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OCR for page 175

Numerical Calculations of the Viscous Flow over the Ship Stern
by Fully Elliptic ancl Partially Parabolic Navier-Stokes Equations
K. J. Oh and S. H. Kang
Seoul National University
Seoul, Korea
T. Kobayashi
University of Tokyo
Tokyo, Japan
Abstract
Two computer codes have been developed to solve the
Reynolds averaged Navier-Stokes equations; namely the fully
elliptic method and the partially parabolic method. These are
applied to simulate flows over the stern of the SSPA model as
a bench mark as well as a multi-purpose ship with a barge
type stern. The numerically generated body-fitted coordinate
system is used to manage the complex geometry of the ship-
hull. A standard form of the k~ turbulence model is
adopted for modelling of the Reynolds stresses.
Simulated results by both methods are nearly identical
when the longitudinal flow reversal does not appear. The
partially parabolic method requires only half of the memory
storage and cuts CPU time by 20% in comparison with the
fully elliptic method. The capability of programs developed
in the present study are confirmed by sucessfully simulating
pressures, skin frictions and mean velocities over sterns of the
two models. The growth of the viscous layer over the stern is
well-simulated and the secondary motion is also captured,
which is usually observed in the experiments. Nevertheless
tl~e standard form of the k~ model is not adequate for
predicting the turbulent kinetic energy over the stern.
Simulated nominal wake fractions show good accordance
with wake measurements. However, values of the outerpart
of the wake are over-estimated, while the trends of the
circumferential variations are consistent with wal;e
measurements. Coefficients of the viscous resistance
predicted by present methods are under-estimated by 10
percent. If further developments on the turbulence model
and numerics are accomplished, this method of numerical
simulations of the viscous flow over the stern would be
promising for the hull form design.
1. Introduction
The importance of the viscous flow simulations around
the ship hull has received wide acknowledgement in the light
of the hull-form design. Predictions of the viscous resistance
are useful in the stage of the bare hull-form design. Ship
forms of good resistance and propulsion performance cannot
be developed without considering the propulsion efficiency
as well as the form factor. Such design and development can
be effectively attained, only if numerical method can
estimate form factors, nominal and effective wakes on the
propeller plane, and thrust deduction factors. These design
parameters can not be reasonably obtained without complex
three-dimensional turbulent flow simulations over the ship
stern and in the wake.
Viscous flows over the ship hull have been calculated by
the three-dimensional boundary layer theory. If free surface
effects are excluded, experiments and calculations indicate
that the first-order boundary layer equations adequately
describe the flow over a large part of a ship hull. But it
begins to break down gradually over the stern, which is
around 10-20 percent of the ship lengtht1,2~. Experimental
information pertaining to the evolution of the flow over the
stern as well as in the near wake has been reviewed by
Patel[2~. Much research has been done for thick boundary
layers over the stern in the past, but they have failed to
provide a designer with valuable information.
The partially parabolic, or the semi-elliptic type, of the
Navier-Stokes equations have been recently employed to
simulate the complex viscous flow over the stern instead of
the full elliptic Navier-Stokes(NS) equations in consideration
of physical phenomena that there is usually no region of flow
reversal in the direction of ship motion. These equations can
be used to describe flows between the thin boundary layer
upstream and the wake far downstream from the ship. The
partially parabolic Navier-Stokes(PPNS) equations have been
first employed to calculate flows and heat transfer in the
straight square duct by Pratap and Spaldingt3~. Abdelmeguid
et al.~4] was the first to have applied to ship hulls. Markatos
et al.~5], Muraokat6,7] etc. have presented further researches
and several paperst8,9,10] appear in the 2nd Symposium on
Ship Viscous Resistance in 1985. Chen and Patel[11] have
adopted the finite analytic numerical scheme and produced
reasonably accurate results for flows exte.rnn1 to an
axisymmetric body of revolution and three-dimensional
mathematical models. A computer program STERN/PPNS
has been developed based on the partially parabolic method
and applied to several models to demonstrate its performance
by Kang and Oh[12,13,14]. The program proved to be
reasonably accurate in describing the pressure distributions on
the hull and the velocity contours.
When there appears flow reversal over the hull, the NS
equations should be solved. A computer code STERN/NS has
been developed in the present study and it's performance has
been investigated by cross checking each of their respective
similated results of flows over the stern. The SSPA 720
model is selected as an bench-mark model and a multi-
purpose ship with a barge type stern for the present study.
The possibility for the program to be used for design purposes
is investigated in the present paper by estimating the viscous
resistance and nominal wakes on the propeller plane of a
barge type ship form. Estimated results are compared with
measured data in the towing tank. Before going further, basic
equations and calculation method are briefly summarized.
At' ''id I' - ~~A ·~1 At/ C111
175

OCR for page 175

2. Governing Equations and Boundary Conditions
2.1 Governing Equation
Geometry of the ship hull is described in the cylindrical
coordinate system (x, r, D) as shown in Fig.1. Governing
equations for the incompressible, steady, and turbulent flow
are given by the continuity and Reynolds averaged Navier-
Stokes equations. Reynolds stresses are modelled by using
the eddy viscosity. In the k~ turbulence model adopted in
the present study, the. eddy viscosity is given by turbulence
kinetic energy k and dissipation rate ~ which are obtained
from their transport equations. In the cylindrical coordinate,
above governing equations can be written in the general form
as follows;
a 1 d 1 ~ d df
- (U’) + - (rV’) + - (W

F2(g, ~, t) = Am r ]~=62
51, 52 denote values of ~ corresponding to upstream and
downstream sections respectively and ~2 denote the value at
the outer boundary. Symmetric Neumann boundary
conditions are used at the water plane (~=oO) and the
center plane (~=90°~. Dirichlet boundary condition is used
at other boundaries.
the fully turbulent layer, and it is assumed that the law of the
wall is satisfied and the velocity vectors.in this region are
collateral. The boundary values at the first grid point are
obtained by assuming the local equilibrium between turbulent
kinetic energy production and dissipation. They are given as
follows.
Kilo
P In (Enp+)
low
(10)
., _
2.3 Transformed Governing Equations C b2 ~ (11)
P
Transformation of independent variables (x, r, D) in
governing equations are considered, leaving velocity 3,4 3,2
components (U. V, W) in the original (x, r, D) coordinate in Cal k
Fig.3. Then governing equations are generally represented as ~ = ~ (1~)
the following formt12]. Kn
[(bluish + b2lV~ + b3lWl)
+(bl2U~ + b22V~ + b3Wl)
+(bl3U4> + b23V~ + b33W’~]
[~r<,,Jgll ~ ~ + d Or Jg22d~ ~
+ d Go Jg33d~ '+ S. (8)
The above equations are still the exact equations in so far as
no approximations have been made beyond those inherent in
the turbulence model. The equation (8) can be rendered
partially parabolic by neglecting the first term which involves
the second order derivative term with respect to 5. Physically
this is not the same as neglecting ~xr nor does it imply that
diffusion in either x or ~ direction is neglected[11~.
2.4 Boundary Conditions
Boundary conditions at each boundaries of the solution
domain are summarized as below.
(1) Upstream A; The position is extended to the upstream as
far as thin boundary layer equations and the potential flow
theory are valid. Then distributions of (U. V, W. k, c) can
be prescribed from boundary layer calculation. If it is placed
over the mid-ship, then distributions may be assumed by
using integral parameters without exerting significant
influences on downstream calculation. The streamwise
velocity profile in the boundary layer is specified by 1/7th
law,
U A t7
= (_)
Ue ~
(9)
and the velocity in the inviscid region is given as the free
stream velocity. The turbulent kinetic energy k and the
dissipation rate ~ are also given by the flat plate
correlations with the boundary layer thickness and skin-
friction coefficient.
(2) Downstream B.; At one ship length downstream from the
stern, zero gradient condition is assumed for the all variables.
In partially parabolic calculation, only the zero pressure
gradient condition is required.
(3) Hull surface S; The wall function is adopted in the
present study. The grid points next to the wall are located in
The magnitude of the velocity at the first grid point np near
the wall is given as V, and n is nomal distance from the
wall.
(4) External boundary Hi; It is placed sufficiently far from
the hull surface so that uniform flows and no turbulence
condition can be assumed there.
dV
U = UO, W = k = ~ = 0,= 0, p = pO (13)
do
(5) Center plane C and water plane W; Symmetric condition
are imposed.
dU dV dk de
W= 0, == = = 0 (14)
dt dt dt d:
(6) Wake center line C; Following conditions are enforced.
dU dk de
V= W= 0, === 0 (15)
din do do
3. Numerical Scheme
Uniform grid spacings are taken in the calculation domain
(1~=/~=~=1) and grid control functions are determined
by specified values on the boundarys. The grid construction is
obtained by solving equation (3), (4) by the finite difference
method.
The Finite Volume Method is applied for discretizing the
governing equations and the hybrid scheme is employed in
the evaluation of the convection terms. The finite difference
equations are obtained by integrating the governing equations
over individual control volumes formed by the staggered grids
system[151. The scalar variables p, k, ~ are located at the grid
nodes themselves, while velocity components are positioned
between the scalar nodes. Such a staggered grid has benifit of
having the velocities at the boundaries of the scalar cells
where they are needed in integrating convective terms.
Furthermore the pressure nodes are located on either side of
the velocity node and it is easy to calculate the pressure
gradient terms in the momentum equations.
Then the final form of the discretized governing equations
are obtained.
amp = aN’N + a5~1'S + aE’E + aWW
+ aUlu + aDID + So (16)
177
The subscript P refer to the grid node to be considered and
the subscript U. D correspond to the upstream and
downstream grid respectively. The other neighbouring grids
in the sections are given by the subscript N. S. E, W. The
a' ~D' AN' AS etc. represent the convection and diffusion
at each corresponding control surface[16].

OCR for page 175

o.,l ~
sly
On. 0.2 0 3
\x s's ~ ~ N
,\\,
AIL
Fig.4 Over-view of generated grids of the SSPA model.
0.2
0.11
When the partially parabolic form of equation (8) is solved, on
the discretized equations are obtained by the similar way.
apt = aNIN + aS’S + ante + avow + Cu4)u + SO (17)
Since the diffusion term in the ~ direction is removed from
the equation (8), only the convection term Cu is included in
the ~ direction. The unknown variables (U. V, W. k, e)
can be obtained by solving the equation (16) or (17) under
the assumed or estimated pressures.
The estimated pressures are indirectly corrected for the
continuity equation to be satisfied. If SIMPLE(Semi-Implicit
Pressure Linked Equation) algorithm is adopted[15i, the
discretized equation for the pressure-correction is obtained.
This equations can be represented by the same form with the
equation (16), which has fully elliptic characteristics.
In the fully elliptic calculation, the flow variables are
iteratively solved and the converged solutions are obtaiined.
The procedures are summarized as:
(1) Construct the coordinate system, and calculate the
metric tensors and Jacobian.
(2) Specify initial conditions at the inlet plane.
(3) Solve the velocities with assumed or previously calculated
pressures.
(4) Solve the pressure-correction equations
(5) Correct pressure distributions and velocities.
(6) Calculate the k, it.
(7) Return to step(3) and repeat step(3~-~7) untill the
residues are reduced by 0.1% of the reference values.
To solve the partially parabolic equation, the marching
procedure along the ~ -direction is employed.
(U. V, W. k, c) at each sections are calculated with
upstream values of each variables and previously calculated
pressure. Pressure-corrections are achieved on each section
without any correction of the upstream and the downstream
pressures during the marching procedure. Pressure of the
whole domain should be stored in the partially parabolic
method and several sweeps in the 5-direction are required to
obtain the converged solutions. The procedure are
summarized as:
(1) Construct the coordinate system, and calculate the
metric tensors and Jacobian.
(2) Specify initial conditions at the upstream boundary.
(3) Calculate velocities at the downstream with the
previously calculated pressures.
.
cp
0.1 r v ~x
._~- , _ of, ~
~~ZI.3
_ STREAK LINE ~ it' I.1 1.2 13
I! ~ :~
.1121.3
1
o.
01
no-
. I ~ I I ~ I
0.5 0.6 0.7 0.8 o.9 1.0 1.1 1.2 1.3
x/L
Fig.5 Pressure distributions over several streamlines
on the SSPA model.
0.4 _
QS ~
Cf pPNS
~ NS
0.4 · EXP(17)
STREAM LINE 7
o' Fat
·:
·\
·
STREAK LINE S
~ ,.
.~% 08 0.9
·~
0.4 _
STREAK LINE 3
0.34 ~ do_ on o.S 1 o
· ,~
0.5 __ · - ~
Q4 :: ~ · ·~,
STREAK LINE I
Ql _
0.0 _
0.5
O.2 ~
O.'
.0
0.6 0'7 0'8 o'g 1. l
X/L
Fig.6 Skin friction coefficients over several streamlines
on the SSPA model.
178
(4) Correct the pressure distributions and velocities.
(5) Calculate k, ~ at the downstream
(6) Marching to the downstream boundary.
(7) Return to step(3) and repeat step(3)-(6) untill the mass
residue are reduced by 0.1% of the reference value.

OCR for page 175

cp
02r
o.
of
o.ll
PPNS
NS
EXP(17)
~ "
. ,,
o too 005 o:lo o Is
2G
Fig.7 Girth-wise variation of pressure coefficient
at x/L=0.9 of the SSPA model.
0.004
n non . it- -.
.~ .
.
PPNS
- - - - NS
· EXP(17)
o.o~ . 1
O. 00 0.05 0.10 0.15
2G
Fig.8 Girth-wise variation of skin-friction coefficients
at x/L=0.9 of the SSPA model.
4. Calculations and Discussion
. .
A computer code, STERN/PPNS based on PPNS
equations has been developed by the method described above
and is applied to flows over several mathematical models.
Calculated results by STERN/PPNS have been discussed by
Kang et al.~12,13,14~. Simulated flow fields and pressure
distributions are generally in good agreement with tested
data. But calculations have a marked trend to over-estimate
turbulent kinetic energies near the stern. Such a trend has
also been pointed out by Chen and Patel[11~.
Another code STERN/NS based on NS equations has
been developed in the present study. General performances
of the code have been checked by simulating flows over
several mathematical models by Oh[163. Performance
characteristics of two programs have been intensively
investigated and compared in the present study by simulating
flows over the SSPA 720 model and a multi purpose barge
type ship. The first one was a container ship model, which
was tested in the wind-tunnel by Larssont17] and used as one
of standard models of II:C-SSPA Workshopt23. The latter
model has a barge type stern, which was chosen to investigate
the possibility of the numerical simulation of viscous flows to
be used for design purposes.
4.1 SSPA 720 model
Part of the numerically generated grid system is presented
in Fig.4. Numbers of meshes in the (5, a, (~-directions are
(58, 25, 14) respectively. They cover the calulation domain
of O.S

°°~1
°~'1
°'°~1
o.o 1
on
0.0_,
cam_
. PPNS
~ ~ NS
i · EXP(17)
. STREAK
_._1 LINE 7 ~
0.05 0.00 ~ 0.5' 1. 0
004R ;1
0.03 ]-
0.02 J
STREAK f
0.01 LINE ~ ~
. r °°° ~ OS I ~
0.04
0.03
0.02
STREAM
0.01 LINE 3 ~
_.__ 0.00 ~ 0 5 0
0.04
0.03
0.02 t
STREAK ·/
0.01 LINE 2 ~
n 0 05 0-00 ~ 0.5 1 0
L o.O~ . ~
0.03 .
0.02
O.C
0.0(
01 STREAM
~ . ~
o.o 0.5 Lo
q
UO
Fig.11 Profiles of resultant velocity at x /L =0.95
of the SSPA model.
distribution of pressure coefficients and skin friction
coefficients calculated in the present study are compared with
measured values in Fig.7 and Fig.8. Secondary flow is
directed away from the keel and the water plane according to
the girth-wise pressure gradient, and the shear layer rapidly
grows thick at the mid-girth. Skin friction coefficients at the
keel show their largest values and decreases along the girth to
the water plane.
The feature of shear layer formation over the stem by
5-l~:RN/NS program is shown in Fig.9 . The contours of the
axial velocity component and the pattern of the transverse
motion at x/L =0.95 are shown in Fig.10. The boundary layer
remains thin along the keel line according to the divergence
of streamlines. The thickness of the viscous layer over the
mid-girth is almost as large as the draft of the model. The
axial velocity contour is well-simulated in comparison with
the measured contour. The bilge vortex, which is a general
feature of the stern-flow, is also observed in the simulation.
Distributions of the total velocity at several points where each
streamlines intersect with the x/L = 0.95 section are compared
with measured data in Fig.11. Here it should be noted that
measurements are obtained nollllal to the hull and
calculations are computed on transverse sections. There are
good agreements between calculations and measurements,
although some error might be involved due to such
differences in the location. Turbulent kinetic energy
distributions are compared in Fig.12, where typical characters
of turbulence in the stern flow appear. Turbulence kinetic
energy shows considerable reduction in the magnitude near
the hull over streamlines 3, 5, and 7, which is quite a unusual
180
~ O!eir
_ __;
0.04
0.0_
PPNS
~ ~ NS
0.0 ~ STREP
0.0 ~ 0.0 ~ LINE 7 .
0.04 0.0 000 0.004 0.0 08
0.03 \ 1
0.o4t
0.02 ~ STREAM
o oc . ; ~
0.000 0.0 4 0.008
nor
0.02
0.0-
~ 1
o.od4
o.o.j
°°;1
n 0.05 on
L 0.04 . 0.Of
0.03 .
0.02 ,
, it. STREAK
01 000 0.004 01 08
-o 1
O.C
0.00
0.000 0.004 0.008
k
STREAM
LINE 3
_
Fig.12 Profiles of turbulent }kinetic energy at x/L=0.95
of the SSPA model.
Fig.13 Body plan of 37K PROBOCON.
feature in the thin boundary layer. It is explained that such
reduction is due to strong flow convergence without enough
generation in the turbulence kinetic energy over the stern.
The k~ model in the present study fails to capture such a
phenomena taking place over the stern. An algebraic stress
model may well be a furture choice for sucessful simulations.
4.2 Barge Type Ship
An object of the present study is to investigate the
potentiality of programs to be used for design purposes. The
selected model, 37K PROBOCON, was originally designed
by KSEC(Korea Shipbuilding and Engineering Co.) and
developed by SSPA through several series tests in the towing
tank. The body plan is shown in Fig.13. Considerable
reductions in the viscous resistance as well as increases in the
propulsion efficiency have been reported in comparison with
conventional stern shapes. Components of the resistance
coefficient and measured nominal wake distributions in the
towing tank are availablet18~. Furthermore pressure

OCR for page 175

o. l
a. 3
Y/L
ztL
0.2 0.3 o s
'I ~ \ ~ ~ \
\ \ \ ~ \ \
- \\ \ ~ \ \
Fig.14 Over-view of generated grids of
37K PROBOCON.
ALL
distributions on the eorresponsing double body have been
measured in the wind-tunnelt193.
Numerically generated grids are shown in Fig.14.
Numbers of mesh points in the (5, a, t)-direetions are (54,
25, 32) respectively. Calculation is performed in the domain
of 0.5

x/L
1.0 ~
~ PPNS
, · ~
0.8 ~ ~ EXP
(I -,
I.0~0.6 ~~
y'L
o. Os , ,,
',,f',,
'!,_
o. to
y/L
Fig.17 Simulation of bilge vortex over the stern of
37K PROBOCON by STERN/NS.
Fig.18 Variation of longitudinal component of vorticity
In the wake of 37K PROBOCON.
,18,'
Uo~U l-0 - 0.6 mu: -
Uo ·N
0.8
O. ~
%\
O. ~ · ~
o 1
o. L
o.
\ 120. 150. 180.
·:
120. 150. 180.
·\ 1
·t
30. 60. 90. 120. 150. 180.
~ (a.)
Fig.19 Variation of nominal wake of
37K PROBOCON (x/L=0.975).
5. Conclusion
(1) Two computer program have been developed in the
present study. STERN/PPNS simulates flows over the stern
by the partially parabolic method, and the SI~ERN/NS by the
fully elliptic method. Simulated results are shown to be
nearly identical. This indicates that the effects of stream-wise
dffusion terms are negligible when the flow reversal does not
appear over the stern. They also cross check each others'
numerical scheme. The partially parabolic method requires
only half of the memory storage and reduces CPU time by
20% in comparison with the fully elliptic method.
(2) The capability of programs developed in the present
study is confirmed by sucessfully simulating pressures, skin
frictions and mean velocities over the stern of the both
models. The growth of the viscous layer over the stern is
well-simulated and the secondary motion is also captured,
which is usually observed in the experiments.
(3) There appears to be some deficiency of the k~
model enough to simulate the turbulence fields over the
stern. The standard form of the model usually over-predicts
the turbulent kinetic energy. It is also investigated that the
model cannot properly account for the reduction of the
turbulent kinetic energy near the wall when the viscous layer
becomes thick over the stern.
(4) The streamlines over the stern of the barge type ship
form show uniform distributions, consequently the gradual
girth-wise variations of the boundary layer thickness and the
pressure distribution are noted.
182

OCR for page 175

Table 1. +, At, and 5~', for the governing equation.
rO
W vat
''ote; G = v,{2[(_)2+( dV )2+( 1 dW + V 32]
ax Or r do r
DU dV 2 1 dU dW 2 1 dV dW W 2
+(+ ) +( + ) +( + - ) }
dr ax r do ax r dD dr r
vat = v + v,, v, = C~k2/e
C '=0~09, CD =1.0, C'=1.44, C2=1.92, (rk =1.0, crc=1.3
54
o
1 dp
p fix
~ dU 1 d dV 1 ~ a4W
+ - v' )+ ~rv' )+ ~V' )
ax ax r or ax r 80 ax
1 ap
p fir
d dU 1 d dV 1 d dW w2
+ - v, )+ ~rv, )+ ~v, )+
ax Br r Br Br r R.q Ear r
1 ~ 2ve aW V V
r2 ~(V,W) 2 do vie 2 v, 2
_ 1 UP
pr 88
d v, dU 1 d dV 1 d v, dW
+ - )+ ~v,)+ (- )
ax r dB r dr do r do r dD
1 d 2 d v, W VW
~v,W)+~(V)ve 2 - 2
v, dW vie dV v dV
+ + +
r dr r2 89 r2 do
GCDE
(C IGC26)
(5) Simulated nominal wake fractions show good
accordance with wake measurements. But values of the
outerpart of the wake are over-predicted, while the trends of
the circumferential variation are consistent with wake
measurements. Setting aside the question of the validity of
the turbulence model, it should be studied further how to
allocate enough meshes locally over the propeller plane, as
well as globally over each ship sections.
(6) The capability of the codes to estimate the viscous
resistance is investigated. A coefficients of the viscous
resistance predicted by the present method are shown to be
under-estimated by 10 percent. If further developments on
the turbulence model and numerics are accomplished, the
numerical simulation of the viscous flow over the stern will
be promising for the hull form design.
References
1. L.Larsson, SSPA-IITC Workshop on Boundary
Layers-Proceedings, SSPA-Publication No. 90., (1981).
2.
13.
V.C.Patel, "Some Aspects of Thick Three-Dimensional
Boundary Layers," Proc. 14th ONR Symp., (1982).
3. V.S.Pratap, D.B.Spalding, 'Fluid Flow and Heat
Transfer in Three-Dimensional Duct Flows," Int.
Journal of Heat and Mass Transfer, Vol.19, (1979).
4. A.M.Abdelmeguid, N.C.Markatos, K.Muraoka,
D.B.Spalding, "A Comparison Between the Parabolic
and Partially Parabolic Solution Procedures for Th-ree-
Dimensional Turbulant Flows around Ship's Hull,"
Appl. Math. Modelling, Vol.3, (1979).
N.C.Markatos, M.R.Malin, D.G.Tatcheel, "Computer
Analysis of Three-Dimensional Turbulant Flows around
Ship's Hulls," Proc. Inst. Engrs., London, Vol.194,
(1980).
6. K.Muraoka, "Calculation of Thick Boundary Layer and
Wake of Ships by a Partially Parabolic Method," Proc.
13th ONR Symp., Tokyo, (1980).
7. K.Muraoka, "Calculation of Viscous Flow around Ships
with Parabolic and Partially Parabolic Solution
Procedures," Trans. West Japen Soc. Naomi Arah.,
Vol.63, (1982).
8. C.E.J~nson, L.Larsson, "Ship Flow Calculations Using
the PHOENICS Computer Code," Proc. 2nd Int. Symp.
on Ship Viscous Resistence, SSPA, (1985~.
9. H.C.Raven, M.Hoekstra, "A Parabolized Navier-Stokes
Solution Method for Ship Stern Flow Calculations," 2nd
Int. Symp. on Ship Viscous Resistence, SSPA, (1985).
10. G.D.Tzabiras, "On the Calculation of the 3-D Reynolds
Stress Tensor by Two Algorithm," 2nd Int. Symp. on
Ship Viscous Resistence, SSPA, (1985).
11. H.C.Chen, V.C.Patel, "Calculation of Stern Flows by a
Time Marching Solution of the Partially-Parabolic
Equations," Proc. 16th ONR Symp, (1986~.
S.H.Kang, K.J.Oh, S.B.Lee, "Study on the Stern Design
by Using Viscous Flow Simulations," RIIS Rept.87-092,
Coellege of Eng., Seoul N. University, (1987).
S.H.Kang and K.J.Oh, "Numencal Calculations of
Three-Dimensional Viscous Stern-Flows by Semi-
Elliptic Equations,"26(1),(1989).
14. S.H.Kang, K.J.Oh,"Numencal Calculation of Three-
Dimensional Using Viscous Flow over a Barge Type
Stern by Semi-Elliptic Equations," Seminar on Ship
Hyrodynamics, Seoul,(1988).
15. S.V.Pantankar, Numerical Heat Transfer and Fluid
Flow, McGrow-Hill, (1980).
16. K.J.Oh, "Numerical Study on the Viscous Flows over
the Ship Stem," Ph.D. Thesis, Seoul N. University,
(1989).
17. L.Larsson, "Boundary Layers of Ships. Part 3: An
Experimental Investigation of the Turbulent Boundary
Layer on a Ship Model," SSPA Rept.46, Goteborg,
Sweden, (1974).
18. SSPA, 37K PROBOCON Wake Measurements,
Rept.3202-12, (1987~.
19. S.H.Kang, J.Y.Yoo, B.Y.Shon, S.B.Lee, and S.J.Bail;,
"Experimental Study on Viscous Flows around Ship
Sterns by Using the Hot-Wire Anemometer in the
Wind-Tunnel,"JSNAK, 11(5), (1987).
20. S.H.Kang and B.S.Hyun, "A Simple Estimation of the
Viscous Resistance of Ships by Wake Survey," JSNAK,
Vol.19, No.2, (1982).
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