requires considerable computer time, cannot be run for all values of parameters so
as to check convergence of the solution for each design. Instead, it may be enough
to check several aspects of parametric changes for a typical case and to assume that
the other cases will reasonably follow the typical case.
In the present work, designing the Model 4717 supercavitating propeller, various
convergence checks have been performed, and the results will be shown in the follow-
up. The design conditions for Model 4717C are shown in Table 1.
The cavity truncation locations were varied from 1.5 to 2.8 chord lengths to
check the cavity model. Figures 5 and 6 show that the truncation at 2.2 chord
lengths and at 2.5 chord lengths produces almost the same pitch and camber distri-
butions (differing less than 1%) ; the pitch distribution is about 3% larger than at
1.8 chord lengths.
The influence of the choice of angular interval on pitch and camber distri-
bution is also shown in Figures 5 and 6. By changing 2-deg intervals to 1-deg
intervals, the camber correction factor c increased about 6% and the pitch diameter
ratio P/D increased about 1%. Since the magnitude of camber is so small, a change
in the correction factor of 10% is within the manufacturing error. If the blade tip
has more than five intervals, 2-deg intervals are sufficient; finer intervals, which
increase costs significantly, appear to be unnecessary.
To check whether there were enough collocation points, we chose 10 points and
created the foil shape by plotting streamlines. The results have been compared with
the case of four collocation points given in Figures 7 and 8. The simple approxi-
mation obtained using Equation (39) is shown to express amazingly close agreement
with the plotted streamlines.
In Equation (28) VвАЮ is the local mean speed, which is not known without exami-
nation, so the value from the lifting line theory has been substituted for it. How-
ever, a more accurate formulation would be Equation (53) (Appendix A) instead of
V /^V\)V\V/V V
The computer results for these two cases were almost the same. The present program
used Equation (47), although slightly more computer time was needed. When G/V is
quite large, it may improve the solution.
To check whether the solution satisfies the boundary conditions well, the left-
hand and right-hand sides of Equation (47) were plotted in Figures 9-12; and the
radial components of velocities on the blades were plotted, Figures 13 and 14. These
calculations were made with and without the hub boundary conditions being satisfied
when the degrees of p and x in the double polynomials in Equations (31) through (33)
were taken as 3 in one case and as 5 in the other case. During this process, we
noticed an interesting phenomenon: an instability occurred in the numerical value
of radial velocity for the solution which did not satisfy the hub boundary condition.
That is, if the hub boundary condition was not specified, a slight change of parame-
ters, such as cavity length, number of intervals, or the degree of polynomials,
produced large changes in radial velocities. Yet, the numerical values of the thrust
and torque coefficients or the pitch distribution did not change too much. This may
be because the linear boundary conditions on a cavity or foil do not include any
constraint on the radial velocity. In the present problem, the only constraint on
the radial velocity is on the hub condition. Thus, the hub boundary condition is
needed not only to find the hub effect on the pitch distribution but also to make
the solution stable.
The specified boundary conditions are well satisfied in general, although when
the boundary condition on the hub is included, the cavity conditions are slightly
less accurate, as shown in Figures 11 and 12. The differences in the radial veloc-
ities occurring for the case with and the case without the hub boundary condition
(see Figures 13 and 14) indicate an instability. The radial velocity satisfying the
hub boundary condition in Figure 13 is the stable solution. In Figure 15 the pitch-
diameter ratio is shown for Model 4717C with hub images. In Figure 16 the camber
correction factors are shown for Model 4717C with and without hub images. When the
computed results obtained without consideration of the hub boundary condition
happened to have radial velocities with small values, the results were very close to
the solutions obtained when the hub boundary condition was considered. The numerical
results reported in the following discussion were obtained without satisfying the
hub boundary condition. In the cases given, the instability phenomenon was not
NUMERICAL EXAMPLES FOR PROPELLER DESIGN AND DISCUSSIONS
Many supercavitating propellers have been designed and tested in the past; of
that number, two propellers, DTNSRDC Models 3770 and 3870 were chosen to determine
if this design program is reasonable. The former propeller has three blades and a
low advance coefficient, and the latter has four blades and a high advance coeffi-
cient. Experiments showed that both propellers had smooth cavities. The experi-
mental results and the previous design calculations are available.
The design and performance characteristics of the two propellers are shown in
It is extremely difficult to compare the present numerical results to the
experimental results for propellers that were designed using an entirely different
method. The present program is intended for design, not prediction. The present
program does not produce data on leading-edge cavity thicknesses, input that is
essential to the design of propellers similar to Models 3770 and 3870. To check
the reasonableness of the present program we guessed at the leading-edge cavity
thicknesses for Models 3770 and 3870; this is presented in Figure 17. Because the
actual cavity thickness was never measured, this is not a scientific estimate. The
leading-edge cavity thickness selected for Model 3770 results in an almost infinite
cavity length at every section of the blade, except near the tip and the hub. How-
ever, because the design cavitation number of Model 3870 is not very small, the
selected leading-edge cavity thickness is not large enough to induce a smooth sheet
cavity all over the blade. Therefore, the leading-edge cavity thickness of Model
3870 was corrected to give a cavity length at least 50 percent longer than the chord.
To do this, an extra leading-edge point drag was added to the cascade theory used in
the present design method, as explained previously. The design thrusts used in the
present lifting-line computations are the experimental values listed in Table 2,
where Case 1 is for cases without angle of attack and Case 2, with preset angle of
attack. It is seen that the efficiencies of the propellers, as predicted by the
present method, are very close to the measured efficiencies, even in the preliminary
design stage of calculations.
Pitch distributions obtained from the preliminary design and lifting-surface
design computations, according to the two design approaches, Case 1 (without angle
of attack) and Case 2 (with angle of attack) are shown in Figures 18 through 21
together with the pitches of the two propeller models. The pitch values obtained
from the preliminary design calculations are higher than those obtained from lifting-
surface calculations because, in preliminary design, the effect of flow retardation
is not considered. The pitch distribution is also related to the leading-edge
cavity thickness. In general, when the leading-edge thickness increases, the pitch
also increases; however, the efficiency decreases slightly. The pitch distributions
for the predictions and for the models are noticeably different. This is because
the optimum lift distribution and the pitch angle, which are influenced by the blade-
cavity interference in the present method are quite different from those of the
models. If these factors are taken into account, all results appear reasonable
compared with those of the models.
The lifting-surface corrections to the source distribution for the two pro-
pellers are shown in Figure 22. Although the correction for Model 3770 is close to
1, it is about 20 percent greater near the trailing edge than at the leading edge for
Model 3870. If the correction factor is unity, this means that the cascade source
strength is the same as the blade cavity source strength. This source strength has
the main influence on pitch, thrust, and efficiency. The calculated efficiency is
close to that obtained in the model experiments, as shown in Table 2.
The lifting-surface camber correction factors for the two propellers are shown
in Figures 23 and 24. The camber correction factor for Model 3870 is much larger
than for Model 3770, as in the calculation of Venning and Haberman. The pitch
distribution and camber correction curves for Model 3870 are quite different from
those for Model 3770. The former has shorter cavities with four blades and large
expanded area ratio (EAR), and the latter has longer cavities with three blades and
EXPERIMENTAL EVALUATION OF THEORY
Finally, the Center conducted an experimental program to evaluate the present
method for designing supercavitating propellers. Two supercavitating propellers
were designed, using the present method, for a 200-ton (181 metric ton) hydrofoil
craft. The propeller design criteria are given in Table 3. The propeller design
characteristics are given in Table 2.
Two model propellers were manufactured from these designs. The geometry of
these propellers is given in Table 4 and drawings of the propellers are shown in
Figures 25 and 26.
The experimental program was divided into two phases. The results of the first
phase, the measurements of blade-cavity shapes, are reported in the following
section, while the results of model propeller performance are described in the last
EXPERIMENTAL INVESTIGATION OF BLADE-CAVITY
A series of experiments was performed to determine how well linear theory
predicted the upper cavity surface location for Propellers 4717C and 4738A. For
these experiments, brass pins of varying lengths were attached to the backs of the
propeller blades. During propeller operation in the 36-in. variable-pressure water
tunnel, one could see when the pins came into contact with the upper cavity surface.
This experimental procedure has already been used to verify the upper cavity
surface location for Propeller 4699, ' and the parent design of Propellers 4738A
and 4717C. In addition, cavity heights have been measured in similar ways for a
supercavitating flat plate and a Tulin-two-term section by Christopher and Johnson.
DESCRIPTION AND LOCATION OF PINS
Number-four brass machine screws with heads cut off were used as pins. These
pins were attached to the blade backs by drilling and tapping holes perpendicular to
the surface at 12 locations. The upper cavity surface location was defined as a
point on a line, perpendicular to the nose tail line, that runs through the center
of the tapped hole at the blade surface; see Figure 27. The holes were drilled and
tapped perpendicular to the back of the blade to cant the pins slightly away from the
reference line (the line perpendicular to the nose tail line) . A slight error was
introduced, but the machining process was greatly simplified. The locations of these
pins were as follows: 10, 30, 60, and 90 percent of chord at nondimensional radii
(r/R) of 0.361, 0.544, and 0.726; see Figure 28. Since the blade at 10 percent of
chord was too thin to tap, the pins at these locations were soldered in place. The
pins at all other locations were screwed in and secured by tiny electrical lock nuts
(see photos in Reference 19) .
Three sets of pins were used for testing Propeller 4717C. When installed, the
first set of pins protruded above the back of the blades to a height that corre-
sponded to three times the distance from the back of the blades to the theoretically
predicted upper cavity surface. The second set of pins protruded by a factor of 1.67
and the third set by a factor of 1.0, the latter being the theoretically predicted
cavity height. However, the pins at 10 percent of chord varied from this order;
their heights corresponded to cavity-height factors of 3, 2.33, and 1.0. The pins at
the 10 percent of chord locations had to be filed by hand to the correct height, and
the factor of 2.33 rather than 1.67 was used to ensure that the pin-height would
exceed the experimental cavity thickness. The experiments have shown, however, that
the theory overpredicted the cavity height near the leading edge, and the pins could
have been filed to a height corresponding to a factor of 1.67.
Four sets of pins were used for Propeller 4738A. The first set of pins pro-
truded above the back of the blade to a height corresponding to a factor of 1.8 times
the distance from the back of the blade to the theoretically predicted cavity sur-
face. The other three sets corresponded to factor of 1.4, 1.0 (theoretical) and
0.6. The pins at the 10 percent of chord locations were filed to a height corre-
sponding to the multiplication factor used for the other pins in the set being
A thin coat of international yellow paint was applied to the tips of the pins
prior to testing to aid in visual observation. The experimenters could thus locate
the pins easily when the propeller was revolving.
The propeller rpm was increased until the cavity enclosed all the pins. This
procedure was begun with the set of brass pins that protruded highest. While the
pressure and velocity in the 36-in. variable pressure water tunnel was held constant,
corresponding to the design cavitation number, a = 0.34, the propeller rpm was
gradually reduced in decrements of 10. Each time the propeller rpm was reduced, a
hand-held strobe unit was used to observe visually whether the pins were in or out
of the cavity. These observations could be made rapidly, although the pins that
barely touched the upper cavity surface required more attention than did the others.
When all pins protruded through the cavity surface, the procedure was repeated with
another set of pins. A depth micrometer was used before and after each test to
measure the height of the pins above the back of each blade. This was done to ensure
that the pins were at the correct height and had not moved during testing.
As the pins began to break through the cavity surface, a furrow or small
groove formed in the surface, accompanied by some spray or cavitation behind the
pin. A directional strobe unit with variable light intensity made these furrows
much more visible. This phenomenon has been recorded in several color photographs.
Large, international-yellow numbers painted on the backs of the blades proved
invaluable during testing. Also, each propeller hub was coded with a series of dots,
the number of which corresponded to the number on each blade.
DISCUSSION OF EXPERIMENTAL RESULTS
Figures 29-31 compare linear theory predictions of cavity height with experi-
mental results for Propeller 4717C. These figures show three experimental upper
cavity surfaces corresponding to three values of J, one of which is the design J
(1.037). Note that the following relationship gives the advance angle, 3, at each
radial blade section,
= tan 1 (V A /2TTrn)
= tan 1 [J/(irr/R)]
From this relationship, one can determine, approximately, the corresponding shifts
in upper cavity surface that result from small changes in angle of attack. For
example, in Figure 29, at the design J (1.037), 3 is determined to be 42.44 deg.
For a J value of 1.0, 3 = 41.40 deg. Therefore, a change of J corresponding to
0.037 has caused, approximately, a one-deg change in angle of attack, which shifts
the cavity surface upwards as shown.
Theoretically, the section lift and cavity thickness for Propeller 4717C are
generated entirely by camber and point drag (note the blunt nose in Figures 29
through 31). That is, no incidence was used in the design to generate lift or cavity
thickness. In Figures 29 though 31, the theoretical prediction of cavity height
agrees fairly well with the experimental data. Near the leading edge, however, the
theory appears to overpredict cavity thickness. Also, visual observations indicated
that the backs of the blades at all radial sections on Propeller 4717C were wetted
to about 2- or 3-percent of chord from the leading edge. At this point, separation
was caused by a locally flat area that was inadvertently machined onto the back of
the blade. Although this local flat was almost microscopic, it effectively caused
separation. Apparently, very near the leading edge some portion of the blade metal
was interfering with the upper cavity streamline.
Figures 32 to 34 compare linear theory predictions of cavity height with experi-
mental results for Propeller 4738A. Note in these figures the large amount of point
drag or blunt nose indicated by the theory. This results because both Models 4738A
and 4717C were designed to have approximately the same full-scale stress levels.
This dictated that the maximum, theoretical, cavity thicknesses for Models 4738A and
4717C would be almost the same. To obtain the same maximum thickness in a shorter
distance, we used a large amount of point drag together with incidence and camber to
generate the theoretical cavity.
Figures 32 to 34 show three experimental upper cavity surfaces corresponding to
three values of J. Note that at r/R = 0.361, Figure 32, the blade was fully wetted
at the design value of J (1.037); therefore, cavity heights for three other values
of J have been shown. At a J value of 0.98, the experimental cavity surface coin-
cides with the predicted cavity surface height at aft locations on the blade. This
J value represents an approximate increase in angle of attack of 1.6 deg over the
design value of J. For the cavity surface, corresponding to a J value of 0.96, the
increased incidence is equal to about 2.2 deg. However, as mentioned before, the
cavity did not spring from the leading edge. It moved down the span as rpm was in-
creased. Since the water-tunnel velocity was held close to 35 fps (10.688 m/sec)
and since the model propeller diameter was 16 in. (40.64 cm), the difference in model
rpm corresponding to the J values of 0.98 and 1.037 was 88. This corresponds to an
increase of about 58 in full-scale rpm. It is also interesting to note that, accord-
ing to performance evaluation experiments, an increase of 29 rpm, over the 1000 rpm
of full-scale design would give the design thrust.
At the two outer radial positions, r/R = 0.544 and 0.726, full cavitation did
occur at the design J (1.037), but the theory overpredicted the cavity surface
height. As with Propeller 4717C, the back of the blade near the leading edge of
Propeller 4738A was wetted to about 2- or 3-percent of chord.
To understand more fully the discrepancy between theory and experiment, the
reader should recall that a point drag is a linear theoretical model of the leading
edge cavity thickness represented by a point singularity. Experimental results indi-
cate that the actual separation point at the leading edge must be carefully chosen,
for example, as a slope discontinuity of the blade surface, to achieve the designed
leading edge cavity thickness; if the predicted leading edge cavity, not just 70
percent of it, had been filled by a material up to 2 percent of chord from the
leading edge, the experimental results would have almost coincided with the theory,
except very near the hub, where the hub effect is important.
CAVITATION PERFORMANCE CHARACTERISTICS OF SUPERCAVITATING
PROPELLERS 4717B, 4717C AND 4738A
Propeller 4717C was originally manufactured as Propeller 4717B. Propeller 4717B
was identical to Propeller 4717C except for the backs of the blades, which had a
shape to conform to the predicted cavity shape at design operating conditions. The
primary purpose of Propeller 4717B was to determine, by observation, how well the
blade section shape represented the blade cavity shape. Following characterization
and observation, Propeller 4717B was finish cut to the final design version, Pro-
Propeller 4738A is a six bladed propeller with the same expanded area as the
previous four bladed propeller, and is designed for the same conditions as the other
Cavitation performance characteristics and cavitation observations were ob-
tained in the 36-in. variable pressure water tunnel. Tunnel water velocities were
measured by the tunnel venturi system. The scope of the experiments is given in
Tunnel pressure and water velocity were set to establish each cavitation number
and then propeller revolution rate was varied to cover a range of advance coeffi-
cients. Propeller thrust and torque were measured at each condition and sketches
were made of the cavitation present. The Reynolds number, R , during the experiments
ranged from 7.5 x 10 to 5.6 x 10 .
PRESENTATION OF DATA AND DISCUSSION
The thrust and torque data were reduced to nondimensional coefficients of K and
K . Propeller efficiencies were calculated from faired values of K and K . The
cavitation performance characteristics of the three propellers are presented in
Tables 6 through 8.
Curves representing the faired data, from Tables 6 through 8, are shown as an
example in Figure 35. Curves of maximum-speed thrust loading (K /J ) have been
added to the performance curves for Propeller 4738A. The intersection of the K J
curve and the K curves at the design sigma (o) determines the predicted operational
point for each propeller. A comparison between the design operational points and
the points predicted by the experimental data is given in Table 9.
Sketches of the back cavitation present on the propellers at two cavitation
numbers are given in Figures 36 through 38. These sketches cover a range of advance
coefficients from partially cavitating to fully cavitating conditions. If advance
coefficients had lower values than those shown, the propellers, at the same cavi-
tation number, would also be fully cavitating. With only one exception, propellers
contained no face cavitation over the range of cavitation numbers and advance co-
efficients covered. The sole exception was at an advance coefficient of 1.2, where
some leading edge face cavitation was observed at cavitation numbers of 0.75 and
At design speed coefficient and design o, Propeller 4717B contains practically
no back cavitation. If advance coefficient is reduced slightly, at design a, the
backs of the blades are covered by sheet cavitation from the blade tip to 50 percent
radius. This indicates that the predicted cavity shape over this part of the pro-
peller blades is quite accurate.
Neither of the designed propellers, 4717C and 4738A, had face cavitation at the
design operational points. Propeller 4717C essentially had full back cavitation and
Propeller 4738A had back cavitation from about 35 percent radius to the tip of the
blades at the design operational point.
The propeller theory slightly overpredicted the available thrust for both pro-
pellers. Propeller 4717C would require 6 percent more rpm and 8 percent more power
than predicted to reach design speed. Propeller 4738A would require 5 percent more
rpm but 4 percent less power than predicted to reach design speed. If the operating
point is defined as the speed and rpm where the propellers absorb the available
maximum power, Propeller 4717C would operate at V = 58.7 knots and rpm = 1016, and
Propeller 4738A would operate at V = 61 knots and rpm = 1054. The propeller
efficiencies at these conditions are 66 percent and 67 percent, respectively.
It has been recognized that the nonlinear effects on lift and drag of cavi-
tating foils are approximately equal to -0.5 C /(1-hj) and -0.5 C C /(1+a). There-
fore, propellers designed according to the linear theory would produce less thrust,
as indicated in the experimental results. However, the leading edge cavity thickness
was slightly smaller than the design thickness due to the unmatched separation point
and 70 percent filling of cavity thickness. Thus the drag should have been a little
less than predicted by the linear theory, as was found experimentally for Propeller