OCR for page 867
curvilinear space (~i, (2) fitted to the body and the
free surface at each time. Free surface elevation,
71~i, t), the two Cartesian velocity components (ui),
the dynamic pressure (p) including gravitational ef-
fects (pit), and turbulent kinetic energy ~3pk) are
the normal dependent unknowns.
Mass conservation is expressed as the classical
continuity equation:
aijuij = 0. (16)
where ai is the contravariant basis. The mean mo-
mentum transport equations, written in the moving
reference system attached to the hull are:
~ ~ + (aid (ui _ of )—fief f f j—air vet, i aft ~ aid
+ ~ at pa,—Fief f gij u~ij—aid v~,iaj~ arm + qi = 0,
(17)
for or ~ 11,2~. pij is the contravariant metric ten-
sor, fi the control grid function, ug the grid veloc-
ity which indicates the displacement of the mesh.
Veff is the effective viscosity obtained by adding
the physical kinematic viscosity to the turbulence
viscosity. Inertia forces due to non-galilean refer-
ence (gyration, accelerated translation) are taking
into account in the A terms.
If a turbulence closure model is adopted, the tur-
bulence term vat needs to be solved. For example,
one might use the classical k—~ model of Wilcox
(19883. The inclusion of turbulence effects is being
pursued. Here, we only consider the special case of
Veff = V, with vat simply taken as zero.
Free-surface boundary conditions are one kine-
matic condition, two tangential dynamic conditions
and one normal dynamic condition. The kinematic
condition, coming from the continuity hypothesis,
expresses that the fluid particles of free surface stay
on it :
9,{ ~
- (hi (0 —09) 9'j) (i In 2}—~ = 0' (18)
where bi are the bi-dimensional contravariant basis.
Dynamic conditions are given by the continuity
of stresses at the free surface. If the pressure is
assumed to be constant above free surface, normal
dynamic condition is:
p—par/ - 2 P ef f ail aji all fit—R = 0'
where ||ai|| is the norm of the contraviant vector
normal to the surface, and ~ is the surface tension
BFFDM Grid for timestep t/T = 4 1/6
Figure 3: Numerical grid for the BFFDM at t/T=4-
1/6, 4/ keels.
coefficient and R the free-surface medium curvature
radius(Ananthakrishnan & Young, 19943. Tangen-
tial dynamic stress conditions are simply given by
a linear combination of first order velocities deriva-
tives:
a2ipj3'uij = o, (20)
where a2i is a covariant vector tangential to the free
surface.
At each iteration, governing equations and exact
nonlinear free surface boundary conditions are dis-
cretized in space and time by second-order finite-
difference schemes on a monoblock structured grid
that is fitted to the hull and the free surface. The
algorithm of solution involves solving the discrete
system of unknowns for the velocities, pressure and
free surface in a fully coupled manner. Transport
equations are first linearized and then discretized
at each grid node i with second-order non-centered
(unsteady and convection terms) or centered (diffu-
sion term) finite difference schemes. The pressure
gradient is interpolated at grid nodes since the dis-
crete pressure unknown P are defined at the cen-
ter of the cells. In order to decrease memory re-
quirements, unknown pseudo-velocities U* are in-
troduced. So, the transport equations for discrete
velocity U at grid node i can be written as:
Ui —Ui + (Mp~i~ Pa = (f~)i~ Or ~ 11, 2), (21)
Meanwhile, the kinematic boundary condition and
the tangential and normal stress conditions on the
5
OCR for page 868
free su face can be manipulated into t..e
discrete form in time:
U\ +(M5~)\) U)~\+(M56),,,ET\ = (f5)\ ~ ~ 11,.
be foil wing
(22)
The use of the discrete ~ al dynamic condition
can lend to the foil wing final form of free surface
boundary conditions or velocities:
Ui U*° (Mix)\ (Mix) Pa = (fix)\ ~ ~ 11, 2}
(23)
where the My, and f are quantities defined in
Alessandrini and Delhommenu (1995) The 'tilde"
notation indicates that interpolation to the grid
point from cell center is necmsarJ The pressure
equation, in matrix form and utilizing the pseudo
velocity, is solved by using the Rhie Ed Oh w
method (1983) so as to woid checkerboard oscil
lotions
~puU + AMP = As,
where lo , and T~ result from the lefi hand
and right hand sides of Eqns (22) and (23),
respectively This full set of equations is solved
at each it ration in n coupled manner using the
Bl CGSTAB algorithm of Van Der Vor t (1992),
conditioned with an incomplete LU decomposition
(24)
Figure 3 illustrates the 0 type mesh fitted
around the cylinder at one instant of time
2.3 Forces and moment
The hydrodynamic forces Ed moment acting
body can be calculated according to
Yw =
r\> = at,) + at,\
In the case of the FSPVM, the only stress on the
body is the pressure which is given by Euler's inte
gral:
ing on the
xw = J (Pant + p~r~)ds,
= / (Pd~2 + p~rz>~)d5,
MW = J [a (Pact + p~rtz~z)
(25)
(26)
Pd~2 + pvrzz~)]ds, Dd (27)
(28)
= + 9~ = f6,2 fu,2 2 Pa (29)
In Yeung and Annnthakrishnnn (1992), the shear
tress was found to be of secondary importance in
trongiJ separated fi w the pressure is the domi
pant dynamic stress contributing to the total fluid
force and moment H waver, BFFDM can provide
the shear tress as part of the solution procedure,
and both component of the stress are used to com
pute the force and moment integrals (Eqns 25 27)
The hydrodynamic coed cients will be dire tlJ ob
twined from these time histories as explained further
in Section 3 2
3 Experimental Setup and
Hydrodynamic Coefficients
The experiment were conducted at the ichmond
Ship Model Is ting Facility of the University of Cal
if ornin at Berkeley A 2 54 x 0 3 x 0 3 m rectangu
Inr acrylic cylinder as shown in Fig 4 was hinged
at the water level by the sides of the tank Bilge
keels of 1 25 and I 90 cm were mounted on the bilge
corner of the cylinder at n 45° angle The model
w ighted 100 kg and displaced 0 11 m3 The rolling
motion was induced by n hydraulic cJ5nder that
can accept n random motion input (Random Mo
tion MechaDi m, Hodges and Webster, 1986) The
cylinder w oscillated from 2 5 to 10 Hz Further
details on the apparatus are explained in Yeung et
cl (1998) in "dry conditions', the model was m
ciliated to oht in its mass moment of inertia (not
al period=1 424 sees) The center of gr wity was
measured by an inclining experiment Force trance
ducers placed at the end of the piston rod and just
above the hinges provide an analog signal to n com
Outer The transducers were calibrated statically
Measured forces and moment were obtained fter
filtering instrument noises and vibrations,
3 1 Measured forces and moment
Figure 5 shows n free 5: d:, di gram of the system
that allow one to nnalJze the applied forces Let
W be the w ight of the cylinder, m the mass, lo the
mass moment of ine tin about 0, G the center of
gravity at (XC,YC) with 0G = To and 0A = rA
Let also Xw, Yw and MW be the forces and moment
exerted -- the fluid on the cylinder, and XA and
'A the forces exerted by the horizontal driving rod
on the cylinder, The subscript o refers to quantities
6
OCR for page 869
Figure 4: Experimental apparatus and setup
MY'
A (Q Drivi g Rod
1
\ W rod
b' ~ o \x
~ E- at' \~L
Figure 5: Free body di gram or
measured at the roller bearing 0,
'the cylinder
and the X' and
Y' refers to the forcm in Ax'y coordinate system
as sh wn in Fig 5 Force transducers are adjusted
to zero and calibrated when the cylinder is installed
on the hinges and in the upright position Their
outputs are zeroed when no motion is present even
though there may be static loads When the cylin
der is in motion, the transducer respoonse are pro
po tional to the additional forces present There
fore, thetotal force applied m --- cylinder at n hinge
can be expressed as
F=F OF,
(3o)
where F can be either X or Y and the second term
'm' is proportional to the output voltage of the
transducer
BJ making use of Eqn (30) and noting that the
F°'s are the static terms computable from the ge
ometric prope ties of the cylinder, w can arrive at
the foil wing equations for the hydrodynamic forces
fit with buoyancy eve ts removed:
Mw =
Xi COS ~ + YA sin c X
To sin 9 Rod
W (cost 1) 2 sick
mrc(c~ cos(c + 9) c2 sin(c + 9
Xi sin c YA COS C Y
W sine 2 (I cost)
mrc(csin(c + 9) + c2 cos(c
XA Ma + Wrc(sin(~ + 9)
sin 9) + 40: pgVL 6 sin a,
))~31)
+ 9))~32)
(33)
where 9 is the angle about 0 between the center of
gravity G and the vertical dire tion when the cylin
der is at rest, i e 9= ntan(xc/yc) Wood is the
w ight of the driving rod, V the submerged section
area, with the last term of (33) representing buoy
once effects We note that even though the meat
sured force carries the dynamic information of the
fluid inertia term, it can be shad wed -- the ho
drostntics term unless the body weight cancels it
This can be done -- making the cylinder h wing n
zero metacentric height
BJ Fourier decomposing the measured time series,
hydrodynamic coed cient are obtained as shown in
Se tion 3 2 We note in passing that the roll mo
tion of amplitude cO is started smoothly by using n
hyperholi~tangent ramp d function of the form
c(t) = C~o sin(htt) tanh(htt)
3.2 Hydrodynamic coefficients
In linear, potential fl w theory, the hydrodynamic
(34)
inertia an d linear damping terms WeEnusen, 1971):
Xw(t)= ~16C~ ~-60,
(35)
Mw(t)= P66~ is60, (36)
e: Xw and Mw are the hydrodynamic way
forces and roll moment as obtained from both the
numerical and experimental simulations Since the
forced roll motion is of the form c(t)=co sin(htt), the
linear hydrodynamic coed cient6 at t=to can be Cal
culatYd by extracting the Fourier coed cient of the
7
OCR for page 870
primarJ frequencJ over n period T HJdrodJnamic 4 Results and Discussions
coefl cient of the WAJ force induced bJ roll motion
cnn be sh wn to be:
1 ,20+T/2
?ts (to) = —J.
1 ,20+T/2
Nt6(to) = 7r~0 J ~ Xw(t)c06(h~t)dt, (38)
while the dingonal term6 nr6:
1 ,20+T/2
7rOOh) J20 T/2 Mw(t)6in(h~t)dt, (3P)
is6 (to) = —J Mw (t) cc6(h~t)dt (40)
Enrlier, we hs:v6 completed som6 experiment6 for
n model without bilge ke616 Yeung st cl (1998) nnd
compared som6 with the reported rmult6 of Vugts
(1968) Her6, for resson of consistencJ nnd ess6
of interpretntion6, we defin6 the snm6 normnlized
cc6fl cient ss befor6:
- ?66
?66 = 4
Xw(t)sin(hlt)dt, (37)
?66 (t
, N66=4 ~, (41)
it6 = 2 ~ Nt6 = 2 ~
Becau66 the fl w is ofl en governed bJ viscou6 sep
nrntion, n nonlinenr formulation for the damping
could nl60 be expr6666d ss:
Mw(t)= ~s6~ N66~ ~ ~ (43)
wher6 N66 is the qundmtic damping cc6fl cient nnd
is n trnditional WAJ of reprmenting the over611 fl w
efl6ct6 H wever, if the prom66 is sssumed to be
sinu60idal, equiv616nt energJ dimipation rnt6 would
impiJ the following formuln fOr N66:
3 ,20+T/2
is6(tO) = 8 J ~ Mw(t)c06(hdt)dt
Thus, under the stnted sssumption, Equntion (44)
is related to Eqn (40) bJ
(42)
(45)
The damping cc6fl cient prmented in Section 4 will
follow Eqn (40) nnd the nonlinenr cc6fl cient cnn
be deduced nccordinglJ H wever, if both linenr nnd
nonlinenr damping nr6 prment, the uniquen666 of
ench of thm6 cc6fl cient cnn be determined onlJ
with more information A formulation for this css6
cnn be found in Yeung st cl (1998)
For n given cJIinder geometrJ nnd with the sssump
tion of prmcribed periodic motion, the problem is
characterized bJ the roll nmplitud6 ~0 nnd the fol
lowing nondimensional parnmeter6:
KD, i = ~V~, 116 = 4~ t2/~ (46)
wher6 KD is the percent ge of bilg+ke61 depth rela~
tive to the benm nnd 1/e is n charn teristic ReJnold6
number similar to Yeung st cl (1996) in the FS
~VM, the ReJnold6 number is taken the snm6 ss the
phJsical one in the InborntorJ, which rang66 from
16,000 to 144,000 TJpicallJ, the number of vortex
blobs umd is of 0(50,000) though n Inrger number
cnn be nccommodated in the BFFDM, the grid
siz6 is ndapted to the medium 1/e to woid remmh
ing nnd consist of 200 x 200 nodm
4.1 Inviscid-fluid results
Viscou6 flow solution6 nr6 emerging becau66 of the
incressing p wer of computer6 Inviscid rmult nr6
till verJ useful in providing not onlJ n guidelin6
but nl60 n vslidation for new numerical method6
The problem of n rectangular section in roll mo
tion hss long been solved bJ n varietJ of meth
ods Yeung (1975) used n HJbrid Integml Equntion
Method, which is verJ powerful for the css6 of nrbi
trarJ geometrJ BoundarJ element method6 bssed
on vsriou6 Green functions is nl60 n pmsibl6 WAJ
for solution An ndv6ntag6 of the FS~VM is that
it Jield6 the inviscid (potentinl flow) solution when
the vortical psrt of the strenm function corrmpond
ing to Eqn (8) is not 'turned on " in Figs 6 to 9,
the inviscid fluid hJdrodJnamic cc6fl cient6 nr6 pr+
mnted The solution for n rectangular crm6 section
(with n smnll bilge rndiu6, s66 Fig 1) is included for
referenc6 purpm66 Thm6 rmult6 c D be considered
ss linenr rmult6, being independent of the roll nm
plitude in the sctunl FS~VM computntion6, ~0 is
taken ss 2 85 degr666 Even though the potentinl
fl w problem solved wer6 nonlinenr, the inviscid
calculation6 wer6 independent of the nmplitud6 of
roll for nngl66 Im6 thnn 10 degr666
The FS~VM inviscid fluid rmult for the 'No
Ke616" css6 nr6 obtnined using n rounded corner
similar to the experimental model wher6 the corner
mdiu6 wss 2 ~o of the benm Note that the FS
~VM for n squnr+cJlinder Jielded the mtablished
8
OCR for page 871
I nviscid Roll Added Momem d Ine ria C efficiem
Figure 6: Added moment of ine tin, inviscid fluid
solution
o~ os os ~
Inviscid RolPSway Coupled Added Momem Coefficiem
032
028
024
020
p15015 ~
012 ~//
008 'i~
004 .<
O ~ 0 5
Figure 7: Added m dS of roll into swnJ, inviscid fluid
solution
Inviscid Roll Damping Coefficieni
2 S 2V M, r n ~ i s c id, 20 K ee l s
252VM, rnViscid' 4t Keels
252VM, rnViscid' 6t Keels
252VM, rnViscid' 0t Keels
\
\\
,,." . " '
/~/',
~ ~"
04 05 08 ~ 10 12 14
Figure 8: Dnmping coeflicient of roll, inviscid fluid
2snM' rnVircid' b Keels 02
<^\ 2snM' rnVircid' 4t Keels
/ ~ 251ZI'M, rnViscid' 6t Keels 024
i_ ~ i~ 2snM' rnVircid' 0t Keel~ 02
{/~/ 'n'\
//,/ ,~'~'., ?~:~Y ~ 015
~,I'' ~ O ~ if ~
0~ ~ ,'T/,'t''
0~ ~
Figure 9: Dnmping coeflicient of roll into wnJ,
9
OCR for page 872
solution of Yeung (1975) nbd was given in Yeung et
cl (1998) As e me ted, the effect of the keels is
to increase the ndded moment of inertia and the rat
diction damping COD idernbly The location of the
peak values, around i = 0 6 for the ndded inertia
and hi = 0 8 for the damping, remain the same as
the cylinder without keels
4.2 Theory versus experiments -
a validation
In Figs 10 and 11, typical time histories of the ho
drodyDamic moment at the condition of hi = 0 8
are sh wn for both numerical methods described in
Sec 2 and the experimental measurements The
I wer figure presents the werage valum of all three
set of time series over three or more periods from
t/T = 0 5 onwards The agreement is seen to be
very good between FSPVM and BFFDM, both so
lotions overpredicting slightly the experimental re
suits Because the phase difference is minimal be
tween all three curves, we expect the ndded mass
and d mping coed cient predicted to be similar, as
will be seen inter
A more detailed comparison of the two methods
of solution is shown in Fig 12 The variation of the
local pressure on the body contour is sh wn, in the
fourth period, as fun tions of the arc length parnm
eter e/! measured in n counter clockwise direction
and with the zero reference point taken at the cen
terline Each subfigure corresponds to an in tant
of time succmsivelJ T/6 apart, with the cylinder
rot tide clockwise The tips of the lefi and right
bilge keels show up as e/! npprcximatelJ equal to
I and +1, respectively Result from both numer
.1 methods agree remark blJ consistently, partic
ularly in view of the drastic difference in formu
lotions F Other, since the boundary condition on
the free surface are treat d difierentlJ, most of the
disagreement is expected to occur at values of e/!
near the waterline The pressure difference between
both faces of the keel lend to n jump in the pressure
from the side to the bottom of the cylinder The
presence of the keel is therefore n major contrib
utor to the hydrodynamic roll moment it should
be noted that during the second half period which
is not sh wn in Fig 12, information = -= .1 to
those presented in the first half is expected since
there is quasi symmetry of the solution about the
T/2 point The pressure distribution will have n
mirror image about the point e/! = 0 to the first
half period
The time series for the way force and roll moment
in this study are very similar to the ones presented
in Yeung et cl (1998), except for an increase in
amplitude as the keel depth or the amplitude of mo
tion increases it is more useful, from the pro tical
tandpoint, to present the Fourier aver bed values
of the force and moment in the form of hydrody
namic coed cients as defined in Eqns (37) to (40)
In Figs 13 to 16, the roll ndded moment of inertia
and roll damping, as obtained experimentally and
numerically using FSPVM, are presented for bilge
keels whose depths are To and 6~7o of the 5 Yam of the
cylinder, and for two amplitudes of roll angle Re
suits for the rounded corners geometry can be found
in Yeung et cl (1998) The diagonal term of the
coed cients Figs 13 16 are grouped separntelJ from
the coupling terms, Figs 17 20 Numerical results
obtained using BFFDM are ndded to Figs 13 and
15, when KD = 470 and coo = 2 85 for comparison
purposes it is grntifJing to see these two very defied
ent and independent methods yield very predictions
close to each other These predictions also tend to
agree well, for the most part, with the experimental
result Generally speaking, FSPVM and BFFDM
do agree better in the diagonal terms than the cou
Fling coed cients in the case of comparisons with
experiments, one should keep in mind that the reli
ability of the experimental measurements decreases
with an increase in frequency because of vibration
of the test apparatus
Detail examination of this e tensive set of data
will suggest the toll wing trends
e An increase in keel depth (ado to Who) w uld in
crease both the ndded inertia and damping for
the entire range of frequency This is expected
intuitiveiJ and from the computations The
inertia measurements indicate otherwise even
though the change is not seen as substantial
e With bilge keels size fixed, experiments and
real fluid theory suggest that an increase in
roll amplitude lead to a decrease in the inertia
coed cients for both di gonal and off diagonal
term, or at least up to the largest angle of 5 75°
investigated here Larger roll amplitude yield
an appreciably larger damping coed cients
e The agreement betw en theory and experi
10
OCR for page 873
oo6r
0.05
0.04
0.03
0.02
Q 0.01
Q O
-
-0.01
-0.02
-0.03
-oo4
-0.05
-0.06
-- BFFDM
- FSRVM
— Experiment
0.1 r
VT = 4 -------a------- BFFDM
---I-- FSRVM
_ i;/; 1/ (i
t/T
Figure 10: Moment MW(t) history, ~ = 0.8 ~
Q
0.05 _
0.025
a)
Q
Q O
-
-0.025
.
~ \
\ \
_\ \
~'\~\
\\ \
~N ~//
—- - - - -- BFFDM
— - - - FSRVM .-
Experiment / .'~'
.......... oc(t) / /~' ~\
..,,. /! ~
/ /'! .,
/ d
/ !/
, ///
~/
-i\ Y .//
. \ ~ ~ .
/
_
;". I ""'i 1 1 1 1
0.75 1
t/T
Figure 11: Steady-state averages of roll hydrody-
namic moments.
0.05
-0 05 _
n1,
0.05
-0.05
—. ~
~ \
-nn.~L
, i, ~,
j d - , ~ -~
~r
l
VT= 4 1/6
~ I
ii
11
::
- ~—~ ~-4 ~ W--~
;-- ~
0.1
0.05
~ O
-
VT=4 1/3
~ -~-- ~ ~ :,-4 W-~ - ':
., ~ ~ 4 ~ ~ ~
-2
T ~ _lI~
-1
1
Figure 12: Local pressure on the body for ~ =
0.8, numerical solutions of two methods.
11
OCR for page 874
contain such dissipation, thus prese ring the d:
tkile better However, FSPVM does not model the
boundary Dyer Flow kVi w 11 kVi BFFDM
5 Conclusions
.
ments is better for the diagonal terms tEkn
for the coupling terms Lkrger discrepancies
betw en theories and experiment exist in the
high frequency regime, in which experimental
measurement are not kVi
e Tkditionki computations baled on i fiscid
fluid theory signiflckntiJ over estimate the iner
tin coefl cients, and under estimate the damp
ing effects Similar observations in this regard
were reported in Yeung et cl (1998) for the
same cylinder without any keels
e The inertia coefl cients approach the i fiscid
fluid theory at high frequency This seem con
sistent with an k alytical viscous fluid theory
-- Yeung and Wu (1991), which modeled only
diffusion e' ts
In the high frequency both a periments
and theory are SUED iting that the damping
are not vanishingly small k i normkilJ expected
from inviscid fluid ccmput tic i This can
he Is impo tant design implicatio i
4.3 Vorticity and flow patterns
Figure 21 details the flow patterns for k period
of one oscillation after k teddy tate is reached
Strong vortices are erekted in the vicinity of the kfl
edge of the bilge keel and are later shed k J k i the
keel move i in the opposite direction During every
half period, the vo tices from the fore keel he f9 k
strong tendency to m fe away sidewkJet, whereki
those from the kfl keel would move drew ward The
phenomenon is not entirely mirror in image for the
keel during each half of one full period This slight
lack of symmetry WE i explained by Yeung and Cer
melli (1998) k i kttributabie to memory effect of the
stk ting swing
In Fig 22, the vo ticity fields obtained using both
numericki models are di played kVi color contour
plot The vortex method predicts larger and more
di tinct vortices, kVi well kVi larger distances from
the 5: d:, once they become ceparkted from the
keel A stronger dissipation apnea i to he f9 taken
place in the BFFDM As in most finite diflerencing
schemes, numericki (krtifleiki) damping is always
present The grid free method of FSPVM does not
direction of roll motion k fly half period, the fore keel be
comes the aft keel, md ic~versa
Experimental and theoretical studies of the forced
roll motion hJdrodJnkmim of k cylinder with bilge
keels are conducted The theoreticki model includes
the use of two free su face Napier Stokes Solvers
(FSRVM & BFFDM)
overkil speaking, the results obtained using the
FSPVM method are well validated -- the experi
mental results in coffee where both numerical so
lotions are k Cable, very acceptable is
observed, at both global and detail d level This
lends much credence to the present methodologies
of predi tions
Added moment of lnertik and (equivalently lin
ekrlzed) damping coefl cient are presented k i fund
tions of frequency and bilge keel depth The compu
tationki and result are compared also
with those from elk isicki methods ha led on an ideal
fluid The computations suggest teat an
in keel depth will increase both the added inertia
and damping T Editions computations ha led on
potential Flow theory will significantly over estimate
the added coefl cients and under estimate
the damping effects P al fluid effects tend to
"round" the shape of the body, thus reducing the if
fe tive ine tik, whereki flmw eeparktlon round the
keels inerekiee the damping Similar oh e ratio i
w re given -- Yeung et cl (1998) for k cylinder
without keels Details of such flown are ekiiiJ vi u
khzable from the solutions presented
For k given bilge keel size, an increase in roll km
plitude reduces the nondimenefonkl lnertik coefl
cients slightly but inerek ies the damping coefl cients
kppreciabiJ The oh ervation appLes to both ding
onki and off diagonal terms Being inve tigated is
the amplitude dependence at Angie i larger than the
ma imum value of 5 75° studied here
The complete treatment of the free motion of
k floating cylinder in Woofs, with the full e' ts
of vi c iity included, can be found in Roddier et
al (2000) Effects of turbulence on the present so
lotions are being examined
12
OCR for page 875
Roll Added Moment d Inertia oeffiaent
olo , .. . . ..
009 /~-\
003 /.' ' ,\
007 /~ ~ '.\
4660o3 ~
oos c<,,_ '_
0 04
003
002 :,
04
Figure 13:
TO = 2 85°
03 03 Jo 12 14
.~: '.,'\
~ +~
j =
, ;~, ,,,`
.-f , .. ..
3
Figure 15: Equi 41ent line 4r d 4mping coef cient
of roll, 4O = 2 85°
=~!
~ ;/ -2
' '. o. 1
Figure 14: Added moment of inerti 4 coef cient,
4O = 5 75°
Figure 16: Equi 41ent line 4r d 4mping coef cient
of roll, 4O = 5 75°
13
OCR for page 876
Roll-Sway Coupled Added Moment d Inedia Coeffiaent
08 ~ 10 12 14
Figure 17: Roll s 4J coupled 4dded inerti 4 coef
ficient, 4O = 2 85°
Ro -Sway Coup ed Added Mom ent of nert a Coeff c ent
030
026 /
/~ \ ~
020 / .\
~ ~ ~ ~ /y '\
u1416 ~N
010 ~ t
06 ~ ~ ~: ~
04 06 OS ~ 10 12 ~4
Figure 18: Roll s 4J coupled 4dded inerti 4 coef
ficient, 4O = 5 75°
14
~ ~ ~
~.. ~
DU
04 oc od ~ Jo 12 14
Figure 19: Roll s 4J coupled d 4mping coefi
cient, 4O = 2 85°
Roll-Sway Coupled Damping Coefficient
~.,.
(: ~
,j,i,
~/,~1, ...
04 od od ~
Figure 20: Roll s 4J coupler
cient, 4O = 5 75°
OCR for page 877
2C
1C
y C
-1 C
-2C
-3C
-4C
2C
1C
y C
-1 C
-2C
-3C
-4Ce —
20
10 _
y O—
-10
-20 _
-30
-40
-40 -30 -20 -10
VorticityVectors,ocO=5.75°, w=0.8
Vorticity Vectors, oc0 = 5 75°, ~ = 0.8
-40 -30 -20 -1 0
VorticityVectors,ocO=5.75°, w=0.8
- ;,~,`~-i-~-- I-.;
-40 -30 -20
t/T = 5 1/6 ~
-10 0
X
VorticityVectors,ocO=5.75°, w=0.8
30 40
-40 -30 -20
t/T = 5 1/2
, 1, . . .
0 10 20
X
VorticityVectors,ocO=5.75°, w=0.8
30 40
20 _
10 _
y O -
-10 _
-20 _
-30 _
-40 _
-40 -30 -20 -1 0 0 1 0 20 30 40
X
VorticityVectors,ocO=5.75°, w=0.8
~~ ' ~ 'a -; . a'.
-10 0
X
30 40
X
Figure 21: Flow visualization: velocity vectors of the vortex-blob field by FSRVM.
15
OCR for page 878
5
o
-5
-10
.'.'.'.F S '.'.'.'.R '.'.'.'.V '.'.'.'.'M.'.'.'.'.
In. 6
a (,.,.t,.,.,) -
3.... s3~]
~35 30 -20 -10 ~ 10 20 30
1-0.60 -0.52 -0.44 -0.36 -0.28 -0.20 -0.12 -0.04 0.04 0.12 0.20 0.28 0.36 0.44 0.52 0.60
'.'.'.'.'.B '.'.'.'.F '.'.'.F '.'.'.'.'.D ''M.'.'.'.'.
t~—.... 4 6
~ (it)
-20
-25
-30
~35 -30 -20
~ .. s3~]
-10 ~
10 20 30
Figure 22: Vorticity contours for a rolling cylinder with 4/ bilge keels at one instant of time, TO = 5.75°
The contour scale is based on a non-dimensional vorticity defined by (3/~.
16
OCR for page 879
Acknowledgement
This resekrch reported hkd been supported in pkUt
bJ the Of ce of Nkykl R,esekrch kgd k Shell Foun
dBtion Grknt of the US kuthom kDd bJ the F7ench
Ministere de Ik R,echerche kDd the French Ministere
de Ik Defense (DG4) of the F7ench eOIIkbOrktOrS
We kre kUpreciktive of the Ecole Centrale de Nkntes
kDd the UniversitJ of Oklifornik kt BerkeieJ for pro
moting the coilkborktive e yironments
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18
Representative terms from entire chapter:
free surface
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