Deift University of Technology
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Experimental and Numerical Study of Semi-displacement
Mono-hull and Catamaran in calm water and incident waves
C. Lugni', A. Colagrossi', M. Landrini', O.M. Faltinsen2
(1INSEAN, The Italian Ship Model Basin, Roma
-
Italy,
2Centre for Ships and
Ocean Structures, NTNU, Trondheim - Norway)
ABSTRACT
There is a broad variety of high-speed vessels. In the
present work, the attention is focused on semi
displace-ment mono-hulls and catamarans both in calm water and in incident head sea waves. Experimental and nu-merical studies have been performed to investigate the
main features of the flow pattern near the bow, along the vessel and downstream the transom.
In the steady model tests, the Froude number has been varied in a wide range and the interaction between
the two catamaran demi-hulls was studied by compar-ing the related flow field with the one generated by a mono-hull identical to a catamaran demi-hull. The ex-periments data have been compared with results of lin-ear 3D and nonlinlin-ear 2D+t computations. In the mono-hull case, for ship-length Froude number smaller than 0.6 the former capture the phasing of the wave pattern
and give more reliable quantitative information. At hi-gher speeds, the latter agree better with the
measure-ments. This is due to two factors: the transverse waves become less relevant and the importance of nonlinear-ities increases. In the catamaran case, the same trend is observed but the 2D+t theory gives globally the best
agreement at smaller speeds than for the mono-hull. For
both the models, the linear code is not able to predict the stern flow correctly at ship-length Froude equal or greater than about 0.5. This is because, from this value on. experimentally the transom stern stays dry during the vessel motion. The 2D+t method was used to
ana-lyze the mechanisms driving the transom flow field and the interaction between the demi-hulls.
In the unsteady experiments, different wave
am-25th Symposium on Naval Hydrodynamics
St. John's, Canada, 8-13 August2004
plitudes and vessel speeds have been considered and the
incoming wave frequency was varied in the heave and
pitch resonance range. This study highlighted the effect
of nonlinearities on the mono-hull and catamaran flow
patterns and showed the relevance of the interaction
be-tween steady and unsteady flows on the wave induced body motions. Such investigation was performed by combining in a synergic manner the experimental data
with 3D linear computations. For the considered speeds and within the resonance range, the measured mean trim
and sinkage of the catamaran are mainly governed by
the steady flow.
INTRODUCTION
The design features of high-speed vessels in use can be very different. The vessel weight can be supported by
submerged hulls, hydrofoils, air cushion, or a combina-tion of these effects. Cavitacombina-tion and ventilacombina-tion on foils,
struts and propulsors limit the speed. Mono-hulls and catamarans, often equipped with foils, trim tabs andlor interceptors to minimize wave induced motions, repre-sent nowadays the most popular concepts. Catamaran designs include the wave-piercing and semi SWATH
(Small Water-plane Area Twin Hull) style hulls. A mono-hull with the same displacement as a catamaran is char-acterized by a lower wave induced vertical acceleration
since its larger length is beneficial from this point of
view. The beam-to-draught ratio B/D of high-speed mono-hulls may vary from around 5 to values larger than 7. Large B/D values result in more limited accel-erations in heave and pitch motions. Faltinsen (1990).
However the roll motions of monohulls need special
natural periods and generally lower vertical excitation loads than a similarly sized catamaran. On the other hand, if control surfaces are not introduced, a
thresh-old Froude number exists beyond which the SWATH is
dynamically unstable in the vertical plane. When
oper-ating in head-sea conditions, its seakeeping behaviour is better than the one of a corresponding catamaran.
How-ever, if the sea state, speed and heading cause resonant
vertical motions, the SWATH may not have a good
sea-keeping behaviour. Surface Effect Ships (SES) use an
air cushion mechanism to obtain the desired cruise
ve-locity and performances. The excess pressure in the air cushion between the two SES hulls lifts the vessel and carries about 80% of its weight. On the other hand, it
reduces the metacentric height and causes wave
genera-tion and addigenera-tional wave resistance. However, the total calm water resistance is smaller than the one of a cata-maran of similar dimensions. A drawback with an SES is that it can suffer more speed loss in waves than for instance a catamaran. Further, the skirts in the how of
an SES are easily worn out. Trimarans and pentamarans with a large center hull combined with smaller outrigger hulls represent interesting new concepts.
In the case of mono-hulls and catamarans, com-prehensive experimental studies have been performed by Molland et al. (1995) in calm water conditions. The model tests showed that the forward speed effect on
trim and sinkage starts to be significant at a ship-length
Froude around 0.35. This has important consequences on the vessel performances since trim and sinkage are directly connected with the resistance and matter also for the wetdeck slamming on multi-hull vehicles.
Doc-tors (2003) conducted calm water tests on five different transom-stern mono-hull models to investigate the
tran-sition from wet to dry transom. A dry transom is gen-erally beneficial and affects the trim and rise as well as
the damping of vertical ship motions. The author
identi-fied the transom-draft Froude number as the parameter governing such transition. Keuning (1988) performed high-speed mono-hull experiments in calm water and head sea waves and analyzed the wave elevation near the vessel and the variation of the hydrodynamic co-efficients along the ship hull. Forced heave and pitch model tests on a high-speed catamaran were presented
in Ohkusu and Faltinsen (1991). A weak interaction
be-tween the demi-hulls was suggested by the fairly good agreement of the measured hydrodynamic forces with numerical 2D+t results based on the assumption of
hy-drodynamically independent demi-hulls.
The numerical studies of wave resistance and
wave induced ship motions on mono-hulls and
catama-rans are mainly based on the assumptions of incom-pressible fluid in irrotational motion. The solution of the more general unsteady Navier-Stokes Equations is still in its infancy and the use of Reynolds Averaged Navier-Stokes equations would lead anyway to uncer-tainties in the turbulence modeling. Linear wave resis-tance analyses based on the Michell's thin-ship theory (Michell 1898) and accounting for the transom effect have been presented by different researchers (i.e. Mol-land et al. 1994. and Doctors and Day 1997). The
de-veloped methods have been applied to high-speed
mono-and multi-hull vessels. In the latter case, the
diffrac-tion of the waves generated by one demi-hull due to the presence of the other demi-hull is not accounted for, i.e.
the waves generated by the separated hulls are simply superimposed. This type of analysis is very efficient and important at a pre-design stage. However, the role played by the nonlinearities becomes more and more relevant as the Froude number increases. The same is true for the interaction between steady and unsteady flows when the seakeeping is of concern. In principle
three-dimensional effects should be accounted for.
Sev-eral numerical codes exist handling them properly for the linear seakeeping case. However, the solution of the fully nonlinear three-dimensional problem is
chal-lenging from the numerical and CPU time requirements
points of views. In terms of nonlinear effects, sim-plifications are often made, for instance, by including nonlinearities associated with hydrostatic and Froude-Kriloff loads. In terms of three-dimensional effects, rather accurate approximate solutions may be obtained by using physical considerations. This is based on that
divergent waves dominate for a semi displacement
ves-sel in deep water. The main 3D effects can therefore be adequately captured by using a 2D+t (also referred to as 2.5D) theory. In this case the longitudinal flow
variations are assumed smaller than the transverse ones, leading to a sequence of 2D problems in the ship
cross-planes. The 3D information travels one way only, that is the solution in each cross-plane is influenced just by
the flow upstream. Faltinsen (2001) pointed out the
rel-evance of 3D flow effects in the close vicinity of both the bow and the transom stern. These aspects represent a limitation for the 2D+t theory since this one assumes that both velocity potential and free-surface elevation
are zero at the bow and it is not aware of what happens downstream. The latter implies a predicted pressure at
the transom stern different from the atmospheric value. These considerations are also relevant in unsteady flow
conditions. Fontaine et al. (2000) accounted for the
bow wave elevation by combining a 3D bow model with
the 2.5D theory. A domain decomposition method (see
i.e. Greco et al. 2002) can be introduced to couple a global 2D+t modeling with a local 3D analysis for the description of the transom-stern flow. The same
strategy can also be applied to handle other phenom-ena characterized by three-dimensional flow features,
i.e. the water on deck and the wetdeck slamming. If
the water-hull interaction is not characterized by
partic-ularly small angles between the impacting free surface and the hull, slamming loads can be modeled as an in-tegrated part of the analysis. Otherwise local hydroe-lasticity will matter. In this case it is not practical to
treat the slamming phenomenon within the global anal-ysis due to the very different time scales involved.
Ob-viously, the global elastic effect of the slamming must be included in the global analysis to describe properly
the occurrence of whipping. Springing. i.e. steady-state
wave induced global resonant hydroelastic vibrations, may be a relevant fatigue issue for larger high-speed
vessels. Since the natural frequencies are high, head sea
are of major concern in this context. Springing may be excited by linear and nonlinear wave effects. The
lin-ear wave excitation corresponds to small incident wave
lengths relative to the ship length. The spatial oscilla-tions of head-sea waves cause strong variaoscilla-tions of the flow in the longitudinal ship direction. As a result, in this case a 2D+t theory is not suitable.
Still many mechanisms are not fully understood and quantified. For instance the relevance of nonlinear effects including the interactions of the steady and un-steady flows. The study of such aspects is more chal-lenging for the catamarans where the interaction
be-tween the demi-hulls represents an additional factor
en-tering the problem. Present research work aims to a
more clear picture of the flow features and of the sea-keeping properties of these vehicles. Consequently a
dedicated experimental investigation of the effect of the vessels forward speed has been carried out both in calm
water and in incident head sea waves by using a mono-hull and a catamaran model. Several Froude numbers, wave amplitudes and wave frequencies have been se-lected. The mono-hull geometry is used to analyze the
interaction between the demi-hulls of the catamaran.
Therefore it has been shaped identically to a demi-hull. implying a much smaller beam-to-draught ratio than the one characterizing the usual mono-hull high speed ves-sels. Detailed measurements of the steady and unsteady wave-field features have been made both along the hull
and downstream the transom stern. Also the flow area
between the two demi-hulls of the catamaran was
stud-ied. The experiments have been compared with nonlin-ear 2D+t and linnonlin-ear 3D Rankine Panel Method compu-tations. The results confirmed the relevance of a
non-linear flow description and the validity of a parabolized
approach in case of sufficiently large Froude numbers. In the next two sections the mathematical models used for the numerical computations are described. Section 3 deals with the experimental set-up. A numerical and experimental study for monohulls and catamarans in steady conditions is then reported in Section 4. Finally in section 5 the unsteady motion of the catamaran in
waves is considered.
i
2D+t THEORYThe 2D+t theory leads to a sequence of 2D problems to be solved in the transverse cross-sectional planes of the
vessel. The 3D effects are partially accounted for since
the generic cross-section is influenced by the flow in the
upstream cross-sections of the hull. The nonlinearities of the problem are retained. For vertical ship motions and steady symmetric flows (straight coarse), this
ap-proach is suitable at sufficiently large Froude numbers,
let us say larger than 0.4, so that the ship transverse
wave system is not dominating. For horizontal wave in-duced motions and steady antisymmetric flows
(maneu-vering), the transverse waves are less important also at
small Froude numbers, i.e. the 2D+t theory can be used for Froude number smaller than 0.4. Faltinsen and Zhao (1991) used the 2.5D method to study the ship motions
of high speed mono-hulls. Nonlinear steady and lin-ear unsteady analyses were considered and the interac-tion between steady and unsteady flows was accounted for. Maruo and Song (1994) retained also the nonlearities in the unsteady problem but assumed linear
in-cident waves. In our case, the 2D+t theory is applied to investigate the steady flow patterns for mono-hulls and
catamarans. In the latter case the interaction between the demi-hulls is accounted for. No correction of the
intro-duced. Damping foils, trim tab and interceptors are not
modeled. In the following the solution method is briefly
outlined under the assumptions of inviscid steady flow
induced by a ship with constant forward speed and fixed trim and sinkage.
Figure 1: Qualitative sketch of the 2D+t
approxima-tion for the steady three-dimensional flow around a ship
with constant forward speed U. Left: 3D ship problem.
Right: equivalent unsteady 2D problem (2D+t). Let us consider a ship moving with constant
ve-locity U (see left sketch in Figure 1). We assume a
beam-to-draft ratio B/D 0(1) and both B and D
individually much smaller than the ship length L. say
r = B/L, D/L « 1. We also assume the Froude
num-ber Fr = U//L = 0(1).
In a ship-fixed frameof reference, the hull geometry is given implicitly as N(x, y, z) = O and the a priori unknown free surface
can be represented as )'V(x, y, z) = z - (xx, y) = O.
Assuming that the fluid is in irrotational motion, the flow field is described by the Laplace equation
com-bined with the kinematic hull (7-1) boundary condition
VVN=O
(1)and the kinematic and dynamic free surface (W) bound-ary conditions
VVW=O
andp=O
(2)respectively. The statement of the problem is completed
by the upstream radiation condition
Ux
as x p
(3)It is convenient to formulate the problem in terms of the
perturbation potential
, linked to 1 by 4 = Ux +
. The longitudinal gradient /3x can be neglectedwith respect to those in the transverse plane. There-fore, the problem for reduces to a sequence of two-dimensional problems in the ship cross planes. The
re-lated boundary conditions imply the generic cross plane
ì2D is influenced by the upstream solution and unaware
of the following cross sections. The problem sequence can be solved once given the conditions at the bow and
the far-field behavior of the solution at each hull cross section. In a fixed frame of reference with respect to
the unperturbed fluid the problem can be re-written as a two-dimensional time-dependent problem. This is fully
equivalent to the unsteady problem of the free-surface flow generated by a body deforming in time (see right sketch in Figure 1). Consistently, the resulting approx-imation is called here 2D+t. The 2D unsteady prob-lem is solved through the Mixed Eulerian Lagrangian
method (Longuett-Higgins and Cokelet 1 976, Faltinsen
1977. and originally suggested by Ogilvie 1967), that is the problem is split in a kinetic and a time evolution step. The kinetic problem for is solved by means of the Green's second identity used as integral representa-tion of the velocity potential. Applying the latter at the domain boundary leads to an integral equation for the unknown boundary data 5z/3n and , on the free
sur-face and body boundary, respectively. The continuity of the velocity potential is enforced at the intersection
points between the body and the free surface. A Bound-ary Element Method (BEM) with linear shape functions
for the geometry and boundary data is then introduced. The relevant integrals are computed analytically, and after some manipulations the discretized integral equa-tions lead to a system of linear algebraic equaequa-tions for the unknowns at the collocation points. The system
in-fluence matrices are only dependent on the geometry of the problem. In the time evolution step, the free-surface
boundary conditions, expressed in a Lagrangian form, and the body velocity are integrated in time to provide
the new boundary configuration and related data for the
next time instant. The time stepping is performed by a
fourth-order Runge-Kutta method. The discretization of
the free surface is controlled through numerical
regrid-ding and the grid refinement is adapted to the evolution
of the solution. If the angle between the body and the
free surface becomes too small, the jet-like flow created
is partially cut to avoid numerical errors (cf Zhao and Faltinsen 1993). Unphysical reflection of the outgoing
waves is prevented by using a damping layer technique (for short waves) and a panel stretching (for longer wave components) toward the edges of the computational do-main. Invariance of the solution under mesh refinement and size changes of the computational domain has been widely checked. Since a BEM is used, bow wave
post-breaking phenomena cannot be studied. The post-breaking
is limited by cutting off the jet flow in the plunging bow
waves. This is not believed to be an important error
Free Swface(x)
nek
/
Thd) Seeth,,. (e) d/dl.p=eone
N
source.
2 LINEAR 3D METHOD
In our analysis of the steady and unsteady mono-hull
and catamaran flows, we used a linear 3D method as
ad-ditional numerical instrument. In the following its
fea-tures are briefly outlined, for more details see i.e. Nakos (1990).
We consider the free-surface flow generated by a ship advancing at constant forward speed U in
reg-ular incoming waves. The problem is solved by
us-ing the potential flow theory and neglectus-ing the non-linearities connected with the wave-body interactions. In case of vessels with transom stern, the latter is con-sidered always wet. The total velocity potential of the fluid is decomposed as the sum of the steady and un-steady wave fields, the latter consisting of the incident, the diffraction and the radiation waves. Then the prob-lem is split into eight sub-probprob-lems: one steady and seven unsteady. Assuming small amplitude of the inci-dent waves and ship motions, we linearize the free sur-face conditions for the unsteady problems around the steady free surface. Then a further linearization is per-formed. Two alternatives are considered: the Double-Model (DM) and the Neuman-Kelvin (NK) lineariza-tion (see i.e. Nakos 1990). These are, respectively, a low-speed and a slender-body approximation. In this
way, the free-surface boundary conditions can be trans-fered on the undisturbed free surface and the steady and unsteady problems may be solved separately. Once this is accomplished, the pressure can be evaluated from the
Bernoulli's equation. Its integration over the ship
sur-face furnishes the added mass and damping coefficients,
the restoring forces and the wave exciting forces. The restoring terms are due to both the hydrostatic pressure and the steady hydrodynamic pressure. The Response
Amplitude Operator (RAO) is obtained by coupling the
fluid dynamic problem with the body motion through the hydrodynamic loads. Finally, the steady wave pat-tern and the radiation and diffraction waves are
calcu-lated from the recalcu-lated kinematic free-surface conditions. Here both the steady and unsteady potentials are represented in terms of source distributions on the body
7-/ and the free siirfare W The nrnhle.m is then snlved
system of algebraic equations for the unknown source strengths at the collocation points. The collocation
po-ints on 7-/ are placed at the centers of the corresponding
panels. The same is made for those along the free sur-face but additionally they are rigidly shifted one panel downstream in order to enforce numerically the radi-ation condition (see i.e. Bertram 1990). In practice, this numerical radiation condition is valid only for r = Uw/g> 0.25, when waves do not propagate upstream the ship. Here We is the encounter circular frequency. An important task is represented by the evaluation of the rn terms in the body boundary conditions for the radiation problems. They represent the interaction with the steady flow and their estimate can lead to relevant numerical errors. Therefore the Tn- terms are often ne-glected. Here, in the DM case, they are estimated by using an extrapolation procedure for the velocity gra-dient on the body boundary. In the NK case the effect of the local steady flow is not incorporated in the rn terms and the evaluation of the non-zero rn terms is
straightforward.
3 EXPERIMENTAL SET-UP
A dedicated and comprehensive experimental investi-gation has been performed to analyze the steady and unsteady behaviour of semi-displacement mono-hul Is
and catamarans. A catamaran model was built
con-sistently with the geometric ratios normally used for semi-displacement catamarans. The main characteris-tics are reported in table 1. The same parameters have
been considered for the mono-hull geometry coinciding
therefore with a catamaran demi-hull. This leads to a
shape finer than those of the existing semi-displacement
mono-hulls. The experimental activity has been car-ried out at the INSEAN basin No. 2: 220 m long, 9 m
large and 3.6 m deep. During the tests, each model was towed by the carriage through a constant force
mecha-nism. Trim and sinkage were free while the center of
rotation was fixed to the center of gravity of the vessel.
The model tests reproduced two main conditions:
for-ward motion in calm water and in head sea waves. In the
former case, several Froude numbers have been
inves-tigated for the mono-hull geometry: Fr = U//L = 0.3 - 0.8 with a step iFr = 0.1. A smaller number of
Figure 2: Body plan of the demi-huit and hydrostatic properties. 2p indicates the distance between the cen-trelines of the catamaran demi-hulls, cf Figure 3.
the hull. This was composed by an array of 40 trans-ducers placed transversally to the basin. The distance between two consecutive wave probes was 4 cm. To refine the grid of measure each run was repeated shift-ing the sensor array of 2 cm transversally to the vessel
axis. With this set-up. the wave field has been recorded within a lateral distance of 0.5L from the model hull. A very detailed picture of the steady wave pattern has been achieved, also including the area between the two
demi-hulls in the catamaran case. Figu-re 3 gives a sketch of
L-4m X
=6.25
L =25 m LCG = 1.7 mKG =3
m = 0.26 L T=2
m BWL = 175m 2p=5
m A =40.48mFigure 3: Sketch: top view of the wave probes (indi-cated by the dots) used along the external profile of a
catamaran demi-huIt and along the vessel central axis.
the wave probe arrangement along the external profile of a catamaran demi-hull and along the vessel central
axis. The external probes were placed to follow the ship
profile at a distance of 3 cm from the hull. The sensors
were fixed to the carriage and therefore moved with the
vessel forward speed. The same wave probe locations
as those used to measure the external wave pattern of
the catamaran have been considered for the mono-hull. In the tests in waves, experimental transfer
func-tions in heave and pitch were estimated both by a
tran-sient test technique and regular waves of different
steep-nesses. More in detail, Response Amplitude Operators (RAO) have been determined preliminarily by a tran-sient test technique. In this way the frequency range characterizing the resonance area of heave and pitch
motions has been identified. Then, for this range of fre-quencies. tests have been carried out in regular head sea
waves with different steepnesses and considering seve-ral Froude numbers. Also in this case we measured the wave profile along the hull and the centreplane of the
catamaran.
Our studies are of fundamental nature and par-ticular care has been taken in performing a dedicated error analysis; for instance each test condition was re-peated between 5 and 10 times to ensure repeatability. The error analysis did not investigate the error bias but just the precision error. Results of such study will be discussed in the following sections for some variables of interest. Due to space limits the unsteady
investiga-tion will be presented for the catamaran only.
4 DISCUSSION: STEADY CASE
Systematic calm water experiments have been carried
out for both the mono-hull and catamaran models. Such study aimed to a better understanding of the steady wave
pattern features and to verify the validity and limits of the numerical methods presented in sections 1 and 2.
The NK linearization is used for the linear 3D (3D RPM)
results, that is the basis steady flow is assumed given just by the uniform flow due to the ship speed. The
re-sults of the repeatability analysis for the model tests are
reported by presenting the experiments as mean mea-sured values and error bars. The latter are given as +a, with a the standard deviation. The wave profiles along the mono-hull are presented in figure 4 for two selected Froude numbers, respectively, Fr = 0.5 (top) and Fr = 0.7 (bottom). Numerically the wave eleva-tion was calculated following the ship profile at a dis-tance of 3 cm from the hull. This was made to be
con-sistent with the measurements (see section 3). Globally the agreement between the experiments and the
0.8 -0.8 1.6 -1.6 -d.4 2.21 -0.2 0 1.28 0.99 020.84 102 04 x/L 0.74 Fr,
Figure 4: Wave profile along the mono-hull for
Fr = 0.5 (top) and Fr = 0.7 (bottom). Experiments
(symbols: mean value and error bar) and numerical re-suits obtained by the 3D RPM (dashed line) and 2D+t
(solid line) methods. Fr = U/J, with x the
longi-tudinal distance from the bow.
can be detected in the bow area between the model tests and the 2D+t theory at the smaller Froude number. The former predict a negative wave elevation, the latter gives
a quite large free-surface rise along the hull. This dis-agreement can be partially explained by ventilation of the sensors in the bow area. Such phenomenon implies
an underestimated wave elevation since the probes stay
more dry than they should be. This is indirectly
con-firmed by a larger experimental error bar near the bow.
The two results fit quite well at the larger speed. In
this case, the 2D+t model is able to capture the bow
splash phenomenon. The stern wave disturbance is
pre-dicted correctly by the nonlinear method for both the
speeds. Also at this Froude number the experimen-tal repeatability is less satisfactory in the vicinity of
the bow, the same phenomena described for the smaller
speed are believed to be the main reason for it. The
3D RPM results agree better with the experiments at the smaller speed. when the nonlinearities are less im-portant and the three-dimensional effects are stronger. At the larger Fr the agreement is stili satisfactory but a
smoother bow rise-up and a deeper trough near the stern
are predicted as a consequence of the linear
formula-tion. The steady wave patterns at Fr = 0.5, 0.6, 0.7
and 0.8 are reported in figures 5 and 6 where the
exper-iments are compared with the 2D+t and 3D RPM re-suits. respectively. For the smallest ship-length Froude
number shown. the linear 3D computations capture
cor-rectly the phasing of the wave pattern and give
reh-able quantitative information in the bow area and along
the ship hull. The stem and downstream flows are not properly described since the numerical transom is as-sumed wet while at this speed the experimental tran-som was already dry. Such aspect will be discussed hater. Concerning the 2D+t results, the agreement with the experiments is just fair along the hull in terms of the phasing of the wave system generated. The
pre-dicted amplitudes are closer to the measurements in the
bow area and tend to underestimate them largely going downstream. A global extension of the lobes (contour lines with constant elevation) is observed and it is not shown by the measurements. The comparison slightly improves behind the stem. This suggests the relevance of nonlinearities in the wake and the need of handling
the dry stern condition. In this context, the 2D+t theory
assumes that the flow leaves tangentially the transom stem in the downstream direction. This approximation appears suitable to capture the transom flow behavior. Ohkusu and Faltinsen (1990) showed that theoretically the 2D+, theory should not be applied for ship-length Froude smaller than 0.4. A reason is that the method neglects the transverse wave system generated by the ship while this is important at such speeds. As Fr
in-creases, the transverse waves become longer and the
re-lated importance reduces with respect to the divergent waves. Therefore a 2D+t formulation can handle sat-isfactorily the features of the wave pattern. According to our results, reliable results can be obtained for Fr at least larger than 0.5. Faltinsen (2000) defined a local
Froude number Fr =
with i thelongitudi-nal distance from the ship bow. He used such a local
number to estimate the goodness of the 2D+t theory for
the ship bow waves prediction. A similar local Froude number should be introduced for the ship waves
gen-erated at the transom. In this case i is the longitudi-nal distance from the ship stern. For a given Fr. the
2D+t results can be considered suitable within the
re-gion where Fr is sufficiently large, say greater than
0.4. Obviously such region enlarges as the ship speed increases. The local bow Froude number is reported in
05 -05 -05 0 Fr. 071 Fr 06 05 -05 -05 0 Fr .' 085 Fr=0.7 o-5 -03 -05 0 Fr .- 099 Fr 0.8 05 -05 Fr = 03 J xIL
fl
041 022 Ezp 05 i xìLJi
0.6 049 042 os J iL Ji 07 057 030 I .1 I I I -os o os ¡ iL Ji Fr, 113 08 065 057Figure 5: Contour lines of the mono-hull steady wave
pattern. From top to bottom: Fr = 0.5, 0.6, 0.7 and
0.8. In each plot the experiments (bottom) are
com-pared with the 2D+t results (top). Fr = U/«/, with
x the longitudinal distance from the bow.
05 -05 05 -05 05 -05 RPM 3D ExperÉnts 05 RPM 3D
Figure 6: Contour lines of the mono-hull steady wave pattern. From to bottom: Fr = 0.5, 0.6, 0.7and 0.8. In each plot the experiments (bottom) are compared with
the 3D RPM results (top). Fr = with i the
longitudinal distance from the bow.
-05
Exs
-05 0 05 J xiL Ji Fr, 113 08 065 057 -os o Fr. 071 Fr = 0.6 05 05 i xiL Ji 0.41 0.32 05 J xiL Ji 06 0.49 042 -03 0 Fr 0.85 Fr- 0.7 05 1 7JLii
07 0.57 050 -05 0 Fr 099 Fr = 0.8Globally, for Fr larger than 0.5 the 2D+t theory gives
better results than the 3D RPM code. This is due to two factors: the transverse waves become progressively less relevant and the importance of nonlinearities increases.
The linear results underestimate the peaks and troughs
along the hull and in the wake. However the dimension
of the lobes of the global wave system is reproduced
quite nicely while the 2D+t results show a wider exten-sion of the lobes even at the largest speed.
The wave profiles along the catamaran are pre-sented in figure 7 for the two largest speeds tested, that
Figure 7: Catamaran: external (top) and centre-line (bottom) wave profile (cf
figure 3) for Fr = 0.5
(left) and Fr = 0.6 (right).
Experiments (symbols: mean value and error bar) and numerical results ob-tained by 3D RPM (dashed line) and 2D+t (solid line).Fr = U/,/gx, with x the longitudinal distance from
the bow.
are Fr = 0.5 (left plots) and Fr = 0.6 (right plots).
The top and bottom plots refer, respectively, to the ex-ternal and centre-line wave profiles, as shown in figure 3. The agreement among the results is generally fairly good. Globally the 2D+t theory compares better with
the model tests, although local discrepancies can be
de-tected, in particular in the external aft hull. Here the experiments show a large error bar for Fr = 0.5 and a
deeper trough than the 2D+t results for Fr = 0.6. At
the latter Froude, in the stern area they are closer to the
linear 3D calculations. An error bar not negligible is also noticed at the aft profile along the centre-line for the smaller speed. The 3D RPM code predicts a lower
external bow splash than the experimental and 2D+t re-sults. Also the free-surface peak at the centre-line is underestimated. Differently larger values for the
exter-nal and inner troughs are predicted. The 3D RPM wave
profiles seem to have a phase shift with respect to the other results at the smaller speed. This suggests a role of the nonlinearities both externally to the catamaran
05
-05
Experimerns
o os J xiL Ji
Fr 0.85 06 049 042
Figure 8: Contour lines of the catamaran steady wave
pattern. Top: Fr = 0.5. Bottom: Fr = 0.6. In each
plot the experiments (bottom) are compared with the
2D+r results (top). Fr = U/'7, with x the
longitu-dinal distance from the bow.
and between the demi-hulls. The steady wave patterns
for Fr = 0.5 and 0.6 are reported in figures 8 and 9
where the experiments are compared, respectively, with
the 2D+t and 3D RPM predictions. The linear 3D code is more able to capture the global picture of the wave
field along the vessel, showing that the transverse waves
play still a role. However, for both the speeds, it does not quantify correctly the free-surface disturbance due to the ship forward motion. This implies that nonlinear effects matter. The 2D+t theory shows a wide exten-sion of the lobes near the bow, similarly to the mono-hull case, and it has a relevant phase difference with
respect to the experiments at the lower Froude.
De-spite this, even at Fr = 0.5 the peaks and troughs are closer to the measured values than the results obtained
with a linear method. Figure 10 shows the same
re-sults for Fr = 0.7 (top) and 0.8 (bottom), as obtained
these larger speeds the nonlinearities are quite impor-tant, therefore the linear results underestimate substan-tially the wave pattern. On the other hand, it is inter-esting to note that the two results agree quite well in terms of phasing. This confirms the unimportance of the transverse waves at such Froude numbers. Top
05 -05 Fr 05 -05 Expermnts -05 0 071 Experiments -1 t 05 1 xiL 15 05 041 0.32 Fr = 0 7 2D.t 05 1 zJL 15 0.7 057 050 .05 0 05 1 JL
is
FrIii
08 0.65 0.57Figure 10: Contour lines of the catamaran steady wave
pattern. Top: Fr = 0.7. Bottom: Fr
0.8. In eachplot the 3D RPM (bottom) are compared with the 2D+t
results (top). Fr =
with x the longitudinaldistance from the bow.
dry. A nonlinear BEM solver, as used in the present study, was adopted to simulate the flow evolution un-til the incipient wave breaking and a Smoothed Parti-cle Hydrodynamics (SPH) method was initialized by the BEM to handle the post-breaking evolution of the
wave system. in the bottom of the figure 11, the re-sults obtained by using just the BEM 2D+t method for Fr = 0.5 are given. As we can see, the 2D+t formu-lation is able to reproduce the flow scenario behind the
transom: hull hollow, rooster tail and incipient breaking divergent wave system. Nevertheless, since the
plung-ing jet is cut to avoid the occurrence of impact on the
underlying water, the energy of the wave system is
fo-cused close to the crest of the divergent wave. Differ-ently, in the physical phenomenon the breaking causes a spatial spread of the wave energy. Figure 12 shows the longitudinal wave cut along the centre-line of the
-05 0 05 J x/L 15
Fr, 085 06 049 042
Figure 9: Contour lines of the catamaran steady wave
pattern. Top: Fr = 0.5. Bottom: Fr = 0.6. In each
plot the experiments (bottom) are compared with the 3D
RPM results (top). Fr = U/,', with x the
longitu-dinal distance from the bow.
picture of figure 11 shows the experimental wave field behind the transom for the mono-hull at Fr = 0.5. The dry transom causes the formation of a hollow just be-hind the vessel. The water reaches a minimum value and then rises to form a rooster tail developing into a divergent breaking wave system. The mono-hull
tran-som flow features have been thoroughly investigated by
Landrini et al. (2001) due to the practical relevance
of the resulting breaking phenomena. These lead to
vortical structures responsible of the visible signature left downstream by the ship. In their study, the authors used the 2D+t theory. The transom was enforced to be
05 -0.3 Fr, -05 RPM 3D o 099
0.1
-0.1
0.2 0.4 .t/L
Fr 1.12 0.79
Figure 11: Mono-hull: transom stern wave field at
Fr
= 0.5. Experimental picture (top) and contour linesof the steady wave pattern predicted by 2D+r theory
(bottom). Fr
= U/J,
with z the longitudinaldis-tance from the transom.
mono-hull transom as obtained by the 2D+r theory. The results show that the rooster tail height is not affected by
the Froude number while both its horizontal width and
the extension of the hollow increase with the speed. The
hollow extension can be measured as the longitudinal distance between the transom position and the location where the free surface becomes zero. The wave
ele-vation downstream a catamaran demi-hull transom (see figure 13) shows a quite different behaviour. Except for
the smaller speeds, showing an increase of the hollow extension with the Froude number, the hollow width is not particularly affected by the speed. The rooster
tail height is lower than the corresponding value for the
mono-hull. It shows a non monotonic but rather lim-ited variation with the Froude number. This demi-hull wave behaviour behind the transom is due to the pres-ence of the other demi-hull. The arrangement of the two demi-hulls causes three rooster tails downstream the catamaran, respectively, in correspondence of the demi-hull transom sterns and of the catamaran
centre-line
(cf 2Dt
contour plots in figures 8 and 10).It is important to understand if the main responsible
02 03 0.4 0.5
Figure 12: Mono-hull: cut of the steady wave pattern a ong the centre-line of the transom stern. z is the
lon-gitudinal distance from the transom.
S
(J J, 07 ,JL
- 6
0. ((2 0.3 0.4 07 ,./L
Figure 13: Catamaran: cut of the steady wave pattern along the centre-line of the demi-hull transom stern. z
is the longitudinal distance from the transom.
mechanisms are connected with the hydrodynamic in-teraction between the demi-hulls or if they are related
to the demi-hulls interference only. The latter means the
diffraction caused by one demi-hull on the waves
gen-erated by the other demi-hull is negligible and the cata-maran wave pattern is just given by the sum of the wave fields produced by each demi-hull as if the other was not
there. To this purpose figure 14 gives the longitudinal
wave cut along the centre-line of the catamaran for
dif-ferent Froude numbers. The 3D RPM and 2D+t
calcu-lations are presented together with the experiments (for
the tested speeds). In the plots, the curves with circles give the 2Dt results obtained as the superimposition of two mono-hulls solutions, that is the interaction be-tween the demi-hulls is not accounted for but just their
interference. From the results, the interference is not the governing mechanism. The interaction between the two
demi-hulls plays a fundamental role. This interaction is mainly nonlinear as evidenced both by difficulties of
the linear solution in capturing the first peak and by the phase shifting existing between the linear and nonlinear results accounting for the demi-hull interaction.
As previously discussed, the 2D+t model
en-forces a dry transom stern condition independently from
the forward speed. In reality the flow at the transom
Frou3.6 -1.8 -1.8 2.8
*0
-2.8 2.8 -2.8 5.6 2.8 o -2.8 -5.6 -d.4 1.6 -0.4 '.9 0.7 o 0.55 04 0.53 0.4 0.63 0.8 0.44 0.8 0.53 12 x 0.38 Fr 1.2 x/L 0.46 Fr -0.4 0 0.4 0.8 1.2 xJL 2.53 1.13 0.84 0.7 0.6 FrFigure 14: Catamaran: cut of the steady wave
pat-tern along the catamaran center-line. Experimental data
(square symbols: mean value and error bar), 3D RPM code (dashed lines) and 2D+s theory (solid lines). The curves with circles give the 2Dt results obtained as the
superimposition of two mono-hulls solutions, that is the interaction between the demi-hulls is not accounted for.
From top to bottom: Fr=0.5,0.6,0.7,0.8.
de numbers, involving partial or full ventilation pheno-mena. Several flow regimes can be distinguished and it
is difficult to identify the critical Froude for the transi-tion from one regime to another. In this context,
analy-tical (Vanden-Broeck and Tuck 1977), numerical
(Scor-Table 1: Moriohull: Dry and wet transom stern
condi-tions as a function of different Froude numbers.
Table 2: Catamaran: Dry and wet transom stern condi-tions as a function of different Froude numbers.
pio and Beck 1997) and experimental (Doctors 2003
and Maki et al. 2004) investigations have been
car-ned out. Vanden-Broeck and Tuck (1977) and Scorpio and Beck (1997) identified FTT
= U/vT =
2.5-2.6as the critical Froude number based on the draft (T) for the occurrence of ventilation behind a 2D semi-infinite body with a flat bottom. The experiments by Doctors (2003) on transom-stern ships confirmed this critical
value. Table 1 gives the Fry corresponding to the speeds tested in our mono-hull experiments. Due to the hull geometry (see figure 2) two different Froude numbers, Fry and FrTtraTIS. are reported based, re-spectively, on the real draft at the transom (including
the trim and sinkage effects) and on the transom draft in still water conditions. The last column in the table gives the transom stern conditions at the different speeds. The
critical Fi-T value is about 2.46. The value has been identified by using top-view pictures from the experi-ments, as reported in figure 15 for the different tested speeds. At Fr =
0.3(FrT
1.76)a partial ventilation of the flow appears. This becomes a full ventilation atFr
=0.4 (Fry
= 2.09). Increasing the model speed(Fr > 0.5, i.e. Fry
2.46) the dead-water region be-hind the transom disappears and the transom stays com-pletely dry.In table 2 the same parameters discussed before are reported for the catamaran hull. The corresponding experimental top-view pictures are given in figure 16. From these, the transition to dry transom condition oc-curs at Fr = 0.5 (Fry = 2.18). The catamaran
config-uration presents a larger value of the trim and sinkage (cf tables 3 and 4) than the mono-hull, This implies
smaller Fry with respect to the mono-hull.
Fr T/L FrT FrTtraThS Transom 03 00290 176 182 wet 04 00364 209 243 wet 05 00413 246 304 dry 06 00424 291 365 dry 07 00444 332 426 dry 08 00438 382 487 dry Fr T/L FrT Frytra Transom 03 00290 176 182 wet 04 00360 211 243 wet 05 00525 218 304 dry 06 00598 245 365 dry 2.2 0.99 0.74 0.61 0.54
Figure 15: Pictures of the monohull transom stern
field. From top to bottom: Fr 0.3,0.5,0.7 (left
col-umn), Fr = 0.4.0.6,0.8 (right column).
Figure 16: Pictures of the Catamaran transom stern field. From top to bottom: Fr = 0.3,0.4, 0.5,0.6.
5 DISCUSSION: UNSTEADY CASE
A second experimental activity was dedicated to the study of the unsteady behaviour of the mono-hull and
Table 3: Monohull: mean value for carriage speed (U), sinkage(s) and trim (8) and related standard deviation (o).
Fr Velocity (mis) Sinkage (mm) Trim (degree)
Table 4: Catamaran: mean value for carriage speed (U),
sinkage(s) and trim (9) and related standard deviation (a).
catamaran models. Preliminarily, the heave and pitch frequency resonance ranges for Fr = 0.3,0.4 and 0.5
have been identified through the transient test technique
(Colagrossi et al. 2001). Then, tests in regular incom-ing waves within such range and with different wave amplitudes have been performed. Also in this case a careful error analysis has been realized. In the
follow-ing the results will be discussed for the catamaran. The RAO experimental data are presented in
fig-ure 17 together with the predictions by the 3D linear RPM code. The standard deviation (a) connected with the transient test technique is also given in the plots showing a good reliability of the experiments. For the
numerics both the NK and DM approximations (see
sec-tion 2) are considered. The NK approach neglects the
interaction with the local steady flow. The DM accounts
for it hut not in a consistent way for a high-speed ship. The numerical results overestimate the pitch motion.
For all investigated speeds. the DM linearization shows the best agreement with the experiments. This is
consis-tent with the conclusions in Bertram (1999) document-ing an important role of the interaction with the steady
flow in the wave induced body motions even at Froude
around 0.2, for a S-175 ship. A strong amplification
of the motions is generally observed near the resonance
due to a small damping level. This suggests the need of proper active control systems and foils. However it should also be noted that the values of the
wavelength-to-ship length ratio giving resonance in heave and pitch
in head sea increase with the Froude number. This
im-Fr Velocity (mis) U a Sinkage (mm) s a Trim (degree) O a 03 1880 0001 -6189 0.110 -0010 0001 0.4 2510 0001 -13974 0057 0543 0093 05 3133 0001 -19637 0.057 1177 0.004 06 3754 0.0001 -16684 0.029 1424 0.003 07 4384 0.0001 -14054 0021 1348 0001 08 5008 0003 -12536 0.413 1328 0022 U a s a O a 03 1883 00005 -8792 0.321 0025 0008 04 2509 00003 -21385 0072 0424 0009 05 3125 0006 -27062 0.223 2652 0021
2.5 03 - Tnrns, TeST Tech 17 Cre-ces. Test) O reg esveOsA=0.0125) Drrgwasr (AA = 00187) * reg, wsvr)kA=0.025( reg wsvrIkA=0.05) Nrn. recetTe INK bese 110w) NeST, reselTs (DM bose liess)
AIL A/L 12 lo <9 12 6 2 A/L I - 2 AIL O AIL
Figure 17: Catamaran: heave (, left) and pitch (9,
right) Response Amplitude Operators. From top to
bot-tom: Fr = 0.3,0.4,0.5. A and k = 27r/A are the
regular incoming wave amplitude and wavenumber. re-spectively. A is the incoming wavelength.
.24
XIL2
3Figure 18: Heave (o Response Amplitude Operator for the mono-hull (dashed) and catamaran (solid) vessels
at Fr = 0.5. A and A are the regular incoming wave
amplitude and wavelength, respectively.
plies that the excitation loads along the ship become stronger in phase as Fr increases. The consequence is
larger excitation loads. The experiments evidence clear nonlinear effects. The regular wave results do not show
a convergence to the transient test results as the wave
amplitude reduces. One possible error source is a varia-tion of the wave amplitude along the track of the model.
This aspect has not been investigated. At high Froude numbers the RAO for the pitch motion shows a double peak behaviour, typical for the multi-hull vessels. So it proves that it is important to account for the hull inter-action in the ship motions calculations. The RAO
re-lated to the heave motion is practically unchanged with
respect to the mono-hull configuration (see figure 18). From the experiments, the mean trim and sinkage are
not influenced substantially by the incident wave
steep-ness, even at a wave frequency equal to the heave and
pitch resonance frequency. as shown in the left and right
plots of figure 19, respectively. It implies that they are
060 034
-24-o
0.4 0.5 r 0.3 04 05 Fr
0.40 (H, 0.55 071 047 S (H?)
Figure 19: Catamaran: mean trim and sinkage in calm water and in regular waves. Regular waves have been generated with frequency u(Hz) close to the heave (,
left) and pitch (9, right) resonance frequencies.
dominated by the steady flow. These results are relevant
for instance for the wetdeck slamming which is sensi-tive to the trim angle (Ge 2002). In the wetdeck slam-ming predictions it also matters an accurate estimate of the relative motions in the impact area. The presented results suggest that the theoretical methods to evaluate
wave induced motions have to be improved for a better prediction of for instance wetdeck slamming.
CONCLUSIONS
An experimental investigation has been carried out to
analyze the flow field around semi-displacement
mono-hulls and catamarans both in calm water and in inci-dent head sea waves. The mono-hull model has been
shaped identically to a catamaran demi-hull to
investi-gate the interaction between the demi-hulls of the
cata-maran and the related influence on the rooster tail
devel-oping from the transom stern. The chosen mono-hull geometry implies a much smaller beam-to-draught
ra-tio than the one characterizing the usual mono-hull high speed vessels. A dedicated error analysis of the tests has
been performed confirming a general reliability of the measurements. The physical investigation was focused on the wave-field features at the bow, along the hull and downstream the transom stern. In the steady ex-periments, very detailed measurements of the wave
pat-tern were performed for both models, including the in-ner region between the two catamaran demi-hulls. The influence of the Froude number has been analyzed by varying such parameter in a wide range. The experi-mental data were compared with the results by a linear 3D RPM code and a nonlinear 2D+r method. For the mono-hull, the 3D RPM simulations are able to
cap-ture the wave pattern along the hull for Froude numbers smaller than 0.6. They do not succeeded in handling the
stern and wake flows for Fr > 0.5. This is because a wet transom is assumed while at those speeds both the experimental mono-hull and catamaran transoms were
dry. At Fr
0.6 the nonlinearities become relevant and the linear method gives only a qualitative informa-tion. The opposite trend is shown by the 2D+t results. These are not satisfactory at the smaller Froude due to the relevance of the transverse wave pattern. For Fr greater than 0.6 they fit well with the experiments. Thestern and wake flows are properly described since a dry
transom condition is enforced. For the catamaran, the same trend is observed but the 2Dt theory gives glob-ally the best agreement at smaller speeds than for the mono-hull. For both geometries this model predicts a
wide extension of the lobes near the bow region. This is
not observed from the measurements. The 2D+t model has been used to investigate the physical mechanisms
causing the flow features downstream the transom, and
to quantify the related influence of the interaction
be-tween the catamaran demi-hulls. In the unsteady
exper-iments, both a transient test technique and regular
in-coming waves have been used to simulate seaway con-ditions. The former method was applied to identify the
heave and pitch resonance frequency regions. Then
reg-ular incoming waves were generated within such areas of interest. This means we studied a frequency range
where nonlinearities matter for the wave-body
interac-tions. During the tests the Froude number and the
am-plitude of the incoming waves have been varied and the
importance of nonlinear effects brought into the
prob-lem has been deduced. The interaction between the
steady and unsteady wave fields was studied by
combin-ing in a synergic way the information from the experi-ments with the results from a linear unsteady 3D code. The results confirmed that the interaction plays a
rele-vant role and should be properly accounted for. For the
considered speeds and within the heave and pitch res-onance ranges. the experiments showed that the mean trim and sinkage of the catamaran are mainly governed by the steady flow. This is relevant for instance for the
wetdeck slamming.
ACKNOWLEDGEMENTS
Present research activity is partially supported by the Centre for Ships and Ocean Structures, NTNU.
Trond-heim, within the "Green Water Events and Related Struc-tural Loads" project, and partially done within the
frame-work of the "Programma di Ricerca sulla Sicurezza"
funded by Ministero Infrastrutture e Trasporti.
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