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OCR for page 839
A New Propeller Design Method for the POD Propulsion
System
Ching-Yeh Hsin1 Shean-Kwang Chou2 Wei-Chung Chen2
([Department of System Engineering and Naval Architecture
National Taiwan Ocean University Keeling Taiwan
~ ~ 7 ~
2United Ship Design and Development Center Keelung' Taiwan)
ABSTRACT
This paper presents a new design method for propellers
used on the POD propulsion system. Two major
features of this method are first the propeller geometry
is designed on a conical coordinate system instead of
the traditionally used cylindrical coordinate system.
Secondly, the inflow of the propeller is simulated by a
coupled viscous/potential flow calculation including
both the POD body and the propeller for the purpose of
designing a propeller in a more realistic condition. In
order to verify the designs, both the potential flow
method and the coupled viscous/potential flow method
are developed to calculate the flow around a POD
propulsor. The numerical calculations for the flow
around a POD propulsor are presented in the paper, and
the computational results are compared to the
experimental data. A design case is also included in
the paper, and the designed geometries are shown to be
different from the traditional method.
INTRODUCTION
Recently, the POD propulsion systems have drawn
more attention than ever, and they are not only used on
cruise ships, but also on semi-submersible heavy lift
vessels, chemical tankers and other types of ships.
The major advantage of a POD propulsion system is
that the propeller inflow is more uniform than that of a
conventional propulsion system, and this can often
improve the propeller blade cavitations and powering
performance. Since the hydrodynamic characteristics
of the POD propulsion system and the traditional
propulsion system are different, it is questionable that
the traditional propeller design method is appropriate
for the POD propulsion system. The object of this
paper is to develop a new propeller design method that
can design propellers used on a POD propulsion
system.
The researches in marine propeller designs are
mainly in two topics, one is to find the blade geometry
1
that produces the desired loading, and the other one is
to find a new sectional geometry that can reduce the
cavitaion. Recently, researchers also begin to use
various optimization methods to find out better loading
distributions (Cho, 1998) (Jang, 2001~. In the
presented method, we would like to emphasize on the
calculation of the inflow, and a different definition of
the blade geometry. The reason to study the inflow is
to understand the performance of the propeller on a
POD propulsor with yaw angles. This is similar to
study the effective inflow problem of a conventional
propulsor. A coupled viscous/potential flow method is
thus used for the computations. The propeller of a
POD propulsor is often installed on two ends of the
nacelle, and it is equivalent to having a conical hub.
The traditionally used cylindrical coordinate system
may no longer be appropriate for this kind of
configuration. Therefore, we will investigate the use
of a conical coordinate system to define the blade
geometry.
In order to understand the performance of
POD propulsors, computational methods are needed to
calculate the flow field. Ghassemi (Ghassemi, 1999)
has developed a potential flow method to calculate the
flow around POD propulsors, and Sanchez-Caja
(Sanchez-Caja, 1999) has applied a viscous flow code
to the calculation of flow field around POD propulsors.
In this paper, we will present two different
computational methods for the computations of flow
around POD propulsors.
BLADE GEOMETRY DEFINITION
Traditionally, the propeller geometry is defined on the
cylindrical coordinate system. The propeller blade
geometry can be considered as a group of
two-dimensional sections at different propeller radii
connected by a centerline, and this centerline is defined
by the rake and skew distributions. When designing a
propeller, an optimized radial loading distribution is
first obtained, and the blade sectional geometry at each
OCR for page 840
conical section
- ----------------- cylindrical section
Figure 1: Blade sections defined on the cylindrical
coordinate system and on the conical coordinate system
3.5
3.0
2.5
2.0
1 .5
1 .0
0.5
~ ~ ~ _
- era
At/
0
~ \
\
\
~ \
At\
At\
Hi\
cylindrical section ~~\
conical sect ion gt
00 ~
0.4 0.5 0.6 0.7 0.8 0.9 1.0
r/R
Figure 2: The calculated circulation distributions for
blade sections defined on the cylindrical coordinate
system and on the conical coordinate system with the
same blade outline.
radius producing the corresponding loading will then be
found. For example, the widely used blade geometry
design program MIT-PBD-10 (Greeley, 1982) is a
lifting surface method for determining the pitch and
camber distributions by giving a prescribed load
distribution. In most design programs like
MIT-PBD-10, the blade sections are aligned with
constant-radius lines, and each section is assumed to be
on the same stream surface. Therefore, the
hydrodynamic characteristics of the two-dimensional
section are assumed to be valid. Since the stream
surfaces are assumed aligned with the cylindrical
surfaces, the thickness and the camber of a section are
defined on the cylindrical surface with the same radius.
The pitch angle (angle of attack) is defined in the same
way in the cylindrical coordinate system. However,
for propellers used on POD propulsion systems, they
are often installed at two ends of nacelles, and the
geometry of that part is often like a cone. The inflow
of the propeller thus aligns with the cone shape
geometry. This is also true for propellers used on any
axis-symmetric bodies such as submarines or torpedoes.
That is, if we still define blade geometries on the
cylindrical coordinate system, the assumption that each
section on the same stream surface can no longer be
true.
It has been observed in the NTOU (National
Taiwan Ocean University) cavitation tunnel that the
root cavitation happened for a well-designed propeller
used on an axis-symmetrical body. A possible reason
for the root cavitation is that the propeller blade
geometry is designed on the cylindrical coordinate
system, and the manufacturer extrapolated the blade
sections near the root to fit the conical body. This may
suggest that a propeller design on the cylindrical
coordinate system may not be a good design for conical
bodies. Therefore, in order to remedy this, and to
more accurately design blade geometry for propellers
on conical bodies, the propeller blade geometry defined
on the conical coordinate system is used in the
presented design method. When the blade geometry is
defined on the conical coordinate system, the blade
sections are aligned with the conical surfaces, not the
cylindrical surfaces. Therefore, blade sections can
still be assumed to be on the same stream surface. The
propeller geometry defined on a conical coordinate
system has been discussed and used for several years,
for example, the design method described by Kerwin et.
al. (Kerwin, 19941.
To demonstrate the difference between blade
geometries defined on a cylindrical coordinate system
and a conical coordinate system, we first calculate the
circulation distribution of a propeller with a conical hub.
With the same blade outline, Figure 1 shows the blade
sections defined on two different coordinate systems.
Figure 2 shows that the calculated circulation
distributions are different when the blade sections
defined on a cylindrical coordinate system and on a
conical coordinate system with the same blade outline,
and the difference is not negligible. Notice that when
the blade geometry defined on the conical coordinate
system, the centerline is vertical to the conical surface;
therefore, there is a negative rake distribution for the
above case (Figure 3~. Another convenience that the
geometry defined on the conical coordinate system has
is that the same blade geometry produces the same
loading distribution even the cone angles are different.
2
OCR for page 841
zero rake center line
~ ~ i~
~ ~~\
3.5
3.0
2.5 l
2.0
1.5
1.0 ~
0.5
Figure 3: In the conical coordinate system, the
centerline is vertical to the conical surface, therefore,
there is a negative rake distribution for a propeller with
zero rake in the cylindrical coordinate system. cone angles.
Figure 4 shows that the calculated circulation
distributions for propellers with the same blade
geometry but different conical hubs (different cone
angles). The computational results show almost no
difference between these propellers.
EFFECTIVE INFLOW CALCULATION
Currently, the inflow of a propeller is assumed to be the
circumferential mean of ship wake velocities in
propeller design procedures. The propeller-hull
interaction is ignored, or considered with an empirical
correction. In the presented method, the effective
inflow is used as the propeller inflow, and it is
calculated by a coupled viscous/potential flow scheme
(Hsin, 2000~. This method follows the method
proposed by Kerwin (Kerwin, 1994), and the
"equivalent body force" concept is adopted.
In this coupled viscous/potential flow
calculation, the propeller flow field is solved by using
either the propeller steady flow analysis program
MIT-PSF-2, or the propeller blade design program
MIT-PBD-10 depending on solving the analysis
problem or the design problem. Both programs are
lifting surface vortex lattice method developed at MIT.
The viscous flow around a ship is solved by a viscous
flow RANS code "UVW" developed at United Ship
Design and Development Center, and a brief
description of this code is shown in Appendix A. In
the following descriptions, we will use "U" to represent
the propeller flow solved by potential flow calculations,
and "V" ship flow solved by viscous flow calculations.
, v
'it.
cone angle= 0 ~
~ cone angle=5 At,
-e cone angle=10 |
00 ,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,I
0.4 0.5 0.6 0.7 0.8 o.s 1.0
OR
Figure 4: Calculated circulation distributions for
propellers with the same blade geometry but different
Considering the propeller inflow, U. it can be
U=Ue+Up (la)
Up = Ui + Ui (lb)
where Ue is the effective inflow, Upis the propeller
induced velocity, and can be divided into Ui, the
circumferential mean of the propeller induced velocity,
and U., the fluctuating part of the propeller induced
velocity. We then consider the ship flow, it also
consists of two parts, which are the effective inflow and
the propeller induced velocity:
V = Ve + Up (2)
Kerwin (Kerwin, 1994) has suggested that the coupling
of viscous and potential flow calculations can be carried
out consistently only under the assumption that both the
ship flow and the propeller flow at the propeller plane
are axis-symmetric. This assumption is reasonable
since that the induced velocity upstream of a propeller
is usually dominated by the circumferential mean value.
Under this assumption, the propeller induced velocity
component in ship flow is also assumed to be the mean
value. The total propeller inflow in ship flow can then
be written as:
V = V = Ve + Up = Ve + Ui ~ ~
3
OCR for page 842
where Ve is the circumferential mean value of the
effective inflow in ship flow. Similarly, the total
propeller inflow becomes:
U =Ue +UP =Ue +Ui (4)
It should be remembered that it is not possible
to separate the effective inflow and the propeller
induced velocity in the ship flow, and therefore a
method to identify each one has to be developed. First,
to couple the solutions from two different flow solvers,
the flow field at the propeller plane has to be the same
Therefore,
V=U (5)
From equation (3) and (4), the following equation can
be then derived:
(2)
U =Ue +Ui =Ve +Ui V (5)
and the effective inflow can be calculated as
Ve = Ue = V Ui
In equation (7), the effective inflow, Ve, is defined as
the difference between the ship flow and the
circumferential mean of the propeller induced velocity.
It thus explains how the effective inflow can be
calculated.
In the solution of the ship flow, the body-force
terms in the Navier-Stokes equation simulate the
propeller effect, and the body forces are calculated by
the potential flow. In propeller analysis program
MIT-PSF-2, the Kutta-Joukowski's law and the
Lagally's theorem are used to calculate concentrated
forces on each vortex element. For example, forces on
each element,f, resulted from the vortex lattice are
calculated as
f = puny (8)
where u is the local velocity, and ~ is the vortex
strength. However, according to the above description,
the local velocity has to be taken as the circumferential
mean. The formula can then be modified as:
f=paxy (9)
The total force on the propeller is the summation of
forces on each element. These forces are then
transferred into the body-force terms in the
Navier-Stokes equation for the ship flow calculations.
The procedure to solve the propeller-hull
interaction is described as follows:
(1) The viscous flow around a ship hull without the
propeller in operation is first solved by the RANS
code UVW;
The circumferential mean value of the flow
velocity at the propeller plane is then extracted
from the solution of UVW, and then used as the
inflow of propeller flow calculations;
For solving the analysis problem, MIT-PSF-2 is
used to calculate the circumferential mean
induced velocities and propeller forces. These
forces are then transferred to RANS code UVW;
(4) The flow around the ship with body forces is
calculated by UVW, and the flow at the propeller
plane is extracted and taken circumferential mean
again, and this is the total velocity V;
The circumferential mean propeller induced
velocity calculated in last iteration will be used as
Ui in equation (7), and it is deducted from the
total velocity calculated in step (4) to become the
effective inflow.
(6) Repeating steps (3) to (5) until the convergence of
the effective inflow is reached.
In the case of simulating a self-propulsion test, step (3)
should be modified. MIT-PSF-2 is still used to
calculate the circumferential mean of induced velocities
and the propeller forces. However, the rotational
speed of the propeller is adjusted to obtain the
self-propulsion point. For solving the design problem,
MIT-PBD-10 is used instead of MIT-PSF-2. The same
procedure can be used to calculate the interaction
between a POD body and a propeller.
Recent computational results have shown that
this coupled viscous/potential flow calculation has
successfully simulated the hull-propeller interaction.
Figure 5 and Figure 6 show the comparison of the
computational results and the experimental data for a
velocity cut at the propeller plane without and with the
propeller in operation. This is the case that the
operating condition of the propeller is known (analysis
problem), and a good agreement can be seen from these
figures. As described above, with some modifications,
this coupled viscous/potential flow method can also
used to simulate a self-propulsion test. Table 1 shows
the comparison of the computational results and the
experimental data of a self-propulsion test. The
simulation of a self-propulsion test is relatively difficult,
4
OCR for page 843
and the computational errors of this case are
satisfactory. Please refer to Chou et. al. (Chou, 2001)
(Chou, 2002) for the detailed configurations of ships
and propellers used in this computation.
1
}0.51
-
~ O
~0 - ~ ~
O ~ ~ O
O\ /
°q~°°
~;~
-05 1 1 1 1111 1 1111 1 1111 1 1111 1 1111 1 11
0.03-0.02-0.01 0 0.01 0.02 0.03
yL
Figure 5: The computational results (lines) and the
experimental data (symbols) of the velocities at the
propeller plane of a bare hull.
1
0.5 t
~~ O ~
- w/U~USDDC-UVW
~ ·~1 ~
-0 5
0.03-0.02-0.01 0 0.01 0.02 0.03
y/L
Figure 6: The computational results (lines) and the
experimental data (symbols) of the velocities at the
propeller plane when propeller in operation.
Table 1: The measured and the calculated forces on the
ship hull and on the propeller of a self-propulsion test
J
Measured 0.852
Calculated 0.863
Error 1.20% I
KT
0.175
0.183
4.39%
KQ
0.0261
0.0281
7.66%
Figure 7: The computer depiction of a POD
propulsor and its calculated pressure distribution.
NUMERICAL COMPUTATIONS
In this section, we will first demonstrate the flow
around a POD propulsor calculated by a potential flow
boundary element method, and then the same flow
calculated by a coupled viscous/potential flow code.
Results from both computational methods will be
compared to the experimental data. Finally, a design
case is shown.
A POD with a pulling propeller is used for the
numerical validation. The experimental data is
obtained from an experiment conducted by Szantyr
(Szantyr, 2002), and the computer depiction of this
POD propulsor is shown in Figure 7. The length of
this POD propulsor model is 16.22 inches (0.412m),
and the maximum diameter of the nacelle is 2.91 inches
(0.074m). The vertical length of the strut (from top to
the centerline of the nacelle) is 5.51 inches (0.14m), and
it has a constant chord length 4.29 inches (0.109m).
The strut has an elliptical section, with a maximum
thickness 1.85 inches. The geometric parameters of
the propeller are listed as follows:
Propeller type:
Diameter:
Hub diameter:
Number of blades:
Area ratio:
Pitch ratio, P/D:
Gawn-Burrill series
7.166 in. (0.182m)
2.52 in. (0.064m)
3
0.8
0.8
. CT The propeller used on this POD propulsor is a
. 3 28*10-3 Gawn-Burrill series propeller, and we first validate the
.
3.32*10-3 computational results of MIT-PSF-2. Figure 8 shows
1.32% the calculated KT and KQ comparing to the
experimental data (Gawn, 1957~.
s
OCR for page 844
0.35
o 0.30
7 0.25
0.20
0.15
0.10
0.05
0.50 _
0~
0.45 _
0.40
A'
_~`
1
-13 - - experiment KT
- - 0- - experiment KQ*10
calculated KT
calculated KQ*10
., . . · .,
-~.40 0.50 0.60 0.70 0.80 o.go 1.00
J
Figure 8: Calculated KT and KQ of a Gawn-Burrill
series propeller comparing to the experimental data.
A boundary element method for analyzing the
flow around a multi-component propulsor (such as
ducted propellers, a stator/propeller combination,
contra-rotating propellers, and a pumpjet, etc.) is
developed for the potential flow computations. A
brief description of this boundary element method can
be seen in Appendix B. Figure 7 shows the pressure
distribution on this POD propulsor at 15 degrees yaw
angle and J=0.8. Figure 9 shows the computed and
measured axial forces on the POD propulsor at different
yaw angles, and the advanced coefficient, J. is 0.8.
One can see that the measured forces are asymmetrical
with respect to the yaw angles; however, the
computational forces are almost symmetrical. The
asymmetrical effect apparently results from the
interaction between the propeller and the POD body.
This asymmetrical effect is not seen in the potential
flow computations due to the interaction between the
propeller and the POD body is taken a circumferential
mean value (Appendix B). Considering a rectangular
strut behind a propeller, and the propeller drifts with the
strut. If the vertical length of the strut is the same as
the propeller diameter, and the chord length and the
thickness form of this strut are constant. Then, the
circumferential mean values of the induced potentials
between the strut and the propeller in the potential flow
computations are the same for the same drift angles but
opposite directions. Although the strut and the
propeller of the POD propulsor used for the
computations are not exactly as described above, the
geometries and relative position are almost the same as
above. Therefore, it is reasonable to see that the axial
force computed is almost symmetrical with respect to
the yaw angles. Figure 10 to 12 show the comparisons
between the computational forces and the measured
forces at different yaw angles. The computational
forces are almost linear with the advanced coefficients,
and the measured axial force has a sudden jump at
J=0.6. One can see that the difference between the
computational results and the measured data is larger
for J smaller than 0.6 due to the sudden jump. For all
the J's, the difference between the computed results and
the measured data is relatively large for-15-degree
yawing angle. The negative yaw angle means that the
direction of the yawing and the direction of the
propeller rotation are opposite, and the measured data
show that the interaction between the propeller and the
POD body is larger at this case. We will then turned
to the viscous flow calculation to see if we can reach
the same conclusion as the experiment.
We then use the viscous/potential flow code to
calculate the same POD propulsor. Figure 13 shows
the calculated streamline in the flow field for J=0.8 and
15 degrees of yaw angle. Figure 14 shows the
effective axial inflow calculated at J=0.7 and J=0.8 for
different yaw angles. From Figure 14, one can see
that the axial velocities are almost the same for yaw
angles 15 degrees and -15 degrees, and the axial inflow
is larger for zero yaw angles. Figure 15 to Figure 17
show the velocity vectors at the propeller plane for
different yaw angles at J=0.7, and it is clear that the
flow filed is asymmetrical. Figure 18 shows the
calculated forces comparing to the experimental data
for J=0.8, and the potential flow calculations are also
included. From Figure 18, we can see that the forces
on the POD propulsor calculated by the viscous flow
code are indeed asymmetrical with respect to the yaw
angles. After studying the computational results, it is
found that the over-prediction of frictional forces
(negative axial force) is the reason why the forces are
under-predicted for yaw angles other than zero degree.
It is still under investigation to understand the cause of
this over-prediction.
Finally, the numerical computation we
demonstrate is a propeller design case. As described
earlier, in this design method, we first get the desired
propeller loading distribution, and then obtained the
effective inflow thru the coupled viscous/potential flow
computation. A modified MIT-PBD-10 using the
conical coordinate system to define the blade geometry
is then used for the blade geometry design. Here, we
will demonstrate the different designs in a cylindrical
coordinate and in a conical coordinate. Figure 19
shows the pitch (P/D) distributions and the camber
distributions designed by two different geometry
definitions, and the difference is obvious.
6
OCR for page 845
0.25:
~ 20 ~
_
Q
X
Ox 0.15
-
~ 0.10
-
x
cr
0.05 _
Jut
~ 1 1 /
_. . ~.00 -10.00 0.00 10.00 20.00
yaw
Figure 9: The calculated and measured axial force on
the POD propulsor at different yaw angles (J=0.8~.
0.30
0.25
Q 0.2C
x
x 0.1~
Cal 0.1 C
x
0.0c
O.OC
YAW=0.0 O Rx(cal)
_ _ - _ - ~ Rx(exP)
I"
ma; ` -
~ . . . . . . . . . . . .
).50 0.60 0.70 0.80 0.90
yaw
Figure 10: The calculated and measured axial force on
the POD propulsor at different J's (yaw angle=0.0~.
0.3C
0.25
Q 0.2C
x 0.15
to O. 1 a
-
x
t 0.05
0.00
0 RX(cal)
_ _ - - - · Rx(exP)
_
_ _
~ YAW=15.0
-v.v~ '0 0.60 0.70 0.80 0.90
yaw
Figure 11: The calculated and measured axial force on
the POD propulsor at different J's (yaw angle=15.0~.
0.30 ~
_ RX(Cal) nest
_ _ - - - · Rx(exP)
. _
`2. 0.20
x 0.1 5
-
~ 0.10
-
~ 0.05
0.00 .
~_.
- 0 Rx(cal)
- - - - - Rx~exp)
-0.08 so 0.60 0.70 0.80 O.gO
yaw
Figure 12: The calculated and measured axial force on
the POD propulsor at different J's (yaw angle=-15.0~.
\
Figure 13: The calculated streamline in the flow field
of a POD propulsors at J=0.8 and 15 degrees of yaw
angle.
1.2t
1 .1
1 .0
0.9
n R
1.2
~1 .1
c) 1.0
0.9
0.8 t
J=0.7, Yaw=O.O
J=0.7, Yaw=15.0
0 J=0.7, Yaw=-15.0
J=0.8, Yaw=O.O
~ J=0.8, Yaw=15.0
0 J=0.8, Yaw=-15.0
......... .............. ............... .
0.3 0.4 0.5 0.6 r°/lk 0.8 0-9 1.0 1.1
Figure 14: Calculated effective axial inflow at J=C
and J=0.8 for different yaw angles.
7
1_7
OCR for page 846
Figure 15: The velocity vectors at the propeller plane
for J=0.7, yaw angle=0.0.
Figure 16: The velocity vectors at the propeller plane
for J=0.7, yaw angle=15.0.
1 ~ w
Ye._15 ,J - .7
__
-
~5J
If~~arm,_
Figure 17: The velocity vectors at the propeller plane
for J=0.7, yaw angle=-15.0.
0.25
In
8
.u>, 0.20
Q 0.15
x
x
'A 0.10
-
-
~ 0.05
-
x _
JO -,,,,1,,,,1,,,,1,,,,1
.0~ .00 -10.00 0.00 10.00 20.00
yaw
Rx(potential)
_ _ - - - · Rx(exP)
O Rx(viscous)
JO
Figure 18: calculated forces comparing to the
experimental data for J=0.8.
1.40 ~
1.35 ..,
1.30 :
125 _
e..20 .
1.15 _
1.10 _
~ no :
............ cylindrical coord.
- Mob_ conical coord.
. .
on
' ~.2 0.4 0.6 0.8 ·.0
r/R
0.030
n non
(' 0.020
=
0.015 _
0.01\ 2
............ cylindrical coord.
~ conicalcoord.
~~
0.4 0.6 0.8 1.0
r/R
Figure 19: With the same given loading distribution,
designed pitch and camber distributions are different
when blade sections defined on different coordinate
systems.
CONCLUSIONS
In this paper, the blade geometry definition
and the propeller inflow are studied for a new,
integrated propeller design method applied to a POD
propulsor. The presented design method can also be
used for propellers installed on any axis-symmetrical
bodies, or for the design of propellers on a traditional
propulsion system.
The computational results show that a given
blade geometry defined in different coordinate systems
results in noticeably different loading distributions.
8
OCR for page 847
For a given loading distribution, different blade
geometries are obtained when geometries defined in
different coordinate systems. Therefore, it is
concluded that the blade geometry defined in the
conical coordinate system is more appropriate for
analyzing and designing a propeller on POD propulsors.
The flow around a POD propulsor is
calculated by both the boundary element method and
the coupled viscous/potential flow code. The results
from the boundary element method show better
agreements with the experimental data; however, they
do not show the asymmetric effect when a POD
propulsor in yaw angles. The results from viscous
calculations show the asymmetric effect; however, the
force predictions are less satisfactory. Results from
two computational methods conclude that the viscous
effect cannot be ignored when calculating the
interaction of a POD body and a propeller especially
when they are in a yaw angle. Therefore, when
designing a propeller on a POD propulsor, the inflow
calculation should include the viscous computation.
For the further research, the boundary element
method used in the POD propulsor calculation should
include the unsteady effect, not just the circumferential
mean values. The wake of a propeller should be
aligned to the inflow when POD in a yaw angle. The
over-predicted frictional forces should be carefully
examined in the viscous flow calculations.
REFERENCES
Baldwin, B.S.
and Lomax, H., "Thin-layer
Approximation and Algebraic Model for Separated
Turbulent Flows," AIAA paper, No. 78-257, 1978.
Benek, J.A., Steger J.L., Dougherty, F.C., and P.G
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Optimization for Efficiency Improvement," Computer
& Fluids, Vol. 27, No. 3, 1998, pp. 407-419.
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"Simulating the Self-propulsion Test by a Coupled
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"Computations of Ship Flow Around Commercial Hull
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of the Chinese Society of Naval Architecture and
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on the Performance of a Series of 16 in. Model
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APPENDIX A: THEORY AND NUMERICAL
SCHEME OF THE RANS CODE "UVW"
In this appendix, we will briefly describe the theory and
numerical scheme of the viscous flow RANS code,
UVW. The 3-D RANS equations with pseudo-
compressibility (Chorin, 1967) are used by UVW, that
iS,
9
OCR for page 848
aq +aF + Do + aH = 1 (&FV + OGv + aHV ) (10)
At ax ay az Re ax By Oz
where
u
v
w
.
IF=
Fv=L
pU
u2 +v
UV
UW
o
T"
T
T~
(u,v,w) denote the velocity components
directions, respectively. fir is the dynamic pressure, Re
is the Reynolds number and ,B is a positive constant
related to artificial compressibility.
On the hull surface, the zero normal gradient
for the pressure is imposed and the velocity components
are all set to zero. The propagation properties of
characteristic variables in the corresponding Euler
Equation
~ pv law
vu , H = wu
vv + ~ we
vw ww +v
O
Tyx
T'
T'
TV =
aq+3F+a6+3H=0 (12)
At ax ay an
are used to construct the far field (inflow, outflow and
outer) boundary conditions.
The application of Finite Volume Method to
the governing equations (10) in a curvilinear coordinate
space leads to the transformed equation in
computational domain. The convective terms are
evaluated by the central difference approximation and a
4th order background dissipation term is added for
stabilization of scheme and elimination of the
non-physical oscillations (Jameson, 1981~. The
multi-stage Runge-Kutta explicit scheme is adopted for
the time-integration. Local time step and residual
smoothing technique are also employed for the
acceleration of convergence. The algebraic
eddy-viscosity model of Baldwin-Lomax (Baldwin,
1978) is used to evaluate the turbulent viscosity.
A composite grid approach is used in this code,
and it mainly follows the method proposed by Benek et
al. (Benek, 1986~. Two types of grids are employed in
the composite grid system: foreground and background
grids, and both grids are structured grids. The
background grids are applied to cover the whole
computational domain, possibly without considering
local geometry for the sake of easy grid generation. The
foreground grids are designed to take care of the local
regions, where complex geometry or high gradient of
physical variable exists. After the foreground and
background grids are created in the computational
domain, cells of background grids resided inside
foreground grids will be removed from the solution
procedure. The removed cells in background grids are
known as hole regions. In order to have enough
coupling between two overlapping grids, sufficient
overlapping region must be assured. Besides, the cells
(11 ) falling inside the body should be also excluded from the
solution process. The hole regions define new inner
boundaries (hole boundaries) of background grids. The
field variables on hole boundaries are interpolated from
the neighboring cells (stencils) in foreground grids. In
in x,y,z the same way, the ones on outer boundaries of
foreground grids are interpolated from the neighboring
cells in background grids. Hence the flow solution is
obtained by alternatively solving background and
foreground grids until the numerical solution
converged.
APPENDIX B: A BOUNDARY ELEMENT
METHOD FOR THE MULTI-COMPONENT
PROPULSOR
In this appendix, we will introduce a boundary element
method for the analysis of flow around a multi-
component propulsor. The detail of the theory and
numerical algorithm of the propeller boundary element
method can be referred to Lee (Lee, 1987) and Hsin
(Hsin, 1990~. We will begin with the governing
equation of a perturbation potential based boundary
element method:
2~(p)=||[~(q)C,; R( )-~(q) R( )]dS (13)
where ~ is the strength of perturbation potentials, or
equivalent to the dipole strength, an is the source
an
strength, ha ~ is the potential induced by a unit
strength dipole, and ~ is the potential induced by a
unit strength source. The discretized form of the
equation (13) is
10
OCR for page 849
N N
Thai Alp = >,bi jaj i = 1,N (14)
j=l j=
where hi and of represent the discrete forms of ~
and an, and al j, bi j represents the discrete forms of
an
a ~ and ~ .
an R R
calculated. We now divide this body into two parts,
and the panel number of these two parts are Nl and
N2 separately. Equation (14) thus can be rewritten as
N is the total number of panels
N. N2 N. N.
i,ai Alp + Thai Ale = >,bi j(;j + 2,bi jaj (14)
j=l j=l j=l j=l
We can solve equation (14) directly to get unknown
potentials, hi, or solve it by an iterative way. That is,
j=,
N
N. N. N. N.
Thai Alp = ~,bi jaj + ~bi joj - Ma
j=l j=l j=1
. ~ N. Nl N.
,ai Alp = 2,bi jaj + ~,bi jaj - Thai Ale
=l j=l j=l j=l
.j~j
(15)
In equations (15), two equations are solved separately,
and an iterative procedure is needed to obtain a
convergent solution. To validate the numerical
method, a wing is used for the computation. We first
solve equation (14) to calculate the loading distribution
on this wing, then divide this wing into two half-wings,
and solve equation (15) to obtain loading distributions
on two half-wings. Figure 20 shows the circulation
distribution obtained from these two solutions, and they
are exactly the same.
If we consider each component of a
two-component propulsor as one part of equation (15),
then the flow around this two-component propulsor can
be calculated. Similarly, we can extend equation (15)
to more than two components for multi-component
propulsors. For a multi-component propulsor with
both rotating and static components, such as a POD
propulsor (the propeller is rotating, and the nacelle is
not), a circumferential mean of the
component-to-component induced potential is taken to
simplify the solution procedure. Equation (15) thus
becomes:
2.5 r
2.0
1.5
1.0
nst
~ --a
I-
~ "
- - - direct solution
~- iter 0.
-~- iter 1.
-e- ~ iter 2.
_
-a- - iter 3.
iter 4.
o n I , I , I I , I I ~ I I I I I I , I ~ , , I , ~ I I , , I , , I I ~ I I I , I I I I I I I I , , I
'~0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
OR
Figure 20: The circulation distribution calculated by
using equation (14) (direct solution) and equation (15)
(iterative solution)..
N. Nl N. _ N2
I,ai j0j = ~bi,j~j +~bjtjaj -Iai,jf
j=1 j=1 j=1 j=1
N2 N. N _ N
I,ai.jfj = ~bi,j~j +~b,,j~j -Zai,jf
j=l j=1 j=1 j=1
(16)
Equation (16) is the equation used to analyze the flow
around a POD propulsor in this paper.
11
OCR for page 850
DISCUSSION
V.I. Krasilnikov
MARINTEK, The Netherlands
I have a number of questions on this extremely
interesting paper. My first question is addressed
to the Figure 19 in the paper where two different
design geometries are shown as obtained using
the cylindrical and conical definitions. They are
different. The question is: which geometry does
meet the required integral performance better?
Did you perform the analysis at both geometries?
The conical geometry definition applied in the
design method for podded propellers allows the
improvement in terms of the blade root
geometry. I wonder if the conical definition
works equally good near the blade tip where the
flow may be far from being conical.
DISCUSSION
Y.L. Young
University of Texas at Austin, USA
1. Wake alignment should be applied, especially
at high yaw angles.
2. Why is the comparison for yaw=0 degrees
better than yaw=15 degrees considering that
wake alignment was not applied?
DISCUSSION
G. Kuiper
Maritime Research Institute, The Netherlands
I am surprised about the large effect of the conical
coordinate system. When we have an oblique
flow the mean thrust does not change (while
there is a radial inflow added periodically at the
same axial velocity). But now the thrust
distribution changes when we change from
oblique to conical inflow, so when a radial
velocity is added everywhere. Did you take the
lift of the (conical) blade sections in the (conical)
axial correction or in the cylindrical axial
correction? What is the physical explanation for
this difference?
REPLY
The authors would like to thank to discussers'
valuable comments. First, we will answer Mr.
Krasilnikov's questions. In Figure 19, both
geometries satisfy the design requirements;
however, the two geometries are different since
they are defined in different coordinate systems.
It's not clear if the propeller blade tip flow can
be beneficial from the conic definition. The
main purpose of the conic definition is to make
the section aligned with the streamline; however,
if this can reduce tip cavitation has to be
investigated.
Dr. Young asked about the wake alignment of
the propeller of a pod propulsion system. In the
computations of pod propulsors in yaw angles,
we take the mean velocities as the inflow of the
propeller; therefore, the wake alignment is not
critical. For propellers in a yaw angle, if we take
the inclined flow as the propeller inflow, then it
is an unsteady flow problem, and the wake
geometry has to be changed at each time step.
We didn't make any conclusion from Figure 9,
and we won't say that the comparison for
yaw=15 degrees better than yaw=0 degrees.
More comparisons between the computed results
and the experimental data have to be made.
Finally, we will try to answer Dr. Kuiper's
question. Figure 2 shows the circulation
distributions of two propellers in an axial inflow
(parallel to the cylindrical coordinate system),
and Figure R1 shows the mean sections of two
propellers. Both propellers have a conic hub,
and one propeller has sections aligned with the
cylindrical coordinate system except the root
section, and one propeller has sections aligned
with the conical coordinate system. We have
carefully recomputed the case, and proved that
the computed circulation distributions have no
mistakes. The thrust and torque coefficients of
two propellers are actually not very different:
KT KQ
Cylindrical 0.3367 0.06157
Conical 0.3207 0.05930
Difference 4.75% 3.69%
Notice that the circulation distributions for two
propellers are at different "cuts", namely,
cylindrical sections and conical sections. We
believe that the different circulation distributions
are due to the radial cross inflow to blade
OCR for page 851
sections, and for propellers with an inflow
aligned with the conic hub (no radial cross
inflow to blade sections), Figure 4 then shows
that the circulations distributions are the same.
· _
cylindrical ,~
Figure R1. The mean sections of two propellers
computed.
Representative terms from entire chapter:
blade geometry
