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OCR for page 687
A Navier-Stokes Solution
of HuN-Ring Wi~hn~ster Interaction
C.-I. Yang (David Taylor Research Center, USA)
P. Hartwich, P. Sundaram, (NASA Langley Research Center, USA)
ABSTRACT
Navier-Stokes simulations of high Reynolds num-
ber flow around an axisymmetric body supported in
a water tunnel were made. The numerical method is
based on a finite-diRerencing high resolution second-
order accurate implicit upwind scheme. Four different
configurations were investigated, these are, 1. bare-
body, 2. body with an operating propeller, 3. body
with a ring wing and 4. body with a ring wing and
an operating propeller. Pressure and velocity compo-
nents near the stern region were obtained computa-
tionally and are shown to compare favorably with the
experimental data. The method correctly predicts the
existence and extent of stern flow separation for the
barebody and the absence of flow separation for the
three other configurations with ring wing and/or pro-
peller.
I. INTRODUCTION
It has been known in marine propulsion technol-
ogy that certain advantages can be achieved by prop-
erly integrating a ring wing arid the propeller. With
a ring wing that accelerates the flow the efficiency of
the propeller remains more or less at a constant level
over a wider range of advance ratios. With a ring
wing that decelerates the flow, the inception of cav-
itation on the propeller can be delayed. In order to
take advantage of the ring wing to obtain the desir-
able benefit, a clear understanding of the role it plays
is important. Thorough water tunnel experiments and
reliable numerical simulations serve as complementary
approaches to gain understanding.
By combining the body, ring wing and propeller,
appropriate configurations can be generated for water
tunnel experiments, and the influence of each individ-
ual component can be isolated. The configurations
studied here are 1. barebody, 2. body with an oper-
ating propeller, 3. body with a ring wing and 4. body
with a ring wing and an operating propeller. Data
collected from the water tunnel tests included the ve-
locity components around the aft erbody and the pres-
sure on the stern. The data show the degree of inter
687
action at the given operating conditions and serve as
benchmarks for evaluating the present numerical sim-
ulations.
One important parameter in hull-propulsor inter-
action is the thrust deduction coefficient t, which signi-
fies the drag augmentation due to the interaction. In
the past, inviscid methods {1,2] have been successful
in computing the coefficient t. The methods become
somewhat inadequate in a situation where the propul-
sor unit is imbedded in the stern boundary layer where
the viscous eRect plays a dominating role. Efforts
have been made to address the problem; in particular,
Falcao de Campos t3] presented an inviscid approach
to calculate the flow on the stern with and without
propulsor based on the Euler equation of motion and
Huang et al t4] developed a numerical technique to
study the interaction between a propeller and unsep-
arated viscous stern boundary layer. To further en-
hance the ability to predict the effect of hull-propulsor
interaction, a Navier-Stokes type viscous analysis is
needed. This is particular true when barebody flow
separation may occur. Previously, Haussling et al t5]
performed extensive numerical simulations of viscous
flow about bodies with appendages using a Navier-
Stokes solver. Here a three-dimensional incompress-
ible Navier-Stokes solver is used to simulate the flow
around a compound propulsor unit on an axisymmet-
ric body supported in a water tunnel with a square
cross-section. The solver is based on a high resolution
second-order accurate implicit upwind scheme 6,7.
The propeller eKect is simulated by imbedding body
forces in a disk located at the propeller plane t8,94. The
experimental data were used to validate the Navier-
Stokes solver.
II. DESCRIPTION OF EXPERIMENTS
The test body is axisymmetric with a length of
139.12 cm and a maximum diameter of 24 cm. Its
radius offsets, y, as a function of axial length, x, are
given in con units by:
OCR for page 688
0.00 _ x < 24.00
y = [12.02 - (12.0-x/2)2]1/2
24.00 < x < 97.81
y = 12
97.81 ~ x ~ 115.84
y = [42.672 _ (x - 97.81)2]1/2 - 30.67
for 115.84 ~ x ~ 129.17
y = (133.0-x) tan25°
for 129.17 ~ x ~ 131.93
y = -0.0214x + 4.554
for 131.93 ~ x ~ 139.12
y = -0.0310x2 + 8.162x - 535.51
The ring wing has a NACA 4415 profile section
with a 5° angle of attack; its chord length is 5.3 cm.
The diameter of ring wing measured at its trailing
edge is 16 cm; its trailing edge is located at 129.7 cm
from the nose of the body. The propeller has four
blades with a diameter of 15.71 cm and was driven
from behind with a Z-drive propelling device. The de-
vice consists of a tapered forebody, a cylindrical mid-
body and an elliptical afterbody. The length of the
device is about 85 cm and the diameter of midbody
is about 10 cm. The propeller plane is located at 135
cm from the nose. The water tunnel in which the ex-
periments were conducted has a square cross-section
with round corners and its dimensions are 90 cm x 90
cm. The model/tunnel blockage ratio is about 5.6 per-
cent. The body was supported from the ceiling of the
water tunnel with two struts located at x = 14.0 cm
and x = 64.0 cm from the nose. A schematic sketch is
shown in Fig. 1.; the propeller drive is not included.
The geometric tolerances are less than 0.5 mm. The
reference length is chosen to be 133.0 cm.
Experiments were conducted at a Reynolds num-
ber of 6 x 106 (based on reference length 133.0 cm).
The flow measurements were carried out on lines lying
on the horizontal plane and perpendicular to the cen-
ter line. The pressures on the body surface were mea-
sured by means of transducers and all signals could
be reproduced satisfactorily within 1.0 percent. For
the flow measurements, the standard deviation of all
signals varied between 0.1 and 2.0 percent.
III. DESCRIPTION OF NUMERICAL
PROCEDURE
Using Chorin's artificial compressibility formula-
tion, the incompressible Navier-Stokes equations are
written in conservation law form for three-dimensional
flow as t10]
Q' + (E -Edit + (F*-Fail + (G*-Gv~z = 0 (1)
where the dependent variable vector
Q = (p,u,v,w)T
represents the pressure and velocity components in a
Cartesian coordinate system (x, y, z). The inviscid flux
vectorsE*,F*,G* and the viscous shear flux vectors
Ev, Fv, Gv are given by
E* = (be, us + p, up, uw )T
F* = (3v, uv, v2 + p, VW)T
G = (~0, UW, VW, W + P)
Ev = Re I(O,T£~,~r~Y,TXZ)T
Fv = Re-1 (O. TYX, TRY, Tyz)T
GV = Re (O,TZ~,TZY,TZZ)T
(2)
The coordinates x, ty, z are scaled with an appropriate
characteristic length scale L. The velocity components
a, v, w are nondimensionalized with respect to the free
stream velocity TOO, while the normalized pressure is
defined as p = (P-poO)/pV2 The kinematic vis-
cosity v is assumed to be constant, and the Reynolds
number is defined as Re = v:. The artificial com-
pressibility parameter ~ monitors the error associated
with the addition of the unsteady pressure term ~ in
the continuity equation which is needed for coupling
the mass and momentum equations in order to make
the system hyperbolic.
Equations (1) can be transferred to a curvilinear,
body-fitted coordinates system (¢, ~ and 71 ~ through
a coordinate transformation of the form
¢ = ¢(x, y, z), ~ = ((x, y, z) and 71 = 71(X, Y. Z)
Eq. (1) becomes
(C)/J)` + (E-Ev)` + (F-FV)( + (G-GV)11 = 0 (3)
with
and
, where
T(E, F. G) = [T] (E*, F*, G*)-t
r7~
(E,F,G) = [T] (Ev,Fv,Gv)
~ ¢ ¢
[T]= (x by (z
fix by Hz
and the Jacobian of the coordinate transformation is
given by
x; y; z;
J-1 = det x~ ye zip
X77 yl7 Z77_
The Cartesian derivatives of the shear fluxes are ob-
tained by expanding them using chain rule expansions
in the ¢, it, and 71 directions.
Defining computational cells with their centroids
at I = ,`~ ( ~ is ¢, (,or 71) and their cell interfaces
at I ~ 1/2, the backward Euler time diFerencing of the
three-dimensional conservation form is
688
OCR for page 689
C] = _ [~(En+1 _ En+l) + ~(Fn+1 _ Fn+l
+ ~ (Gn+1 _ Gn+l )]
(3)
where At is the time step, i\Qn = Qn+~ In and
i\~( ) = t( )~+~/2 - ( )~-1/2~/~8 · Superscript denotes
the time level at which the variables are evaluated.
Linearizing Eq. (3) about time level n, we obtain
~ ~ + ( {lEn JEW )~ In dGn
(0G" GO )A Amen
In _ Fan) + /(-FUn) + i\~(Gn _ Gn)]
(4)
where I is the identity matrix.
The left hand side is the implicit part and the
right hand side is the explicit part of the formulation.
The explicit part is the spatial derivatives in Eq. 2
evaluated at the known time level n; its value di-
minishes as the steady state solution is approached.
Hence, it is also called the residual. The L2 norm of
the residual is often used as a measure of convergence
of a solution. Letting the flux Jacobians A, B arid C
be defined as follow
A i3E" B OF" C JIG"
- Em' - Ad' - 'am'
discretize the inviscid and viscous fluxes according to
upwind differencing scheme and central differencing
scheme respectively in (, ~ and r1 coordinate direc-
tion independently and then assemble them together.
Equation (4) becomes
tt /\~J)
-(A- + X)i+ ~ i\i+ 2 ~ (A+ + X)i- 2 ~i- 2
-( B- + Y jj+ ~ i\ j+ ~ + ( B+ + Y jj _ 1 i\j _
-(C + Z)k+ 1 ink+ ~ + (C + Z)k-2 ink-1 ~ AQ
=-RES(Qn)
where i,j, and k are spatial indices associated with
the (, ~7 and ~ coordinate direction. At,B~ and
Cal are flux matrices split from the flux Jacobians
A, B and C according to the signs of their eigenval-
ues. The residual RES(Qn) is evaluated with a TVD
technique together with Roe's t11] flux-difference split-
ting scheme, the discretization is third-order accurate.
Conventional second-order central differencing is ap-
plied to obtain the viscous flux matrices X, Y and
Z. Equation(5) is solved by an implicit hybrid algo-
rithm, where a symmetric planar Gauss-Seidel relax
ation is used in the streamwise direction ~ in combina-
tion with approximate factorization in the remaining
two coordinate directions ~ and y. It is used to avoid
the i\t3 spatial splitting error incurred in fully three-
dimensional approximate factorization methods. This
scheme is unconditionally stable for linear systems and
offers the advantage of being completely vectorizable
like a conventional three-dimensional approximate fac-
torization algorithm. As a result, Eq. (5) becomes
tM-(B- + Y jj+ 1 /'\j+ 1 + (B+ + Y jj_ 1 i\j_ ~ ]~Q
=-RES¢Qn Qn+i ~
[M-(C + Z)k+t Ok+ + (C + Z)k-2 k-2] Q
=Mi\Q
Qn+1 = Qn + Akin
with
IM = [,\~ J + (A + X)i+ ~ + (A+ + X)i_ ~ ~
(6)
and the residual on the RHS indicates the nonlinear
updating of the residual by using Qn+i whenever it be-
comes availabe while sweeping in the ~ direction back
and forth through the computational domain.
For laminar flow computations the coefficient of
molecular viscosity ,u = ,`~ is obtained from Suther-
land's law. Turbulence is simulated using the Baldwin-
Lomax algebraic turbulence modelt12~. For turbulent
flow laminar viscosity coefficients are replaced by
~ = p1 + lit
The turbulent viscosity coefficient lit iS computed by
using the isotropic, two-layer Cebeci type algebraic
eddy-viscosity model as reported by Baldwin-Lomax.
Modifications proposed by Degani and Schiff t134 and
Hartwich and Hull t14] were implemented.
IV. DESCRIPTION OF BODY FORCE
PROPELLER MODEL
The principle of the body force model is to in-
troduce the body force terms into the Navier-Stokes
equation to include the effects of the propeller. The es-
sential parameters that define the propeller effects are
the thrust coefficient CT, the torque coefficient CQ' the
advance coefficient J and the radial circulation distri-
bution G(r). The same parameters were used to define
the bud force for the propeller model. The thrust and
torque coefficients are defined as follow:
T C = Q
~ pV2 Ii D2 ~ Q ~ pV2 It D3
where T and Q are thrust and torque, respectively. D
is the diameter of the propeller. The axial and circum-
ferential body force per unit volume are obtained from
689
OCR for page 690
the following equations:
f be- CTR2 G(r)
fRh ~ ~
CQRpG(r)
2r/\X iR P G(r)rdr
where fob and fbe are the body forces per unit
volume in the axial and circumferential directions, re-
spectively, Rh and Rp are the radii of propeller hub
and blade tip, respectively, and /\X is the thickness
of tl~e disk. The computed body forces are then incor-
porated into the right hand side of Eq. 5 and form a
part of the residual. Only a slight modification to the
flow solver is needed to accommodate the body force
type propeller model and there is no need for special
"ridding.
fbe =
In reality, the blade circulation distribution G(r)
depends upon the inflow at the propeller plane which
in turn is influenced by the blade circulation. This mu
tual dependency implies that the body forces f be and
f be which are functions of G(r) should be obtained
by an iterative procedure. To complete this procedure,
knowledge of propeller-induced axial and tangential
velocities, pa and us is needed. A propeller program
based on the vortex-lattice lifting-surface method de
veloped by Greeley and Kerwin t15] can be used for
this purpose. The iterative procedure can be described
as follow:
1. Calculate the nominal inflow with the Navier
Stokes solver.
2. Obtain the circulation distribution G(r), the
induced-velocities Ha and us, the thrust and torque
coefficients CT and CQ by using the calculated nominal
inflow as input to the propeller program t15~.
3. Compute the body forces fbX and fob
by using the calculated circulation distribution G(r),
thrust and torque coefficients CT and CQ.
4. Obtain the total velocities at the propeller
plane by using the Navier-Stokes solver with the infor
mation obtained in step 3.
5. Compute the effective wake by subtracting
the propeller-induced velocities obtained in step 2 from
the total velocities obtained in step 4.
6. Obtain an updated circulation distribution,
propeller-induced velocities, and thrust and torque
coefficients by using the newly computed effective wake
as input to the propeller program.
7. Repeat the process from step 1 to step 6 until
the total velocities, the body forces and the propeller
induced velocities are unchanged.
It has been shown that this procedure converged
after two iterations t84. For the purpose of illustration,
the results presented in following are obtained by usingOn the solid boundaries such as surfaces of tunnel
measured thrust and torque without any iteration. Inwall, ring wing, body and propeller drive the no-slip
spite of its simplicity, it was able to predict the flowcondition is applied, in addition, the normal gradient
pattern around the propeller disk and the stern region
rather accurately [8,94.
V. GRID GENERATION
Based on the configuration shown in Fig. 1, which
models the experimental setup described in Section II
above, a 180° sector of the tunnel needs to be mod-
elled in order to resolve the effect on flow due to the
supporting struts. This requires a large amount of
grid points and extensive computational resources. Af-
ter one computation for the barebody configuration,
it was found that the struts produced an influence
around and directly behind them with no significant
eRect on the horizontal plane on which measurements
were made. Therefore, the struts were eliminated al-
lowing the numerical simulations to be performed ac-
curately in a 90° sector of the tunnel. Also included,
due to its proximity to the stern, is the propeller drive.
To include the ring wing geometry, two block C-
O type grids were generated. The grid points were
matched at the branch cut that separated the two
blocks. Computations were performed on a CRAY-
YMP machine which has eight processors. Compu-
tational efforts on the two blocks can be performed
on two processors simultaneously. For the purpose of
synchronization between the processors, it is more ef-
ficient if the number of computations is the same for
each block. For this reason, each block has the same
number of grid points. The grid system was generated
by a transfinite interpolation technique. Several grid
systems with different number of grid and distributions
had been generated and were used for computations on
barebody configuration in order to investigate the re-
lationship between the convergence and grid density
and distribution. The grid systems examined include
1. a two block 25 x 49 x 110 (r,B,x) grids, 2. a
two block 25 x 25 x 90 grids and 3. a two block
25 x 13 x 98 grids. Grid points were clustered near
the boundaries such as tunnel wall and body and ring
wing surfaces where the viscous eject dominates. The
minimum spacing normal to the body surface for the
three grid systems mentioned above is 5.0 x 10-4 . The
differences between the solutions based on the grid sys-
tem 1 and 3 are about two percent which is within the
reported experimental accuracy. The results presented
below are based on a two block 25 x 13 x 98 (r, 8, x)
coarse grid system. Convergence is achieved when the
L2 normal of all residuals is reduced by three order of
magnitude with CFL=10. Computational CPU time
is 40 ,usec per node per iteration. For each compu-
tation, over 500 iterations were carried out to ensure
convergence.
VI. BOUNDARY CONDITIONS
690
OCR for page 691
of the pressure is assumed to be zero. Freestream con-
dition is applied as inflow condition. Zero-order ex-
trapolation is used to obtain the outflow conditions.
During the computation process, the variables on
the branch cut that separates the two blocks are not
computated. At the end of each iteration, these vari-
ables are updated by averaging the values at the adja-
cent grid points from each block. The averaging Dro-
cess is linear. The updated values are then used as
boundary conditions for both blocks for the next iter-
ation.
VII. RESULTS Ai~D DISCUSSIONS
The four configurations that were investigated are
1. flow over the barebody, 2. flow over the body with
an operating propeller, 3. flow over the body with a
ring wing and 4. flow over the body with a ring wing
and an operating propeller. Computational results are
presented in the form of velocity profiles and pressure
contours in the stern region; they are compared with
available experimental data. In addition, to facilitate
the flow visualization and discussion, computed par-
ticle trace are also included. The length scale was
normalized with reference length L = 133cm.
Case 1. Flow over the Barebc~dy
In this test case, the configuration is simple but
the flow is rather interesting. At a Reynolds num-
ber of 6.0 x 106, flow separation was observed in the
stern region. Figure 2 shows the velocity vectors in the
stern region from both computation and experiment.
The correlation is good except at the axial location
X/L = 0.89 where the experiment shows a some-
what fuller profile near the body. Figure 3 shows the
particle traces and clearly depicts the separation bub-
ble. The predicted separation location is about 1.5 cm
(1.13 percent of body length) ahead of where it was
observed experimentally. Figure 3 also indicates that
the size of the bubble is predicted correctly. Figure 4
shows the predicted and measured pressure distribu-
tions on the surface of the stern. The surface pressure
begins to recover as the flow passes the shoulder of the
afterbody at X/l = 0.78 . The recovery levels off
where flow separation takes place. Figure 5 shows the
pressure contours in the stern region.
Case 2. Flow over the Body with
an Operating Propeller
In this test case, the propeller was operating at
J = 0.47 and V = 4m/~; CT and CQ were measured as
2.052 and 0.247, respectively. To apply the body force
propeller model, the circulation G(r) was assumed to
be distributed over the disk according to:
G(r) = r(1-r j]/2
with r = (Y-Yhub)/(Rp-Chub)- The propeller was
located at X/L= 1.015.
Figure 6 shows the comparison of predicted and
measured velocity vectors in the stern region. The pre-
diction confirms that the separation bubble is removed
due to the propeller suction. Experimental data in-
dicate stronger downward radial velocities at all four
stations. In addition, at axial locations X/L = 0.89
and X/L = 0.93 the predicted velocity profiles are less
full than those that were measured near the body. It is
found from computations that the predictions behind
the propeller are sensitive to the circulation distribu-
tion over the propeller disk and the axial locations at
which the predictions are made. Figure 7 shows the
particle traces, indicating that flow contraction takes
place immediately in front of and behind the propeller
disk. Figure 8 shows the predicted and measured pres-
sure distributions on the surface of the stern. The dif-
ference between the pressures presented in Fig. 8 and
Fig. 4 gives the amount of the pressure drag on the
barebody due to propeller action. The added drag con-
stitutes the major part of the thrust deduction fraction
Ail. The agreement between prediction and measure-
ment is very good. This is an indication that thrust
deduction (1-t) can be predicted correctly with this
numerical method. Figure 9 shows the pressure con-
tours in the stern region; the pressure jump across the
propeller plane is clear.
Case 3. Flow over the Body with a Ring Wing
An intended function of the ring wing was to ac-
celerate the flow and produce thrust. It was also ex-
pected that the flow separation over the stern would
be removed once the ring wing was in place. Figure 10
shows the velocity vectors in the stern region from both
computation and experiment. In comparison with Fig.
2, it can be seen that the ring wing achieved its func-
tion in accelerating the flow and removing the stern
separation. The predicted and the measured velocity
vectors are in good agreement. Due to lack of details,
the experimental data failed to resolve the wake struc-
ture behind the ring wing. Figure 11 shows the parti-
cle traces and there is no detectable flow separation on
the surface of either the body or the ring wing. Figure
12 shows the pressure on the stern surface. Note that
the predicted surface pressure immediately upstream
of the ring wing's leading edge at X/L = 0.93 is some-
what higher than that measured. Figure 13 shows the
pressure contours in the stern region. Clearly shown
are the locations of the pressure and suction peaks on
the ring wing surface.
691
OCR for page 692
Case 4. Flow over the Body with a Ring Wing
and an Operating Propeller
In this test case, the propeller was operating at
J = 0.47 and V = 4m/~; CT and CQ were measured
as 2.081 and 0.250, respectively, and are about 2 per-
cent higher than those shown in Case 2. Figure 14
shows the comparison of predicted and measured ve-
locity vectors in the stern region. The agreement is
quite good. Figure 15 shows the particle traces. It
can be seen that flow is accelerated as it passes the
ring wing and is contracted as it passes the propeller
disk. Figure 16 shows the pressure on the stern sur-
face. The discrepancy between the predicted and mea-
sured pressure immediately upstream of the ring wing
at X/l = 0.93 as discussed in Case 3 is present here
also. The difference between the pressures presented
in Fig. 16 and Fig. 12 gives the amount of the pres-
sure drag on the body due to the propeller action, the
agreement between the prediction and measurement is
very good, despite the discrepancy mentioned above.
Figure 17 shows the pressure contours in the stern re-
gion.
VIII. COi~CLUSIOt~
A numerical method based on a finite-differencing
high resolution second-order accurate implicit upwind
scheme was used to discretize the three-dimensional
Navier-Stokes equations. Simulations of flow over an
axisymmetric body with a compound propulsor sup-
ported in a square water tunnel were performed. A
body force type propeller model was used to simulate
the propeller action. Results indicate that this nu-
merical method is eFective and accurate. Observed
flow phenomena such as separation, acceleration and
contraction were realistically predicted. The pressure
drags due to propeller action were computed correctly.
A ring wing may affect the circulation distribution on
the propeller, depending on its proximity to the pro-
peller. In order to improve the predictions further,
an iterative procedure described in section IV will be
explored in a future study.
ACKi\:OWLEDGEMEN:T
This study was supported by The Office of Naval
Technology under David Taylor Research Center Work
Unit 1-1506-060-40. The computing time of CRAY-
YMP was provided generously by NASA Ames Numer-
ical Aerodynamic Simulation (NAS) Program. The ex-
perimental data presented in this paper are of foreign
origin, obtained via informal correspondence.
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pp 173-196.
14. Hartwich, P-M., Hall R.M., "Navier-Stokes So-
lutions for Vortical Flows over a Tangent-Ogive
Cylinder", AIAA Paper No. 89-0337, Jan. 1989
15 Greeley, D.S., Kerwin, J.E., "Numerical Meth-
ods for Propeller Design and Analysis in Steady
Flow", Trans. SNAME, Vol. 90, 1982, pp. 415-
453.
,~'
~;;:
Figure 1. Schematic Sketch of Test Configuration
n~ARF,P'OnY | ------> Experimental dots l
l I ~ Computational results |
o
Figure 2. Velocity Vectors in Stern Region for Case 1
PRPT!CLE FARCES
l
{~.45 1.00
1 1 1
0.80 o Bit, 0.90
X/L
Figure 3. Particle Traces for Case 1
o.eo
0.
0.60
0.50
0.40
0.30
0.20 _
0.10 _
0.00 _
-0.10 _
_ ._
-0.20 _
-0.30 _
-0.40
-0.50 _
-0.60 _
-0.70 _
-0.80
0.60
' l · i , , i
BAREBODY
| ~ Experimental data |
| Computational results |
0.65 0.70 0.75 0.80 o.ss 0.90 0.95 1.00
X/L
Figure 4. Pressure Distribution on Stern Surface
for Case 1
HAREBODY \
v . _
0.25 _
v.2 _
0. 15
0.1 _
\,
At,
L . 5.95.5511.5
00. 85 B.7 e.7s 0.B B.B5 B.9 B.9S 1 1. 35
,Y/L
Figure 5. Pressure contours in Stern Region
for Case 1
693
OCR for page 694
n~Or)Y WITH PROPF,I.I.F.R ~> Experimental data
:e Computational results
8
o
o
0.3r
BODY W1TI! PROPELI,ER
0.25
-
\ I.''
I'
\
0.2
`< 0.15
\\
.
Figure 6. Velocity Vectors in Stern Region for Case 2
HODY WITH PROPF2l,f,F,R
PARTICLE TRACES
_
1 '
().85 0.90
Xll,
Figure 7. Particle Traces for Case 2
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.1 _
_
D.05 _
O. _
it ~,~\'l)
5 0.7 0.75 0.0 0.85
XlL
0.9 0.95
Figure 9. Pressure contours in Stern Region
for Case 2
~ > Experimental data
BODY WITII mNG WING ~ Computational results
~ ==
~ =
\
Figure 10. Velocity Vectors in Stern Region
for Case 3
BODY Wll'll PROPEI,LER
0 Experimental data
Computational results
\
\ ,?
1 _
1 l l l l
0.60 0.65 0.70 0.7S 0.80
X/L
0.85 0.90 0.95 1.00
Figure 8. Pressure Distribution on Stern Surface
for Case 2
Figure 11. Particle Traces for Case 3
694
OCR for page 695
~ptulTlCLE ~ CE I
07OT r~O,,~v''Hlu~ ~0 , ~ __= ~
I;_ :: _~) ,;~
_o 20 ~c: ~
070 '7 X/t ion On Stern surface p rtiCle Traces for CaS
12 pressure Distribut Figure 15
-----| ~ ~ ~~ '=
2 - j\ /: 0~"~-~\ ~
= '1' ~ ~ N~\ 1 / ~
~ ~ ·~N ~m
pegi°n re DiSuibu0~° ~ °0
3 pressure contours It Is; Fig for case 4
Figure 1 for Case 3 031-- ~'~YWlylln\~ I' \~]p~ln l /
; '~ ~ ': ~
OCR for page 696
DISCUSSION
Philippe Genoux
Bassin d'Essais des Carenes, France
It is well known that Navier Stokes solvers may have difficulties at
high Reynolds numbers. How does your code behave at full scale
Reynolds numbers?
AUTHORS' REPLY
The differencing scheme presented in this paper is based on
hyperbolic formulation, assuming that at high Reynolds numbers the
behavior of Navier-Stokes equations becomes hyperbolic-like. At
present, the solver has been used to simulate flow at Reynolds
number 15 X 106. The agreement between computation and
experiment is good. No convergence problem was experienced.
DISCUSSION
Fred Stern
The University of Iowa, USA
The results presented display some interesting features for a dueled
propulsor; however, it would be of greater interest if the authors
would include the propulsor-hull interaction, i.e., male full use of
the viscous-flow approach to propulsor-hull interaction and
demonstrated in Reference [8].
AUTHORS' REPLY
We appreciate Prof. Stern's comments and suggestions. We are using
the method described in Reference [8] for our future work.
696
Representative terms from entire chapter:
stern region
