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OCR for page 191
Numerical Simulation of Ship Waves and
Some Discussions on Bow Wave Breaking & Viscous
Interactions of Stern Wave
K.H. Mori, S.H. Kwag, Y. Doi (Hiroshima University, Japan)
Abstract
Numerical calculations are carried
out to simulate the freesurface flows
around the Wigley model and S103 Inuid
model. The NS equation is solved by a
finite difference method where the
bodyfitted coordinate system, the wall
function and the triplegrid system are
invoked. The numerical scheme being
examined for the scheme to be accept
able for discussions, the calculations
are extended to the turbulent high
Reynolds number flows with the aid of
the Oequation model to discuss the
Reynolds number dependency of the
waves. The wave elevation at the
Reynolds number of 104 is much less
than that at 10 6 although the Froude
number is the same. The numerical
results are referred to predict the ap
pearance of the subbreaking waves
around bow and stern. The prediction
is qualitatively supported by the ex
perimental observation. They are also
applied to study precisely on the stern
flow of S103 as to which extensive
experimental data are available. A1
though it is not yet made clear about
the interaction between the separation
and the stern wave generation, the ef
fects of the bow wave on the develop
ments of the boundary layer flows are
concluded to be significant. The sub
breaking of the stern wave is also
discussed.
1. Introduction
_
Freesurface flow around ship is
one of the most complicated flows where
various nonlinear phenomena exist such
as wave breaking, viscous interactions,
freesurface tension and so on. Not
only for the practical hull form design
but also for academic interests, it is
important to make clear their flow
mechanism. They are worthy to be
studied more intensively.
There are pretty many experimen
tal studies even about the wave break
191
ing such as Duncan[1], Mori[2],
Maruo[3], Grosenbaugh[4] and so on.
In spite of their extensive experi
ments, however, the freesurface non
linear phenomena still remain unclear.
Theoretical investigations are also at
tempted to explain the phenomena or to
provide a suitable model. Dagan[5],
Tanaka[6] and Mori[7] applied the in
stability analysis to predict the
breaking. Some models for breaking
waves are proposed after experiments.
There are few studies on the free
surface tension; Maruo[3].
The stern flows with the free
surface show also important phenomena
in ship hydrodynamics. Although Doi[8]
and Stern[9] studied extensively about
them, very little are made clear.
Despite the viscous interactions are
essential there, theoretical approaches
are so limited and most of the ap
proaches are based on the simple flow
models.
On the other hand, there are some
researches by the direct numerical
simulations such as Miyata[10],
Grosenbaugh[4], Shin[11] and so on.
They have tried to make clear the
mechanism of the NavierStokes equa
tions directly. Because the free
surface flows of our interests are
strongly nonlinear and viscous effects
are primary, the simulations by solving
the NavierStokes equation can be a
desirable tool for the study. They can
provide any necessary data for the
study once a calculation is carried
out.
However, the important point is on
whether their codes are accurate enough
for such studies. It may be possible
to draw misleading conclusions from the
results calculated by an unproved
code. The use of an insufficient grid
scheme is likely to bring forth misun
derstanding for the phenomena. The as
sumptions of 2dimensionality or
laminar low Reynolds number flows are
also possible sources for misun
derstanding.
The present paper is a study along
OCR for page 191
the approach lastly mentioned; the wave
breaking and the viscous interaction of
the stern waves are studied by making
use of the results of numerical simula
tions. The numerical scheme for the
simulation i s based on the MAC method
where the bodyfitted coordinates and
the nonstaggered grid system are
used. The convection terms are
presented by the thirdorder upstream and
differencings. The wall function is
invoked to follow a steep veloci ty
changes close to the hull in high
Reynolds number flows. The turbulent
stress terms are presented by a O
equation model.
Bearing in mind the above
ment i oned danger s, the computing code
is validated first by carrying out
var ions computations to be convinced
with, although it may not be enough due
to the limitation of the computer. The
f r e e  s u r f ac e f l ows around the Hi gl ey
model at the Reynolds numbers of 104
and 1 o6 are used for this purpose .
Although the theory is 2
d imen s i anal, the r e sul t s are us ed to
predict the appearance of the sub
breaking wave s . The vi s cons inte rac
tion of the stern waves is also dis
cussed .
Hn = WAt + ( Rle + Vt ) V2w
 ( un ax + vn aw + wn aw )
 ax{vt(aU +~Wx)} ay{Vt(az + ay)}
 az{vt(2 aaZW)}
an = p+ z
Fn2
53)
All the variables are on the car
tesian coordinates system(x,y, z) where
x is in the uniform flow direction, y
in the lateral, and z in the vertical
direction respectively; u, v and w are
the velocity components in the x, y,
and zdirections, respectively. They
are normalized by the model overall
length L and the uniform velocity Uo.
Subscripts denote the differentia
t ions wi th respect to the referred
variables and superscripts the values
at the referred timestep. The term At
stands for the time increment, p the
pressure and At the eddy viscosity. Rn
and En are Reynolds and Froude numbers
respectively based on L and Uo, and
2. Numerical Simulation of Shim Waves v2 a + a + a (4)
2.1 Basic Equation
Numerical simulations of 3D free
surface flows are carried out by solv
ing the NS equation basically follow
ing to the MAC method. The velocity
components u, v and w at (n+1 ) time
step are determined by
un+1 = ( En An ) At
vn 1 = ( On ~ ) At ( 1 )
wn+1 = ( Hn An ) At
where
At + ( Re + at ) V2u
 ( us ax + an ay + w" aaz )
 aX{vt(2 ax) } t{Vt(a.y + ax) }
az~Vt(az + axw)}
n _
G
At + ( Re + at ) V2v
un aaX + V" aV + wn av )
ax{vt(aay + a3XV)} aay{Vt(2~} (2)
 a {vt(av + aw)}
192
Differentiating ( 1 ) with respect to x, y
and z, we can have
V ~ = F + G + H
 ( u n+1 + v n+1 + w n+1 ) / AL (5)
x y z
The last term in ( 5) is expected
to be zero to sat i sty the cant inui ty
condition. ( 5) can be solved by the
relaxation method. The new freesurface
at the (n+1 ) th timestep is calculated
by moving the marker particles by
n+1 n n
x = x + u At
n+1 n n
y = y + v lit
n+1 n n
z = z + w At
(6)
The oncoming flow is accelerated
from zero to the given constant
velocity. Thirdorder upstream dif
ferencing is used for the convection
terms wi th the fourthorder truncation
error, and for the central differenc
ings, 4 or 5point central differenc
ings are used.
It is desirable to introduce
numerical coordinate transformations
which simplifies the computational
domain in the transformed domain and
facilitates applications of the bound
OCR for page 191
ary conditions. In the present study, a
numericallygenerated, bodyfitted
coordinate system is used,
~ = ((x,y,Z), n = ntx'Y,
and ~ = (tx,y,z) (7)
It offers the advantages of
generality and flexibility and, most
importantly, transforms the computa
tional domain into a simple rectangular
region with equal grid spacing.
Through transformations, (1) can
be written for the velocity component q
as
qt+Uq;+Vqn+Wq; (8)
=(Rl +vt)v2g KREYSF(:,n,`)
where,U,V and W are the contravariant
velocities and K is the pressure
gradient. Their full expressions can be
found in [12]. REYSF((, n, ~ ) repre
sents the terms transformed from the
last three terms on RHS of (2).
2.2 TriPlegrid Method
It is common to use a single grid
system for the whole computation whose
minimum size is determined for the
numerical diffusion to be less than
that by viscosity. However, the grid
size for the calculation of the free
surface elevation must be determined by
a different scale, the minimum wave
length[131. In our simulation, three
mesh systems are used whose sizes are
different from each other depending on
the characteristic of equations. We
call it triplegrid method here. The
first one is for the convective terms
in the NS equation, the second is for
the Poisson equation, and the third is
for the freesurface equation. The
third grid system requires the finest
one; a quarter of the first one in each
direction. Because it is used only on
the freesurface which is two
dimensional, the increment of the
memory is modest. On the other hand,
the second one can be coarser than the
first one; here half of the first one
is used. According to the results, the
development of the freesurface eleva
tion is strikingly improved. The CPU
time and the memory size of the present
computation are rather reduced owing to
the use of coarse meshes of the second
mesh system for the Poisson equation
and the diffusion term.
2.3 Body Surface Condition
In the numerical solution for vis
cous flows, the noslip condition for
the solid surface is used by discretiz
ing the region fine enough to the im
posed condition. This method, however,
requires a large number of grid points
to resolve the large gradients in the
nearwall region especially for the
high Reynolds number flows. This is
the main barrier in the high Reynolds
number calculations. In view of the
complexity involved in resolving the
nearwall flow, it is preferable to
employ a simpler wallfunction approach
for the velocity profile which can be
valid for the velocity profile in the
near wall region.
In the present study, the two
point wallfunction approach is
employed as by Chen and Patel[14]. The
numerical solution is that the velocity
at n=3(~=1 is on the wall) is provided
and the wallfriction velocity us is
updated with some iterations by requir
ing this velocity to satisfy the law
ofwall equation. The iterative two
point wallfunction approach for a
threedimensional flow means that it
provides the updated boundary condi
tions for the numerical solution and
the procedure is iterated until the
solution converges.
Fig.1 Perspective view of grid
scheme
2.4 Computational Results and Discus
sions
Computations are performed for the
flow fields around the Wigley model
with freesurface at Rn=1o4 and 108 .
Fig.1 shows the perspective view of the
grid scheme used. For the numerical
stability and efficiency, the grid
scheme near the hull is required to be
orthogonal to it and the grid size
should change smoothly. The grid number
is 74x29xl9. During the computations,
the location of grids between the free
surface and the bottom is re
distributed proportionally to the free
surface elevation. By this scheme, it
is expected that the freesurface con
193
OCR for page 191
dition, i.e., the constant pressure
condition can be directly applied
without any interpolation. For the high
Reynolds number flow, the twolayer al
gebraic BaldwinLomax model is used to
make the eddy viscosity. The numerical
results are compared with the ex
perimental data [15].
Drag
coeff.
(x103)
7.O
5.0
3.0
1.0 ~
'4igley Model
Rn = 106
En = 0.316
Of
Schoenherr's ~]
friction line /
at Rn=10**6 /
~ _ = = _ , _,
~~C Measured Cw
(assuming K=0.07) /
, . . .
1.0 1.5 2.0 2.5 3.0
Fig.2 Time history of drag
coefficients
0.5ol
rr
0.50
n 50 .
194
In order to check the convergence
of the computations, the wave patterns
and drag coefficients of Cp, Of and Cw
are compared along the marching time
step as shown in Figs.2 and 3, where
Cp, of and Cw are the pressure, fric
tional and wavemaking resistance coef
ficients respectively. Schoenherr fric
tion line and the measured wavemaking
resistance coefficient are shown for
comparison. Although the wave seems
still developing further, it can be as
sumed to be converged at T=3.0 where T
is the nondimensional time, which can
be supported by the results shown in
Fig.3. The calculated frictional resis
tance, which is directly derived by the
difference of the velocities at the
two points, is still larger than the
Schoenherr.
Fig.4 shows the logarithmic plot
of velocity at x/L=0.835 in the format
of the lawofwall (q+ versus y+) using
the friction velocity. Some plots are
drawn at several points in the girth
and depthwise directions; they are
generally in good agreement.
Fig.5 shows the comparison of the
velocity distributions near the wall
between that obtained by making use of
AD
,~ ~ //O.2,,J,,~ _
AP
Fig.3
Time history of wave
patterns for Wigley
at Rn=106 and Fn=0.316
OCR for page 191
20.0
.
u
1o.a
3 ~ ~ ~ 3 8 ~ 8 ~ g
~ : z/D~O. 09
O : z/D=O. 52
O : z/O=O. 15
, . .. t. _
1 10 1o2 103 104
y
Fig.4 Log plot of velocity for
Wigley model(x/L=0.835)
the lawofwall and directly. Because
the expression by the wallfunction is
not valid any more for the separated
flows, it is not applied in the stern
5% where the separation is suspected.
There~the scheme is switched into the
direct method. We can see that the
0.4
2y/B
0.2
0.4
2y/B
0.2
~3 ~atW~ orI ~1
FP AP
AP
Fig.6 Velocity vectors on free
surface
(a) Laminar flow at Rn=104
(b) Turbulent flow at Rn=106
0 _ 0
Go Q O
Wigley0 I O O O
Rn=lo6~ . ~_
Fn=0.316 ~x x x
. .
~Go cn
LO ~
cut
0 0 0
_ _ _
i
~ x x x/
it_
0 _ O
O Q Go
~ ·
o ~ o
ll ~ 11
J ·~ _
, CX ,
J
O . l
~Cal ~ CO
~cn 0 _` 0 0 0
o o o ~ ~ ~
X X _ XJ ~
Fig.5 uvelocity distribution on freesurface
(a) without wall function
(b) with wall function
tallfunction approach removes much of
She dependency of the numerical solu
~ion on the location of the two mesh
points and the steep velocity changes
are well followed even by limited size
of grid. The usual logarithmic law of
the wall can be reasonably used in
favorable pressure gradients.
Fig.6 shows the comparison of
velocity vectors on freesurface be
tween the two Reynolds numbers: the one
(b)
(a)
(b)
iS 104 which is laminar flow while the
other 1oB , turbulent. The laminar flow
is subject to separation in the stern
region and wider boundary layer thick
ness, while the turbulent flow is to
larger velocity gradient near the hull
by which we can guess a larger wall
friction on the body surface. Fig.7
shows the comparison of the wave pat
terns at Rn= 104 and 106. We can clearly
see that the Reynolds number dependency
195
OCR for page 191
0.50
(a)
0.50~
/°~/~°"/"~'' ,~
_
AP
cat
~ °
0 i.
~ 0
/ ~0
~0
~O
q
FP \ 0
~o°/
Wi g1 ey Model
Rn=1 0**6
Fn=0.316
i6( \~;AP
 : Cal cul ated
O: measured
Fig.8 Wave profiles on hull surface
0.2
a
SWL
Fig.7
Wave patterns of
Wigley model at Fn=0.316
(a) Laminar flow at Rn=104
(Mesh size: lOOx25x15)
(b) Turbulent flow at Rn=10 6
(Mesh size:70x25x15)
o.o
1
l
AP
Fig.9 Pressure contour on Wigley
hull surface at Fn=0.316 
(a) Laminar flow at Rn=104
(b) Turbulent flow at Rn=106
of the wave. It may confuse us that
even the second wave crest differs much
in height, for we usually expect the
Reynolds number effect on wave is not
so significant there.
Fig.8 shows the wave profile along
the hull surface. The Reynolds number
of the measurement is 3.59X106. It is
well presented around the bow, but
slight discrepancies are still observed
in the aft half of the hull. Fig.9
shows the pressure contours on the hull
surface at Rn=104 and 106. The pressure
around the stern region is much
recovered at Ace than the low Reynolds
number flow. The pressure distribu
tion on the hull surface shows some
wiggles in appearance at the bow and
stern parts due to the use of still
coarse mesh. However, the wave height
on freesurface, which means the pres
sure here, shows no serious wrinkles
because finer grid is used there.
Fig.10 shows the comparison of
wave patterns between the calculated
and the measured. Although they can not
be compared exactly due to difference
of Reynolds number, we can say that
the computed patterns are qualitatively
reasonable for both the bow and stern
waves.
Fig.11 shows the velocity vectors
on the transverse section near A.P. in
which the vertical motion can be ob
served around the keel. It seems not
so easy to calculate the cross flows
accurately at the stern part consider
ing some aspects in the numerical point
of views. First, the assumption of
the symmetry or the steady flows should
be pointed out. The unsteadiness and
nonsymmetry observed in experiment
should be taken into account in numeri
cal simulation. Of course the grid
196
OCR for page 191
Fig.10 Comparison of wave patterns at Fn=0.289
(a) Computed contour(Rn=lxlO 6 )
(b) Measured contour(Rn=3x106)[15]
_. ~
 `` ~ _
_~` ~ i<
i...~N
,~,,`~N
,. `~N ~ ~\
\~\~4
And
As
XiittitY ~ ~
t~ttti ~ ~ t ~ '
T=3.O
Fig.11 Velocity vectors of Wigley
model(Rn=106, Fn=0.316 and
x/L=1.02)
Ctx1 03
 4.0
: measured [ 15 ]
X : computed ( Rn=l o6 )
 2.0 : measured(corrected at Rn=lO6)
0.25 0.28 0.31 F
n
Fig.12 Comparison of total drag
coefficients between the
computed and measured.
197
(a)
used in the computing domain is still
coarse and can be a reason of not being
able to capture completely the details
of the fluid motions.
In Fig.12, comparison is made for
the total drag coefficients between the
calculated at Rn=1o6 and the measured.
The Reynolds numbers of measurement are
2.84x106, 3.28x106 and 3.59X106 for the
corresponding Froude number of 0.25,
0.289 and 0.316, respectively. For the
more direct comparisons, the measured
results are corrected at the same
Reynolds number of 1o6 by use of the
PrandtlSchlichting's friction formula.
The computed drag is still greater than
the experimental data; We can not men
tion the reason for the difference con
clusively, but the accuracy of the
velocity calculation close to the hull
may not be enough which resulted in
poor agreement in the frictional resis
tance. The computing time is abt. 90
hrs for one case by Apollo DN10000
(abt.13 MFLOPS).
3. Detection of Subbreaking Waves
3.1 Appearing Condition of Subbreaking
Waves
Computed results are applied to
detect the appearance of the appearing
condition of subbreaking waves around
bow. The critical conditions for their
appearance were studied in Mori[7].
There the Creakings at their infant
stage are concluded as a freesurface
turbulent flow. The flow mechanism is
OCR for page 191
supposed that the surplus energy ac
cumulated around the wave crest by the
increment of the freesurface elevation
is dissipated through the turbulence
production and freesurface could even
tually maintain itself without any
overturnings or backward flows. An
instability analysis fog 2dimensional
flows provides a critical condition for
their appearance;
M aM_ auS_ 1 anz , O (9)
UshOs has nz has
where M is the circumferential force
given by
k} = ( ~ Us  nz g ) nz, (10)
s is the stream line coordinate along
us
M/Us
a)
cat
to
.
o
1
the freesurface and h is its metric
coefficient, while n is the normal;nz
the direction cosine of n to z. Us is
the velocity component of basic flow in
the sdirection; ~ is the curvature of
the freesurface and g the gravity ac
celeration. Limiting ourselves to a
narrow proximity to the wave crest, we
assume nz=1 and a /h~s.a/3x ; then
(9) can be reduced approximately into
uS a M
__ > 0
where,
to
o
l
M/U ~
2
M = lC Us ~ g
us 0
o 1 Fn=0.316 . . °

~. I
X 0 . 4
(FP) (I. s. 9)
0
0
0
l
O Fn=0,25 OO
to
·.
M/U S
· 
LO
O
.
to
us
cat
to
· ,
to
~,
0. 5
(FP)
0.4
( s . s . 9)
u,
cat
0,
.
to
us
to
lo
1
it
M/Us
M/U s
l
0.3
x (S.S.8)
o
up
~` E'n=0.20
1 ~
~hi_
0 . 5  0 . 4  0 . 3  0 . 2
(FP) X (s.s.7)
Fig.13 Variation of M/Us and freesurface elevation
and lines analyses for bow wave breaking.
198
(11)
(12)
M/U s
lo
rut
lo
o
to
l
OCR for page 191
Because M is always negative, the
negative gradient of M/Us to x suggests
the possibilities for the freesurface
flow to be unstable.
3.2 Numerical Application for Bow Waves
The appearing condition is numeri
cally simulated to predict the ship
wave subbreakings by (11). Although
the flow for the Wigley model is not 2
dimensional, 3dimensionality may not
be so strong that we can expect it is
applicable without serious errors.
Fig.13 shows the variations of M/US vs
x around the first bow wave crest. The
analyses are made at three speeds of
Fn=0.20, 0.25 and 0.316 along the
curved lines indicated there; ~ is the
freesurface elevation.
At Fn=0.20, no steep negative
gradient is seen, but the gradients at
Fn=0.25 and 0.316 are significantly
negative behind the wave crest. It may
be suggested that the freesurface
flows at Fn=0.25 and 0.316 are likely
to be unstable behind the wave crest
while that at Fn=0.20 is stable. Fig.14
shows the photographs of the free
surface flows taken at three cor
responding Froude numbers. There can be
seen wrinklelike waves behind the
diverging waves at Fn=0.25 and 0.316;
those at Fn=0.316 are much more inten
sive than those at Fn=0.25. On the con
trary, no such waves can be observed at
Fn=0.20. This observation supports the
instability analysis for the bow wave
breaking.
4. Discussion on Stern Waves
The stern wave of S103 is studied
to make clear the flow mechanism espe
cially on the viscous interaction by
referring the computed results. S103
is an Inuid model with the beam/
length ratio of 0.09 and extensive ex
periments have been carried out by
Doi[8]. All the experimental data are
referred from there.
4.1 Computation for S103 Model
Fig.15 is the computed wave con
tour at Fn=0.30 and Rn=106. The result
is that at the time T=4.0, when the
convergence is well assured. The grid
size is 74x29x30; computations are
carried out on another finer grid
scheme to find no significant dif
ference in the resistance. The comput
ing domain is 1.4~x/1~2.0 and
0.0
l
JO
 '\\` on
0.05
0.03
 compu ted
 measured
0.01
,,'/
 1 '
,r~
.  O .f Hi_ ,,
/y
 O . () 3
Fig.16 Comparison of wave profiles of
S103 model between the
calculated and measured on
hull surface at Fn=0.30
profile shows a good agreement with the
measured to conclude that the present
numerical scheme works well and the
results may endure for our purpose to
discuss on the flow mechanism.
4.2 Review of Experiments
Now let's refer the wave contours
of S103 from [8], shown in Fig.17. We
can notice significant difference in
the stern wave patterns although the
Froude number changes modestly from
0.26 to 0.30; at Fn=0.27 no significant
stern wave is observed compared with
those at Fn=0.26 or 0.28. On the other
hand, a wide "wake" zone is observed
behind the hull at Fn=0.30. A careful
observation of the "wake", as shown in
'I ~ / / l
\ //// //1/~
Fig.15
Wave pattern of S103
at Fn=0.30 and Rn=106
(Contour interval is
0.02x2g;/Uo and
dotted lines show
negative values)
Fig.17 Wave patterns of S103 at four
different Froude numbers
Fig.18, tells us that the freesurface
fluctuates intensively there. The free
surface of Fn=0.27 is completely dif
ferent where such freesurface fluctua
tion is not observed. This fluctuation
of the freesurface is subbreaking.
It is reported in Doi[8] that the
starting points of the stern waves
could be easily and definitely deter
mined
along the hull at Froude numbers other
than 0.27. This is because the wave
profile at Fn=0.27 is a little dif
ferent from that at other speeds.
from the observed wave profiles
4.3 Discussion on Viscous Interaction
Fig.19 shows the computed stern
wave patterns at the three Froude num
200
OCR for page 191
~
 .
Fig.18 Stern wave pictures of S103
at Fn=0.27 and 0.30
Fig.19
Stern wave patterns
of S103 at Rn=lo6
(a) Fn=0.27
(b) Fn=0.28
(c) Fn=0'.30
(Contour interval is
0.02x2g;/Uo and
dotted lines show
negative values)
hers of 0.27, 0.28 and 0.30. Comparing
the first stern wave crests, we can see
that the result of Fn=0.27 looks dif
ferent from the others; not so sharply
developed. It differs from that of
Fn=0.28 although the difference in the
speed is not so large. The stern wave
crest of Fn=0.27 is not clear. This
may agree qualitatively with the ob
served. On the other hand, the crest of
Fn=0.30 is rather sharp and large.
The modest elevation of the stern
wave at Fn=0.27 may be much related to
the development of the boundary layer
and separation. Fig.20 shows the
velocity profiles in the boundary layer
around the stern and close to the
freesurface. The separation of Fn=0.27
takes place at more upstream position
than that of Fn=0.30. This situation
can be seen more clearly in the limit
ing streamlines shown in Fig.21. The
separated region of Fn=0.27 is sig
nificantly wider than that of 0.30. The
experiments by twin tufts show
similar tendency; the separated region
of Fn=0.30 close to the freesurface is
due to the freesurface subbreaking
which has been shown in Fig.18. It is
quite natural that a dull pressure
recovery by separation may bring forth
" '' '~'/~//,rK',','>~
AP
201
I(a)
(b)
OCR for page 191
modest wave elevation. On the other
hand, at Fn=0.30, separation region is
so limited that a steep pressure
gradient may generate strong waves.
Here we should remind that all the
computations are carried out at the
same Reynolds number of 106. This means
that the flow fields are exactly the
same in the sense of the viscous ef
fects. Then why such a difference in
separation? The bow wave may be
responsible; the phase of the bow wave
can be a key factor for the separa
tion. The wave contour lines change
peculiarly in the boundary layer and
wake at all the speeds. A careful ob
servation of the freesurface, shown in
Fig.18, suggests us complicated flows
in the boundary layer, which may cor
o.R
2y/B
2y/B
n 4
n R]
0.4 ~
Fig.20 udistribution for S103
on freesurface
(a) Fn=0.27 (b) Fn=0.30
~t t
t t
t t
Z/l
 O _
_ _
_ 0.02
_ 0.04 _
<20mr~
`r
,: BEAD
\~\
TWIN TUFT
respond to the computed peculiar con
tour curves.
Fig.22 shows the velocity vectors
at the two yz sections. Compared be
tween the two Froude numbers, it is ob
viously seen that the viscous region of
Fn=0.27 is much wider than that of
Fn=0.30. A wider wake region made the
stern wave elevation modest. The
peculiar changes of the wave contour,
pointed out in Fig.19, is assumed to
appear in the viscous region.
4.4 Detection of SubBreaking
Fig.23 shows the comparison of
wave profiles and velocity vectors of
Fn=0.30 between the measured and the
calculated. The measured freesurface
fluctuates intensively around the crest
(indicated by I there). This fluctua
tion corresponds to the subbreaking
seen in Fig.18. Because no special at
tention to the subbreaking is paid
in the calculation, the calculated
freesurface is steady, of course. It
should be pointed out that the measured
wave crest is in upstream compared with
the calculated. As seen Fig.18, the
crest angle of Fn=0.30 is much larger
than the calculated. If we remind the
good agreement in the wave profile
along the hull surface, shown in
Fig.16, we can say that the appearance
of the freesurface fluctuation, i.e.,
subbreaking makes the crest shift for
ward which is commonly observed in ex
periments.
The detection of the appearance of
subbreaking waves is made for the
stern waves at Fn=0.27 and 0.30. The
values of M/US are calculated along
FREE SURFACE
~ , _ I,
; ; ; ~_ _ _ _ _ 
_ _ _
1  
, ., . ~ 1 _ _ _
x/lo.g x/13.0 x/l=o.g x/ll.o
FREE SURFACE
FREE SURFACE
_ _
Xtl=O.9 X/l=1.0 X/l=O.9
A.P
Fn=0.27
202
T..
Xll=1.0
Fig.21
Calculated(above)
and observed(below)
limiting streamlines
at Fn=0.27 and 0.30
OCR for page 191
y
x/1=0.95 ~0.15 x/1=1.05
x/1=0.95
(b)
y/l=O.O9 which are shown
Similarly to Fig.13, the negative
gradient of M/US to x suggests the oc
currence of unstable freesurface
flows. A steep negative gradient is
seen in Fn=0.30 but not so much in
Fn=0.27. This means that the wave
crest of Fn=0.30 can be subject to sub
breaking but not at Fn=0.27.
it\ \ \ ~
~ i~, t`~W \ ~ ~
>\4 ~ ~
We can point out that the bow wave
affects much on the separation and
eventually the stern wave generation
appreciably. The appearance of sub
breaking waves makes the flow field
completely different and it may be
necessary to include them in the com
putation.
Me asured
all
o.o4
0.02
0.02
Intensive Fluctuation ~
\ 1 _ o_
JO
0.04,
1.O 1.1 1.2
Z/1
0.04
0.00
0.04
Calculated Free Surf ace
._
Fig.23 Wave profiles and velocity
vectors on xz plane
at y/l=0.15 for S103
\ ~z
~\
Lr
red
O _
.
o
_ z
Fig.22
Velocity vectors
in yz planes
of S103
(a) Fn=0.27
(b) Fn=0.30
1 ·  . .
/
/
Fn=0.30
. · 
/
M/Us
M/U s
\
(A
0_
O I I r I
1~2 1~4 1~6 x/l
o
O
up
o
up
r.
o
O
o
o
o
_ ~
o
1
M/Us
M/U s
Fig.24 Variation of M/Us and free
surface elevation and lines
analyses for stern wave
breaking of S103
203
OCR for page 191
5. Concluding Remarks
A computational code is developed
to simulate the high Reynolds number
freesurface flows around ship by solv
ing the full NavierStokes equation.
After validating the scheme, it is ap
plied for the studies on the Reynolds
number effects, the detection of wave
breaking and viscous interaction of the
stern wave. Findings through the
present study are summarized as fol
lows.
(1) A numerical scheme is developed to
simulate 3D freesurface flows at
high Reynolds number by which a steady
state solution can be obtained with
monotonic convergency. The triple grid
method is applied to get welldeveloped
freesurface waves within a moderate
computer's memory. The wall function is
used to overcome still larger minimum
grid spacing, which is confirmed to
work well.
(2) The Reynolds number effects on
waves are significant especially on the
stern wave and pressure distribution in
the aft part of the hull.
(3) The calculated resistance is
greater than the measured. The estima
tion of the frictional resistance is
still a source of overestimation. The
use of much finer grid is a possible
improvement.
(4) The criterion for the appearance
of subbreaking waves works well, and
the present numerical scheme can be ap
plied to detect the occurrence of
breaking waves of ships. The results
are supported by the observation.
(5) The stern wave is much affected
by the separation of the boundary layer
flow which may be under the influence
of the bow wave.
(6) The occurrence of subbreaking
changes the flow fields drastically,
especially for the stern waves. It is
important to introduce a model into the
numerical calculation which is capable
for the breaking.
All the calculations are carried out by
Apollo DN10000 Computer at N.A.& O.K.
Dep't of Hiroshima University.
Acknowledgments
The second author Mr.Kwag, who is now
on leave from HMRI(Hyundai Maritime Re
search Institute, Korea), expresses ap
preciation to HMRI to allow him to make
research at Hiroshima University.
References
[1] Duncan,J.H.: "The Breaking and Non
breaking Wave Resistance of a
Twodimensional Hydrofoil", Jour. Fluid
Mech.,Vol. 126,1983
[2] Mori,K.,Doi,Y.:"Flow Characteris
tics of 2Dimensional SubBreaking
Waves, Turbulence Measurements and Flow
Modeling", Hemisphere Pub. Co.,1985
[3] Maruo,H.,Ikehata,M.:"Some Discus
sions on the Free Surface Flow around
the Bow", Proc. of 16th sump. on Naval
Hydro.,1986
[4] Grosenbaugh,M.A.,Yeung,R.W.:"Non
linear Bow Flows  An Experimental and
Theoretical Investigation",_Proc. of
17th Symp. on Naval Hydro.,1988
[5] Dagan,G.,Tulin,M.P. : "Two Dimen
sional FreeSurface Gravity Flow past
Blunt Bodies", Jour. Fluid Mech.,
Vol.51,Part3, 1972
[6] Tanaka,M.,Dold,J.W., Peregrine,D.
H. : "Instability and Breaking of a
Solitary Wave", Jour. Fluid Mech.,
Vol.185,1987
[7] Mori,K., Shin,M.S. :"SubBreaking
Wave:Its Characteristics, Appearing
Condition and Numerical Simulation",
Proc. of 17th Symp. on Naval Hydro.,
1989
[8] Doi,Y., Ka~itani,H., Miyata,H.,
Kuzumi,S.:"Characteristics of Stern
Waves generated by Ships of Simple Hull
Form (1st Report)", Jour. of Soc. of
Naval Arch. of Japan, Vol. 150,1981
[9] Stern, F. :"Influence of Waves on
the Boundary Layer of a Surface
Piercing Body",Proc. of 4th Int'l Conf.
on Numerical Shin Hydro.,1985
tlO] Miyata,H.,Ka;itani,H.,Shirai,M.,
Sato,T.,Kuzumi,S.,Kanai,M.:"Numerical
and Experimental Analysis of Nonlinear
Bow and Stern Waves of a Two
Dimensional Body(4th Report)", Jour. of
Soc. of Naval Arch. of Japan, Vol.
157,1985
Ill] Shin,M., Mori,K.:"On Turbulent
Characteristics and Numerical Simula
tion of 2Dimensional SubBreaking
Waves", Jour. of Soc. of Naval Arch.,
Vol.165,1989
[12] Kwag,S.H.,Mori,K.,Shin,M. :"Num
erical Computation of 3D Free Surface
Flows by NS Solver and Detection of
Subbreaking", Jour. of Soc. of Naval
Arch. of Japan, Vol.166, 1989
[13] Xu,Q.,Mori,K.,Shin,M. : "Double
204
OCR for page 191
Mesh Method for Efficient Finite
Difference Calculations", Jour. of the
Soc. of Naval Arch. of Japan
Vol .166, 1989
[143 Chen,H.C., Patel,V.C. : "Cal
culation of TrailingEdge, Stern and
Wake Flows by a TimeMarching Solution
of the Part iallyParabolic Equations",
IIHR Report No. 285 1985
, ,
[15] IHI, SRI,UT,YNU : "Cooperative Ex
periments on Wigley Parabolic Models in
Japan ", p repare d f or the ITTC Re s i s 
tance Committee, 2nd edition, Dec., 1983
DISCUSSION
Fred Stern
The University of Iowa, USA
The authors' treatment of the freesurface boundary conditions is
unclear. It appears that a MAC type method is implemented using a
fixed grid that does not conform to the free surface since the terms
necessary for a moving grid have not been included in the governing
equations. A similar approach was presented by Hinc at the 5th
International Conference on Numerical Ship Hydrodynamics. Please
explain the differences between the present approach and that of
Hino. The grids used for each of the triple grids should be clearly
stated. It appears from the results that the grid for the convective
terms in the NS equations is much too coarse to accurately resolve the
viscous flow, e.g., Fig. 4 indicates very few points within the
boundary layer and virtually no resolution of the logarithmic and
outer wakelike region of the boundary layer. The reliability of the
discussions of the results is uncertain due to these issues.
AllTHORS' REPLY
1. The freesurface boundary conditions consist of the pressure and
kinematic ones. Here, the pressure condition can be directly applied
because the uppermost moving grid at each time step corresponds
always to the freesurface. The kinematic condition, the fluid
particles on the freesurface keep staying on it, is used to determine
the freesurface elevation at each time step in which the velocities u,
v, and w are extrapolated equally from the value at the lower grid
points. The viscous condition, the tangential stress is zero on the
freesurface, is not considered here.
2. In Hino's case, there are several grid points above the still water
surface and the pressure boundary condition is applied at the
intermediate point between grid points as used in the SUMMAC. But
in our case, the pressure condition is directly applied without any
interpolation. The time dependent term is, of course, taken into
account when the grid is rearranged.
3. On the Triple Grid Method: We used three different schemes A,
B. and C for the convective terms, the diffusion terms, and the free
surface condition respectively; A, B. and C are 74~29~19, 37~29~19,
and 296~116xl in x, y, z directions. The point of the triple grid
method is that the fourth order central difference scheme in the
Poisson equation does not so much improve the accuracy even if finer
grid is used, while, as you comment, the fine mesh system is
necessary for the convective term. This is because the truncation
errors which come from the dissipation terms, gradient of pressure
and Poisson equation give little influence on the accuracy of the
computations. The move of the freesurface particle to satisfy the
kinematic freesurface condition requires fine grid.
4. Due to the limitation of computer, about 15 points are used
around A.P. within boundary layer. It is the reason why we have
introduced the wall function to compensate for the smaller number of
grid points near the hull surface.
DISCUSSION
Hoyle Raven
Maritime Research Institute Netherlands, The Netherlands
When studying Reynolds number effects with a numerical method,
one must be sure to have negligible numerical viscosity. This
requires that the hull boundary layer is well resolved. How many
grid points in normal direction did you use inside the boundary layer?
It seems that your grid is not adapted to the boundary layer thickness
and may therefore have no grid point at all inside the boundary layer
near the bow.
AUTHORS' REPLY
Thanlc you for your kind interest in our paper. Fifteen grid points
are used within boundary layer around A.P. In CFD, the grid
requirement is not easy to satisfy in case of calculating the boundary
layer. Especially around the bow, the boundary layer thickness is so
thin that it is very hard to catch the larger velocity gradient with two
or three grids. In order to overcome that problem, we applied the
empirical wall function approach to the scheme except for some stern
region where the separation is suspected.
DISCUSSION
Ronald W. Young
University of California at Berkeley, USA
The instability criteria mentioned in the paper applies to, or was
derived on the basis of, 2D flow. I find it rather skeptical that it can
be used for oblique waves along the side of the ship. Perhaps some
additional justification is helpful.
AUTHORS' REPLY
As the discusser pointed out, the criteria used here is that derived
from 2D instability analysis. Therefore, exactly speaking, it may
not be applied to the 3D ship waves. However, the obliqueness of
the present case is so small that we expected to have some qualitative
detection of subbreaking; dependency of Froude number and so on.
Through the present application, we have been much encouraged to
provide a criteria based on 3D analysis.
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