-.jrumal'of Ship Research, Vol. 37, No. 2, June 1993, pp. 87-101
On Seakeeping of Conventional and High-Speed Vessels1
Odd M. Faltinsen2
The focus is on numerical and theoretical seakeeping problems for conventiOnal and high-speed
ves-sels. Wave-induced motions, accelerations and added resistance in waves of nonplaning high-speed monohulls and catarnarans are discussed. Emphasis is on the interaction with the steady flow and
transom stem effects. The "cobblestone" effect and the speed loss of a surface-effect ship (SES) in a seaway are discussed. The importance and possibility of predicting the influence of flow separation on the vertical motiodf conventional ships are studied. Numerical methods that accurately describe slam ming on hull sectiáns are discussed. Correlations with published model test data for bow flare slamming are presented. TECHfLIISCIFE UNfl1ELS1TUT Laboratorlum voor Scheepshydromechanlcft Archief Mekelweg 2,2628 CD Deift Tel.: 015. 788373 Fa 016.781829
Fifteenth Georg Weinbium Memorial Lecture
Journal of
Ship Research
Introduction
WE:NBLUM AND ST. DENIS stated in 1950 in their SNAME
paper "On the Motions of Ships at Sea" that "It is, therefore,
quite understandable that in spite of the efforts of many in.
vestigators, the problem of the motions of a ship in a seaway
has not been solved in its entirety." Even if there has been great progress since 1950 in seakeeping predictions, for in-stance, with the development of the strip theory, it is felt that Weinblum and St. Denis's statement can still be used.
We will try to focus on some areas of seakeeping predictions that need improvement.
Generally speaking, strip theories are still the most suc essful theOris for wave-induced motions of conventional
:;hips at forward speed. However, from a theoretical point of
7iew one can question striptheories. A strip theory is based on linearity. This means, for instance, that the ship motions are small relative to the cross-sectional dimensions of the
thip. In practice one "forgets" the linearity assumptions and
applies strip theory programs when parts of the ship come out of and into the water or in predicting green water on deck. Due to the linearity assumption, only hydrodynarnic
effects of the hull exist below the mean free-surface level. A
strip theory program will not distinguish between alterna:
tive above-water hull forms.
LThe annual Georg Weinbium Memorial Lectures were inaugurated in 1978 by two groups of German and American colleagues and friends of the late Georg Weinbluin, in continuingtribute to his many valuable contributions tO ship hydrodynamics. The 15th lecture was presented by Professor Faltinsen in Berlin on November 18, 1992 and in Washington,
D.C., to the National Acadámy of Sciences, on April 14, 1993. 2DivisiOn of Marine Hydrodynamics, Norwegian Institute of Technol. cgy, Trondheim, Norway.
Manuscript received at SNAME headquarters December 17, 1992.
Strip theory is based on potential flow theory. Lifting ef-fects on the hull can be included consistent with a
low-as-pect-ratio Tlifting surface assumption. This is accounted for by the end terms in the hydrodynamic coefficients. Potential flow theor means that viscous effects are neglected. The most
severe co sequence of this is in the prediction of roll at res-onance. In practice, viscous roll damping effects are ac-counted for by empirical formulas. The eddy-making roll
dampingdile to sharp corners such as bilge keels represents an Important contribution. The effect of flow separation may also matter in the prediction of vertical ship motions, in par-ticular when the ship cross seqtions have sharp corners (Beukeiman 1980,1983). This is discussed in the main text; also, a twc,-dimensionai nunierical method that combines
free-surface effects and vortei shedding is presented and com-pared with experimental results.
The method for solving forced motion problems in strip theory cannot be justified when the frequency of encounter is low as it might be in following ancl quartering seas. The
Seakeeping Committee of the 16th ITTC reports substantial
disagreement between calculated results and experimental
investigtiOns of vertical wave loads in following waves. Strip tl4eories account for the interaction with the forward speed in a simplistic way The effect of the steady wave sys-tem around the ship is neglected. The free-surface conditions
are simplified so that the unsteady waves generated by the
ship are propagating in directions perpendicular to the
cen-terplane.In reality the wave systems may be far more com-plex. For instance, for high Froude numbers unsteady "di-vergent" wave systems become important. This effect is neglected in strip theory and is discussed in the main text. Since strip theory is a slender body theory, it has ques-tionable application for ships with low length-to-beam ra-tios. On the other hand, the Seakeeping Committee of the
18th ITTC concludes that strip theory appears to be
markably effective for predicting the motions of ships with
length-to-beam ratios as low as 2.5.
There have been different attempts to improve strip the-ories by a rational approach. Nakos & Sclavounos (1990) presented a linear three-dimensional frequency-domain
so-lution based on a Rankine panel method. Their results are
promising. A difficulty is to include the effect of the so-called
mrterms in the body boundary conditions. These are diffi-cult to compute in areas on the body surface with high
cur-vature. If the body surface is not close to vertical at the free
surface, difficulties occur also with the mrterms at the in. tersection between the body surface and the free surface.
Another difficulty is to properly include the effect of a transom stern. It is necessary to model high-speed flow
sep-aration at the transom stern, where the pressure is atmo-spheric. This is further discussed in the main text.
Committee 1.2 of the 10th ISSC has reviewed the state of
the art in predicting strong nonlinear wave loads on ships. Presently there are no rational methods available. Experi-mental results of wave bending moments, shear forces and torsional moments in steep waves can show strong influ-ences of nonlinearities. The same is true in predicting large relative vertical motions between the ship and the waves, bow flare forces, slamming loads and the effect of green water on deck. Committee 1.2 of the 10th ISSC recommended
prac-tical methods to calculate nonlinear wave loads. Important
parts of the methods are exact calculations of Froude-Kriloff
and hydrostatic pressure forces on the wetted hull surface. The incident wave field is described by linear theory. The
nonlinear hydrodynamic forces due to ship motions and the diffraction of the incident waves correspond closely to
con-ventional strip theories in the limit of small ship andwave motions. The nonlinear parts of the hydrodynamic forcesare
questionable. In the main text the water entry part of this
problem in connection with slamming is discussed. It is
im-portant to accurately describe the flow at the intersection
between the free surface and the body surface, including the
rise up of water. There are many seakeeping problems that require more extensive studies. Emphasis here will be
con-centrated on high-speed vessels, the effect of the flow sep-aration, and slamming. Faltinsen (1990) has discussed
sea-keeping problems both for ships and offshore structures.
High-speed vessels
In the following, some of the challenging seakeeping prob-lems for high-speed surface-effect ships (SES) and
nonplan-ing monohulls and catamarans are presented. Slammnonplan-ing is a problem of particular importance for high-speed vessels.
This will be dealt with later in the section on nonlinear ship motions.
Nonpianing monohulls and catarnarans
Details about a theoretical and numerical method usedto
analyze the steady and linear unsteady flow about high-speed
nonplaning mono- and multihulls in calm water and waves have been described by Faltinsen & Zhao (1991a,b). It is as-surned that the hulls are hydrodynamically independent of each other. This is a reasonable assumption at high speed as long as the hulls are not too close and the waves from
one hull do not propagate to the other hull. The consequence of this is that one can concentrate on the study of monohulls.
The problem is formulated in terms of potential flow the-ory. The unsteady motions of the ship and the fluid are as-sumed to be small so that the unsteady body boundary and
free-surface conditions can be linearized. The boundary-value problem is simplified by introducing the slenderness param-eter e. This expresses the order of magnitude of the beam or
the draft to the ship length. It is assumed that the
longi-tudinal derivatives of any flow variable f caused by the body
in some region near the ship are 0([C') and that the trans-verse derivatives off are 0(fC'). This implies that the three-dimensional Laplace equation for the velocity potential is
approximated by a two-dimensional Laplace equation in the cross-sectional plane. The Froude number based on the ship length is assumed to be 0(1). Nondimensionalized frequency
of encounter with respect to the ship length is assumed to
be 0(). The form of the body boundary and free-surface
conditions depend on the order of magnitude of the
longi-tudinal component n1 of the unit normal vector n to the
wet-ted part of the ship surface.
If n1 = 0(e), which is normal to assume in slender body theory, it leads to the classical linearized free-surface con-ditions with forward speed. It means that thereare no
in-teractions between the unsteady flow and the local steady flow. If n1 = 0(eh/2), it leads to nonlinear free-surface
con-ditions in the steady flow part. In the unsteady free-surface
and body-boundary conditions there are interactions with the
local steady flow. This involves calculating second
deriva-tives of the local steady flow velocity potential 4,. This causes problems at sharp corners on the body surface andat the
intersection between a non-wall-sided body surface and the free surface. The interaction terms with the local steady flow occur in the body-boundary conditions because the steady flow satisfies the body-boundary conditions on the mean oscilla-tory position and not the instantaneous position of the ship. Faltinsen & Zhao (199 Ia) have presented a procedure to avoid the numerical problems in evaluating the second derivatives
of & in the body-boundary conditions near sharp corners. However, the treatment of the intersection between the free
surface and a non-wall-sided body surface needs improve-ment. It may actually have been better to also handle the
unsteady free-surface conditions in a nonlinear manner. This
means one cannot operate in the frequency domain. If the frequency of oscillation is small, there is a simple way to avoid the difficulties in calculating second derivatives of 4,
at the free-surface intersection with the body. A quasi-steady approach can then be followed. The solution for each motion
mode can be obtained by solving the steady flow problem
when the ship is displaced in different motion modes relative
to the mean oscillatory position of the ship. By calculating the forces and moments on the ship in the different posi-tions, one can derive those parts of the forces and moments that are hydrodynaniic restoring terms in the equations of motions. However, it is always mathematically possible to
rewrite a restoring term as an added-mass term. This means
it is important to define what the restoring terms represent when comparing with model tests and other numerical
cal-culations.
Both when n1 = 0(E) and n1 = 0(E½) a numerical solution for the flow is found by starting at the bow. The free-surface
conditions are used to step the solution in the longitudinal direction of the hull. The velocity potential for each
cross-section is found by a two-dimensional analysis. Transom stern
effects are accounted for by assuming that the flow leaves the transom stern tangentially in the downstream direction so that there is atmospheric pressure at the transom stern. In close vicinity upstream of the transom stern the predic-tion method will be in error. The reason is that the solupredic-tion has no information that the pressure should change to at-mospheric pressure at the transom stern.
The wave resistance, the steady vertical forces and pitch
moments are found from the steady flow analysis. The latter
can be used to calculate the vertical position and trim. The transom stern has an important effect on the steady longi-tudinal force on the ship. A reason for this can be seen by
integrating the hydrostatic pressure force over the body
sur-face below the mean free-sursur-face level. Since there is
at-88 JUNE 1993
rspheric pressure at the transom stern, the hydrostatic rpressure force causes a longitudinal force on the vessel.
The wave excitation forces in regular waves and the
fre-quency-dependent added-mass and damping coefficients are
found from the unsteady flow analysis. By combining this. with information about mass distribution and hydrostatic
considerations the equations of motion in six degrees of free-dom can be solved'.
Figure 1 shows the steady wave elevation according to lin-ear theory around a parabolic strut with length 1 m, breadth
0.1 m and draft 0.25 in. The Froude number was 1. A com-parison is made with thin-ship theory. The agreement be-tween the two methods is reasonable. Since the method by
Faltinsen & Zhao (1991a) neglects the transverse wave
sys-tem [see Ohkusu & Faltinsen (1990)1, it indicates that the
transverse wave system is not important for high-speed ships. Similar comparisons for the unsteady problem are presented in Fig. 2. A strut with parabolic waterplane area and
wedge-shaped cross-sectional area in unsteady heave motion with
forward speed is
studied. The length of the strut
is I in, the beadth 0.05 rn and the draft is 0.2 in. The draft is constant along the whole length of the strut. The Froudenumber is 1.0 and the circular frequency of osciflation w 8 rad/s. The thin-ship theory calculations by Hoff (1990) are
based on distributing three-dimensional sources over the
centerplane of the ship. The sources satisfy the classical
free-sirface condition with forward speed. The agreement with the high-speed theory is reasonable, but not as good as for steady flow' problem. It should be noted that the thin-ship theory is also an approximate theory. What the com-parison indicates is that neglect of the transverse wave sys-tem is reasonable at high Froude number.
The bodies"studied in Figs. 1 and 2 are relatively easy cases.
Figure 3 shows a more difficult model with transom stern
and nonvertical body surface at'the intersection between the
free. surface and the body surface inthe forebody. Figure 4
shows thesteady vertical force per unit 1ength'along the ship. The ship was divided into seven segments. The experimental
values are averaged values per unit length over a segment. The hydrostatic force due to.the pressure terms "pgz" on
the body surface z 0 is not included. The agreement be-tween theory and experiments is good except in the aft part of the ship. The nonlinear theory agrees better than the
lin-ear theory. Nonlinlin-ear- theory. means that the nonlinlin-ear
free-surface conditions were used. However, some
approxima-Ag. 2 Amplitude of wave elevation around a vertical strut with parabolic waterplane in unsteady heave motion with forward speed. Fn = 1.0, w = 8
rad/s. Unit heave amplitude. Strut length 1 m. Results from thin-ship calculations by Hoff (19901 in upper half, and in lower halt results from
high-speed theory by Faltinsen & Zhao (1991a) using classical free-surface conditions with forward speed
tions were made. The velocities at the free-surface segment
closest to the intersection points between the free surface
and the body surface -were set equal to the velocities
calcu-- lated at the neighboring freecalcu--surface segment. There is no
theoretical justification for doing that. However, if we used
the calculated velocities at the free-surface segment closest
to the intersection point, it resulted in numerically unstable.
predictions of the intersection points. The reason for this is that it was difficult to accurately predict the high velocities
that occur around the intersection points. In a following sec-tion on slamming amore proper handling of the intersecsec-tion
problem will be presented. At the transom stern the flow
Fig. I Wave elevation around a vertical strut with parabolic waterplane area in steady forward
motion. Fn = 1.0. length = 1 m. Results by Hoff (1990) based on thin-ship theory are shown in
upper half, and results in lower half are based on linear high-speed theory by Fattinsen & Zhao (1991a)
FIg. 3 Body plan of the model used in experiments by Keunirig (1988)
should leave the transom stern tangentially in the
down-stream direction so that there is atmospheric pressure at the last section. This means that the vertical force per unit length
on the stern of the ship is opposite to the vertical force due to the pressure term -pgz". This asymptotic value is shown in Fig: 4. The theory is not able to predict this value. The reason is that what is happening at a crosssection is
influ-enced only by upstream effects.
If we draw a line in Fig. 4 between the asymptotic value ofF3 at the transom stern with the. numerically predicted
value ofF3closest to the transom stern,-we get a reasonable
agreement with the experimental force value at the aft
seg-PL2 0.02-0.01. -0.25 0.00 a25
4f
I,
2H
*
0.50 1.Aaymptodc vabo FIg. 4 Measured and computed steady vertical force distribution along modelpresented in Fig. 3.Experiments by Keuning (1988). = nonlinear theory, linear theory, F3 = vertical forde per unit length. (Buoyancy force in calm
water excluded.) L = ship length, X = longitudinal coordinate, X = -L/2 at
the bow, p mass density of water. Fn = 1.14, trim = 1.62 deg
ment of the ship. This indicates that there must be a rapid change of pressure back to atmospheric pressure in a small neighborhood of the transom stern. This change is stronger than assumed when the boundary-value problem has been
approximated.
The inability to properly describe the transom stern flow
will have an influence on the predictions of the vertical
mo-tions. This is illustrated by Fig. 5. The classical free-surface condition with forward speed is used and no interaction
be-tween the unsteady and the local steady flow is accounted for. There are two types of theoretical results. In one case.
there are included transom stern effects. This was done by
using the normal approach up to the station next to the
transom stern At the transom stern it is assumed that the pressure must be atmospheric. This value was used for the whole last station. There is no theoretical justification for doing this for the whole station. The main purpose is to
il-lustrate a possible effect from the transom stern on the ship motions and accelerations. No special treatment of the local
transom stern flow was made in the other case. The ship model. is the same as presented in Fig. 3. The pitch radius
of gyration is 25% of the ship length.
The experimental values presented in Fig. 5 were given
by Blok & Beukelman (1984). Note that the theory is in good agreement with experimental values. The description of the
local flow around the transom stern has a small effect on
the heave motion, while there are some effects on the pitch and the vertical accelerations in the bow. Including
tran-La Heave co_=.._ Pttch Acceleration 11131 Heave: -Pitch: Acceleration at station 19 (in bow) a3 L 50g 0.750 AAA ODD 000 experiments
theory (flow at the transom Stem gives no difference in results)
experiments
theory without transom stem effects theory with transom stem effects experiments
theory without transom stem effects theory with transom stern effects
Fig. 5 Heave, pitch and vertical acceleration amplitudes for model presented
in Fig. 3 in head sea regular waves. Fn = 1.14, trim = 1.62 deg.
Experiments by 810k & Beukelman (1984). ( = wave amplitude of incident waves, k = 2ir/X wave nUmber of incident waves, a3 = vertical acceleration
amplitude, L ship length Length of test waterhne 2.00 m
Beam of test waterline 0.25 m
Draft 0.0624 m Block coefficient 0.396 0.00 -0.01--0.02 -0.50
om stern effects" in the numerical predictions improves the
agreement with experimental results.
The interaction between the unsteady and the local steady
flow is important at high forward speed. It may not influ-ence the vertical ship motions that much, but it can be im-portant in the predictions of added resistance in waves.
Details about how added resistance in waves of high-speed mono- and multihulls can be calculated are given by
Faltin-sen et al (1991). It is based partly on a direct pressure in-. tegration method using exptessions based on the linear
un-steady flow analysis. In regular waves the problem is solved to second order in wave amplitude. Transom stern effects are
included in the expressions and are important. The inter-action with the local steady flow is accounted for in an
ap-proximate way. The linear theory used in the expressions for
added resistance did not include interactions between the unsteady and the local steady flow. This is not consistent with saying that interactions with the local steady flow are important in calculating added resistance in waves.
How-ever, if this had been done consistently, it would likely have
led to numerical problems with higher-order derivatives of the velocity potential due to the local steady flow. The
in-teraction with the local steady flow was therefore evaluated
by a quasi-steady approach where the steady longitudinal
force on the vessel was calculated in different oscillatory
po-sitions of the ship. The expressions were then time-aver-aged. The difficulties in consistently handling the interac-tion between the local steady flow and the unsteady flow imply that one should investigate the possibility of using a time-domain solution. An obvious drawback will be the re-quired CPU time relative to a frequency-domain solution.
In Fig. 6 comparisons are presented with experimental
re-sults for added resistance in regular head sea waves. The ship model is presented in Fig. 3. Transom stern effects in
the added resistance expression were found to be important.
We note. that the interaction with the local steady flow is important for the high Froude number presented in the fig-ure For lower Froude numbers the importance will be less.
15.0 10.0 -5.0 0.0 00 pg B3 0.5 1.0 Fn.1.14 1.5 FIg. 6' Added resistance RAW for model presented in Figure 3 in head sea
regular waves. Fn = 1.14, trim = 1.62 deg. = wave amplitude of incident
waves; X wavelength; L = ship length; B = beam; 000 experiments by Blok & Beukelman (1984); *- theory [Faltinsen et al (1991)1 with interaction
with local steady flow; ...theory (Faltinsen et al (1991)] without interaction
with local steady flow
A reason for the large influence from the interactiOn with the local steady flow is believed to be that the hull side is
not close to vertical at the waterline. In another unpublished example with a hull form that has vertical sides at the
waterline, the influence from the local steady flow was not important. Even if the formulas are only approximate, note that there is good agreement between theoretical and
ex-perimental values for added resistance. It should also be kept
in mind that it is difficult experimentally to determine the
added resistance in waves because the added resistance is a
small fraction of the total resistance for high Froude
num-ber.
Th.e high-speed theory presented above can easily be used
for catamarans since no hydrodynamic interaction between the hulls is assumed. Ohkusu & Faltinsen (1990) showed reasonable agreement with experimental values for heave and pitch added-mass and damping coefficients of a cata-maran. Faltinsen et al (1992) presented comparisons with model tests for motions and global loads in the centerplane of a catamaran in oblique regular waves at Froude number 0.49. No interactions with the local steady flow was ac-counted for in the numerical calculations. In general, the
agreement between theory and experiments is good for heave,
pitch, vertical bending moments and pitch connecting mo-ments. The agreement is not so good for roll. This also
in-fluences the predictions of vertical shear forces. Surface-effect ships
A major deficiency with the SES concept is the "cobble-stone" effect that occurs in small sea states. It is caused by resonance effects in the air cushion. This results in vertical
accelerations of the vessel that can represent a comfort
prob-lem for passenger transportation. Kaplan et a! (1981) have studied the "cobblestone" phenomena. The dynamic part of
the excess pressure in the cushion is oscillating with the same
amplitude all over the cushion according to their analysis. The compressibility effects of the air in the cushion are
es-sential. The oscillations are excited because the waves change
the enclosed air cushion volume. A typical resonance fre-quency for a 35-rn-long (115 ft) SES is around 2 Hz. The resonance period is approximately proportional to the ship length. This implies that model tests based on Froude scal-ing will not describe this resonance phenomena properly. Historically, most attention has been focused on the reso-nance phenomenon described by Kaplan et al (1981), and
ride control systems have been designed to increase the damping of those resonance oscillations. Sørensen et al (1992)
have also studied the uniform pressure resonance
phenom-ena. They point out that a traditional linear theory will
overpredict the dynamic pressure variations and the
result-ing heave accelerations of the vessel if comparisons are made with full-scale measurements.
SØrensen et a! (1992) recommended that acoustic reso-nance in the air cushion should be studied; full-scale mea-surements document this. They presented a simplified
the-oretical model for the acoustic effect for an SES with a rigid
rear planing seal. The resonance frequency of interest for a 35-m-Iong SES is around 6 Hz. The corresponding mode shape for the pressure variations along the length of the cushion
has a node midships and the largest amplitudes at the seals.
This means that it is most effective to minimize the
reso-nance oscillations by placing louvers close to the seals of the
cushion. For similar-shaped vessels the acoustic resonance
periOd will increase linearly with the ship length. This means
that model tests based on Froude scaling will not describe this phenomena properly. The acoustic resonance causes a
pitch moment on the vessel that excites pitch accelerations.
SØrensen et al (1992) reported qualitative agreement with
-full-scale measurements in the acoustically dominated fré-quency range. They discuss the sensitivity of the results to the sea state and stress the importance of accurate mea-surement of the sea conditions. Since the significant wave heights of interest are low, typically 0.1 to 0.2 rn for a 35-rn-long vessel, special wave measurement devices are nec-essary. If the SES is equipped with an airbag as an aft seal, unpublished full-scale measurements indicate that the
low-est resonance frequency is lower than with a rigid rear
plan-ing seal.
Air leakage from the cushion in waves has an important effect on the added resistance of an SES in waves. The air
leakage causes the SES to sink and the stiliwater resistance
components to change. For instance, the altered excess
pres-sure in the cushion changes the wave resistance due to the air cushion. Further, the increased wetted surface area of the hulls changes the frictional and wavemaking resistance due to the hulls. In addition, there is a contribution to the
added resistance in a similar way as described previously for mono- and multihulls. This is due to second-order nonlinear
interaction between the dynamic vessel oscillations and the incident waves. The air resistance on an SES due to wind and the vessel's own speed is also important. This is not so
much the case for a catamaran. Reasons for this are the
presence of the skirt on an SES and a lower hull resistance on an SES relative to a catamaran. A method to predict the added resistance in waves of an SES is presented by
Faltin-sen et al (1991). This is based on finding the mean air
leak-age in waves. The expected value for the drop in pressure in the cushion is found by using the characteristics for the cushion fans in combination with an expression for the
ex-pected value of the dynamic change in the leakage area. The
fan characteristic gives a relation between the excess
pres-sure and the volume flux for constant rpm of the fans. When the pressure drop in the cushion has been found, an estimate of the sinkage is found by balancing the weight of the
SES with the vertical forces due to the excess pressure in
the cushion and the buoyancy forces on the hulls. Due to the
increased sinkage of the SES, there occurs a change in the stiliwater resistance on the hulls. Due to the change in the
excess pressure in the cushion there occurs also a change in
the stillwater wave resistance due to the cushion pressure. The results depend on the condition of the skirts and how the rpm of the fans is regulated.
Faltinsen et al (1991) presented calculations of the
invol-untary speed loss of a 40-m-long (131 ft) SES and a
40-rn-long catamaran as a function of significant wave height H,., in head sea waves. The SES shows a much more rapid drop in speed with increasing H,,., than the catamaran. HOwever,
the SES uses less power and keeps a higher speed than the catamaran for nearly all sea states with H,3 less than 2 m. An SES with a flush wateijet inlet can easily be exposed to air ventilation in the waterjet in a seaway. The calcula-tions of Faltinsen et al (1991) indicate that problems may
occur for a 40-rn-long SES when H,,, is larger than 1 m. The
air ventilation can lead to significant engine load fluctua-tions, which result in increased thermal loads and the
pos-sibility of engine breakdown. Meek-Hansen (1990,1991)
pre-sented service experience with a 37-rn-long (121 ft) SES equipped with diesel engines and wateijet propulsion. An example with significant wave height around 2 m in a head sea at 35 knots speed showed significant engine load fluc-tuations at intervals of 3 to 5 seconds.
Nonlinear ship motions
In order to develop physically based numerical tools for nonlinear ship motions, many fundamental physical prob-lems have to be better understood. We will concentrate on
two aspects: the effects of flow separation and the water en-try (or slamming) problem. In addition, we may mention that the water exit problem and the modeling of steep (including
breaking) irregular waves need to be addressed.
Effect of flow separation on motions of conventional ships It is well accepted that flow separatiOn matters in describ-ing the roll motions of conventional ships around resonance,
but it is common practice to neglect flow separation in the prediction of heave and pitch motions of a ship. However,
Beukelman (1980,1983) presented experimental results that
suggest flow separation can have an influence on vertical
ship motions. This was evident in his studies of a ship model
with rectangular cross sections (see Fig. 7) in regular head sea waves. The Froude numbers were 0.16 and 0.26. As a part of his studies he presented experimental heave
damp-ing coefficients as a function of Froude number, Fn = U/v'L, nondimensional circular frequency of oscillations,
WCVZ7, and amplitude of forced heave oscillations,
r. Here
L is the ship length and g is the acceleration of gravity. Partof the damping coefficient is due to linear wave radiation damping. The nonlinear effects can be interpreted in terms
of a drag coefficient CD. This means one may write the
ver-tical force due to flow separation on the ship as F1,3 = CDAW
where p is the mass density of the water, A the waterplane area, and d'q3/dt the heave velocity. By equivalent linear-ization it follows that
F3 = - PCDA
WIT dt
The drag coefficient depends on the geometrical form, the free surface, the Reynolds number and, the Keulegan-Car-penter numberKC = irq3jD. Here D is the draft. The free-surface effect is a function of wVZ7 and the Froude
num-ber. For cross sections like a rectangular cross section where separation occurs from sharp corners, Reynolds number de-pendence is not expected to be important as long as viscous
shear forces do not matter. The latter may be true for small Reynolds and Keulegan-Carpenter numbers and can matter
for small models and laminar boundary-layer flow.
Figure 8 shows Beukehnan's experimental results when the nonlinear part of the heave damping coefficient is
in-terpreted in terms of a drag coefficient. There is a clear fre-quency and Froude number effect. The experiments were done for /D = 1/15, 2/15, The data did not show any impor-tant KC-number dependence. The Reynolds number depen-dence is not known. Figure 8 also shows the numerical value
of CD obtained by the two-dimensional vOrtex tracking method presented by Faltinsen & Pettersen (1987) and Braathen &
Faltinsen (1988). No effects of Reynolds number and viscous shear fOrces are included. The midship cross section was used
in the calculations and the Froude number was zero. The
two-dimensional vertical drag force was nondimensionalized
by the beam when the drag coefficient was calculated. In
practice, three-dimensional end effects should have been ac-counted for. This will result in lower CD values. The
nu-merical results show a similar frequency dependency as the experimental values. The frequency dependency implies that the free-surface waves influence the vortex shedding.. There are no experimental results for zero Froude number, but the
numerical values for Fn = 0 are reasonable relative to the experimental values for Fn = 0.16.
The vortex tracking method can be described by means of
Fig. 9. The problem is solved as an initial value problem. dq3
dt di13
dt (1)
(2)
2.0-0,25
The vorticity is concentrated in thin boundary layers and free shear layers (So). The separation points are assumQd known. Outside the vorticity domain a potential flow prob-lem is solved at each time instant by means of Green's Sec ond identity. Across the free shear layer S there is a
veloc-ity potential jump ( - 4,) Here + and - refer to the two sides of the free shear layer. Since the pressure is continuous
across the free shear layers it follows that 4 4, is
con-vected with the average velocity between the two sides of S. In order to shed vorticity from the. boundary layer it is necessary that the flow leaves the separation point parallel
VORTX tErHoo Co Fn 0. 3/D= 1/15 EXPERIMENTS (BEUKELMAN (1983)) Fn 0.16 EXPERIMENTS
3.0-
(BEUKELMAN (1983)) Fn 0.26FIg. 7 Form and dimensions of ship model with rectangular cross-sections
(Beukelman (1980,1983)] A A A 0 A 1.0 - Do 0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 RECTANGULAR CYLINDER
Fig. 8 Drag coefficientsC0obtained from forced heave oscillation tests of ship model with rectangular cross sections shown in Figure 7. Data are
presented as a function of nondimensionalized frequency of oscillation
w,v'E7 for different Froude numbers
2.50 m
2.50 m
0,251 --i__
0,25
=1°"5
to the body surface on One of the sides of the free shear layer
at the separation point. Which side the flow leaves
tangen-tially depends on the sign of the shed vorticity. It is required
that 4
- 4
be continuous at the separation point. Since the velocity jump is known at each time instant along thefree shear layer; this is a constraint on the velocity potential along the body surface. The starting of the solution requires
some comments. In the case of sharp corners a similarity
solution may be used for corners as a starting solution. This implies starting with a, discrete vortex at time equal to zero.
The discrete vortex is connected to the free shear layer by
a Rieman cut SR (see Fig. 9).
On the instantaneous position of the body boundary it is
required that there be no flow through the body surface. On the free surface SF inside y = b(t) (see Fig. 9) the exact dynamic and kinematic free-surface conditions according to potential theory are used. As long as the body surface is nearly
vertical at the waterline, there are no numerical difficulties in describing the flow at the intersection between the free surface and the body surface. For
II
> b(t) where b(t) is a large number dependent on time, the flow is approximated by a vertical dipole in infinite fluid with singularity at y = 0, z = 0. This implies that all waves are insideiI = b(t).
Faltinsen (1977) has shown in detail how the free surfaceproblem can be handled.
The viscous forces in the numerical model are due to
pres-sure forces and can be related to the vorticity distribution in the free shear layers and the motions of the free shear
layers. Both the vOrticity distribution and the motiOns of the
free shear layers depend on the presence of free-surface waves.
This is why the CD values presented in Fig. 8 are dependent
on wVL/g. However, the presence of the free shear layers
do not have an important influence on the free-surface waves
when the KC-number is. small. This implies that the linear wave radiation damping is not influenced by flow
separa-tion.
The method described above has a clear advantage, in ana lyzing separation from sharp corners at small KC-numbers. The effect of the free surface can be included in an easy way.
However, the method has disadvantages in long-time
sim-0.25 2.00 m 0.25
10.25
>L
0.25 2.00 m 025
-ulations and in describing flow separation from continuously
curved surfaces In the latter case itis better to use a vortex in cell method or a Navier-Stokes solver.
Figure 10 shows numerical simulations of the free surface and free shear layers at three time instants during forced
heave oscillations of the rmdship cross section of the ship
presented in Fig 7. Figure 11 shows more details-of the free
V'3
FIg. 9 Plow situation around a two-dimensional cross section performing forced heave
mOtion with effect of free surface waves and vortex shedding
t 1/2
FIg. 10 Numerical simulations of free-surface and free-shear layers during
forced heave oscillations of ship presented in Figure 7. w0Vt7 = 3.9. KC =
0.21. The free-surface elevation is multiplied by 10
94 JUNE 1993
shear layer around one of the corners of the cross section. The heave oscillation is gwen as
l3 = '13a sinWet (3)
where w/Z7 =3.9 and KC 0.21. The oscillations started
at t = 0. The time step used in the numerical simulatiOns was T/400, where T= 21r/W. The dots on the body surface
are endpoints of segments used in the numerical analysis. The highest dot on a vertical side is the intersectionpoint with the free surface. It should be noted that the free,surface elevation has been multiplied by 10. Figure 11 shows details
of the ireeshear.. layer after the free shear layer starts to
leave tangentially from the bottom surface. In the beginning
it leaves tangentially from the vertical surface.
The added-mass and damping coefficients of the rectan--gular cross sections -were obtained from the time record-of
the vertical hydrodynamic force in the time interval0.5T T. It was controlled so that the results analyzed by the force history from T < t < 1.5T were nearly the same.
Fig-ure 12 shows the heave damping coefficient for different
am-plitudes of oscillations. The damiing- force was written as
the sum -of a linear term and a quadratic drag term [see equation (1)]. The linear potential flow damping due to wave
radiation was found to agree with a frequency domain so lution based on the Frank close-fit method. The results for
= 1/5 were used to derive the C0values presented in
Fig. 8. However,, it should be noted that the numerical
re-sults presentedin Fig. 12 show KC-number dependence. The
figure also illusttates that the effect of flow separationcan
be large relative to wave radiation damping, in particular
t 105 T/400
FIg. 11 Details of free shear layer during numerical simulation explained in Fig. 10
JOURNAL OF SHIP RESEARCH
--t= 1/4
SaD
0.3 0.2 -0.1 -(20) £ B33 P.B.D 115a
ka
1.0 0.5 00 1.0FIg. 12 Numencally cSiculated two-dimensional heave damping coefficients B for midship cross section of ship model presented in Fig. 7. Data are
presented as a function of nondimensionalized frequency of oscillation wV7 for different oscillation amplitudes q. Fn = 0, B = beam, D= draft
for high frequencies. In a practical context it is frequencies around the natural frequency that is of prime interest. For the two-dimensional body analyzed in Fig. 12 the nondi-mensionalized natural circular frequency of heave
oscilla-tions wV7 is 0.76 according to the numerical method.
The ratio between viscous drag damping at i/D = '/ and
linear potential flow damping is 0.09 and 0.29 for wVb7j = 0.588 and 0.835, respectively. This indicates that this ra-tio can be large when is the order of magnitude of D.
However, when i becomes that large, nonlinear potential flow effects may also matter The ratio between linear po-tential flow damping and the critical damping is 0.13 and
0.066 for wVD/g = 0.588 and. 0.835, respectively. This sug-gests that the effect of flow separation matters in predicting heave (and pitch) motions of the ship model presented in Fig. 7 when the frequency is close to resonance. However, in ana
lyzing the vertical motions of the vessel in a head sea, one should also account for the effect of flow separation due to
pitch motion and the incident waves. Beukelinan (1980,1983)
showed experimentally that viscous effects also matter for
pitch damping, heave and pitch motion. An example on pitch results for Fn = 0.16 are presented in Fig. 13. The influence
a = 0.02m
= 0.03m a = 0.04m
Fri = 0.16
LINEAR POTENTIAL FLOW DAMPING o DAMPING AT A DAMPING AT
.1.
0 0 0 . 0 1 2 3 4FIg. 13 Pitch amplitudes ii in regular head sea waves of Ship model
presented in Fig. 7. Model tests by Beukelman (1983). L = 2.333 m, = incident wave amplitude, k = wave number of incident waves
of scous effects is largest around resonance. For instance,
from the results in Fig. 13 we find that
1.09 - 0.34 for tue\/L7 = 2.85
= 11.15 - 1.35 for = 3
(4)
For shIps with cross sections without sharp corners the effect of flow separation will be less important, while the presence
of bilge keels can make the effect of flow separation more
important. Weinblurn & St. Denis (1950) presented
experi-mental results for vertical motions that showed the
influ-ence of bilge keels.
In order to numerically describe the influence of viscous effects on heave and pitch motion it is necessary to gener-alize the method presented above to include
three-dimen-sional and forward speed effects.
Slamming
In order to describe slamming properly it is necessary to account for nonlinear effects. A great difficulty is to accu-rately describe the flow at the intersection between the free
surface and the body surface. Zhao & Faltinsen (1993) pre-sented similarity solutions for wedges that show how the flow looks at the intersection. Figure 14 presents the angle 13 be-tween the free surface and the body surface at the
intersec-tion. The wedge is forced with a constant vertical velocity
through an initially calm free surface. Particularly for small deadrise angles a, the angle 13 becomes small and is numer-ically difficult to determine. Cointe (1991) proposed an asymptotic formula for 13 valid for small values for a. Figure 14 shows that the asymptotic formula is in reasonable agreement with the similarity solution for a < - 10 deg. The flow at the intersection has the character of a jet flow. Zhao & Faltinsen (1993) used this information to simplify the
de-scription of the flow at the intersection. They represented the velocity potential of the flow by using Green's second
identity over an instantaneous fluid domain (1 that does not
contain the whole jet flow The surface S enclosing. fi
con-sists of A.B, CD, S8, SF and is a control surface far away from the body. AB is shown in Figure 15. The angle
between the body surface and AB is. 90 deg, while the angle between .AB and the free surface is close to 90 deg. The line
AB is in an area where the jet starts and where the pressure can be approximated by atmospheric pressure. CD is sym-metric with AB about the z-axis. S is the wetted body
sur-face between the points A and C. SF is the free surface
out-side the points B and D and inout-side A one-dimensional
flow is assumed at AB and CD. The nonlinear free-surface
conditions without gravity are used at SF, AB, and CD. The
effect of gravity is neglected compared with the large fluid
accelerations. (However, including gravity would not cause
difficulties.) The pressure is set equal to a constant atmo-spheric pressure on the free surface The nonlinear
bound-ary element method by Zhao & Faltinsen (1993) can handle the water entry of a broad class of two-dimensional bodies. The body velocity and the undisturbed fluid velocity can be general. However, the following limitations should be noted.
The water is assumed incompressible and the flow
irrota-tional. This means that the impact velocity and pressure are not so high that compressible effects in the water matter. In many cases this is' not a severe limitatiOn. It is assumed that no air pocket is created during the impact, which means that the deadrise angle a has to be larger than 2-3 deg. Flow separation is not accounted for. This occurs, for instance, at
the immersion of a knuckle and must be accounted for to
properly describe the loads in this part of the flow. The local deadrise angle should not be larger than 90 deg.
5.1O 4-10 0.75 0.00 -0.75 -1.50 --- 0.00 0.75 0.50
I
Asymptotic theory (Cointe (1991)) z 1.50 Similarity solution a 10° 200 300 400 500 500 70°FIg. 14 Angle between free surface and body at intersection between free surface and body.. Calculations are based on similanty flow- of wedges that is
forced with a constant velocity V through an initially calm free surface. a =
deadrise angle
tic behavior is not accounted for, but is probably not
signif-icant for hull sections of conventional material.
Zhao & Faltinsen (1993) presented an asymptotic formula for the pressure distribution during the water entry of a two-dimenEional body with local small deadrise angle. The body
has a vertical symmetric line, the body velocity V is
verti-cally downwards, and there is initially no fluid motion. The work was based on Cóinte (1991), who used matched
asymp-totic expansions to find the velocity potential. The flow is divided into an inner and outer flow domain. In the inner flow the details of the jet flow at the intersection between the free surface and the body surface are studied by means
of Wagner's (1932) local jet flow analysis. In the outer flow the body-surface condition and the free-surface conditions are
transferred to a horizontal line. The body corresponds to a flat plate of length 2c. The velocity potential 4) is set equal to zerO in the surface conditions. The kinematic
free-surface condition is used in the same way as Wagner (1932)
to find c as a function of time. This involves solving an in-tegral equation, which can be done either by a power series expansion as Wagner showed or directly by a numerical method. For a general body shape that is neither a wedge
nor a circle, a direct numerical solution of the integral equa-tion is recommended. In the descripequa-tion of the pressure
dis-tribution in the outer flow, the quadratic velocity term in Bernoulli's equation is neglected. In the inner flow, Wag-ner's expression for the pressure distribution is used. The inner and outer flow pressure distributions are matched, which enables one to set up a composite solution for the pressure distribution. Figure 16 shows the pressure on a wedge with a deadrise angle of 15 deg. The outer pressure distributiOn goes to infinity at the edge of the flat plate in the outer flow, If the quadratic velocity term in Bernoulli's equation (which is a higher-order term in the outer flow) had been included in the outer flow pressure distribution it would have caused the outer flow pressure to go to
-
atthe edge of the plate. Further, the outer and inner flow pres-sure distribution would not match. The inner solution, which is based on Wagner's local jet flow solution, gives the max-imum pressure. It is common to referto this maxmax-imum
pres-sure as the Wagner's slimming prespres-sure. However, since it is the total pressure distribution that matters, one should be careful in only referring to the maximum pressure. It is not obvious how to generalize the asymptotic theory to a finite
deadrise angle. As an example, let us consider a wedge,
sat-isfy the body boundary condition at the exact body surface
in the outer flow, and use the same free-surface. condition as
in the outer flow asymptotic theory: The. velocity potential will then be proportional to
r"2' -
in the 'vicinity of theintersection point between the body surface and the free sur-face. Here
r
means the radial distance from the intersectionpoint between, the body surface and the free surface. In order
to match the outer flow solution with the local jet flOw so-lution by Wagner it is essential that the inner expansion of
the velocity potential in the outer flow be proportional to r2.
This is only possible when a - 0.
Watanabe (1986) has also provided a solution basedon matched asymptotic expansions and local jet flow analysis. However, his analysis of the jet flow, the matching, and the
final results are- not the same as Zhao & Faltinsen (1993).
This is evident from Fig. 17, which also shows 'that the
asymptotic method, similarity solution and boundary
ele-ment method by Thao & Faltinsen (1993) are in good agree-ment. At small deadrise angles the pressure is sharply peaked
close to the jet flow domain. Calculation of the pressure in
Ag. 15 Definitions of coordinate system and control surfaces used in numerical solution of water entry of a wedge by a boundary element method (Zhao & Faltinsen
(1993)]. a = deadnse angle. 310
210
1 Q-2
this area requires high accuracy both by the similarity so-lution and the nonlinear boundary element method. A rea-son is that the. pa/at term and the velocity square term in Bernoulli's equation are of different signs and have large
and nearly the same absolute value in the jet flow area. This
Ia illustrated ,inFig. .18 for a = 20 deg. When. the deadriae angle a-of the wedge is larger than about .45 deg, the
max-imum pressure occurs at the apex ofthe wedge (see Fig. 19).
When a is large, the pressure shows a rapid change around the apex. Wagner's formula for maximum pressure is often
-used, in practical calculations of maximum slpmming pres-sure for any value of a. However, it has no ratiOnal basis for
very large a-values where 'it clearly underpredicts the
max-imum pressure (see Fig. 20).
One should be careful in applying results for wedges to
other cross sections. The local deadrise angle is not the only
important body parameter. For instance, the local curvature also matters. Further, the assumption of constant body
ve-locity does not. account. for accelerations that may have im
portance, in particular for drop test experiments. We will discuss this further by studying slamming loads on a bow
flare section, which have been experimentally examined by
Yamamoto et al (1985). The bow flare section was inclined at a constant angle during the drop tests. to account for, in
an approximate way, the rolling of 'the corresponding vessel. Figure 21 shows the tested cross section as well as two
sec-tions used in the numerical studies. These two secsec-tions will
be referred to as the symmetric and the asymmetric section.
Only the half-part Of the model where the pressure was
re-corded was correctly numerically modeled. The reason is that
the boundary element method with the jet flow approxi-mation cannot handle cases where the local deadrise angle
is larger than 90 deg. The numerical simulations showed that
the pressure level in the bow flare region was sensitive to small inaccuracies in the hull form. The numerical
simu-lations started with a calm free-surface vertical position that was 13% of the vertical distance from the bottom of the
sec-tion tO the knuckle. The influence on the pressure level in
the bow flare part was not important if the simulation started
with a cairn free-surface vertical position that was twice as
125.0 0.0 O.5pV2 p-pa 0-Spy2 a - 4° Vt Vt
Fig. 17 Prediction of pressure (p) distribution during water entry of a wedge
with constant vertical velocity V. Deadrise angle a = 4 deg. Pa = atmospheric pressure, p = mass density of water. The z-coordinate is defined it Figure 15; (= time variable. The boundary element method, similarity solution and
asymptotic solution is by Zhao & Faltinsen (1993)
high relative to the bottom of the section. Starting the
sim-ulation from the bottom of the section would require special starting conditions. The reason is that the local deadrise
an-gle is zero there,, which implies infinite jet velocities (and pressure) at the initial time of impact. In reality the com-pressibility effects will limit the pressure at the time of the
first impact. This means that the pressure will not be higher than the acoustic pressure PVCe, where ce is the velocity of
sound in water.
Figure 22 presents comparisons between experimental and
numerical values of the pressure for the pressure gages P-2, P-3 and P4 as a function of the time. The asymmetric section in Figure 21 is used in the numerical calculations. The vertical velocity during the experiments and in the
nu-merical simulations are shown in Figure 22. The difference 500.0 375.0 I
I.,
I. 250.0 a 40 , I / 125.0 BOUNDARY ELEM. SIMILARITY SOL - ASYMPTOTIC SOL WATANABE (1988) 0.0 500.0 375.0 SIMILARITY SOL -. - ..- ASYMPTOTIC SOL 250.0 BOUNDARY ELEM.FIg. 16 Prediction of pressure (p) distribution during water entry of a wedge with constant vertical velocity V by means of asymptotic method described by
Thao & Faltinsen (1993). Deadrise angle a = 15 deg.p = atmospheric pressure, p = mass density of Water. The z-coordinate is defined in Figure
15; t time variable p-pa
-1.00 -0.50 0.00 0.50 1.00 30.0 - 20.0- 10.0-0.0 -1.0 40.0- P-Fi 0.5 pV2 = 15° Outer / solution
____-ç
solution Inner solutiofl I U U I 0.5 0.0 0.5 1.0 0.500 0.525 0.550 0.575 0.600p 16.0 - 12.0- 8.04.0 -o.5_p V2 -1.00 -0.50 0.00
FIg. 18 Contributions to pressure distribUtion frOm the pä4/at-term and the velocity square term in Bernoulli s equation dunng water entry of a wedge with deadnse angle 20
deg and constant vertical velocity V. Calculations based on similarity solution. C =
pressure coefficient = z-coordinate of intersection point between free surface and body surface. Other-variables explained in Figure 16
in the numerical and experimental velocity in the first part
of the time record is of no importance; The large retardation
(about 3 g) that occurs later on is of importance The small discontinuity in the n rierical pressure calculated is due to
added mass effects connected with the sudden retardation of
3g The numerical simulations are limited in time relative to the experiments In the last time instant the free surface
elevation is Similar to Figure 23 Since flow separationfrom
the knuckle will matter later on and this feature is not in
-corporated in. the numerical-method, the numerical compu-tation had to stop. We note from Figure 22 that the
riumer
-I,''I
/ I 0.0 fl-i: ... -1 0 -0.5 0.0 0.5 0.50 1.00a-20°
_a=25°
-.-.-.-.- a=
300 a = 400a=45°
a=
50°a= 60°
---..-.--
CL= 700a=
810 1.9955FIg. 19 PredictiOns of pressure distribution during water entry of a wedge with constant vertical velocity V by means of the similanty solution. Variables explained in Figure 16
Vt
ical method predicts well when the pressure starts to deviate
from atmospheric pressure at P 2, P 3 and P 4 In calculat
mg the time we have accounted for the varying velocity and
the difference in starting time of the experiments and the
numerical simulations The magmtude of the pressure is well
predicted for P-2 and P3,. while -the numerical predictions are- too large fOr P-4. An error source is that the complete cross section was not correctly modeled Further there may
be 3 D effects in the experiments The model was not equipped
with ezid. plates to reduce 3-D effects. Arai & Matsunaga
(1989) have -also made numerical comparisons with the Ya.
98 JUNE1993 JOURNAL OF SHIP RESEARCH
z
20 10 10.0 5.0 0.0 p 0... -,go o 3° 400 5b°6o° 7O°8O°
FIg. 20 Prediction of maximum pressure coefficient Cp, during water entry of a wedge with constant vertical velocity V by means of similanty solution
and Wagner's jet flow solution,a = deadrise angle
Asymmethc body
150
...Original
- Symmetric bodybow flare section inclined 22.50 Wagner jet flow solution
Similarity solution
FIg. 21 Cross section of model used by Yamarnoto et al (1985) in their drop tests together with two models used in numerical solution
p-9
- Measurements
Theory (Zhao & Faltinsen) Theory (Arai & Matsuriaga)
TIME
FIg. 22 Comparisons between numerical and experimental pressure mètitiurerrients on bow flare section. Experiments are drop test results by
Yamamoto et al (1985). The sections are presented in Figure 21
mamOto et al (1985) drop test experiments. Their results are also presented in Figure 23 anl show good agreement with the results by Zhao & Faltinsen. The effect of gravity as well as separation from the knuckle was incorporated. The finite-difference method based on the "volume of fluid" method by Nichols et al (1981) was used to solve the time-dependent Euler equations. The symmetric body shape shown in Figure 21 and the measured body velocity were used.
Figure 23 shows the free-surface elevation and the pres-sure distribution at the final time of the numerical simu-lation by the nonlinear boundary element method. Three different cases are presented One case is with a symmetric. body and constant downward velocity of V = 4.05 rn/s. The two other cases are with an asymmetric body. The free-sur-face elevations are very close in all three cases. We note that it does not matter significantly that the' body is symmetric or asymmetrc. It is more important to account for the vary-ing velocitythat means the large retardation of the body that occurs in the end of the simulation The retardation causes pressure reductions due to "added-mass" effects. This means that drop tests require accurate velocity measure-ments of the model. Further, the scaling of drop test results to full-scale pressure on a vessel iS questionable when large retardations of the model occur. The lower the pressure, the higher will be the retardation of the model. Since a vessel will have Smaller retardations than the drop test model, un-conservative full-scale predictions for a vessel can occur as a consequence.
Figure 24 shows numerical predictions of the pressure dis-tribtition by the asymptotic method presented by Zhao & Faltinsen (1993). The time instant corresponds to when the spray root is at the knuckle. The symmetric body is used and a cOnstant body velocity is assumed. The results can be
corn-JUNE 1993 JOURNALOFSHIP RESEARCH 99
p J'Vdt 0.375 0.0 0.375 -10.0
Fig. 23 Numencal predictions of free-surface elevation arid pressure distribution on the cross
sections with bow flare presented in Figure 21 at final time of simulation. Numerical results are by boundary element method presented by Zhao & Faltinsen (1993)
10.0 5.0 Cp Asymptotic theory V = 4.05 m/s, symmetric body 0.0 -10 -0.5 0.0 0.5 1.0
FIg. 24 Numerical predictions of pressure distribution on symmetric cross
section presented in Figure 21 at the time instant when the spray root is at
the knuckle. The asymptotic theory presented by ZhSo & Faltinsen (1993) has been used
pared with the results in Figure 23. The value of max mum pressure is in good agreement, while we see differences in the pressure distribution and where the calm water surface is relative to the knuckle. The position of the calm water
surface relative to the knuckle shows that the jet moves faster according to the asymptotic theory.
Zhao & Faltinsen (1992) have studiednumerically wet deck slrnming on a high-speed catamaran inhead sea waves. They
generalized the asymptotic method presented by Zhao &
Faltinsen (1993) for water impact of 2-D bodies. This means
that the wetted area and the maximum pressure are found by a method similar to Wagner (1932). Thao & Faltinsen
(1992) point out that the curvature of the wave at the impact
position is an important parameter. In general, the results are sensitive to how and where the waves hit the wet deck. Zhao & Faltinsen (1992) stated that slamming loads on the wet deck cannot be estimated by a theory that neglects the
effect of slamming loads on the wave-induced motions.
How-PB
0.0
-10 -0.5
C with V= 4.05 rn/s
Asymmetric body, Varying Velocity
- - Asymmetric body, V = 4.05 rn/s Symmetric body, V 4.05 rn/s 5.0 0.0 Free surface elevation 0.5 z
ever, this was based on the assumption of a rigid' wet deck. It is possible that local hydroelastic effects can lower the im-pact loads and the effect of the slamming loads on, the global motions and accelerations of the vessel.
Conclusions
This. paper' focused on. numerical' and theoretical seakeep-ing problems of conventional and high-speed vessels.
For high-speed monohulls and catarnarans it is pointed out that the most important ship-generated waves are due to the
"divergent" wave system. More accurate descriptions of transom stern effects and interaction between unsteady and
local steady effects are needed. The interaction between steady
and unsteady flow is particularly important for hull forms that are not close to.vertical at the intersection between the
free surface and the body sUrface.. The influence can be par
ticularly large for added resistance in waves.
The importance of acoustic resonance in the air cushion of
an SES and the "cobblestone" effect are pointed out. Also
discussed is the fact that an SES can have a substantial drop
in speed due to air leakage in waves and that air ventilation
in the watezjet inlet can easily occur for an SES in a seaway.
In order to develop physically based numerical tools for nonlinear ship motions, many fundamental physical prob-lems have to be better understood. This paper concentrated on the effect of flow separation and the water entry
(slam-ming) problem.
It is indicated that the effect of flow separation can matter in the description of vertical motions of conventional ships, in particular for hull forms'with sharp corners like bilge keels.
A numerical two-dimensional method that accounts for the
interaction between free-surface waves and flow separation
is presented. It gives reasonable predictions for Fn = 0.16, but notfor Fri = 0.26. The effects of flow separation are both frequency and Froude number dependent.
Simming on hull cross sections is discussed. Verified
an-._J
,
gles; a similarity solution, and a nonlinear boundary
ele-ment method with jet flow approximation are presented. The
intersection problem between the free surface and the body
surface requires high accuracy. The nonlinear boundary ele-ment method is a useful tool for a broad class of hull forms. However, it is necessary in the future to incorporate the
effect of flow separation, for instance from knuckles. The "added-mass" effect on the slamming pressure due to large retardation of a drop test model can give false effects and unconservative results in slamming predictions The local deadrise angle is not the only hull parameter influencing
the slpmming pressures.
Acknowledgment
J. V. Aarsnes of Marintek did the viscous flow simulations and R. Zhao of Marintek the slamming simulations reported
in this paper.
References
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'CoINvs, R. 1991 Free surface close to a.sufface-piercing body. Math-ematical Approaches in Hydrodynamics, T. Miloh, Ed., SIAM, 319-333.
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