Lift on a bent, flat plate


Material Information

Lift on a bent, flat plate
Series Title:
Physical Description:
15 p. : ill. ; 27 cm.
Keune, F
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


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


The lift on a bent, flat plate is calculated exactly by the use of conformal mapping. Results are presented in terms permitting direct determination of the angle of zero lift, the lift coefficient, and the lift-curve slope for any flap-chord ratio, flap-deflection angle, and angle of attack.
Includes bibliographic references (p. 5).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by F. Keune.
General Note:
"Report date February 1955."
General Note:
"Translation of "Auftrieb einer geknickten ebenen Platte." (Bericht der Aerodynamischer Versuchsanstalt Göttingen.) Luftfahrtforschung, March 20, 1936, Annual Volume."

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003778991
oclc - 86087910
sobekcm - AA00006190_00001
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Full Text
icA- mi 1390




By F. Keune

For the theoretical treatment of fin and rudder, or of a wing with
flap, the profile is simplified in such a way that a bent flat plate may
be considered. The aerodynamic forces on it can be calculated by means
of known approximations due to Munk (ref. 1) or Glauert (ref. 2). The
problem was recently treated exactly by the Russian work of Chapligin
and Arjanikov (ref. 3). The calculations below will be carried out fol-
lowing the latter work, and the difference between them and the usual
approximations will be determined.1


1. Theory
2. Results
3. Summary
4. References


The lift coefficient of a profile at angle of attack a can be
written in the form

Ca = c sin (a ao) (1)

The problem is to determine the lift-curve slope c and the angle of zero
lift so. For the flat plate, the theoretical lift-curve slope is c = 2n;
for other profiles, it deviates only a little from this value. The angle
of zero lift for the flat plate is, moreover, 0o = 00. With the deflec-
tion of the flap, both values vary as a function of the chord ratio tR/t
and the deflection angle B.
*"Auftrieb einer geknickten ebenen Platte." (Bericht der Aero-
dynamischer Versuchsanstalt G6ttingen.) Luftfahrtforschung, March 20,
1936, Annual Volume, pp. 85-87.
11 am especially grateful to Dr. I. Lotz for many suggestions in
carrying out the analysis.

2 NACA TM 1340

In order to calculate c and ao we must determine the flow around
the bent plate and in particular the circulation needed for smooth flow
at the trailing edge B (fig. 1).

With the aid of conformal mapping, the region outside of the profile
(fig. 1) in the z plane is mapped, according to the Schwarz-Christoffel
procedure (ref. 4), onto the lower half of the A plane (fig. 2b), so
that the border of the profile (see fig. 2a) on the h axis and the point
at infinity in the z plane transform to the point

= -ik

The mapping function has for this case the form

z = c f-- ) (A )n d (2)
0 (2 + k2)2 (A 2)n

where the exponent n is related to the deflection angle 0 of the flap
(see fig. 1) by

Snn (3)

In order to be able to evaluate the integral (see eq. (2)) one carries
out an appropriate linear transformation, by which the ? plane goes into
the plane. The p axis is thereby mapped onto the axis. If we
consider now the point-by-point matching of the z plane with the t
plane (fig. 3), there results from this transformation: the leading edge
A (fig. 1) and the point A* (C = 1i); the location of the bend on the
suction side and the zero point. 0*; the trailing edge B and the point
B* ( = r2); the location of the bend on the pressure side of the profile
and the infinitely distant point on the 5 axis. The point at infinity
in the z plane falls at the point (3 = 1 i5. For these positions of
the designated points the integral of the Schwarz-Christoffel formula (2)
can be calculated. If we introduce the abbreviation K2 = 1 + 82 (1 n2),
we obtain the mapping function
( "n -n
K- 1 + n
z = 2(t tR) (4)
1 n2 1 + n ( 1)2 + 52

The position of the point t3 = 1 i6, the image of the point at
infinity of the z plane, depends on the ratio of the flap chord to the
forward part of the wing and on the deflection of the flap. The position
is determined by the equation

t = o = K 1( + n) (5)
t- tR K + 1 K n

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In the z plane a parallel stream prevails at infinity; on this
stream (on account of the circulation around the plate) is superposed a
vortex whose strength depends on the angle of attack. To the parallel
flow of the z plane there corresponds a flow in the t plane which is
produced by a dipole at the point %3 = 1 ib; and to the circulation
flow there corresponds a vortex flow around the same point. The strength
of the circulation is given by the condition that in the z plane the
flow is smooth at B; thus at B* the velocity is zero. If we give the
position of the stagnation point t4 = 4(a), then the velocity must have
the form

d 2(t t(vK, l -1 + n n 2)(t + t4)
d ( 1 n 2 ,1 + n [( 1)2 + 2]2 (6

In this equation the quantity N is a constant which is essentially
dependent on the angle of attack a.

We determine the circulation from the residue of the velocity func-
tion w (see eq. (6)). One obtains this by resolution into partial frac-
tions. According to lengthier calculation, which can be seen in Chapligin
and Arjanikov (pp. 5-7, ref. 4), one has for the lift

Ca = (1 + o)(< + 1) 1 + n,1 ,-n.
ca = (1i )(K + + 1) ( j ) (+ n'(n) sin(o ) (7)

where the angle of zero lift (pp. 7 and 8, ref. 4) is calculated as

+0o = arctan n nr + n arctan j-2 (8)
KI F- n2 n/

By equating formulas (1) and (7) there is obtained for the lift-curve

( 1t + ) + (i+n)/2 K (- -n)/
= ( ( (9)
(1 + + 1) n, 1 n,

In the evaluation one next determines, for a prescribed deflection
angle 0 and ratio a = tR/(t tR), the value K according to

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equation (5).2 With this equation the calculation of the angle of zero
lift a. (eq. (8)), of the lift-curve slope c (eq. (9)) and of the lift
ca (eq. (1)) are feasible.3


The dependence of ao, the angle of zero lift, on the ratio tR/t
for fixed-deflection angle B is seen in figure 5 (eq. (8)). Since for
values n = p/i > 0.1 the curves for tR/t > 0.5 become approximately pro-
portional to one another, the curves for 1 = 450 and 600 are therefore
not plotted any further.

If one plots the angle ao of zero lift for different chord ratios
against the deflection angle 3 (fig. 6), one obtains almost straight
lines. The dashed lines representing Glauert's approximation deviate
from these curves only for small chord ratio and there one has somewhat
smaller values.

Figure 7 gives the ratio of the lift-curve slope c of the plate
with flap deflection to the flat plate value of 2n. The lift coefficient
ca is referred to the original wing chord t. For the actual chord of
the wing t', which becomes smaller with increasing flap deflection,4 we
obtain somewhat larger deviations from c/2n = 1 in the reversed sense
(c/2n > 1). For all curves 3, they are the largest with chord ratio
tR't = 0.5 in both cases (eq. (9)). Glauert assumed that the quantity
c/2n, referred to the original chord t, varies as the ratio t'/t. This
assumption results in values that are too small.

Figures 8 and 9 show the dependence of the lift coefficient ca on
the ratio tR t for fixed deflection 1 with an angle of attack of the
profile of a = 5 or a = 100 (eq. (1)).

Figure 10 is a cross plot of figure 8 and points out that the depend-
ence of lift on the deflection angle B for values 0 > 150 does not
remain linear any longer. The linear dependence assumed by Glauert thus
holds only for deflections to 150. Glauert has replaced the sine of the
angle of attack, referred to the zero-lift direction, with a ao itself.
The lift becomes too large, therefore, for large a ao. Up to deflection
of 1 = 150 this error is counterbalanced in practice by the too-small
value of the lift-curve slope.
2For calculation of the quantities c and ao for chord ratios
tp/t > 0.5, one employs the values 0 < tR/t < 0.5, while imagining the
wing surface and flap exchanged, as shown in figures ha and b, and also
the potential correspondingly changed.
3The specified formulas hold for positive and negative deflections;
only the sign of ao and therefore the lift ca are changed.
4t'2 = (t tR)2 + tR2 + 2(t tR)tR cos P, (see fig. 1 and the
sketch in fig. 11).

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Figure 11 gives the circulation distribution for the chord ratio
tR/t = 0.5 and the deflection angle 0 = 300 with the angle of attack
a = 70; Glauert's approximations are shown by dashed lines. For this
case the lift, referred to the original chord, is ca = 3.25 and accord-
ing to Glauert ca = 3.33, referred to the same chord.


The lift on a bent, flat plate is calculated exactly by use of con-
formal mapping, and the results for the angle ao of zero lift, the
lift-curve slope c, the lift coefficient ca and the circulation dis-
1 dr
tribution are compared with those obtained by Glauert's approxi-
vo dx
mation. This approximation suffices for deflection angles of the flap
p < 150, when the angle of attack a is so large that the small inexact-
itudes of the value of ao of Glauert's approximation in the term
(a cro) can be neglected: otherwise, one obtains values somewhat too
small in this region. The results for the lift-curve slope are practi-
cally the same for 0 < 9.

Translated by Paul F. Byrd
National Advisory Committee for Aeronautics
Ames Aeronautical Laboratory
Moffett Field, California


1. Munk, M.: Elements of the Wing Section Theory and of the Wing Theory.
TNACA Rep. 191, 1924.

2. Clauert, H.: Theoretical Relationship for an Aerofoil with Hinged
Flap. ARC Rep. and Mem. lo. 1095, 1927.

3. Chapligin, S. A., and Arjanikov, N. S.: On the Forward Aerofoil and
Flap Slot Theory. Trans. Centr. Aero-Hydrodyn. Inst., l1o. 105, 1'31.

4. Courant, R., and Hurwitz, A.: Funktionentheorie, Springer-Verlag,
Berlin, 1922.

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Figure 1.- The bent or hinged flat plate.


a) r



A 1,.

Figure 2.- Mapping of the outer region of the bent, flat plate onto a
half plane.

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Figure 3.- Correspondence of the z plane (fig. 1) to the 5 plane.


S (t- t)-
b) -,J .

Figure h.- Exchange of the wing surface and flap.

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Figure 5.- Angle of zero lift a,,, as a function of the chord ratio for
various flap deflections.

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-30 0 o P0,75




0O 15O 30

Figure 6.- Zero lift angle as a function of the flap deflection 3 for
various chord ratios. Glauert's approximation is shown by dashed

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7_\ _--- /

0,98 ----- -

q98 \IN


o q 5 1o
Figure 7.- Ratio of lift-curve slope c to flat plate as a function of
the chord ratio for various flap deflections.

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4- ----^


.o5 1 0

Figure 8.- Dependence of the lift coefficient ca on chord ratio for
various flap deflections at an angle of attack a = +5o.

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2 --. gO



0 0,5 ,0

Figure 9.- Dependence of lift coefficient ca on chord ratio for various
flap deflections at angle of attack a = +100

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40- S

315 / 0o5



0 150 300 o 600
Figure 10.- Lift coefficient ca as a function of flap deflection for
various chord ratios at angle of attack sr = +50.

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0 0,1 q2 0,3 0,4 0,5 0,6 0,7 0,8 9 1,0

Figure 11.- Circulation distribution with tR/t = 0.5, a. = 7, 3 = 300.
The dashed lines give Glauert's approximation.

NACA-Langley 2-21-55 1000


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