Wind tunnel test data for falling sphere drag coefficient determinatio

Heinrich, H. G.

Noreen, R. A.

English

NASA Technical Reports Server

AN ASSESSMENT OF SPHERE DRAG COEFFICIENT DATA
By Helmut G. Heinrich and Robert A. Noreen
University of Minnesota
INTRODUCTION
In order to reliably determine atmospheric conditions from
the descent velocity of a sphere, the drag coefficient of the
sphere as it falls through the air must be known to a high degree
of accuracy. This paper covers wind tunnel studies which estab-
lished drag coefficients from 360,000 ft. altitude down for a one
meter sphere with a given weight, when ejected from a rocket at
an altitude of 450,000 ft. The wind tunnel test conditions were
adjusted for simultaneous duplication of the Mach and Reynolds
Numbers as they occur during the descent of the sphere in a
standard atmosphe_re. The range of the test conditions was wide
enough to cover the expected atmospheric deviations.
TEST FACILITIES
The drag coefficient of a perfect sphere moving through air
is a function of the Mach and Reynolds Numbers and under certain
conditions also of the Knudson Number which is, however, a com-
bination of Mach and Reynolds Numbers. These relationships re-
quire that meaningful drag measurements must be made with simul-
taneous duplication of the respective Mach and Reynolds Numbers.
In standard atmospheric pressure wind tunnels, this condition is
usually difficult to fulfill. Therefore, the University of
Minnesota used for the required measurements a subsonic and a
supersonic variable density wind tunnel, because the density
variation provides the third parameter necessary to establish the
required Mach-Reynolds and Knudson Number simulation.
The low density subsonic wind tunnel is of the closed hori-
zontal return type, Fig. i (Ref. i), with a mechanical vacuum pump
used to evacuate the circuit and a centrifugal compressor to move
the air around the circuit. Continuous operation at Mach Numbers
from nearly 0 up to approximately 0.9 can be obtained by using
various nozzles and a throttling valve. Wind tunnel operating
93
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pressures range from 200 torr down to approximately 0.2 torr, de-
pending upon the flow Mach Number. The nozzles exhaust into an
open jet test section and have outlet diameters varying from
12 in. for lowest Mach Numbers to 3 in. for highest Mach Numbers.
The low density supersonic wind tunnel, Fig. 2 (Ref. 2),
has a standard "blow down" configuration, operating between a
5000 psig high pressure source and a i0 millitorr vacuum reser-
voir. For these tests the supersonic wind tunnel was equipped
with five axisymmetric nozzles for operation at Mach Numbers of
1.2, 1.5, 2.0, 2.5, and 3.2 at static pressures in the order of
I torr. All five nozzles have a core flow region 4 in. in dia-
meter at their design operating pressures and exhaust into an
open jet test section. For the pressures of interest in the
sphere drag tests, the 33,500 cu. ft. vacuum reservoir could sus-
tain nearly constant pressures for run times of a minute or
longer.
MODELS
The sphere models used in both test sequences were 0.50 in.
Teflon spheres with diameter tolerances of 0.002 in. and
sphericity of 0.001. Teflon was selected because its surface
roughness characteristics approximate those of an inflated
spherical Mylar balloon.
MEASUREMENTS
The primary measurements made during the tests were flow
pressures and the drag forces of the spheres. Naturally the
size of the models and flow temperature were also measured, but
these measurements presented little or no difficulty and could
be made with sufficient accuracy to be neglected in an error
analysis.
The pressure measurements in the required range from 20 torr
to i0 millitorr presented the most difficulties and created the
largest portion of the total error. For both the subsonic and
supersonic tests measuring two pressures enabled determing the
flow conditions. In the subsonic tests, total pressure and the
difference between total and static pressure were measured,
while in the supersonic tests, total pressure and static pressure
were determined. These measurements were made using Bourdon tube,
strain-gaged diaphragm, and thermocouple gage_ all having dif-
ferent ranges, accuracies, and reliabilities (Refs. i and 2).
Total pressures were detected with a total probe rake located in a
near stagnation region, and static pressures were obtained from
static pressure taps in the nozzle wall near the exit.
94
Drag forces were measured using the force balance shown in
Fig. 3. A drag force acting on the sphere twists the torsion
member causing the core of a linear variable differential trans-
former to deflect witMin the coil, thus producing a measurable
change in coil current. Two permanent magnets placed near a
copper "paddle", also attached to the torsion member, produced
sufficient eddy currents to damp any oscillation of the balance
system. The balance was calibrated by hanging weights on the
damping support strut and relating the moment produced by the
weight to a force on the sphere by the ratios of lengths to the
moment center. Changing torsion members allowed selecting nearly
any required sensitivity of the force balance. Repeated cali-
brations indicated that the balance performed quite well and was
more accurate than the electrical read-out equipment used to
measure the coil currents.
For each data point two types of drag measurements were made,
one with the sphere mounted on its sting support, and another one
of the support alone with the sphere rigidly mounted slightly
ahead of the support (Fig. 4). Subtracting the support drag from
the total drag yielded the drag of the sphere. The drag of the
support was, of course, considerably smaller than that of the
sphere.
RESULTS
The results from the supersonic and subsonic tests are shown
in Figs. 5 and 6, respectively, as plots of drag coefficient vs
Reynolds Number with Mach Number as a parameter. One notices
that the higher speed subsonic data is somewhat scattered and
that there are a few questionable points among the supersonic
results. In view of this uncertainty, the data was replotted in
Figs. 7a and 7b as CD vs Mach Number with Reynolds Number as
parameter. The critical review of both types of presentations
provides guide lines for the data interpretation and for the
establishment of the final conclusion. In this view, the data
in the previous figures was carefully analyzed by members of the
staff of the University of Minnesota and USAF Cambridge Research
Laboratories. The results of this joint effort are the curves
shown in Figs. 8 and 9 which are considered to be the final
results of this study. The two dashed curves in Fig. 8 show the
spread of data obtained at Mach Numbers between 2 and 4 in wind
tunnel tests by Ashkenas who performed wind tunnel tests with
spheres mounted on thin wires (Ref. 3).
The agreement between the different supersonic data can only
be termed approximate at best, and even though Ashkenas (Ref. 3)
presents no error estimate, there is no principal reason that
accurate results could not be obtained with his methods.
95
Figure 9 includes several data points from tests conducted
with moving spheres in a ballistic range (Ref. 4). The agreement
between these data is again approximate, and not within the esti-
mated error range of either set of measurements. However, it is not
known if drag measurements taken from a sphere which is decel-
erating to various degrees as it travels along the ballistic
range can be compared to steady state measurements. In a flow
field that changes with time, the flow pattern may be quite
different from the one under steady state conditions at the same
Mach and Reynolds Numbers. The kinetic energy in that flow field
varies definitely, and apparent mass effects may have to be
accounted for. Therefore, it is at this time not known if these
effects involved in the ballistic tests are or are not signifi-
cant, but it is a possible explanation for the difference be-
tween results obtained under steady state and non-steady conditions.
ERRORS
References I and 2 give extended analyses of the random and
instrument errors encountered in the measurements performed at
the University of Minnesota. The analysis follows the standard
concept of expressing the error in drag coefficient as a total
differential considering all the terms measured to obtain the
drag coefficient. The results of this analysis predict errors
from i% to 5% for the subsonic measurements, and from 2% to 28%
for the supersonic measurements; these numbers represent the
maximum possible errors due to random or instrument errors, and
the possible error for a particular point is a function of the
Mach and Reynolds Number at that point. For both the subsonic
and supersonic data, the highest errors occur at the lowest
Reynolds Numbers of the range, and in both cases, the highest
possible error is due to the pressure error term.
CONCLUSIONS
Sphere drag coefficients have been measured over the range
of Mach and Reynolds Numbers encountered by a falling sphere
density sensor. An error analysis of the data shows that
generally the data should be accurate to within about 5%; agree-
ment with other measurements is within about I0_ and
results from actual tests show that the data is at worst very
reasonable and at best highly accurate. Of course, since the
drag coefficient data is one basic part of a measuring system
where greater and greater accuracy is needed, the drag coeffi-
cients must be critically checked for possible inaccuracy and
improvement. Reviewing the measurements from this aspect,
several possible areas for improvement arise.
96
i. Pressure Measurement
Since the measurements were conducted fairly significant
improvements in pressure measuring instrumentation and techniques
have been made. Realizing that the highest errors in the data
presented above were no doubt caused by pressure errors, it seems
probable that at least in some ranges significant improvements
could be made in the accuracy of the coefficients.
2. Sphere Temperature
Recent publications have shown that sphere temperature does
have an influence on drag coefficients in this low density flow
regime (Ref. 5). Although this effect is not large and the model
spheres should have been at a temperature very near wall tempera-
ture, the sphere temperature was not measured. If temperature
differences existed, this may have caused some unknown error.
3. Sphere Surface Roughness
Only one type of sphere was used in the tests at the
University of Minnesota, hence the effects of roughness were not
determined experimentally. It is possible that at least part of
the deviation from measurements at other institutions is due to
different surface roughness of the models.
SUMMARY
It appears that the drag of the sphere is known reasonably
well with an accuracy usually expected from aerodynamic measure-
ments. Yet an over-all improvement of the accuracy of the
pressure sensing system offers greater certainty in the process
of data acquisition and probably a significant improvement of the
accuracy of the drag coefficients.
A new effort to measure the sphere drag under well simulated
steady state flow conditions with the best instrumentation and
facilities available appears to be justified and desirable.
97
_D_D_
During the discussion following this and other presented papers_ questions
were raised about the ori@in of the M _ 0.39 sphere drag coefficient data
between Reynolds Numbers of 1,600 and 25,000 presented in Ref. i. The uncer-
tainties center around Fig. 19 of Ref. i, added here as Fig. i0, which shows the
results of measurements by the University of Minnesota. All of the data in
Fig. i0, except the M _ 0.39 data, has been superseded by the data shown in
Fig. 9 (Ref. 2), which were obtained from tests conducted with better instru-
mentation at a later date. The drag coefficients shown in Fig. i0 and identi-
fied as belonging to Mach Number =<0.39 have not been changed because newer test
cases with improved instrumentation did not show significant deviations from
the earlier established data.
The M J 0.39 curve is based on seven series of measurements at various
Mach Numbers-between Reynolds Numbers of 1.5 × 103 to 3 × 104. This is also
stated in Ref. i. Detailed results of these measurements are shown in Fig. ii.
The data points presented in Fig. ii were not shown in Fig. 19 of Ref. i in
order to avoid overcrowding the presentation.
Figure i0 shows an additional curve representing drag coefficients in the
incompressible flow regime for Reynolds Numbers less than 103. This information
is taken from Fig. I0 in Chapter Ill of Hoerner's "Fluid-Dynamic Drag," as well
as a graphical transposition can be made. Admittedly, Fig. 19, Ref. I, can
easily be misunderstood to the extent that the entire curve representing the
sphere drag coefficients in the incompressible range is credited to Hoerner.
This is, however, not true and the right-hand segment represents the average
values shown in Fig. ii. Merely the left-hand portion is from Hoerner.
Comparing the University of Minnesota curve M _ 0.39, one finds as mini-
mum drag coefficient CD = 0.427 for Reynolds Numbers between 3.5 and 4.0 x 103 •
Trying to extract the minimum CD value from Hoerner's curve one finds for the
same Reynolds Number range CD = 0.407 with a possible reading inaccuracy
of ±0.025. This fact led to the remark in Ref. I that the University of
Minnesota measured data agree well with Hoerner's curve covering this Reynolds
Number range.
Furthermore, it will be noticed that Fig. ii also contains data points
taken from numerical values of Ref. 6, which is one publication listed in
Hoerner. Figure 12 is a photographic reproduction of Fig. 26, Ref. 6.
This figure is particularly interesting since it shows different drag coef-
ficients for the same Reynolds Numbers obtained by a different experimental
technique, namely dropping steel spheres in water.
Figure 13 which is reproduced from Ref. 7, Fig. 178, is also included and
shows the variation of drag coefficients measured in the same institution under
the supervision of the same individuals but in different wind tunnels. Of
course, it must be considered that Fig. 13 shows the drag coefficients in the
98
critical subsonic rangej and the degree of wind-tunnel turbulence is in this
region particularly influential.
Summarizing then all matters of accuracy_ it is the opinion of these
authors that the accuracy of the drag coefficients measuredat the University
of Minnesota is about as good as can be expected from wind-tunnel experiments
at that time. Repeating these measurementsand utilizing instrumentation
improved over that available in 1960 to 1962 may provide somewhatdifferent
results_ but the deviations cannot be large. Furthermore_ different measuring
techniques such as measuring the descent speed of spheres in water or their
deceleration in a ballistic range may again produce slightly different numer-
ical values.
99
REFERENCES
i. Heinrich, Helmut G.; Niccum, Ronald J.; and Haak, EugeneL.: The Drag
Coefficient of a Sphere Corresponding to a "One Meter ROBINSphere"
Descending From 260,000 ft Altitude (Reynolds Nos 789 to 23,448, MachNos
0.056 to 0.90). Research and Development of ROBINMeteorological Rocket
Balloon, Vol. II, Contract No. AF 19(604)-8034, Univ. of Minnesota, May
1963. (Available from DDCas AD480309.)
2. Heinrich, H. G.; Niccum, R. J.; Haak, E. L.; Jamison, L. R.; and
George, R. L.: Modification of the Robin Meteorological Balloon.
Vol. II - Drag Evaluations. AFCRL-65-734(II), U.S. Air Force, Sept. 30,
1965. (Available from DDCas AD629775.)
3. Ashkenas, H. I.: Sphere Drag at Low Reynolds Numbersand Supersonic Speeds.
Res. SummaryNo. 36-12, Vol. I (Contract No. NAS7-100), Jet Propulsion
Lab., California Inst. Technol., Jan. 2, 1962, pp. 93-95.
4. Lawrence, W. R.: Free-Flight RangeMeasurementsof Sphere Drag at Low
Reynolds Numbersand LowMachNumbers. AEDC-TR-67-218_U.S. Air Force,
Nov. 1967. (Available from DDCas AD 660_45.)
5. Ashkenas, Harry: Low-Density Sphere Drag With Equilibrium and Nonequilibrium
Wall Temperature. Rarefied Gas Dynamics, Vol. II, J. A. Laurmann, ed.,
AcademicPress, 1963, pp. 278-290.
6. Ergebnisse der AerodynamischenVersuchs_nstalt zu Goettingen. II. Lieferung,
Verlag yon R. Oldenbourg, 1923, pp. 28-30.
7. Ergebnisse der AerodynamischenVersuchsanstalt zu Goettingen. IV. Lieferung,
Verlag yon R. Oldenbourg, 1932, pp. 106-108.
100
a_
i
101
t(_ HIGH PRESSURE AIR SUPPLY
(_ PRESSURE REGULATOR
(_ INLET VALVE
STILLING CHAMBER
(_ NOZZLE SECTION
(_) TEST SECTION
Q DIFFUSER
RESERVOIR
Figure 2.- Schematic representation of the low density supersonic wind tunnel.
LINEAR
TRANSFORMER OSCI LLOGRAPH
CORE
.22 in. SHIM
STOCK
SECOND MAGNET
OMITTED FOR
CLARITY
Figure 3.- Schematicof dragbalanceused in low density supersonicwind tunnel.
102
NOZZLE
MODEL
(a) Total force arrangement.
TO BALANCE
/,,,,--SHIELD
FLOW
NOZZLE
MODEL
BALANCE
IELD
FLOW
(b) Tare force arrangement.
Figure 4.- Schematic representation of suspension arrangement for force measurements.
103
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SE E FIG 7a,
a__ , ]-
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-- D, Re,2000 ----
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(b)0.66=<M =<0.86.
Figure 7.- Drag coefficient of a sphere as a function of the free-stream Mach number with Reynolds number as a parameter.
106
1.7
i
m4
I
'7
I <
1._ -- _E)_
i
1,5 i
COlA -- C
\ (
1.3 __ -- iX \
1.0
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Figure 8.- Drag coefficient of a sphere in supersonic flow as a function of the free-stream Reynolds number.
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I
_ ,-,.,. ,,!.,,,.,.j./ , _- .__
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R,
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23OO
Figure g.- Drag coefficient of a sphere in subsonic flow as a function of the free-stream
Reynolds number.
107
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x ×
× ×
y
x
;,<
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s REE 1 , FIG. 19[
2.103 5.103 104 2-104
REYNOLDS NUMBER
×
×
5,,104
Figure ]l.- Low subsonic drag coefficients measured at University of Minnesota compared with data taken from Ref. 6.
C D
03
O,C
, ,, |
STEEL SPHERES
IN WATER
- -3 3 ' 13
10 2-10 5,113 104 2,104
REYNOLDS NUMBER
5.104
Figure 12.- Drag coefficients of a sphere in subsonic flow at Reynolds Number between 103 to 5 x 104. (Reproduced from Ref. 6.)
109
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110


An assessment of sphere drag coefficient data

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