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OCR for page 149
Effect of Viscous Damping
on the Response of Floating Bodies
M. brownie, I. Graham2, X. Zheng~
(~University of Newcastle upon Tyne, United Kingdom)
(Imperial College of London, United Kingdom)
ABSTRACT.
The prediction of hydrodynamic damping flue
to vortex shedding on a threedimensional body
in waves is discussed. A method of matching
precalculated results for separated oscillatory flow
past an isolated edge at each vortex shedding
region of the outer potential flow field of the body
in waves has been extended to threedimensional
floating bodies by two different methods. In the
first method the pressure field due to a local
separated flow computed by a discrete vortex
method, is applied at all vortex shedding edges of
the hull. In the second method the effect of
vortex shedding at an edge is represented directly
in the frequency domain panel method by special
edge panels. A dipole density on each of these edge
panels is evaluated by making a comparison
between the far field effect of the panel an.` the
far field of the appropriate separated vortex flow
modelled by the vortex method.
NOTATION.
A,B,C,M added mass, damping, restoring and
mass coefficient matrices
vortex force
body motions = ~je~it~t
KeuleganCarpenter number =
211U/COLB
matching length scale
LB, LV body and vortex length scales
l barge beam
1V p vortex panel length
q velocity amp' itude
s surface dist.rlce
U velocity scale
z complex coordinate in crosssectional
plane = x + iy
vortex strength (circulation)
internal edge angle
complex coordinate in the transformed
plane
dipole strength
density
source density
velocity potential, sum of component
potentials fje~i(0t, j=1...6, for surge, sway,
heave, roll pitch, and yaw.
frequency of motion
Hj
Kc
TRODUCOl'lON.
Prediction of the forces and responses induced
by waves on floating bodies is usually based on
linear potential flow theory (1), which is quite
adequate for the prediction of many types of
response . The damping arising from linear
potential flow is associated with the waves that
radiate out from the body. However certain types
of motion for which the first order radiation
damping is fairly small are not well predicted. In
these cases nonlinear effects must be included in
the analysis. The most important nonlinearities
may be accounted for by second order potential
flow effects and separation. Both of these are
significant for example in low frequency sway
motion. In the present paper we are concerned
with incorporating separation effects alone into a
linear potential method. This is found to greatly
improve the prediction of, in particular, toll
motions of many types of floating body. The effect
of neglecting separation is conspicuously evident
in the case of roll, particularly near resonance,
when the response can be greatly overpredicted
by linear theory (2), see figure 1.
In engineering application of linear
potential theory it is usual practice to incorpc'~ ate
empirical coefficients into the calculation to
represent the effects of viscous damping. Good
agreement between predicted and experimental
results can be obtained for most standard hull
sections. The results of the semiempirical method
(3) to predict wave and viscous damping are also
shown in figure 1. However prediction of the
damping for nonstandard hull shapes is more
difficult. Other semiempirical meth ads include
experimental determ ination of the individual
components of the roll damping (4). use of the
concept of crossflow drag on a barge hull and its
appendages (5), and work described in references
(6  10). In the case of a barge or ship hull the
dominant nonlinear component at toll resonance
is due to viscous, i.e. separation, effects. These
effects, often described by the general term
'viscous damping' are primarily due to vortex
shedding from the hull and its appendages. A two
dimensional model incorporating theoretically
the effect of separation has been developed for
barge roll (6). This model used a local solution of
separated flow past an oscillating edge which was
representative of the bilge of the hull and its
relative motion. This local flow field was computed
149
OCR for page 149
l ~
81
6~
go
a
,
11a
11 \\
·t \e
./ ~
W(Hx)
Fig. 1. Roll amplitude against frequency for a rectan
gular cylinder in beam waves (Salvesen et al.(2)~.
wave and viscous damping;   , wave damping only;
·, experiment.
as a universal semiinfinite edge flow dependent
only on the edge angle (typically 90° ) and was
then applied by a matching process to the flow
field around the hull. The method used to compute
the flow field was the discrete vortex method (11)
which gives good flow and force predictions for
vortex shedding from sharp edges in oscillatory
flow. The flows induced on floating bodies are
essentially high Reynolds number and oscillatory.
In many cases separation is fixed by the geometry
of the body. Flows of this nature have a strong
vortex structure that is easily modelled by the
discrete vortex method. Numerical solution of the
NavierStokes equations by finite differences or
elements could also be used but is generally m ore
expensive because unless a very fine grid is used
these methods do not capture the thin shear lay ers
in the flow field. The discrete vortex method
provides a time domain solution for the separated
flow. The KuttaJoukowski condition is used in the
inviscid case to specify the strength of vorticity
shed from the edges. Later refinements of the
method have shown that rounded edges, bilge
keels and viscous effects can also be incorporated
successfully,( 12).
Since prediction of the response of a floating
body requires a large range of input amplitudes,
directions and frequencies to be computed, for
practical use it is essential to have a method which
minimises the computer time for each case. The
use of a matched inner flow field as above
provides this economy since the separated flow
need only be calculated once for all edge sections
of the body having the same included angle. This '
inner' flow is then matched to the ' ou er'
linearised potential flow at the bit be or salient
region of separation so that one time consuming
vortex calculation can be used to provide results
for a complete range of motions for any given
body. The outer flowfield associated with the
incident wave field and general motion of the
body is solved with a full bound ary integral
method. Since this r.`ethod is based on inviscid
potential flow theory which cannot model flow
separation, the flow field is singular along the
lengths of bilges and other shedding 'edges. In the
case of a threedimensional body in a general
incident wave field a quasitwodimensional
assumption is made for the vortex shedding
locally. The outer flow field is fully three
dimensional but it is assumed that the separated
flow is generated within a formation region
sufficiently close to the edge on the scale of the
body that it may be considered to be locally two
dimensional. Hence the inner separated flow field
is matched on a strip theory basis. This approach
is justified at long continuous edges but clearly
breaks down at those points where sudden
changes in geometry occur. The method also
assumes that the flows associated with different
edges do not interact and also that the wave
making effects of the vortices are insignificant.
The first assumption is justified for sufficiently
small amplitudes of motion, although the results of
the method are in reasonable agreement with
measured data up to moderately large roll angles.
The second assumption is less certain, see for
example ( 13). It should be noted here that the
entire method is theoretically based and does not
require the input of any empirical data. Also the
linearised free surface boundary condition is
applied so that the method is not limited to the low
frequency approximation.
The sep arated flow field inducer! by
oscillatory flow past an edge commonly consists of
pairs of vortices, one pair shed during each flow
cycle. These vortices are of alternate sign and
often form a closely coupled pair moving away
from the edge under their mutually induced
velocity fields. Other modes of shedding also occur.
A single vortex may convect along a surface
under the induced velocity field of its image c Id
for rounded edges two vortex pairs may be shed
per cycle, ( 12).
This method which matches the vc tex
calculations to the outer flow past a general body
has been described in detail in ( 14). The main
feature is that the singularity at each edge in the
potential flow past the body crosssection is
evaluated by applying a conformal
transformation to the section which opens the
perimeter out into a straight line, figure 2. The
force induced by the ~ ortex shedding on the body
is calculated from an integral over this
transformed crosssection. This procedure is
reasonably straightfc rward to implement for
simple hull shapes but becomes complicated and
difficult to specify for many other type; of
floating structure. For this reason an alternat ive
approach which is the subject of th s paper has
been developed.
In the new method the effect of vortex
shedding at an edge of the body is mode:led
directly in the potential flow calculation by dipole
panels representing the vortices along each edge.
The dipole panels satisfy the free surface
boundary condition and may also be considered as
representing the force induced by vortex
150
OCR for page 149
Real planeTransformed plane
Infinite wedge
:_ _,_
z = L~]
L= all
~us matched to US
1 ~(all)~' ~
_ _
_
UB
Rolling barge
Fig. 2. Matching the local and exterior flows.
shedding at an edge. Because of the convenience
of working in the frequency domain, as with the
matching method, the nonlinear vortex force is
Fourier analysed into a fundamental component at
the input frequency and harmonics which are all
considerably smaller (10% or less) than the
fundamental. It is therefore justifiable to neglect
them when primary motions such as roll
resonance are being considered. However the
same would not necessarily be true where second
order potential effects are important such as in
low frequency response to waves. The dipole
panels are therefore evaluated in the same way as
the body surface singularity panels in terms of a
complex amplitude at the input frequency. The
length of the panels however depends on the
amplitude of the motion or waves and is calculated
by considering the separation condition at the
edge.
COMPUTATION OF THE FLOW F1~;LD.
Oscillatory flow about a body ma, be
characterized by the KeuleganCarpenter
number, Kc. At small Kc the maximum
displacement of the fluid particles in the
undisturbed flow is small in comparison with the
scale of the body. Thus vortices may only move
away from its edges under the influence of the
velocity field of other vortices shed from those
same edges, and hence the shedding from any one
edge may become independent of the shedding
from other edges. In these circumstances, the
local flow becomes analogous to the local flow
about an infinite wedge. The discrete vortex
analysis of shedding from an isolated edge (11)
carried out for a series of infinite wedges of
varying internal angle showed that the complex
force due to vortex shedding could be related to
the vortex strengths and positions by
FV = i P d/dt { ~ JO ( (j ~ ~ J ) }
= 1/2 p u2 L KC(26~)I(3~23) Aft) (1)
where ~ is a dimensionless timedependent force
function which can be Fourier analysed into a
fundamental and higher harmonics of the input
frequency and U and L are length scales defined
in the matching process.
Abode Surface Panels.
The flow field is described by a velocity
potential which satisfies Laplace's equation. For
each wave frequency the total potential may be
written in terms of the separate potentials for the
undisturbed incident wave, the wave scattered by
the body considered to be fixed and rigid, and the
waves radiated by the body in its six degrees of
freedom, where it is understood that the real part
only of the potential is considered:
6
do e + ~ (j Hj + ¢7 edict (2)
j=1
The potential satisfies the linearised free surface
condition
`,,2 ¢,j + g 6~/by = 0, j = 0 ... 7, Y=0 (3)
where y is the vertical axis. A radiation condition,
and a normal velocity condition on the body and
sea bed are also satisfied. In the new method the
potentials were represented by a mixture of
sources and dipoles:
tj(q)=l/4nl{oi(q) G(q',q)  2 fj(q) dG(q~q)/dn}ds
B
j = 1... 7
(4)
The integral is over the points q lying on the
wetted surface of the body B with normal n. ~ ( q )
are the source densities and G(q',q) are suitable
Green's functions ( 15). The potentials are
evaluated by solving the boundary integral
equations on the surface of the body numerically
( 16). The equations of motion of a freely floating
body can conveniently be expressed as:
~ {  c32 ( Mjk + Ajk )  i°Bjk + Cjk } Ilk = f; (5)
k
The force coefficients and the exciting forces fj
can be calculated from the linearised form of the
Bernoulli's equation using the potentials given I,y
the boundary integral method.
_ Vortex (Dipole) Panels.
We assume that the body has a number of
well defined edges from which vortex shedding
will take place. Cases where separation is taking
place from a moderately curved continuous
151
OCR for page 149
surface will not be considered in this analysis.
Then the solution for the above potential flow will
have the local behavi our:
With this assumption the complex potential W
takes the form, at large distances ~ from the edge:
'`:, ~ (2) (6) W >  ( i / 2~( ) £ Id ( (j c
where ~ is the included angle of the edge and s is
the surface distance measured perpendicularly to
the edge. For simplicity we will now consider the
case of a body for which all the edges are right
angles so that ~ = ~/2 Hence the surface speed in
the vicinity of the edge is
q = c I s I 1/3 + O(1) terms (7)
where c can be evaluated from the two adjacent
panels (1 and 2 ) as:
c = 2/3 ( ~2  Of ) i ( Is2 12/3 + 1 s1 12/3 ) t;8)
The viscous flow response to this
singularity in q is for separation to occur
resulting in the shedding of vortex pairs from the
edge each flow cycle.
As in eqn.(1) vortex shedding of this type
has been shown ( 17) to exert a force on the body
in twodimensional flow:
FV = iPd/dt { ~rj((j~(Ij) }
where Fj is the circulation of each shed vortex and
(j is its location in a plane obtained by apply ing
the local SchwartzC:hristoffel transformation,
Z = k `2~/~
(10)
in the plane z = x + i y normal lo the edge and
local to it, figure 2. (Ij is the image of (j in; the
plane surface which is the transformed Goody
surface in the ~ plane.
The complex potential in the ~ plane in the
vicinity of the edge due to this array of shed
vortices is
W = i/2 £rj { log()  log() } (11)
We now make the assumption, as in the prey ous
matching method ( 17), that the amplitude of the
body motion or of the incident waves is
sufficiently small for the vortex shedding to be
considered as a local phenomenon at each edge.
That is the length scale Lv associated with the
vortex shedding ( for example the typical distance
of the centre of a vortex from the edge when it
detaches and a new vortex starts to form ) is much
smaller than the typical length scale of the body
crosssection LB. This is a formal restriction, but
having made it, the method will be used
practically for amplitudes as large as can be
justified by the results.
for LV2/3 << ~ << LB2/3 (12)
This is the potential of a vortex dipole of strength:
pL = _ Ad, Fj ( (j _ (IJ ) (13)
located at the edge ~ = 0.
The shed vortices are therefore modelled by
a dipole panel in the physical plane whose
potential at large distances in the transformed ~
plane including the effect of its image is equal to
that of eqn.(13). It is convenient in the present
panel method to represent the dipole by a
piecewise constant distribution of source dipole
density aligned in the direction normal to the
panel surface, and hence to the vortex dipole
direction, with the panel lying along the external
bisector of the edge angle. This dipole may also be
considered through eqns. (9) and ( 13) as being
proportional to the impulse J. FV dt exerted by
the vortex shedding at the edge. Because of this
the total dipole strength may be obtained directly
from the vortex force computations in the form of
eqn. ( 1). However the length of the vortex panels
remains as an apparently disposable parameter in
the numerical method. Since the intention is to
derive a method which can be applied in the
frequency domain, the dipole density is taken to
be proportional to the component of the impulse
at the fundamental frequency, which as discussed
earlier is the dominant part. The length of the
panel is kept constant through the flow cycle. [he
length can be assigned in a number of ways and it
is not yet clear which is optimal. It could, for
example, be made equal to the characteristic
distance LVof the vortices from the edge during
the formation process . This is not always easy to
specify and in the results described here the
length is prescribed instead by requiring the
KuttaJoukowski separation condition to be
satisfied in the mean sense over the flow cycle at
each edge. Hence we obtain for a right angle edge,
after some algebra, the source dipole density ,u'
length 1Vp given by:
~ = x2 ( b1 cos cot  al sin lot ) / 4Cd 1Vp2/3 (14)
and
1Vp = ~ X ( al2 + bl2 )1/2 / 4~,~ ~3/4 (15)
where X = 3/2 Icl, c being defined in eqn.(8) in
terms of the potential 4? on panels either side of
the edge. al and hi are the Fourier coefficients
of the fundamental component of the vortex force
152
OCR for page 149
FV, defined by:
2
a1 = ~/~ ~ FV COS At dt and 14
o
12
2~
b1 = W/~ FV sin of at (16) 10
o
Having specified each vortex panel in this 8
way in terms of the attached flow potential on the .
adj acent panels to each segment of an assumed 6
shedding edge, the surface potential calculation is
repeated with the vortex effect incorporated. The
pressure field can then be computed from 4 .
Bernoulli's equation over the panels on the
surface of the body only, ie. excluding the vortex
panels. 2
RESULTS.
O
Both methods have been tested for cases of O
barges for which the hulls take the simple form of
boxes with plane rectangular sides and right
angle edges. Experimental data contain ing
sufficient information to compute these cases
however are scarce. Three cases have been
studied, the first and second being cases of barge
response in beam seas and the third a forced roll
case. The first case was a barge with a beam to
draught ratio of 10.0 and a length to beam ratio of
approximately 3.3. Experimental data was obtained
for a model of the barge, at scale 1:30, with
rounded bilge corners with radius of curv ature
equal to 18~o of the draught. The results (18) for
roll in beam seas ate compared with computed
predictions using the first (matching) method in
figure 3. In the second case study (8), a barge
with a beam to draught ratio of 7.6 and a length to
beam ratio of 3.0 was used. Experimental results
for both sharpedged and rounded bilge corners
were obtained. The rounded ones had a radius of
curvature of 38% of the draught. The damping was
significantly reduced when the bilge corners
were rounded. These experimental results and
those computed by the first method for the sh~.rp
edged case are shown in figure 4. In the third
case results of a forced roll experiment ( 19) on a
barge of beam to draught of 21.0 are compared
with the computed results of the second numerical
method. The experiment was on a barge spanning
the test section of a tank whereas the computation
was for a barge of the same beam to draught ratio
and a length to beam of 6.14. The results for the
damping coefficients for different wave
amplitudes are shown in figure 5 and for the
added mass in figure 6
DISCUSSION.
The vortex force for each degree e of freec~c!m
is predicted to have components in phase with the
velocity and with the acceleration of the body.
Therefore, the method accounts for vise ous
contributions to both the damping an ~ added mass
terms, including the effects of crosscoupling.In
;1*\~<
~ . . . · ~
16 20
· ,
4 8 12
Fig. 3. Case study 1; Roll RAG against wave period.
, wave damping only; viscous damping included,  0
, Hw = 3 m, , Hw = 5 m; experiment,  /\  , Hw
= 3 m,x, Hw = 5 m.
16 i
1
. so
_
12 · O
lo:
_
0
8
4
o
Ir
y~
, . .
O 4
~At
8
T (B)
· · · · _
12
Fig. 4. Case study 2; Roll RAO against wave period
in beam seas. , wave damping only; viscous damp
ing included,  ·  , Hw = 0.90 m,  , Hw = 1.74
m; experiment, sharp edged bilges,~, Hw = 0.90
m,x, Hw = 1.74 m. round edged bilges,  0  , Hw
=0.9Om +OH,,, = 1.74 m.
153
OCR for page 149
3~
2
1
O .
0
x
cob ~
/@
lX: e~\
~ ~ \
\
'A \
\
\
a,' . Wy2 ~
0 0~5 1 0 1 5
Fig. 5. Forced roll damping coefficient against fre
quency of motion.    wave damping only; viscous
damping included,  ·  ,8 = 0.05 red.,  +  , ~ =
0.10 red.,x, ~ = 0.20 red.; experiment,(~), a =
0.05 rad.,E, ~ = 0.10 rad.,O, o~ = 0.20 red.
6
4
2
o
°x~
I Eve \L
N
~~
In X~X
`e, e ~ ~
. . . .
O 0 5 1~0
Fig. 6. Forced roll added mass moment of inertia
against frequency of motion.    potential flow only;
viscous effects included,  ·  , ~ = 0.05 red.,  +  ,
= 0.10 red.,x, ~ = 0.20 red.; experiment,(D,
= 0.05 rad.,ffl, ~ = 0.10 rad.,~, ~ = 0.20 red.
the first roll response case the results of the first
method show good agreement between the
predicted and measured roll responses through
the resonance region. In reality the computed
results are slightly on the high side because ~ the
model barge had rounded edges. This s evident in
figure 4, where the model barge for the second
roll response case was fitted alternately with
rounded and sharp ed<~,,es. However, both figures
show a dramatic improvement in the prediction of
roll when viscous effects are included in the
calculation in comparison with the predictions of
potential theory alone. Both sets of results also
show that the nonlinear dependence of the roll
amplitude on the wave height is well predicted.
The fact that the vortex forces are critic ally
dependent on the relative velocity of the fluid in
the immediate vicinity of the shedding edges,
being very sensitive to the position of the roll
gentle and the barge geometry has been
demonstrated before (6). For the first barge,
which has a relatively high centre of gravity and
less rounded bilge corners, the potential flow
calculation greatly overpredicts the roll response.
For the second barge, with its centre of gravity
approximately at the level of the mean free
surface and with well rounded bilges, the
potential flow calculation gives results which are
in quite good agreement with experiment. In this
case the effects of vortex shedding from a rounded
edge are small and a relatively large loll
amplitude is obtained. However when this barge
was fitted with sharpedged bilges, vortex
shedding became important and the response at
resonance was considerably reduced to a value in
reasonable agreement with the theory including
vortex shedding effects and much low' r than that
predicted by potential theory alone. The
assumptions made abe at the location of the roll
centre when using forced roll data are therefore
very important.
Figure 5 shows the comparison of damping
and figure 6. the added mass for forced roll
predicted by the second ( vortex panel ) method
compared with experimental data ( 19). This data is
for a rectangular barge of finite length. 'Jajce
potential theory result without vortex damping is
shown calculated by the present thriee
dimensional potential panel method. the
agreement between the predicted and measured
damping coefficients is very good when the
vortex panels are incorporated in the
computation. The added mass is less well predicted.
Disagreement here is somewhat unexpected
because the vortex force contributes relatively a
considerably smaller part to the overall ad led
mass than it does to the damping. Any discrepancy
/ ~would therefore be expected to be more clearly
w\/ 29 apparent in the damping rather than added mass
coefficients. However Vugts has commented t 19)
15 that in this particular experiment it was difficult
to measure the added mass very accurately. It
should also be emphasised that the computation
was carried out for a threedimensional barge, but
without computing vortex effects from the edges
around the two ends, whereas the experiment was
for a barge which spanned the tryst tank. She
vortex damping is likely to be less effected by this
difference than the added mass.
4
OCR for page 149
RAGES.
6
4
O _
cat
to
x
1~
Axe
WY2
0 05 10 15

Fig. 7. Vortex/dipole panel length against frequency.
·  , ~ = 0.05 red.,  +  , ~ = 0.10 red.,x, ~
0.20 red.
Results for three different forced roll
amplitudes have been plotted and it is apparent
that the vortex panel method does predict the
increase in hydrodynamic damping with
amplitude quite accurately. Figure 7 shows how
the vortex panel k;~gth 1V p v aries with
frequency and amplitude. The fact that the length
changes very little with frequency is not
unexpected since it is only the effect of the free
surface condition which will cause such a change.
The variation of 1V p with amplitude is
approximately proportional to (amplitude)] /4
which would be expected from the variation of the
vortex length scale Lv, (17).
CONCLUSIONS.
It has been shown that the motions of
sharpedged rectangular body floating freely in
waves are well predicted by the first ( matching )
method. The results suggest that the n,on
linearities in response are largely due to vortex
shedding from the body. The second ( vortex panel
) method has been used to predict damping of a
barge due to forced roll and the results are
similarly in good agreement with the measured
data. The second method is the easier to set up and
apply to general shapes of floating body and is
therefore being developed into a full motion
program.
ACKNOWLEDGh/ENTS.
The authors gratefully acknowledge the
financial support of the SERC through the Marine
Technology Directorate Ltd.
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Press. 1977.
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'Ship motions and sea loads.' Trans .C',NAME. !78.
1970, p421.
Tanaka, N. 'A study on bilge keels. (Part 4. On
the eddy making resistance to the rolling of a
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Ikeda, Y., Himeno, Y. and Tanaka, N. 'A
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Kaplan P., Jiang C.W. and Bentson J.
'Hydrodynamic analysis of barge platform
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'Effect of vortex shedding on the coupled rol.
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a 14.
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OCR for page 149
DISCUSSION
Choung M. Lee
Pohang Institute of Science and Technology, Korea
It is a very interesting work to simulate the bilge vortices by doublet
distribution. It may work very well for the zerospeed case;
however, when the body as a forward speed, the bilge keel acting as
a lowaspect ratio wing might have a quite different types of vortex
sheddings. How would you approach such a problem?
AUTHORS' REPLY
We agree that the vortex shedding is different in the case of a body
undergoing the combined motion of roll and forward speed, since the
vorticity is now convected along the hull surface as well as away
from it. Work is in progress to extend the model to this case and we
hope to be able to report further on it at a future date.
DISCUSSION
Targut Sarpkaya
Naval Postgraduate School, USA
It will be appreciated if the authors would comment on the
similarities and the fundamental differences in the boundary
conditions of the physical experiments and the numerical simulations?
What is the actual motion of the barge model in the experiments.
Thank you.
AUTHORS' REPI,Y
The calculations were carried out for two basic flow configurations.
The first was a barge freely floating in beam waves, and the second
was a barge undergoing forced roll in otherwise quiescent water. In
both cases the potential flow part of the calculation was for a three
dimensional barge. The matched edge technique has been applied to
both cases, the forced roll calculations being reported in a previous
paper. The viscous edge panel technique has so far been applied to
the case of forced roll only. The results of the calculations were
compared with experimental results, in the first case, obtained using
a model barge floating in waves, and in the second case, with two
dimensional results obtained with a rectangular section spanning a
channel and undergoing forced roll.
There are of course some important differences between the computed
and experimental flows, particularly in the case of forced roll. The
computations have assumed perfectly two dimensional flow. In the
freely floating case there will be some end effects which are not
accounted for by the theory. Similarly, in the case of forced roll, the
vortex shedding in the experiment will be affected by the presence of
the end walls and will not be entirely two dimensional. Also, in both
cases, the vortices approaching the free surface may be subject to
strong three dimensional instabilities. Nevertheless we believe that
the results are reasonably representative of the major effects of vortex
shedding in the flow regimes considered and demonstrate a
considerable improvement on those obtained using potential flow
theory alone.
156