m the wave maker, and moored to the sides o f the tank by four lines w i t h elastics at the ends. The mooring lines were prepared i n a manner which allowed the model to undergo some swaying motions.
The motions o f the model were measured by a Selspot system, f o r w h i c h t w o light emitting diodes were mounted on each side o f the model deck. The signals emitted by the diodes were received by a camera fixed on the side o f the tank. I n order to measure the incident wave heights three resistance type wave probes were mounted across the breadth o f the tank, between the wave maker and the model. I f B is the breadth o f the tank, the wave probes were installed at B/2, B/3, and B/4. F i g . 12 shows the outhne of the model i n the tank set-up.
A l l the signals, detected by the wave probes and by the camera, were collected at a rate o f 60 samples per second. Collecfions started when it was observed that the model was i n steady o s c i l l a ü o n state. The signals were processed by specific systems and then passed through the Data CoUect-ing System by seven channels and recorded i n a
micro-M o o r i n g ] ^
Processor and collect data system
Micro-computer
Numencal and experimental study of waves
Fig. 14. Experimental results, = 2.0 cm and OJQ = 5.03 rad/s.
Fig. 15. Experimental results, f" = 1.8 cm and COQ = 7.54 rad/s.
ft
1
i
lllllll
1111111
1 1 j 1
i
111
5 10 1 _ -i . -i 5 -i •hó Time (sec) 11
11)11
nil u
III
1
II
1
11
Fig. 16. Experimental results, f" = 3.0 cm and COQ = 5.65 rad/s.
O QA
Fig. 18. Experimental results, f ° = 3.1 cm and cop = 3.77 rad/s.
computer. The channels recorded i n the computer contained horizontal and vertical motions o f the port and starboard diodes and the free surface elevations collected by the three wave probes.
A series o f experiments were conducted i n order to mea-sure the heave and r o l l motions o f the body subjected to set o f regular beam waves w i t h different frequencies and heights. Seven wave frequencies were used f r o m 3.77¬ 8.80 rad/s, and six wave heights f r o m 2 . 0 6 . 0 c m . A l t o -gether 17 different cases were tested.
Figs 1 3 - 1 5 show the type o f experimental results obtained f o r the heave motion. A l l three graphs correspond to incident wave amplitudes o f approximately 2 c m , but different wave frequencies. W h i l e the results i n Figs 13, and 15 correspond to wave frequencies, respectively, below and above the r o l l resonance, the results i n F i g . 14 correspond to a frequency close to the r o l l resonance.
Observing the graphs, a non-linear characteristic can be detected i n F i g . 14 since the positive amplitudes o f the motion are greater than those o f the negative ones. A s men-tioned before this is due to the coupling through the hydrostatic forces between the heave and roll motions. I n fact, as can be observed i n the other two heave graphs, this difference between the amplitudes o f heave is more pronounced f o r the wave conditions i n w h i c h the roll amplitudes are large. I n general, the same heave non-linear behaviour was obtained i n the numerical results calculated by the non-linear method.
Figs 16 and 17 represent measurements o f r o l l motion, corresponding respectively to an incident wave frequency close to the r o l l resonance and to a high frequency. A non-linear behaviour can be observed i n both graphs since the negative amplitudes are greater than the positive a m p l i -tudes.
Comparing w i t h the con-esponding numerical simulations represented i n Figs 9 and 10 i t is observed that the tendency is coiTectly predicted f o r the lower wave frequency but not f o r the higher wave frequency.
For the lower frequencies the r o l l experimental results are essentially linear. F i g . 18 presents measured r o l l results corresponding to a l o w wave frequency. One may observe that the measured motion is not regular during the first f e w cycles, but this is a transient period after w h i c h the r o l l motion becomes stable.
5 C O R R E L A T I O N B E T W E E N T H E O R E T I C A L A N D E X P E R I M E N T A L R E S U L T S
In order to compare the experimental results against the theorefical model solutions, all the results are condensed i n the f o r m o f "pseudo transfer funcfions". The heave is non-dimensionalised w i t h respect to the wave amplitude, w h i l e the r o l l is non-dimensionalised w i t h respect to the wave steepness. s X 1.5 . X X 1 - O 0 a - - - x - - - D - - i - • y . . . • • - . , X 0.5 . - - • - . . linear 0.5 . X non-linear o experin^nts 0 1 1— —
.
1.5 3 4 5 6 7 8 9 frequency (rad/s)Fig. 19. J ^ / r , non-dimensional heave positive amplitudes for
fa = 2.0 cm. 0.5 1 - • • O X
-.
X . . X linear non-linear experln^nts 1 1 1 1 1 3 4 5 6 7 8 9 frequency (rad/s)Fig. 20.
êsVr,
non-dimensional heave negative amplitudes for r = 2.0 cm.Numerical and experimental study of waves 45 X X 1.5 -X 0 1 - X • S 0 . . f t . . . X X •0 0.5 - . linear X non-linear 0 experiments 0 1 - 1 1 10.0 3 4 5 6 7 8 9 frequency (rad/s)
Fig. 21. ^ " V r , non-dimensional heave positive amplitudes for = 3.0 cm.
As the negative amplitudes are different f r o m the positive amplitudes, two transfer function graphs are presented f o r each case. I n Figs 19, 20, 23 and 24 the wave amplitudes are approximately equal to 2 cm. I n Figs 2 1 , 22, 25 and 26 the wave amplitudes are approximately equal to 3 cm. The experimental results are represented by the black balls, the linear results by the dashed line w i t h the squai-es and the non-linear results by the asterisks.
The heave m o t i o n graphs (Figs 1 9 - 2 2 ) show that the linear frequency domain solution has a very smooth depen-dency on the frequency while both the experimental results and the time domain results exhibit some variability o f s i m i -lar pattern.
The results o f the non-linear time domain simulation compare w e l l w i t h the experimental results except f o r one or two points i n the frequency around 6 rad/s where the numerical results lead to overprediction o f the positive amplitudes.
I t was f o u n d that the hydrostatic coupling w i t h the r o l l displacements is responsible f o r these inaccurate predic-tions. I f the hydrostatic forces are kept non-linear, but the r o l l m o t i o n is neglected i n the evaluation o f the
7.5 J-i 5.0 2.5 0.0 linear X non-linear d 1 1 9 experiments ^ stability arm X \ 3 4 5 6 7 8 9 frequency (rad/s)
Fig. 23. ^T'/ko^", non-dimensional roll positive amplitudes for f° == 2.0 cm.
hydrostatic heave force then the discrepancies o n the results disappear.
Figures 1 9 - 2 2 indicate that, although the linear model is not able to represent the slight non-linear behaviour f o u n d i n the experiments, the hnear results compare w e l l w i t h the experiments. I n order to obtain heave results w i t h more pronounced non-linear behaviour i t is required that the rela-tive m o t i o n is higher than that obtained i n the experiments. I n fact, it was found that, f o r the wave conditions of the experiments, the vertical motion of the body f o l l o w s the ver-tical m o t i o n o f the incident wave. I t is believed that this results f r o m a combination of a low displacement and a too large flare o f the triangular section, which does not allow a large relative motion between the body and the waves.
I n the case o f the r o l l m o t i o n the predictions o f the non-linear (Figs 2 3 - 2 6 ) are encouraging since they compare very w e l l w i t h the experiments, even near the resonance frequency where the linear model fails completely. For the higher wave amplitudes there was one result at a frequency o f 4.4 rad/s (Fig. 25) i n w h i c h a too large prediction o f the non-linear program was observed.
1.5 10.0 1 i 0.5
4-0 X • X . . . . . linear non-linear experiments 8
1
1 1 H 3 4 5 6 7 irequeney (rad/s)Fig. 22. ^I'/i;", non-dimensional heave negative amplitudes for r = 3.0 cm. . . . Q . . . linear X non-linear 0 experiments stability arm 3 4 5 6 7 frequency (rad/s)
Fig. 24. ^47^:or, non-dimensional roll negative amplitudes for f" = 2.0 cm.
10.0
5 6 7 frequency (rad/s)
Fig. 25. ?4"^/A:o^^ non-dimensional roll positive amplitudes for
r 3.0 cm.
Some modern ship motion programs use the stability arm curve i n order to calculate the ship r o l l restoring moment at large r o l l angles. This is certainly an improvement i n rela-t i o n rela-to rela-the use o f rela-the linear merela-tacenrela-tric rela-theory, rela-thus rela-the same method was implemented to evaluate the restoring moment. Here the hydrostatic moment was computed neglecting the coupling w i t h the heave motion and the free surface eleva-t i o n effeceleva-ts. A eleva-t each eleva-t i m e inseleva-taneleva-t eleva-the momeneleva-t is evaluaeleva-ted considering the cylinder i n still water w i t h the actual r o l l displacement and zero heave displacement. I n the r o l l m o t i o n graphs, i n addition to the symbology already pre-sented, the new solutions are represented by solid lines w i t h triangles.
Observing the graphs, two characteristics o f the new model can be pointed out, first the restoring term i n the equations o f m o t i o n is increased thus the resonance f r e -quency is higher, consequently the amplitudes o f m o t i o n f o r frequencies near the new resonance are higher than those predicted by the other models. The resonance f r e -quency f o u n d i n the experimental results compares better w i t h the other two models.
10.0 . - Q . . . linear _ X — non-linear 9 experiments - stability arm 5 6 7 fi-equency (rad/s)
Fig. 26. ^Xlkfs^'^, non-dimensional roll negative amplitudes for
= 3.0 cm.
Secondly i t seems that the new model tends to under-estimate the r o l l amplitudes corresponding to higher waves, perhaps because i n this case the neglected effects o f the free surface elevation become more important. The results analysed are l i m i t e d to support general conclusions, but i f these tendencies are verified f o r other cases, especially f o r cylinders w i t h cross sections w i t h less steep sides, one may have to start taking into account the effects o f the free surface elevation i n order to obtain accurate predictions o f the r o l l motion o f ships.
6 C O N C L U S I O N S
Experimental results o f the motions o f a floating prismatic body w i t h triangular cross section shape, subjected to incoming regular waves, are presented and compared w i t h the predictions o f two theoretical models. One o f the models is linear and the other one is based on a time domain non-linear approach. I n the time domain model the radiation forces are represented by infinite frequency added masses and convolution integrals o f memory functions, the incident wave exciting forces are linear, and the hydrostatic forces are non-linear since they are evaluated over the "exact" instantaneous wetted surface o f the body, where the free surface elevation due to i n c o m i n g , d i f f r a c t e d and radiated waves are taken into account.
Non-linearities were identified i n the experimental results, w h i c h i n some cases are coiTectly predicted by the non-hnear theoretical model. However f r o m the comparison o f the results o f the linear and non-lineai' theories, i t was concluded that f o r the non-linear effects i n heaving to be more pronounced, it is required to have experiments w i t h larger relative motions between the body and the free sur-face. I n fact, it was f o u n d that, f o r the wave conditions o f the experiments, the vertical m o t i o n o f the body f o l l o w s the vertical m o t i o n o f the incident wave.
The r o l l motion and the associated hydrostatic moment is strongly non-linear even f o r the wave amphtudes used i n the experiments, and f o r this reason the linear solution is very inaccurate near the resonance. I n this case the predictions o f non-linear theory were very good. I n addition, the effects o f the free surface elevation seem to be important f o r the pre-diction o f the hydrostatic moment.
The l i m i t e d experimental w o r k conducted has indicated that the non-linear theory described here has the potential to provide a good description o f non-linear motions. However the intensity o f the sea states i n w h i c h the motions become non-hnear depends on the type o f motion.
A C K N O W L E D G E M E N T S
The first author is grateful to J N I C T , Junta Nacional de Investiga^ao Cientifica e Tecnologica, f o r having provided h i m a scholarship w h i c h has supported h i m i n the course o f this w o r k .
Numerical and experimental study of waves 47
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