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1
Propeller Inilow Corrections for Improved Unsteady Force ant!
Cavitation Calculations
T. S. Mautner
Naval Ocean Systems Center
San Diego, USA
Abstract
An existing propeller design method was modified and
used to calculate the spatial variation of propeller perfor
mance, velocity components and blade pressures for use in
determining unsteady forces and cavitation. The calculations
showed only small changes in the magnitude of the velocity
components and blade pressures when compared to typical
counterrotating propeller design results. Approximate agree
ment was found between the absolute magnitude of both the
harmonic coefficients and the total unsteady forces obtained
using the calculated axial velocity and the measured wake.
However, the unsteady force distributions associated with
the calculated axial velocity, which includes propeller
effects, resulted in small reductions in the magnitude of the
total unsteady forces on the propulsor. The calculated blade
pressures and cavitation index also showed only small varia
tions in magnitude with circumferential position.
Nomenclature
an, bn Fourier coefficients
AF Vehicle frontal area
c*
C
CD
CL
Cp
Cp
CQ
CT I  r D
D Propeller diameter
DB Vehicle diameter
Fx, Fy Unsteady side forces
F. (n)
/`hBL
Hop
J
k
it(k)
L
LB
m
M
n
Nb
p
Complex Fourier coefficient = an—ibn
Propeller blade chord
Vehicle drag coefficient = Drag/~pV2AF
Blade section lift coefficient
Pressure coefficient
Power coefficient = QQ/~hpV3'TR2
Torque coefficient = Q./ ~hpV2xR3
Thrust coefficient = T/~pV2'rR2
Unsteady thrust for the nth harmonic
Change in energy from freestream to local
Change in pressure through propeller disk
Index taking on values = 1, ,P
Advance ratio = or V~/S2 R
The reduced frequency = ~Cw/vs
Sears' function
Lift force on an airfoil or blade section
Vehicle length
Index taking on values = nNb
Moment/Torque on a blade element
Order of the propeller force harmonic
Number of propeller blades
Pressure
p
PC
Q
r
Ar
R
RB
SL
t
t/C
T
T. T.
x, Y
T. (n)
t(k)
u
vO
vi
vt
rev
V
Ve
TV
we
wt
x,y,z
x
y
(x
n
p
r
Subscripts
Number of blade elements having width Or
Propulsive coefficient
= (Thrust (1—r) Vs)/(Torque a)
Propeller torque
Radial coordinate
Width of the jth blade element
Propeller radius
Vehicle radius
Propeller stacking line location
Time
Blade thicknesstochord distribution
Propeller thrust
Unsteady moments
Unsteady torque for the nth harmonic
Horlock's function
Measured inflow velocity
Axial velocity with.propellers present
Axial component of the interference velocity
Axial inflow velocity with propellers not present
Tangential component of interference velocity
Change in axial velocity due to propellers
Resultant velocity of blade section and fluid
Free stream velocity or vehicle speed
Overvelocity due to thickness and lift
Axial component of selfinduced velocity
Tangential component of selfinduced velocity
Rectangular coordinates
Radial position = (rrh)/(Rrh)
Nondimensional chordwise coordinate
Angle of attack of a blade element
Blade section pitch angle (radians)
Propeller efficiency = J CT/7r CQ
Frequency
Angular velocity of the propeller
Potential function
Midchord skew angle
Fluid density
Cavitation number
Thrust deduction factor = ~ Drag/Thrust
Angular coordinate in the direction of
propeller rotation
Bound circulation
Forward propeller
A After propeller
701
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Introduction
One current and important issue in the design of marine
propellers is the reduction of propulsor generated noise due
to both the transmission of unsteady propeller forces through
shafting into the vehicle and propeller noise radiated into the
near and far fields. The unsteady blade forces/pressures and
cavitation calculations to be discussed in this paper result
from the passage of a propeller blade through the spatially
varying wake generated by upstream appendages. It is
known that when a propeller blade passes behind an append
age, unsteady blade pressures and loadings occur and, with
an appropriate velocity and pressure field surrounding the
propeller blade, periodic propeller blade cavitation will
occur.
While there are a wide range of techniques available to
calculate unsteady forces and blade pressures, these methods
require accurate knowledge of the wake incident upon the
propeller. One method of determining the incident flow field
required in propeller design is to make wake measurements
at the proposed propeller stacking line locations. Typically,
these measurements are made without a propulsor present,
and only simple corrections, if any, are made to determine an
effective wake. For single propellers, one could reasonably
assume that the appendage generated wake is incident upon
the propeller without further distortion or modification by
the propeller induced velocity field. However, for compound
propulsors, the wake incident upon the aft blade row now
passes through the forward propeller (or stator) and is modi
fied by propeller induced velocities and interactions.
In recent years measurements have been made detailing
propeller velocity fields. For example, Thompson's [24] fig. 6
shows the change in harmonic content of the axial velocity
distribution for a body with and without a propeller. In gen
eral, the results show a reduction in the magnitude of the
harmonic components when a propeller is operating.
Another example is the work of Blaurock and Lammurs [1]
which, for three values of thrust coefficient, illustrates the
significant changes in axial, radial and tangential velocity
components before and after an operating propeller. Addi
tional examples of propeller flow studies can be found in
refs. 7, 10, 11, and 22.
The above mentioned experimental results and uncer
tainties in the velocity field used in calculation of unsteady
forces, blade pressures and cavitation performance provide
the motivation to explore, analytically, the effect of a spa
tially varying wake on propeller forces and cavitation. The
continued success of the propeller design method of Nelson
[1720] suggests that if the liftingline portion of the design
method can accurately predict propeller performance using
circumferential mean data, the possibility exits of using the
same calculation techniques, with some modification, to
explore the effect of spatial variations in the wake. The
selection of Nelson's liftingline method was also based upon
its availability and ease of modification. Certainly, there are
many liftingsurface methods, panel methods and blade pres
sure calculation techniques [2, 5, 8, 9, 23] which might be
used for this purpose.
The discussion to follow will present a description of the
propeller design method and the geometry, velocity fields,
harmonic content, unsteady forces (bladerate) and blade
pressures associated with a counterrotating propeller set.
Measured Velocity Field
In the design of wakeadapted propellers, it is important
that the inflow velocity distribution at the propeller stacking
lines be properly specified. Even though circumferentially
averaged velocity profiles are sufficient for propeller design
calculations, the calculation of unsteady forces and pressures
requires that both the radial and circumferential distributions
of the wake be considered.
The velocity data used in this study was obtained from
wind tunnel tests [21] where boundary layer measurements
were made on a 0.6 scale model. Pitot tubes, oriented approx
imately parallel to the afterbody surface, were used to obtain
both the static pressure and the total head over a Reynolds
number range of 1.32.4x106. To avoid strut interference
effects and to utilize body symmetry, measurements were
made over the top 90° of the body where the zero degree
point coincides with the centerline of the fin trailing edge.
The measured wake at the forward and after propeller stack
ing line locations is show in fig. 1.
1.0OT
o
,, 9.60
a 0.48 ~ =~
e.20
~ $~ 0.422
(a) , . . . . . . .
0.00
50.0 30.0 10.0 10.0 30.9 50.0
CICUMFERENTIRL ANGLE (BEG) OF
1.00~
~ o.ao
o
_.,
,, 0.60
z
~ 0.40
o
He
0.20
1 (b)
0.928
—  r~ ace
~,~
\/ 0.498
in =
. ~ I
o.~o5 .0 30.0 10.0 10.0
CICUMFERENrIAL ANGLE (BEG) eA
. .
30.0 50.0
Fig. 1. Circumferential variation of the measured inflow
velocity u/V,, at the a) forward and b) after measurement
locations.
702
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Propeller Geometry
The propeller geometry used in this study is that for a
counterrotating propeller set designed using the method
developed by Nelson [1720]. The design utilized the param
eters given in table 1, and the circumferential mean inflow
velocity, u/V., and static pressure, Cp, formed from the
measured data [21]. Thrust deduction calculations were
made using the circumferentially averaged, threedimensional
potential flow velocity profiles calculated using a 3D body
coordinate generator [13] and a 3D panel method [3, 4]. The
details of the input velocity and pressure profiles, the circula
tion distribution and thickness distributions are given in refs.
1516.
Using the inputs described above, a counterrotating pro
peller design was performed. The calculated performance
parameters are given in table 1, the radial distribution of
chordtodiameter, local pitch angle and resultant blade sec
tion velocity are listed in table 2 and the geometry is
sketched in fig. 2. In addition to calculating the aft propeller
circulation distribution required for tangential velocity can
cellation, the design method determines the minimum value
of C/D required to meet cavitation (cr=0.75), blade stress (40
ksi maximum) and lift coefficient (CI,)MAX=05) require
ments. A distribution, which also satisfies the given hub and
tip values of C/D, is fit about this value of C/D and has the
shape shown in fig. 2.
Vehicle Velocity  V', (knots) 40
Drag Coefficient  CD 0.1 165
Vehicle LB/DB 11.77
Propulsive Coefficient PC 0.929
Propulsive Efficiency  ,7 1.08
Thrust Deduction  1r 0.860
Advance Ratio  J 2.12
Thrust Coefficient  CT 0.212
Torque Coefficient  CQ 0.133
Power Coefficient  Cp 0.196
Torque Ratio —QA/QF 1.00
Thrust Ratio —TA/TF 1.02
Blade Surface Cavitation  or 0.75
Maximum Stress (ksi) 40
Parameter
Forward
Propeller
After
Propeller
Blade Number
SL/DB
R/RB
rhub/RB
RPM
CTIP/CMAX
CHUB/CMAX
(C/D)"Ax
(t/C)HUB
(t/C)T~p
(CL)MAX
VIIP/V8
6
11.46
0.781
0.3280
1400
0.4
0.6
0.255
0.18
0.09
0.346
1.733
4
11.67
0.705
0.1823
1400
0.4
0.6
0.380
0.16
0.08
0.435
1.606
Table 1. Summary of Counterrotating Propeller Design
Inputs and Calculated Results Using Circumferential
Mean Inflow Data
RF R.] C/D)F
0.3393 0.1586 0.5567 0.7199
0.3619 0.1698 0.5924 0.7922
0.4072 0.1906 0.6411 0.9300
0.4525 0.2094 0.6797 1.0503
0.4978 0.2259 0.7102 1.1566
0.5431 0.2397 0.7101 1.2567
0.5884 0.2501 0.6894 1.3526
0.6337 0.2548 0.6622 1.4457
0.6790 0.2448 0.6190 1.5378
0.7243 0.2064 0.5657 1.6273
0 7696 0 1286 0.5217 1.7133
OF
V/VS)F
RA Rim C D)A 
0.1953 0.2367 0.7779
0.2215 0.2540 0.7593
0.2738 0.2864 0.7562
0.3261 0.3154 0.7708
0.3783 0.3405 0.8100
0.4306 0.3611 0.8260
0.4829 0.3757 0.8074
0.5352 0.3783 0.7724
0.5875 0.3575 0.7117
0.6398 0.2981 0.6218
06920 0.1883 05503
PA
V/V0A

0.6411
0.6924
0.8236
0.9701
1.1155
1.2384
1.3372
1.4177
1.4818
1.5334
1.5919
Table 2. Propeller Geometry and Operating Characteristics
Determined Using Circumferential Mean Inflow Data
y
STACKING LINE
Tfly,,
~: ~
R r i
DIRECTION
OF ADVANCE
Fy' Ty
////
Art
AXIS OF ROTATION
/ 0, S2 DIRECTION
V'')/ ~ LIZ
\ I I I
~ l I
\'J
~W
FX ~ X
Fig. 2. Description of a typical propeller and its geometry.
Unsteady Force Calculation Method
The method used in this paper to calculate unsteady
blade forces was developed by Thompson [24] and extended
by Mautner [14]. The method divides the propeller blade
into strips which are considered twodimensional airfoils,
and twodimensional unsteady airfoil theory is used to con
sider sinusoidal velocity fluctuations normal and parallel to
the inflow velocity. Corrections to the blade lift force due to
the presence of adjacent propeller blades, the inclusion of
camber and the calculation of the total force and moment on
the propeller have been included. The expressions used to
calculate the unsteady thrust and torque are
703
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PF (n)=Nb ~ Lj(~,Nb~ einNb~j (1) and
j=1
T(n)=Nb ~ Mj(,lNb) e b j (2)
j=1
where the lift and moment are determined using
and
Lim = L p l jV V C
[ ]i
{~(klru)coski—~ji(knU)sin~j }Arjcos§;
(3)
Mim = Lim tank rj (4)
Examination of the unsteady force equations (1)(4) and
a Fourier analysis of the current four cycle wake where the
velocity is represented by
vi(r,8) = aO(r) + ~ (an(r)cos(n8) +bn(r) sin(n8)1 (5)
V~ _
~ ~ n=1
show that the harmonic numbers of interest, for a 6x4 pro
peller set, are nNb= 12, 24, 36, for the forward propeller
and nNb= 4, 8, 12, 16, for the after propeller. Thus, only
the unsteady thrust and torque require calculation.
Blade Pressure and Cavitation Calculation Method
The starting point in calculating blade pressures is the
unsteady Bernoulli equation
p = p ~  ~pV2 ~ c(t) (6)
at
applied in a constant total head stream annulus located in an
inertial frame of reference. After evaluating the potential
function, i, and determining the constant c(t), one obtains
the following expression for the pressure coefficient along
the blade~s suction surface
where
~, p v2 = H~ ~ H2 (7)
H~ =—1 — ~ V2 ~PV2 + ~ V, V,, V,,
2 V. [V,+V,~+ V, +~V,~
(8)
~hBL [ V ~ + C 1 (9)
~pV2 = 2 ~V, ~ [ V, ~ (10)
704
H2=2 ~ ~ ~ ] + ~ ]. 1 (D ~ (11)
In the above expression for the blade pressure, ~hB~,
represents the change in energy per unit volume from far
upstream to where the boundary layer is measured, ~hp
represents the change in pressure through the propeller disk
and AV accounts for the increase in velocity along an airfoil
section due to thickness and lift. In the calculation we varies
along thc airfoil while the other induced velocities are con
sidered to be constant. Both w~ (y~ and AV(y~ are functions
of the radial distribution of circulation and the chordwise
loading. The overvelocity ~V due to thickness is determined
from the experimental data for NACA 001064 airfoil
adjusted for local blade section thickness. If the pressure p
on the suction surface is defined as the vapor pressure, a cav
itation number, based upon free stream conditions, can be
defined as ~=(poopva`>or)/~hpv2 It should be noted that,
although not shown here, the pressure equation has been
extended to include the axial, radial and tangential com
ponents of the inflow velocity f~eld.
Velocity Field Components
During the design of wakeadapted propellers, one must
account for velocities arising from several sources, and these
velocities are shown in the velocity diagram given in fig. 3.
The resultant relative velocity (V) between the blade section
and the fluid is determined from the tangential velocity due
to propeller rotation (Slr), the axial inflow velocity with pro
pellers not present (v0, the axial and tangential components
of the selfinduced (we, w~) and interference (va, v~) veloci
ties and the change in the axial inflow velocity (Av) due to
the presence of the propellers.
Development of expressions for the velocity components
are given by Nelson [1720] and have been summarized by
Mautner [16]. BrieQy, in determining the radial variation of
~v, one considers an axisymmetric Qow having a radial vari
ation of total head as it passes through a single propeller. As
/ ~Wa
/ ~:v!
jVj
Fig. 3. Relative flow velocity diagram at the liftingline.
OCR for page 701
the boundary layer flow moves through the propeller, the
induced velocity field of the propeller rearranges the vorti
city in the boundary layer and consequently alters the boun
dary layer profile. Using a potential flow model, an approxi
mate calculation is made to determine ~v.
The interference velocities are calculated using an exten
sion of the work of Hough and Ordway [6]. Using classical
vortex system representation, Hough and Ordway developed
expressions, in terms of Fourier coefficients, for the induced
velocities of a finite bladed propeller with arbitrary circula
tion distribution. The zeroth harmonic, or steady com
ponent, of their expressions have been extended by Nelson
to the case of a moderately loaded, wakeadapted propeller
with nonzero circulation at the hub.
The selfinduced velocities are calculated using an
extended version of Lerbs' [ 12] induction factor method.
Lerbs' original method was restricted to circulation distribu
tions which go to zero at the hub. For wakeadapted, coun
terrotating propellers this restriction is not desirable because
circulation distributions having nonzero circulation at the
hub are more efficient (more work done on slower moving
fluid near the hub) and the after propeller may be used to
remove the tangential velocity from the forward propeller so
that the hub vortex can be avoided. Thus, Lerbs' method was
extended by Nelson to include nonzero circulation at the
hub.
The blade pitch and the relative flow angle are related by
r tank = V~/Q and this expression has been extented to the
moderately loaded, wakeadapted propeller case by replacing
V~/Q with its equivalent r tank. The result is
~ t I v + va + wa (12)
From this expression it can be seen that the calculated velo
cities and propeller geometry are not independent but must
be determined in an iterative fashion to account for the
effects of both the forward and after propellers.
Propeller Calculations
As stated before, one problem inherent in the calculation
of unsteady forces involves the use of wake data obtained
without a propeller present. While the propeller design
method of Nelson calculates changes in the circumferential
mean inflow velocity field due to the presence of a propulsor
the uncorrected, spatially varying inflow has been used to
determine the unsteady forces acting on the propeller blades.
It is known that the presence of a propulsor will cause
changes in streamlines due to acceleration of the flow that
there may be additional unsteadiness due to the relative
motion of the blade rows and that the propeller will change
the amplitude and phase of the incident flow distortions.
From these few facts it is apparent that the measured wake
should be "corrected~to reflect propulsor induced velocity
field and then be used in unsteady force, pressure and cavi
tation calculations.
Calculation Procedure
Before velocity or unsteady force calculations could
begin, complete specification of the input velocity and static
pressure profiles was required. Due to the fact that the
design method requires the velocity and static pressure at the
hub surface (numerical requirements specify a slip condition
at the wall), the measured radial distributions of u/V,, and Cp
were extrapolated to the wall for 45°<~<+45°. Since the
3D potential flow distributions, for the body with append
ages, show only minor deviations from the circumferential
mean, (max =0.3% of V8), the circumferential mean profiles
will be used for all cases. The measured wake data was
specified in Look up. table format.
In an attempt to account for, at least first order, propeller
effects in the calculation of unsteady forces and pressures,
the liftingline portion of Nelson's design method was modi
fied to provide the calculation of the circumferential varia
tion in velocity components required to form the resultant
velocity V/V (fig. 3) at each blade section. The technique
used fixed the velocity profile, u/V',, at a particular forward
propeller angle, OF, and then calculations were performed as
the after propeller angle PA was varied from—45° to +45°.
In the region of large inflow velocity changes,
10° < ~ < +10°, single degree increments were used, and
outside this region calculations were made at increments of
46 degrees.
Velocity Components and Performance Parameters
For comparison purposes, the counterrotating propeller
design results obtained using circumferential mean inflow
data are presented in fig. 4 and tables 1 and 2. Referring to
fig. 4, it is seen that, for the forward propeller, the constant
magnitude of va is small compared to wa over the central
portion of the blade, while at the after propeller these veloci
ties are similar in magnitude. Furthermore, it is found that
the maximum value of i\v (change in v;) is of comparable
magnitude to the maximum value of we. This infers that Av
plays an equal role with wa in determining the radial distri
bution of blade pitch. The data also shows a small, constant
value of vie on the forward propeller while the after propeller
vie and both the forward and after propeller values of we have
comparable magnitudes. The radial variation of a indicates
that the region on the blade most prone to blade surface cavi
tation occurs at x~0.75, and the CL. profile (not shown)
reflects the change from large loading at the blade root to the
unloading of the blade tip region. It should be noted that
while these results are for a specific design, they are typical
of counterrotating propeller designs obtained using Nelson's
design method.
Next, the results obtained by variation of the forward
and after propeller inflow will be presented. During the cal
culation procedure, 49 values of x were used to determine
the radial variation of parameters. An example of the calcu
lated variation of propeller parameters with both OF and PA iS
the variation in the change in the axial inflow velocity (rev).
The results, given in fig. 5, show that the most significant
variation in rev occurs in the region of ~F=O, ~A=0 and near
the hub surface (x=0). As one moves away from the hub
surface the region of large parameter change narrows from
d:20° at x=0 to only a few degrees at x=1.0. Although not
shown here, similar variations are found for v and ,B while
the spatial distribution of the induced and interference velo
cities, lift and drag coefficients have a nearly constant magni
tude except for very small changes in a narrow region cen
tered about ~F=0 and ~A=O
The spatial distribution of propeller performance param
eters (ref.. 16) indicates that, with the design constraint of
constant Cat the performance parameters PC, 1~, ,7 and CQ
show only small variations in the region of ~F=0 and ~A=O It
should be remembered that the circumferential variation of
potential flow was not included. Thus, due to the fact that
705
OCR for page 701
~ /
o
o
.
.
!l / ~
~. / X
.
. .
1'
,. ..
, , , , , i
0.2 o 0.2 0.4 0.6 0.8
V/VS OELV
r~
o
o _
o . .
.
o.s 1 1.5
CflP V/VS
Q
o
CD
ry o
~:
X ~
o
~_
~';~
1,
.../~`r;
. ;/~,
. i,,
: ~
.! I O_
_
o
CD
.
G
X . _
o
C~
o
C
0.4

a'
o~ .
(D _
~ o 
G
X . _
o
C~
o~
1 C /
2 0.4 0.6
.'.~W 1:'.^
\,)'
~"'
, _ , ·, , o
o.oo 0.14 0.
VR/VS WR/VS
o.o 0.1
VT/VS WT/VS
1
/
/
BETR
"~
)
0.8 1
,
0.2 o o.s
SIGMR
Fig. 4. Velocity components and geometric parameters for the forward and
after propellers calculated using circumferential mean inflow data
v~s , forward; ~ , after. /~v/V~,: . , forward; · · · · ·, after.
V/V,,: , forward, , after. v'~/V',: , forward; , after.
Wa/VB:—· , forward; · · · · · ·, after. v`/V~: , forward; , after.
wt/V,,:  · , forward; · ~ , after. ,8: , forward; , after.
a: , forward; , after.
~,
NS:'
X BRR  0.0000 X BRR  0.2500 X BRR  0 .5000
~°°'~0
WO 1 o~ ~ o(G)
X BRR  0.7500
(st o
Fig. 5. Circumferential variation of the calculated change
in axial inflow veIocity, Av/V,,, for the forward propeller
at various x . ~F FWD (DEG). 6A  AFT (DEG).
706
~ :~,45U
Pkb °~ [orG)
X BRR  1 .0000
OCR for page 701
::
l..
i
1
1
I
0.2 o
~ o
G
X A_
Cat
I of o~
G .
Xo._ i; l,
A
o—
__ ..
o
,,;
, I, , ,
0.2 o.4 0.6 0.8
V/VS DELV
..
It'd
):
,'.!
.
_ . /~?
of ,.,, (.''
o. 14 o.oo
VR/VS WR/VS
:/
o ~ ~
G ~ ~
Xo ~ Xo_
o ,' / o_
,,'
o ., , , o
o.s 1 1.5 
CRP V/VS
~ o—
0.14 0.1
G · ~ \ , ·.
=0 ;~~~~
O {it,
to
l .\ ·.
o.o 0.1
VT/VS WT/VS
2
0.2 o
~ / '" 1
o.4 0.6 0.8
BETH
_ %%,%~
CD
.
to °
G
X,=
0.5
SIGMR
Fig. 6. Velocity components and geometric parameters for the forward
and after propellers. ~F=0 and ~A=14°.
v/V8:  , forward; , after. Av/Vs:—. , forward; · · · · · ·, after.
V/V,,: ~ , forward, , after. va/V8 , forward; , after.
walVa ·—~ forward; · · · , after. vt/V8: , forward; , after.
wt/V8: ·—, forward; ·  , after. ,6:  , forward; , after.
cr: , forward; , after.
PC is a function of the thrust deduction factor, one can con
clude that there would be additional small changes in PC
with ~F,A if the circumferential variation of the potential now
were included; however, calculations made for the forward
propeller indicate that changes in PC would be on the order
of 0.5%.
To further illustrate the results obtained by varying
OF and ~A, the radial distribution of parameters for the case
when ~F=0° and bA=14° is given in fig. 6. Comparison of this
data set with the circumferential mean data presented in fig.
4 indicate that the magnitude and radial distribution of the
profiles are very similar. However, the complete set of calcu
lated data shows the sensitivity of the results to the relative
angular position of the propellers and to the lack of total
symmetry in the measured wake.
Unsteady Forces
The axial velocity component of the measured wake and
the calculated axial inflow velocity were used in both
Fourier analysis and unsteady forces calculations. The data
was supplemented with the geometric parameters found in
tables 1 and 2. First, Fourier analysis and unsteady force
calculations were made using u/V8, which due to the meas
urement procedure, contains both axial and radial com
ponents. Figure 7 presents the radial distribution of Fourier
coefficient magnitude and unsteady thrust and torque for the
forward and after propellers. For the forward propeller, the
results show the dominance of the 12th harmonic (an), espe
,,;
By)
l
cially in the region of r/RF<0.6, and the rapid approach
toward zero of the 24th and higher harmonics. The har
monic content distribution for the after propeller shows the
dominance of the 4th harmonic and, when compared to the
forward propeller, a slower decrease in magnitude of the 8th,
12th and higher harmonics. As in the forward propeller case,
the large magnitude of an is located in the region of
r/RF<0.6. For both propellers, the magnitude of bn is nearly
zero for all harmonic numbers, and the magnitude of an
becomes nearly constant for r/RF> =0.6. Results for higher
harmonics can be found in ref. 14, and, in general, harmonic
component magnitude changes are in agreement with previ
ous work [24].
The calculated radial distributions of Fit and Tz associ
ated with the forward propeller's 12th harmonic (fig. 7) show
that regions of large forces occur in both the inner and outer
portions of the propeller blade. Also, there is a distinct
minimum force region, located at r/RF ~ 0.7, which coin
cides with the minimum velocity defect/excess region of
r/RB~ 0.45 shown in fig. la. For the after propeller, the
unsteady forces for the 4th harmonic show a minimum point
at r/RF ~ 0.38 which coincides with the region of greatest
velocity excess at r/RB ~ 0.3 (fig. lb). Even though the 4th
harmonic has a small and nearly constant magnitude over the
outer region of blade, the force and moment distributions
have large magnitudes in both the inner and outer regions of
the blade. With increasing harmonic number, nNb=8 and 12,
the minimum force point moves outward along the after pro
707
OCR for page 701
petter blade's span to r/RF~ 0.52 and coincides with the
region of minimum velocity excess/defect, r/RB ~ 0.48 (fig.
lb). Finally, for nNb=8 and 12, the shape and magnitude of
the force and moment distributions become more like that
found for the forward propeller.
Next, data will be presented to illustrate the unsteady
forces obtained by varying the propulsor inflow. Figures 8
and 9 present the radial distribution of harmonic coefficient
magnitude and unsteady forces for the calculated axial velo
city fields obtained by specifying (a) bA=0° and (b) ~F=0°
while the other propeller ((a) forward, (b) after) used wake
data over the range—45°<8A<~45°. Fourier coefficients and
unsteady forces were calculated using the measured axial
velocity vi and the calculated axial velocity, v=vi+^v. For
~A=0° (fig. 8), the harmonic coefficient distributions show
total removal of the small magnitude of bn previously
obtained in the Fourier analysis of u/Vs . Also, the sign of an
has been reversed while its (absolute) magnitude remains
nearly the same. The radial distribution of Fz and Tz,
obtained using vi are nearly equal to that calculated using
u/V8. For the calculated axial velocity distribution, v/V',,
the magnitude of an for the 12th harmonic has been reduced
in the region of r/RF<0.5 where the peak magnitude has
been reduced by =25%. The unsteady forces associated with
v show a reduction in magnitude over the inner radii, an
increase in magnitude over the outer radii and movement of
the minimum force point to a smaller radii, r/RF ~ 0.65.
When ~F=0 and HA iS varied, the measured axial velocity
field results in a sign change for the 4th and 12th harmonics
while the magnitude of an remains approximately the same.
As in the previous case, the small magnitude of bn has been
removed. The characteristics of the unsteady force distribu
tions are nearly the same as those obtained for u/V~; how
ever, they show a lower magnitude of Fit and To at the other
radii. The harmonic content obtained from the Fourier
analysis of v/V8 reveals a reduction in an, for all harmonic
numbers, over the after propeller blade's inner radii. Also,
the minimum force point has been shifted to a smaller radii,
r/RF ~ 0.35 for the 4th harmonic and to r/RF ~ 0.57 for 8th
and 12th harmonics. Only small deviations from the results
given above were found for other combinations of OF and ~A.
In addition to the radial distribution of unsteady forces
given above, the total forces on the forward and after pro
pellers were calculated. The results for the measured and
calculated wakes are given in table 3. It can be seen that
removal of the radial component from the measured inflow,
u/V~, results in the introduction of unsteady forces associ
ated with the 6th and 18th harmonic components, on the for
ward propeller, which were zero for the measured inflow
case. The magnitudes of these forces are substantially lower
(3.44.9 times) than those obtained for the 12th and 24th har
monics and are probably an artifact of the computation pro
cedure. The data in table 3 also shows a general reduction in
unsteady force magnitude for vi/V,, data on both the forward
and after propellers when compared to the u/V`, results.
While the unsteady forces obtained using v/V8 show a reduc
tion in magnitude for most harmonic numbers, small
increases in magnitude, over the u and vi data, are found for
T~)F at the 12th harmonic and TEA at the 8th harmonic.
Cavitation and Blade Pressures
An integral part of Nelson's propeller design method is
the ability to determine the blade geometry (C/D) required
to satisfy a given blade surface cavitation requirement. The
counterrotating propeller geometry detailed in tables 1 and 2
reflect the the chordtodiameter ratio required to meet the
cavitation requirement of a=0.75. In all subsequent cavita
tion and blade pressure calculations, the blade geometry was
held constant, and, since cavitation is the parameter of
interest, only suction surface pressures will be calculated.
The variation of a with both forward, IF, and after, ~A, pro
peller angles is presented in figs. 10 and 1 1. The results show
that a has the same type of variation from the circumferential
mean data as did the various velocity components, blade
pitch and performance factors (see fig. 5 and ref.. 16). The
variation of a is greatest in the region of 20° < ~F,A < 20°.
As shown by equations (6)(11), a and the blade section Cp
distribution depend on all the velocity components compris
ing the local blade section velocity diagram (fig. 3). However,
as mentioned before, both wb and TV, are functions of the
chordwise trapezoidal loading distribution and the radial dis
tribution of circulation. Thus even with corrections for
thickness, lift and circumferential variation of velocity com
For Use Fz for Nb =
PA Vel 6 12 18 24
u/Vs)F O 32.3 O 12.4
045 vi/V~)F 3.9 25.5 3.7 12.1
0 45 v/V.)F 4.8 23.3 3.2 10.8
For Use Fz for Nb =
F Vel 4 8 12 16
u/V8)A 79.1 24.4 20.7 8.1 32.9 7.3 7.5
045 vi/V8)A 60.4 24.7 17.5 7.4 26.7 7.4 6.0
045 v/V8)A 63.3 20.8 14.8 7.2 26.6 7.7 5.6
=
For OF or HA = 040, the values of Fz (lb)and Tz (ftlb) are representative
OF or HA was fixed while the other propeller angle was varied
Tz for Nb =
6 12 18
o
1.5
1.3
24
4.9
4.9
4.4
_
0 10.8
2.0 8.0
22 9.8
4
Tz for Nb =
8 12
16
3.0
2.8
2.9
Table 3. Calculated Total Unsteady Forces for the Forward and After Propellers
708
OCR for page 701
~o
o
o ~
I NB
~o
o
o
G o ,, o~ ~:~
C~ ~ o
NB  12 · NB  12
1 1 O 1
0.4 0.6 0.8 1 0.4 0.6 0.8
R/RF R/RF
a,
o \
o \ ~U,
\  
C o \ ~ _
. , ~
L~ ~ ~ o 
o ~ o
C: o NB  4
o_ ~, o
0.2 0.4 0.6 0.8
R/RF
~ ,~ ~%
0.2 0.4 0.6 0.8
R/RF
Fig. 7. Variation of Fourier coefI;cient magnitude (an, ten) and unsteady thrust (Fz) and
torque (Tz) with radial position (r/RF) for the forward and after propellers using the
measured inflow u/V.. a) forward propeller, nNb=12. b) after propeller, nNb=4.
~0
o
co
R/RF
o
CD
~ ~1
.= ·—
~o
,`K ~ o
/ `~]
~ mC_
r. Ln I
. .
0.4 0.6 0.8
.
o
o
\~
. / , ~ \\ {/
, ~ ~,
. . . .
0.4 0.6 0.8
R/RF
,\ ,.''''
I NB  12
o.4 0.6 0.8 i o.4 0.6
R/RF R/RF
Fig. 8. Variation of Fourier coefficient magnitude (an, ten) and unsteady thrust (F~)
and torque (Tz) with radial position (r/RF) for the forward propeller.
~A=O nNb=12. a) vi~s; b) v~.
 F.  T.
~ Z. ~ b
709
OCR for page 701
a,
o
o
G o
. o
rid
L"
° a'
o_
a'
lo
o
G °
. o
~1
lo
C) lo
o _
' T
0.2 0.4 0.6
R/RE
/
/
0.8 1 0.2 0.4 0.6
R/RF
_ ~—
~ U.
, .
—_
~ U)
o
o
NB  4
LO o_
0.2 0.4 0.6 0.8
R/RF
me
l
0.8 1
_~
~ ,
l
0.2 0.4 0.6 0.8
R/RF
Fig. 9. Variation of Fourier coefficient magnitude (an, ten) and unsteady thrust (Flu)
and torque (To) with radial position (r/RF) for the after propeller.
8~0. nNb=4. a) vi/V~; b) v/V8. , Fz, , T.,
portents, only small changes in a are noted. The variation of
a in figs. 10 and 11 also represent the maximum (negative)
pressure on the blade's suction surface, and, if the blade
geometry were allowed to vary as required by the calculation
at each 8, one might obtain larger changes in a.
Figures 12 and 13 are representative of the variation in
pressure along the suction surface of the blade at various
radial locations. First, fig 12 gives the values obtained for
the counterrotating propeller set using circumferential mean
inflow data The data shows that the shape and magnitude of
suction surface pressure coefI;cient are typical of that found
on various airfoils and are in general agreement with the
results of other researchers (for example refs. 2, 5 and 9). It
can also be noted that the magnitude of Cp is nearly the same
for both the forward and after propellers. Figure 13 gives
the results for the case when ~F=0° and FA=14° where the Cp
calculation uses the data presented in fig 6. Comparison of
figs. 12 and 13 indicate very small differences in CF This
result is consistent with the a data presented in figs. 10 and
11 and is typical for the range of ~ used in the study. When
the equation for the suction surface pressure is derived from
the Euler equations (see for ref. 9) in a rotating frame of
reference, an additional term equal to Hip (eqn. 10) is
obtained. The blade pressure results for this formulation
reject small changes in magnitude due the additional Ahp
however, the magnitude and shape of the suction surface
pressure remain in general agreement with previous work
[2,5,9].
Conclusion
An existing propeller design method was modified and
used to calculate the spatial variation of propeller perfor
mance, blade pressures and velocity components for use in
determining changes in bladerate forces and cavitation per
formance. The calculations showed only small changes in the
magnitude of the various velocity components, forces and
blade pressures when compared to the counterrotating pro
peller design results. In general, there was approximate
agreement between the (absolute) magnitude of both the har
monic coefficients and unsteady forces obtained using the
axial component of the inflow velocity and the measured
wake data. However, the unsteady force distributions associ
ated with the calculated axial velocity, which includes pro
peller effects, showed an increase in magnitude at the inner
radii with minimal change in its general shape. Overall, there
was a small reduction in the magnitude of the total unsteady
forces on the propulsor. The circumferential variation of both
the cavitation index and suction surface pressure distribu
tions showed only small variations with blade position. Also,
the difference between the forward and after propeller blade
pressure distributions were small.
While the simple approach used in this paper did not
reveal large changes in the inflow velocity field, cavitation or
unsteady forces, it did show that the design method has the
tendency to modify the measured inflow velocity field in
such a way that both the forward and after propeller "see.
nearly identical velocity fields. This result indicates a
"smoothing. of the incident velocity field. From a design
standpoint this is desirable since one would not want pro
peller performance to be highly sensitive to small perturba
tions in the inflow. However, these results can only be con
sidered first order and require extension of the calculation
procedure to account for additional effects due to the
unsteady flow field such as blade interaction terms and flow
acceleration due to the moving blade rows.
Acknowledgments
This work was supported by the Naval Ocean Systems
Center's Independent Research and Independent Explora
tory Development Programs.
710
OCR for page 701
k~ (0 tore)
rWO ~ torG)
~~ it'
X BRR  0.0000 X BRR  0.2500 X BRR  0.5000
: ' :35
W torG
X BRR  0.7500
Fig. 10. Circumferential variation of the cavitation index
a, for the forward propeller at various x locations.
OF FWD (DEG). PA  AFT (DEG).
X BRR  1 .0000
~n
X BRR  0.0000 X BRR  0.2500 X BRR  0.5000
WO 'o ~or§]
X BRR  0.7500
Fig. 11. Circumferential variation of the cavitation index
~, for the after propeller at various x locations.
~F FWD (DEG). ~A  AFT (DEG).
711
WO 1 torG}
X B8R  1 .0000
OCR for page 701
~ o 
o
BRR  0 .0000
0.5
Y BRR
_
~ 1
~ o~
" _
~  e
X 8RR  0.2500
,
1 o 0.5
Y BRR
~_


 ~_
X BRR  0.5000
l
'':_
 1
~ 0 0.5 1
Y BRR
o ~
X BRR  I .0000
X BRR  0.7500
, , _
u 0.5 1 0 0.5 1
Y BRR Y BRR
Fig. 12. Variation of suction surface pressure coefficient,
Cp = (pOOp)/~ p V2, calculated using circumferential mean
data for the forward and after propellers at various x locations.
. , forward propeller; o , after propeller.
BRR  0.0000
,_ ~ ,
0 0.5
Y BRR
~ o 
1~
_ X BRR  0 .7500
1
 `~
 `~
, , .

0 0.5
Y BRR
~,
X BRR  0.5000

~o
, _
o 0.5
Y BRR
I X BRR  1 .0000
_ 1 _
1 0.5 1 ' ~ 0.5 i
Y BRR
Y BRR
Fig. 13. Variation of suction surface pressure coefficient
Cp = (pOOp)/ ~ p V2, calculated for the forward and
after propellers at various x locations with ~F=0° and ~A=14°.
· , forward propeller; 0 , after propeller.
712
OCR for page 701
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