Below are the first 10 and last 10 pages of uncorrected machineread text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapterrepresentative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 211
Numerical Simulation of Viscous Flow
around Practical Hull Form
A. Masuko and S. Ogiwara
IshikawajimaHarima Heavy Industries Co.
Yokohama, Japan
Abstract
This paper deals with numerical simulations
of viscous flow around ships having practical
hull form under propeller operating condition.
The Reynoldsaveraged NavierStokes equations
for threedimensional flow are discretized by
finitedifference approximation and solved
with SIMPLE algorithm. The k ~ turbulence
model and the standard wallfunction are
adopted. A propeller is simulated by giving
pressure jump at its position.
In order to eliminate skew grid around
practical hull form, which violates the
computational result, adjustment of grid angle
is applied to the grid generated by solving
the elliptic partial differential equation.
Computational examples for the cargo ship
Series60 model and the practical tanker
models under propeller operating condition are
presented and compared with experiments.
1. Introduction
In the ship building industries,
computational fluid dynamics ( CFD ) technique
is being int roduced to design and develop the
ship hull form under the concept of "numerical
tank". Although quantitative accuracy of the
prediction is not sufficientat at the present
stage, qualitative prediction of flow field
can be applied to evaluate the propulsive
performance and to design a new hull form. It
is considered to become a useful design tool
from the point of efficiency and cost for the
development.
The authors have been developing the
numerical code for calculation of viscous
flow' around ship hull aiming at the practical
use in ship design. The preliminary studies
using mathematical ship models2 showed that
the method has a sufficient robustness of
convergence for iteration number and the
variation of computational grid. The
comparisons between calculation and experiment
demonst rated that the method is effective to
predict the wake distribution, viscous
pressure drag and propeller effects on them,
although some discrepancies, for instance in
strength of bilge vortices, are found in
detail.
In this paper, this method is applied to
simulation of the flow around the practical
hull form with complicated shape.
Computational grid becomes often extremely
skew for such form and calculation by the
author's method diverges due to skew grid. In
order to stabilize the calculation, the
computational grid generated by solving the
partial differential equations is modified to
eliminate the skew grid. The above flow
calculation method is applied to the cargo
ship Series00 model and two types of tanker
model, one is the ordinary type and the other
has the IHI B.O. ( Bulbous Open ) stern4. The
calculated wake and p ressure are compared with
the experiments and propeller/hull interaction
is numerically investigated.
2. Calculation method
2. ~ Basic equation
The governing timeaveraged equations for
threedimensional turbulent flows in Cartesian
coo rdinates are :
Pa (pZtj ) = 0
( ~ )
axe (Pa; hi ) =  Bali ~ All t,4 ( Hi ~ AL hi (2)
where u i is the velocity component, p is the
fluid density and p is the pressure. ,tl~ is the
effective turbulent viscosity and given by:
~ = JO ~ at, = ~ ~ Cow
where ~ is the laminar viscosity, ,u~ is the
turbulent viscosity, CD is the constant, k is
the turbulent kinetic energy and £ iS the
dissipation rate of k.
In the k ~ model of turbulences, k and
ate governed by the following equations:
(3)
a3(P~ik) = a mak ~ pi pa
aa (PORE) =
211
Alp ~ axj Car P; k  C2 p ok ( ~ ~
OCR for page 211
where Pk is the production rate of k and given
by:
pa = pt Maui ( ~Xj ~ Wi )
Standard values for the constants in Eqs.(4)
and (5) are as follows6:
C., = 0. Og C] = 1.44 C2 = ~ . g2
c;r~ = 1 . O ~e = ~ .3
Eqs.(1), (2), (4) and (5) are represented
in the following general form:
pa (paid) = sax ( P~ flex ) ~ S' (8)
where ~ is a general dependent variate le.
=( l,u i ,k, c). When a general curvilinear
coordinate system ($ I, $2, $3 )=($, A, () is
introduced, Eq. (8) is transformed to the
following equation:
As
`j(~i~) = Jar: ( P' A;~; ) ~ S' ' AP
= ,~a'~j ( ~ Ant ) ~ S¢* as)
where J is the Jacobian of transformation, So'
is the transformed source term corresponding
to Eq. (8), and S. * is the modified source term
including So' and the cross derivatives in the
diffusion terms. G I is the contravariant
velocity components without the metric
normalization, and defined as follows:
Gi = a, ju;
Ai j and ai j are the metric coefficients for
transformation:
a;%, = 9~ J
DXJ
Production term Pk in general curvilinear
coordinate is given by:
p ~ a, Du; ( ~ ale ~ a, ,'~uj ) (~3'
Table 1 shows a, F. and S. * in Eq. (9).
2.2 Finite difference equation
Variab les u i and p are set at staggered
location to avoid spurious error7. Integrating
Eq. (9) over a control volume (Fig. 1), the
finite difference equations are given as
fol lows:
pop !t,];v ~ ~t,2 ~~ ~ ~ ~G2 Gas n ~~] ~3
~ [pG3~J6~2
~~ A:. and e ~~ ~~ ~ [~' Aims n at. At~
~ [~ J43~;d~3 ~ At, J5~2 ~ By ~ }fit,] ~2 ~3
(6)
Further discretization by the Hybrid schemer
leads to the to flowing algebraic equation :
AD ~ =A,' ~ BAD ~ CAN ~N HAS As CAT ~T +~ MOB
USE (I, . . . ,~, . . ., etc. ) (15)
where Su denotes a source term which involves
the diagonal terms (~E, etc.) and offdiagonal
terms (AN E ,etc.). Coefficients AE ~ AP are as
follows :
AE = [0, D20 ~0.5C10 ~ ~C10
Ad = [0, Did +0.5C~, Cede
A.~, =
Al =
As =
=
and
~ 0 , D2 n ~ 0. 5C2 I7 ~ ~C?
{0, D2S +0.5C25 ~ C25 ~
[0, D~0.5C3e, Calf
[0, Dad +0.5C3~, C36]
As ~ An ~ AN ~ As ~ Ar ~ ~  Sp
(16)
_ Al ~ Ajar ^83
_ r' Jell
[, 23 C2 = pG2~~dt3 (~7)
JAI2
Aids At] at2
 JAIL
A sign [,, ] in Eq.(16) means the maximum
( ~ O.) value in [ ]. Su and Sp are given by the
following equation:
SO ~ S.' UP = JS, ~ d(' d(~ ~3 (18)
(11 ) The SIMPLE algorithms links the pressure
to the velocity through the pressure
'2) correction equation. When pressure p is
supposed to be summation of previous value p*
and correction p', the pressurecorrection
equation is obtained from the continuity
equation and momentum equations. The
discretized pressurecorrection equation has
the same form of Eq.(15). In the present
calculation, offdiagonal terms in Su of
pressurecorrection equation are assumed to be
small compared with diagonal terms and
neglected for saving computing time.
In consequence, the sequence of solving the
governing equations is as follows:
(1) Solve the momentum equations with the
initial or previous pressure field to obtain
the intermediate velocity components u j * .
(2) Calculate the intermediate contravariant
ve] ocity components G j with u i * .
(3) Solve the pressurecorrection equation
using G j
(4) Obtain the new values of G j and u i which
( 14 ) satisfy the continuity equation using pressure
212
OCR for page 211
correction.
(5) Correct the pressure by the pressure
correction.
(6) Solve the governing equations for k and c.
(7) Iterate steps (1 ) to (6) until the
solution converges.
As a solving algorithm of algebraic
equation, a "checkerboard" method with SUR
(Successive Under Relaxation method) ~ ~ is
employed to vectorize the calculation on a
supercomputer.
2.3 Computational domain
In this paper, a body fixed Cartesian
coordinates are adopted whose origin is
settled at the bow on the still waterplane, x
axis in positive direction of uniform flow and
zaxis downwards. In the body fitted
coordinate system ( $, r7, ~ ), constant ~
planes are chosen as correspond to constantx
planes, 71axis in radial direction from hull
surface and (axis in girth direction.
Calculation is carried out in the domain
surrounded by the following boundaries ( see
Fig.2 ):
Hull Surface
Center plane
Water plane
Upstream boundary
Downst ream boundary
Outer boundary
: x= 0.0 ~ L
.Y= Ye
z= 0.0 ~ d
: y= 0.0
: z= 0.0
: x=0.5L
: x= 2.0L
: r= 0.5L
where L, d and ye denote ship length, draft
and half b readth of a ship respectively.
2.4 Boundary conditions
As for flow field around a fixed model in
the uniform flow U. the boundary conditions
are given as follows under neglect of free
surface disturbance :
On hu l l su rface :
a, v, a, k, ~ = 0
On center plane:
flu law ak
v = 0, and and An,
On water plane:
flu 3D ak
w = 0, an' an' fin'
At infinity:
= 0
= 0
~ = U. v, a, p, k, ~ = 0
where a/an is normal derivative to th e
boundary su rface.
When the finite difference equation is
solved in the computational domain shown in
Fig.2, the boundary values subscribed with p
are determined by the following manners
considering the boundary conditions.
On the hull surface:
up =0, UP =0, ~P =0,
~P=O, EP=0
On the center plane:
UP=~N ~ vP=0, w~=m~
kin Ok, eP =~
On the water plane:
UP =ur, vP =~T, wP =0,
kP =~T, EP =EN
On the upst ream boundary :
uP =U,
P. =0,
On the downstream boundary:
uP=O, wP=O,
icP=O, EP=0
~,~ =U`, VP =m, lo =W~,
PP =PU . UP =iC~, EP =ES}
On the outer boundary:
(20)
UP=U, VP=O, WP=O,
ps=O, , iCP=O, EP=0
Since the standard k £ model cannot be
applied in the viscous sublayer and
transition layer around the hull, the standard
wallfunctions6 are adopted. In the present
calculation, the effective exchange
coefficient rat is modified at the wall
boundary so as to make the velocity profile
fit to that from the loglaw. Therefore the
grid spaces in r'direction adjacent to the
hull surface have to be set to satisfy the
following criterion.
20 ~ ye = d7Jm i r; pU+/p < ]00 (21)
where IBM in denotes the minimum spacing of ~
direction ( distance from the hull surface to
the nearest grid point ) and u+ means the
frictional velocity.
2.5 Propeller model
In order to simulate the propeller effect,
the pressure jump model is employed in which a
propeller is replaced by a accelerating
disc'. The pressure jump is assumed uniform in
the disc and its value is derived from the
measured thrust of the selfpropulsion test.
Fig.3 shows the grid configuration at the
section of the propeller. The grid is not
Age fitted to the propeller disc and the uniform
pressure jump is applied at the grid points
indicated by circles.
Fig.4 shows the pressure distribution
calculated by this method for a propeller
which is operating in open water. The
calculated pressure connects smoothly with
the given pressure jump at the propeller
position. The velocity at the propeller
position and far behind the propeller
coincide with the values given by the momentum
theory.
213
OCR for page 211
3. Computational grid for practical hull form
In the present calculation, the
computational grids are generated by solving
the elliptic partial differential equations,
however these grids do not always suitable
for practical hull form. In general,
computational grid for viscous flow
calculation requires some characteristics
such as orthogonality, smoothness, adequate
concentration, configuration like streamlines
and so on. Above method does not guarantee
orthogonality condition of the grid and often
provides the extremely skew grid.
The present flow calculation method has a
characteristic to be sensitive to the skew
of the grid and the convergence of the
calculation is violated by highly skew grid.
Fig. 5 shows an examp le of the grid
configuration including skew grids and the
results of flow calculation which is violated
by skew grid. The grid in Fig.5 is generated
by Kodama's methods, which satisfy the above
characteristics necessary for computational
grid by geometrical manner. This is one of
the sophisticated method to generate the grid
for arbitrary hull form and the flow
calculation method proposed by Kodama gives
satisfactory results using these grids.
However there are some skew grids near stern
region because the grid line is chosen to
correspond to the end profile and when these
grids are applied to the flow calculation by
the present method, the solution diverges.
The computational results shown in Fig.5 is
the velocity vector near hull surface just
before calculation breaks down. It is found
that the calcu ration is getting to be
vio lated around the skew grid. The reason why
this breakdown occurs is considered that the
all offdiagonal terms are treated as the
source term and they are ignored in the
pressurecorrection equation. Although it is
possible to treat these terms more precisely,
it brings enormous increase of computational
time. Therefore, from the practical point of
view, it is much convenient to generate the
computational grids which restrain the
breakdown of calculation.
In order to restrain this breakdown,
computational grid generated by Thompson's
method is modified so as to improve the shape
of the grid in the transverse plane. For
convenience of expression of computational
results, constant ~ stations are chosen to
correspond to the transverse section.
3.1 Grid points on the hull surface
The coordinates of grid points on the hull
surface are given by interpolation technique
from the offset data. The method of a circular
arc approximations is adopted for the
interpolation. The procedure of the
interpolation is as follows ( see Fig.6 ).
First, an angle of tangent of each data
point ( P i ) is determined as follows. Using
four data points near the point P i, make three
circular arcs ( arc Pi2P Pi, arc PA PiP:+,
and arc P i P j + ~ P i + 2 ). Each arc has an ang le
of tangent at point Pi( (,), (5i)2 and (hi)
), and then a mean of these three angles is
taken for an angle of tangent of point P I.
Next make a triangle PiPi+i P' using a
segment PiPi+~, and tangents at Pi and Pi ~ i,
where P' is a cross point of two tangents. An
interpolated point Q is taken at a inner
center of this triangle.
By treating an interpolated point as a new
data point, these process are repeated until a
length of the segment becomes sufficiently
short. These process are carried out along
the waterlines and frame lines. The
coordinates of grid points on the hull surface
are determined by choosing a point from these
sequential points.
3.2 Generation of grid
The coordinates of grid points in the
computational domain are generated by
Thompson's method. They are solutions of the
following Poisson equation:
Up,
(22)
Exchanging the independent variables and the
dep endent variate les in Eq. (22), fo 1 lowing
partial differential equation is obtained:
hi, ~~ '~: ~ J~Pj 3~' = 0 (23)
It is not necessary to solve the equation for
x, in Eq.(23), because the constant$ stations
are chosen to correspond to the transverse
sections.
This method has some problems when
generating grid for arbitrary hull form.
Since this method does not guarantee the
orthogonality of the grid, extremely skew grid
is often generated depending on the hull form.
When sufficient number of grid points cannot
be taken for the limitation of computer
storage, grid lines often break into the ship
hull. Fig.7(a) shows a typical example of such
case. This is the grid at the section of
bulbous bow which has extremely convex
configuration. At the side of bulb, high] y
skew grids are found and some grid lines
break into the bulb.
Since the stability of the solution by the
present flow calculation method is very
sensitive to the skew of the grid, a method of
grid modification which adjusts the angle
between grid lines is adopted. Fig.8 shows
the way of this adjustment. The two segments
on constant ~ lines which close to the hull
surface are adjusted normal to the constanta
lines. For the other segments on constant ~
lines, the direction is adjusted to the angle
between 45 deg. and 135 deg. When this
adjustment is carried out directory, the
change of the grid direction is too large and
the computational grid breaks down. So the
above adjustment is carried out iteratively
using relaxation method. After obtaining
constantr' coo rdinates, smoothing of constant
r~ line is carried out by Lagrange
214
OCR for page 211
interpolation.
At the same time, the minimum spacing of
the grid Ar7m in iS set constant and determined
using Of of the PrandtlSchlichting's formula
at the aft end of equivalent flat plate as
follows:
30~ 30~
Hum; rat pus ~/~ p
T~ = — pU2 Of
2 PU2O.455(!Og~N )~2 SS (1 4~10gR2 ~ (25)
Fig.7(b) shows the modified result of
Fig.7(a) by this method. Fig.9 and Fig.10 show
the computational grids for tanker forms with
normal stern and IHI B.O. stern respectively.
In every case, orthogonality condition is
almost satisfied.
4. Convergence property
Convergence histories of pressure at three
monitoring points, midship, after
perpendicular (A.P.) and in the wake, close
to the keel line, are shown in Fig.11 in the
case of Series60 model (Cb=0.6) at the
Reynolds number based on ship length
Rn=9.22x106. The number of grid points is
94x25x21 and the iterative calculation is
repeated 300 times. The calculation seems to
converge at about 100 times.
Fig. 12 shows the convergence histories of
variate les u, w, p, k and ~ near the keel line
at A.P. The values are nondimensionalized by
the differences between maximum and minimum
values of each variable. Convergence of
variables excepting pressure is considerably
slow. This may be attributed to the lack of
grid smoothness.
Fig. 13 shows the variation of mass
imbalance with the number of iteration. SSUM
means a sum of massimbalance for all the grid
points normalized by inlet mass flow rate.
SSUM decreases rapidly with the number of
iteration and this verifies that the equation
of continuity is satisfied.
From the above results, 200 iteration
steps is chosen for practical use. It takes
about 20 minutes of CPU time to calculate 200
steps using the supercomputer FACOM VP50. In
the following calculations, the iteration
is stopped at 200 steps in every case.
5. Computational results and discussions
5.1 Wigley model
The Wigley hull is defined by the following
parabolic equation:
ye = BL1 ~ ( r2X_ 1 )2 ~ [] ~ ( d A] (26)
where B is the ship breadth.
Calculations are carried out using 93x25x19
grid with L=~.Om, B=0.6m, d=0.375m, U=0.85m/s
and RN =4. 5X1 0G, and compa red with two kinds of
experimental data. One is obtained in the
towing tank of Ishikawaj imaHarima Heavy
Industries Co., Ltd. (IHI) with em length
model, where resistance and hull surface
pressure are measured at RN=4.2X1O69 and the
other data is from the wind tunnel test by
Sarda' ~ using a double model at RN=4.5X1O6.
Fig. 14 shows the isowake contours at the
(24 ) aft end station compared with Sarda's results.
The calculated result shows qualitative
agreement with the experimental results,
however, it is a little diffusive due to the
numerical diffusion of the Hybrid scheme.
Fig.15 shows the comparison of hull surface
pressure distributions, and reveals good
agreement between the calculation and
experiment. In this figure, inviscid results
obtained by HessSmith method are also shown.
The results of the present method agree with
the inviscid results in the fore part and the
displacement effect of boundary layer is
simulated near the aft end.
Fig. 16 shows the comparison of the local
skinfriction coefficient Cf. Calculated skin
friction is given by the following equation:
ply /2
_ 1 ~cqC~ i ~ ~ kit 2 (27)
Up /2 l ~t( E,oC3 ~ 4 k~ ~ d7?m; n /~)
where K iS Karman constant and E is constant
The figure shows good agreement between the
calculation and experiment except slight
discrepancy at the station x=5.9m close to the
aft end.
Agreement of the turbulent kinetic energy
between calculation and experiment is poor as
shown in Fig.17. Calculation does not simulate
the sharp peek of the turbulent kinetic energy
in the region close to the hull near the after
end. As the same tendency appears in Sarda's
calculation which also adopts k E model, this
seems to be due to the defect of the
turbulence model.
The pressure resistance and the frictional
resistance are calculated by integrating the
hull surface pressure and local skinfriction
respectively on the hull surface. The
calculated results are as follows compared
with the experimental values given by th ree
dimensional analysis of the resistance tests.
Calculation
r
Total
resist.
(rT)
Fric.
resist.
(rF)
Press.
resist.
(rp ) _
( RN =4.2X1O6 EN =0.1043 )
12.6xlO 3
10.8xlO 3
1.8xlO3
215
Experiment
Total
resist.
(rT)
Fric.
resist.
(rF0)
Residual
resist.
(rig)
13.2xlO 3
12.7xlO 3
(Schoenherr)
0.5xl 0 3
OCR for page 211
where r=R/,oU2 V2 ~ 3, R is the resistance and
is the disp lacement of the model. Calculated
value of the p ressure resistance is larger
than residual resistance which does not
include wave resistance because of very low
Froude number. This may result from the fact
that the calculated pressure of after part is
a little lower than measured one as shown in
Fig.15 observing in detail. However the order
of the total resistance comparatively agrees
with experiment.
5.2 Series60 model (Cb=0.6)
The first application to a practical hull
form is made for prediction of flow field
around Series60 model (Cb=0.6) under
propeller operation. The calculation is
carried out with L=7.0m and U=1.5m/s,
corresponding Reynolds number is RN=9.22xlO6.
In the calculation of propeller operating
condition, pressure jump Ap=412.47N/m2 which
is equivalent to the measured thrust T=22.119N
( propeller diameter is .2613 m ) is set on
the propeller disc. Total number of grid
points is 94x25x21.
Fig.18 shows the hull surface pressure
distribution, where measured data is from VEB
towing tank using 5m model at U=1.54m/s
(RN=7.7X108)] 2. Calculated and measured
patterns of pressure contour without propeller
resemble each other. In the propeller
operating condition, suction effect of
propeller is appeared in the stern region.
Fig. 19 shows a comparison of wake patterns
at three different transverse sections.
Experimental data is obtained in IHI towing
tank using 7m model. The unit of calculated
vector is twice of the measured one in order
to make clear the direction of the flow. The
calculated contour of wx=O.l is a little
diffusive compared with the measured one,
however, the pattern of the isowake contour
is well simulated as a whole.
Fig.20 shows the effect of propeller on the
isowake contours. It can be seen that the
effect of propeller is restricted in the
propeller disc. The measured isowake
contours with propeller at A.P. section (just
abaft the propeller) show the asymmetrical
feature about center line due to the rotating
flow ( Fig.21 ). As the present method does
not deal with the rotating flow, calculated
results are compared with the measured contour
taking the mean of the contours in starboard
and port side. It is found that the present
method can simulate well the propeller effect
on the contour.
5.3 Practical tanker form
'I'he second example of the application to
practical hull form is the simulation of the
flow around two kinds of tanker form shown in
Fig.10 and Fig.11.
Fi g.22 shows the comparison of isowake
contour of ordinary tanker form ( Fig.10 ) at
propeller position in towing condition at
R~ =7.8xl 0~; . The vortical motion can be
simulated, however, it is smaller than that
of experiment and an islandlike contour
of vortical motion is not found in the
computed resu its.
The flow around tanker form with IHI B.O.
stern ( Fig.11 ) is calculated at RN=4.94X106.
IHI B.O. stern is developed aiming at both
merits of wake gain by bulbous stern and low
thrust deduction by open stern. The
configuration of B.O. stern is so complicated
that the flow calculation does not succeed by
the o rdinary method of grid generation.
However the present method of grid
modification makes the flow calculation
possible.
Fig. 23 and Fig. 24 are the comparison of
hull surface pressure distribution and wake
pattern in towing condition respectively.
Calculated pressure distribution agrees well
with the experiment except near the stern end
where calculation gives lower pressure. B.O.
stern gives uniform wake in propeller disc
compared with ordinary stern shape. Present
calculation simulates this feature of wake
pattern, however the correspondence with
measured results is not good because the bilge
vortex is not simulated well. In order to
improve the accuracy of the prediction,
further examinations are necessary for finite
difference scheme, grid generation, adoption
of wallfunction, turbulence model and so
forth.
Fig.25 shows the velocity vectors near the
hull surface for both cases of with and
without propeller. In the propeller operating
condition, the pressure jump of 691.67N/m2
which corresponds to measured thrust of 17.6N
is set on the propeller disc of diameter
0.18m. Applying this propeller model, the
accelerated flow afore and abaft the
propeller can be simulated as well as
decelerated flow just above the propeller. The
present method predicts the boundary layer
flow into the propeller around such a
complicated stern form.
6. Conclusions
Present studies are summarized as follows:
(1) In the present flow calculation method,
the offdiagonal terms in source term in
pressurecorrection equation are ignored as
small quantities in order to save the
computational time. It is found, however, that
this leads to the breakdown of computational
results when there are skew grids in
computational domain around practical hull
form.
(2) In order to stabilize the calculation for
practical hull form without increase of
computational time, improvement of the grid
shape generated by solving the elliptic
partial differential equation is carried out
by adjustment of grid angle.
(3) Using this grid, the calculations of
viscous flow around practical hull form
(Series6O and tanker forms) under propeller
operating condition are carried out and the
216
OCR for page 211
results are compared with experimental
results. This method is applicable for hull
form examination at the initial design stage.
In order to improve the accuracy of the
prediction, further examinations are necessary
for finitedifference scheme, grid generation,
turbulence model, adoption of wallfunction
and so forth.
The final goal of the present study is to
build a design code which can evaluate self
propulsion factor of a ship taking account
rudder effect.
Acknowledgements
The authors are indebted to Dr.Y.Kodama of
Ship Research Institute for providing us the
computational grid of Series60 generated by
his own method. They also express their thanks
to Dr.Y.Ando, Mr.M.Kawai, Dr.T.Tsutsumi,
Dr.R.Sato and Mr.Y.Shirose of Ishikawajima
Harima Heavy Industries Co.,Ltd. for their
support and advice, and all the staff of IHI
towing tank for their help with the
experiments.
References
1. Ando,Y., Kawai,M., Sato,Y. and Toh,H.,
"Prediction of threedimensional turbulent
flows in a dump diffuser", AIM 2Cth Aerospace
Sciences Meeting, Reno, Nevada, (1988).
2. Masuko,A., Shirose,Y. and Ishida,S.,
"numerical simulations of the viscous flow
around ships including bilge vortices",
Proceedings of the 17th ONR Symposium on Naval
Hydrodynamics, The Hague, (1988).
3. Thompson,J.F., Warsi,Z.U.A. and
Mastin,C.W., "Numerical grid generation,
foundation and applications", NorthHolland,
New York, Amsterdam and Oxford, (1985).
4. Koshiba,Y. and Mori,M., "How to design the
stern form"( in Japanese ), Ishikawajima
Harima Engineering Review, Vol.27, No.5,
pp288pp293, (1987).
5. Launder,B.E. and Spalding,D.B.,
"Mathematical models of turbulence", Academic
Press, London and New york, (1972).
G. Launder,B.E. and Spalding,D.B., "The
numerical computation of turbulent flow",
Computer Method in Applied Mechanics and
Engineering, Vol.3, (1974).
7. Patanker,S.V., "Numerical heat transfer and
fluid flow", Hemisphere Publishing, (1980).
8. O rli, S. and Haraguchi,M., "Efficient
implementation of a fluid simulation algorithm
on the FACOMVP 100/200( in Japanese ), The
28th National Convention of ISPJ, Information
Processing Society of Japan, (1984).
9. Kodama,Y., "Threedimensional grid
generation around a ship hull using the
geometrical method", Journal of the Society of
Naval Architects of Japan, Vol.164, pp.916,
(1988).
10. Oki,Y., Ochi,M. and Ohgane,E., "Ship lines
design system"( in Japanese ), IshikawaJima
Harima Engineering Review, Vol.21, No.5,
pp .422427, ( 1 98 1 ) .
11. Sarda,O.P., "Turbulent flow past ship
hulls  An experimental and computational
study ", Ph.D. Thesis, the University of
Iowa, (1980).
12. "Flow examination on a model of Series 60
with Cb=0.60 Model No.675", VEB Report,
(1983, Applied to the ITTC Cooperated
Experimental Program).
Table 1 Effective exchange coefficients and
source terms of equation (9)
1
1
r~ ~I s,
0 11 0
u; ~ ~ I _ ai ~P t ~ a,, auk
~ ~ mAfk3~t
f k J9t, J3tk
k ~ ~^ ,
~ 1 ~
'CJ'
_ pi _ pe ~ ~ ~ ~Aj~3k
C] Pk k~  C2 p k
~ ~ ~ ~Aj'3E
~  ~ J3t,, ~E J3tk
T ( ~
Cont ro I Vo I ume 
B ~
Fig. 1 Grid points and a control volume
217
OCR for page 211
y/L
0.5 0.0
FP
n n
0.5
3~ X/L AP
l.o 1.5 2.0
0.5 0.0 0
FP
. _
l ~ ~ = = ~ ~ ~
_ _ _ _ _ _ _
_ _ _ _ _ _ _
_ _ _ _ _ _ _
,
.5 1.0 1.5 2.0
X/ L AP
Fig.2 Computational domain and grid configuration
~:
O Grid point where pressure jump is imposed
Fig.3 Grid configuration at the propeller
position
cp = P
0.6
0.4
0.2
o.o ,
1 .o
—0.2
0.4
—0.6
x(m)
_  I 1
U—U VI'V2 Prediction with momentum theory
U  Propeller position
0.4 _ Vl ~
/i x(m)
0.0 , ~/ 1 1 1
1 n o.o l.o
1.0 2.0
Fig.4 Computational examination of pressure
jump model
218
OCR for page 211
Fig.5 Influence of skew grid
(a) Before modification
(b) After modification
Fig.7 Modification of the grid
219
x

\\W
(~:
P,:
X;"
K,"""
to,
ig.6 Generation of interpolated point
i=]
2 (Hull Surface)
I\\
\ \
\ ~ \
id,' \
*: Angle adjusted to be between 45°~135°
Fig.8 Adj ustment of grid angle
OCR for page 211
Bow Part
Stern Part
Stern Section
Fig.9 Generated computational grid
( Tanker form with normal stern )
Row Fort
_~
Stern Part
Stern Section
Fig.10 Generated computational grid
( Tanker form with IHI B.O. Stern )
220
OCR for page 211
P (Njm2)
50C
Hi
—500
Max
SSUM
Fig.11 Convergence history of pressure
/
/
/
~ , , w 300 ISTEP
150/ 200
~_
u

Cal.
—— Exp. ~)
(Sarda) ~
Fig. 14 Calculated and measured wake pattern
( Wigley model, RN=4.5X1O6 )
Cp  PU2/2 '
Fig. 12 Convergence history of u, w, p, 0.2
k, £ ~ at A.P.
~o
10'1(
iot
1n.
10' ~
~\\



10'
O SO 1W lSO
200 250 3W
ISTEP
Fig. 13 Convergence history of massimbalance
0 1~
0.0 ;
z/d  0.2
iC,
..
. .$ . , .
FP ..
~ D
 ,. _
· Cal.
O Exp. (IHI, R,,~.2X106)
Inviscid (Calculation)
.e
..
.
:.
, ~
,. _
. . ~ ~ , ~ I
5 ° AP
. .~
Fig. 15 HU11 surface pressure distribution
( Wigley model, RN=4.5X1O6 )
221
OCR for page 211
5.0
C'X103
x =3.0m
(Midship)
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ao ~ ~e
' ' z/d
0 0 0.5 1.0 . Cat.
5 . 0 x = 5 . 4 m 2 ( yyO)/B 0 Exp. (Sarda)
(S. S. 1 ) O x5.7m x6.0m x 6.6m x 9.Om
~ 1.0  (S.S. 1/2) 1.0  (A.P) 1.0 1.0
C'X103 ~ ~ ~ ~d.~ 0.8~ 0.8~ 0.8t 0.8
0.6~ 0.6~ 0.65 0.6~
1 ~ z/d 0 4 Oo 0 4  Oo e 0 4 °O. 0.4  0 e
0 0 0.5 0.2 .2 0.2
~ (S.5~/2) . ~°~
C, X103 1 ~ o · °'° k/U7XIO' 5'00'0 k/U2 103 5'00'0 / ~ , 5.00.0 5,0
· ~ ~O · · Fig.17 Comparison of turbulent kinetic energy
l 1;0 z/d ( Wigley model, RN =4.5xlO6, z/d=O.O )
0.0 0.5
5.0 x =5.9m
 · Cal.
C~ X103  0 Exp.(Sarda) °
· · ·o · · 
~ I
' ' z/d
0.0 0.5 1.0
Fig.16 Comparison of local skinfriction coefficient
( Wigley model, RN=4.5X1O6 )
~ ~ j ~ ~ ~
Cal. (without propeller)
Cal. (with propeller)
C) ~_
. .. , . ... .... .. .. . _
Fig.18 Hull surface pressure distribution ( Series60 Cb=0.6 )
222
2_0~075 ~=
OCR for page 211
x  ~ ~ ~ ~ ~ ~ 
~ o ooo o o o o
~ ::: S~:~
S.S.1
Cal.
1 ~ ~ u,
x . . ~
~ 0 0 0 0
/` `~` `: i_
~'NJ''~
/ N/' N/'
//~//~:
\ ~ ~
/ \ ~ ~ \ ~
~/1 \ \ ~ \
1 1 ~ ~ \ ~
.,
.~I Exp.
O.SU
. . .
O o 0
/ ~ /
_/ _ )` ~ ~ (` ~ ~ ~
~ ;'/~/`J~ 1'
. t\~/~0.5~U:'
Cal. Exp.
~/
n su
t
/ \1
t
1
/1i'
.,., ~
A.P.
Cal. Exp.
1 1
Fig.19 Calculated and measured wake patterns
( Series60 Cb=0.6, RN=9.22X1O6 )
11
x ~ r~ ~ Ln
_ . . . .
5 o o o o
~ \N
without propeller
 with propeller
S.S. 1/4
Cal. Exp.
0.1 '~
without propeller
 with propeller
A.P.
Cal. Exp.
Fig.20 The effect of propeller on the
calculated and measured wake patterns
( SeriesGO Cb=0.6, RN=9.22X1O6 )
o.
without propeller
 with propeller
Fig.21 The effect of propeller on the
measured wake pattern
( Series60 Cb=0.6, RN=9.22X1O6 )
223
OCR for page 211


/ /
/ ~~/\~
Cal.
~,,~J ~ /  
· , /~ ~ \ ,
,/1("~1/—\ Am)' \
K'\4 1\ /\~N
~ \ '\
''_)t / 1 1 ~ t t ~ ~ ~
0 5U
Prop. Position
Exp.
1 '
)'
Fig.22 Calculated and measured wake patterns ( Tanker form with normal stern, RN=7.8X1O6 )
~ '\~"'\"\ V'\'\\~

/ ~ _11 ~11
Exp.
Fig.23 Hull surface pressure distribution ( Tanker form with IHI B.O. Stern, RN=4.94X1O6 )
~.~\
Prop. Position
Exp.
Fig.24 Calculated and measured wake pattern
Cal.
1 ~~
224
Fig.25 The effect of propeller on the velocity
vector near hull surface