A flat wing with sharp edges in a supersonic stream


Material Information

A flat wing with sharp edges in a supersonic stream
Series Title:
Physical Description:
48 p. : ill ; 27 cm.
Donov, A. E
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


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


A basic treatment is given for the approximate solution of the problem of two-dimensional supersonic flow past a thin wing at small angles of attack. The pressure distribution at the surface, the lifting force, and the wave drag are determined.
Includes bibliographic references (p. 41).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by A.E. Donov.
General Note:
"Report date March 1956."
General Note:
"Translation of "Izvestiia-Akademia, NAUK, USSR, 1939."

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Source Institution:
University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003874043
oclc - 156913584
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Full Text
IWkck -rP'\ 091




By A. E. Donov

In this work there is given an approximate solution of the problem
of a two-dimensional steady supersonic stream of ideal gas, neglecting
heat conduction, flowing around a thin wing with sharp edges at small
angles of attack. (Determination of the law of distribution of pressure
along the wing, lifting force and head resistance of the wing.)


The problem of the investigation of the mechanical action of a
moving gas on an immiovable win-g appears as a special case of the ;ome-
what more general problem of the Investigation of the mechanical action
of a moving- gas on a11In imovable fixed wall constraining the motion of'
the gas. In our own explanastion we begin with the formulation of this
last problem in which we confine ourselves only to the consideration of
the steady two-dimensional forces of ideal gases not subject to the action
of gravitational forces. In the plane of motion of thle gas we shall
arrange an immovable rectan~gular coordinate system in such a mann~er that
it is situated as in figure 1. We introduce three functions v, p, and p
of the independent variables x and y defined, respectively, as the veloc-
ity, density, and pressure. The vector frictions v will be determined by
a pair of scalar functions of the independent variables. For these fune-
tions we shall agree to take either the functions vx, vy defined as the
projections of the velocity of the axis x and y, respectively, or the
functions v and p, defined, respectively, as the absolute value of the
velocity and its angle with respect to the positive direction of the x-axis,
measured in the counterclockwise sense. In what follows we limit yourselves
to the consideration only of flows for which thle function P satisfies the

2 2

*Izvestiia-Akad~emia, NIAUK, USSR, 1959, pp 603-626.

2 N~ACA TM 1334

As is well known, the study of the gas motion under consideration
leads to the investigation of the following system of differential

ax avx 1_ ap
v, -+v -+ =o
ax bx y p ax

avy a7y 1 ap
vx + Vy + =0O
ax ay P ay

a(pyx) apvy)=O
ax ay

vx axl + "y = oI

Here k is the adiabatic exponent (for air k = 1_.k05). If the motion
of the gas is constrained by an immvable frictionless fixed wall in the
plane XGY, the gas will be adjacent to it along some curve. We shall
call this curve~ the "contour K."

Consider the unit vtector t tangent to the contour K directed in
such a manner that its projection on the x-axis is positive. Denote
by Pk the angle which it makes with the x-axis. Clearly Bk may be
regarded as a function of the abscissa x of that point of the contour K
associated with the vector t. We denote this function by Bk(x) and
assume that it is continuous. If the function Pk(x) is prescribed and,
moreover, the coordinates of any point of the contour K are given, the
form and position of the contour is completely determined. We agree to
take as origin the left edge of the contour K. Then the: equation, of
this contour will have the form

y = tan Pk(x)dx (5)

NACA TM 1594

We can write this equation more briefly if we designate its right-hand
side by, yk(x)

y =yk(x) (4)

Since ini the' flows under consideration the direction of the velocity on
the c-ontour K must coincide with the vector t, the condition on the flow
along an immorovable fixed frictionless wall may be written in the following

at y- = ;/k(x). The condition (5) must be added to the system of equa-
tion3 (2) as a qualifying boundary condition. Much work has been dedicated
tor the investigation of solutions of the system (2) subject to the con-
dition~ (9). Of these we are interested here only in those in which the~
f'low is ;upersonic, i.e., flows at every point of which the following

v >a (6)

is ~sti-fied, where a is the local speed of sound

The investigations contained in these works divide in two fundamental
directions. The first direction is represented in works in which solu-
tions of the problem are achieved with the help of numerical or graphical
processes permitting the step-by-step calculation of a system of parti-
cuilar values of the desired functions. (Works of Busanann, Kibelia,
anid Frankll.) The fundamental achievements of the methods represented by
these worKs consist of the fact that by their use many actual practical
probllem-- ray be solved quantitatively of which the solution by other
methods would present great difficulties. In particular these methods
solve thoroughly corner-nonpotential problems. The chief defect of
these methods is that the solutions obtained are numerical so that it
is impossible to obtain a general qualitative estimate of the phenomena

4 NACA TM 1594

under investigation. The second direction is represented by the works
of Meyer, Ackeret, Prandtl, and Busemann, which are confined to a culti-
vation of an exact theory of irrotational flows. The results are based
on the fact that in the case where vorticity is absent the character-
istic system of differential equations (2) adn~it of integrable combina-
tions. This theory leads to series of approximate results of any desired
accuracy, giving a complete qualitative and quanitative picture of' the
flo~w. Since our investigation is mostly connected with the theory of
irrotational flows we give below a brief introduction to the fundamental
methods and results of this the~ory.

We introduce the stream function Jrdefined by the following relations

-9 p
ax P"


As is well known from equations (2), ("1), and (8) the following relations
follow without difficulty

e (9)

v2 a2
--- + -tO (10)
2 kr 1

Where 6, tO denote quantities which display the~ flow once and for all
as a function only of J. With the help, of equations (2), (9), and (10)
it is easy to obtain the two equations

by, aVx
0 (ll)
ax ay

Here ml* m2 denote the following expressions

-v2sin B cos B + a\ 2 3
ml =
a2 2cos2B

NACA ?TMl 194

(a2 vx2) Yx+ (a2 q2)bi _
ax ay


ay j


where R denotes a quantity defined as


Equations (11) and (12) represent linear relationships between the first
partial derivatives of the functions vx, vy With respect to x and y.
Since every flow under consideration is supersonic, the entire region
of the flow may be covered by a pair of families of characteristics. 'The
differential equations of these characteristics are obtained easily by
the use of equations (11) and (12). For one family of characteristics,
which we shall agree to call the first family, we obtain the equations



dy = mldx

S(a2 v2cos2p)ml + v~sla P cos B x

d(v cos 0) + m2d(v sin P) =

and for the other, which we
have the equations

v2cos 2P a

shall agree to call the second family, we


dy = m~dx

sinp) 0(a2 v2cos2B)m2 + v2sin B cos p x
v2cos2B a2

d(v cos p) + mld(v



d~ k(k 1)a~ &n

NACA TMr 1394

-vfsin p cos B a\/v2 a2~
n2: = (1.9)
a2 v2cos2p

We now consider a supersonic stream with constant hiyird;redynamical
elements (i.e., functions v, p, p, p, a). We shall call th~is flow th~e
undisturbed flow. The values of the functions v, p, p, s in the unidis-
turbed stream will be denoted by v, pO' p0, aO respectively,, aInd
the ratio w/a0 by M. Since the stream under consideration is super-
sonic, M > 1. We shall choose the direction of the velocity of the:
undisturbed stream to correspond, to the direction, of th-e x-axiS.

We assume that thLe undisturbed stream strikes anl Lmmovable, fixe~d,
frictionless wall (contour K), inclined in such a mar-ner that in flowing
around this wall the stream never detaches from it adi~ remains suiper-
sonic evreryvbere. We~ may distinglluish two cases of -flo1ws of thijS tyrpe.

Case I.- The contour K is situated in such a manner1 that the
condit ion

Pk(0) (O (20)

is fulfilled. .In this case, as is well known, there appea~rS a ciirve of
weak discontinuity OC (figs. 2(a), and 2(b)) procee~dingl from thie origin
and dividing the entire flow in two parts. On one slide ofr the cui~rve~ of
weak discontinuity OC extends the reg~ionl containing thie uCdisJturbe~d
stream and on the other the region of flow around the wall. In the region-
of flow around the fixed frictionless wall the hydrodynamical elements of
the stream, generally zpestingv, are not constant but vary. In what fol-
lows we shall call this part of th~e stream the disturbed stream. In the
entire region of the flow undr consideration the functions v, P, p, p, a
are continuous but their partial, derivatives with respect to -x and y
(all or only some) exhibit jump discontinuities, at least on the curve of
weak discontinuity OC. The same curve OC appears as a characteristic of
the second fam;ilyr since the hydrodynamical, elements of the stream. are con-
stant. On this line the following relationships will hold in th~e entire
region containing the stream

,2 aO2
tO -f + (21
2 k 1

NdACA 1TM 1394


9 = 80

where 903 del-otes a quantity defined as


Froim equadtions (13), (21), and (22) we easily obtain

0 = O



i.e., thel flo)W under consideration is irrotational. By virtue of rela-
tion (24) the right-hand side of equations (15) and (1'1) vanish and these
equations canl be integrated. As a result of integration of equation (15)
we obtain the relationship

p + cp(v) = constant


satisfied along any characteristic of the first family, and as a result
ojf integra~ting equation (171) we have

p rp(v) = constant


satisfiedj along a characteristic of the second family. cP(v)
function defined as

cp(v) --- are tan \va.2- are tan \r

denotes a

2a- a (27)

Since on! the curve of weak discontinuity OC the quantities v and P have
the values w and O, respectively, the following relation is satisfied
along every characteristic of the first family intersecting this line
and consequently in the entire region of the disturbed stream:

NACA TM1 1394

s +t r(v) = rp(w) (28)

From equations 16, 26, and 28 it immediately follows that the char-
acteristics of the second family (the curve OC being am~ong these) are
straight lines since along each of these characteristics the hydrojdynamu-
ical elements are constant.

Making use of these circumstances it is not difficult with the aid
of equations 28, 26, 22, 21, 16, 10, 7, and 5 to construct expressions
for the functions v, p, p, p?, a in the region of the disturbed flow.
HoweJver, thne construction of these expressions is not of great interest
since our chief interest is centered on the construction of an expres-
sion for the pressure on the contour K: which may be accomplished without
the use of these expressions for the hydrodynamical elements of the flow.
Actually equation 28 allows us to determine the velccity v as a function
of the angle of inclination of this velocity with the x-axis at every
point of the region filled by the disturbed flow. Since by virtue of
equation 5 the angle of inclination of the velocity with respect to the
x-axis is a given function of x on the contour K th-r'e is thle possibility
of using equations 22, 21_, 10, 9, and 7 to determine the pressure p as a
function of x on the flow around a contour. If we limit ourselves to
the consideration of slightly disturbed flows, i.e., flows whose hyjdro-
dynamical elements differ but little from the~ hydrodynamical elements of
the undisturbed flow, the expression for the pressure on the flow around
a contour K may be written in the form of a series. This series has the

p = pO + q ali(x) + a28k2(x) + a pk5(x) + a pk(x) + .. .] (29)

NACA TM 1394



p ~w2
2 =


a2 = (MB -1)-~2 2 ~2+ 4 2 4 4 +1

3 36 6

ak = (M?2 1)-5 1 2 M2 + + 19k M4 + -21 43k + 18k2 M6 +
\3 3 6 12

12 48 48

Case II.- The flow around a contour K is situated in such a manner
that the following inequality is satisfied

Pk(0) > O


In this case, as is well known, a line of shock discontinuity OD appears
(fig. 3) proceeding from the origin 0 and dividing the entire flow under
consideration in two parts. On one side of this line is the region of
the Lundisturbed stream and on, the other the region in which the fluid
flows around the fixed frictionless wall. Just as in case I we call the
flow in the region in which the stream around the fixed frictionless wall
is accomplished the disturbed flow. In the present case, in contrast to
case I, the functions v, B, p, p, a exhibit jump discontinuities on the
shock-line OD.

10 HIACA TM 1396

In, the region of the disturbed flow these functions must, satisfy,
not only equations 2, 5, and 7 but also the dynamical conditions across
the shock line. Considering the flow to be only slightly distuirbed, these
conditions may be written in the following form

v2 a2 w2 &0O
-- + (31)
2 k 2 k- 1

v = w(1+ bly+ b202 b pi + b48 + .) (2


b1 = -(M2 1)- 2

+-l M"

b2 = (M2 -

1)- 21 +

-5 1
1) --+

- 1)M4 + Sk2 12k + 5 Mo


5 2


-1 27k + 12k2 1

-17 + 29k e

b4 -(M2 -

+-5 k Sk2t +h MkpD)

5 + 5k k2 + k-3 M8

O +zg +2p

(' 3~)

(1 + 1)2 M

NACA TM 1594


k(k2 )
11 = ~M (M- 1)2

34 = 2(k2 1) N (M2 -

= c + elB + esp2 + ...



eO = (M2 1)- 2

el k+ IM (M2 1)-2

Condition 31 shows that, disregarding the presence of jump discon-
tinuities in the functions v, p, p, p, a, equation 21, just as in case I,
is valid throughout the entire region filled by the flow under considera-
tion. BHowever, condition 22 is not, generally speaking, fulfilled in
the case now under consideration. However, there is the possibility of
speaking of satisfying this condition approximately. In fact, consider
equation 33. Its right-hand side does not contain terms in the first
and second powers of B. Therefore, for slightly disturbed flows, equa-
tion 22 may be regarded as approximately satisfied on the line OD and con-
sequently throughout the entire region filled by the flow under considera-
tion. From this it follows that in the region of disturbed flow equa-
tion 24 may be regarded as approximately satisfied, which means that

1) 4 + 2(k 2)M2 (k 1)M4

12 NACA TM 1394

equations 25 and 26 hold on characteristics. For values of B and v
near 0 and v, respectively, equation 28 may be written in the form
of a series

v = v(1 + bl'p + b2 2~ + b 'p) + b 'p4 + .)



bl' = -(M2 1) = bl

b2' = -(M2 1) -2

k 1 b)= b2


b = -(M2 1)

+-12 M2 + 2k -1)M

+ 19k + 16k2 M6 +

24 8

+-17 + 29k p l

b =-(M2 -

+-3 + 8k 7k2 + 2k59 M10

j 2kr 5k2 + 4kk M8

Comparing equations 32 and 35 we see that for slightly disturbed
streams the first may be substituted for the second with good approxi-
mation. Consequently, for slightly disturbed flows, equation 28 will be
approximately satisfied along the line OD. Since, on the other hand,
along each characteristic of th~e first family equation 2) is approximately
satisfied, equation 28 will be approximately satisfied throughout the
entire region of disturbed flow. The approximate expressions for the
functions v, B, p, p, a are constituted exactly like the accurate expres-
sions for these functions in case I. Substituting the approximate expres-
sion for the function B in the right-hand side of equation 34, we obtain

+2k2 5k + 3 "]

NACA TMI 1394 13

a differential equation of the first degree for the approximate deter-
mination of the form of the shock line. Summing up our considerations
we can deduce that the accurate results contained in case I can serve as
approximate results for case II, and further that expression 29 can serve
as an approximate expression for the pressure on the flow around a contour
in case II. These same considerations show that there is no sense in cal-
culating all terms in this expression. It is sufficient to limit ourselves
to the first two or three terms.

From all that has been said about cases I and II one may conclude
that the form of the contour K may be made up in such a manner that art-
fully constructed shocks may be caused to appear in the region of flow
around th~e fixed frictionless wall. In such cases when we pay attention
to this phenomenon, the results we have obtained are valid, not for the
entire region of flow around the fixed frictionless wall, bu~t only for
that part in the neighborhood of the front side of the flow around a
contour. The fundamental problem of the present work is the3 construction
of approximate expressions for the pressure on the flow around a contour
in case II, with the calculation of the circulation of the flow occasioned
by the presence of the shock discontinuity OD. In spite of the fact that
in the case of the presence of circulation it is impossible to integrate
equations 15 and 17l, there is the possibility, however, of making up such
combinations of differentials from equations 14, 15, 16, and 17, adding
to these equations expressions for differentials of the streak function,
that with the aid of these combinations it is possible to construct expres-
sions which we shall integrate. Investigations concerning the preceding
construction constitute the contents of the following section.


Suppose we have a flow corresponding to case II of the proceeding
section. Assume that in this flow the hydrodyna~nical elements in the
region of the disturbed stream differ infinitely little from the hydro-
dynamical elements in the region of the undisturbed flow. We revamp
somewhat our notion of the region of disturbed flow. Shortly before we
agreed to apply this name to the region bounded by the curvelinear
triangle made up of the curve OC2 (contour K), the shock line OC1, and
the characteristic of the first family CIC2 emerging from the lowest
point of the contour K (fig. 4). Taking into consideration equations 5,
14, 18, and Sk, it is not difficult to conclude that with the assumptions
made just now relative to the hydrodynamical elements the curvelinear
triangle OCIC2 differs infinitely little from the isoceles straight-line

triangle O'Cl'C2' (fig. 5) where the equal sides O'Cl' and Cl'C2' are
parallel to characteristics of the second and first families in the

NACA TN 1394

undisturbed flow. As for the functions Bk(x), a, v, P, p, a we assume
that they all have the properties of differentiability and continuity to
as many degrees as may be necessary to insure le~gitim~acy of operations
which are performed upon them. Moreover, we assume that in the flow
under consideration the infinitesimal quantities Bk(x), Pk'(x), Bkrl(x),
B, v w, p pO' P 0 a O aO have the same3 order of magnitude.

Taking this last group of infinitesimals as fundamental (having Lun-it
order of magnitude) we shall agree in, what follows to adhere to the fol-
lowing system of notations appearing in investigations involving infinitely
small quantities. By Em (m being any positive integer) let us denote
an infinitesimal whose order of magnitude is not less than m. Clearly
such a mode of notation does not exclude the possibility of several dif-
ferent infinitesimals being denoted by the same symbol_, and, vice versa.
The same infinitesimal may be denoted by several different symbols. Thus,
for example, if an infinitesimal a is denoted by E4, the infinites-
imal 2a may also be denoted by E4, and, moreover, the infinitesi-
mals a and 2a may be denoted by E E E .

On an arbitrary characteristic of the second or first family the

d~ = py(sin pdx cos Bdy) (36)

will be satisfied by virtue of equation 8 throughout the entire region
filled by the flow. Eliminating dx and dy from equations 14, 15,
and 36 and taking into account formulas 13 and 21, we arrive at the

d(v cos p) + m2d(v sin. 8) = (9ld In 9 (ST)

which is satisfied on any characteristic of the first family. Here 61
denotes the quantity

a2 viesin B cos p (v~cos2p a2)ml

k(k 1)v(v2cos2p 2)(ml cos B sin P)

NACA TM~ 15394 15

On the other hand, having the integral 25 of the equation

d(v cos B) + m2d(v sin P) = 0 (39)
it is easy to find an integrating factor L1 of this equation, such that
after multiplying by L1 it may be written in the form

ap + ,(v) = o (4o)

In order to determine L1 we have the obvious relationship

L1 d(v cos p) + m~d(v sin B) = d[B + q(v) (41)

from which we obtain without difficulty

L1 m2 cos P sin P)vap = dB (42)


v(m2 cos p sin p)

If now we multiply both sides of equation 37 by L1, this equation takes
the form

a~ +[p +(v)] = Hia In e (44)

where H1 denotes the quantity

H (v~coseg a2)ml v2sin p cos P (~
k(k 1)v2


We! denote by R10 the value of B:1 at v = w, P = O. We have

10 1 (M2 -1)1/2
k(k 1):M2


Equation 44 may be rearranged in the following fashion

di a+ q(v)= KlOdIn -+ (H1 E10)d


Nov choose an arbitrary point S in the region of disturbed flow and lead
a characteristic of the first family through it. We denote the point of'
intersection of this characteristic vith the shock line by A (fig. 6).
Integrating both sides of equation 47 along the above characteristic from
point A to point S we obtain

- Ba -q~va)= 8100S In +

S3(q1- 810)d in

Ps + cp(vs)


where ps, vs> es denote, respectively, the values of B, v, 9 at
the point S and p,, v,, 63 denote the values of these quantities at
the point A. Taking account of equation (32) we have

va =w( + blia + b2Pa + b pg$ + b4Ba4 )

We introduce the quantity val defined with the help of the expansion 35
in the following fashion

val = v(1 + b1 a + b2 a2 + b '8a3 + b4'Pa4 + .)

(cf eq. (35) -Tr.)


NACA TK 1594 17T

By this definition of thet quantity val we~ have

Ps + rp Yal) = cP(w) (51)

With the help, of formulas 49, 50, and 51 we rearrange~the expression
a, + rpYa) in the following manner

Pa + cP(va) = Pa + CP(val) + 9(PVa) cplval)

= cpr) + cp(va) cp(val)

= 9(v) + 9'(w) (va V) -(val ')

tp"'(w) (va u Y#- (Val ')2] i

= q(w) + vy'(v) (b J b )paJ + (b4 b4')sdI +

wip"(v)bl(by b ')Ba + E5 (e)

Calculating cp'(w), cp"(w) we obtain

cp'(w) =1(55)

9 (() = 2b 2

where Bp denotes the value of p at the point p. Assuming that the
mean value theorem is applicable to the integral arising from the right-
hand side of equation (48), we easily find

IInstead, take a slightly more general assumption admitting the part
AS of the characteristic under consideration to be divided in the same
finite number of parts in such manner that on each part the mean value~
theorem can be applied to the integral under investigation.

NACA TM 1594

Using formulas 52, 53, and 54 we easily find

Ba + cp(va) = cP(w) --(bj

- b~')Ba3 +

Now pass a stream line through the point S and denote by P the point of
intersection of this line with the shock line. Since the stream function C
is constant along this line we have

9, = Bp (56]

where 8p denotes the value of 6 at the point p. Taking logatrithms of
both sides of equation (33) we obtain

In 0 3jP3 + 1 'p+ 3 'Pi + .

Since the values of the coefficients 14', 19', ...
in what follows, we shall not calculate the~m.

Using formulas (56) and (57) we easily see that

will not be needed


in ea 3(BP,- ba3)+ 24 '(Pp B ) + E

~(bs- bj')b4 'b4 8a4+E5

IJACA TM~ 13'94


(Hi 81)d In -e-= Z(Hy 10)(p p a,) E


(In consequence of this equation one must keep in mind that Kl' 10 = E ).
He re HI de~notes the~ value of HI1 at some point on the characteristic
uinder consideration between the points A and S.

Using relations (BB), (58), and (59) we write equation (48) in. the
following~ form

Ps + qp(vc) = cp(w) lb3 -(y b ')Pa3 +

1023g(Pp3 p, ) + H1034'(pp4 a 4) +

35(H1~ R10) ip paJ) + Sy


We denote by B the intersection of the characteristic of the first
family under consideration with the contour K. Applying formula (60) to
the point B (which is possible, since the point S was chosen arbitrarily)
we obtain

Pb +' Vb) = q(w) -(b3 b ')Pa$ + I-(b3 b ')

3 884+

H10 5 0o3 a 3) + lo34'(Po~ ak4) +

23(H1 Hio) 803 Ba3) + E


(b b ') -1 (b4 b4' ) pa+

(v2cos28 a2)m2 2sin p cos B

k(k 1)v2

NACA TM 1594

where Bb, vb denote, respectively, the vaus of a and v
~point B and SO denotes the3 value of B at the point 0.

We now proceed to the3 derivation of an expression for pa.
formula (60) WE? have

Ps + cP 's) = (P') +

From equation (62), using formula (35) we obtain

vs = (1 + blBs + b2Bs2) +E

at the




and denoting by m2s the value of m2
using formulas (19) and (63)

at the point S we obtain, by

mes = (M2 1)-1/2 k-

= eO + 2eliS + E2

b (M2 1)-2 s, + E2


Analagous to the derivation of equation (47), which holds on character-
istics of the first family, we may deri~ve equation

dp q(v) = H20d I 0- + (112 B[20)d InO


which is valid on characteristics of the second family. Bcere B2 denotes
the function defined as


Hf2 =

and H20 denotes the value of this function at a = 0, v = v.

HACA TMr 1594

Ilow pass a characteristic of the second family through the point S
and denote by Q its intersection with the contour K. Integrating both
sides of equation (65) along this characteristic from the point Q to the
point S we obtain


(H2 H20)d In


where P vq, 6q denote respective the values of P, v, 9 at the
point Q. Since the contour K is a stream line we have

Gg = (0)


where e( ) denotes the value of 9 a~t the point 0. Assuming that the
mean value theorem can be applied to the integral arising from the right-
hand side of equation (67) we easily find, with the aid of formulas (56),


Applying~ formula (62) at the point Q we have

Pg + 9(vq) = cp(w) + E


E~liminating cq(w) from equations (62) and (7O)
lowing equation

we arrive at the fol-


P, Pq (vs) I;- 979 = 2 I -In +

Ps-Pg`C'p(v,) m(vg~l= Ej

Ps P, +[rp(v,)-cp(vq~l=~j

the slope of the characteristic of the second family is greater than the
slope of the shock line at the point F.2

2It is easy to show that if the shock line is unbroken andJ moreover
condition (50) is satisfied the inequality P < O is impossible on this
linze. As a matter of fact, in the opposite case the shock line is broken
since with F < 0 condition (34) must be replaced by the~ following con-
dition in virtue of Tsemplen's theorem


Families (69) and (71) give


a, = Pq + 63

On the shock line we take an arbitrary point F (fig. 7) and pass
through it a characteristic of the second family in the region of the
disturbed flow and we denote by Pf, m2f, :respectively1, the values P
and m2 at the point F. Applying formua (64) at the point F we obtain


m2f = eO + 2el f j 2

We denote by '~ the slope of the tangent to the shock line at the
point F. From equations (74) we have

=r eO + el f + Eg

(7 )

Comparing formulas (73) and (74)
second family passing~ through F
make an infinitesimal angle with

we see that the characteristic of' thte
and the shock line a~t this intersection
each other moreover, if


Br > o

dy -e l 2p .
d = -(O -ep+er"""*

HIACA TM4 1394

Denioting by L the intersection of the characteristic of the second
family under consideration with the contour K and by x2 abscissa of this
point, we have


xZ = E

L~et PZ denote 'the value of p at the point L.
and (76) and MacLauren's formula we obtain

Using equations (5)

Pt = Pk(0) + Bk'(0)xZ + f


Applying7 formula (72) at the point F ve obtain


Bf = BI + E

As a consequence of equations (77) and (78)

Pc = Pk(0) + Pk'(0)xZ + E


Since the- point F was chosen arbitrarily on the shock line by use of
equitions (74), (76), and (79) we can obtain the following differential
Iequ~at~ion for the shock line

-eO + el k(0) + Eg


Conse~~qucntly the equation of the shock line may be written in the fol-


y = eO+ el k(0 x + E2

NACA TM 1394

Applying formlulats (64) and (72) to acn arbitrary point situated on the
characteristic of the second family FL weF easily obtain. the differential
equation of this line from the following form

-eO + 2el 3 + E


Employing formulas (76) and (77) this equation may be writer.

-eO + 2elpk(0) + E2


Consequentlyr the equation of the characteristic FL;may be written in the


where y- denotes the ordinate of the -point L.

On the other hand, taking account of formulas (3) and (76) we have

yZ =


tan Pk(x)dx = E

Emnploying: formulas (85) and (76) we may write equation (84) in. the form


Applying formulas (81) and (86) at the point F and denoting by xf, yf
the coordinates of this point we obtain

y =yZ R) + 2e~l k(0 (x xZ) + Ep

y = e(O +2li()x egx1 + E2

N~ACA ITMl 1594

Yf. =eO t elik(0)]* xy +


yf = eO + 2el k(0) xr egxy + E2

From equation (giT) we easily obtain


Replacing x2 in the right-hand side of equation (79) by the expression
in formua (88) we obtain


We denote by xa, y, -121e coordinates of the point A and by xb, yb
the coordinates of the point B. Applying formula (89) at the point A
we ar2rive at the following result

pa, =k(0) + elxaik(0)Bk'(0) + E, (90)

We now express xat in terms of xb. To this end, using formnulas (1 )
an~d (18), we write the differential equation for characteristics of the
seco~ndi family in the following fashion

eO 1 E (91)

26 NACA TM~Y 1394

Emplo)ying formula (91) we~ write the~ equation for the characteristic AB
of the~ first family in the form

Y = ;Yb eO(x xb) + E1 (92)

Taking account of formula (3) we have

ybJ x tan Pk(x)dx = El (95r)

consequently equation (92) may be written.

y = -eO(x xb) 1E (9 )

On the other hand, equation (81) for the shock line ma-y be written in
the form

y = eO x + el Si

and applying formulas (94) and (95) at the point A we obtain

ya = -eO(xa xb) + EI(6

ya = egxa + El

From equation (96~) we easily final

xa + El 97)

HAlrCA TM4 1594

And consequently

Pa = Pk(0) + 1e xbik(0)Pk'(0) + q (98)

Substiturting this expression for Pa in the right-hand side of equa-
tion (61) and substituting P0 for Pk(0) in the fundamental formula (5)
we obtain

Pb + cPvb) = cPY) 1 (b3 bj')Pki(0) -

-L(b4 b ') 2-tb12(bJ b3')j pk (0)-

Sel b -b 'v + 310 3 x k()Pk'( 0)+ E ~ 99)
2e00 bl

Alpi;loyig relation (35), we easily obtain from equation (99)

b b4' ~2(b1 b.') pk (0) + 'u(byb b ')Pk5(0)Pb +

be r-b'+ 1 bxk(0k() + 1)

28 N~ACA TM4 1594

Substituting vb, Pb, xb for v, Bk(x) and x, respectively~, in
formua (100) we arrive at the following final expression for the velocity
on the contour K:

v 1 + bl k(x) + b28k2(x) +- by' kJ(x) +b' k (x + b- )k50

b by'- ( b ') pk4 (0) +2(b3 b ')Pk3(0)8k~x) +

2 b3 b + HIfO }bl x k (0)Pk(0)0) + E5 (101)

We now proceed to the derivation of formulas from which the pressure
on the contour K can be calculated. Clearly

a2 = k pO(102)

and maoreover, on the contour K the following equation holds

AL= e(0) (1os)

Employing formulas ('1), (10), (21), (25), (102), and (105) we easily obtain
the following expression for the pressure on the contour K

1 k

P k-1 h-l k- k-1

lIACA TM4 1394

On the other hand, by virtue of equations (5) and (33) the following
equation holds

e(0)-1+op50)+ 4pk (0) + .. (105)

Substituting the expressions for v and obtained in formulas (101)
and (105), respectively, in the right-hand side of eggsation (104) we obtain
after elementary transformations the desired formula for the calculation
,of the pressure on the contour K

P = 0 +2 alk~x + gg2() +a30k(x)+ a4k (x) + ald k3(0) +

asd k () + a 4pk5(0)Sk(x) + aggPk5)(0)Pk'(0)x + Eg (106)

whe re

ald = -2(b3 b ')
k(k 1)M2

= M (M2 1)- + M2 S24 M

4b2 214
Sad =-2(b4 b') + (by b ')
bl k(k 1)M2

Sk + 1 + 3k 2k;2 2 -10 Sk + 6k2 kj M4
2 4 8

16 32


agd = (b3 b ') 2421 -21--- --

S6M2 1-5 jk + 1 7 + 2k 5ik2M
6 24

-4 + Sk+ 6r~ k2 kS 5- Tk 7k2 + k5
24 96

ag = e ~(bj b + .5103 bl)

(k + 1)2 -kk-
= M8(M2 1) + M2k + M
16 2 8

For x = 0 the formua (106) takes the form

P = Po + q Ial k(0) + app (0) + a 'DkJ(0) + ag'pk i(01 + Ej (107)

where a = a3 + ald, ag = a4 + a~d + a .

Formula (107) may be used for the calculation of the pressure on a
flat plate which is inclined at an angle Pk(0) to the undisturbed flow.

In order to single out of the right-hand side of equation (106) those
terns which depend exclusively on the presence of the shock in front of
the contour K, we add to the contour K under consideration an arc O'0
of finite length in such a manner that this are is tangent to K at the
point 0 and is parallel to the x-axis at O' (fig. 8). Since the flow
around such an additional contour is accomplished without the appe~iarace
of shocks (we suppose that the angle between the direction of flow and the
x-axis and the derivative of this angle with respect to x aire both
infinitely small), formua (29) may be employed in the calculation of the
pressure on this contour. Comparing formls (29) and (106~) and denoting
by Cpstoss the pressure resulting fran the presence of the shock front,
we obtain

NACA TM4 1394

CPstoss = q aldki(O) +a2d k l(0) + a~di5(0)8k(x) +

aggppi(0)p,'(o)x + eg (108)

We mnay, in turn single out of the expression for aCpstoss the term
depending solely on the vorticity caused by the presence of the shock.
In order to do this we add to the contour K(OC2I) a straight-line segment
tangent at the point 0 (segment O'O in fig. 9). With the contour O'OC2
a shock is formed, but the shock line O'Cl is straight so that vortex
formation is absent. Calculati~ng the pressure on the portion OC2 of the
contour O'OC2, we obtain

P. = 9 + q algg(x) + appk2(x) + a pk5(x) + aqpk (x) +

a.1dPk~(0) + a2d k(0) + a 4pk5(0)Pk(x ) + 6 (109)

Comparing formulas (106) and (109) and denoting by Cprot the pressure
due to vortex formation caused by the shock, we obtain

rot = g~aggk (0)Pk'(0)x + E5 (110)


We now. apply the results obtained to the calculation of the lifting
force and head resistance of a flat wing with sharp front and rear edges
placed in a supersonic stream having constant hydrodynamical elements.

We place the origin 0 at the front edge of the wing and arrange the
coordinate system so that the positive x-axis- corresponlds to the direction
of the velocity of the undisturbed flow and measure angles in the manner
used heretofore. Segment -OC2 connecting the front and rear edges (fig. 10)
will be called the chord of the wing as in the theory of wihgs. The length
of this curve will be denoted by T and the angle it makes with the x-axis
by P.

NACA TMu 13596

The form of the~ wings we are investigating is dlefined by a pair of
contours like that investigated in the preceding section, possessing a
pair of common points O, C2. Comparing ordinates of points on these
contours having the same abscissa, we call the upper contour Ku that
contour of which every point on the ordinate is greater t'han the corre-
sponding point on the ordinate of th other contour, Adhell we call the
lower contour K2. The function Pk(x) for the upper contour we denote
by Pku(x) and for the lower by 13kl(x).

We choose an arbitrary point A on the chord of the wing and denote
the distance OA by t. Through A we pass a straight line perpendicular
to the chord of the wing and denote by An and At, respectively, the
intersections of this straight line wit the~ upper and lower contours.
With the point Ag we associate a unit tangenrt vtector ti, and at the point At
a unit tangenrt vector t'l Thne vectors tu and t2 will be directed in such
a manner that their projections on the direction OC2 are positive. We
denote by Bu and P3, respectively, thne angles these vectors makre with
the vctor OC2. Clearly pu, Pt may be regarded as functions of t.
We denote by BuO and B20 the vaue of Pu and Pl at the point 0
and the values of the derivatives of pu and St with respect to t
at the point 0 by Bu0' and PZOr

The abscissa x of Au may be calculated from the formula

x = t cos p sin 5 tan pudt (111)

and that of the point At from the formula

x = t cos p ,sin tan Pgdt (112)

The value of the functions Pku(x) at the point Aul is determined by the

eku(x) = + pu

NACA 1TM 1394

and the value of the function PkZ(x.) at the point A2 by the relation

PkZ(x) = + pl P


Moreover we have the relations



tan Pudt = 0


tan P~dt = 0

We alssulme that and also pU and pZ and
respect to t are infinitesimal quantities.
(116:) we easily obtain

other derivatives with
From equations (115) and

+~ E~ 117)

+ E5 (118)

SgTudt =

u at


pgVdT I


Pzt = -

Procreeding now to the calculation of the lifting force and head
of th-e wing under consideration, we remark that on the top side
wing6 a shock appears when and only when

of the

S+ Puo > o


anld at the bottom side when and only when

S+ P10 < O


Sk NACA TMl 1594

We introduce the quantities alu, a2us a5u, a4a defined as follows

alu = &ld

a2u = a2d
aju = a54

abu = agg

alu = a2u = aju = abu = 0



In an analogous way we define' the quantities all, a223, aj~ &4,

all = ald

a;23 = a2d

a32 = a3d

a42 = aba

if S+ BzO < 0


if + B30 0 O


az2 = a23 = )3t = a42 = 0

Denoting by pu the pressure on the~ upper contour Ku and by p2
the pressure on the lower contour K2 we easily obtain, with the~ hel~p of
formulas (106), (121), (122), (123), (124)

Pu P 9 + 9 aliku(x) + a2 ku2(x) + a Pku(x) + aggku (x) +

aluPku(0) + a2uiku ()+auk50pux b u()k()

B + Bu0 > O

if B + Pz0 < 0

HApCA, Tpr 1396

P2 PO + -4 ~ 1kl(x) + a2 32(x) a Pk;#(x) + a Pk1 (x)

~alhIS(0) + a2l3 k/(0) + a 7Pk35(0)P k3(x) + a 3pk3(0)p3 '(0)xll(~x + E


LetP denote the resultant vector of the hydrodynanical force
acting on a unit length of the wing under consideration. We have then

P =pads


where n denotes a unit vector normal to the contour of the wing and
directed inwards.

We introduce the dimensionless coefficient of the lifting force Cy
and the dimensionless coefficient of head resistance Cx. These coeffi-
cients are defined by the formulas

cy 7


where Py and Px denote the projections of the vector P on the



xr- and y-axes, respectively. From formulas (127), (128), (129) we have

CY = 1T s udx + -r sop x


NACA TM! 1394

Pu tan Pk~u(x)dx S


pZ tan Pk3(x)dx


where 2 denotes the abscissa of the point C2. With the aid of formu-

and (131) after a few elementary transformations we obtain

Cy = Cyl + Cy2 + Cy3 + Cy4 + 9 (1J2)


= -

- 032 d



- altl@ + Pz30) +

Cy5 = (a 2a7)8 aluf@+ puo)

So (u2 +

+1 al




q5 8dt

l(al Sa3)

Cyk = -a2ulf + Buo)4 + a22($+ p2o)4 a~u ( + Buo)3 +

a uP(P + Bzo)5 _-t~ ak u + Buo)p + a1(P + BZo)3P Zo' +

as 6ay)(

(B~u2 pl )dt + P as la4)~ ju


(BuZ BIS)dt -


- 1 adt

u ,

NACA TM 1594


Cx = Cx2 + ox} + Cx4 + 65


SOT pu 312 at

Cx2 = 2alp +

- pl2 dt +OgT

C,~ =

cxb = 2a3

)T (u2

(u "

8 dt

~ p + aluB i~ uo)+ al P(

+ to)J +

+ a3



6a3 5

- 'I

u + p at


Let us consider a numerical example.


k =1.405,

Po = 1.o53~ kg/cm2

M = .5>


q 1.6~3) kg/cm2

al =2(M2- 1)-1/2 = 1789
a, = (M2 1)- (2 ME + 1.20544) = 2.2196
5 a = (M2 1)-7/2(1.533 BR + 4.oo8R~ 1.8194 +
0.4008M~8) = 5.082
ag = (M2 1)-5(o.6667 o.6667M2 + 5.616M4 3.824M6
2.96948 0.786aglo + 0.079924I12) = 8.290


(Equations continued on next ~page).

. 3d t +

NACA UK 1594

ala = 1.203M4(42- 1-7/2(-0.5553 + 0.26584 0.o3271M4) = 0.2766

ad = M (M2 1)(-5~1.203 + 1.51'7M2 0.645mbI + ook556M6 +

0.0485=J18) = 0.448

a~d = (g6~2 1)-5(-0.4oo8 0.oo25M2 + 0.386944C 0.1285M6) = 0.5318

akd = 0.3615M8(M2 1)-5(-1 + 0.7975M2 0.09813M4) = 0.9035

Let us take as the functions Pus St

Bu = -28 + -- llt

Bz = o


Moreover we assume that

p < (137)

The form and position of the profile~ of thre wing, determined by equa-
tions (136) and condition (157), is shown in figure 11. It is easily
seen that the straight line S182 drawn perpendicular to the wing through
its mid point is the axis of symmetry of the profile under consideration.
From equations (1-36) and condition (137) we have

S+ Bu0 = -P > 0

S+ Bzo = B< 0



NJACA TM~ 15394


alu = all =: ald = 0.2766

a2u = "23 = a2d = 0.4448

a~u = ap2 = azd = 0.5318

aku = an1 = age = 0.9035

(155), (135), (156), and (139) we obtain

42 2 i 1 6aj)8 +


Using formulas (152),

Cy = -2a~l

(lo 56 -
as2 s+25d+2 + E5

= -3.5788 3.061p2 14.327 82.7381 + >

10-~91~ 26 3 al 6 -4

= 5.965P + 9.1845p3 + 40.8p~ + ..


T = 100 em



NACA TM~ 1394


c, = 0.2936

Ox = 0.04168 12

Knowing Cy, Cx, T, and q, we easily obtain

Py = qTCy = 47.9 kg/cm 13

Pg = qTCx = 6;.81 kg/cn

In the~ present work the problem of a flow of stream of ideal gas
around a thin wing at small angles of attack is investigated, this stream
being supposed to be two-dimensionatl, stationary, supersonic and deprived
of heat-con~duction.

In the initial part of the work, the problem is stated, and the well-
known results obtained by Ackeret, Prandt1, and Busernann are cited. These
results, as known, are obtained on the basis of the potential supersonic
streams theory, which is founded on the existence of integrable combina-
tions of charracteristics of differential equations concerning this problem,
and in which some peculiarities of the dynamical conditions on the line of
the shock wave are utilized.

In, the second part the approximate solution of the problem is given
with an allowance for vortex-formation caused by the change of entropy
along the shock wave, when receding from the leading edge of the wing,
near which this shock wave is formed. For this purpose differential
equations of characteristics non admitting integrable combinations are
to be dealt with. The solution is obtained by means of' a special method,
which enables us to find the approximate int;-grable- co~mbination~s of dif-
ferential equations of the characteristics. Thie ob~tained combinations
let us receive the approximate formula of pressure in any point of' the
contour of the~ wing investigated. From this formla the term is easily
segregated depending exclusively on the vortex formuilation, caused by


the change of entropy along the shock wave. The characteristic dis-
tinction of this term of the obtained formula of pressure from. the other
ones, is that it includes the curvature of the wing contour at the leading
edge and the distance from this edge up to the element of the wing for
which the pressure is calculated.

In the third part of the work the expressions for lift and drag
coefficients of the wing are given, on the base of the formula of pres-
sure obtained above. In conclusion a numerical example is studied.

TraLnslated by R. Shaw
Institute of Mathematical Sciences


1. Meyer, Th.:l Uber zweidimensionale Bevegungsvorg~inge in einem Gas,
das mit Uberschallge schwindigkeit strijmt. Forsch. -Arb., Ing.-Wes.,
62, 1908.

2. Prandt1, L., and Busemann, A.: Nabrungsverfahren sur zeichnerischen
Ermittlung von ebenen Strijmungen mit ijbers challge schwindigkeiten.
Stodola Festschrif-t, Ziirich 1929.

3. Ackeret, I.: Gasdynamik. B~and~bu~ch der Ph~ysik, 19, B3. VII.

4. Busemann, A.: Aerodynamischer Auftrieb bei tjberschallge schwindigkeit.
Luftfabrtforschung, 1355.



Figure? 1.

MI I m


Ol~" nn1

NIACA TM 15394

Figure 2(a).

Figure 2(b).


Figure 3.

Figure 4.

/ \ c'


Figure 5.

Figure 6.

46 NACA TM 1394




Figure 7.


/ C2

Fig are 8.

NrACA TM 139 4


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Figure 9.





Figure 10.

NACA TM 1394





Figure 11.

NACA Langley Field, V1

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