Airfoil theory at supersonic speed

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Title:
Airfoil theory at supersonic speed
Series Title:
NACA TM
Physical Description:
54, 4 p. : ill. ; 27 cm.
Language:
English
Creator:
Schlichting, H
United States -- National Advisory Committee for Aeronautics
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National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
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Subjects / Keywords:
Drag (Aerodynamics)   ( lcsh )
Aerofoils   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
A theory is developed for the airfoil of finite span at supersonic speed analogous to the Prandtl airfoil theory of 1918-19 for incompressible flow. In addition to the profile and induced drags, account must be taken at supersonic flow of still another drag, namely, the wave drag, which is independent of the wing aspect ratio. Both wave and induced drags are proportional to the square of the lift and depend on the Mach number, that is, the ratio of the flight to sound speed. In general, in the case of supersonic flow, the drag-lift ratio is considerable less favorable than is the case for incompressible flow.
Bibliography:
Includes bibliographic references (p. 54).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by H. Schlichting.
General Note:
"Report date June 1939."
General Note:
"Translation of "Tragflügeltheorie bei Überschallgeschwindigkeit." Jahrbuch 1937 der deutschen Luftfahrtforschung, pp. I 181-97."

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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM NO. 897


AIRFOIL THEORY AT SUPERSONIC SPEED*

By H. Schlichting


A theory is developed for the airfoil of finite span
at supersonic speed analogous to the Prandtl airfoil theory
of 1918-19 for incompressible flow. In addition to the
profile and induced drags, account must be taken at super-
sonic flow of still another drag, namely, the wave drag,
which is independent of the wing aspect ratio. Both wave
and induced drags are proportional to the square of the
lift and depend on the Mach number, that is, the ratio of
the flight to sound speed. In general, in the case of
supersonic flow, the drag-lift ratio is considerably less
favorable than is the case for incompressible flow. Among
others, the following examples are considered:

1. Lifting line with constant lift distribution
(horseshoe vortex).

2. Computation of wave and induced drag andthe twist
of a trapezoidal wing of constant lift density.

3. Computation of the lift distribution and drag of
an untwisted rectangular wing.


I. INTRODUCTION


The basic principles for the following computation of
airfoil flow at supersonic speed are presented in the paper
of Professor Prandtl (reference 1), and a detailed expla-
nation of the method may therefore be dispensed with here.

The potential 1T of a lifting line at supersonic
speed may be derived in a simple manner from the potential
P-Q of a stationary source in the presence of a supersonic
flow.

*"Tragfligeltheorie bei Uberschallgeschwindigkeit." Jahr-
buch 1937 der deutschen Luftfahrtforschung, pp. I 181-97.







N.A.C.A. Technical Memorandum No. 897


If IQ denotes the source potential of strength 4-r,
the potential ET of the lifting line element dy with
circulation F about the y axis is given by

xl = X

T = 2 / d x (1)
4 8z
X' = -_0

The potential of a source at the point x = y = z = 0 in
the presence of a flow with velocity uo > c in the di-
rection of the positive x axis is


Q = 1_ __ (2)
2 (u 1 (y- + z2)
S L c J

The potential (2) is real within the double cone with half
cone angle c, the axis of which cone is parallel to the
direction of flow (sin a = c/uo). Outside of this cone
the potential, according to the formula, is imaginary.
Actually, kQ is there to be taken identically equal to
.zero. The potential has physical reality only in the "af-
ter cone" of the point x = y = z = 0. In the "forward
cone" it is similarly to be taken identically equal to
zero.

The potential Q is the starting point for con-
structing the airfoil potential. We shall first derive
from it the potential of a line source of finite length,
then with the aid of the operation indicated in equation
(1) we shall obtain the potential of a lifting line of
finite length for various lift distributions. From the
lifting line, there is finally obtained by the familiar
method, the lifting surface. In this manner, a theory
of the iairfoil of finite span for supersonic speed is ob-
tained that forms the counterpart of the Prandtl airfoil
theory for the incompressible flow case (reference 2).









N.A.C.A. Technical Memorandum No. 897


II. CONSTANr LIFT DISTRIBUTION (LIFTING LINE)


We shall now assume a line source of length b
(later = span of wing) which lies in the direction of the
y axis and extends from y' = -b/2 to y' = +b/2 (fig. 1).
Let g(y') be the initially given local source intensity
(later = the lift distribution). Further, let x, y, z
be the coordinates of a point in the flow and 0, y', 0,
the coordinates of a source point. Then from equation (?)
the potential of the line source is


b

t I= ,j


yl b
y -_


g(y' ) cd y'


(3)


o c


- 1 (- y') + z 2


Se introduce nondimensional coordinates by dividing all
lengths by the half-length b/2 of the line source and
accordingly set

2 x 2 y 2 z 2 7'
b b b b

Further, we introduce the abbreviated notation

u 0
--1 =

or (4)

tanc = 1 = -

1
C2

where a denotes the Mach angle.

The potential of the line source then becomes


ri =+ 1

-in1=-1


g(l') d ,1t

V2- K[En *n,)2 + t2)


(5)








N.A.C.A. Technical Memorandum No. 897


We shall now carry out the integration in equation
(5) for the simplest case of the line source with constant
source density, that is, g(il) = const. = 1. This gives

'n= +1
T, d '

: -i / -..E(n T') + 3]


Writing T 1' = h; t' = -1: b = 1 = 1 + 1
n' = +1ti = b3 = n 1

and (t/)j K) = a2

equation (6) becomes


S= -- = -- arc sin- arc sin- (7)
K, aK c a a

By the operation in equation (1) there is then obtained
the potential cT of the lifting line with the constant
lift distribution Fo, setting


2d (a)


The first step of the above operation, differentiation with
respect to S, may be carried out immediately but the inte-
gration requires a somewhat longer computation. There is
obtained

a a 2 s a2 2 a 2 2
-. a2 a = -. /9)


In integrating with respect to I, 42 and 1, are con-
stant. For the'first term, th6re is obtained








N.A.C.A. Technical Memorandum No. 897


/ d -'

a2 1 -2 2 2 4 2


where there has been set

t* = t)2

The evaluation of the integral gives



* a2 2 2
a a e


K
= arc tan
2



= arc tan
2


2 2 ~2 + (?2 2)

2t3 2 + It, 2 ," 2


w .~ ( 1)

2 (n i) (./-u-


(11)


where


S= 2 [(n 1)2 + t2]


(12)


Since the integral (10) outside of the Mach cone,
at the cnd point n = 1 of the lifting line with axis
parallel to the x axis,

t 4 [(n 1)2 + C] = o

is imaginary, i.e., is to be taken equal to zero, the in-
tegration with respect to E' need not be extended from
= -* but only from the cone surface along lines paral-
lel to the Y axis. For the lower limit of integration,
we have thus the constant arc tan m TT/2, which we may
suppress. There is thus found from (8), (9), and il) the
required potential of the lifting line with constant cir-
culation


(10o)







N.A.C.A. Technical Memorandum No. 897


r E r I t3 d r t2 W so t2( 1)n
K = -- = -- -- arc tan
2T w asa / 2 4T 2 ( ( 1)( JL"

ao (2 1)
= -- arc tan + similar term for the cone at
S= -1 (13)


This potential is different from zero only within the two
Mach cones arising at the ends of the lifting line (w > 0)
while in the entire remaining space it is equal to 0. For
a complete circuit about each of the cone axes 1n = 1,
= 0, the arc tan increases by 2 Tr. The enclosed vortex
filament therefore has the circulation Fo. The lifting
line assumed to extend from y = -b/2 to y = +b/2 with the
constant circulation Po along the span continues behind as
a free vortex line in the two axes of the Mach cones. Equa-
tion (13) thus gives the potential of a "horseshoe vortex"
at supersonic flow. As in the case of the incompressible
flow, this simple horseshoe vortex becomes the starting point
for more complicated lifting systems.

In order to obtain an idea as to the appearance of the
supersonic flow in the neighborhood of a horseshoe vortex,
we differentiate the potential (13) to find the induced ve-
locities


cx = ; cy = cz T
ax a y z

and obtain

c = K ---
T b ( K K ) ,-7


c = -- (14a,b,c)
TT b [ + (n 1)] ]

r0 =( r- 1) (w K22)
TT b [2w+ (T l)2t2]

See footnote on next page.








N.A.C.A. Technical Memorandum No. 397


rhe field of these velocities exhibits a number of singu-
larities. On the cone surface all three velocity compo-
nents become infinite. On the cone axis cx = 0, but
cy and cz become infinite as 1/r (where r is the
distance from the axis). In the neighborhood of the cone
axis, Cy and cz thus behave exactly as in the neigh-
borhood of a vortex filament in the incompressible flow.
The field of the induced velocities gives a motion which
encircles the vortex filament traveling downstream from
the end of the lifting line r = 1, o = t = C, as may
be seen immediately from (14).

In the plane r 1 = 0 through the end of the lift-
ing line cz = 0 and

> 0 : cY < 0

< : Cy > 0

In the plane 0 = 0, which contains the lifting line,
c,, = 0 and
i 1 > 0 : cz > 0

r 1 < 0 : cz < 0

The flow picture in the cone, however, in its detail is
essentially different from that in the neighborhood of a
vortex filament in the incompressible case. Figure 2 shows
the flow picture of the y and z velocities in a plane
perpendicular to the cone axis downstream of the lifting
line. The figure was obtained by computing the isocline
field cz/Cy = const. On the cone surface, as has been
said, cg and cy are infinite, although for the slope
of the streamlines cz/Cy there is here obtained the sim-
ple value

cz

cy T 1


'A check for the correctness of this solution is obtained
b.y substituting in the linearized continuity equation
Scx 8 cy 8 c z
-- .+ +
8x 8y z
which must be identically satisfied.







N.A.C.A. Technical Memorandum No. 897


The direction of the streamlines is therefore radial to
the center. The flow consists partly of the closed stream-
lines which circulate about the vortex filament and partly
of the streamlines that enter on one side of the cone and
leave it again on the other side.

In addition to the two Mach cones that arise from
each of its ends, the lifting line generates two plane
waves, which enclose a "wedge space" and which appear in
the streamline picture as the common tangents of the two
cones.

For the downwash distribution in the plane = 0
through the cone center, there is obtained from (14c) the
simple formula
2 TT x /l 2
--- tan a cz = (15)
o
where
b
y -
2
S= (15a)
x tan a

This downwash distribution is shown in figure 3.

In order to study the processes on an airfoil of fi-
nite length at supersonic speed, particularly the induced
drag, the replacement of the wing by a lifting line with
constant circulation as in the case of the incompressible
flow, appears inadmissible since on account of the infi-
nite velocity at the end of the lifting line an infinite
induced drag would be obtained. This difficulty in the
case of the incompressible flow is avoided, as is known,
by allowing the circulation to drop to zero in a suitable
manner toward the wing tips. The induced drag is then
computed by the formula.

y = +b/a

Wi= P co(y)r (y) d y (16)

y = -b/2


(where czg(y) is the induced downwash velocity at the
place of the lifting line, and P the density).








N.A.C.A. Technical Memorandum No. 897


In the case of the supersonic flow, the relations are
complicated by the fact that in spite of the assumption of
a lift distribution decreasing to zero toward the wing tips,
there are obtained singularities at the lifting line posi-
tion of such a character as to make the computation of the
induced drag by formula (16), which maintains its validity
for supersonic flow, impossible. As closer investigation
shows, this is due to the fact that the lifting line is the
geometric locus of the vertices of all the Mach cones that
pass down behind. This difficulty may be overcome by pass-
ing from the lifting line to the lifting surface.


III. WAVE RESISTANCE (DRAG)


Before proceeding to the corresponding computations,
we shall discuss briefly the supersonic flow about an in-
finitely long airfoil (two-dimensional problem), a problem
that had been considered by J. Ackeret in 1925 (reference
3).

The simplest and at the same time the ideal supersonic
profile is that of the infinitely thin flat plate of chord
t set to a small angle of attack o (fig. 4). For such
a plate the lift per unit span is

A = 2 tan a po t P u02 (17)

or

A
= ca = 4 tan a 0o (18)
2 2
uot

On account of A = p u Fo, the relation between the an-
gle of attack of the wing and the circulation is

1 Fo
o uo tan a (18a)
2 t

From the incompressible flow, the supersonic flow
about the airfoil differs in that, for the latter case,
even if the fluid friction is neglected, there is always
associated a drag that originates from the plane waves
which start out from the lifting surface and are inclined
to the latter by the Mach angle and which.therefore may







N.A.C.A. Technical Memorandum NTo. 897


*be denoted as the wave drag. For the-flat plate, the wave
drag per unit span is

Wwave = Po A = 2 tan a Pos t p uo (19)
or


SWwave = 4 tan a o0 (19a)
Cwave


The resultant of the lift and the drag is here at right
angles to the plate. This comes from the fact that at su-
personic flows there is no suction force at the leading
edge of the plate. From equations (1) and (19a), there is
obtained for the polar of the wave drag

ca2
wwave 4 tan a

which is thus a parabola as in the case of the incompres-
sible flow.

Plane waves start out from the leading and trailing
edges of the inclined flat plate (fig. 4) and in the space
between them the induced downwash velocity is


c = -Pn 0 1 (20)
wave o2 t tan a

The wave drag,on the other hand, can also be computed from
this downwash velocity induced by the plane waves, accord-
ing to the formula

Wave = P e Cwave (21)


as may be seen by comparison with (19) and (20). In the
next section it will be shown that,for a lifting surface,
the velocity induced by the tip vortices like zwav is
proportional to ro/t tan a. It then follows from equa-
tions (21) and (16) that the.wave drag behaves in exactly
the same way as the induced drag from the tip vortices.
For practical applications it is therefore of no inter-
est to consider the induced drag alone, but it is the sum
of the induced and wave drags that must be considered.








N.A.C.A. Technical Memorandum No. 897


For an airfoil of finite span and constant chord with
circulation that is constant along the chord and variable
along the span F(y) = t 7 (y) the total lift and wave
drag are given by


y = +b /


rT= +1


A = p U0


y = -b/a


r dy = ~ uo b t


'Y (7)) d r


r = -I


n = +1


(d


c r d y = b t
z~ow 2


cz W a"d n (23)
ow


1 ta
Stan
3 tan a


(24)


is the induced wave velocity. Accordingly

n = +1


wave 4 t
4 tan a ,


S2 d


(25)


By comparison of equations (22) and (25), there is found
the relation between drag and lift


Z b A a
Wave = 2 Z ( --
P t tan a uo b-


(26)


In the above equation Z is a nondimensional coef-
ficient that depends only on the lift distribution


TI = +1


z =- -I


Ya 2 / d \d


From equation (26), it follows that:


w
wave


(22)


where


(27?)


czow
o'








N.A.C.A. Technical Memorandum No. 897


wave = tan Ca
tan a


(28)


The numerical values of Z are given in table I for sev-
eral simple lift distributions.


TABLE I Values of the Coefficient Z for Various

Lift Distributions (Wing of Rectangular Plan Form)

Number Lift distribution Z


rectangular

ellipse




trapezoidal



b' = b/2

parabola


triangle


1/4

8
3 TT


= 0.250.

= .271


b


= .370

= .300


= .667


IV. LIFTING SURFACE WITH CONSTANT LIFT DISTRIBUTION


For the successful computation of the induced drag
for supersonic flow, according to section II, the simul-
taneous assumptions must be made of a suitable drop in
lift toward the edges and a surface distribution of the
bound vortices. This. twofold extension means naturally
a considerable swelling of the computation of the field
of induced velocities as compared with the incompressible
flow where the computation involves mostly a lifting line.
In order to be able to recognize more clearly the effect
of each of these two extensions, we proceed in two steps.








N.A.C.A. Technical Memorandum No. 89?


We first maintain the lift distribution constant along
the span and consider only the transition from lifting
line to lifting surface. The fieli of induced velocities
thus obtained for a wing with constant spanwise circula-
tion distribution and constant chord, while it does not
enable as yct the computation of the induced drag never-
theless furnishes many aseful results so that we orocecd
first to compute this field.

We assume therefore the circulation Fo constant
along the span b as uniformly distributed over a rec-
tangular lifting surface of chord t and extending from
x = 0 to x = t (fig. 5). The circulation for a strip
of the lifting surface of unit width is therefore o =
fo/t. It would be most convenient to make the transition
from the lifting line to the lifting surface directly on
the potential (13). On account of the integration diffi-
culties that arise, however, the transition will be made
on the velocity components (equation (14)), first for the
z component since the latter is the most important for
the computation of the induced drag.

A strip of the lifting surface of width dx' at a
distance x' from the leading edge contributes to the
induced z component c. at the point x, y, z, if the
point lies within the Mach cone arising from the end of
the strip the amount

S = ', ) = -- d f( -
T b 2 rT

where, according to equation (14)


f(( ) 1)(w K t2) (29)
[2W + ( 2 V
If the point x, y, z lies outside the cone, the amount
contributed is zero. The contributions from the plane
waves starting out from the lifting surface will be sepa-
rately considered. Integration over the wing chord there-
fore gives for the downwash velocity induced by the lifting
surface


2 TT z = Yo ft (',r|)d U


or written out in full









N.A.O.A. Technical Memorandum No. 897


0




CO
o


I--I

cla





-Q^-
+






CO
+


LO
le
ca
I 0





L%1
i -





A- A


I -


I
C-.' 103 r











Ia





I





0.r
r-iF













II



lo
11 ^J
,gf> ^_


we have

2 z
2 TT -
YO


T = T2
T 1 / (.T- K a2 ) d T

a 2
a 2 J T VT K (1) 1)
T -T=


a- 1
T= T


d T

VT -K (T)


T =T


K2
2 I
T= T


d T


T /f n2 (j _1)2


The upper integration limit C' = ri. is different
according to whether the point lies within the
Mach cone II arising from the end point of the
trailing edge (fig. 5) or between the latter and
cone I arising from the end point of the leading
edge. The corresponding limits will be

Ei = 2t/b = (within cone II)

= K V(- 1) + t (between cones I(
and II) C

as may be easily seen after some consideration.
Introducing the new variables of integration

2- K t2 = T

and writing for briefness

a 2= (n --i) +








N.A.C.A. Technical Memorandum .No. 397


The evaluation of the integral gives


1



2- r)2 +
0 A ,) L i)0 + \

Kl arc tan
L K( 1)



Taking account of the different upper integration limits
according to whether the point considered is within cone II
or between cones I and II, equation (31), and setting for
briefness

Wc, = ( ) 1)2 + (32)

there is obtained as tie final expression for cz:

For cone II
-- = ----- 1
0o (, 1)2 + t



K {arc tan arc tan (3a)
<(T 1) K(n 1)

Between cones I and II


2 T 1 arc tan VW (33b)
eo ( i)2 + t(n I)

where
S(T 1)
1 < < + 1 : I < arc tan <
2 2

From these formulas it follows that c on the surface of
cone I is equal to zero and on account of W = 0 is con-








N.A.C.A. Technical Memorandum No. 897


tinuous in passing through cone II. On the common axis of
cones I and II, there still occurs the same singularity as
in the case of the lifting line, Fz, there becoming in-
finite as 1/r.

In order to obtain the total downwash velocity, there
is still to be added to equations (33a) and (33b) the por-
tion contributed by the plane wave. This contribution is
different from zero only between the plane waves starting
from the leading and trailing edges (fig. 4). According to
equation (20)

Zwave 2 o wave Cxwave = (=3)

The expressions for the two remaining components of
the induced velocity y and -x are found by similar in-
tegrations. We shall only indicate the results:
Cone II:


o (- i)3 + (n


ex -( g)( 1) l)
2 = arc tan arc tan


Between cones I and II: >(35)

2T- -- V(U
0o ( 1)2 + 3

2 TT = arc tan -1)


In the above equations, the arc tan is to be taken -TT/2
and +TT/2. As may be seen from equations (33) and (35) by
comparison with equation (14) in passing from the lifting
line to the lifting surface, the difficulty of the infi-
nite velocity at the cone surface has been set aside. The'
singularity of Cy and cz on the cone axis (infinite as
1/r) still remains, however, and prevents the computation
of the induced drag for this lift distribution.








N.A.C.A. Technical Memorandum No. 897


For the downwash distribution in the plane z = 0,
at 1) the location of the wing x < t, and at 2) behind
the wing x > t, taking account of the ,plane wave, there
is obtained the following:


1) For x < t:


- i< K < 0:-- = + arc tan
KY 0


2 o 1 j 1 z P 1 2
0< U< + 1:--- = arc tan -
'Yo 0 U


a)



(36a)


where for the arc tan the same values are to be taken as
in (33) and 6 is given by equation (15a).


2) For x > t: The plane waves do not contribute
anything but the formulas obtained differ according as
region considered is within cone II or between cones I
II (fig. 5).


the
and


t t 2cz0
-I- )< 5 < i t- ;: -o
x K Y7


rr v/ x
S(y7 2 ( 1- 2-


- 1 (arc tan U -arc tan -
TT b /


- 1 < t < (1 )


1 l< < + 1


-.


1 (. t- 0- o--\
-arc tan-
Tn T


(36b)


2cz
0








1.A.C.A. Technical Memorandum No..897


The downwash distribution..for x:< t and for x = 2 t
computed by the above equations is shown in figure 6.

Further, we have in the same.manner as for the lift-
ing line determined for the lifting surface the stream-
line field of the y and z velocities in a plane at
right angles to the cone axis. At the location of the
wing (x < t, fig. 7) there is obtained outside of the
cones springing from the wing tips a constant downwash,
due to the plane waves, along the span. The streamline
picture within the Mach cone in the outerhalf is similar
to that of the lifting line (fig. 2); the inner half how-
ever is entirely changed by the additional downwash ve-
locity from the plane wave.

The streamline picture behind the wing (x = 2t, fig.
8) has, outside the Mach cones springing from the wing
tips, a constant downwash velocity due to the plane waves
in two strips symmetrical to the plane **! = 0. These two
strips are limited by the plane waves starting out from
the forward and the trailing edges of the lifting surface.
Within the Mach cone the streamline picture in the outer
ring is the same as for x < t and is changed only in the
inner region.

We shall yet consider briefly the question, what the
form of the wing surface must be that corresponds to the
assumed lift distribution. The wing plan form we have as-
sumed as rectangular. Angle of attack and twist are ob-
tained from the consideration that at the wing, i.e., in
the plane z = -0 in each section parallel to the flow di-
rection, the direction of the streamlines must be parallel
to the wing tangent.. Let z = z(x, y) be the equation of
the wing surface and z(O, y) 0, i.e., straiht leadin-
e1te. T.en we have,

d z czo(x, y)
d x u

where co includes the induced velocities from both the
plane waves and the edge cones. There is thus obtained
for the wing surface


z(x, y) = czo(x', y) d x' (37)

X' = 0








N.A.C.A. Technical Memorandum No. 397


so that a further quadrature is required to compute the
form of surface wing.

For the case considered of constant lift distribution
there is obtained for the region outside of the two Mach
cones at the wing tips, from equations (37) and (20):

z(x, y) = -o x

that is, a flat surface with angle of attack po. Within
the Mach cone the surface bends downward more and more
strongly as the edge is approached. The edge itself (y =
jb) is bent infinitely downward, i.e., actually the rec-
tangular surface with constant spanwise and chordwise lift
distribution is not possible. For this reason we may dis-
pense with the further computation of the wing-surface
shape.

V. TRAPEZOIDAL WING WITH CONSTANT LIFT DISTRIBUTION


We consider now a trapezoidal wing with constant sur-
face density of the lift 70 (fig. 9). If the wing is
cut away behind (taper angle T, fig. 9) in such a manner
that the Mach cone at the tip of the leading edge does not
overlap the wing (7 > a), the induced drag is obviously
equal to zero and only the wave drag exists (reference 4).

T> a:Wi = 0

The trapezoidal wing with constant surface density of the
lift 'o is plane outside the Mach cone and has the angle
of attack p0 where

0o = 2 o u0 tan a

The trapezoidal flat surface with constant lift distribu-
tion whose cut-away angle T is greater than the Mach
cone angle may be looked upon as the "ideal supersonic
wing with finite span" since for it the ratio of drag to
lift is no greater than for the wing of infinite span.

The computation of the induced drag for T < a is
possible in a simple manner from the above results. By a
lifting element we shall mean a strip of the lifting sur-
face of chord d x and therefore with circulation Y d x.






N.A.C.A. Technical Memorandum No. 897


Such a lifting element at x = 0 generates at a lifting
element of chord d x' at x = x'
y=yi(x)


d2 Wiox = P = o d x' f d c (OX') d y

y=yo (x)


(38)


(ox')
where d czo


denotes the downwash velocity induced by


the lifting element x = 0 at the position x = x'. The
integration limits are the surface of the Mach cone aris-
ing from the tip of the wing leading edge and the side
edge of the plate. For the downwash velocity
c(ox') in the plane 0 = 0, we have according to equation
zo
(14c)


d c(ox') c= 2 _o K 2(T 1)2
zo 1 b( l)


d x


(39)


with the aid of which equation (38) becomes


d2 Wiox. = -- 70 d x d x'
2 TT


oI 1
or with = K------


. 1
/
J./
T)_\~


S_2(n 1)2

((9 1)


, according to equation (15a) and


8 .= tan T
tan a


as .the -reduced angle of taper


Wiox 70o2 d x d x'




P- 70 d x d x'
2


Ji1 -
Jd 4
a=6


g (e)


(40)


(41)








N.A.C.A. Technical Memorandum No. 897


The evaluation of the definite integral gives


S() = log 1- (42)
e

According to equation (41) the induced drag from the lift-
ing element x = 0 at the position x = x' is independ-
ent of the distance between the two elements. All ele-
ments lying between x = 0 and x = x' accordingly pro-
duce the same drag, so that the total drag induced at x =
x' amounts to

p 2
d Wix, 70 x' d x' g (6)
2 n

The drag for the entire wing is obtained from the above
by integrating over x' between the limits x' = 0 and
x' = t and multiplying by two (both ends)

x'=t

Wi = -- 0 (6) d xI
n .1
x'=0


= t g (6) S g (6) (43)
2 2 n

The minus sign is explained by the fact that with our
choice of coordinate system the drag component of a force
is in the direction of the negative z axis. Formula (43)
for the induced drag of a trapezoidal wing with constant
surface density of the lift is of the same structural form
that is found for the incompressible flow. For triangular
lift distribution (lifting line) in the case of incompres-
sible flow, we have, for example,

log 2 2
Wi = ----- p Fo


where 1o is the circulation at the wing center. For
equal total circulation Fo' iTi according to equation
(43) is independent of 6, i.e., the ratio of the tangent








N.A.C.A. Technical Memorandum No. 897


of the angle T to the tangent of the Mach angle (equa-
tion (40)). In passing to the rectangular wing, e ---> 0,
the induced drag according to equations(42) and (43) be-
comes logarithmically infinite, in agreement with our re-
sults of the previous section.

Actually, we are not interested so much in the value
of the induced drag alone as in the sum of the induced and
wave drags. For the wave drag, according to equation (21)
we have

wave = F 0o Czwave

where F = b t ( t tan T \
where = t 1 tne area of the wing

(44)
c 0
Zwave 2 tan a


hence Wwave = t 02
2 tan a

For the lift we have, on account of T, = 2 po uo tan a:

A = P To uo F = 2 P uo2 Fo tan a (45)

or
A
c = = 4.ao tan a (6)
P F2


For the wave drag we obtain from (44)

Wavee= 2 P F uo2 Ao2 tan a

Wave =w 4 Po tan a = po ca (47)
O wave
2


and for the induced drag from equation (43)








N.A.C.A. Technical Memorandum No. 897


Ni =----
2 u2




uo 2
^----oe F
J^


t2 4 P 0o tan2 a g (6)


1 4 t o02 tan" a g (6)

S b t tan T
b


cwi = 4 Po tan a -----g
7r 1 6 X


t tan a
where X = -
b


is the "reduced aspect ratio" of the


wing. For the total drag there is thus obtained from (47)
and (48)


(cW)wave + ind = 4 po tan a 1 + g
*n 1 6 A


Ca2 X g (! )
(cw)wave + ind = a I + C)
4 tan a n 1 6


(49)


It follows therefore from the above that for supersonic
speed the wave plus induced drag, like the induced drag
in the incompressible flow case is proportional to the
square of the lift. Equation (49) is analogous to the
a 2 F
well-known formula cwi _- of the elliptic lift
TT b
distribution for the incompressible flow. The essential
difference lies in the fact that for the supersonic flow
the drag parabola for small aspect ratios t/b is to a
first approximation independent of the aspect ratio. The
manner in which the drag increases with increasing reduced


aspect ratio A = t tan a and decreasing 6


is shown in


figure 10 where


c/ Ca

w/4 tan a


is plotted against X for


various values of 6. Our formulas are valid only for


(48)







N.A.C.A. Technical Memorandum No. 897


1
< -, i.e., for the case in which the Mach cones do not

overlap on the wing.

In order to be able to predict what the wing-shape
must be sc that our assumed lift distribution may be pos-
sible, we must first compute the field of the induced
velocities. For this purpose equation (39) is to be in-
tegrated over the trapezoidal area. The value

dcto) according to (39) gives the downwash velocity in-

duced at the position | by a lifting element 7o dx
starting at t = 0 and ending at rT = 1. A lifting ele-
ment which starts at t = 1.' and ends at | -, I' thus
produces at the position t, T, t = 0 the downwash veloc-
ity


d c = )2 -
zo 2 TT (n


For the velocity induced by the entire surface there is
thus obtained



0 2. (f (50)


In order to evaluate this integral we introduce the new
integration variables


K =) 2L- = K tan%)" (51)


.(See fig. 9.) Since the end-points of the lifting elements
lie on the wing contour there exists the relation

T' = |' tan f (52)

The upper integration limit (' = in equation (50) is
obtained from the condition. (See fig. 9.)







N.A.C.A. Technical Memorandum No. 397


tan 0J = tan a: = 6 : &' = 1

The lifting elements whose ?' is greater than the t
thus determined give no contribution at the points t, r,
= 0. For the lower integration limit

= 0: '' = 1: @'=u U =


From equations (51) and (52) there is obtained

d g' d d '
e

where 0 is the abbreviation introduced in equation (40).
There is then obtained from equation (50)


czo (iY) 1


--=1


J/


-- --d 7' 2- F(O,6) (53)
-1( 6) 2 TT


In evaluating the above integral the following three cases
are to be distinguished:


1. O< 6 < o ;


2. 0 < < 6e;


3. < 0 < 6


In case 1 the point P(o) lies within, in cases 2 and 3
without the trapezoidal wing. In case 1 the integrand is
regular over the entire range of integration; in case 2
it possesses a singularity at 6' = 6; and in case 3, two
singularities at b' = 0 and o1 = E. In cases 2 and 3
the principal values are to be taken, namely,


0 < b < e: '=9-b / -
l [ V I ,b
F(,G) =lim { I 0' +
'=o 0 '
^ '=,


S4=1


0'=e+e








N.A.C.A. Technical Memorandum No. 897


and


I-f ,


F( 6) =lim / .---- d a +
c- -o J ^l ^. 6


*~5 1=6-E

/
.1
ti
t


..---- =


.' -:-


3'=6+e


The integral


, (1 2



. '=.


(55)


may be obtained by elementary methods. We set


l = t 1

where t is the new integration variable so that (55) be-
comes


t= t


t
t =1


1


(1 ta)2 dt


t(l + t2) t2 t 6 (1 + t2)


1 +4 -
ti = 'V =


(56)


By breaking up into partial fractions there is obtained


(54b)


where


a < 0 < 6:


E' -








N.A.C.A. Technical Memoranduam o. 897


(1 t 2)


t(1 + t2-)


t 6(1 + t2) >


2 1

1 + t2 E t


6 % 1
+ t- t t
6 't- to t tz


where


1 1 62


rerforraing the integration, there is obtained


F( ,6) = I 2
L


arc tan t lo t + _


lo-r (t t ) log ( t tI ) t
% t = 1


For 0 < b < b there is therefore obtained directly


F(6,6) 2 arc tarn iJ -
2 6


6
+1
6


S+ i + 62

b i 6 + 1 62 /


while the formation of the principal value according to
equation (54a,b) gives


(57)








N.A.C.A. Technical Memorandum No. 897


0 < b < 6:

F(6,6) = 2 arc tan log
2 6


1 6 l
- log
6


S< 0< 6:
log (-_)
F(4,6) 2 arc tan 6 -
2 6


'*Yi- 68
+ log


There


W 6 i +' .- 92
4/ e 1 1 p8/


- < arc tan V < -_
2 2


There is thus found the downwash distribution in the entire.
Mach cone springing from y = b/2, x = 0. For F(.,6) we
have

S= :F(l; 6) 5 0


3 = 6, = 0:F(6,6) = F?(, e) = as log 3 at b = 0


(58)


On the two rims of the cone (0 = -xl) the induced velocity
is thus zero and on the edge of the trapezoidal wing
(D = 6) and on the cone axis (b = 0) it is infinite.
In figures 11 and 12 for the particular case 6 =


1 (tan a = &T, tan T = ---
3


there is shown the induced


downwash velocity in a section parallel and perpendicular,
respectively, to the principal stream direction.

To the above velocity field of the tip vortices there
is still to be added the velocity field due to the plane
wave. The latter in the plane z = 0 within the wing area
is


4e-1 + e6
e_ 1 -l 1 6








N.A.C.A. Technical Memorandum No. 897


CZo.wave


S- P0
2 tan a


and outside the wing area

Czowave 0

From the velocity field it is now possible to compute
the form of the trapezoidal wing surface that has constant
lift distribution. Outside of the Mach cone we have, ac-
cording to equation (37)

x'I x

z(x, y) = wave (x', y) d x = 0 x
u0 wave
x, = 0

that is, a flat surface with the angle of attack 0 giv-
en by equation (18). The twist of this flat surface within
the edge region of the trapezoidal area that is overlapped
by the Mach cone is given by


XI = X
I 1
z(x, y) = coi d
Uo .I

x'=(b/2-y)/tan a

and according to equation (53)
x'I=x


x'


x'=x


z(x,y) _-- 10o K
27 uo J
x'=(b/2-


F(a', 6 ) d x' = jo F(', 6 ) d x'

y)/tan a x'=(b/2-y)/tan a


S(b- y)
On account of 6 = there is obtained




z(x, y) = Lo y8) d '
T 2 ba
S M-


(59)







N.A.C.A. Technical Memorandum No. 897


Since the function F(6, 8) is known from equation (57),
it is possible from the equation above to compute for a
given 8 the profile sections of the surface at various
distances (b/2 y) from the edge. The ordinate of the
obliquely cut-away edge of the trapezoidal area for b/2 -
y < t tan T:
*'=8

z(xR, y) = K Y) F y ~ d .' (60)
S 2=1

The integrand becomes infinite for 'l = 8 (equation (58)).
The integral exists, however, and may be evaluated by spe-
cial computation. There is obtained
,'=6

/ F(t', 9) nd = are sin 8 1 1-- (60a)
d'=1

(The evaluation of the integral was performed by Dr. F.
Riegels.)

For 6 = 1/3, we thus have



6 = 1/3: /" d =-d 4.92



The ordinate of the rear edge point x = t, b y = t tanT

for 6 = 1/3 is thus z = 1.522 Po t. (Flat surface
z = -Po t, twist z = -0.522 po t.)

For the special case 6 = 1//3 (tan a = v tan T =
-1) the profile sections have been computed and are

given in figure 13. If the trapezoidal wing were flat
there would be a drop of the lift toward the edge down to
zero. In order that full lift be maintained up -to the
edge, the wing must be bent downward. The twist of the








N.A.C.A. Technical Memorandum No. 897


wing directly at the edge is very strong as may be seen
from the "elevation contour lines" (fig. 14).


VI. COMPUTATION OF THE LIFT DISTRIBUTION

FOR THE UNTWISTED RECTANGULAR WING


The examples thus far considered are all in connec-
tion with the so-called first principal problem of the
airfoil theory where the lift distribution is given and
it is required to find the drag and the wing shape. Of
greater practical importance is the second principal prob-
lem where the wing shape being given it is required to
find the lift distribution and the drag. As in the case
of the incompressible flow, so also in the case of the
compressible flow the first problem, which leads only to
quadratures, is considerably more simple than the second,
which requires the solution of an integral equation.

In what follows there will now be given an example
of the second principal problem, namely, the computation
of the lift distribution for a plane rectangular wing
(span = b, chord = t), that is to say, the same problem
that was first considered by A. Betz (reference 5) for the
case of incompressible flow. In the treatment of this
problem we can utilize to a large extent the results we
had obtained in the previous section for the trapezoidal
wing with constant surface density of the lift. We con-
sider a rectangular flat plate which extends from x = 0
to x = t and from y = -b/2 to y = +b/2 and is set at
the small angle of attack 0o to the undisturbed veloc-
ity uo (fig. 5). Within the region bounded by the plane
waves starting out from the leading and trailing edges and
the two Mach cones there is the constant downwash velocity
due to the plane waves

czo uo --Q- (61)
Wave o2 tan a

Outside the region of the flat plate overlapped by the
Mach cones at the tips there thus exists the constant lift
distribution 7o. At the tips y = =b/2 the lift must
vanish, that is, 7 = 0 at y = b/2. There is required
the lift distribution 7 = (x,y) within the region


overlapped by the Mach cones.


The problem is considerably








N.A.C.A. Technical Memorandum.No. 897


simplified by the circumstance that, as will immediately
become apparent, .7 does not depend on the two independ-
ent variables x, y, but only on one of the variables
b
= (51)
x tan a

(fig. 11). For the required lift distribution

'Y ) = 'Yo f () (62)

of the rectangular wing there then exist the boundary con-
ditions

6= 0 : f(3) = 0
>. (63)
i= 1 : f(.) = 1 |

In order to be able to set up the integral equation for
V7(6) we must first compute the field of the downwash ve-
locities w(,) induced by a rectangular wing with the cir-
culation distribution 7Y() in the plane z = 0. The in-
tegral equation for 7Y(b) is then obtained in the known
manner from the consideration that for each position of
the wing the sum of the effective angle of attack
1 Y)W
=#() 1 () (64)
2 u. tan a

and the induced angle of attack w.-1) must be equal

to the geometrical angle of attack' .o


p(-) (t)l (65)
uo

The velocity field w(.) induced by the edge vortices is
obtained by considering the rectangular wing with the var-
iable lift distribution 'Y(b) = 'Yof (W) as built up by the
superposition of trapezoidal wings with various taper an-
gles each of which wings possesses a constant lift distri-

bution. Again, let 6 = -- be the "reduced taper angle"
tan a
(equation 40), then the lift distribution Y = Y f (4) may








N.A.C.A. Technical Memorandum No. 897


be obtained by the superposition of trapezoids with angles
6 and lift densities 7of'(6) d 6. Each of these trape-
zoids produces, according to equation (53) the velocity
field

d w (j) = f () F (N,6) d 6
2 TT tan a

and integration over 6 from 6 = 0 to 6 = 1 then
gives the induced velocity field over the rectangular wing

t=1

w (j) o / f'(6) F (,,6) d 6 (66)
2 n tan a -/
6= o

By substituting the above expression for w(t) in equation
(65), there is finally obtained, taking account of (61) and
(64) the required integral equation for f(0):

6=1

f() + / f'(6) F (,,6) d 6 = 1 (67)

b=o


to which are added the boundary conditions (53). This
integral equation for the lift distribution has the same
structural fornas that for the incompressible flow. It
differs from the latter, however, by the different core
F(%,6), which is given by equation (57), and for the super-
sonic flow is of a much more complicated form that for the
incompressible flow. Equation (67) also exhibits the nota-
ble property that neither the aspect ratio of the wing nor
the iach number appears explicitly, whereas in the incom-
pressible case the characteristic value of the integral
equation depends on the aspect ratio. The dependence of
the lift distribution on the Mach number appears in the
introduction instead of the geometric angle cp (fig. 9)
tan cp
the reduced angle 6 = as the variable. It is neces-
tan a
sary to solve the integral equation (67) only once to ob-
tain the lift distribution of the rectangular wing for all
aspect ratios and all Mach numbers.







N.A.C.A. Technical Memorandum go: 897


The solution of the integral equation (67) appears
at first sight quite difficult, particularly on account
of the complicated structure of the core F(O,6). (See
equations 56 and 57.) By a simple transformation of
equation (67) it is possible, -however, to simplify the
problem considerably.* The -equation is a nonhomogeneous
integrodifferential equation for f(b). Instead of it
we shall consider the equivalent equation for f'(4).
Taking account of the singularity of the core, equation
(67) may be written


6= d
/ f'(#) F(4,.) d e = 1


T


Differentiation with respect to b gives


f,(b) + 1 { f'(b) F(6,4) +
Tr


f, (6) .! d6
d b


8=o


6=1


- f' (6) rF(b,) +


f,(6) d ed F = 0
d -


j
0=-b


and because


d Fl

d i s( 6)


according to equation (53):


6=0
iT .1O


(68)


*For this suggestion I.am indebted to Doctor Lotz and for
carrying out the numerical solution of the integral equa-
tion to Mr. Pretsech.







N.A.C.A. Technical Memorandum No. 897


The above is the equivalent integral equation for f'(C)
which, however, is now homogeneous. The solution of this
integral equation for f' (0) is possible by building up
f' () in n steps and solving the corresponding system
of linear equations

f'" U 1) -


1 i _b2 n 2 : X
Si V+1 I f (2x+1) / d -o
+i 2 V ?o .j 2V+i =+

(v = 0, 1, .....n 1)

(69)

This is a system of n homogeneous equations for the n
unknowns f'( ,mu+1) (u = 0, 1 ..., n 1). Since, as
closer investigation shows, f'(0) = a, f'(2 ) is suit-
ably chosen not constant but equal to

f'('1) = a b b

There is then obtained in place of equation (69) a
nonhomogeneous system of equations of the nth order for
b 1
the n unknowns -, -f'(baV+1) (v = 1 ..., n 1). The
a a
further unknown a is obtained in the numerical integra-
tion for f(L) from the condition



f(1) = a Z 2 = 1 (70)
V=o a

In carrying out the numerical process there were first
taken five steps (Cau+1 = 0.1; 0.3; 0.5; 0.7; 0.9), then
ten steps (basu+ = 0.05; 0.15; ...; 0.95). It was found
that the ten-step approximation gives an improvement over
the five-step process only in the interval 0 < 2- < 0.2.
In the third approximation therefore only the interval
0 < -4<0.2 was again subdivided (zsvu+ = 0.025; 0.075;
0.125; 0.175). The values obtained inthis manner for
f' ( ) and f(2) are given in table IIand the function
f(6) plotted in figure 15. At b = 0 the function







N.A.C.A. Technical Memorandum No. 897


f(6) possesses a singularity since f'(b) there becomes
infinite. The mathematical nature of this .singularity
could not as yet be determined.

We shall now compute the lift, wave drag, and induced
drag as well as the moment about the transverse axis of
the rectangular flat surface.

The lift A, of that portion of the surface which
lies outside the two Mach cones is

t t tan a
AI = P Uo o b t 1 -


while the lift of the two triangular portions overlapped
by the Mach cones is

*=1

AII = P uo t2 tana / Y7 d 6 = p uo Y0 t2 tan a E

'=o

where


K = / f(t) d i = 0.684 (71)

b=o
The total lift of the rectangular plate is therefore


A = P uo Y b t {l (1 K)X

or, according to equation (20)

A = 2 P uO2 Po F tan a {l (1 K) } (72)

For the lift coefficient there is thus obtained

ca = 4 3o tan a l (1 K)\} (73)

For the wave drag outside of the Mach cones there is
obtained simply

= p b t t tan a -
Wwave I = o AI = ol _t (74)
tan ac b








N.A.C.A. Technical Memorandum No. 897


The wave drag of the two triangular portions overlapped
by the Mach cones is

Wwave II = 2 P f 7 cave d f


where


c Zwave


2 tan a 2 tan a


and


1
d f = t d (tan c)
2


Table II

Lift Distribution of the Untwisted Rectangular Wing

f(B) and f'(4)


I f d- I __ J__f


0
0.025
.075
.125
.175
.25
.35
.45
.55
.55
.75
.85
.95


CC,
4.49
1.86
1.39
1.24
1.10
.958
.850
.753
.655
.546
.417
.225


0
0.05
.1
.15
.2
.3
.4
.5
.6
.7
.8
.9
1.0


0
0.219
.312
.381
.444
.554
.349
.734
.810
.875
.930
. 971


We then have


o t2
Twave II = /
2 tan a J


2B d (tan m) = t _ry
2


'Wwave II = 72 t2 K1
2


0=1

f d

4=o
(75)







N.A.C.A. Technical Memorandum.No. 897


where

K J' f2( ) 0 d

*= o
Similarly there is obtained for the induced drag in
the two triangular regions overlapped by the Mach cones

Wi = 2 P / 7 c. d f

where from equations (62) and (65)


cz- 1 -o (1 f())
2 tan a

We then have


Wi = t o f(t) [1 f(4)] d
2


-i t2 Yo2 (K K) (76)
2

For the total drag

W = Wwave I + wave II + Wi

there is thus obtained from equations (74), (75), and (76)

P b t
W = ---- (1 K)A
2 tan a.

or from equation (20)

W = 2 p uo02 o0 F tan a {l (1 K), I (77)

and for the drag coefficient

C, = 4 po0 tan a {I (1 K) ?I (78)

From equations (72) and (77) there is obtained between the
lift and the drag the simple relation


W = o A


(79)








N.A.C.A. Technical Memorandum No. 897


There is thus obtained for the plane surface of
finite span the same simple result as for the infinitely
long flat plate, namely, that the ratio of the total drag
for a frictionless flow to the lift is 0o : 1. This may
also be explained by the fact that in contrast to the in-
compressible flow no suction force arises at the leading
edge in the supersonic case and the resultant air force
is therefore at right angles to the plate.

For the relation between the drag and lift coeffi-
cients, there is obtained finally from equations (73) and
(78)

ca2 1 c a 1
cw = (80)
4 tan a 1 (1 K) A 4 tan a 1 0.316 A

The above formula has the same structural form as
formula (49) for the trapezoidal wing with constant lift

distribution. In figure 10 cw/ ca has been plotted
14 tan a
against the reduced aspect ratio \ (dotted curve). It
may be seen that the rectangular plane wing for the same
lift has the same drag as the trapezoidal wing with con-
stant lift distribution with the reduced taper angle 6 =
tan T
-ta = 0.27. For the reduced aspect ratio ;A = 0.3 the
tan a
rectangular plane wing has, for the same lift, about 10
percent and for A = 0.5, 19 percent more drag than the
ideal trapezoidal wing whose taper angle is greater than
the Mach angle.

With the above results the theoretical polar and
moment curves for the plane rectangular wing may be given
for various aspect ratios and Mach numbers. For the mo-
ment MH about the transverse axis in the wing leading
edge, there is obtained

MH = 2 P uO2 tan a b t{ -2 (1 K)
o_ 2 3

* It is interesting to note that the constant 1 K = 1 -

/1f(6) d e is equal to 1/T within the computational accu-
0 /
racy. That this is exactly so has as yet not been shown.
For this it would be necessary to know the exact solution
of the integral equation (67).







N.A.0,A. Technical Memorandum iTo. 897


MH
and for the moment coefficient Cm = P
Cuo

<1 -
mH = 4 Po tan a (1 K)
2 3

(81)
1
0.211 A
cH = ca ----
cH 1 0.316 A


Through equations (73), (80), and (81), the polar and mo-
ment curves not considering the frictional drag, are com-
pletely determined. In figure 16, the polars are given

for the aspect ratios = 0, and I and for the Mach
b 5 2
numbers = 1.2, 1.5, 2.0, and 3.0. The drag differences
C
between wings with var ious aspect ratios are considerably
smaller in the case of the supersonic flow than for the
incompressible flow since in the first case the greatest
part of the drag is contributed by the wave resistance,
which is independent of the aspect ratio.

The plane rectangular wing at supersonic flow is one
with constant center of pressure position, if the fric-
tional drag is disregarded. The position of the center
of pressure depends only to a slight extent on the re-
t tan a
duced aspect ratio X = For the infinitely long
b
wing, the center of pressure lies at the midchord position
and with decreasing aspect ratio it moves forward somewhat
(table III).

Table III

t tan a 0 1/5 /2


cmH 1
0.489 0.469
Ca








N.A.C.A. Technical.Memorandum No. 897


Formula (80) for the rectangular flat plate is the
analogy to the familiar Cwi = cy2 F/n b2 of the incom-
pressible flow. Like the latter it enables the recompu-
tation of the drag from one reduced aspect ratio


t tan a,
S = -
1


t2 tan a2
to another A = .From equa-


tions (73) and (80) there is obtained for the new angle
of attack and the drag


a = + Ca


tan a (1-0.316 2)


= caa 1+
1C = 4c +--- (-0.316
tan a (1-0.316X )


- --1
tan a (1-0.316Nx)
S (81a)
1

tan a (l-0.316XY'j
.1


VII. TRAPEZOIDAL LIFT DISTRIBUTION

a) Lifting Line

As a further example we now compute the induced drag
and the velocity field for trapezoidal lift distribution.
for both the lifting line and the lifting surface (fig.
17). Let the lift distribution therefore be given by


P(r~') 1 -n'
-- for n < n' I 1
o I ~


r(r') = r0 for


(82)


- 1 11' < TIi

- ~1 1 _< + :h


where bn = b'/b, according to figure 17. The field of
the induced velocities and induced drag for variable lift
distribution may be obtained in the familiar manner from
the lift distribution by superposition. On account of
integration difficulties, however, this computation can
not directly be made on the potential but must be carried
out separately for the three velocity components. From
equation (14c) we have for the induced downwash velocity
of a lifting line ending at n = r' with circulation Fo








N.A...A. Technical Memorandum 17o..897


. (.-71 ')[_ aK2( .)( + 2 2} )
.(rb (t .- V (AA-_- Ka


(83)


From the above there is obtained by superposition
the downwash velocity cz for variable circulation Fr'):


1 .

c' o


(84)


C2 (n,)) -d Tn
d i


For the trapezoidal lift distribution according to
equation (82) we have therefore if, on account of symmetry,
we restrict ourselves to the half-wing y > 0


1

1 01 J
TI


cz('r') d 'n'


or., according to equation (83)

2l1 (i i')){g- K2[(T'i)2 +4 2t2]}d d'
rb cz 1



or with

K2 (I r)2 + = T


b cz 1/2 d/ 1/2 1/2 /i dT
T --



Performing the integration there is obtained for
points ., r, p within the Mach cone at the wing tip T = 1


be 1 1 +
--- = + log
ro 1 i t, K-2 t 2-


(85)








N.A.C.A. Technical Memorandum No. 897


and a corresponding expression with reversed sign for the
Mach cone at Ti = TI. The value of w is here given by
equation (12). In the cone T = 1, cz > 0 so that there
is upwasi velocity. In the cone n = 1 there is a
downwash velocity of the same absolute magnitude (c- < 0)
and outside of the two cones c. = 0, a result which is
also to be expected from reasons of symmetry since, on ac-
dr
count of = const., all separating vortices are of
d r'
the same strength. With

K(T 1) K(TI T)
=a nd =


there is obtained for the downwash distribution in cone I
and III respectively in the plane z = 0.

b cz 1 2 1 1+ 1TTT
S--- = + log -- ---- (86)
o 1 2 1 J ~

On the cone surface according to equations (85) and
(86) c, = 0 ani is thcrefore continuous in passing
through the cone. On the cone axis c now becomes log-
arithmically infinite, whereas with the rectangular lift
distribution (horseshoe vortex) cz becomes infinite on
the axis as r-1. The logarithmic singularity of cg is
no longer a disturbing factor for the computation of the
induced drag.

For the sake of completeness there will also be given
the remaining two components of the induced velocity.
There is found for the cone at T = 1:

b c 1

o 1 1 n ga
(87)
b c -1 -fT
T arc tan
ro 1 T t(I 1)








N.A.C.A. Technical Memorandum.No; 897


and corresponding expressions with reversed-signs for the
cone.at T = TI For the arc tan there is to be taken the
principal value 0 < arc tan < Tr. For the outer cone -
(n = 1) the arc tan is zero in the upper half plane on
the outer quadrants of the cone surface and equal to +rT
on the inner quadrants. In the wedge-shaped space be-
tween the two cones cy is constant, being equal to


<0: Cy= 2 ; Cx = 0 (88)
b b'

In passing through the plane 0 = 0, therefore
there is a discontinuous increment in cy by
Po
2 -- The region of the t plane limited by the cone
b- b'b b-
axes T = I and In = 1 distance -- is thus a vor-
\ 2
tex surface with constant circulation density the total.circula-
tion of which is equal to the circulation F of the
bound vortex in the region of the constant lift.

A streamline picture of the y and z velocity
components for a plane x = constant that intersects both
cones is drawn in figure 18. Like the streamline picture
for the constant lift distribution (fig. 2) it was ob-
tained by computing the field of isoclines. On the outer
halves of the cone surfaces cy and cz are equal to
zero but the directions of the streamlines c/Cy have a
value different from zero. In this case, too, not all
streamlines are closed, part of the streamlines entering
from the undisturbed region into the one cone and coming
out from the other again into the undisturbed region.


b) Lifting Surface

In order to compute the induced drag for the trape-
zoid-shaped lift distribution, we must, as in section IV,
make the transition from the lifting line to the lifting
surface. A rectangular lifting surface will therefore
now be assumed of span b and extending from x = 0 to
x = t. The chordwise circulation distribution is assumed
to be constant of density F/t, while along the span the
distribution is that given by equation (82). For the com-
putation we may here restrict ourselves to the region be-







N.A.C.A. Technical Memorandum No. 897


tween the cones springing from the leading and trailing
edges of the lifting surface, since only this region en-
ters into the question of the computation of the induced
drag. Te likewise need carry out the computation only
for the cone at T = 1; for the downwash in zone T = T
there is obtained the corresponding expression with re-
versed sizn.

For tnc induced z component cz of the lifting sur-
face, thErc is found, according to equation (85), with


1
2 1 / (- ,>A(i- ')2 [(v,- l)2 + 2]
1 1 / ,)2- L -





+ ----- log ---- -- -=--------- i_ t




w i: er e

+i' = ?- + J 1) +


according, to equation (31). With the new integration var-
iables = *, the above equation becomes


12 2 + l2 2
o 1 ./ E*2 K 2e
^=0
1/2 1 + K2
-lo (t* + .... ) d
i r 1 = /


1/2
+ -/ log (2* ....) 2 *
1 V.=j


where







46 N,A.C.A. Technical MemornLdum.No. 897


,Th =Kth ai) + ol .

The three integrals are evaluated as follows:


Setting


*-. _- K = 7, we have for J


1/2 / T d.( 1).
j .- d 7



J = -- K ( 1) arc tan
1 '-


~(~-l)


With


... ( 1)2+ 2 = ais


and -


S* +4*-Kala =


there is obtained for J2


J 4 1- j1 T, 2 a 1 log
4 1 T2


Td T


After a brief intermediate computation we have


Ja = -- l (- [) a] log O1 ) + )

log (+ Wa) + .-})

and similarly

S= + 1/2 ')a + s3 log ( 1 /j7- 1) + 2

t log Q +) +








T.A.C.A. Technical Memorandum No. 897


By adding we obtain


2 In ~1-u + <( 1) arc tan
o i n K(n l)

1 L U }
+ loo (89)
2 -

A corresponding expression with opposite sign is obtained
for the cone T = n The arc tan in equation (39) lies

within the range --< arc tan < + as follows from the
2 2
fact that must be symmetrical in ( 1) since the
same holds for c, according to equation (85).

The induced z component thus found for the rectangu-
lar liftin? surface with trapezoidal lift distribution
has the same singularities as the correspondin; formula
(85) for the lifting line. On the cone surface c 0
ani on the cone axis logarithmically infinite. For the
downwash distribution at the location of the wing in the
plane G = 0, there is obtained

co /- r 1777
2 = 2 + b arc tan ----- +



+ log ---+---- (

1 1 -

K(, 1) K(rV )
were =- for coed I and = f-or


cone III, the upper sizn holding for cone I and the lower
for cone III. Equations (89) and (90) include only the
downwash velocity induced by the edge vortices. In order
to obtain the field of the total downwash motion, there is
still to be addel the induced downwash velocity due to the
clanc wave. In the wedge-shaped space between the leading
and forward edges of the wing (fig. 4), this induced veloc-
ity component is








N.A.C.A. .Technical Memorandum.No. 897


Kr 1 r I/=
= Z -a 0 1/2 0
wave 2 t 2 1 ~I 1


For the total downwash velocity in the plane
is thus obtained from equations (90) and (91)


z.= 0, there


For cone I:.

1 < ?3 < 0:


S< < + 1:


For cone III:

1< < 0:


0 < 7 < +1:


2(1 Tl) g ()
0 WCzo = + (
Yo TT7
g ( Tr


2(1 -, Ti) -
-Yo


g (4)
- ={ -1 -

= -l- Co)


The downwash distribution'thus computed is plotted
figure 19.


We-are now in a position to compute, for the wing
with trapezoidal lift distribution, the induced drag. In
order to avoid special complications, we shall assume that
the Mach cone springing from the leading edge at T = I
does not extend beyond the wing tip and does not overlap
the region of dropping circulation of'the other half-wing.
The first is identical with the condition that the cone
springing from T = 1 does not extend into the region of
the.wing where the circulation is constant.! This gives
for the Mach angle the two conditions


b b'
tan a < ----- and
2t


ab '
tan a <-t
t


The induced drag of one half-wing -lWi is composed ad-


ditively of the drag of half of.cone I, .Wi
I1


and the


(92)









in


(91)








N.A.C.A. Technical Memorandum No. 897


drags of the two half-cones of cone III,

Ii. 2 (fig. 17).


lvi
'III,


and


i i = wi + wIII + WiI (93)

Si:-ce in cone I, in the plane = 0, there is upwash
velocity, '7i gives a forward thrust which in absolute
ii
value, however, is smaller than the back thrust in cone
III, since the circulation is greater here. We have


x=t


y=b/2


Wl = P / d x
1 -.-/ J7


X=0


- Czo a y, ( = x tan
t


Y=Y,


where co is known from equation (90).


We thus have


0 =-t/b
1 P Fob
Wil /
S. pn s T P t2 K
=o
In cone I for -1< 1 < 0:



1 -

and therefore


l ,


1 P P 2 b \2

(1 ) 8 K t


=0o

/ d r() g (-) (54)

i,'=- i


-K
1 1


b


0-.
Ho


3=0


6=-1


For briefness we set


1

j- g (o) as = jg (C) d 6 = K1
0=-i i0=o


(95a)








N.A.CA. Technical Memorandum No. 897-


o +1
g ( d) d = g (0) = K2

4=-_ 4=-'o


These integrals may be exactly computed. There is obtained


3
S=---8
8


7
Kg = --
18


so that finally

p r t
Wi = 2
I (1 ,) Kb


The portion WiII

substituting -g(b)
that

WIIl1


is obtained from equation (94) by

for g(4) and taking r = F so

1 K Pt (98)
= -- n- ----- (98)


3 1 .


Finally, Wi
ITin


T K b


is obtained by putting in equation (94)


r = o -11 )
S o I ----

and substituting -g(6) for g()). By comparison with
equations (97) and (98) this gives


i i + wi
IIIa il llli


and therefore
Wi
1
S Wi = a Wii + 2 Wi 2 { +
iII1


(95b)


(96)


(97)







N.A.C.A. Technical Memorandum No. 897


Substituting the values from (97) and (98) the induced
drag of the entire wing is found to be

4 K3 P 10o' t tan a 1 3 K2 t tana (
Wi ------ )(99)
3 1 'n b 1 2 K b
1 1 3
If, in place of Fo, there is now substituted the lift A
of the entire wing

A = P b ro u0 1
2

we have
16 E3 1 1 / A t tan a
3 n (1 ri) ( + Tr)2 p b uo b

1 3 K, t tan a
1 T K3 b


2 1 ( A )2 t tan a

(1 )( + r2 p u b b


S- 1 14 t tan ac
1 l 9 TT b

Thus the formula has been found for the induced drag with
trapezoidal lift distribution. To this must be added the
wave drag. The latter according to equation (26) and
table I is
2(2 + 'r) b 1 AuA
'Twav e b (101)
3(1 + .1) 2 t tan a P o

If c, denotes the coefficient of the wave plus induced
drag then from equations (100) and (101)

Sca2 4(2+ 1) 4 hX 1 14
Cw + 1- (102)
4 tan a 3(1+T' ) (l-n )(l+r ) 1_-ti 9 Tv








N.A.C.A .Technical. Memorandum No. 897


The above formula differs from the corresponding formulas
for the rectangular flat plate (equation (80)) and the trape-
zoidal wing with constant lift distribution (equation (49))
in that for small X the induced. portion of the drag is
proportional to .a -whereas for.the other two cases it is
proportional to X. In figure 20 the coefficient
c 2
cx/-a i- i.s plotted against the red.uc.ed aspect ratio
4 tan a
t tan a = X for various trapezoid shapes b'/b. It may
b
be seen that by far the greatest portion of the drag is
contributed by the wave resistance. The portion contrib-
uted by the induced drag, within the range of validity of
our formulas, amounts to a maximum of 11 percent of the
wave resistance for AX = 0.5 and b'/b ="1/2. It is
therefore smaller than for the rectangular flat plate where
for the same aspect ratio it amounts to 19 percent (fig.
10).

VIII. SUMMARY


With the aid of the expressions given by L. Prandtl
(reference 2) a theory is developed of the airf-oil of fi-
nite span at supersonic speed. As in the case of the
Prandtl airfoil theory for the incompressible flow, it is
a first order approximation theory. The airfoil is first
replaced by a "horseshoe vortex" and the induced velocity
field of the .latter computed. This field is considerably
different from that of the incompressible flow. From the
horseshoe vortex there are obtained in the familiar manner
by superposition more complicated lifting systems. The
computation of the induced drag., :in 'contrast to the incom-
pressible case, is for the compressible flow possible only
if there is first assumed a surface vortex distribution
and secondly a suitable dropping off of the lift toward
the wing tips.

As an example of the "first principal problem" there
are computed the induced drag and the wing surface shape
for a wing of trapezoidal plan form with constant surface
density of the lift. The induced drag, as in the case of
the incompressible fldw, is found to be proportional to
the square of the lift and depends on the Mach number as
well as on.the aspect ratio. In addition to the frictional
and induced -drag there is present in the- supersonic case
also the wave drag, produced by the sound waves; which


52- ...








N.A.C.A. Technical Memorandum No. 897


varies as the induced drag. It is therefore only the sum
of the wave and induced drags that is of practical inter-
est.

As an example of the "second principal problem" there
is computer the lift distribution and induced drag for the
rectangular flat plate (untwisted rectangular wing). Out-
side the two Mach cones springing from the leading edges
of the wing tips the lift density is constant; within
these cones the lift drops from the full value at the cone
rim to the value zero at the lateral wing edge. The inte-
gral equation that arises is independent of the aspect
ratio and of the Mach number and may be solved numerically
by approximate methods. In general for airfoils of normal
aspect ratios at supersonic flows the greatest portion of
the total drag is contributed by the wave resistance while
the induced drag contributes only a small proportional
part.

Finally, there is considered che lifting line with
trapezoidal lift distribution and the lifting surface of
rectangular plan form whose lift is constant along the
chord and trapezoidal along the span. For these cases the
downwash distribution and induced drag are computed.


Translation by S. Reiss,
National Advisory Committee
for Aeronautics.








N.A.G.A. Technical Memorandum No. 897


REFERENCES


1. Prandtl, L.: Theorie des Flugzeugtragflugels im zusam-
mendrickbaren Medium. Luftfahrtforschung, vol. 13,
no. 10, Oct: 12, 1936, pp. 313-19.

Prandtl, L.: General Considerations on the Flow of
Compressible Fluids. T.M. No. 805, N.A.C.A., 1936.

2. Prand'l, L.: Tragflugeltheorie, 1. u. 2. Mitteilung.
Nachr. von der Kgl. Gesellschaft der Wissenschaften.
Math. Phys. Klasse (1918) S. 451 u. (1919) S. 107.
Wider abgedruckt in Vier Ahhandlungen zur Hydrody-
namik und Aerodynamik. Gbttingen 1927.

3. Ackeret, J.: Air Forces on Airfoils Moving Faster, than
Sound. T.M. No. 317, N.A.C.A. 1925.

4. Busemann, A.: Aerodynamischer Auftrieb bei Uberschall-
geschwindigkeit. Luftfahrtforschung, vol. 12, no. 6,
Oct. 3, 1935, pp. 210-20.

5. Betz, A.: Beitrage zur TragflUgeltheorie mit besonderer
Bericksichtigung des einfachen rechteckigen Fligels.
MWnchen 1919.








N.A.C.A. Technical Memorandum No. 897


Figure 3.-
Lifting line with
constant lift
distribution.
Downwash distrib-
ution in Mach
cone.


Potential of the
lifting line.
i I I I


Figure 2.- Lifting line with constant
lift distribution (horshee
vortex). Streamline picture of the y-
and a- velocities in a plane at right
angles to the axis of the Mach cone.

fl



I I 4 1"

,--, Figu
i / i I "

Figure 5.- Rectangular dist
wing as lifting the
surface with constant lift x< t
distribution, for


Wave of rarefaction Compression
I -- ,shock
I I




S' N-Wave of
Compression shock rarefaction
Figure 4.- Plane sound waves
at a flat plate.


1 s
I
P) I -


re 6.- Rectangular wing as lifting
surface with constant lift
ribution. Downwash distribution in
wing plane. Continuous curves for
(at location of wing) dotted curves
x = 2t (behind the wing).


Figure 1.-


Figs. 1,2,3,4,5,6







N.A.C.A. Technical Memorandum No. 897


I /
S > '',


,"-, / \ / /










Figure 7.- Rectangular wing as Figure 8.- Rectangular wing as
lifting surface with lifting surface with
constant lift distribution, constant lift distribution.
Streamline picture of the y- and 5- Streamline picture of the y- and t-
velocities in a plane x angles to the axis of the Mach cone. x = 3t at right angles to the axis
of the Mach cone.


---------- -





I I I


Figure 9.- Trapezoidal wing with
constant lift ,
distribution.




Figure 10.- Trapesoidal wing with constant lift distribution.
Coefficients of the wave plus induced drag c / a
as a function of the "Oiduce aspect ratio" X = t tand/b for
various trapesoid shapes 0 s4;na/tand.


Figs. 7,8,9,10







N.A.C.A. Technical Memorandum No. 897


b _

Figure 11.- Trapezoidal wing
with constant lift
distribution. Induced downwash
velocity in section AB (in
direction of flow) (tanT" 1/13;
tan c( =- -3).


Figure 12.- Trapezoidal wing
with constant lift
distribution. Induced downwash
velocity in section CD (at right
angles to flow direction)
(tanr= 1/V3; tand-V-).


/ ----" -.." -


Figure 13.- Trapezoidal wing
with constant lift
distribution. Profile sections.
(tancf = : tan T= 1/t/).


Figure 14.- Trapezoidal wing
with constant lift
distribution. Elevation contour
lines. (tand = 'FV tan = 1/V3).


Figs. 11,12,13,14







N.A.C.A. Technical Memorandum No. 897


/' / r / \


Figure 17.- Rectangular surface
with trapezoidal
lift distribution.


Figure 16.- Polars of plane
rectangular wing
for various aspect ratios.


/ 7fS


'1
/ \"
- ) 1

/ ////


Figure 18.- Lifting line with trapezoidal lift distribution.
Streamline picture of the y- and s- velocities in
a plane at right angles to the axis of the Mach cone.


Figs. 15,16,17,18








N.A.C.A. Technical Memorandum No. 897


Figure 19.-


Rectangular wing as lifting surface with trapezoidal
lift distribution. Downwash distribution for x< t.


A ---
I
dri


J ~u I


Figure 20.- Lifting surface with trapezoidal lift distribution.
Coefficient of wave plus induced drag eCw/t a s a

function of the "reduced aspect ratio" A for various values of b'/b.


Figs. 19,20










UNIVERSITY OF FLORIDA

3 1262 08106 286 0III
3 1262 08106 286 0




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