Concerning the flow about ring-shaped cowlings

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Material Information

Title:
Concerning the flow about ring-shaped cowlings
Series Title:
NACA TM
Physical Description:
41 p. : ill ; 27 cm.
Language:
English
Creator:
Küchemann, Dietrich, 1911-1976
Weber, Johanna
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Airplanes -- Motors -- Cowlings   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
The investigations carried out in a previous report (NACA TM 1325) concerning the flow about ring-shaped cowlings were extended by taking a circulation about the cowling into consideration. The present second report treats bodies of infinite length with approximately smooth entrance. The circulation was caused by distributing vortex rings of constant density over a stream surface extending to infinity. Furthermore, the influence of a hub body on such cowlings was dealt with. The examples treated are meant to give the designer a basis for his design.
Bibliography:
Includes bibliographic references (p. 11).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Dietrich Küchemann and Johanna Weber.
General Note:
"Report date November 1951."
General Note:
"Translation of Über die Strömung an ringförmigen Verkleidungen. II Mitteilung: Ringkörper unendlicher Tiefe mit Zirkulation bei stossfreiem Eintritt." Zentrale für wissenschaftliches Berichtswesen der Luftfahrtforschung des Generalluftzeugmeisters (ZWB) Berlin-Adlershof, Forschungsbericht Nr. 1236/2, Göttingen, November 11, 1940."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003779492
oclc - 86222046
sobekcm - AA00006210_00001
System ID:
AA00006210:00001

Full Text
(Afr- 131










NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL MEMORANDUM 1326


CONCERNING THE FLOW ABOUT RING-SHAPED COWLINGS

PART II ANNULAR BODIES OF INFINITE LENGTH

WITH CIRCULATION FOR SMOOTH ENTRANCE

By Dietrich KUchemann and Johanna Weber


ABSTRACT:

The investigations carried out in a previous report (reference 1)
5-; concerning the flow about ring-shaped cowlings were extended by taking a
Circulation about the cowling into consideration. The present second
r.:I report treats bodies of infinite length with approximately smooth entrance.
it The circulation was caused by distributing vortex rings of constant density
|::::over a stream surface extending to infinity. Furthermore, the influence
-' of a hub body on such cowlings was dealt with. The examples treated are
:. meant to give to the designer a basis for his design.

OUTLINE:

SI. STATEMENT OF THE PROBLEM AND METHOD
IT. RESULTS
1. Annular Bodies without Hub
2. The Influence of a Hub Body as Seen in the Example of Annular
SBodies without Circulation
3. Annular Bodies with Hub
4. Mass Flows and Thrust Forces
III. SUMMARY
IV. REFERENCES


I. STATEMENT OF THE PROBLEM AND METHOD


In the first of this series of reports concerning the flow about
-ring-shaped cowlings of finite thickness (reference 1), it was shown how
...to obtain in a simple manner the forms and pressure distributions of ring-
shaped bodies from existing tables of functions according to the customary

"Uber die Stromung an ringformigen Verkleidungen. II. Mitteilung:
k Ringkorper unendlicher Tiefe mit Zirkulation bei stossfreiem Eintritt."
SZentrale fur wissenschaftliches Berichtswesen uber Luftfahrtforschung des
Seneralluftzeugmeisters (ZWB) Berlin-Adlershof, Forschungsbericht
',.Nr. 1236/2, Gottingen, Nov. 11, 1940.
1Mfl:r.l






NACA TM 1326


method of superposition of flow due to singularities and parallel flow.
Whereas in reference 1, a circulation about the cowling which would
cause a velocity in the interior of the ring different from that in the
exterior was not considered, the present report will deal especially
with the influence of such a circulation. We shall, however, limit
ourselves to cowlings of infinite length and, thus, obtain an insight
as to the flow conditions about such cowlings for smooth entrance. We
herein utilize the advantage of the singularity method of yielding body
shapes and velocities, which in the case of given singularities is
relatively simple. However, the inverse problem determining the
singularities for a given body shape or velocity distribution is
considerably more difficult and troublesome. We hope to be able later
on to treat arbitrary bodies of finite length with circulation for non-
smooth entrance as well.

In the present report, the starting point is simple singularities.
It will be shown that even the simplest singularities yield shapes and
pressure distributions that are absolutely usable in practice. Moreover,
it will be possible to recognize fundamentally, for instance, the
influence of a hub body; however, it must again be stressed that we
obtain cowlings which the flow always approaches in the most favorable
manner only (approximately smooth entrance), and that we do not investigate
a certain form for various operating conditions. The examples described
are meant, above all, to give the designer a basis for the design of
aerodynamically favorable ring-shaped cowlings.

We shall first discuss the singularities.

We obtained a thick ring-shaped body in the simplest manner by
placing a source ring into a given main flow. For the bodies of infinite
length treated here, one such source ring will be sufficient. The
simplest main flow is the parallel flow. We then obtain the annular
bodies without circulation treated in reference 1. A hub body is
produced by placing a three-dimensional single source on the symmetry
axis. Finally, a main flow with different velocities within and outside
of a ring is obtained by distributing vortices over a stream surface.
We select vortex rings, the centers of which lie on the symmetry axis.
Since we want to investigate the entrance, the distribution must start
at a certain point and may extend to infinity. It will be shown that
it is sufficient to assume the density of the distribution constant in
the direction of the symmetry axis. For the numerical calculation, the
values of the stream function and velocity components of source and
vortex ring are taken from the tables of functions (reference 2) where
the pertinent formulas also are to be found. The accuracy of the calcu-
lations corresponds to the slide-rule accuracy.

In view of the great number of parameters, we must necessarily
limit ourselves to a few characteristic values. The thickness d of the








NACA TM 1326


cowling (compare fig. 1) is referred to the radius R at the infinity
of the streamline on which lies the source ring, and all results are
given only for the values d/R = 0 (mean camber stream surface) and 0.2.
Intermediate values may be linearly interpolated. The radius RN of
the hub body is accordingly referred to the radius ro of the source
ring by which the distance a in axial direction between source ring
and single source on the axis also is standardized. The shapes and
pressure distributions are indicated, in general, for the values
RN/ro = 0, 2/3, 1, and a/ro = 0, 0.5, 1. We characterized'the strength
of the circulation by the ratio between the velocity U. which ulti-
mately results within the ring and the undisturbed velocity of the
oncoming flow Uo in the outer space, and assume the values Ut/Uo
as 0.2, 0.4, 0.6, 0.8, 1.0, and in a few cases up to 2.0.


Ir. RESULTS

1. Annular Bodies without Hub


We investigated, first, annular bodies with circulation without
hub. We have already noted that we want to produce the velocity differ-
ences by a constant distribution of vortex rings over a stream surface.
The shape of this stream surface is unknown at first. If, for instance,
an axially parallel cylinder with a distribution of vortex rings is put
into a flow in the direction of the axis, the cylinder does not remain
a stream surface. The originating streamline pattern is drawn in dashed
lines in figure 2. An iteration method can be carried out by placing
the distribution on that stream surface which has the same starting
point (x = 0; r = ro). Step by step, we thus approach the stream surface
which coincides with the locus of the distribution for the prescribed
vortex strength. Figure 2 shows the result (Uw/Uo about 0.3) in the
solidly drawn streamlines. The position of the distribution is heavily
outlined. It could be made the wall and would be stream surface when
the prescribed velocity ratio is produced. From this consideration it
follows that a certain wall is obtained for each velocity ratio. These
walls are plotted for various U,/Uo in figure 3. For U/Uo = 1
naturally a cylinder results (no circulation). For values smaller than 1,
the walls are bent inward toward the front; for values larger than 1,
they are bent outward.

The originating flow conditions are of interest. In figures 4
and 5, we plotted once more the velocity components of the vortex flow
alone (at the same time as illustration of the dashed-line streamline
pattern, fig. 2) for the case that the distribution is located on a
cylinder. We can see that very soon parallel flow, stemming from the







4 NACA TM 1326


vortices, develops in the interior of the ring. The corresponding fig-
ures for the case where the distribution is a stream surface have not
been drawn again. The deviations may be understood with the aid of
figure 2. Figure 5 shows, furthermore, that the vortices at the initial
point of the distribution (x = 0; r = ro) cause an infinitely high
radial velocity.1 Our simple assumption becomes noticeable in the
streamline pattern by a vertical inflectional tangent of the respective
streamline at this point; however, this fact cannot be expressed in the
diagram. Since, furthermore, such cowlings in practice always will be
of finite thickness, this singular point'remains without decisive
influence. If later on in the pressure distributions at this point, an
infinity appears for d = 0 and a local suction peak for d / 0, this,
too, is caused by the simplification mentioned. Nevertheless, refine-
ment of the theory in the manner indicated above was foregone since its
effects are immediately clear and we are here concerned only with
fundamentals.

The continuity equation readily gives information about the mean
velocities in any cross section within the ring, since cross-sectional
area and velocity at infinity are known. Accordingly, the ratio between
the mean velocity U in the entrance cross section (x = 0) and UO
is plotted against the velocity ratio in figure 6. This quantity also
represents the ratio between the quantity of fluid flowing in the
interior of the ring in the presence of circulation, and the quantity
which would flow through the entrance cross section without circulation;
thus, the ratio Qo = nro2Uo. The mean velocity U in the entrance
cross section is therewith shown to be not yet the final velocity 1,;
however, if one proceeds somewhat further into the cowling (x/R = 0.5),
the mean velocity has come considerably closer to the ultimate one, as
figure 6 shows. The end velocity is attained to the same degree as the
curves in figure 3 approach the asymptote. In figure 6, the contraction
ratio ro/R is plotted once again more accurately. Moreover, one
recognizes from this figure that the velocity decrease or increase as
well as the pressure recovery takes place neither completely inside nor
completely outside of the cowling, so that one is dealing neither with
a nose inlet nor with a side inlet according to P. Ruden (reference 3)
but with an intermediate form. More accurate information can be gained
from figure 7 where the pressure on the symmetry axis is plotted for
U /Uo = 0.2. For comparison, we see the pressure which would be caused
by a vortex distribution over a cylinder. One recognizes that the
pressure recovery takes place mainly ahead of the cowling so that one

lIn order to avoid this, the distribution would have to start as Vx,
for instance, as x/r + x. For the sake of the simplicity of the calcu-
lations, a constant vortex distribution was maintained.








NACA TM 1326 5


may count on relatively small losses inside of the cowling. From fig-
ure 8, we can see the pressure at the cowling itself for the different
velocity ratios. The appearance of an infinitely large pressure at the
leading edge has already been discussed. Further inside, however, a
very favorable pressure variation becomes evident which is caused by the
constriction or extension, respectively, of the stream surface of the
S distribution. Thereby, one may subsequently justify the selection of
the special vortex distribution.

If one now places into this basic flow a source ring at the start
of the distribution, one obtains body shapes as indicated in figure 9.
The source strength is determined in such a manner that a thickness of
0.2R results at infinity. One obtains, corresponding to the various
mean camber lines (fig. 3), a certain body shape for each velocity ratio.
In order to calculate the behavior of a certain body, for instance that
of Uc/U = 0.4 in case of other velocities, one would have to determine
the additional source and vortex ring distributions which make the
prescribed body again a streamline; this calculation would exceed the
scope of the present report.2 Since the body shapes are, however, not
too different, it may be assumed that the permissible velocity range for
a certain form will not be too small. Figure 10 also shows the bodies
of the thickness d/R = 0.1 for several velocity ratios and simultaneously,
the mean camber streamline (dashed) from which one may see the position
of the body contour with respect to the mean camber line. It is note-
worthy that the inward curving of the mean camber line near the entrance
opening is filled up by the source fluid so that a relatively straight
body inside results.

The velocity and pressure distributions at the surface of these
bodies have not been calculated. One may attain an approximate idea
by superimposing the velocity distributions of annular source bodies
without circulation (U./Uo = 1, compare also fig. 13) resulting from
the first report (reference 1) on those on the mean camber lines (fig. 8).


2. The Influence of a Hub Body as Seen in the Example

of Annular Bodies without Circulation

It was shown in the first report what important effect a hub body
may have on the flow conditions at the cowling. It could be shown that
it is possible to attain, by favorable selection of position and thick-
ness of the hub body, the absence of suction peaks, and only very slight
incremental velocities for the pressure distributions at the inside and
outside of the cowling. The radius RN/ro and the position a/ro of
the hub, being the most influential parameters in this respect, will be

2Compare section I.







NACA TM 1326


treated first in somewhat more detail than in the first report (in this
section without taking a circulation into consideration). The influence
of circulation is investigated in section II, 3.

Figures 11 and 12 show the investigated forms of hub and cowling,
partly to be found in the first report. The considerable thickening at
the Leading edge of the annular source body for RN/ro = 1 and a/ro = 1
is noteworthy. Figures 13 to 18 show the pertaining pressure distribu-
tions on the surface of the cowlings. Figure 13 may be used as comparison
for RN = 0. The influence of the hub is shown particularly clearly by
the pressure distribution on the mean camber line d = 0. Let us
consider first the hub bodies of the radius RN/ro = 2/3 in their
various positions (figs. 14 to 16). No essential changes in pressure
distribution are caused by a hub body of this thickness. The maximum
negative pressures stay within moderate limits, and the maximum incre-
mental velocities fluctuate between the values 0.16Uo and 0.25Uo. The
pressure distribution over the forms investigated shows the smallest
incremental velocities and the smallest pressure rise for the value
a/ro = 0.5. The thicker hub body RN/ro = 1 (note the curves for d = 0)
has considerably greater effort. For a/ro = 0.5 we have, for d/R = 0.2,
no longer a suction peak on the inside of the cowling and only slight
incremental velocities; for a/ro = 1, we have the same favorable condi-
tions also on the outside of the cowling. A hub of this position and
thickness obviously exerts a favorable influence on the flow at the
cowling.

The pressure distributions on the hub body itself are plotted in
figures 19 to 23. The cowling always affects the hub in an unfavorable
sense though only to a small extent. The position a/ro = 0.5 is again
shown to be a favorable position for RN/ro = 2/3, just as is the position
a/ro = 1, recognized as favorable before for d/ro = 1.

The examples of this section have once more demonstrated in detail
what effect a hub may have on the flow at the cowlings. It must be noted,
though, that no statement can be made on the behavior of these bodies if
the oncoming flow is not smooth.


3. Annular Bodies with Hub

We now turn to the investigation of the influence of circulation for
annular bodies with hub. Figures 24 and 25 show the mean camber lines
for various velocity conditions in the presence of the thin hub body
RN/ro = 2/3 once for a = 0 and then for a/ro = 1. In these examples,
we investigated only positive circulations (Uw/Uo < 1) since the






NACA TM 1326


calculations without hub bodies showed, true to expectation, that the
outward bending for UJ/Uo > 1 is not significant and does not lead to
greatly altered bodies.

Figures 24 and 25 indicate the following new effect of the hub body.
Since the mean camber line for U/Uo = 1 now is also curved and
narrowed, whereas, the one for U/Uo = 0.2 has remained practically
unchanged compared to its form without the presence of a hub body, these
stream surfaces now move considerably closer together. This is particu-
larly noticeable for the position a/ro = 1. One may assume that this
effect will be clearest if the respective streamline has, even without
circulation, a form similar to the one in case of a strong circulation.
Thus, we are again led to arrangements with a relatively thick hub body
in the interior of the cowling an arrangement found advantageous in
other respects as well as in the previous section.

Before investigating such an arrangement, we want to point out the
pressure distributions on the mean camber lines shown in figures 24
and 25 (figs. 26 and 27). Their variation is favorably similar to that
of the corresponding pressure distributions without hub. In this respect,
too, the rearward position a/ro = 1 appears advantageous among the
examples investigated. Figures 28 and 29 show the thick cowlings with
hub pertaining to figures 24 and 25. One can see clearly that the
separate contours are now considerably less different from each other.

We now investigate the arrangement RN/ro = 1 and a/ro = 1
treated without circulation in the previous section (figs. 12, 18, and 23),
for the velocity ratio Um/Uo = 0.4. The mean camber line for (Uo/Uo = 0.4
(fig. 30) is shown to differ little from the one for U/Uo = 1.) In
figure 30, the cowlings without circulation with d/R = 0.2 and
d/R = 0.123 have been plotted additionally for comparison. One recog-
nizes from figure 30 that for a certain form under various operating
conditions, the travel of the stagnation point will not be as extensive
as in the absence of a hub bbdy. (Compare fig. 9.) This fact may be
explained by a certain guiding of the streamlines by the hub body. This
guiding may be additionally aided by a different shaping of the hub body.
In regard to the simplicity of the singularities selected, it is, of
course, not to be expected that we should have obtained the most favorable
forms. The pertaining pressure distributions are shown in figures 31,
32, and 33.

4. Mass Flows and Thrust Forces

So far, we have indicated the velocity ratio Um/Uo as the laram-
eter characteristic for the circulation; however, this makes sense only
: --
3The pertaining pressure distributions may be found in the first
report in this series (reference 1).







8 NACA TM 1326 ,


for our particular selection of infinitely long bodies. We therefore
show additionally the connection between U4/Uo and the quantity of
fluid Q flowing through the entrance cross section Tro2, for the
examples treated, in the following figures 34 to 37. Qo there signifies
the quantity which would flow through the entrance cross section in the
absence of a cowling. With a hub body present, this quantity is smaller
than rro2Uo (compare fig. 38). The quantity Q ro2Uo represents the
ratio between mean velocity in the entrance cross section and undisturbed
free-stream velocity Uo, and may be taken from the figures 34 to 37.
(Compare also fig. 6.) At this point, it should be mentioned that the
thickness of the cowling, too, reduces the mass flow as has been discussed
in detail in the first report of this series (reference 1).

Furthermore, we want to investigate what axial forces are absorbed
by the entire arrangement and by the cowling, in particular, for the
different operating conditions. A detailed calculation by means of the
theorem of momentum for the two-dimensional case may be found in
P. Ruden's report (reference 3). A corresponding deliberation for our
rotationally symmetrical problem leads to a simple formula for the forqp
S exerted by the flow on the total arrangement:


S=- uo22 U 2 (1)


According to equation (1), one always obtains a thrust on the combined
elements. The component of the force absorbed by the hub may be deter-
mined according to A. Betz (reference 4) in a simple manner by presup-
posing the single source replacing the hub body to be situated so far
inside of the cowling that the velocity induced by the vortices and
sources of the cowling has, at the locus of the source, already reached
the final value




This condition is approximately satisfied shortly behind the entrance
opening as has been shown before in section II, 1. The relation

S= n- uE (2)

is then valid for the force at the hub with
E RN2
E = URN UA

VI







NACA TM 1326 9


signifying the source strength. Therewith SN becomes


22 _1\ Uo22 e (3)
SN =-_ o2tR22 .\U


.According to this, the hub is subject to a drag for Ut/Uo < 1, whereas,
it takes over part of the total thrust for Ug/Uo > 1. The axial force
SV at the cowling is obtained as the difference between total thrust and
hub force. In figure 39

Sv/ Uo2iR2

is plotted against U,/Uo for various values of RN/R. One recognizes
that the cowling, if it contains a hub body, must, for UU/Uo < 1,
absorb thrust forces. For

/Uo > 1 and RN/R>

the cowling always experiences a drag which increases with U,/Uo. For


Uo/Uo > 1 and RN/R < 1/2

we find at the cowling at first drag, later on, however, again thrust
forces.

If the hub is pulled forward so far that at the locus of the single
source which replaces it, a velocity stemming from the vortices and
sources of the cowling practically no longer exists, the cowling takes
over the entire thrust given by equation (1) exactly as in the case where
no hub is present.

Since the equations (2) and (3) essentially depend on the selection
of the singularities, these relations as well as the figure 39 calculated
from them are to be rated only as limiting cases; however, we thus obtain
a convenient survey of the forces to be expected. Equation (1) is valid
generally with the one presupposition: that the thickness of the cowling
Sfor x -- oo does not continuously increase.

Finally, we want to point out a problem not yet treated in the
present report: that of oblique free-stream direction. Our results
apply only to axial free-stream direction. It may, however, be assumed
that a slightly oblique direction will not essentially alter the results.






NACA TM 1326


III. SUMMARY


After the statement of the problem and the separate assumptions
had been discussed in the first section, the results were represented
by a series of examples in the second section. First, the form of the
surfaces resulting as stream surfaces for different strength of circu-
lation is determined. The pressure distribution indicated shows that
the pressure recovery to a great part takes place ahead of the cowling.
The influence of thickness and position of a hub body is represented
in the example of bodies without circulation. It is further shown that
in the presence of a circulation by suitable selection of thickness and
position of the hub body, the cowlings are more similar for the various
ratios between internal and external velocity with than without hub
body. The body shapes and the pertaining pressure distributions are
given for each of the calculated examples. Finally, the axial forces
are investigated. The direction of these forces is always opposite
to that of the oncoming flow.


Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics







NACA TM 1326


REFERENCES


1. Kuchemann, D.: Ueber die Stromung an ringformigen Verkleidungen
endlicher Dicke. (I. Mitteilung). Forschungsbericht Nr. 1236,
1940. (Available as NACA TM 1325.)

2. Kuchemann, D.: Tafeln fur die Stromfunktion und die
Geschwindigkeitskomponenten von iuellring und Wirbelring. Jahrbuch
1940 der Deutschen Luftfahrtforschung, p. I 547.

3. Ruden, P.: Ebne symmetrische Fangdiffusoren. Forschungsbericht
Nr. 1209, 1940. (Available as NACA TM 1279.)

4. Betz, A.: Singularititenverfahren zur Ermittlung der Krafte und
Momente auf Korper in Potentialstromungen. Ing. Archiv Bd. III,
1932, p. 454.






12 NACA TM 1326


R N



N0


Figure 1.- Over-all sketch for the symbols.








NACA TM 1326 13











I I QI
.I l oll



-I ,-- a



II Cd
ii I




C? ^0

x 0

o 011
lt 0



I I I .


I I
I \
II l l \o 3
\ \- '-l"




HCZ\ \ ", \S
w\\ I \ 'J





S-Cd
C I C'

I~ lit I I I
il -- I h
II II







NACA TM 1326


1.4


,u ----- -

0.2 d = 0 R 0
N ._r/R

.4



0 0.4 0.8 1.2 1.6 -x/R R



Figure 3.- Distribution stream surfaces for various velocity ratios.







NACA TM 1326


K/!o


-0.2 0 0.2 09 0.6 80 1.0
-.U/U .


Figure 4.- Axial velocities stemming from a distribution on a circular
cylinder in various cross sections.






NACA TM 1326


\ 0.8
\ I
\ 0.


\ I .4
\ I'
x/rU= 0 ML-- 12





-AID -0.8 -06 -0.4 0:2 0
-fw V/U.O


Figure 5.- Radial velocities stemming from a distribution on a circular
cylinder in various cross sections.


"ii
/ It


I/ .2
I


Tr/r


/'
/y




w w


NACA TM 1326


(IU./x /R= 0.5


=0
/R=0 0o


Figure 6.- Mean velocity values, mass flows, and contraction ratios for
the distributions shown in figure 3.






NACA TM 1326


U//Uo=0.2


Ir----

r/R


Figure 7.- Pressure distribution on the axis of symmetry for a
distribution on a circular cylinder (dashed) and on a stream
surface (solid line).







NACA TM 1326


Figure 8.-


Pressure distributions on the outside and inside of the
distributions shown in figure 3.


PPO


d=O;RN=0


-- Inside
Outside







NACA TM 1326


I 0








% z




isN-


a

8


qt
0 0


;

;ii.


- o





I,
















I


I 1 c, o o



\










q' z
I I I
cq
























I \
I \




I I
I I
_ I I _


NACA TM 1326


0
*-.

C,

0
-4
O



-.
o
>



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i'-
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-11 .





o 6
Cd




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NACA TM 1326


0)
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0
a



SLOc



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0 0










0
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0
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r-i



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NACA TM 1326 23












N \ I' I 8



o 0

I .0
SO-C




I \ O





I 0"
I I\ \ 8uS











S .a
aD ;











I E
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I E I ~











O-d
o4-'-E
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-oo
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NACA TM 1326


Fig.13


Fig. 14


Figures 13 and 14.- Pressure distributions on the cowlings.


P-P
4i Inside
0.6 -=- Outside

0.4 U../U.R/r.N/LqaI;



S.0d/R o



-0.4
S0,2 0.4 0,.e O 1.0I -X/R
I\/o o. iqn os






NACA TM 1326


Fig.15


Fig.16


Figures 15 and 16.- Pressure distributions on the cowlings.


oI


Inside
Outside
I U_/U.= I; RN/r,=2/3a/ro=0.5
0.4 ------



0 a/R0 -


=0,4
o.4 ( 1__ 0 x/R
0 0.2 Q4 o6 Ora.o-, /R







NACA TM 1326


,r 1 1 1 r
* -- Inside
--- Outside
0.4 U UoI. ; RN/r.I; o/r 0.5-



















-9 --- -- --
o-- Outside
sa o .._--







1Fig. 17



SInside d/ =










q 0.2 0.4 oQ Qs I.o-*x/R


Fig.18


Figures 17 and 18.- Pressure distributions on the cowlings.







NACA TM 1326


Fiq.19


Fig. 20


Figures 19 and 20.- Pressure distributions on the huos.







NACA TM 1326


Fig. 21


Fig. 22

Figures 21 and 22.- Pressure distributions on the hubs.






NACA TM 1326 29


Figure 23.- Pressure distributions on the hubs.






NACA TM 1326


Figure 24.- Mean camber lines and hub bodies for various velocity ratios.






NACA TM 1326 31


Figure 25.- Mean camber lines and hub bodies for various velocity ratios.






NACA TM 1326


- Outside
--.- Inside
RN/r.= 2/3;ak.o= 0; d=O


Figure 26.- Pressure distribution on the mean camber lines of figure 24.






NAA TM 1326
NACA TM 1326


- Outside
Inside
d :=; RN/ro: 2/3; a/r,=l


Figure 27.- Pressure distribution on the mean camber lines of figure 25.



















C4 0 8







cuo
I -:













0-
S.d
II

0











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a\ a\d \0
Or-


NACA TM 1326


I



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0



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0




















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O



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'-4























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.0


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w
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sa
ho













0L







NACA TM 1326 35



















94
I I
I --- i I I r -- I ^ v I E 8 1 "


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cd


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---
o







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S0*c *. b






NACA TM 1326 '







NACA TM 1326


Outside
RN/r.= I,; ao= I

Fig. 31


Inside
RN/r.=, I o /r. I

Fig. 32


Figures 31 and 32.- Pressure distributions on the inside of the bodies
represented in figure 30.






38 NACA TM 1326


Figure 33.- Pressure distributions on the hub bodies of figure 30.






NACA TM 1326 39


Fig.34


d/R:
0
0.1
10.2


0.6 -

04

0.2
Qo
r.= z0.78; R/r2/3,a=0
0 0.2 0.4 0.6 0.8 LO
-- UI/U.

Fig. 35


Figures 34 and 35.- Dependence of the mass flow on the velocity ratio.


I- 1 1L1A






NACA TM 1326
*i


Q/Q


IAJ









0 2
-0. 94; 2/.3a/r.=l
0 02 0.4 0.6 .
-iUoO/Uo


d/R:
0
0.I
0.2


FIG. 36


Q/Q.
t/a


:U--r rzI-
I I^__


d/R=
0

0.I
0.2


o'- =0.85 RN/ro--I;aro=

0 02 0.4 06 1.0
Uco/Uo


FIG. 37


Figures 36 and 37.- Dependence of the mass flow on the velocity ratio.


. :. ". .. ."


Ar


4^


1.0






NACA TM 1326


IN-

Tr Rue

n,


.20.4 0.6 a/r


0 02 0.4 0.6 o/r, 1.0


daO


Figure 38.-


Dependence of the mass flow on the position and the thickness
of the hub body for cowlings without circulation.


Figure 39.- Thrust coefficient on the cowling as a function of the velocity
ratio.


NACA-Langley 11-30-51 1000


RN/rO=
0
2/3
I


SV
AeuVR2--
A RN/RO.75


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UNIVERSITY OF FLORIDA