ARCHIEF
A PRELIMINARY STUDY OF THE EFFECT OF THE FREE SURFACE ON THE DRAG OF A TRtJNCAThD 2:1 ELLIPSOID
by
Arthur D. Newsharn
Institute of Hydraulic Research University of Iowa
Iowa City
y.
Sci eepsouwtund
Techsche
Hogschoo
Deft
Contract Nonr l5O(O5) August
l96-°f1c- ot Naval
esarc
A PRELIMINARY STUDY OF TitE EFFECT OF T}tE FREE SURFACE ON TItE
DRAG OF A TRUNCATED 2: 1 ELLIPSOID
Introduction
This phase of the study of the drag force on truncated, axisymmetric,
supercavitating bodies near a free surface has involved the design,
construc-tion, and checking of the apparatus and instrumentation necessary to conduct
the experiments. A truncated 2:1 ellipsoid at zero angle of yaw was chosen
for the initial investigation. The drag of this body at zero angle of yaw
has been measured and compared with existing data for a
2:1
ellipsoidcomplete-ly submerged [1, 2, 3]. In addition, some preliminary work concerning the
effect of the proximity of the free surface has been performed. This work was
supported by BuShips, Code
421,
under Contract Nonr 1509(05).Nomenclature
d diameter of the body
g acceleration of gravity
h depth of submergence
D drag of the body
Q. air-supply rate
V velocity
kinematic viscosity mass density
PC pressure difference between the cavity and a point in the undisturbed
flow
at the same elevationcp
¶d2pVz/8
drag coefficientIF -
Y
Froude numberIR=
Reynolds numbera
dimensionless air supply
Trd2V/4
cavitation coefficient c
-2-Experimental Equipment
The primary facility for this study is the uHR towing tank. This tank, which is 10 feet wide, 9 feet deep and 300 feet long, is equipped with a cable-driven carriage capable of traveling at speeds up to 24 feet per
second.
For towing the bodies the apparatus (Figs. 1 and 2) is mounted on the carriage and is designed to support axisymmetric bodies 5 inches or more
in diameter. The body may be submerged a maximum distance of 45 inches below
the free surface with angles of yaw varying from -15 to +15 degrees.
A 4-cubic-foot tank placed on the carriage and equipped with a
pres-sure regulator was used to provide air for the cavity. The hollow strut and
sting are used to deliver the air to the rear of the body at a constant rate, This air is introduced into the cavity in the direction of travel, and is
de-flected downstream without exerting an extraneous force on the body. The
sys-tem is capable of producing steady discharges somewhat in excess of 1.0 cubic
foot per second. To determine the rate of air supply for each run, the time
the supply valve is open and the resulting isothermal pressure drop in the
storage tank are measured. The steady discharge is then computed using the
ideal gas law.
In order to measure drag directly, the body is connected to the
sting by two coaxial disk springs (Fig. 3) which allow a single degree of
free-dom of motion in the direction of travel. The deflection of the center of
the springs with respect to the sting is measured with a differential trans-former that will operate in either air or water with a negligible change of
sensitivity. Calibration is performed by applying known weights
gravimetri-cally. The response is linear over a range of approximately 0.010 inch on
either side of null, and the mechanical hysteresis at the origin is of the
order of 0.5 ounce. Forces from 2 to 50 pounds can be measured accurately
and the upper range can be extended by simply increasing the spring thickness.
The drag device is sensitive to carriage vibrations. A study of
this problem has shown that the error in obtaining the mean drag by filtering the signal is negligible for the conditions of this study.
-3-.
A Statham PG22TC 0-10-psig subminiature strain-gage pressure cell placed at the same elevation as the centerline of the model and vented to the atmosphere was used to measure the pressure in the rear of the body. This cell was positioned at the elevation of the model to prevent errors due to
the presence of air-water mixtures in the vertical tubing. Pressure is
re-corded simultaneously with the drag on a two-channel recorder mounted on the carriage.
It was found necessary to take profile photographs of the body and
cavity to determine the shape and extent of the cavity near the head. The
camera was mounted on the carriage almost directly over one carriage rail.
When the camera passed over a stationary glass-bottomed tank (Fig. 4), it was tripped electrically to record the image of the body as it appeared in a
sta-tionary mirror. The mirror was located directly below the glass-bottomed tank
at the same elevation as the body and oriented at 45 degrees to the vertical.
Results and Discussion
Tests have been conducted with a 2:1 ellipsoid truncated at 1/4 of
its length, having a diameter of 6 inches at its base. The preliminary
re-sults for the body set at zero angle of yaw are shown in Fig. 5 for three
depths of submergence. Photographs of the cavity profile, using the
arrange-ment shown in Fig. 4, have been taken for a number of runs. Figure 6 shows
the photographs for four runs using a camera shutter speed of 1/loo second
and an air-supply rate of approximately 1 cubic foot per second. For a
sub-mergence of h/d = 1 the image of the body and cavity is reflected internally by the water surface and appears inverted at the top of the photograph.
Variations of the dimensionless air-supply rate from 0.6 to 0.10
had no significant effect on the results. The magnitude of the air-supply
rate can be misleading, however, since its effect depends upon the method used to introduce the air and the system used to guide the air downstream.
The experiments were conducted at Reynolds numbers between 4 x 1O and i x io6, corresponding to velocities between 6 and 20 feet per second,
with no apparent effect on the results for deep submergence. At shallow
-4-less than 11 feet per second, and Froude numbers below 2.0, lay well above
the plotted curves. These results at low velocities and reduced submergence
are probably due to improper cavity formation; consequently, they were not
included in Fig. 5. It may be seen from the photographs that the cavity is
affected much less by the depth of immersion than by the velocity.
The effect of submergence on the drag of a truncated 2:1 ellipsoid
is apparent in Fig. 5. The scatter of the results is due primarily to the
low sensitivity of the pressure cell, which was operating over only the lower
5 percent of its nominal range. It is seen that for each depth the data may
be fitted by a straight line with an equation of the form CD = A(l + K0) This result is in agreement with theory and with previous studies of the drag of axisymmetric bodies in which free-surface effect was not considered [4]. The three lines drawa in Fig. 5 have the equations
CD 0.22 (1 + 3.4 I(e), h/d = 4.0
CD = 0.26 (i + 2.6 Ka), h/d 2.0
CD = 0.24 (i + 2.6 K0), h/d 1.0
The line for h/d = 1.0 has been extended as a broken line, because the re-sults are dependent upon only one point for relatively large values of K The equations for the lines show a significant decrease in slope when the
sub-mergence is decreased. The intercept with the line K = 0, however, does not
show a definite trend.,
The results of water-tunnel studies at the lIER and the DTMB were recomputed using the dimension equivalent to that used in this study rather
than the maximum diameter of the complete ellipsoid. The results of the
pres-ent work agree quite weil with the previous IIJR work but differ markedly from
those obtained at the DTIIB. Although there is a slight difference in slope,
the primary difference between the results is the value at the intersection of an extrapolated line through the data with the axis K = 0. This dis-crepancy may be partly attributable to the presence of an ambient adverse pres-sure gradient in the test section of the water tunnel in which the DT data were obtained.
Conclusions and Recommendations
The results obtained were found to conform to the usual linear rela-tionship between C and K , but results have not yet been obtained for low
cavitation numbers. However, improvements of the equipment will permit
exten-sion of the study in this direction. These lower values, as well as some
re-peated higher values,will be measured with a more sensitive pressure cell. It
is hoped that this
rill
also reduce the scatter caused by errors in themeas-urement of the cavity pressure. This will permit a more accurate evaluation
of the effect of the free surface.
These preliminary results indicate that the effect of the free sur-face is principally to change the slope of the curve of drag coefficient versus
cavitation numbers. Additional tests are necessary to determine more accurately
the shape of this curve. In order to determine whether these results are
typi-cal for other truncated bodies, it is planned to study the effect of the free surface on radically different forms such as a supercavitating disk and cone.
References
-5-.
Eisenberg, P., and Pond, FI. L., "Water-Tunnel Investigation of Steady State
Cavities," DTIvIB Report 668, 1948.
Rouse, H., and McNown, J. S., "Cavitation and Pressure Distribution - Head Forms at Zero Angle of Yaw,' Bulletin 32, Studies in Engineering, Univer-sity of Iowa, 1948.
Macagno, M., and Hsieh, T. Y., "Drag Coefficients of Supercavitating Bodies of Revolution at Various Aníles of Yaw," Report by the Institute of
Hydrau-lic Research for Contract Nonr 1509(05), Task NR 062-271, March
1963.
Gilbarg, D., "Jets and Cavities," pp. 311-438 of Encyclopedia of Physics, vol. IX, Edited by S. Fligge, Springer-Verlag, Berlin, 1960.Fig. 1.
Photographs of assembly for supporting
Fig. Ea. Exploded view of head and sting.
Fig. 2b. View of head and sting in towing tank showing pressure
Truncated z:! Ellipsoid
Air Deflector
Coaxial Disk Springs
r Linear - Variable D
if fere n fiai
Transform e r
Fig. 3.
Cuta'.4
sketch of the head.
¿tin
g
L.V.D.T. Cable
V
)
-Line of View
Tow/n q-TanX Water Surface
Towing-Tank Wall
Fig. 4.
Arrangement for photographing tored body.
Carriage-Mounted Cornera
Stationary Glass-Bottomed Tank
Stationary Mirror
I
/
0.6
0.5
0.4
0.3
0.2
0-I
o
Fig. 5.
Drag coefficients of a supercavitating, truncated 2:1 ellipsoid at zero angle of yaw.
h/dI.O
h/d:2.0
O h/d4.0
OD D.T.M.B.
:uHR Bulle/in 32
,
d
o
___o
.
h/d=2.0
)
h/d4.0
- -
-Qe
s
h/d/.O
o
0
0.05
0./O
0.15
0.20
0.25
0.30
035
C.40
h/d = i K = 0.15 V = 13.7 Í'.p.s. h/d = 4 K = 0.38 V = 8.2 f.p.s. h/d = = 0.18 V = 14.0 f.p.s.
