ELSEVIER
Applied Ocean Research 26 (2004) 256-273Applied Ocean
t ? © s © o i r c ï iwww.elsevier.coni/locate/apor
Validation of a time-domain strip method to calculate the motions
and loads on a fast monohull
N. Fonseca, C. Guedes Soares*
Unit of Marine Technology and Engineering, Technical University of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Abstract
The paper presents a comparison between experimental data and numerical results of the hydrodynamic coefficients and also of the wave induced motions and loads on a fast monohull model. The model with 4.52 m length was constructed in Fibre Reinforced Plastic (FRP), and made up of 4 segments connected by a backbone i n order to measure sectional loads. The objective of the investigation was to assess the capability of a nonlinear time domain strip method to represent the nonlinear and also the forward speed effects on a displacement high speed vessel advancing in large amplitude waves. With this objective i n mind the experimental program included forced oscillation tests in heaving and pitching, for a range of periods, three different amplitudes and several speeds of advance. In head regular waves comprehensive ranges of wave periods, wave steepness and speeds, were tested in order to measure heave, pitch and loads in three cross sections.
The numerical method assumes that the radiation and diffraction hydrodynamic forces are lineai- and the nonlinear contributions arise from the hydrostatics and Froude-Kiilov forces and the effects of green water on deck. The assumption of linearity of the radiation forces is validated by comparing calculated hydrodynamic coefficients with experimental data for thi'ee different amplitiides of the forced oscillations. Both global coefficients and sectional coefficients are compared. The motions and loads i n waves are compared i n terms of fii'st and higher harmonic ampUtudes and also in terms of sagging and hogging peaks.
© 2005 Published by Elsevier Ltd.
1. Introduction
There is a tendency to move f r o m empuical procedures to methods based on the first principles to define the criteria f o r refiabitity based sffuctural design o f ships [ 1]. The approach relies on aprobabitistic model of sti-uctural strength and o n a long-term probability d i s ü i b u t i o n o f wave induced loads and i t identifies the conditions o f larger probabitity of failure. W h e n designing f o r ultimate snength these conditions tend to occur i n heavy weather and thus the wave induced loads are dominated by nontinear effects.
A s wave induced loads are dependent o n the speed o f the vessel, a comprehensive study on wave loads i n fast ships has been pursued, as described i n Ref. [ 2 ] .
T o properly account f o r the nonhnear effects that develop i n heavy weather i t is necessaiy to consider time domain codes, since frequency domain formulations are i n most cases based i n linearity assumptions. A time domain seakeeping code was proposed by Fonseca and Guedes Soares [3,4] to calculate the vertical motions and global stinctural loads i n large amptitude
* Corresponding author.
E-mail address: guedess@inai-.lst.utl.pt (C. Guedes Soares). 0141-1187/$ - see front matter © 2005 Published by Elsevier Ltd. doi:10.1016/j.apoi-.2005.08.003
waves. The method, which is based on a sttip theory approach and accounts f o r the most important nonlinear effects, is appropriate f o r practical engineering applications and f o r conventional ships w i t h relatively low speed. Comprehensive comparisons between experimental data and numerical results f o r a containership advancing i n regular and itregular waves [5,6] have shown that the method is able to represent quatitatively a l l the nonfineai-effects identified i n the experimental data and is an improvement compared to the tinear solution.
W i t h the present investigation, the authors want to validate that method o n a ship w i t h slender f o r m s and higher speed and to test the Umits o f application o f the present f o r m u l a t i o n regarding the speed of advance. I n fact a strip theory approach is i n principle valid f o r small Froude numbers, since the boundary conditions o f the boundary value problem are h i g h l y simplified w i t h respect to Üie f o r w a r d speed effects. I n practical terms, i t is not clear what the timit Froude number should be. B y systematic comparisons o f numerical results w i t h exper-imental data f o r different speeds o f the ship i t is possible to i d e n t i f y the limitations o f this nonlinear strip theory.
The experimental data refer to a model of a fast and slender monohuU which has a sei-vice speed that corresponds to a Froude number of 0.4. This cannot be considered high speed, but rather a moderate speed, which however, is certainly above the 'theo-retical' Hmit acceptable f o r strip theories. The expeiimental data
N. Fonseca, C. Guedes Soares /Applied Ocean Research 26 (2004) 256-273 257
relate only to the vertical ship responses (associated with heave and pitch motions) and includes: hydrodynamic coefficients f o r the whole ship hull and for isolated hull segments, ship motions i n regular- waves, and cross sectional loads.
2. Time domain seakeeping code
(/55 + A 5 " ) ? 5 ( 0 + [ A ' ^ ^ ( r - r ) ? 5 ( r ) ] d T + CS?5(0
[K^',it-T)^,iT)]dr + C ' ^ , m
T h e t i m e d o m a i n seakeeping code, is based on the f o r m u l a t i o n proposed b y Fonseca and Guedes Soares [3,4]. The method assumes that the nonlinear contribution to the vertical responses is dominated by hydrostatic and F r o u d e -K i i l o v forces, w h i c h depend o f the instantaneous h u l l wetted surface. The exciting forces due to the incident waves are decomposed into d i f f r a c t i o n part and F r o u d e - K r i l o v part. The d i f f r a c t i o n part, w h i c h is related to the scattering o f the incident wave field due to the presence o f the non-oscillatory ship, is kept linear. Since this is a linear problem and the exciting waves are k n o w n a p r i o r i , i t can be solved i n the frequency domain and the resulting transfer functions be used to generate a t i m e history o f the d i f f r a c t i o n heave f o r c e and pitch moment. The F r o u d e - K i i l o v part is related to the incident wave potential and results f r o m the integration at each time step o f the associated pressure over the wetted surface o f the h u l l under the undisturbed wave profile.
The radiation forces are represented i n the t i m e domain by i n f i n i t e frequency added masses, radiation restoring coefficients and convolution integrals o f memory functions. The convolu-tion integrals represent the effects o f the w h o l e past history o f the m o t i o n accounting f o r the memory effects due to the radiated waves. B o t h the radiation and diffi-action coefficients i n the frequency domain are calculated b y a strip method.
The vertical forces associated w i t h the green water o n deck, w h i c h occurs when the relative m o t i o n is larger than the freeboard, are calculated using the m o m e n t u m method [ 7 ] .
A c c o r d i n g to the classification o f the Committee V I . 1 o f the International Ship and Offshore Structures Committee [ 8 ] , this code is based on a 'partially nonlinear m e t h o d ' . T h i s means that the equations of motions and loads combine linear and nonlinear terms. T h e mentioned Committee has reviewed the methods available to calculate nonlinear ship motions and loads i n large amplitude waves and concluded that, f o r practical applications, the methods that were more appropriate are the ones based on approaches similar to the one described here.
The heave and pitch equations o f m o t i o n i n the time domain are given bellow:
( M + A 3 " 3 ) ? 3 ( 0 [ ^ : ^ ^ ( r - r ) f 3 ( r ) ] d r + C^'|,?3(0
[ 7 f S ( f - r ) f 5 ( r ) ] d r + C S ? 5 ( 0
F^(t)-Mg+Fl"'{t)
= F^(t)+Fht) (2)
= F^(t) + F f i t ) (1)
where ^3 and ?5 represent, respectively, the heave and p i t c h motions and the dots over the symbols represent differentiation w i t h respect to time. M i s the ship mass, g is the acceleration o f gravity and 755 represent the ship inertia about the y-axis.
T h e hydrostatic f o r c e and moment, F^ and F^, are calculated at each time step by integration o f the hydrostatic pressure over the wetted h u l l under the undisturbed wave profile, as weU as the F r o u d e - K i i l o v contribution o f the exciting forces . The diffi-action forces are represented by
i'k-The radiation forces are represented i n the t i m e domain by i n f i n i t e frequency added masses A^-, r a d i a t i o n restoring coefficients C f j , and convolution integrals o f memory functions
K'f}j{t). The radiation restoring forces, associated w i t h the
restoring coefficients, represent a coirection to the h y d r o d y n -amic steady forces acting on the ship due to the steady flow. The memory functions and the radiation restoring coefficients are obtained by relating the radiation forces i n the t i m e domain and i n the frequency domain by means o f Fourier analysis.
Finally, the vertical forces associated w i t h the green water on deck, F f \ t ) , w h i c h occurs when the relative m o t i o n is larger than the freeboard, are calculated using the m o m e n t u m method. This way the hydrodynamic pressure on the deck includes thi'ee terms: the hydrostatic pressure, one term that accounts f o r the variation o f mass o f water on the deck, and one t e r n associated w i t h the acceleration o f the deck. T h e height o f water on deck is given by the difference between the relative m o t i o n at the b o w and the freeboard, where the relative m o t i o n depends o f the undisturbed free surface elevation. The mass o f water on the deck is proportional to the height o f water o n the deck.
The wave induced structural dynamic loads at a cross section are given b y the difference between the inertia forces and the sum o f the hydrodynamic forces acting o n the part o f the h u l l f o r w a r d o f that section. The f o r m u l a t i o n to calculate a l l contributions f o r the loads is consistent w i t h the f o r m u l a t i o n applied to solve the unsteady t i m e domain motions problem. The convention f o r the loads is such that the sagging shear force and hogging bending moment ai-e positive.
3. Experimental set-up
The tests were conducted at the L a b o r a t o r y o f Ship Dynamics o f the E l Pardo M o d e l Basin ( C E H I P A R ) , i n M a d r i d . The Laboratory o f Ship Dynamics is made up o f thi-ee
258 N. Fonseca, C. Guedes Soares / Applied Ocean Research 26 (2004) 256-273
basic f a c i l i t i e s : t h e tank, the wave generator and the computerized planar m o t i o n carnage. The tank is 145 m long f r o m the face o f the wave generator to the intersection o f the beach w i t h the water free surface, 30 m wide and 5 m deep. A d d i t i o n a l l y there is a sectional p i t w i t h 10 X 10 m w i t h an additional depth o f 5 m f o r specific tests requiring large depths. The beach has a dual slope profile and is composed o f stainless steel shavings f o r m i n g a rough and porous layer 50 c m thick. The wave generator is along one o f the 30 m sides o f the tank and i t is composed o f 60 elements that operate independently. B y adjusting the phase of the almost sinusoidal m o t i o n o f each o f the elements, the direction o f the waves can be adjusted w i t h i n certain limits. Each element is o f the flap type, hinged at t w o meters f r o m the tank bottom. I n this way the wave generator is able to produce regular waves and
i i T e g u l a r waves, both long crested and short crested.
Regarding the Computerized Planar M o t i o n Carriage ( C P M C ) , i t can f o l l o w any path i n the horizontal plane either t o w i n g a captive model, as i n the case o f the experiments presented here, or tracldng a free-innning, self-propelled model. I t can be used both f o r seakeeping and maneuvering smdies.
The experiments were canied out w i t h a model o f a fast m o n o h u l l , w h i c h has a length o f 134 m and a service speed o f 27 knots ( F n = 0 . 4 0 ) . F i g . 1 presents the h u l l bodylines and Table 1 the ship m a i n particulars. The model was constructed i n Fiber Reinforced Plastic (FRP), the length is 4.52 m between perpendiculars (scale o f 1:27) and i t is made up o f four segments connected by a r i g i d a l u m i n i u m backbone. The cuts between segments are located i n sections 5 , 1 0 and 15. V e r t i c a l cross sectional loads were measured at these positions. The height o f the rigid backbone is approximately at the waterline l e v e l . F i g . 2 shows the m o d e l at the workshop under preparation f o r the tests. Overall the model may be considered r i g i d , meaning that the first natural frequency o f flexural vibration is very h i g h and the related stmctural response was not detected i n the time records o f the structural loads.
7.0 5 . 0 -7.0 5 . 0
-1
1
\
ill
1 :
ill
0I k
W1Ijs
- 5 . 0 'TtO-\
i
Fig. 1. Fast monohull bodylines.Table 1
Fast monohull main particulai s
Length overall L„, (m) 133.7
Length betw. Perp. Lpp (m) 122.0
Breadth overall B ( m ) 15.19 Depth £ ) ( m ) 9.2 Draught r ( m ) 4.66 Displacement A (ton) 4329.9 Service (max) y ( k n ) 27 Long, posit, of CG LCG (m) - 1 . 3 4 7 Vert, posit, of CG VCG (m) 4.374 Block coefficient Cb 0.49
Pitch rad. of gyr. Ky-i/Lpp 0.253
Table 2 shows the weight and inertial characteristics o f the complete h u l l and o f each segment (values f o r f u l l scaleship). l y y ' represents the longitudinal inertia w i t h respect to the centre o f gravity o f the whole slrip, Xg is the longitudinal position o f the centre o f gravity relative to midship and Zg is the vertical position o f the centre of gravity w i t h respect to the baseline. The order o f the segments numbering is f r o m stem to bow. Each of the segments was caUbrated i n order to obtain the desired global weight, longitudinal position o f the centre o f gravity and longitudinal inertia.
The experimental p r o g r a m i n c l u d e d two groups o f tests: f o r c e d m o t i o n tests i n c a l m water and f r e e response tests i n head regular waves. Four d i f f e r e n t speeds were considered f o r b o t h groups o f tests. T h e o b j e c t i v e o f the f o r c e d m o t i o n tests was to obtain the h y d r o d y n a m i c c o e f f i c i e n t s o f the v e r t i c a l m o t i o n s f o r the w h o l e frequency o f o s c i l l a t i o n range, their dependence o f the o s c i l l a t i o n a m p l i t u d e and d i s t r i b u t i o n along the ship length. The tests i n regular waves gave results o f the v e r t i c a l m o t i o n s and structural loads and their dependence o f the a m p l i t u d e o f the i n c o m i n g waves.
A detailed description o f the forced motions experimental program including the preparation of the model, instrumen-tation, and analysis o f the experimental data is presented i n Ref. [ 9 ] .
4. Hydrodynamic coefficients
The hydrodynamic coefficients, meaning added masses and damping coefficients, are associated w i t h the radiation problem and represent the hydrodynamic forces due to harmonic f o r c e d motions w i t h the ship advancing i n still water (without incoming waves). A d d e d masses represent the forces i n phase w i t h the acceleration of m o t i o n normalized by the acceleration and damping coefficients represent the forces i n phase w i t h the velocity normalized by the velocity.
A s described i n Section 2, the time domain m o d e l does not use directly frequency dependent hydrodynamic coefficients but rather i n f i n i t e frequency added masses, radiation restoring coefficients and convolution integrals o f memory functions. However, these time domain coefficients and functions are obtained f r o m Fourier analysis o f the frequency dependent coefficients, thus i t makes sense to compare the numerically calculated added mass and damping coefficients w i t h the
N. Fonseca, C. Guedes Soares/Applied Ocean Research 26 (2004) 256-273 259
Fig. 2. Model at the workshop. experimental ones to assess the merits o f the f o r m u l a t i o n used
f o r the radiation forces. Eqs. (3) and (4) present the relation between time domain m o t i o n coefficients and functions and frequency domain hydrodynamic coefficients:
B 33 ^33 d x
A 35
u
Wt)cos wr}dt (3)
0/A;^i6j)=6//^-C^-0J {K',"it)sin ojt}dt (4)
Bi5 = A-fl33 d v - —5 3 3 A-fe33 dx + UA°^
u
X(333 c k H 2 ^ 3 3 xbj2 dx — UAwhere i n addition to the symbols already defined, Bfcj(w) and
Af;j((x>) represent, respectively, the ship damping coefficients
and added masses. There are sinular relations f o r the radiation contributions f o r the wave induced stractural loads.
A d d i t i o n a l l y , i t is o f interest to include the strip theory formulas f o r the vertical motions hydrodynamic coefficients, since i t w i l l help the interpretation o f the results. These formulas represent frequency dependent coefficients i n teims o f two-dimensional cross sectional added masses and damping coefficients i n heave (055(03) and bssiw)):
^ 5 5 = 2 t / n X (733 d x + ^ A 3 3 B 55 x%,dx + ^ B l , (5b) (5c) (5d) (5e) ( 5 f ) (5g) (5h) A,3 = Table 2 033 d x (5a)
I n addition to the symbols already defined, A33 and B33
represent the heave added mass and damping coefficient o f the whole ship w i t h o u t speed. There ai'e similar relations f o r the radiation contributions f o r the wave induced structural loads. The f o r m e r equations neglect the "end terms" associated w i t h the transom stem. This is because the area o f the immersed transom stern is very small compared w i t h the whole h u l l .
Inertial properties of the model (translated to f u l l scale)
Description Segm. 1 (stem) Segm. 2 Segm. 3 Segm. 4 (bow) Ship
Weight IV (ton) 888.6 1373.8 1214.1 852.9 4329.6
Long. pos. centre of (m) - 4 3 . 4 8 3 - 1 4 . 8 2 4 13.747 42.697 - 1 . 3 6 3
gravity
Lat. pos. centre of (m) 0.000 0.000 0.000 0.000 0.000
gravity
Vert. pos. centre of Zj, (m) 4.374
gravity
lyy (ton m")
Long, inertia with lyy (ton m") 1634378 376532 410800 1701199 4122909
respect to model centre of gravity
Pitch radius of RyyiLpp 0.253
260 N. Fonseca, C. Guedes Soares / Applied Ocean Researcli 26 (2004) 256-273
H o w e v e r i f one wants to calculate the h y d r o d y n a m i c coefficients o f a segment o f the h u l l , then this segment w i U have flat ends w i t h the shape o f the aft and f o r w a r d cross sections o f the segment and the area o f these flat ends cannot he neglected anymore.
A c c o r d i n g to the theory f r o m Salvesen et al. [10] the "end terms" arise f r o m the application o f the Stokes theorem and they are related to a line integral along the cross section that defines the end o f the immersed body. I n case the immersed body has two flat ends, like segments 2 and 3 o f the model being studied, then one needs to account f o r t w o end terms w h i c h w i l l have opposite signs. Segment 1 has one end term corresponding to its f o r w a r d end and segment 4 has one end term corresponding to its aft. The hydrodynamic coefficients f o r the segments are:
A33 = U "ft _ rM B
33
b3^dx +u(at-4
'33 "33 (6a) (6b) A 35 xa.^dx-^B'i, + 4 Uftbt-Xf,,„y^^' Ü) 6) \ (^«33-033 ) (6c) 535 = xbj^dx + [M?3 - U33
."ft. aft _,üvd\ '33 ^^33 )As3 = xassdx+^B°3 + (x„ftbf^-Xfl,.,y;^
B
53
xb^j,dx — UA•U{x„ftat-xj,,,4^')
(6d)
(6e)
( 6 f )
Ass = X a^^dx +2 U'^ 0 ^ A 3 3
I aft fwd OJ \ U 0) 2 V ' V 3 3 ^fi<'dl^33 B (6g)
55
x%,dx+~B',, +OJ \U(xlflat-xl,4';'
u
^ L. h"ft-,- iJ^'''' (6h)I n addition to the symbols already presented the superscripts and subscripts aft andftvd stand f o r the aft end and f o r w a r d end o f the segments. The longitudinal coordinates o f the a f t and f o r w a r d ends o f the segments are represented by x^ji and Xfi^.^.
For the hydrodynamic calculations, the complete h u l l was ideahzed by 30 cross sections. Each h a l f cross section is represented b y 10 segments under the static waterline. The next
pages present comparisons o f the hydrodynamic coefficients associated w i t h the vertical f o r c e d motions. Results include the global coefficients f o r the whole h u l l and also results f o r isolated segments. The experimentals were carried out f o r three amplitudes o f the forced motions i n order to assess the nonlinear effects on the radiation forces, and to investigate the possible presence o f viscous damping. The experimental data cover a wide range o f conditions, w h i c h include ( f u l l scale):
Forced heave m o t i o n w i t h three amplitudes: 0.466,0.932 and 1.398 m
Pitch forced motions: 1, 2 and 3 degrees
Four speeds o f advance: 0, 13.45, 20.18, 26.90 knots W i d e range o f periods o f oscillation between 3.6 and 14 s
The results are presented i n Figs. 3 to 11. I n the graphs the symbols represent experimental data, w i t h different symbols f o r d i f f e r e n t f o r c e d m o t i o n s amplitudes, and the lines numerical results. The added masses are given i n ton and the damping coefficients i n k N , as a f u n c t i o n o f the oscillation period.
Observation o f the experimental data f o r different forced m o t i o n amplitudes does not show any clear nonlinear tendency o f the hydrodynamic coefficients f o r this h u l l . I n fact the conclusion to take is that the hydrodynamic coefficients ai^e basically hnear w i t h respect to the forced motions amplitude. This conclusion seems to validate one o f the most important assumptions o f the t i m e domain f o r m u l a t i o n presented i n Section 2, namely, the assumption that the nonlinear effects associated w i t h the vertical motions ai'e dominated by the F r o u d e - K r i l o v and hydrostatic components and the radiation forces may be kept linear. O n the other hand, since the experimental hydrodynamic coefficients are basically linear, i t makes sense to compare the calculated linear results although the paper deals w i t h a nonlinear time domain code. I n spite o f the conclusions taken f r o m the previous graphs, recent investigations show that although the hydrodynamic coeffi-cients are linear, the radiation forces may present important higher order effects [ 9 ] .
Figs. 3 and 4 present the heave added masses and p i t c h added inertias f o r different ship speeds. The added mass numerical results slightly overestimate the experimental data, a tendency that increases f o r higher ship speeds and higher periods o f oscillation. The added inertia numerical results compare very w e l l w i t h the experimental data f o r the lower ship speeds; however, they largely overestimate the exper-imental data f o r the higher speeds and oscillation periods. This set o f comparisons clearly show the limitations o f the strip theory approach; firstly i t is a h i g h frequency formulation, and i n f a c t the agreement is always good at this frequency range, secondly i t is a l o w speed f o r m u l a t i o n and this is reflected i n the bad correlation between experiments and numerical results f o r the higher speeds. The f o r w a r d speed term i n the p i t c h added inertia f o r m u l a (Eq. 5g), w h i c h is derived assuming l o w speed, is m a i n l y responsible f o r the very large numerical results at high speed and l o w frequency (last graph o f F i g . 4 ) .
N. Fonseca, C. Guedes Soaies/Applied Ocean Researcli 26 (2004) 256-273 261 < 12000 10000 8000 6000 4000
Heave Added Mass (V = 0 Kn) Heave Added Mass (V = 13.45 Kn)
1
J / • Xa ° „ . 1 X Q ° ^ 1-y
Calculated ° A = 0.466m ^ A = 0.932m ° A = 1.398m 1 _ „ Calculated ° A = 0.466m ^ A = 0.932m ° A = 1.398m Calculated ° A = 0.466m ^ A = 0.932m ° A = 1.398m Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m 10 T(s) 15 20 12000 10000 8000 6000 a a i r ^ ^ A CO 4 0 0 0 + - - F - 1-^ 2000 0 -2000 + -4000 1b 115 A f -2Ö 20000 18000 16000 14000 g 1 2 0 0 0 m 1 0 0 0 0 < 8000 6000 4000 2000 0Heave Added Mass (V = 20.17 Kn) Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m • Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m ^ Q_ ^ 0 o ÏAKTAA "a A A A A
1
16000 14000 12000 ^ 1 0 0 0 0 CT 8000 6000 4000 2000 0 T(s)Heave Added Mass (V = 26.90 Kn) — Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m 10 T(S) 15 20 10 T(s) 15 20
Fig. 3. Heave added mass for the whole ship and four speeds. Regarding the coupled added masses i n F i g . 5, the
agreement between the two types o f results is reasonable and w i t h a tendency o f worsening f o r higher speeds. Since the calculation (and measurement) o f the coupling coefficients is usually more d i f f i c u l t to p e r f o r m the correlations observed i n F i g . 5 are satisfactory.
F i g . 6 presents the damping coefficients f o r the whole h u l l and f o u r speeds. The agreement between numerical and experimental data is very good, even f o r the higher speeds and lower frequencies. One may say that the stiip theory limitations are not reflected i n the heave damping coefficients f o r the whole h u l l . I n fact f o r w a r d speed effects are less important f o r the hydrodynamics o f pure vertical motions than f o r angular motions. However, f o r l o w e r frequencies, there is a tendency o f the experimental damping coefficients to increase w i t h the speed o f the ship. This tendency is not represented by the numerical results, since the strip theory damping coefficient is speed independent (see E q . (5b).
W i t h respect to the p i t c h damping coefficients ( F i g . 7), the numerical results compare worse w i t h the experimental data then the coiTesponding heave coefficients. B o t h types o f results converge f o r h i g h frequencies o f oscillation, but i n the middle and high frequency range the predictions overestimate the experiments. I n the middle and h i g h frequency range the experimental results tend to increase w i t h the ship speed. The numerical model also represents this tendency. One observes that the numerical results
overestimate the experiments f o r l o n g periods and high speed, but not so drastically as i n the case o f the pitch added inerda.
F i g . 8 presents the heave into p i t c h coupled damping coefficients. A l t h o u g h the numerical resuhs represent w e l l the tendencies o f the experiments along the periods range, the agreement is poor and clearly worsening as the speed increases. One should note that the pitch moments associated w i t h these coupling coefficients are not small compared to the other damping moments i n the p i t c h equation o f m o t i o n . For instance, i f one selects an intermediate period o f oscillation, then consider a heave forced m o t i o n o f 0.5 m and a pitch forced m o t i o n o f 1°, then the coupUng damping moment induced b y the heave m o t i o n is around 25% o f the damping induced by the pitch m o t i o n . I t is therefore important to have good estimates o f the coupling coefficients.
F i g . 9 shows the heave added masses f o r segment 1 w h i c h is the stem segment and represents the aft 1/4 o f the h u l l . A g a i n one observes that the n u m e r i c a l m o d e l represents w e l l the experimental data f o r l o w speeds, however, i t overestimates the experiments as the speed increases. Figs. 10 and 11 present, respectively, the added masses and damping coefficients f o r the f o u r segments and the m a x i m u m ship speed. I t is interesting to observe that, f o r both coefficients, the agreement between experiments and numerical results is m u c h better f o r the segments i n the middle than f o r the b o w and stem segments. This probably
262 N. Fonseca, C. Guedes Soares/Applied Ocean Researcli 26 (2004) 256-273 1 9 8 7 ^ 4 3, 2. 1. O, OE+07 OE+06 OE+06 OE+06 OE+06 OE+06 OE+06 OE+06 OE+06 0E+Ü6 OE+OO
Pitch Added Inertia (V= O Kn)
Calculated 1 __ ° A = 1'' A A = 2° - _ _ ^ ^ ^ _ _ n_J. 1 ° A = 3° j - ^ ^ - i ^ l ^ j
1
! 1 10 T(s) 15Pitch Added Inertia (V= 20.17 Kn)
20 2.5E+07 2.0E+07 <^ 1.5E+07 ^ 1.0E+07 + 5.0E+06 O.OE+00
Pitch Added Inertia (V = 13.45 Kn) Cateulated ° A = 1° A A = 2° ° A = 3° Cateulated ° A = 1° A A = 2° ° A = 3° 1 O A A O i 2.5E+07 2.0E+07 <^ 1.5E+07 5 1.0E+07 5.0E+06 O.OE+00 10 T(s) 15 20
Pitch Added Inertia (V = 26.90 Kn) Cabülated ° A = 1° A A = 2° ° A = 3° L Cabülated ° A = 1° A A = 2° ° A = 3°
i
J . < : : : : ^ . o ° I A A ^ 10 T(s) 15 20Fig. 4. Pitch added inertia for the whole ship and four speeds.
250000 200000 -g- 150000 CO < 100000 50000 0 450000 400000 350000 _^ 300000 I . 250000 I 200000 150000 100000 50000 0
Heave into Pitch Added Mass (V = 0 Kn) Calculated ° A = 0.466m A A = Q.932m ° A = 1.398m 10 T(s) 15 20
Heave into Pitch Added IVIass (V = 20.17 Kn) Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m --arB-P--10 T(s) 15 20 350000 300000 250000 E" 200000 I 150000 100000
Heave into Pitch Added Mass (V = 13.45 Kn) Calculated " A = 0.466m A A - 0 . 9 3 2 m ° A = 1.398m
1
Calculated " A = 0.466m A A - 0 . 9 3 2 m ° A = 1.398m r X"!% \
Calculated " A = 0.466m A A - 0 . 9 3 2 m ° A = 1.398m 1 — '/A
/ è1
1iIZAA:
7 i , ~ l 1 iy 0 " 1 10 T(s) 15 20 600000 400000Heave into Pitch Added Mass (V = 26.90 Kn)
300000
5
Calculated1
" A = 0.466m L i _ " A = 0.466m A A = 0.932mi / i
° A = 1.398m1
' / [ 0- "I " " 1/ i
.tè-o
1
] / _ êA* ' r U T ~ ] [ 10 T(s) 15 20N. Fonseca, C. Guedes Soares /Applied Ocean Research 26 (2004) 256-273
Heave Damping Coefficient (V = O Kn)
1 1 1 i „ _ 1 > i / 1 O . 1 O 1 Calculated " A = 0.466m A A = 0.932m " A = 1.398m 1 1 1 1 U - , 1 Calculated " A = 0.466m A A = 0.932m " A = 1.398m 1 1 Calculated " A = 0.466m A A = 0.932m " A = 1.398m 10 T(s) 15 20
Heave Damping Coefficient (V = 20.17 Kn)
1 L _ - l ^ _ a a A _ J
/ °
- l ^ _ a a A _ JiTt
1 1 0 1 • 1 0 1 Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m i Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m 1 Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m ( 1 Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m 10 T(s) 15 20 8000 7000 6000 1 5000 " 4000 3000 2000 1000 0Heave Damping Coefficient (V = 13.45 Kn)
CD 8000 7000 6000 f 5000 •5) S 4000 ^ 3000 2000 1000 0 Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m 10 T(S) 15 20
Heave Damping Coefficient (V = 26.90 Kn)
1 1 _ _ J 1 & 1
a.-A^-B-(j
a
Vf °V
a
I 1 1 Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m 10 T(s) 15 20Fig. 6. Heave damping coefficient for the whole ship and four speeds.
6.0E+06 5.0E+06 „ 4.0E+06 m E 1 3.0E+06 in ID 2.0E+06 1 .OE+06 O.OE+00
Pitch Damping Coefficient (V = 0 Kn) Pitch Damping Coefficient (V = 13.45 Kn)
8.0E+0& 7.0E+06 6.0E+06 l " 5.0E+06 S 4.0E+0& ^ 3.0E+06 2.0E+06 1.OE+06 O.OE+00-0 1 / A ; U 1 1 J L • 0 1 1 % 1 1 A 1 1 è 1 " ] SS {• Calculated A ° 1
"~'r"i'^\'\
° A = 1° A A = 2° ° A = 3° 1 1 1 1 1 ) 5 10 15 20 T(s)Pitch Damping Coefficient (V = 20.17 Kn)
• • . . i A ' ' ^ 4__ A - A A - ^ ê - ^ D A 0 o 4__ A - A A - ^ ê - ^ D O Calculated ° A = r A A = 2° ° A = 3° Calculated ° A = r A A = 2° ° A = 3° Calculated ° A = r A A = 2° ° A = 3° Calculated ° A = r A A = 2° ° A = 3° 10 T(s) 1.2E+07 1.OE+07 —. 8.OE+06 E S 6.0E+06 ^ 4.0E+06 2.0E+06 O.OE+OC
Pitch Damping Coefficient (V = 26.90 Kn) 1 - ^ - T „ ° ° ° ' ^ | A ^ A 1 " 0 1 1 Calculated ° A = 1° A A = 2° ° A = 3° Calculated ° A = 1° A A = 2° ° A = 3° Calculated ° A = 1° A A = 2° ° A = 3° 15 20 10 T(s) 15 20
264 N. Fonseca, C. Guedes Soai es / Applied Ocean Research 26 (2004) 256-273 70000 60000 50000 40000 30000 20000 10000 O
Heave into Pitcli Damping Coefficient {V = O Kn)
1 1 — — 1 I / k\ AA . -A"! ^ " • - - A - i • D 0 O A ° J _ _ _ _ _ V _ _ ^ - _ - J -^-^a T A 1 • 1 Calculated ° A = 0.466m ^ A = 0.932m ° A = 1.398m • 1 Calculated ° A = 0.466m ^ A = 0.932m ° A = 1.398m 1 1 Calculated ° A = 0.466m ^ A = 0.932m ° A = 1.398m 10 T(s) 15 20
Heave into Pitcfi Damping Coefficient (V = 20.17 Kn) 60000 -40000 20000
I
0 S -20000 m -40000 -60000 -80000L ' ^ ^ A
°A 1 ° fl " i l _ . . è A 2 . . - j . _ • 1 A J - ^ 1 1 , I A_J - -/ -| > • \ 1 t )5 X t O lis 2 Calculated 1 \ ° A = 0.466m A A = 0.932m A A = 0.932m 1_ _ _ 1_ _ _ ° A = 1.398m 1 1Heave into Pitcii Damping Coefficient (V = 13.45 Kn)
60000-—Calculated ° A = 0.466m
A A = 0.932m
° A = 1.398m
Heave into Pitch Damping Coefficient (V = 26.90 Kn) 60000 -0 --: T{s) S -40000 m -60000
Fig. 8. Heave into pitch damping coefficient for the whole ship and four speeds.
Heave Added fVlass - Segment 1 (V = 0 Kn) 4 5 0 0 - 4000- 35003000 -CO CO
<
2 5 0 0 -2000 - 1500- 1000500 0 -Calculated ° A = 0.466m A A = 0.932m " A = 1.398m Calculated ° A = 0.466m A A = 0.932m " A = 1.398m • Calculated ° A = 0.466m A A = 0.932m " A = 1.398m Calculated ° A = 0.466m A A = 0.932m " A = 1.398m o Xr
A - A ^ ^ 10 T(s) 15 20 7000 6000 5000 S 4000 CO < 3000 2000 1000 0Heave Added Mass - Segment 1 (V = 20.17 Kn) Calculated ° A = 0.466m A A = 0.932m / Calculated ° A = 0.466m A A = 0.932m
/
° A = 1.3£ )8m / o / A O A ° A * 5t 1 ^ 10 T(s) 15 20 6000 5000 4000 CT 3000 < 2000 1000 0Heave Added Mass - Segment 1 (V = 13.45 Kn) Calculated ° A = 0.466m A A = 0.932m " A = 1.398m Calculated ° A = 0.466m A A = 0.932m " A = 1.398m
A
Calculated ° A = 0.466m A A = 0.932m " A = 1.398m / a/ .
/ a A o a a A • 10 T(s) 15 20 8000 7000 6000 5000 CT 4000CT
^ 3000 2000 1000 0Heave Added Mass - Segment 1 (V = 26.90 Kn) Calculated ° A = 0.466m A A = 0.932m » A - 1.39Bm / Calculated ° A = 0.466m A A = 0.932m » A - 1.39Bm
/
Calculated ° A = 0.466m A A = 0.932m » A - 1.39Bm/
/
o ^ °. ° ° a a a 10 T{s) 15 20N. Fonseca, C. Guedes Soaies/Applied Ocean Research 26 (2004) 256-273 265 8000 7000 6000 5000 : 4000 3000 2000 1000 O 10000 8000 6000 3 4000 CO < 2000 O -2000 -4000
Heave Added Mass - Segment 1 (V = 26.90 Kn) Calculated ° A = 0.466m A A = 0.932m ° A - 1.398m / Calculated ° A = 0.466m A A = 0.932m ° A - 1.398m
/
Calculated ° A = 0.466m A A = 0.932m ° A - 1.398m/
/
a & a 10 T(s) 15 20Heave Added Mass - Segment 3 (V = 26,90 Kn) Calculated ° A = 0.466m A A = 0.932m " A = 1.398m bA A IB 20 4500 4000 3500 3000 S 2500 <p 2000 1500 1000 500 500
Heave Added Mass - Segment 2 (V = 26.90 Kn) Calculated " A = 0.466m A A = 0.932m ° A = 1.398m Calculated " A = 0.466m A A = 0.932m ° A = 1.398m Calculated " A = 0.466m A A = 0.932m ° A = 1.398m
/
Calculated " A = 0.466m A A = 0.932m ° A = 1.398m a • n ft a • n ft & • n n j - - ^ a^a 10 T(s) 15 20Heave Added Mass - Segment 4 (V = 26.90 Kn)
-500 -1000 -1500 -2000 Calculated •= A = 0.466m A A = 0.932m ° A = 1.398m 1Ë 2Ö T(s) T(s)
Fig. 10. Heave added mass for the four segments and the service speed (26.9 kn).
Heave Damping Coefficient - Segment 1 (V = 26.90 Kn) 2000-1500 1000 CÜ 500 -500 — Calculated ° A = 0.466m A A = 0.932m " A = 1.398m 1^ 15 2b T(s)
Heave Damping Coefficient - Segment 3 (V = 26.90 Kn) 3000 -2500 2000 1500 1000 500 0 A A
AA ^
A ^ a è 0/
• A ^ o A ^ o C/alculated ° A = 0.466m A A = 0.932m ° A = 1.398m C/alculated ° A = 0.466m A A = 0.932m ° A = 1.398m 10 T(s) 15 20Heave Damping Coefficient - Segment 2 (V = 26.90 Kn) 2500-2000 1500 1000 500 0 Z ^-J^^ o " • A/ ^ 1 no ^ ' A | o A
/
7 Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m 10 T(s) 15 20Heave Damping Coefficient - Segment 4 (V = 26.90 Kn) 1400-1200 •a 1000 CT 800
CT
^ 600 400 200 0 a ° ^ n a "o^^
A Op A • Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m • Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m • Calculated ° A = 0.466m A A = 0.932m ° A = 1.398m 10 T(s) 15 202ÓÓ A'. Fonseca, C. Guedes Soares /Applied Ocean Research 26 (2004) 256-273
means that the thi'ee dimensional effects, w h i c h ai^e not accounted by the numerical model, are more strongly f e l t at the ends o f the h u l l .
The first graph i n F i g . 11 shows negative d a m p i n g coefficients obtained f r o m calculations w h i c h do not agree w i t h the experimental data. I n fact the damping coefficients are not w e l l represented by the numeiical m o d e l f o r the higher f o r w a r d speed. A l t h o u g h the n u m e r i c a l heave d a m p i n g coefficients f o r the whole h u l l compare very w e l l w i t h the experimental ones (Fig. 6), discrepancies are observed when one compares the results f o r each o f the segments (Fig. 11). W h i l e f o r the bow segment (segment 4) the numerical results overestimate the experiments, f o r the stern segment they underestimate the experimental data. The global numerical result f o r the whole h u l l becomes good.
Regarding the negatively computed heave damping coeffi-cients at long periods i n F i g . 11, they correspond to the stem segment o f the h u l l . These results arise f r o m the f o r w a r d speed effects i n the f o r m u l a o f the damping coefficient (Eq. 6b). The f o r w a r d speed term is —Ua'^'^'', where 0^3'' represents the added mass o f the f o r w a r d end cross section o f the stem segment, w h i c h when m u l t i p l i e d by the high f o r w a r d speed (U) the result becomes quite large. The other term i n the Eq. 6a is an integral o f the sectional damping coefficient w h i c h f o r very l o w frequencies is close to zero. For these reasons the calculated heave damping coefficient of segment 1 is negative f o r long periods o f oscillation.
5. Motions and sectional loads in waves
T h i s section presents comparisons between n u m e r i c a l results and experimental data, of the wave induced motions and loads on the fast-monohuU. The numerical results ai'e calculated b y the nonlinear seakeeping program described i n Section 2. The experiments were carried out i n head regular waves f o r a w i d e range o f conditions covering:
Four Froude numbers coiTesponding to 0., 13.45, 20.17 and 26.89 knots
Wavelengths w i t h ratios VLpp between 0.24 and 2.43 (A is the wavelength and Lpp is the length between perpendicu-lars)
Three different wave steepnesses coiTesponding to the ratios A/2(^„ = 80, 50 and 35 (i^n is the wave amplitude)
Table 3 lists the characteristics o f the f u l l set of regular waves tested, where / / ^ represents the wave height and is the wave period.
D i f f e r e n t wave steepnesses were used i n the tests i n order to assess the influence of the wave amphtude on the nonlinearity of the responses. I n fact it is f o u n d f r o m the analysis of the experimental data that nonlinear effects are present in a l l responses and they are especially important in the case of the vertical loads. A m o n g the several ship responses that were measured and also calculated, the heave and pitch m o t i o n together w i t h the vertical shear force and vertical bending moment at midship, are chosen to compare results. The
Table 3
Characteristics of the regular- waves tested Wave I D H„ (m) T-.v (s) 4 0.371 4.361 0.24 5 0.743 6.168 0.49 6 1.114 7.554 0.73 7 1.485 8.723 0.97 8 1.856 9.753 1.22 9 2.228 10.683 1.46 10 2.599 11.539 1.70 11 2.970 12.336 1.95 12 3.341 13.084 2.19 13 3.713 13.792 2.43 14 2.228 8.446 0.91 15 2.525 8.991 1.03 16 2.673 9.252 1.10 17 2.822 9.506 1.16 22 3.182 8.446 0.91 23 3.606 8.991 1.03 24 3.819 9.252 1.10 25 4.031 9.506 1.16
calculated time histories are Fourier analyzed i n order to extract the harmonic contents o f the signals. The average o f positive and negative peaks is also calculated.
5.1. Wave induced motions
Figs. 12-15 present results o f wave induced motions, w h i c h include the first and second harmonics as f u n c t i o n o f the nondimensional wavelength X/Lpp. T h h d harmonics o f the vertical motions are very smaU, as weU as the asymmetry o f the positive and negative peaks. For this reason these results are not presented f o r the motions.
The first harmonic amplitudes and the second harmonic amplitudes are represented, respectively, by and where
7 = 3 stands f o r the heave m o t i o n and7 = 5 f o r the pitch m o t i o n . B o t h heave and pitch amplitudes are normalized b y the wave amplitude The experimental results are represented b y the solid symbols and the nontinear calculated results by the lines w i t h open symbols.
Figs. 12 and 13 present the first harmonic amplitudes o f heave and pitch normalised b y the wave amplitude as f u n c t i o n o f the nondimensional wavelength. Each graph corresponds to one Froude number and includes experiments and numerical results f o r the thi-ee wave steepnesses. The analysis o f the results is similar f o r heave and pitch: the numerical results compare very w e l l w i t h the experiments f o r the lower speeds, however, they overestimate the heave and pitch response peaks f o r the higher speeds.
The l i m i t a t i o n s o f the strip method to represent the hydrodynantics at the lower frequency range are not reflected i n the ship motions and this is because at this frequency range the dynanrtcs are mostly goTerned by F r o u d e - K i i l o v and hydrostatics and inaccuracies i n the radiation forces are not important. The linntations o f the strip theory are i n f a c t reflected i n the nuddle frequency range m o t i o n responses. The interpretation o f the results here is not clear also. I f one observes the p i t c h m o t i o n a r o u n d the n o n d i m e n s i o n a l
N. Fonseca, C. Guedes Soares/Applied Ocean Research 26 (2004) 256-273
Heave 2nd harmonies - Fn = 0.0 Heave 2nd iiarmonics - Fn = 0.2 0.08- 0.06- 0.04-0.02 0.00 exp. (Lw/Hw=80) - n u m . (Lw/Hw=80) Heave 2nd iiarmonics - Fn = 0.3 0.08-exp. (Lw/Hw=80) - n u m . ( L w / H w = 8 0 ) 0.04 H 0.02- 0,00-0.08 0.06 ,1.2) 0.04 •] 0.02 0.00 0.080 0.060 Heave 2nd harmonics - Fn = 0.4 f(2) S3 0.040 0.020 0.000 exp. (Lw/Hw=80) - n u m . (Lw/Hw=80)
Fig. 14. Heave motion. Amplitudes of the second haimonics for several speeds.
Pitch 2nd harmonics (7m) - Fn = 0.0 Pitch 2nd harmonics (7m) - Fn = 0.2 0.12-, 1 0-12-i
Lpp '-PP
N. Fonseca, C. Guedes Soares/Applied Ocean Research 26 (2004) 256-273 269
wavelength o f 1.25 f o r the higher speed, then the numerical predictions clearly overestimate the experiments. This is a
'strip theory p r o h l e m ' that has been identified before f o r other ships. The results seem to indicate that the pitch m o t i o n is under-damped or over excited. However, looldng back to F i g . 7, one observes that the numerical p i t c h d a m p i n g coefficients are larger than the experiments. The explanation may i n fact l i e i n the numerical heave into pitch damping coefficients (see F i g . 8) w h i c h is m u c h smaller (negative) than the measured one (positive). Another possibility is that the numerical model overestimates the pitch excitation. A more i n -depth investigation is needed to c l a r i f y these aspects.
Regarding nontinear effects induced by increasing wave amplitudes, they are very small both i n the experiments and the numerical results. The numerical results f o r the higher speeds show a small tendency to decrease the normalised m o t i o n amptitudes w i t h increasing wave amptitudes. Apparenfly the experimental data has the same tendency, but i t is not clear since larger wave steepnesses were tested only f o r a f e w wavelengths below the most interesting range. A n y w a y , nonlinear effects on the m o t i o n amplitudes are very small f o r this h u f l , w h i l e relatively large nontinear effects on the vertical motion amplitudes were detected before f o r a containership model [5,11]. Those resuhs showed a marked reduction o f the amplitude o f the transfer functions w i t h the wave amplitude.
Figs. 14 and 15 present the second harmonic amplitudes of the heave and pitch motions normalised by the wave amplitude. One observes that higher harmonics are small compared to the first harmonics, up to around 4 % f o r both motions, and that the numerical model is able to represent these effects, although not accurately i n some cases.
5.2. Wave induced loads
This section presents graphs w i t h the results o f wave induced vertical shear force at cross section 15 {llALpp f r o m the f o r w a r d peipendicular) and vertical bending moment at midship. The results include the amplitudes o f the first and second harmonics o f the time records, as w e l l as the sagging and hogging peaks. A l l results ai-e presented as f u n c t i o n of the nondimensional wavelength, XlLpp.
V e r t i c a l shear' forces are represented by and vertical bending moments b y M5. The superscripts 'a' stand f o r the first harmonic amplitude, (2) f o r the second harmonic amplitude and max, m i n f o r the magnitudes o f the positive and negative peaks.
The convention f o r the loads is such that sagging is positive f o r the shear force and negative f o r the bending moment. The wave amplitude normalizes a l l vertical shear force and bending moment results. V e r t i c a l shear force results are given i n k N / m , and the bending moment results i n k N m / m . The experimental results are represented by the solid symbols and the nonlinear calculated results b y the lines.
The graphs i n F i g . 16 show the first harmonic amplitudes o f the vertical bending moment at midship f o r the different Froude numbers. The numerical results tend to overestimate the peak o f the experimental transfer f u n c t i o n , and the
difference between results increase w i t h the ship speed. The tendency is the same that was observed f o r the vertical motions, and the largest differences between experiments and numerical results occur, again, f o r wavelengths slightly longer than the ship length. W h i l e the experimental amplitudes are approximately independent of the ship speed, the numerical amplitudes increase w i t h the ship speed around the peak of the transfer function.
The second harmonic amplitudes o f the vertical bending moments at midship are presented i n the graphs o f F i g . 17. Several conclusions can be d r a w n . C o m p a r e d to the corresponding fu'st harmonic amplitudes, second harmonics ai'e very small f o r the stationary ship, but they increase very m u c h w i t h the ship speed up to around 12% of the first harmonics. T w e l v e percent is anyway a relatively small contribution; however, one should note that the wave steepness considered here is relatively small. The higher harmonic content of the vertical bending moment increases also very m u c h w i t h the wave steepness, as demonstrated by Fonseca and Guedes Soares f o r a containership [5,11]. Regarding the numerical results, they compare reasonably weU w i t h the experimental data since both the speed dependence and frequency dependence are coirectly captured.
Fig. 18 presents the steady vertical bending moments, at nudship and section 15, measured w i t h the model advancing i n calm water at different speeds. One observes that the steady moment increases drastically w i t h the f o r w a r d speed and, as w i l l be shown i n the f o l l o w i n g figures, i t reaches magnitudes w h i c h cannot be neglected.
Fig. 19 presents the sagging and hogging peaks o f the vertical bending moment at midship f o r the f o u r speeds. These values are determined by selecting a large number o f cycles o f the response time history (more than 10 cycles), i d e n t i f y i n g the m a x i m u m and m i n i m u m values over each cycle and finally averaging a l l the m a x i m u m and m i n i m a . The lines represent numerical results and the solid circles experimental data as measured. The crosses represent expeiimental data, but where the measured sagging and hogging peaks are subtracted b y the steady bending moment measured w i t h the m o d e l advancing i n calm water w i t h the same speed (only f o r F « = 0.3 and 0.4).
Fu'stly, and comparing the two sets o f experimental data f o r the higher speed, one concludes that the steady bending moment at this h i g h speed is quite large and cannot be neglected. A s a quantitative example, the peak o f the transfer f u n c t i o n f o r Fn = Q.A i n F i g . 16 is 50,500 k N m / m , thus 50,500 k N m f o r regular waves w i t h 1 m o f amplitude. This value is comparable to the steady bending moment i n c a l m water i n F i g . 18 w h i c h is 4 9 , 5 0 0 k N m . Its influence on the m a x i m u m bending moment is obviously relatively smaller f o r larger wave amplitudes, nevertheless i t should always be considered b y a seakeeping code to predict the vertical bending moments i n this type o f fast monohuUs.
The graphs i n F i g . 19 also show that the numerical sagging and hogging peaks compare quite w e l l w i t h the experimental ones f o r l o w speeds, including the asymmetry o f the peaks w i t h respect to zero. For the two higher speeds, the n u m e i i c a l results compare w e l l w i t h the experimental ones w h e n the steady
270 N. Fonseca, C. Guedes Soares / Applied Ocean Research 26 (2004) 256-273 "1^ 80000 60000 40000 VBM at SS10 (kNnVm) - 1st harm. - Fn = 0.0 • evp. (Lw/Hw=80) - ° — num. (Lw/Hw=80) Mg 80000 60000 40000 20000 VBM at SS10 (kNm/m) -1 st harm. - Fn = 0.2 • exp. (Lw/H\v=80) ^ = ^ n u m . (Lw/H\v=80) / • \ / • O O VBM at SS10 (kNm/m) - 1st harm. - Fn = 0.3
'5
60000 40000 A 20000 • exp. (Lw/Hw=80) num. (Lw/Hw=80) # • \ * / © j * \ Al 80000 VBM at SS10 (kNm/m) - 1st harm. - Fn = 0.4 60000 H 40000 20000 1 2 3 0 1 2 '-pp l-ppFig. 16. Vertical bending moment at midship. Amplitudes of the first hannonics for different speeds.
VBM at SSt O (kNm/m) - 2nd inarm. - Fn = 0.2 /W 10000 (2) 5 6000 VBM at SS10 (kNm/m) - 2nd tiarm. - Fn = 0.0 4000 4 2000 1 exp. (Lw/Hiv=80) - n u m . (Lw/Hw=80) 8000 i w f 6000 4000 2000 • exp. {Lw/Hw=80) - 0 — n u m . (Lw/Hw-aO) / e 0 O • looro VBM at SS10 (kNm/m) - 2nd tiarm. - Fn = 0.3 8000-j Mf 6000
'A
40CO exp. (Lw/Hw=80) -num. (Lw/Hw^80) 10000 8000 VBM at SS10 (kNm/m) - 2nd tiarm. - Fn = 0.4 Mf 6000A
4000 exp. (Lw/Hw=80) -num. (Lw/Hw=80}N. Fonseca, C. Guedes Soares / Applied Ocean Research 26 (2004) 256-273 271
Vertical bending moments in calm water
-20000 -60000 —
i \
1
A section 10 —•—section 15 - 11
-I 1 'l 1 1 1 O 0.1 0.2 0.3 0.4 0.5 Froude numberFig. 18. Vertical bending moments in calm water.
moment is removed f r o m the sagging and hogging peaks. Thus one may say that the time domain strip method is able to correctly represent the wave-induced component o f the expeiimental sagging and hogging peaks. However, since there are several discrepancies i n the comparisons w i t h experimental data o f hydrodynamic coefficients, motions amplitudes and bending moment amplitudes, one cannot assure that the good correlation o f the moment peaks can be repeated f o r other headings or other ship types.
Finally the first harmonic amplitudes o f the vertical shear forces at cross secdon 15 are presented i n Fig. 20. The analysis
is similar to the one described f o r the bending moment; the correlation is good f o r l o w speeds, however, as the speed increases the numerical results tend to overestimate the experimental data around the peak o f the transfer f u n c t i o n .
6. Conclusions
The paper presents comparisons between experimental data f r o m tests w i t h a model o f a fast m o n o h u l l i n a seakeeping laboratory and numerical results f r o m a time domain seakeeping code. The invesdgation focuses i n the vertical responses and the comparisons include: global and sectional hydrodynamic coefficients heave and pitch motions and vertical stmctural bending moments and shear forces. The main objectives o f the study ai'e to test the tinuts o f application o f the strip method w i t h regard to the f o r w a r d speed and also to assess i f the nonlinear effects on the vertical responses are coirectly represented b y the time domain model.
The experimental hydrodynamic coefficients are basically tinear w i t h respect to the amplitude o f the forced motions, w h i c h validates one o f the important assumptions o f the time domain model, namely, that the nontinear effects on the vertical responses are dominated by the Froude-lCrilov and hydrostatic components and the radiation forces may be kept linear. However recent investigation is showing that although the hydrodynamic coefficients are linear, the radiation forces may present important higher order effects.
M:
10000 0
5000 0
VBIVl at midship - IVlax. and Min. - Fn = 0.0
{kNm/m) 0 -5000 0 -10000 0 -15000 0 exp. (Lw/Hw=80) -num. (Lw/Hw=80)| ^ M" 10000 0 50000
VBM at midship - IVlax. and Min. - Fn = 0.3
{kNm/m) 0 -5000 0 M -10000 0 -15000 0 2^ ^ Lpp ., with s.w. vbm + exp., no s.w. vbm — num. 10000 0 5000 0
VBM at midship - Max. and Min. - Fn = 0.2
(kNm/m) 0 -50000 -10000 0 a -15000 0 ) \ 1 ''PP • exp. (Lw/Hw=80) —B—num. (Lw/Hw=80) 10000
0-VBM at midship - Max. and Min. - Fn = 0.4
5000 0-{kNm/m) o -50000 • exp., with s.w. vbm[ + exp., no s.w. vbm — num. -10000 0 -15000 0
272 N. Fonseca, C. Guedes Soares/Applied Ocean Research 26 (2004) 256-273 1800 1200 600-^ VSF at SS15 (kN/m) - 1st harm. - Fn = 0.0 VSF at SS15 (kN/m) - 1st harm. - Fn = 0.2 9 exp. (Lw/Hw=80) - • ^ n u m . (Lw/Hw=80)| 1800 1200 600 • exp. (Lw/Hw=80) • ^ n u m . (Lw/Hw=80)| 1800 VSF at SS15 (kN/m) - 1st harm. - Fn = 0.3 12001 600 1800 1200 600 VSF at SS15 (kN/m) - 1 st harm. - Fn = 0.4 9 exp. (Lw/Hw=80) - ^ n u m . (Lw/Hw=80)|
Fig. 20. Vertical shear forces at station 15. Amplitudes of the first harm for different speeds. Regarding the comparison o f calculated and experimental
hydrodynamic coefficients, i n general the agreement is good f o r the heave added mass and damping coefficients and f o r the pitch added inertia. However, large deviations are observed f o r higher ship speeds and l o w e r frequencies o f oscillation, w h i c h reflects the limitations o f the strip theory approach. The numerical model fails to predict the heave into pitch coupled damping coefficients. A l t h o u g h the numerical results represent w e l l the tendencies of the experiments along the periods range, the agreement is poor and clearly worseiung as the speed increases. The resuhs o f the h y d r o d y n a m i c coefficients separated along the four h u l l segments show that the numerical model has more d i f f i c u l t y to represent the experimental data at the ship's ends than at the segments i n the middle. This is probably because the thi'ee dimensional effects are more important at the ship ends.
Regarding the first harmonic amptitudes o f the vertical motions and structural loads, the numerical model compares very w e l l w i t h the experimental data f o r the lower ship speeds, however, i t overestimates the experimental data around the peak of the ü-ansfer functions as the ship speed increases. The higher harmonic content o f these responses detected i n the experimental signals is coiTecÜy captured by the t i m e domain model.
Finally, an analysis o f the experimental sagging and hogging peaks o f the vertical bending moment showed that the steady bending moment, w h i c h is measured also w i t h the m o d e l advancing i n c a l m water, reaches large
magnitudes f o r higher speed w h i c h are o f the same order o f the fir'st harmonic amplitudes measured i n regulai' waves. The numerical method does not represent this effect. I f this effect is removed f r o m the experimental data, then the numerical sagging and hogging peaks compare w e l l w i t h the experimental data.
T h e general c o n c l u s i o n s are that the t i m e d o m a i n seakeeping model represents w e l l the vertical ship responses f o r l o w and moderate speeds, and i t overestimates the motions and structural loads as the speed increases. This is due to the limitations o f the strip theory to represent coirectly the f o r w a r d speed effects.
Acknowledgements
The presented w o r k was performed w i t h i n research project BE-4406, Advanced M e t h o d to Predict Wave Induced Loads f o r H i g h Speed Ships ( W A V E L O A D S ) . This project was p a r t i a l l y f u n d e d b y the C o m m i s s i o n o f the E u r o p e a n C o m m u n i t y under the B R I T E / E U R A M program; contract BRPR-CT97-0580.
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