repeatability of an identical condition.
An effort was made to set experimental
conditions identically on repeat runs;
however, some variation was unavoidable
due to manual setting of the model speed
and propeller rotation speed, random
zero shift in data channels, and minor
variations in basin water conditions.
This resulting error is approximately +2
percent of the measured first harmonic
loading component. A more detailed
error analysis based on the repeata
bility of the hydrodynamic, centrifugal
and gravitational loads will be presented
in a future DTNSRDC Report.
EXPERIMENTAL RESULTS
Loading Components
The basic loading components are
shown in Figure 1. For a righthand
propeller the sign convention follows
the conventional righthand rule with
righthand Cartesian coordinate system.
For a lefthand propeller all the loads
are the same, but for this case the sign
convention follows a lefthand rule with
a lefthand Cartesian coordinate system.
Each component of loading is
generally represented as a variation of
the instantaneous values with blade
angular position, e, and as a Fourier
series in blade angular position in the
following form:
F,M(0) = (F,M)+ Z < F ' M >n COS { n< '" (0 F.M>n} (D
In general, the loads consist of
hydrodynamic, centrifugal and gravita
tional components. However, in this
Fig. 4  Experimental Variation of Loading Components with Blade Angular Position
paper only the hydrodynamic component of
blade loading is presented. Centrifugal
and gravitational loads were measured
soley to permit the hydrodynamic loads
to be determined by subtracting the
centrifugal and gravitational loads from
the total experimental loads.
Centrifugal and Gravitational Loads
Centrifugal and gravitational loads
were determined from airspin experi
ments with each flexure over a range of
rotational speed n for * = 0, 10, 20,
and 30 deg . The centrifugal load, which
is a timeaverage load in a coordinate
system rotating with the propeller,
should vary as n^ and be independent
of i(i. The timeaverage experimental
data followed these trends closely. The
gravitational load, which is a first
harmonic load in a coordinate system
rotating with the propeller, should vary
with i> and be independent of n. The
first harmonic experimental data
followed these trends closely.
The centrifugal and gravitational
loads measured during these experiments
agreed with the values previously
reported in References 3 and 4 and the
gravitational loads agreed with values
deduced from the weights of the blades
and associated flexures. Therefore,
these results will not be repeated here.
Variation of Loads with Angular Position
Figure 4 shows the variations of
the loading components with blade
angular position for Propeller 4661 for
operation with 10 degrees shaft
inclination with no screen and for
operation behind the wake screen with no
shaft inclination. These plots present
typical results obtained in inclined
flow and behind the wake screen. For
other conditions evaluated in inclined
flow and behind the wake screen the
trends are basically the same as shown
in Figure 4, but the magnitude of the
unsteady loading varies with
experimental conditions.
The variations of the primary load
components , F x and M y with blade
angular position in inclined flow and
behind the wake screen follow patterns
approximately similar to the tangential
and the longitudinal velocity profiles
of the respective wakes shown in Figure
3. The variation of the loads behind
the wake screen implies that the high
velocity region covers a smaller portion
of the propeller disk than the low
velocity region, which agrees with the
measured wake data.
Effect of Plate Clearance
Figure 5 shows that the periodic
blade loads in inclined flow are fairly
insensitive to the presence of a nearby
flat boundary parallel to the flow for
tip clearance to propeller diameter (tip
clearance ratio) as small as 0.1. The
ratios of the periodic to the time
average loading components at tip
clearance ratio of 0.1 are no greater
than 10 percent larger than the
CLEARANCE/PROPELLER DIAMETER
Fig. 5  Effect of Plate Clearance on First Harmonic Loading
in Tangential Wake
. \\
1 1
PROPELLER 4661
*  20 DEG
NO SCREEN
\\
 \\
\ \\
\ \ \

1 1 1
1 1 1
1
1 1
1
OPELLER4661
0OEG
My
TH SCREEN
"^

^
\
/
/:
(F. M) 1 nV A
\J^S
i
1
1
1
i i 1
In Figure 6 the force and moment
components are nondimensionalized as
follows:
Fig. 6 â€” Experimental Variation of First Harmonic Blade Loads
On Propeller 466 1 with Advance Coefficient
corresponding values at very large tip
clearance ratios for Propeller 4402 with
an inclination angle \f> of 10 degrees.
Effect of Advance Coefficient
Figure 6 presents the trends of the
variations of the first harmonics of the
primary load components F x , My,
Fy, and M x with J for Propeller 4661
at the three primary experimental
conditions described in the section on
Experimental Conditions and Procedures.
'(Fx.yh
(Mx.y)l
< F x.y>,
pnV A D 3
< M Â»,y>1
pnV.D 4
(2)
(3)
where the subscript 1 represents the
first harmonic component, and V& =
V(Iwvm) is tne speed of advance based
on the volume mean wake. This form of
nondimensionalization was used in part
to verify Wereldsma's argument (12) that
for a given propeller in a given wake
pattern the circumferential variation of
the hydrodynamic loading varies
approximately as nV^; i.e., for a
given value of nV& the circum
ferential variation of hydrodynamic
loading fs insensitive to J. This form
of nondimensionalization is different
only by a factor of the advance coeffi
cient, J&, from the conventional form
of nondimensionalization; i.e.,
F Xf y/pn 2 D 4 and M Xf y/pn 2 D 5 .
The coefficients *(Fx,y)l
and K (Mx,y)l ^ n Figure 6 are all
normalized by the respective
coefficients at design J to illustrate
the relative sensitivity of the loading
coefficients to J, for different wakes.
The results presented in Figure 6
show that in tangential flow *(Fx,y)l
and ^(Mx y) 1 generally decreases with
increasing J. K (My) 1 i s tne most
sensitive to J, and *(Fy) 1 is the
least sensitive to J. Except for the
Fy component, these data do not
closely follow the trends indicated by
Wereldsma (12). For the longitudinal
wakes behind the screen, the slopes
of K(p m) with increasing J are less
negative than in inclined flow. The
relative slopes among the four
components are generally the same for
each propellerwake combination, except
for some cases at greater than design J.
Figure 7 compares the variations
of K (Fx)l with J obtained in the
present experiment in inclined flow with
variations obtained in previous exper
iments in which the unsteady blade loads
were determined over a range of J in
inclined flow (13,14,15,16). Each of
these sets of experimental results show
that K(Fx) 1 decreases substantially
with increasing J.
Figure 8 compares the variation
in *(Fx)l with J obtained in the
present experiment behind the wake
screen with the blade frequency *jFx)n
obtained in previous experiments in
which the unsteady bearing forces were
measured in longitudinal wakes produced
by screens in a closed jet water tunnel
(17,18,19). The results in References
17, 18, and 19 are for the blade fre
quency component of k fx' whereas in
the present investigation the shaft
frequency component of k Fx was
measured. However, the mechanism for
generating the unsteady blade loading in
axial flow appears to be independent of
the harmonic of shaft rotation. Each of
the three previous sets of experiments
shows that K(fx) n decreases with
increasing J to some minimum value near
design J, and then increases with
further increase in J. Figure 8
illustrates that variations of *(Fx) 1
with J obtained in the present
experiment generally follow the trend of
the data in References 17, 18 and 19.
In general, *(Fx)n i s somewhat less
sensitive to J in longitudinal wake
patterns than it is in tangential wake
patterns.
"
REFERENCE
HGURE
tld.,1
V,
m
'
v\
2.0
1.8
V
A
O
A
â–¡
EEL
â€ž.*,
I
Â°T,
Z
;
E
1.6
1.4
1.2
1.0
A
;X \

0.8
I
1
1
1
1
â€” 
0.4 0.6
REFERENCE
PROPELLER
FIGURE
P/0
2
v\
V
A
O
D
â„¢Â«
E
"
Z
E
\
E
1 1 1 1 1
0.8 1.0
Jref
1 .4 1 .(
Fig. 7  Comparison of First Harmonic Blade
Loading Coefficients in Tangential Wakes
Fig. 8  Comparison of Various Experimental Blade
Loading Coefficients in Longitudinal Wakes
CORRELATION BETWEEN EXPERIMENTAL RESULTS
AND THEORETICAL PREDICTIONS
Theoretical Methods
The experimental results were
correlated with predictions based on the
following methods:
1. The quasisteady method developed
at DTNSRDC by McCarthy (5) ;
Computer Program QUASI.
2. The procedure developed at Davidson
Laboratory by Tsakonas, et al (6,7)
based on lightlyloaded unsteady
lifting surface theory; Computer
Program PPEXACT.
3. The procedure developed at MIT by
Kerwin and Lee (8) based on
moderatelyloaded unsteady lifting
surface theory; Computer Program
PUF2.
4. A refinement of the method of
Kerwin and Lee (8) developed at MIT
by Kerwin (9) , to allow the axis of
the propeller slipstream to depart
from the propeller axis for opera
tion in inclined flow; Computer
Program PUF2IS.
The procedure developed by McCarthy
(5) is a simple quasisteady procedure
utilizing the open water characteristics
of the propeller. It is assumed that
the thrust and torque developed by the
propeller blade at any angular position
in a circumf erentially nonuniform wake
is the same as would be produced by the
propeller blade if it were operating
continuously at the advance coefficient
J and rotational speed n based on the
local wake at the angular position
corresponding to the midchord of the 70
percent radius. It is further assumed
chat the instantaneous thrust and torque
can be adequately estimated by entering
the propeller open water characteristics
at the values of J and n based on a
weighted average over the propeller
radius of the wake at the local blade
angular position. This simple method
can be expected to yield reasonable
results only if the reduced frequency of
interest is low, the propeller projected
skew is small relative to the wave
length of the pertinent wake harmonic,
and the wake harmonic corresponding to
the force harmonics of interest does not
vary substantially in amplitude or phase
with radius. These conditions are met
for propellers and wakes being evaluated
in the present paper.
The procedure developed by
Tsakonas, et al (6,7), is based on the
linearized unsteady lifting surface
theory for a lightlyloaded propeller
using an acceleration potential. The
numerical procedure applies the mode
approach and collocation method in
conjunction with the "generalized lift
operator" technique. This procedure
based on the frequency domain assumes
that a given harmonic of blade loading
depends upon the corresponding harmonic
of the wake velocity normal to the blade
chord line, independent of whether the
normal velocity results from axial or
tangential components of the wake.
The blade load does not depend upon the
radial variation of the circumferential
mean wake. The shed and trailing
vortices are assumed to lie on an
"exact" helicoidal surface of constant
pitch extending to downstream infinity
determined by the propeller rotational
speed and a single axial inflow
velocity. The axis of this helicoidal
surface coincides with the propeller
axis, regardless of the inclination of
the propeller shaft to the incoming
flow. This method does not consider the
contraction or rollup of the propeller
slipstream. All geometric characteris
tics of the propeller are considered,
except rake, camber, and thickness which
are assumed to be zero.
Valentine (20) developed a
refinement to the method of Tsakonas, et
al (6,7) for operation in slightly
inclined flow. This refinement, in
effect, replaces the "exact" helicoidal
wake whose axis coincides with the
propeller axis with a slightly distorted
one in the direction of the inflow
velocity. The refinement, which is
incorporated as a perturbation for small
inclination angles, relates the unknown
loading at the first harmonic of shaft
frequency with the loadings at the mean
and second harmonics of shaft frequency
evaluated without the distorted helicoi
dal wake. All other assumptions of the
method of Tsakonas et al , are retained.
Calculations made by Valentine (20)
showed that his modification to the
method of Tsakonas et al , did not
significantly improve the poor
correlation between this method and
experimental blade loads in inclined
flow. Further, with the modifications
by Valentine the disagreement in phase
between predicted and experimental blade
loads in inclined flow were even worse
than the agreement obtained with the
method of Tsakonas et al. Valentine
concluded that the effects of shaft
inclination required a moderatelyloaded
propeller theory, and could not be
adequately calculated based on the
lightlyloaded formulation of Tsakonas
et al. Therefore, the theory of
Tsakonas et al , as modified by Valentine
was not correlated with the experimental
results in the present paper.
10
The numerical method developed by
Kerwin and Lee (8) is based on a
linearized liftingsurface theory in the
time domain. The propeller blades are
represented by a spanwise and chord
wise distribution of discrete line
vortex' and source elements located on
the exact camber surface of the blade.
Thus the geometric complications of
skew, rake and radial variation in pitch
are readily accommodated. The trailing
vortex wake is permitted to contract and
roll up, and the effect of vortex sheet
separtation from the blade tip is taken
into consideration. The inflow velocity
to the propeller may have radially and
circumf erentially varying axial,
tangential, and radial components, and
may therefore give rise to both steady
and unsteady blade loading. This
method, like the method of Tsakonas et
al , assumes that the axis of the
propeller slipstream coincides with the
propeller axis.
Kerwin (9) developed a refinement
to the method of Kerwin and Lee (8) for
operation in inclined flow where the
slipstream is not axisymmetric about the
propeller shaft. This refinement
entails a more realistic representation
of the path of the propeller
slipstream. In this method the axis of
the slipstream coincides with the
propeller axis immediately behind the
propeller, and coincides with the
direction of the mean inflow far
downstream in the ultimate wake. A
simple function is assumed for the
slipstream axis in the wake region
between the propeller and the ultimate
wake. Due to the asymmetry, the
position of the trailing vortex wake
relative to a blade oscillates with a
onceperrevolution fundamental fre
quency, thus giving rise to unsteady
induced velocities normal to the blade
surface and thereby unsteady blade
loadings of the same frequency. The
strength of the vorticity in the wake,
and thus the induced onceperrevolution
variation of loading on the blades, is
dependent upon the timeaverage load
ing of the propeller. All other
characteristics of the method of Kerwin
and Lee except for rollup are retained,
including the flexibility for the
trailing vortex wake to contract, and
allowance for the effect of vortex sheet
separation from the blade tip.
Correlations in Tangential Wakes
Figures 9 and 10 present the
amplitudes and phases, respectively, of
the first harmonic loads on Propellers
4661, 4710, and 4402 operating in the
various tangential wakes over a range of
advance coefficient J. These figures
present both experimental results and
predictions based on the various
theoretical methods described in the
preceding section. For Propeller 4661
with 10 degrees shaft inclination, the
first harmonic components of F x , My,
Fy, and M x are presented. For other
conditions only the F x component is
shown. The results in Figure 9 and 10
are summarized in Table II for design J.
In the tangential wakes, a
consistent variation from the
experimental data occurred in the
theoretical predictions for the three
propellers evaluated. With few isolated
exceptions, all of the theoretical
methods underpredicted all of the
loading components throughout the range
of conditions evaluated. In general,
the correlations are better for the F x
and My components than for the Fy
and M x components . The F x and My
components are the more important
components since, in general, these are
larger than the Fy and M x components.
The quasisteady method of McCarthy
(5) underpredicted the amplitude of
(F x ) ]_ by approximately 10 to 30
percent of the experimen tal values with
closer agreement at higher values of J.
The quasisteady predictions main
tained a similar trend to the
experimental data for F x and M x
components for all conditions in
tangential wakes; i.e., both the
experimental and predicted values of
"(Fx) 1 an< 3 *(Mx) 1 decrease with
increasing J. For a given propeller the
predicted slope of the *(F X )i and
*(Mx)l curves (not shown) with J
increases with increasing amplitude of
the tangential wake. No predictions of
(Fy) i and (My) i were made with
the quasisteady procedure since this
method does not predict the radial
center of the load. These components
could be predicted by the quasisteady
method by assuming a radial position of
application of the load.
The phases presented in Figure 10
show that the quasisteady method of
McCarthy predicts that the maximum
values of the loading components will
occur at approximately 10 to 20 degrees
of blade angular position before the
experimentally determined angles.
Predictions by the unsteady theory
of Tsakonas et al (6,7) did not agree
well with experimental results in
tangential flow. This theory predicts
that the amplitudes of all four loading
coefficients increase with increasing J,
11
PROPELLER 4661
J/  10DEG
NO SCREEN
/
/
i I I L
4 0.6 0.8 10
16 1.8 2.0
V Hw VM l/nD
PROPELLER 4661
V  10DEG
NO SCREEN
I I III I I 1_
0.4 06
1.0 12 1.4 16 1.8 2.C
' Hw VM l/nD
PROPELLER 4661
*  10DEG
NO SCREEN
0.4 06 0.8
V (1Ww M IMD
\ ' '
1
1 1 1 1
PROPELLER 4661
NO SCREEN
O OO \
Â°o
\o o
Expenman, Â°
6
\
o
O o Â° Â°
^
5
â€” "'.^
^/"
 .
^~_^^ .
^
F Â«
Kerwin & Lee
 (PUF2)
McCarthy
(QUASI)
i
Tsakonasetal
â€¢ "" (PPEXACT)
3
^
â– ^
2
/
"
1
i
1
1
1
1
1
II 1 1 1
0.4 0.6
1.0 1.2 1.4 1.6
J = Vllwwâ€žVnD
Fig. 9  Amplitudes of First Harmonic Blade Loads in Tangential Wakes; Correlation Between Experiment and Theory
12
T 1 1
i i i i i
PROPELLER 4661
ill = 20 DEG
NO SCREEN
\ A
\ A

\ \ A /
Kerwin IPUF2ISI
X \ A ^
(PUF2I >
A
A
a A E ; nm,m
Cv^ A A A A A A A "
McCarthy (QUASI \S
^^  """"
^Tsakonai et al IPPEXACT)
i i
Design J
I
I
in i i i
0.4 0.6 0.8 1.0 1.2 1.4 16 1.8 2.0
J" V(1wâ€žâ€ž)/nD
PROPELLER 4710
* â– 10 DEG
NO SCREEN
04 0.6 0.8 1.0 1.2 1.4 16 1.6
Tsakonai at al IPPEXACTI
J I l_L
0.4 0.6 8 1.0 1.2 l.i
J = V(1Â» VM )/nD
PROPELLER 4710
v = 20 DEG
NO SCREEN
Kerwin IPUF2ISI
0.4 0.6 0.8 1.0 1.2 1.4
â– V l1Â«i VM )/nD
Fig. 9  (Continued)
13
l 1 r
PROPELLER 4402
*  10 0EG
NO SCREEN
McCarthy {QUASH
Tsakonai et al IPPEXACT)
Â°T"
I
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
J = V(1w v â€ž)/nD
Fig. 9  (Continued)
Villi
\
1 I 
PROPELLER 4402
Â« â– 10OEG
NO SCREEN

Â°\
\ Â°


\ Expetime
\ Â°
O

^VKerwm
IPUF2IS)
_
/ 

,, â€” " Kerwin & Lee (PUF2I
Tsakonai et al (PPEXACT)
McCarthy "*
IQUASII
Design J
1
1 1 ill
1 1
Fig. 10  Phases of First Harmonic Blade Loads in Tangential Wakes
Correlation Between Experiment and Theory
which is opposite to the trend of the
experimental data. Since this is a
lightlyloaded linear theory, it was
anticipated that the predicted unsteady
loading coefficients Â«(p m)1Â» would be
essentially independent of J . An
earlier version of the unsteady lifting
surface method developed by Tsakonas and
his associates (21) at Davidson
Laboratory, in which the propeller
helical wake was approximated in a
staircase manner did predict that the
periodic loading coefficients K (F,M)
in axial wakes were essentially
independent of J (17) . It is not clear
why the method of Tsakonas et al (6,7)
predicts that k, f M t ]_ increases
substantially with increasing J.
In general, the predictions by the
method of Tsakonas et al are closer to
the experimental results at high advance
coefficients than they are at low
advance coefficients. The predicted
amplitudes are approximately 65 percent
of the experimental value at design J
and 30 to 35 percent of the experimental
values at the lowest values of J
evaluated. The finding that this method
predicts periodic loads in inclined flow
that are significantly less than the
experimental values is consistent with
the results presented in References 1
through 4.
The unsteady theory of Tsakonas et
al predicts that the maximum values of
the loading components will occur at
approximately 15 to 20 degrees of blade
angular position before the
experimentally determined angles; see
Figure 10.
The unsteady theory of Kerwin and
Lee (8) showed, in general, somewhat
better agreement with experimental data
over a range of advance coefficient than
either the quasisteady method of
McCarthy or the unsteady method of
Tsakonas et al. This applies to both
amplitudes and phases. The trends of
the predictions of the amplitudes over a
range of advance coefficients were very
similiar to the trends of the predic
tions by the method of McCarthy. The
method of Kerwin and Lee underpredicted
the periodic loads, by approximately 5
to 20 percent of the experimental
values, with closer agreement being
obtained for the F x and My
components at the higher values of J.
The unsteady theory of Kerwin and
Lee predicts the phases to within 5 to
15 degrees of the experimental values
14
\
McCarthy (QUASH
I I I L_J I I L
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
J = V(1Â» UM l/nD
PROPELLER 4661
* = 10DEG
NO SCREEN
J I I L_l I I L
0.4 0.6 0.8
1.0 1.2 1.4 1.(
J = VI1w VM l/nD
PROPELLER 4661
i, Â» 10DEG
NO SCREEN
J I I L
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
J = V ll.w UM )/nD
PROPELLER 4661
* > 10DEG
Kerwin (PUF2ISI NO SCREEN
 70
McCarthy (QUASI)
Design J
0.4 0.6 0.8 10 1.2 1.4 1.6 1.8 2.0
J = V (1w v â€ž)/nD
Fig. 10  (Continued)
15
1
i i i i i
PROPELLER 4661
a A
b â– 20 DEG
NO SCREEN
A \
a2
A
\6
Experiment A ^

yV

^KerwinS Lee(PUF2)
_
â€¢>Â»
\
^
McCarthy (QUASH "
â€” â– Â» â€” ^ s * *^^ Tsakonai et al
^"^(PPEXACT)
1
i
Design J "â– Â»
1
1
ill i i i
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
J = V(1w VM )/nD
\ 1 1
1 1 1 1
PROPELLER 4861
tf 30DEG
NO SCREEN
Experiment
Â°Â°oooÂ°y

/' N Kerwin (PUF2ISI
_ i ^Â»' \ Kerwin & Lee (PUF2)
â€”.___ .
â€” .
\
McCarthy (QUASH
1 1 1
\ Ttakonai et al (PPEXACTI
Design J *"""
ll 1 1 1
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
J= V ,1w VM )/nD
I I I I I I I
\ PROPELLER 4710
\ 4, = 10 DEG
\ NO SCREEN
CQ^ \ Kerwin (PUF2IS) //
V /
\ Â° A
\ o E """â„¢"' /I
\ /' Â°
\ /
 McCarthy (QUASH \v^^">V ~~
Kerwin & Lee IPUF2)
^\ T.Konaseta,
_ ' ^^^ (PPEXACTI _
1 III 1 1 1 1
 Kerwin (PUF2ISI
PROPELLER 4710
i = 20 DEG
NO SCREEN
*^ McCarthy (QUASH
Tsakonasetal (PPEXACTI
0.4 0.6 0.8 1.0 1.2 1.4 1.6
J = V(1w VM )/nD
J L_l L
0.4 0.6 0.8 1.0 1.2 1.4 1.6
J= V llw UM )/nO
Fig. 10  (Continued)
16
TABLE II  COMPARISON OF EXPERIMENT AND THEOR> IN INCLINED FLOW  AMPLITUDES AND PHASES