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is shown in Figures 18 and 19. As expected, increased sensitivity of the pressure

coefficients to J occur towards the leading edge of the blade. From two-dimensional

theory, the sensitivity should be zero at the trailing edge and monotonically in-

crease towards the leading edge. This generally occurs except on the suction side

of both propellers at the 0.5R and 0.7R radius positions, where the point of insensi-

tivity to J occurs forward of the trailing edge. In each of these cases, this point

is shown by the location at which a reversal occurs in the direction of C variation

P

with J at the 90 percent chord position. This could be due to viscous effects or by

induced velocities from a variable "tip" vortex and the trailing-vortex sheet.

The results for Propeller 4718 appear consistent with expected trends, with the

exception of the 50 percent chordwise position at the 0.5 radius on the pressure

side of the blade. A reversal in the direction of the C variation with J occurs

P

there, completely inconsistent with surrounding gage results. A polarity error in

the gage output would seem obvious, except for the proper polarity of the measured

C at design J. Also, this gage has no speed effect or loading correction, and the

measured unsteady pressures, as explained later, support this result. All this

supports a real flow effect, but given the expected behavior over the rest of

propeller, this seems unlikely.

26

Propeller 4679 showed behavior similar to Propeller 4718, with some noticeable

variations. At the 0.7 radius on the suction side, the suction peak that occurred

at the leading edge at design J remained even at the high J condition, implying not

a simple angle of attack cause, but perhaps a local effect due to blade geometry at

the leading edge. Also, the dip in -C at the 50 percent chord corresponds to a

questionable surface mounted gage. The bracket at that position represents the

range of pressures recorded in earlier tests, before the surface gage was installed.

On the pressure side of the blade, a constant sensitivity of C to J was observed in

the aft chord region. This appeared only at the 0.9 radius and could be related to

tip-vortex separation and rollup occurring on the opposite side of the blade.

At the 0.9 radius of Propeller 4679, dramatic tip effects appeared to dominate

the variation of pressure coefficient with J. On the suction side, variations with

J occurred to a greater degree than for Propeller 4718. Also, from Figure 7c, the

measured pressure coefficients at design J were larger than the theoretical predic-

tions, contrary to data for Propeller 4718. As J was reduced, large decreases in

C occurred near the leading edge. On the pressure side, variation in C was small,

and the data were mostly uniform across the chord. This behavior was substantially

different from the expected sensitivity of C to J occurring over most of the blade

sections, including the 0.9 radius of Propeller 4718.

It is hypothesized that the formation and position of the tip vortex on

Propeller 4679 produced the unconventional pressure distributions at the tip.

Figure 20 shows Propeller 4679 operating in uniform flow at advance coefficients of

1.077, 0.8, and 0.6. The carriage speed in this preliminary test series was slightly

greater than test values reported herein, causing a visible tip vortex even at design

J. All three conditions shown were at approximately the same Reynolds number. As J

was reduced, a thicker vortex core formed on the back of the blade migrating forward

along the broad tip. At J = 0.6, the tip-vortex formation seemed to begin close to

the leading edge at the 0.7 radius. Increased tip-vortex separation may have

occurred also, but this is unclear from the photographs. If the tip vortex formed

well ahead of the 0.9 radius, the tip vortex would have induced higher velocities

along the 0.9 radius, causing a decrease in the pressure coefficients on the suction

side which are strongly dependent on J. The pressure coefficients on the pressure

27

side seem little affected by the vortex, and perhaps are desensitized to J by its

position on the back. This effect at the tip occurred only on Propeller 4679 which

is characterized by swept-back blades with wide, swept tips.

An attempt was made to compare the sensitivity of C with J along the chord

30 P

with the two-dimensional theory used to predict the pressure distributions at

design. From Figure 17, the slopes of the first-order curve fits of C versus J

were plotted against chord position in Figures 21 and 22. The magnitude and sign

of the slope are proportional to the magnitude and direction of the sensitivity of

C to J. At the 0.5 and 0.7 radii, the pressure coefficients on the pressure side

of the blade were more sensitive to J than those on the suction side, while at 0.9

radius, the pressure coefficients on the suction side appeared more sensitive on

both propellers. Also, the sensitivity reversal at the trailing edge on the suction

side can be seen as a negative slope.

Similar slope distributions along the chord were approximated from the two-

dimensional theory. With the same propeller blade sections, pressure distributions

were calculated over a range of assigned angles of attack a. Slopes of these

approximately linear relationships between C and a were calculated . The predicted

slopes on each side of each section were then normalized by a constant factor so

that the predicted and experimental slopes were equal for the gages nearest to the

leading edge. This procedure was used to make simple approximate predictions of the

slope or sensitivity distribution of C to J along the chord, because no simple

relationship between effective two-dimensional angle of attack and advance coeffi-

cient is known. The predictions show roughly similar distributions of slope, but

do not predict the differences between the measured slopes on the suction and

pressure sides. Also, as expected, the predictions do not indicate any sensitivity

reversal near the trailing edge. One might conclude from the gross similarity

between prediction and measurements, that the effective three-dimensional camber

distribution is similar to that of the equivalent two-dimensional model. More

accurate comparisons with a lifting surface model should be made to confirm this

hypothesis.

28

ACCURACY OF MEASURED DATA

At the time of earlier evaluations, attempts to quantify the accuracy of the

measured mean pressures were hindered by small variations in advance coefficient.

Because the carriage speed and the propeller rotational speed are set manually, a

precise value of J cannot be set. The prescribed test matrix produced a series of

runs at values close to the specified test conditions. The dependence of certain

gages on speed or Reynolds number further hindered an quantification of the accuracy

of the measurement system.

To overcome these problems, an error analysis was conducted based on the C

P

versus J curves in Figure 17. The mean pressure coefficient represented in these

curves had been corrected for Reynolds number dependence, as described earlier.

Therefore, this accepted Reynolds number effect, whether being an instrumentation

error or a real-flow phenomenon, had been eliminated in these figures. First- and

second-order curves were least squares fit to these speed-corrected pressure coeffi-

cients over a range of J, and then a standard error for each curve was calculated.

The standard error represents the standard deviation of the measured pressure

coefficients from the least squares curve-fit values. The standard error was

multiplied by 1.96 to represent the standard error at a 95 percent confidence level.

This implies that, if one assumes a normal distribution of the variation of measured

pressure coefficients from the curve fit values, then 95 percent of the measured

pressure coefficients fall within plus or minus the value of the standard error from

the curve-fit result. This procedure permitted the use of the entire test matrix,

over a range of J and carriage speed, in calculating a statistical error band. Also,

small variations in J, for a given test condition, were properly accounted for. The

resulting nondimensional error bands in +C , are shown in Table 7.

These results were extended to provide a dimensional error band in terms of a

dimensional pressure. The standard error process was modified to calculate dimen-

sional pressures and arrive at a 95 percent confidence level error band in psi that

could be compared to the approximated error band of the measured pressures during

calibration. These results, shown in Table 8 for the two propellers tested, indi-

cate, in the best case of Propeller 4718 in uniform flow with a second-order curve

fit, an average error band very close to the predicted error from the calibrations.

Most other cases indicate a test error band up to twice the predicted error based on

calibration error.

29

These results, especially in the best case, are very encouraging, indicating an

observed test accuracy similar to the expected accuracy of the gages. The only

discrepancy in the overall result is the Reynolds number effect, which when corrected

for, produces test accuracy similar to the expected accuracy of the instrumentation.

The runs conducted with Propeller 4718 in inclined flow produced noticeably

larger error bands than the uniform flow runs. Some of the increase was due to the

inclusion of one or two questionable runs in the inclined flow case. The general

policy was to remove bad runs from results if justifying errors where found. If

errors were concluded to be random for given gages, then the result was not removed.

This type of error can be seen in Figure 13. Another possible error in the inclined

flow runs was the use of speed corrections generated from the uniform flow runs.

Any difference in the speed dependence between inclined and uniform flow runs would

show up as an error in the inclined flow result. It appeared that on some gages with

large speed corrections, for example, Gage 25, the C values in inclined flow did not

collapse onto the fitted curve as well as in uniform flow. Another possible source

of error in inclined flow could have resulted from instrumentation problems asso-

ciated with maintaining and measuring carriage speed that occurred at the beginning

of the inclined-flow measurements with Propeller 4718.

The average error bands generated from runs of Propeller 4679 in inclined flow

and uniform flow are both noticeably larger than the best case. Table 8b indicates

many gages having numerous bad runs that were not removed from the error analysis,

implying no obvious gage malfunction. It was generally felt that the gages on

Propeller 4679 were less reliable due to previous use on two other tests. These

gages were more prone to zero shifts during a given run, which would cause random

errors in the pressure measurements. Fortunately, most gages performed properly

in both uniform and inclined flow so that C versus J measurements were available.

P

Speed correction problems did not occur in the error analysis due to the small

speed dependence of most of the gages.

Generally, error bands were reduced on both propellers when the second-order

curve fit was used. From Figure 17 it is obvious that certain gages displayed a

nonlinear behavior that was better fitted by the second-order curves. Where no

improvement occurred using a higher order fit, then the C versus J relationship

could be assumed linear.

30

Accuracy in the measurement of carriage speed and propeller rotational speed

would also affect the overall accuracy of the pressure measurements. No determina-

tion was made to evaluate the accuracy of these measured quantities, but given the

good results of the best-case test error, it was felt these measurements were

accurately made.

The only remaining assumption in the measurement process that could be ques-

tioned was the equating of the carriage speed to the advance speed V through the

propeller disk. This assumption is always made in basin testing; however, with the

large size of the dynamometer, small amplitude, low-frequency standing waves were

setup after a few runs. These standing waves caused small additional velocities in

the basin. It was assumed that this effect would average out over a run, and given

the accuracy of the best case, was neglected.

The accuracy of the measured fluctuating pressures was generally good. Repeat-

ability was the only indication of accuracy in this case because no consistent

governing trend existed for unsteady pressures. Error bands with a 95 percent con-

fidence level were calculated for first harmonic amplitude C 1 and first harmonic

phase (J> 1 from the repeat runs conducted at each given test condition in inclined

flow. Propeller 4718 produced an average error band of AC = +0.002 and A(j> 1 =

+4 deg, while Propeller 4679 produced expected larger average error values of AC - =

+0.005 and A(j> 1 = +8 deg. This average error band was relatively small for typical

first harmonic amplitudes in a range greater than C 1 = 0.0150, but in some cases on

the pressure side of the propeller blade, values of C ^ were less than 0.0050, thus

causing uncertainty in the measured amplitude and also in the measured phase.

FLUCTUATING PRESSURE MEASUREMENTS AT DESIGN J

Periodic pressure measurements were obtained when the propellers were operated

in inclined flow. The 7.5 deg shaft inclination produced a first harmonic, once per

revolution variation in the measured pressure. A typical variation of pressure with

gage angular position is shown in Figure 23. As expected, the pressure variation

was primarily first harmonic, with negligible higher harmonics observed, attributed

to noise. The fluctuating pressure is represented as the first harmonic pressure

31

coefficient amplitude C .. and the corresponding lagging cosine series phase, ()> 1 :

C (9) = C cos (e-fj).)

p p j. x

This result at design J is presented for Propellers 4679 and 4718 in Figures 24

and 25. Included in the figures are fluctuating and quasi-steady predictions. The

small effect of speed or Reynolds number is depicted by the similarity in C 1 and (j> 1

at two speeds. There appears to be no correlation between the Reynolds number de-

pendency of certain pressure gages measuring mean pressure and the same gages

measuring unsteady pressures.

The corrections to the fluctuating pressure measurements due to loading are

shown in Figures 9 and 10 for Propellers 4679 and 4718 at design J. Note that no

corrections due to loading occur at r/R = 0.9 on either propeller attributed to the

use of the coverplate gage installations. The locations of the largest corrections

are the 0.5 and 0.7 radius positions on the suction side of Propeller 4718. These

loading corrections were determined from a quasi-steady analysis of the measured

mean load corrections in uniform flow. This approximation places some uncertainty

on the unsteady measurements associated with gage positions with large corrections,

and the difference between the corrected and uncorrected pressure measurement could,

conservatively, provide an envelope for the actual measured result.

Before correlating the measured fluctuating results to the unsteady and quasi-

steady predictions, a detailed description of the quasi-steady technique is

necessary.

QUASI-STEADY PROCEDURE FOR PREDICTING FLUCTUATING

PRESSURE DISTRIBUTIONS

The quasi-steady analysis for predicting the fluctuating pressures was an

32

adaptation of a quasi-steady procedure by McCarthy for predicting fluctuating

thrust and torque on a propeller. The procedure predicts the fluctuating propeller

loads from the steady open-water propeller performance characteristics. The proce-

dure is applied to predict unsteady pressures using the C versus J curves in

Figure 13. The procedure is identical to the technique used earlier to approximate

32

the fluctuating load correction. Fluctuating pressure is produced by the variations

in local advance coefficient, J(6) and resultant inflow speed V^CB) as the propeller

blade rotates through a spatially nonuniform wake.

In inclined flow, the quasi-steady procedure is relatively simple due to the

simple nonuniform wake. The flow inclination, as seen in Figure 26, produces a uni-

form downward component of tangential velocity V . This tangential velocity compo-

nent adds to the propeller's angular rotational speed when the blade is moving upward

at = 270 deg, and subtracts from its rotational speed at 8 = 90 deg as shown

in Figure 26b. This variation in rotational speed produces a variation in local

advance coefficient J (6), with a maximum value at 9 = 90 deg, and a minimum value at

= 270 deg, as shown.

V A V A

Jmax = J(90) = Hn _^ /2llr) Jmin = J(270) = Hn+ ^ /2lJr)

where Vâ€ž, is V sin (7.5 deg), and V. a V

T c 6 A c

The sinusoidal variation in J(0) produces a sinusoidal variation in pressure in

the blade based on the C versus J curves in Figure 17 as shown in Figure 26c. Also,

P

J . and J produce corresponding pressure coefficients, C . and C . The

mm max pJmxn pjmax

maximum and minimum pressures calculated from the pressure coefficients are,

(p-p ) T = C _ â€¢ l/2pV 2 (90)

r *o Jmax pJmax R

(p-p ). . = C . â€¢ l/2pV_ 2 (270)

r *o Jmin pJmax R

where V 2 (90) = V 2 + [2TTr(n-V T /2wr)] 2

K C 1

V R 2 (270) = V c 2 + [27rr(n+V T /2Trr)] 2

33

The first harmonic pressure coefficient is approximated by,

^"Vjmax (P-^Jmin

J pl

l/2p[V 2 +(2TTrn) 2 ]

This information produces a lagging cosine series phase angle <(>.. , as defined by

Equation (1), of 270 deg if C . is negative, and 90 deg if C - is positive. Sub-

stituting into the previous equations, C 1 can be represented as

~ pjmin

pi ~ 2

V 2 +(27Trn+V ril ) 2

c T

V 2 +(2TTrn) 2

c

"pJmax

V 2 +(27Trn-V T ) 2

c 1

V +(2irrn)

The first harmonic pressure coefficient C 1 , can be seen to depend upon two effects.

One is the local variation in J producing the C T . and C T terms. The other is

pjmm pJmax

the speed correction of those terms due to the local variation in speed V,,, repre-

sented by the ratios inside the brackets. Term C T . will always be increased by

' pjmin J J

the speed correction by a constant ratio, dependent upon radial position for a given

operating condition. In a similar manner, C will always be decreased.

From this result, trends can be observed in the predicted quasi-steady first

harmonic pressure coefficients. Figure 27 demonstrates typical quasi-steady calcu-

lations on the suction and pressure sides of the propeller blade. Note that the

magnitude of the slopes of the C versus J plots for the suction side and pressure

side of the blade are roughly similar, but the pressure side has a negative slope

while the suction side has a positive slope. This slope polarity difference will

produce an opposite effect of the quasi-steady speed correction in calculating the

first harmonic pressure coefficients. The speed correction will tend to decrease

the first harmonic pressure coefficient on the pressure side of the blade, and

increase it on the suction side. This trend is due to only the difference in local

velocities at J , and J . , and the signs of slopes of the C versus J curves,

max mm p

34

The speed correction term produces a dependency of the first harmonic pressure

coefficient on the magnitude of the mean pressure coefficient, C . Since the ve-

P

locity correction terms are constants multiplied by C , . , and C . increased

pJmxn __ pJmax

values of C _ . and C _ will produce an increased value of C , . This trend is

pJmin pJmax pi

important when observing C .. over a range of J, and when considering the accuracy of

the first harmonic pressure coefficients, C 1 generated from values of C with large

speed effects.

The quasi-steady analysis represents an intuitive description of the fluctuating

pressure, excluding any unsteady effects. It provides a good base for comparison of

33 34

the measured data for the two propellers, and the unsteady theory by Tsakonas. '

The correlation between the measured and quasi-steady results can also be compared

to similar correlations of fluctuating blade loads performed by Boswell and Jessup.

CORRELATION OF FIRST HARMONIC PRESSURE COEFFICIENTS WITH THEORY

The measured first harmonic pressure coefficient in Figures 22 and 23 generally

tend to decrease in amplitude from leading to trailing edge. This trend was gen-

erally approximated by the quasi-steady approach, but with an amplitude 30 percent

to 50 percent less than the measured result. This result matched similar correla-

tions of quasi-steady and measured fluctuating blade loads by Boswell and Jessup. '

Intuitively, the observed trend from leading edge to trailing edge was reasonable due

to the higher sensitivity of the leading-edge pressures to angle-of -attack variation.

Good correlation with the quasi-steady predictions was due partially to the shaft-

rate frequency of the nonuniform tangential wake. Fluctuating effects will be small

for low-frequency, shaft-rate variations in the wake. Therefore, with small fluc-

tuating effects, a quasi-steady analysis should provide close agreement to the

measured result. Also, good correlation may be due to the incorporation of measured

mean results in the quasi-steady procedure, avoiding possible errors by the predic-

tion of mean pressure variation with advance coefficient. The unsteady theory by

33 34

Tsakonas et al. ' produced a reduction in the first harmonic pressures in the

first quarter chord at each radial station. The extreme nature of this trend as

compared to both the measured and quasi-steady results produced little confidence in

the accuracy of the method of Tsakonas et al.

35

The quasi-steady method and the experiment both indicate that the first harmonic

amplitudes of the pressures are larger on the suction side of the blade than on the

pressure side of the blade. This trend was consistent for both propellers except

for the measurements nearest to the leading edge at the 0.5 and 0.7 radius stations,

where the results on the pressure side were larger than the results on the suction

side. This variation in fluctuating loading between the suction and pressure sides

of the blades did not occur in the theoretical prediction method of Tsakonas et al.,

further supporting the hypothesis that this method does not adequately predict the

distribution of pressures.

On the suction side of Propeller 4718 at the 0.5 and 0.7 radius positions, the

quasi-steady analysis at certain chordwise positions over -predicted the measured

first harmonic amplitudes. These over-predicted values were partially due to the

strong dependence of the quasi-steady result on the magnitude of the mean pressure

coefficient, C . Some of the C measurements from the 40 to 70 percent chordwise

P P *

locations exhibited a relatively strong speed dependence. The mean pressure coeffi-

cients were corrected upward for the speed effect, leaving -C values greater than

average over the range of speeds conducted. This would artifically increase the

quasi-steady results. A calculation of the quasi-steady results with reduced C

values did not reduce the quasi-steady first harmonics enough to match the general

trend completely, possibly implying inaccuracies or over-simplifications in the

quasi-steady analysis.

CORRELATION OF FIRST HARMONIC PHASE ANGLE WITH THEORY

The measured first harmonic phase angles were generally in the range expected.

On the suction side of the blade, most phase angles ranged from 60 deg to 120 deg,

while on the pressure side of the blade, values ranged from 270 deg to 333 deg.

There was no specific variation in phase angle over the chord. Cases of gradual

phase angle increase, decrease, and consistency occurred over the chord, with some

instances of sudden drops in phase angle near the trailing edge. No overall trend

in phase angle occurred, but certain blade sections produced similarities between the

two propellers tested.

36

On the suction side of the blade, at the 0.5 radius position, the phase angle

of the first harmonic on the two propellers was strikingly similar. Values of

120 deg at the leading edge dropped slightly below the quasi-steady phase of 90 deg,

remaining constant over most of the chord. In each case, the phase dropped substan-

tially at the trailing edge, to approximately 100 deg. This drop in phase angle at

the trailing edge was not predicted in the quasi-steady analysis, but justification

coefficients to J occur towards the leading edge of the blade. From two-dimensional

theory, the sensitivity should be zero at the trailing edge and monotonically in-

crease towards the leading edge. This generally occurs except on the suction side

of both propellers at the 0.5R and 0.7R radius positions, where the point of insensi-

tivity to J occurs forward of the trailing edge. In each of these cases, this point

is shown by the location at which a reversal occurs in the direction of C variation

P

with J at the 90 percent chord position. This could be due to viscous effects or by

induced velocities from a variable "tip" vortex and the trailing-vortex sheet.

The results for Propeller 4718 appear consistent with expected trends, with the

exception of the 50 percent chordwise position at the 0.5 radius on the pressure

side of the blade. A reversal in the direction of the C variation with J occurs

P

there, completely inconsistent with surrounding gage results. A polarity error in

the gage output would seem obvious, except for the proper polarity of the measured

C at design J. Also, this gage has no speed effect or loading correction, and the

measured unsteady pressures, as explained later, support this result. All this

supports a real flow effect, but given the expected behavior over the rest of

propeller, this seems unlikely.

26

Propeller 4679 showed behavior similar to Propeller 4718, with some noticeable

variations. At the 0.7 radius on the suction side, the suction peak that occurred

at the leading edge at design J remained even at the high J condition, implying not

a simple angle of attack cause, but perhaps a local effect due to blade geometry at

the leading edge. Also, the dip in -C at the 50 percent chord corresponds to a

questionable surface mounted gage. The bracket at that position represents the

range of pressures recorded in earlier tests, before the surface gage was installed.

On the pressure side of the blade, a constant sensitivity of C to J was observed in

the aft chord region. This appeared only at the 0.9 radius and could be related to

tip-vortex separation and rollup occurring on the opposite side of the blade.

At the 0.9 radius of Propeller 4679, dramatic tip effects appeared to dominate

the variation of pressure coefficient with J. On the suction side, variations with

J occurred to a greater degree than for Propeller 4718. Also, from Figure 7c, the

measured pressure coefficients at design J were larger than the theoretical predic-

tions, contrary to data for Propeller 4718. As J was reduced, large decreases in

C occurred near the leading edge. On the pressure side, variation in C was small,

and the data were mostly uniform across the chord. This behavior was substantially

different from the expected sensitivity of C to J occurring over most of the blade

sections, including the 0.9 radius of Propeller 4718.

It is hypothesized that the formation and position of the tip vortex on

Propeller 4679 produced the unconventional pressure distributions at the tip.

Figure 20 shows Propeller 4679 operating in uniform flow at advance coefficients of

1.077, 0.8, and 0.6. The carriage speed in this preliminary test series was slightly

greater than test values reported herein, causing a visible tip vortex even at design

J. All three conditions shown were at approximately the same Reynolds number. As J

was reduced, a thicker vortex core formed on the back of the blade migrating forward

along the broad tip. At J = 0.6, the tip-vortex formation seemed to begin close to

the leading edge at the 0.7 radius. Increased tip-vortex separation may have

occurred also, but this is unclear from the photographs. If the tip vortex formed

well ahead of the 0.9 radius, the tip vortex would have induced higher velocities

along the 0.9 radius, causing a decrease in the pressure coefficients on the suction

side which are strongly dependent on J. The pressure coefficients on the pressure

27

side seem little affected by the vortex, and perhaps are desensitized to J by its

position on the back. This effect at the tip occurred only on Propeller 4679 which

is characterized by swept-back blades with wide, swept tips.

An attempt was made to compare the sensitivity of C with J along the chord

30 P

with the two-dimensional theory used to predict the pressure distributions at

design. From Figure 17, the slopes of the first-order curve fits of C versus J

were plotted against chord position in Figures 21 and 22. The magnitude and sign

of the slope are proportional to the magnitude and direction of the sensitivity of

C to J. At the 0.5 and 0.7 radii, the pressure coefficients on the pressure side

of the blade were more sensitive to J than those on the suction side, while at 0.9

radius, the pressure coefficients on the suction side appeared more sensitive on

both propellers. Also, the sensitivity reversal at the trailing edge on the suction

side can be seen as a negative slope.

Similar slope distributions along the chord were approximated from the two-

dimensional theory. With the same propeller blade sections, pressure distributions

were calculated over a range of assigned angles of attack a. Slopes of these

approximately linear relationships between C and a were calculated . The predicted

slopes on each side of each section were then normalized by a constant factor so

that the predicted and experimental slopes were equal for the gages nearest to the

leading edge. This procedure was used to make simple approximate predictions of the

slope or sensitivity distribution of C to J along the chord, because no simple

relationship between effective two-dimensional angle of attack and advance coeffi-

cient is known. The predictions show roughly similar distributions of slope, but

do not predict the differences between the measured slopes on the suction and

pressure sides. Also, as expected, the predictions do not indicate any sensitivity

reversal near the trailing edge. One might conclude from the gross similarity

between prediction and measurements, that the effective three-dimensional camber

distribution is similar to that of the equivalent two-dimensional model. More

accurate comparisons with a lifting surface model should be made to confirm this

hypothesis.

28

ACCURACY OF MEASURED DATA

At the time of earlier evaluations, attempts to quantify the accuracy of the

measured mean pressures were hindered by small variations in advance coefficient.

Because the carriage speed and the propeller rotational speed are set manually, a

precise value of J cannot be set. The prescribed test matrix produced a series of

runs at values close to the specified test conditions. The dependence of certain

gages on speed or Reynolds number further hindered an quantification of the accuracy

of the measurement system.

To overcome these problems, an error analysis was conducted based on the C

P

versus J curves in Figure 17. The mean pressure coefficient represented in these

curves had been corrected for Reynolds number dependence, as described earlier.

Therefore, this accepted Reynolds number effect, whether being an instrumentation

error or a real-flow phenomenon, had been eliminated in these figures. First- and

second-order curves were least squares fit to these speed-corrected pressure coeffi-

cients over a range of J, and then a standard error for each curve was calculated.

The standard error represents the standard deviation of the measured pressure

coefficients from the least squares curve-fit values. The standard error was

multiplied by 1.96 to represent the standard error at a 95 percent confidence level.

This implies that, if one assumes a normal distribution of the variation of measured

pressure coefficients from the curve fit values, then 95 percent of the measured

pressure coefficients fall within plus or minus the value of the standard error from

the curve-fit result. This procedure permitted the use of the entire test matrix,

over a range of J and carriage speed, in calculating a statistical error band. Also,

small variations in J, for a given test condition, were properly accounted for. The

resulting nondimensional error bands in +C , are shown in Table 7.

These results were extended to provide a dimensional error band in terms of a

dimensional pressure. The standard error process was modified to calculate dimen-

sional pressures and arrive at a 95 percent confidence level error band in psi that

could be compared to the approximated error band of the measured pressures during

calibration. These results, shown in Table 8 for the two propellers tested, indi-

cate, in the best case of Propeller 4718 in uniform flow with a second-order curve

fit, an average error band very close to the predicted error from the calibrations.

Most other cases indicate a test error band up to twice the predicted error based on

calibration error.

29

These results, especially in the best case, are very encouraging, indicating an

observed test accuracy similar to the expected accuracy of the gages. The only

discrepancy in the overall result is the Reynolds number effect, which when corrected

for, produces test accuracy similar to the expected accuracy of the instrumentation.

The runs conducted with Propeller 4718 in inclined flow produced noticeably

larger error bands than the uniform flow runs. Some of the increase was due to the

inclusion of one or two questionable runs in the inclined flow case. The general

policy was to remove bad runs from results if justifying errors where found. If

errors were concluded to be random for given gages, then the result was not removed.

This type of error can be seen in Figure 13. Another possible error in the inclined

flow runs was the use of speed corrections generated from the uniform flow runs.

Any difference in the speed dependence between inclined and uniform flow runs would

show up as an error in the inclined flow result. It appeared that on some gages with

large speed corrections, for example, Gage 25, the C values in inclined flow did not

collapse onto the fitted curve as well as in uniform flow. Another possible source

of error in inclined flow could have resulted from instrumentation problems asso-

ciated with maintaining and measuring carriage speed that occurred at the beginning

of the inclined-flow measurements with Propeller 4718.

The average error bands generated from runs of Propeller 4679 in inclined flow

and uniform flow are both noticeably larger than the best case. Table 8b indicates

many gages having numerous bad runs that were not removed from the error analysis,

implying no obvious gage malfunction. It was generally felt that the gages on

Propeller 4679 were less reliable due to previous use on two other tests. These

gages were more prone to zero shifts during a given run, which would cause random

errors in the pressure measurements. Fortunately, most gages performed properly

in both uniform and inclined flow so that C versus J measurements were available.

P

Speed correction problems did not occur in the error analysis due to the small

speed dependence of most of the gages.

Generally, error bands were reduced on both propellers when the second-order

curve fit was used. From Figure 17 it is obvious that certain gages displayed a

nonlinear behavior that was better fitted by the second-order curves. Where no

improvement occurred using a higher order fit, then the C versus J relationship

could be assumed linear.

30

Accuracy in the measurement of carriage speed and propeller rotational speed

would also affect the overall accuracy of the pressure measurements. No determina-

tion was made to evaluate the accuracy of these measured quantities, but given the

good results of the best-case test error, it was felt these measurements were

accurately made.

The only remaining assumption in the measurement process that could be ques-

tioned was the equating of the carriage speed to the advance speed V through the

propeller disk. This assumption is always made in basin testing; however, with the

large size of the dynamometer, small amplitude, low-frequency standing waves were

setup after a few runs. These standing waves caused small additional velocities in

the basin. It was assumed that this effect would average out over a run, and given

the accuracy of the best case, was neglected.

The accuracy of the measured fluctuating pressures was generally good. Repeat-

ability was the only indication of accuracy in this case because no consistent

governing trend existed for unsteady pressures. Error bands with a 95 percent con-

fidence level were calculated for first harmonic amplitude C 1 and first harmonic

phase (J> 1 from the repeat runs conducted at each given test condition in inclined

flow. Propeller 4718 produced an average error band of AC = +0.002 and A(j> 1 =

+4 deg, while Propeller 4679 produced expected larger average error values of AC - =

+0.005 and A(j> 1 = +8 deg. This average error band was relatively small for typical

first harmonic amplitudes in a range greater than C 1 = 0.0150, but in some cases on

the pressure side of the propeller blade, values of C ^ were less than 0.0050, thus

causing uncertainty in the measured amplitude and also in the measured phase.

FLUCTUATING PRESSURE MEASUREMENTS AT DESIGN J

Periodic pressure measurements were obtained when the propellers were operated

in inclined flow. The 7.5 deg shaft inclination produced a first harmonic, once per

revolution variation in the measured pressure. A typical variation of pressure with

gage angular position is shown in Figure 23. As expected, the pressure variation

was primarily first harmonic, with negligible higher harmonics observed, attributed

to noise. The fluctuating pressure is represented as the first harmonic pressure

31

coefficient amplitude C .. and the corresponding lagging cosine series phase, ()> 1 :

C (9) = C cos (e-fj).)

p p j. x

This result at design J is presented for Propellers 4679 and 4718 in Figures 24

and 25. Included in the figures are fluctuating and quasi-steady predictions. The

small effect of speed or Reynolds number is depicted by the similarity in C 1 and (j> 1

at two speeds. There appears to be no correlation between the Reynolds number de-

pendency of certain pressure gages measuring mean pressure and the same gages

measuring unsteady pressures.

The corrections to the fluctuating pressure measurements due to loading are

shown in Figures 9 and 10 for Propellers 4679 and 4718 at design J. Note that no

corrections due to loading occur at r/R = 0.9 on either propeller attributed to the

use of the coverplate gage installations. The locations of the largest corrections

are the 0.5 and 0.7 radius positions on the suction side of Propeller 4718. These

loading corrections were determined from a quasi-steady analysis of the measured

mean load corrections in uniform flow. This approximation places some uncertainty

on the unsteady measurements associated with gage positions with large corrections,

and the difference between the corrected and uncorrected pressure measurement could,

conservatively, provide an envelope for the actual measured result.

Before correlating the measured fluctuating results to the unsteady and quasi-

steady predictions, a detailed description of the quasi-steady technique is

necessary.

QUASI-STEADY PROCEDURE FOR PREDICTING FLUCTUATING

PRESSURE DISTRIBUTIONS

The quasi-steady analysis for predicting the fluctuating pressures was an

32

adaptation of a quasi-steady procedure by McCarthy for predicting fluctuating

thrust and torque on a propeller. The procedure predicts the fluctuating propeller

loads from the steady open-water propeller performance characteristics. The proce-

dure is applied to predict unsteady pressures using the C versus J curves in

Figure 13. The procedure is identical to the technique used earlier to approximate

32

the fluctuating load correction. Fluctuating pressure is produced by the variations

in local advance coefficient, J(6) and resultant inflow speed V^CB) as the propeller

blade rotates through a spatially nonuniform wake.

In inclined flow, the quasi-steady procedure is relatively simple due to the

simple nonuniform wake. The flow inclination, as seen in Figure 26, produces a uni-

form downward component of tangential velocity V . This tangential velocity compo-

nent adds to the propeller's angular rotational speed when the blade is moving upward

at = 270 deg, and subtracts from its rotational speed at 8 = 90 deg as shown

in Figure 26b. This variation in rotational speed produces a variation in local

advance coefficient J (6), with a maximum value at 9 = 90 deg, and a minimum value at

= 270 deg, as shown.

V A V A

Jmax = J(90) = Hn _^ /2llr) Jmin = J(270) = Hn+ ^ /2lJr)

where Vâ€ž, is V sin (7.5 deg), and V. a V

T c 6 A c

The sinusoidal variation in J(0) produces a sinusoidal variation in pressure in

the blade based on the C versus J curves in Figure 17 as shown in Figure 26c. Also,

P

J . and J produce corresponding pressure coefficients, C . and C . The

mm max pJmxn pjmax

maximum and minimum pressures calculated from the pressure coefficients are,

(p-p ) T = C _ â€¢ l/2pV 2 (90)

r *o Jmax pJmax R

(p-p ). . = C . â€¢ l/2pV_ 2 (270)

r *o Jmin pJmax R

where V 2 (90) = V 2 + [2TTr(n-V T /2wr)] 2

K C 1

V R 2 (270) = V c 2 + [27rr(n+V T /2Trr)] 2

33

The first harmonic pressure coefficient is approximated by,

^"Vjmax (P-^Jmin

J pl

l/2p[V 2 +(2TTrn) 2 ]

This information produces a lagging cosine series phase angle <(>.. , as defined by

Equation (1), of 270 deg if C . is negative, and 90 deg if C - is positive. Sub-

stituting into the previous equations, C 1 can be represented as

~ pjmin

pi ~ 2

V 2 +(27Trn+V ril ) 2

c T

V 2 +(2TTrn) 2

c

"pJmax

V 2 +(27Trn-V T ) 2

c 1

V +(2irrn)

The first harmonic pressure coefficient C 1 , can be seen to depend upon two effects.

One is the local variation in J producing the C T . and C T terms. The other is

pjmm pJmax

the speed correction of those terms due to the local variation in speed V,,, repre-

sented by the ratios inside the brackets. Term C T . will always be increased by

' pjmin J J

the speed correction by a constant ratio, dependent upon radial position for a given

operating condition. In a similar manner, C will always be decreased.

From this result, trends can be observed in the predicted quasi-steady first

harmonic pressure coefficients. Figure 27 demonstrates typical quasi-steady calcu-

lations on the suction and pressure sides of the propeller blade. Note that the

magnitude of the slopes of the C versus J plots for the suction side and pressure

side of the blade are roughly similar, but the pressure side has a negative slope

while the suction side has a positive slope. This slope polarity difference will

produce an opposite effect of the quasi-steady speed correction in calculating the

first harmonic pressure coefficients. The speed correction will tend to decrease

the first harmonic pressure coefficient on the pressure side of the blade, and

increase it on the suction side. This trend is due to only the difference in local

velocities at J , and J . , and the signs of slopes of the C versus J curves,

max mm p

34

The speed correction term produces a dependency of the first harmonic pressure

coefficient on the magnitude of the mean pressure coefficient, C . Since the ve-

P

locity correction terms are constants multiplied by C , . , and C . increased

pJmxn __ pJmax

values of C _ . and C _ will produce an increased value of C , . This trend is

pJmin pJmax pi

important when observing C .. over a range of J, and when considering the accuracy of

the first harmonic pressure coefficients, C 1 generated from values of C with large

speed effects.

The quasi-steady analysis represents an intuitive description of the fluctuating

pressure, excluding any unsteady effects. It provides a good base for comparison of

33 34

the measured data for the two propellers, and the unsteady theory by Tsakonas. '

The correlation between the measured and quasi-steady results can also be compared

to similar correlations of fluctuating blade loads performed by Boswell and Jessup.

CORRELATION OF FIRST HARMONIC PRESSURE COEFFICIENTS WITH THEORY

The measured first harmonic pressure coefficient in Figures 22 and 23 generally

tend to decrease in amplitude from leading to trailing edge. This trend was gen-

erally approximated by the quasi-steady approach, but with an amplitude 30 percent

to 50 percent less than the measured result. This result matched similar correla-

tions of quasi-steady and measured fluctuating blade loads by Boswell and Jessup. '

Intuitively, the observed trend from leading edge to trailing edge was reasonable due

to the higher sensitivity of the leading-edge pressures to angle-of -attack variation.

Good correlation with the quasi-steady predictions was due partially to the shaft-

rate frequency of the nonuniform tangential wake. Fluctuating effects will be small

for low-frequency, shaft-rate variations in the wake. Therefore, with small fluc-

tuating effects, a quasi-steady analysis should provide close agreement to the

measured result. Also, good correlation may be due to the incorporation of measured

mean results in the quasi-steady procedure, avoiding possible errors by the predic-

tion of mean pressure variation with advance coefficient. The unsteady theory by

33 34

Tsakonas et al. ' produced a reduction in the first harmonic pressures in the

first quarter chord at each radial station. The extreme nature of this trend as

compared to both the measured and quasi-steady results produced little confidence in

the accuracy of the method of Tsakonas et al.

35

The quasi-steady method and the experiment both indicate that the first harmonic

amplitudes of the pressures are larger on the suction side of the blade than on the

pressure side of the blade. This trend was consistent for both propellers except

for the measurements nearest to the leading edge at the 0.5 and 0.7 radius stations,

where the results on the pressure side were larger than the results on the suction

side. This variation in fluctuating loading between the suction and pressure sides

of the blades did not occur in the theoretical prediction method of Tsakonas et al.,

further supporting the hypothesis that this method does not adequately predict the

distribution of pressures.

On the suction side of Propeller 4718 at the 0.5 and 0.7 radius positions, the

quasi-steady analysis at certain chordwise positions over -predicted the measured

first harmonic amplitudes. These over-predicted values were partially due to the

strong dependence of the quasi-steady result on the magnitude of the mean pressure

coefficient, C . Some of the C measurements from the 40 to 70 percent chordwise

P P *

locations exhibited a relatively strong speed dependence. The mean pressure coeffi-

cients were corrected upward for the speed effect, leaving -C values greater than

average over the range of speeds conducted. This would artifically increase the

quasi-steady results. A calculation of the quasi-steady results with reduced C

values did not reduce the quasi-steady first harmonics enough to match the general

trend completely, possibly implying inaccuracies or over-simplifications in the

quasi-steady analysis.

CORRELATION OF FIRST HARMONIC PHASE ANGLE WITH THEORY

The measured first harmonic phase angles were generally in the range expected.

On the suction side of the blade, most phase angles ranged from 60 deg to 120 deg,

while on the pressure side of the blade, values ranged from 270 deg to 333 deg.

There was no specific variation in phase angle over the chord. Cases of gradual

phase angle increase, decrease, and consistency occurred over the chord, with some

instances of sudden drops in phase angle near the trailing edge. No overall trend

in phase angle occurred, but certain blade sections produced similarities between the

two propellers tested.

36

On the suction side of the blade, at the 0.5 radius position, the phase angle

of the first harmonic on the two propellers was strikingly similar. Values of

120 deg at the leading edge dropped slightly below the quasi-steady phase of 90 deg,

remaining constant over most of the chord. In each case, the phase dropped substan-

tially at the trailing edge, to approximately 100 deg. This drop in phase angle at

the trailing edge was not predicted in the quasi-steady analysis, but justification

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