Impingement of water droplets on an ellipsoid with fineness ratio 10 in axisymmetric flow

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NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS

TECHNICAL NOTE 3147

IMPINGEMENT OF WATER DROPLETS ON AN ELLIPSOID

WITH FINENESS RATIO 10 IN AXISYMMETRIC FLOW

By Rinaldo J. Brun and Robert G. Dorsch

Lewis Flight Propulsion Laboratory-
Cleveland, Ohio

Washington

May 1954

LIBRARY
MAY 13 1954

AMERICAN HELICOPTER CO,

MANHATTAN S^A""'-' " * ' '

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL NOTE 3147

IMPINGEMENT OF WATER DROPLETS ON AN ELLIPSOID WITH
FIKEHESS RATIO 10 IN AXISYMMETRIC FLOW
By Rinaldo J. Brun and Robert G. Dorsch

SUMMARY

The presence of radomes and instruments that are sensitive to water
films or ice formations in the nose section of all-weather aircraft and
missiles necessitates a knowledge of the droplet impingement character-
istics of bodies of revolution. Because it is possible to approximate
many of these bodies with an ellipsoid cf revolution, droplet trajecto-
ries about an ellipsoid of revolution with a fineness ratio of 10 were
computed for incompressible axisymmetric air flow. From the computed
droplet trajectories, the following impingement characteristics of the
ellipsoid siirface were obtained and are presented in terms of dimension-
less parameters: (l) total rate of water impingement, (s) extent of
droplet impingement zone, and (3) local rate of water impingement. These
impingement characteristics are compared briefly with those previously
reported for an ellipsoid of revolution with a fineness ratio of 5.

INTRODUCTION

The data presented herein are a continuation of the study reported
in reference 1 on the impingement of cloud droplets on a prolate ellips-
oid of revolution. The calculations discussed in reference 1 for an
ellipsoid with a fineness ratio of 5 (20 percent thick) were extended
for this report to an ellipsoid of fineness ratio of 10 (lO percent
thick). As mentioned in the reference cited, a prolate ellipsoid of
revolution is a good approximation in the determination of cloud-droplet
impingement for many body shapes used for radomes, rocket pods, and
bombs. The data presented herein, along with the results presented in
reference 1, permit the estimation of impingement characteristics on
many of these bodies.

The trajectories of atmospheric water droplets about a prolate
ellipsoid of revolution with a fineness ratio of 10 moving at subsonic
velocities at zero angle of attack were calculated with the aid of a
differetial analyzer at the NACA Lewis laboratory. From the computed

1M3A TN 3147

trajectories J the rate^ the distribution^ and surface extent of impinging
•water were ohtained and are sunuaarized in this report In terms of dimen-
sionless paraiaeters.

SYMBOLS
The following symbols are used in this report;
d droplet diameter^ microns
Ejj^ collection efficiency^ dimensionless

f fineness ratio, dimensionless

K inertia parameter, 1.704X10"-'-^ d%/|j.L, dimensionless (eq^. 1)

L major gixis of ellipse, ft

ReQ free- stream Reynolds number with respect to droplet,
4.813X10"° dp u/|j., dimensionless

r,z cylindrical coordinates, ratio to major axis, dimensionless

Tq staxting ordinate at z = - ob of droplet trajectory, ratio

to major gixis, dimensionless

rQ ^g_j^ starting ordinate at z = - w of droplet trajectory tangent
' to ellipsoid surface, ratio to major axis, dimensionless

S distance aJ-ong surface of ellipsoid from forward stagnation
point to point of droplet impingement, ratio to major axis,
dimensionless

Sjj^ limit of impingement zone, ratio to major axis, dimensionless

U free- stream velocity, or flight- speed, mph

u local air velocity, ratio to free-stream velocity

WjQ total rate of Impingement of water on surface of ellipsoid,
Ib/hr

Wn local rate of impingement of water, lb/(hr)(sq ft)

w liquid-water content in cloud, g/cu m

to

lACA TN 3147

^0 ^0

P local Inipliigement efficiency, — dq~> dlmensionless

M. viscosity of air^ sl'ugs/(ft)(sec)
pg^ density of air^ sliigs/cu ft

Subscripts:

r radial coniponent

z axial component

CALCULATION OF DROPLET !IEAJECTORIES

The equations that describe the motion of cloud droplets about an
ellipsoid axe given in reference 1. A solution of the differential
equations of motion was obtained with the use of the mechanical analog
(described in ref . 2) based on the principle of a differential, analyzer.
The procedure for calculating the trajec«urles of cloud droplets with
respect to the ellipsoid of revolution with a fineness ratio of 10 (10
percent thick) is the same as that described in reference 1. As shown
in figure 1^ the ellipsoid orientation in the coordinate system used in
reference 1 is retained herein. Since a flow field is axisymmetrlc
around an ellipsoid of revolution oriented at 0° between its major axis
and the direction of the free stream, the droplet impingement on the
elliptical section of all meridian planes is the same, and the impinge-
ment chaofacteristics of the ellipsoid of revolution can be obtained from
trajectory calculations in the z,r plane of figure 1.

The air-flow field around the body, required for the solution of
the equations, was obtained with the use of digital ccanputers from the
mathematical expressions presented in reference 1. The values of the
air-velocity components u^, and u^ as functions of r and z are

given in figure 2 for comparison with the flow-field velocities given
in reference 1 for the 20-percent- thick ellipsoid. In figure 2(a),
Ug is given as a function of z for constant values of r, while u^.
as a function of r for constant z is given in figure 2(b). The ve-
locity components in figure 2 are dimensionless, because they are ratios
of the respective local velocities to the free-stream velocity^ the
distances r and z are also expressed as ratios of actual distance
to the major axis length L.

The position of impingement of cloud droplets on the s\irface of
the ellipsoid is given in subsequent sections in terms of the surface
distance S, which is measured along the siirface from the forward

MCA TS 3147

stagnation point. The vaJLues of S axe the ratios of the actual sur-
face distances to the major ajcis length L. The relation between S
and z is shown in figure 3. The relation between S and r can be
obtained from figiire 4 from the ciorve for 1/k = 0.

The equations of motion were solved for various values of the pa-
rameter 1/K between 0.1 and 90. The inertia parameter K is a measure
of the droplet size^ the flight speed and size of the body of revolution,
and the viscosity of the air^ through the relation ^

H

K = 1.704X10"^"^ dTlS/^L (1) "^

The density of water and the acceleration of gravity, which are expressed
as part of the conversion factor, are 62.4 pounds per cubic foot and
32.17 feet per second per second, respectively. For each value of the
parameter i/k, a series of trajectories was computed for each of several
values of free-stream Reynolds number ReQi 0, 128, 512, 1024, 4096,
and 8192. The free-stream Reynolds number is defined with respect to
the droplet size as

Reo = 4.813X10"^ dpg_U/iJ, (2)

In order that these dimensionless parameters have more physical signifi-
cance in the following discussion, some typical combinations of K and
Reo are presented in table I in terms of the length and velocity of the
ellipsoid, the droplet size, and the flight pressure altitude and tem-
perature. Equations and graphical procedure for translating the dimen-
sionless parameters used in this report into terms of flight speed, major
a^xis length, altitude, and, droplet size are presented in appendix B of
reference 3.

RESULTS MD DISCUSSION

The results of the calculation of droplet trajectories about the
ellipsoid of revolution of fineness ratio 10 (lO percent thick) at zero
angle of attack are summarized in figure 4, where the starting ordinate
at Infinity rQ of each trajectory is given as a function of the point
of impingement on the surface in terms of S. Each of the solid lines,
except for i/K = 0, is obtained from a series of calciilated trajecto-
ries. The curve for 1/K =0 is obtained from the nKithematical relation
between S and r for an ellipse. The dashed lines in figure 4 axe
the loci of the termini of the constant i/k curves. As was the case
for the ellipsoid with a fineness ratio of 5 (ref . l), these loci were
foxind to be the same within the order of accuracy of the computations
for all values of ReQ (figs. 4(a} to (f)). From the data presented

mCA TS 3147

in this figixre^ the rate^ the area^ and the distribution of water-droplet
impingement on the surface of the ellipsoid can he determined for given
values of Rbq and K.

Total Rate of Impingement of Water

In flight through clouds composed of droplets of uniform size^ the
total amount of water in droplet form impinging on the ellipsoid is de-
termined by the amount of water contained in the volume within the
envelope of tangent trajectories (fig. l). lEhe total rate of impinge-
ment of water (ib/hr) can be determined from the following relation
derived in reference 1:,

¥„ = 1.04 r^ ^ wL% (3)

m 0,tan ^ '

where the flight speed U is in miles per hour, the liquid-water content
w is in grams per cubic meter, and L is in feet.

The value of Tq + for a given combination of ReQ and K can

be obtained from f Igijre 4 by detennining the value of Tq that corre-
sponds to the maximum S (S^^) for the constant K curve of interest.
The values of rQ ^g_^ fall on the dashed termini curves of figure 4.
In order to facilitate interpolation and extrapolation, the data are
replotted in the form of r§ . as a function of K for constant

Reg in figure 5.

The accuracy in the determination of rQ ^^^ from the calculated

trajectories is about the same for the 10-percent- thick ellipsoid as for
the 20-percent- thick ellipsoid discussed in reference 1. For values of
rQ , greater than 0.015, r_ , was determined with an accuracy of

the order of ^0.0003. For combinations of i/k and ReQ that result
in values of rQ -^an between 0.015 and 0.01, the accuracy of rQ ^^^^
is within iO. 0007. For reported values of rQ ^g^^<0.01, the accuracy
in determining the tangent trajectory is somewhat indefinite, but appears
to be within jrO.OOl.

The effect of body size on the value of r^ ^ for selected cloud-

0,tan

droplet sizes and flight conditions is Illustrated in flgiire 6. The

calculated values given in figure 6 are for ellipsoids with a fineness

ratio of 10 axid major axis lengths between 3 and 300 feet, for flight at

50, 100, 300, or 500 miles per hour through uniform clouds composed of

droplets of 10, 20, or 50 microns in diameter. Pressiore altitudes of

5,000, 15,000, and 25,000 feet, and temperatiores (most probable icing

C\

6 lACA TN 3147

temperature given in ref . 3) of 20°, 1°, and -25° F^ respectively, were
used for the caJ-culatlons . For example, consider a 40-foot-long ellips-
oid with a fineness ratio of 10 moving at 500 miles per hour at zero
angle of attack at a pressure altitude of 15,000 feet through a uniform
cloud composed of droplets 20 microns in diameter. From figure 6(b), o

r§ , is 0.000056. If the liquid- water content of the cloud is assumed !~
u, "Can

to be 0.1 gram per cubic meter, then (from eq. (3)) the total rate of
impingement of water W^^^ is 4.7 pounds per hour.

Extent of Droplet Impingement Zone

The extent of the droplet impingement zone on the stirface of the
ellipsoid is obtained from the tangent trajectories (fig. l). Ihe point
of tangency determines the rearward limit of the impingement zone. The
limit of impingement S^ for a particular Reg and K condition can

be determined from the maximum S value of the constant K curves of
interest in figure 4. Again, to facilitate Interpolation, the data axe
replotted in the form of S^ as a function of K for constant Rbq

values in figure 7. The data of this figure indicate that the maximum
extent of impingement increases with increasing K but decreases with
increasing ReQ*

Because of the difficulty in determining the exact point of tan-
gency on the surface of the ellipsoid of each tangent trajectory, the
accuracy in determining Sj^ is of the order of +0.005. The accuracy

in determining the value of S for the intermediate points of Impinge-
ment (between S^ and forward stagnation point) given in figure 4 was

much higher, because the points at which the intermediate trajectories
terminated on the ellipsoid surface were much better defined.

The effect of body size on the value of Sj^ for selected cloud-
droplet and flight conditions is illustrated in figure 8. For example,
consider a 20-foot-long ellipsoid with a fineness ratio of 10 moving 300
miles per hour at zero angle of attack at a 5000-foot pressiire altitude
through a uniform cloud composed of 50-micron droplets. From figure 8(a),
Sjj^ is 0.102 J that is, the impingement zone extends 2.04 feet reajrward

(measured along the surface) from the forward stagnation point.

Local Rate of Impingement of Water

The local rate of impingement of water in droplet form
(lb/(hr)(Bq ft)) on the surface of the ellipsoid can be determined from
the expression

HACA TN 3147

^0 ^0

Wo = 0.33 Uw ~H — ^ = 0.33 Uw P (4)

^ r dS

where p is the local impingement efficiency. The values of P as a
function of S for combinations of ReQ and K are presented in

figure 9. Ihese curves were obtained by multiplying the slope of the
cxirves in figure 4 by the corresponding ratio ^q/t at each point.
Because the slopes of the curves of tq as a function of S (fig. 4)
in the region between S = and S = 0.01 are difficult to determine,
the exact values of P between S = and S = 0.01 are not known.
The dashed portions of the P curves are extrapolations that were main-
tained consistent with a seemingly reasonable pattern.

Comparison of 10- and 20- Per cent- Thick Ellipsoids

Collection efficiency . - The collection efficiency of an eU.ipsoid
is defined as the ratio of the actual amount of water intercepted by the
ellipsoid to the total amount of water In droplet form contained in the
volume swept out by the ellipsoid. In terms of the ellipsoid fineness

ratio f and r^ , the collection efficiency may be defined as

^ = ^^0,tan^' <5)

For the 10-percent-thick ellipsoid (f = 10), the collection efficiency
may be obtained from the curves of figure 5 through the expression

Ejjj = 400 r^ , . The results of the application of this equation to the

data of figure 5 are shown in figure 10(a). Some curves for the collec-
tion efficiency of a 20-percent-thick ellipsoid (f = 5) are also shown
in figure 10(a) for comparison with those for the thinner body. The
values of collection efficiency for the 20-percent-thick ellipsoid were
obtained from figure 6 of reference 1 through the relation

^ = 100 4, tan-

A comparison of the collection efficiency for the two fineness
ratios indicates that the collection efficiency for the two ellipsoids
is nearly the same for ReQ = 0, except for small and large values of K.

For large values of ReQ, the collection efficiency of the 10-percent-
thick ellipsoid is greater than that of the 20-percent-thick ellipsoid.
The effect of differences in collection efficiency on the rate of im-
pingement is best discussed after the total rates of impingement of
water have been examined.

WACA TN 3147

Total rate of impingement of water . - For a given set of atmospheric
and flight conditions and a major eixls length L, the rate of water im-
pingement is proportional to r§ , (eq. (s)), A comparison of the

rate of water impingement (given as r^ ) is made in table I between

the two ellipsoids of different fineness ratio. The data for the ellips-
oid with a fineness ratio of 5 are obtained from table I of reference Ij
whereas^ the resiilts for the ellipsoid with a fineness ratio of 10 are t^

obtained from figure 5 of this report. ^

to

The resoolts presented in table 1 show that, under all comparable
flight and atmospheric conditions, the rate of water impingement is less
on the thinner ellipsoid (major axes of two ellipsoids are the same).
The difference in rate of impingement between the two ellipsoids is much
larger under combinations of flight and atmospheric conditions resulting in
high rates of inipingement than under conditions in which the impingement
is limited to a region around the stagnation point. For example: at an
altitude of 5000 feet, a speed of 500 miles per hour, and in a cloud
composed of droplets 50 microns in diame"^ci., a 20-percent-thick ellips-
oid with a major axis of 3 feet would be subjected to a rate of water
impingement of over 3.7 times the rate of impingement on a 10-percent-
thick ellipsoid under the ssune conditions; whereas, if the flight speed
is reduced to 50 miles per hour and the droplet diameter to 10 microns,
the same 20-percent-thick ellipsoid would be subjected to only 1.2 times
the rate of impingement intercepted by a 10-percent-thick ellipsoid of
the same length.

The use of collection efficiency in comparing impingement character-
istics of aerodynamic bodies is very often misleeiding, as shown by an
example. The collection efficiency of a 3-foot ellipsoid moving 500
miles per hour at a 5000 -foot altitude through a cloud composed of 50-
micron droplets is 0.636 (fig. 6(a) and eq. (5)) for the 10-percent-
thick ellipsoid as compared with 0.590 for the 20-percent-thick ellips-
oid (ref. l). Although the collection efficiency for the thinner
ellipsoid is 1.08 times that for the ellipsoid with twice the diameter
(major ajces same), the thicker body was shown in the preceding paragraph
to intercept over 3.7 times as much water under these same atmospheric
and flight conditions. This example illustrates the need to account for
the differences in the projected frontal area of the bodies when the
concept of collection efficiency is used in comparing the impingement
characteristics of two ellipsoids.

Surface extent . - The surface extent of impingement is compared in
figure 10(b) for the two ellipsoids by comparing some ctirves of S^

given in figure 7 with curves for equivalent conditions for the 20-
percent-thick ellipsoid presented in figure 8 of reference 1. The ctirves
for low values of Rbq are very nearly coincident. For large values of

NACA TM 3147

Rbq^ the rearward limit of impingement is slightly greater on the 10-
percent-thick ellipsoid. In terms of usual flight and atmospheric
conditions, this means that the rearward extent of impingement is some-
what greater for the thinner ellipsoid when flying through rain or driz-
zle (table l), but is very nearly the same when cloud droplets are
encountered.

Konuniforffl droplet size . - The data presented in all the figures
and discussion are based on flights in clouds composed of droplets that
are all uniform in size. As was shown in reference 1, the total rate
of water impingement when a distribution of droplet sizes is assumed
may be different from that obtained on the basis of the volume-median
size only. A comparison of the total rate of impingement is made here
between the 10-percent-thick ellipsoid and the 20-percent-thick ellips-
oid, using the droplet-size distribution given in figure 11 of reference
1 and the computational procedure described in that reference. Hie
volume-median droplet size is again assumed to be 20 microns, the ve-
locity 200 miles per hoiK-, the ellipsoid length 10 feet, the pressure
altitude 5000 feet, and the temperature 20° F. On a weighted basis and
for the particiilar droplet-size distrib'-tion assumed (fig. 11, ref . l),
the thicker ellipsoid intercepts about 3.1 times as much water. On a
volume-median basis (wherein a uniform droplet size equal to the volume-
median droplet size of the distribution is assumed), the thicker ellips-
oid intercepts 2.8 times as much water.

CONCLUDING REMARKS

Because the droplet trajectories about the ellipsoid were calculated
for incompressible fluid flow, a question may arise as' to their appli-
cability at the higher subsonic flight speeds. As was discussed in
reference 1, the ellipsoid impingement results shoiild be applicable for
most engineering uses throughout the subsonic region (also see ref. 4).

The data of this report apply directly only to ellipsoids of revolu-
tion with a fineness ratio of 10. The cautions presented in reference 1
regarding the extension of the data presented in that report for the 20-
percent-thick ellipsoid to bodies of other shapes are reemphasized here.
In some cases, where the body is of different shape, it may be possible
to match its nose section physically with the section of an ellipsoid of
selected length and fineness ratio for which data are available herein
or in reference 1. p", in such a case, the contribution of the after-
body to the air-flow field in the vicinity of the nose of the body is
small (as it often is), then the impingement data for the matching por-
tion of the s'urface of the ellipsoid can be used for determining the
impingement characteristics of the nose region of the body. In other

10 KACA TS 3147

cases ^ where the body shape differs from that of an ellipsoid but the
fineness ratio is the same, the air-flow field may be similar enough
that an estimate of the total catch can be obtained from the ellipsoid
data. In this case, no details of the surface distribution of impinging
water could be obtained.

Lewis Plight Propulsion Laboratory

National Advisory Committee for Aeronautics

Cleveland, Ohio, February 25, 1954 h

to

REFERENCES

1. Dorsch, Robert G., Brun, Rinaldo J., and Gregg, John L. : Impingement

of Water Droplets on an Ellipsoid with Fineness Ratio 5 in Axi-
symmetric Flow. NACA TN 3099, 1954.

2. Brun, Rinaldo J., and Mergler, Harr^'- H.: Impingement of Water Drop-

lets on a Cylinder in an Incompressible Flow Field and Evaluation
of Rotating Milticylinder Method for Measurement of Droplet- Size
Distribution, Volume-Median Droplet Size, and Liquid-Water Content
in Clouds. MCA TE 2904, 1953.

3. Brun, Rinaldo J., Gallagher, Helen M. , and Vogt, Dorothea E.: Im-

pingement of Water Droplets on HAGA 65-[_-208 and 653_-212 Airfoils

at 4° Angle of Attack. NACA TH 2952, 1953.

4. Brun, Rinaldo J., Seraflni, John S., and Gallagher-, Helen M. : Im-

pingement of Cloud Droplets on Aerodynamic Bodies as Affected by
Compressibility of Air Flow Around the Body. KACA TN 2903, 1953.

lACA Tl 3147

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13

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lACA Tl 3147

"r

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Figure 2. - Concluded. Flow field for lO-percent-thick ellipsoid.

HACA TI 3147

15

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16

MCA TE 3147

6

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17

0.T <

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18

MCA TN 3147

\

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19

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20

MCA TH 3147

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21

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MCA TE 3147

PR" —

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MCA TW 3147

23

.0018

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t

ph

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Droplet
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microns

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w

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Major axis length, L, ft

(a) Pressure altitude, 5000 feet; temperature, 20° P.

Figure 6. - Square of starting ordinate of tangent trajectory as functlor Jor
axis length of ellipsoid. ^

24

MCA TH 3147

.0018

.0016

.0014

.0012

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0, tan

.0008

.0006

.0004

.0002

Cj

30 50 100

Major axis length, L, ft

(b) Pressure altitude, 15,000 feet; temperature, 1° F.

300 500

Figure 6. - Continued. Square of starting ordinate of tangent trajectory as func-
tion of major axis length of ellipsoid.

lACA TE 3147

25

0,tan

,0020

.0018

.0016

.0014

.0012

.0010

.0008

.0006

.0004

.0002

10 30 50 100 300

Major axis length, L, ft

(o) Pressure altitude, 25,000 feet; temperature, -25° F.

Figure 6, - Concluded. Square of starting ordinate of tangent trajectory as
function of major axis length of ellipsoid.

26

MCA TH 3147

H

T3
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m

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(U

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4-5

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MCA TN 3147

27

\

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velocity.

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mi

h

1
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500*

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diameter.

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1

microns

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10 30 50

Major axis length, L, ft

(a) Pressure altitude, 5000 feet; temperatiH'e, 20° P.

Figure 8. - Maximum extent of Impingement zone as function of major axis,
of ellipsoid.

as

HACA TN 3147

10

30 SO 100
Major axis length, L, ft

300 500 1000

i4

(h) Pressure altitude, 15,000 feetj temperature, 1 P.
Figure 8. - Continued. Maximum extent of impingement zone as function of major axis of elllpaold.

MCA TW 5147

29

30 SO 100 300 500

Major axis length, L, ft

(c) Pressure altitude, 25,000 feet; temperature, -25° P.

Figure 8. - Concluded. Maximum extent of impingement zone as function of major
axis of ellipsoid.

30

MCA TN 3147

,M

^^

^

H— —

1/

1/

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r

1

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MCA Tl 3147

31

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32

MCA TSr 3147

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MCA TN 3147

33

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34

HACA Tl 3147

1

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mCA TN 3147

35

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NACA-Langley - 5-5-54 - 1000