Concerning the flow about ring-shaped cowlings of finite thickness

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Title:
Concerning the flow about ring-shaped cowlings of finite thickness Part I
Series Title:
NACA TM
Physical Description:
24 p. : ill ; 27 cm.
Language:
English
Creator:
Küchemann, Dietrich, 1911-1976
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:
It is shown how one may obtain, in a simple manner, the forms of ring-shaped bodies from existing tables of functions according to the customary method of superposition of flow due to singularities and parallel flow. A number of examples of the forms and pressure distributions of annular source bodies with and without hub body are given and the inlet conditions of such ring-shaped cowlings are investigated. Furthermore, the annular bodies of finite length are indicated that correspond to Joukowsky profiles for the two-dimensional case. The examples are to give a basis for the design of a cross-sectional forms of ring-shaped cowlings and a survey of the flows to be expected.
Bibliography:
Includes bibliographic references (p. 9).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Dietrich Küchemann.
General Note:
"Report date January 1952."
General Note:
"Translation of Über die Strömung an ringförmigen Verkleidungen endlicher Dicke." Zentrale für wissenschaftliches Berichtswesen der Luftfahrtforschung des Generalluftzeugmeisters (ZWB) Berlin-Adlershof, Forschungsbericht Nr. 1236, Göttingen, June 13, 1940."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003779497
oclc - 86222527
sobekcm - AA00006192_00001
System ID:
AA00006192:00001

Full Text
t-pr TI2-













NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1325


CONCERNING THE FLOW ABOUT RING-SHAPED

COWLINGS OF FINITE THICKNESS*

PART I

By Dietrich Kiichemann


ABSTRACT:

S. It is shown how one may obtain, in a simple manner, the forms of
ring-shaped bodies from existing tables of functions according to the
customary method of superposition of flow due to singularities and
parallel flow. A number of examples of the forms and pressure distri-
butions of annular source bodies with and without hub body are given,
and the inlet conditions of such ring-shaped cowlings are investigated.
Furthermore, the annular bodies of finite length are indicated that
correspond to Joukowsky profiles for the two-dimensional case. The
S examples are to give a basis for the design of cross-sectional forms of
ring-shaped cowlings and a survey of the flows to be expected.


OUTLINE:

I. GENERALITIES CONCERNING RING-SHAPED COWLINGS AND PURPOSE OF
THE REPORT

II. METHOD AND RESULTS

1. The Method
2. A Ring of Singularities in a Parallel Flow
3. Annular Source Body with Hub
4. Ring-Shaped Cowlings of Finite Length

III. SUMMARY

IV. REFERENCES

*"Uber die Str6mung an ringf6rmigen Verkleidungen endlicher Dicke."
Zentrale fur wissenschaftliches Berichtswesen fiber Luftfahrtforschung
(ZWB), Berlin-Adlershof, Forschungsberlcht Nr. 1236, G6ttingen,
June 13, 1940.







2 NACA TM 1325


I. GENERALITIES CONCERNING RING-SHAPED COWLINGS AND

PURPOSE OF THE REPORT


In techniques, annular bodies or ring-shaped cowlings are used
mainly for two different purposes: In the cooling problem (cowlings of
radial engines, nozzle radiators) and in the shrouding of propellers (Kort
nozzle, axial blower). In both cases the cowling serves for regulating
the flow and its velocity at the location of the cooling block or of the
propeller, respectively, although in a different manner. One speaks in
this connection mostly of cowled radiators and shrouded propellers, and
links with this designation the following conception:

For the cooling problem, it appears useful to cool at minimum flow
velocities; therefore, a diffuser is placed ahead of the cooling block
and the through flow is regulated by an adjoining nozzle of variable
opening. For the propeller, on the other hand, it is desired that the
greatest possible mass of fluid be reached and accelerated, and that the
propeller operate in a region of increased velocity. Thus the nozzle
is arranged ahead of the propeller and the diffuser behind it. The
theoretical calculations of the flow conditions are therefore so far
based almost entirely on the conception of nozzle and diffuser.

Recently, in the reports of Horn (reference I) and Dickmann
(reference 2), another interpretation of the physical phenomena, parti-
cularly for ship propellers, is adopted and investigated. The cowling
is regarded as a ring-shaped wing. For propellers, the shrouding is
selected in such a manner that a (negative) circulation is produced which
increases the velocity in the interior of the ring. This interpretation
has already led to successes in the calculation of the Kort nozzle, and
it may be surmised that a similar approach may be used for the cooling
problem. Especially the question of regulating the mass flow thereby
assumes new aspects. The ring-shaped wing has, compared to the ordinary
wing, the advantage that a "two-dimensional" wing theory is sufficient
since no tip vortices are present, due to the rotational symmetry. Thus
it is necessary to extend the two-dimensional profile theory, which
replaces the wing section by source, sink, and vortex arrangements, to
include a ring-shapei wing; however, it becomes evident that difficulties
arise in calculating the velocities induced by source and vortex rings;
these velocities are no longer simple functions of the distance, but are
composed of elliptic integrals in a complicated manner. Dickmann's
theory represents an approximation in which the elliptic integrals are
developed into exponential series, an approximation valid only for rings,
the diameters of which are large in proportion to the length. In calcu-
lating rings of greater length, one must use numerical treatment. The
numerical treatment is facilitated by setting up tables of functions for
the velocity components of source and vortex rings (reference 3).







NACA TM 135 5 3


Comparisons with tests in shrouded ship propellers showed, moreover, that
Dickmann's theory of the ring-shaped wing (in which the profile thickness,
too, was neglected) cannot come fully into play if the flow conditions
at the cowling, which, in many cases, lead to separation, are not
completely under control. Thus consideration of the finite thickness
of the cowling also seems necessary. Naturally, this influence of the
finite thickness is of like importance in the fairing of radiators. In
addition, any large incremental velocities at the cowling for high flight
velocities must be avoided here.

The present report is intended as a first step in the direction
of determining the influence of the finite thickness of annular bodies.
We shall, at first, consider the annular body alone and disregard the
propeller or the cooling block itself. In a few cases, the influences
of a hub body will be investigated. First of all, we shall find out how
the forms of the semibody (single source) and of the Joukowsky profile
(source-sink distribution), known from two-dimensional flow, appear in
the rotationally symmetrical case where the singularities are distributed
on circles. Pressure distributions will then give us information on the
flow conditions to be expected for inflow into ring-shaped cowlings.
We are not treating the problem of finding for a prescribed cowling the
substitute singularity and hence the flow, but the opposite and simpler
one of determining the form for prescribed singularities.

II. METHOD AND RESULTS

1. The Method

We use the method of superposition. Since we want to treat flows
free from circulation only, we select as singularities distributions of
source and sink rings, and superimpose a parallel flow. We execute the
calculation numerically, and take the stream function and the velocity
components of the source rings from the tables mentioned above
(reference 3). Reference 3 also contains the formulas for the velocity
components and the stream function of vortex rings.

In the composite flow, we look for the streamline which gives the
body contour. This streamline is characterized by the stagnation point
lying on it. The thickness of the originating body is linked with the
source strength E in the following manner:

We consider first an annular source body produced by a single source
ring in a parallel flow. Designating the cross-sectional area at
infinity by F and the velocity of the undisturbed parallel flow by U,
we have, according to the continuity equation


E = FU







4 NACA TM 1325


From reasons of symmetry F is annular


F = r22 r12) = 2w rl + r2(r = 2nR
2 )2

with rl designating the inner radius, r2 the outer radius, R the
mean radius, and d the thickness. Thus we obtain

E = 2rRUd (2)

The pressure distribution is obtained, as customary, by means of
Bernoulli's equation from the velocity distribution at the body surface.

In order to obtain a hub body, we superimpose on this flow that of
a single source on the symmetry axis. Let the latter have the strength
Eo. The thickness do of the originating hub body is linked with E0
and U by the relation

2
=do
Eo U (3)
4

Thus we may write in this case the equation (2)

Rd
E =8 Rd E (2a)
do

For a given value of Rd/do2 the ratio of the two source strengths is
fixed. If, in addition, we express do as a function of the radius r'
of the source ring, for a given value of do/r' the quantity Rd/r'2
characterizes the ratio of the source strengths.

In a further series of examples, we place a distribution of source
and sink rings on a coaxial circular cylinder (stream surface of the
parallel flow). We then obtain, if the combined strength of all source
and sink rings is zero, a closed profile, a streamline, which is in a
flow without circulation (ca = 0). We select the intensity of the
distribution in the form


E = E' 1- x/r' + 2 x (4)







NACA TM 1325 5


(x = coordinate in the direction of the symmetry axis, r' radius of the
source and sink rings). For the corresponding two-dimensional problem,
we obtain a Joukowsky profile from this source sink distribution. The
method of determining the contour corresponds in every detail to that
used for the single source ring except for the fact that one now must
integrate over the entire distribution.


2. A Ring of Singularities in a Parallel Flow

As a preliminary remark to this section, we want to show as an
example the appearance, in the rotationally symmetrical case, of a flow
about a circular cylinder, obtained (as is well known) by superposition
of a parallel flow on a plane dipole. The result is shown in figure 1
which originated by superposition of a dipole ring and a parallel flow.
One sees clearly the deviation from the two-dimensional analogue, also
from that of a flow about a sphere in the proximity of a wall2. It is
particularly striking that the two stagnation points no longer lie
diametrically opposite each other.

The following figures show the conditions for a flow about a source
ring. 'Figure 2 gives the contours of annular source bodies of various
thicknesses. The straight line d/r' = 0 is the streamline of the
parallel flow which passes through the location of the source ring
(r = r'). The contours of the bodies of a thickness different from zero
do not lie symmetrical to this straight line, and again the stagnation
points do not lie directly ahead of the source ring. Since for all
x-values the same mass flow, which at infinity has the velocity U of
the undisturbed parallel flow, must flow through between this stagnation
point streamline and the axis, this streamline (the interior of the body)
rises, for large positive x-values, behind the narrowest place of the
cross section again to the same distance from the axis it has had
infinitely far ahead of the body. The mean radius R of the annular
body, that is, the arithmetic mean of outer and inner radius at infinity
is therefore larger than r'. In figure 3 the ratio R/r' is plotted
against the thickness. From this state of affairs it follows that the
"mean camber line" of the body is not a coaxial straight line, but a
line pulled inward toward the leading edge. In practice, where smooth
entrance flow is desired, one always uses this "lip" for ring-shaped
cowlings even though conditions there may be slightly different, for
instance, due to the presence of the cooling block. For the body shapes
indicated, the pressure distribution on the outside is given in figure 4,
that on the inside in figure 5. A negative-pressure region with slight

1Compare F. Keune (reference 4).
2Compare Prandtl-Tietjens, Hydro-and Aeromechanics, Vol. 2, page 125.







6 NACA TM 1325


pressure increase results on both sides shortly behind the stagnation
point so that for such body shapes probably no separation phenomena will
occur. Figure 6 shows how contour and pressure distribution of the
annular source body differ from that of the two-dimensional semibody.
It is noteworthy that the pressure distribution on the outside is hardly
different from that of the two-dimensional semibody whereas on the
inside, for understandable reasons, a greater negative pressure prevails.


3. Annular Source Body with Hub

The following examples show the body shapes resulting if, aside
from a source ring in a parallel flow, in addition, a three-dimensional
single source is assumed on the symmetry axis. We then obtain a ring-
shaped cowling with a hub. The most essential result of these calcu-
lations is the fact that noteworthy differences appear according to
whether the hub is inside or outside of the ring. Figure 7 shows in the
heavily drawn lines the streamline pattern about the hub body alone,
the well known three-dimensional semibody flow. If we put into this flow
a source ring so that its x-coordinate is equal to or larger than that
of the single source, there results the annular body cross-hatched
throughout. The third streamline from the axis, in the undisturbed flow,
forms its contour (fig. 7). This stagnation-point streamline of the
composite flow is plotted as a dashed line. Far ahead of, and far behind
the annular source body, it approaches the previous undisturbed stream-
line. If one now places a source ring of the same strength at a certain
distance ahead of the single source in such a manner that again the same
streamline of the undisturbed flow becomes the body contour, an annular
body results which, it is true, at infinity has the same thickness and
the same distance from the axis as the previous one, but an entirely
differently formed front part. The form of this body is plotted in
figure 7 partly cross-hatched and shows at the entrance a thickening
caused by the stagnation due to the hub body, in contrast to the annular
body behind the hub, the form of which does not differ essentially from
the afore-treated annular bodies without hub. What additional conclusions
may be drawn from this state of affairs will be seen later. The annular
bodies with outward protruding hub will now be investigated somewhat
more closely. Such bodies of various thicknesses are plotted in figure 8.
The thickness do of the hub is kept constant at 1.38 r', and the
x-coordinate of source-ring and single source is equated. The rearward
widening which occurred before may again be observed; the form of the
hub body also is slightly changed with increasing thickness of the cowling.
The pressure distribution over the outside of the body (fig. 9) does
not essentially differ from that of the source-ring body without hub,
even though the pressure increase behind the minimum is somewhat reduced
due to the hub, so that the flow conditions on the outside appear slightly
more favorable with respect to separation in the presence than in the







NACA TM 1325 7


absence of a hub. In contrast, true to expectation, on the inside
(fig. 10) a considerably higher negative pressure results than without
hub; however, these conditions improve greatly if the cowling is extended
beyond the hub. Figure 11 shows again, more accurately, the resulting
body shapes for the special position of the ring r' = do/2; x' = -do/2,
figures 12 and 13 show the pressure distributions over the outside and
inside of the body surface. It is a particularly striking fact about
these pressure distributions that the suction peaks are almost entirely
eliminated, due to the incremental excess pressure ahead of the hub body.
For the body of the thickness d/r' = 0.5, one no longer has any suction
peak at all. When this case will arise depends, of course,"on the
special position of the source ring with respect to the single source;
for other positions, it could be attained for lesser thicknesses of the
cowling. Thus, one may assume that for cowlings of this type, an inflow
particularly free from losses is insured.

The pressure distributions on the hub body itself are represented
in figure 14 for both cases. It can be seen that on the hub body, too,
the effect is more favorable when the cowling is extended beyond the
hub.

All these results apply, at first, only to the special case where
no circulation develops about the cowling; however, they will permit a
survey of the conditions to be expected in the presence of a circulation
as well, since the flow which is free from circulation may always be
regarded as the initial state. Thus these results may offer a basis for
the construction of ring-shaped cowlings.


4. Ring-Shaped Cowlings of Finite Length

Whereas only ring-shaped bodies extending to infinity have been
treated so far, one may cause the body contour to close again by a
distribution of source and sink rings on a coaxial circular cylinder in
a parallel flow. We choose the distribution given by reference 4 which
results, for the two-dimensional case, in the Joukowsky profile drawn
in dashed lines in figure 15. The corresponding annular body (drawn
solidly in fig. 15), however, has a quite different cross section. In
the region of the sources, the mean camber line rises; in the region of
the sinks, it drops off again. Thus the profile obtains a camber with
the maximum rise at about 1/4 of the length counted from the leading
edge. Profile forms of various thicknesses are plotted in figure 16. All
of them show the characteristic criterion of these annular-body profiles.
The camber increases with the thickness of the profile.

3By x' one designates the distance of the source ring from the
single source in x-direction.








NACA TM 1325


This result becomes significant if one wants to obtain a circulation
by means of the cowling. If one wants to produce the circulation by
giving the profile a camber, one must not (for ca = 0) start from the
symmetrical section but from a section cambered in the indicated manner.
Thus, the profiles for producing a positive circulation (reduction of
the velocity in the interior of the cowling) must have a large positive
camber whereas negative circulations are obtained with uncambered
profiles. From these results there follows a new two-dimensional inter-
pretation for the well-known phenomena on annular-shaped cowlings so
far explained by a one-dimensional nozzle theory.


III. SUMMARY


The present report represents a first step in recognizing the
influence of the finite thickness on the flow about annular-shaped
cowlings. It is shown how one may obtain in a simple manner the forms
of ring-shaped bodies from existing tables of functions according to
the customary method of superposition of flow due to singularities and
parallel flow. Knowledge of these forms is important for evaluating
the entrance phenomena into ring-shaped cowlings, particularly for the
fairings of radial engines, for ring-shaped radiators and shrouded ship's
propellers. A series of examples show the cross sections of such
annular source bodies as well as their pressure distributions without
and with the presence of a hub body. It is shown that, among the forms
investigated, the entrance of cowlings slightly thickened at the leading
edge with a hub body within is particularly favorable and free from dan-
gerous separation. Furthermore, the profiles of annular bodies of
finite length are given to which Joukowsky profiles correspond for the
two-dimensional case. These profiles show a camber dependent on the
thickness, with the maximum rise at about 1/4 of the length counted
from the leading edge. All calculations have been performed with the
assumption that the flow about the body is free from circulation; how-
ever, they may serve as a guide for the flow conditions to be expected
also in case of the existence of a circulation or of disturbance bodies
(cooling block, propeller).


Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics







NACA TM 1325 9


IV. REFERENCES


1. Horn, F.: Beitrag zur Theorie ummantelter Schiffsschrauben. Schiffbau
XLI, 1940, p. 2 and p. 18.

2. Dickmann, H. E.: Grundlagen zur Theorie ringformiger Tragflugel (frei
umstromte DUsen). Ing. Archiv, Bd. XI, Feb. 1940, pp. 36-52.

3. Kuchemann, D.: Tafeln fur die Stromfunktion und die Geschwindigkeits-
komponenten von Quellring und Wirbelring. Jahrbuch 1940 der deutschen
Luftfahrtforschung, p. I 547.

4. Keune, F.: Die ebene Potentialstromung um allgemeine dicke Tragflugel-
profile. Jahrbuch 1938 der deutschen Luftfahrtforschung, p. I 3-I 26.
(Available as NACA TM 1023.)










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NACA TM 1325


I.U3 -- -
in -^_-----


S04 0.5
--- d/r'


Figure 3.- The mean radius at infinity of annular source bodies as a
function of the thickness of the body.


Figure 4.- Pressure distribution over the exterior of annular source
bodies of various thicknesses.


1.15
R/r'

1.10


______t


0 0.1 02 Q3







NACA TM 1325


Figure 5.- Pressure distribution over the interior of annular source
bodies of various thicknesses.







NACA TM 1325


---bx/r'


Annular source body
Outside
*inside

Isionol semibody


1.5-- Annulr source body
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O5 -Two- dimensional semibody



-o x/r'
!- I
0.5 1.0 1.5

Figure 6.- Comparison between form and pressure distribution of the
two-dimensional semibody and the annular source body.


-0.5






NACA TM 1325


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Figure 7.- Heavily drawn, streamline pattern about a three-dimensional
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the hub; partly cross-hatched, annular source body ahead of the hub.


~----~







NACA TM 1325


. r/r
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o
dlr--0

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-- x/r'

------- --


Figure 8.- Annular source bodies of various thicknesses with hub outside.







NACA TM 1325


Figure 9.- Pressure distribution over the outside of the bodies represented
in figure 8.







18 NACA TM 1325


Figure 10.- Pressure distribution over the inside of the bodies represented
in figure 8.





NACA TM 1325 19



















r/r'


2.0 I

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0.2-- x/r'

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Figure 11.- Annular source bodies of various thicknesses with hub inside.


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20 NACA TM 1325









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