Finite span wings in compressible flow

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Title:
Finite span wings in compressible flow
Series Title:
NACA TM
Physical Description:
130 p. : ill. ; 28 cm.
Language:
English
Creator:
Krasilʹshchikova, E. A
United States -- National Advisory Committee for Aeronautics
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NACA
Place of Publication:
Washington, D.C
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Subjects / Keywords:
Airplanes -- Wings -- Testing   ( lcsh )
Aerodynamics -- Research   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: Equations are developed using the source distribution method for the velocity potential function and pressure on thin wings in steady and unsteady motion. Closed form solutions are given for harmonically oscillating wings of general plan form including the effect of the wing wake. Some useful examples are presented in an appendix for arrow, semielliptical, and hexagonal plan form wings. The paper is essentially a summary of previous work by the author.
Bibliography:
Includes bibliographic references (p. 98-99).
Statement of Responsibility:
by E.A. Krasilshchikova.
General Note:
"From scientific records of the Moscow State University, Vol. 154, Mechanics No. 4, 1951, with appendix condensed from a document "Modern Problems of Mechanics," Govt. Pub. House of Tech. Theor. Literature, (Moscow, Leningrad) 1952."
General Note:
"Report date September 1956."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003807553
oclc - 127120673
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AA00009202:00001


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[~A~A ~M-17gj









NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMOjRANDUM 1383

FINITE SPAN WIlfGS IN COMPRE~SSIRLE FLOW*

By E. A. Krasilshchikova


This work is devoted to the study of the perturbations of an airstream
by the motion of a slender wing at supersonic speeds.

A survey of the work related to the theory of the compressible flow
around slender bodies was given in reference 14 by F. I. Frank and
E. A. Karpovich.

The first works in this direction were those of L. Prandtl (ref. 4)
and J. Ackeret (ref. 23) in which the simple problem of the steady motion
of an infinite span wing was studied. Borbely (ref. 25) considered the
two-dimensional problem of the harmonically-ojscillating norndeformable
wing in supersonic flow by using integrals of special types for solutions.

Schlichting (ref. 24) considered the particular problem of the flow
over two-dimensional rectangular and trapezoidal wings. To solve this
problem, he applied Prandtl's method of the acceleration potential which
he looked for in the form of a potential of a double layer. However, as
shown later, Schlichting made an error and arrived at an incorrect result.

In 194), Busemann (ref. 26) proposed the method of solving the prob-
lem of the conical flow over a body by starting from the homogeneous
solution of the wave equation. This method was modified by M. I. Gurevich
who, in references 11 and 12, solved a series of problems for arrow-shaped
and triangular wings when the flow, perturbed by the wing motion, is
conical. The work of E. A. Karpovich and F. I. Frankl (ref. 15) was
devoted entirely to the problem of the suction forces of arrow-shaped
wings.

In 1942, at a hydrodynamics seminar in Moscow University, Prof. L. I.
Sedov proposed the problem of the supersonic flow over slender wings of
finite span of arbitrary plan form.

In response to this proposal of L. I. Sedov, there appeared in
1946-47 a series of works by Soviet authors on the question of the super-
sonic flow over wings of finite span.

The first work in this direction was our candidate's dissertation
(ref. 5), in which we found the effective solution for a limited class

~Scientific Records of the Moscow State University, Vol. 154,
Mechanics No. 4, 1951, PP. 181-239.

The appendix represents a condensation made by the translator from a
document I'Modern Problems of Mechanics," Covt. Pub. House of Tech. Theor.
Literature, (Moscow, Leningrad) 1952, pp. 94-112.






NACA TM 1585


of harmonically-oscillating wings. In reference 6 we solved the problem
for wing influences by "tip effect." Later works refss. 15, 16, and 17)
were devoted to the same problem.

In reference 6, using an idea of L. I. Bedov as a basis, we reduced
the problem of the influence of the tip effect on harmonically-oscillating
wings to an integral equation.

The question of the flow over wings of finite span remained open for
some time.

At the start of 1947, there appeared works in which different methods
were proposed for solving the tip effect problem which would be applicable
to any particular wing plan forms. In reference 18, M. D. Khaskind and
S. V. Falkovich solved the problem, in the form of a series of special
functions, for a hannonically oscillating triangular wing. Later,
M. I. Gurevich generalized this method (ref. 19). In reference 20,
L. A. Galin reduced the problem of determining the velocity potential of
an oscillating wing to the problem of finding the steady-motion velocity
potential and gave a solution, in series, for the velocity potential of
a rectangular, oscillating wing cambered in the direction of the oncoming
stream.

The methods, proposed by different authors, for solving the problem
of the flow over wings of finite span do not permit the solution of the
problem for any finite-span wing and may only be applied to a limited
class of wings.

Parallel developments in this direction were made by the foreign
authors Puckett (ref. 21) and Von Karmin (ref. 22) who solved the problem
of the steady flow over finite-span, symmetricall wings at zero angle of
attack. As is known, such wings produce no "tip effect" and the study
of the perturbation of the airstream by their motion presents no mathe-
matical difficulties .

In references 6, 7, and 8 we proposed a method of solving the finite-
span wing problem by constructing and solving an integral equation which
considered the wing plan form in both steady motion and oscillating
harmonically. In reference 9 we generalized the problem to more general
forms of unsteady wing motion by the method of retarded source potentials.

Introducing characteristic coordinates we solved the integral equa-
tion for wings of arbitrary plan form and represented the solution for
steady wing-motion in quadratures and for the harmonically-oscillating
wing in a power series of the parameter defining the oscillation frequency.

The present work is a detailed explanation and further development
of ur apes (efs 6 o 9 whchwere published in the Doklady, Akad.










Nauk, USSR. In this work we propose an effective method of solving
aerodynamic problems of slender wings in supersonic flow.

All the results and problems explained in this paper were reported
by the author in 1947-48 to the USSR Mechanics Institute, V. A. Steklov
Mathematics Institute, Moscow University, etc.

In the first part of the work we find a class of solutions of the
wave equation, starting from which we obtain the solution to the problem
of determining the velocity potential of some wing plan form in unsteady
deforming motion. The obtained solution contains the solution of the
two-dimensional problem as a special case. In the same part of the work,
we solve in quadratures the problem of steady supersonic flow over a
wing of arbitrary surface and plan form. The effective solution for
wings of small span is similarly given. We obtain formulas determining
the pressure on the wing surface in the form of contour integrals and
integrals over the wing surface.

The author thanks L. I. Sedov for reading the manuscript.


PART Il


1. SETTING UP THE PROBLEM


1. Let us consider the motion of a thin slightly cambered wing at
a small angle of attack.

We will consider the basic motion of the wing to consist of an
advancing, rectilinear motion at the constant supersonic speed u. Let
be superposed on the basic motion, a small additional unsteady motion in
which the wing surface may be deformed.

Let us take the system of rectangular rectilinear coordinates Oxyz
moving forward with the fundamental wing velocity u. The Ox-axis is
directed opposite to the wing motion and we take the x,y-plane such that
the z coordinates of points on the wing shall be small (figs. I and 2).

We will consider the normal velocity component on both sides of the
wing surface to be given by

vn = AO + Alf~t + al (1.1)


LResults of Part I, sections 6 and 7 were found by the author in
May, 1947 at the Mathematics Institute, Akad. Nauk, USSR.


NACA TM 138j






NACA TM 1585


The first component defines the wing surface


(1.2)


AO = -up0


where B0 is the angle of attack of a wing element. The second compo-
nent defines the additional unsteady motion of the wing. The functions
AO and AL and a are considered given at each point of the wing
surface.

We will assume that the fluid motion is irrotational and that there
are no external forces.

The velocity potential of the perturbed stream cp(x,y,z,t) is
represented in the form


q(x,y,z,t) = c90(x,yiz) + 91(xiyiz,t)


(1.5)


where the potential (p0 corresponds to the basic steady motion of the
win and the potential cl corresponds to the additional unsteady motion.

Thus the projections of the velocity v of the fluid particles on
the moving Oxyz coordinates are determined by


"(0 1
Vy -- + ,
dx ox )


VY = + -,
by ey


z z b


The functions c0
first-order quantities
With these assumptions


and q? and their derivatives will be considered
and second-order quantities will be neglected.
it is known that the potential (l satisfies the


wave equation which in the moving axes is


a291
ax2


+ ,2a ~
y2


2 a429
+ a
az2


4291
at2


a2r1
S2u = 0
atdx


(a' u2)


(1.4)


and the potential cp0 satisfies


(,2 2) a2rp0
ax2


4 2 a2p0
a2


,2 42 0 0
dz2


(1.5)


where a is the speed of sound in the undisturbed stream.

A vortex surface, called the vortex sheet, trails from the side of
the wing surface opposite to its motf~on. Just as on the wing surface
the velocity potential undergoes a jump discontinuity on this sheet.






NACA T1M 138)


We represent the projection of the vortex sheet on the x,y-plane as the
semi-infinite strip Z1 (fig. 1) extending along the~ x-axis to infinity
from the trailing edge of the wing.

Let us establish the boundary conditions which the functions c90
and 91 satisfy.

Let us transfer the boundary conditions on the ving surface parallel
to the z-axis onto the projection E of the wing on. thie x,y-plane,
which is equivalent to neglecting second-order quantities in comarison
with first-order ones. Therefore on the basis of equation (1.1) we~ obtain
the streamline condition


= AO(x~y),
dz


d1=A (x'y)f t + a(x,y)
dz


(1.6)


which must be fulfilled on both the upper and lower sides of C.

The kinematic condition, which expresses the continuity of the normal
velocity components of the fluid particles, must be fufilled on the dis-
continuous surface of the velocity potential and on the vortex sheet.

We transfer the condition on the vortex sheet parallel to the z-axcis
onto its projection ZL on the x,y-plane which is again neglecting second-
order quantities. Therefore we have the conditions


(1.7)


to be fulfilled on El"

Furthermore, the dynamic condition which the potentials cp0 and
qg satisfy must be fulfilled on the vortex sheet.

Since the pressure remains continuous on crossing from one side of
the vortex sheet to the other, then from the Lagrange integral


P = -u -12


2


390 90
- = -
Oz z=+0 dz z=-0


91 91
- = -
bz z=+0 SE z=-0


2~(, IPSE






MACA TM 138J


Keeping equation (1.5) in mind and neglecting second-order quantities,
we obtain


z-t ,=0 1 +-O-a u + u (1.8)


which must also be fulfilled on C1-

After boundary conditions (1.6~) and (1.7) are established, we
correctly consider that, to the same degree of approximation, the surface
of discontinuity of the velocity potential the vortex surface lies
entirely within the x,y-plane. Therefore, the functions 9O and cq
are odd functions in z


(P0(x,yI-z) = -0(Fx'y~z), 91(x'y,-zrt) = -(Pl(x',yz,t) (1.9)


Combining equations (1.8) and (1.9) we conclude that the functions
CP0 and cp1 satisfy the respective conditions

--=0, + u on El (1.10)
ox it ax

Since th motion of the wing is supersonic, the medium is disturbed
only in the region bounded by the respective disturbance waves represent-
able by a surface enveloping the characteristic cones with vertices at
points of the wing contour. Ahead of this surface in front of the wing -
the medium is at rest, therefore, the velocity potential is a constant
whih we assume~ to be zero. Hence we have the condition on the disturb-
ance wave


D(POxIy~Z) = O, rp1(xiyiz,t) = 0 (1.11)


Th potentials g0 and (l are continuous functions everywhere
outside the tw dimensional region C + El and, as was established, are
odd in z, therefore, in the whole x,y-plane outside of the region E + 51
where the medium is perturbed, the following conditions are satisfied:


PO(x'y,0) = 0, 9 (x',y,,t) = 0


(1.12)









The region where equation (1.12) is satisfied is denoted in figure 1
by C2 and C~
Thus the considered hydrodynamic problem is reduced to the following
two boundary problems:

I. To find the function cPl(x,y,z,t) which satisfies equation (1.4)
and boundary conditions (1.6), (1.10), (1.11), and (1.12).

II. To find the function rp0(x,y,z) which satisfies equation (1.5)
and boundary conditions (1.6~), (1.10), (1.11), and (1.12).

Since the functions cp0 and 91are antisymmetric functions rela-
tive to the z = 0 plane, it is sufficient to solve the problem for the
upper half plane. From the solution of boun~dary problem I it is possible
to obtain the solution of II if the function f in the first be considered
a constant equal to unity, and AO replaces Al*


2. VELOCITY POTENTIAL OF A MOVING SOURCE WITH VARIABLE INTENSITY


1. Let us construct a solution of equation (1.4) as the retarded
potential of a source moving in a straight line with the constant velocity
u and having an intensity which varies with time according to fl i)*
Let us consider the infinite line along which, at each point from Left
to right, sources with velocity u start to function one after the other
with the variable intensity q = fO~ tl)fl(t). The law of variation
of the function f0 is the same for all the sources if the initial
moment of each source is considered to be the moment when it came into
being.2

The function fl has the same value for all the sources at each
instant. Let a source at an arbitrary point of the O'x'-axis be acting
at time tl fg ) The retarded potential of the velocity at the
point M as a result of such a system of sources is represented in the
fixed coordinates by


01 7 >y'z',t) = A r dtl



r = \(x' + ut)2 + ,2 + z.2 (2.1)

2Prandt1 (ref. j) considered an analogous problem with q = fO L t13-


MAC.A TM 158)






8 NACA TM 1583


where A is a constant with the dimensions of a velocity. The limits
of integration tl' and tl" take into account those sources which
affect M at time t. The origin of the fixed coordinates O' is
placed at the point at which the source started at t = 0.

Introducing the new variable of integration T = a(t tl) r and
transforming to the coordinate system x = x' + ut, y = y', z = z'
which is moving forward in a straight line with the velocity u, we
transform equation (2.1) into








If it is assumed that u > a then the velocity potential at M(x,y,z)
is the sum of the expressions (2.2), with the minus sign in front of the
radical taking into account the effect of the sources in the strip AC
on M a~nd with the plus sign taking into account the sources on CB. The
smaller root of the radicand is taken as the upper limit of integration
7l. It is easy to see that in this case both roots are real, positive
quantities (fig. 5).

On the basis of expression (2.2) we now construct a velocity potential
at M from the sources moving with speed u > a which have an intensity
.which varies with time as fl(t). The derivation remains valid if the
additive constant al is added to the argument t of the function fl*
Putting the sources at the origin, we find the velocity potential from
equation (2.2) bDy considering the interval of integration from 0 to rL
to be vanishingly small. Then, neglecting the term ( I) and putting

A o,1. 0 &7 = C where C is a constant, we obtain the desired solu-

tion for equation (1.4) in the general form


1 1 "u2 uxa,2 u2 a2 x2_u-1)y 2
9 ~(x,y,z,t) = C I
x2-2-12 422


flt a ux a x2 -u2a 1) y2 7
.iit a 'u2-' a2 u2 a2 g


(2.5)


x2 -(a 1 y2 + z2






MACA TM 1585


Let us note that each component of the arbitrary function fl as
well as the constant C and all in equation (2.3) is separately also
a solution of equation (1.4).

In equation (2.)) putting al = 0 and the velocity of motion of the
source u = O, we arrive at the well-known solution for a spherical wave.

If the velocity of motion of the source is u < a then to obtain
the retarded potential of a moving source the right side of equation (2.3)
must be limited to the first component.

Considering the function fl in equation (2.5) to be constant, we
arrive at the Prandt1 (ref. 3) solution for the retarded potential of a
moving source of constant intensity

Cl



2. It is possible to obtain, by the same method, the velocity
potential of a source with the variable intensity fl(t) moving
arbitrarily.

For example, in the case of rectilinear motion of the source when
the motion is given by X = Fl(t), Y = O, Z = 0an whn a
at
that is, the motion of the source is supersonic, the velocity potential
of the source at the origin of a coordinate system moving with the source
is






[x + F ( t) F~llt *- + y2 + z2 [x + F1(Z 1 Flt citldt1










a~t tl) lx+ Fl(t) Fl t1)2+ y2+ 2 = (2.5)





10 MACA TM 1585


If dbFl(t)/dtl < a, i.e., the source velocity is subsonic, then to obtain
th~e velocity potential one must be limited to the one component in equa-
tion (2.4) which corresponds to the smaller of the values of the parameters
tl and ty*.
Th function expressed by equation (2.4) satisfies the linear equa-
tion with variable coefficients


2
2


dFt a2p
2
dt axat


42F,( ) b
d2 ax


ax2


+ a2 I
a2


+a2
a ,2


(2.6)


If the source moves with constant acceleration as Fl(t) = -ut bt2
(where b is a constant) then equation (2.0) is an algebraic equation
of the fourth degree in ti with two real roots.

Formula (2.4) contains the Lienard-Weigert (ref. 27) formula as a
special case when the source intensity is constant.


j. DERIVATION OF TEE BASIC VELOCITY POTENTIAL FORMULA


1. We apply a solution of the form (2.3) of the wave equation (1.4)
to the above-mentioned boundary problem I.

At each point of the x,y-plane let us place sources with the poten-
tial 9*. Hence, we will consider C and al in equation (2.3) functions
of points of the x,y-plane and we will replace al by a and fl by f.

As a consequence of the linearity of equation (1.4), its solution
is a flunction cpl expressed by


u(x CI h X -~12 kC(3 91 IC-.2
f ;t c aiSr'll -111_ ~2 UI _.1 I
U~S~i x drpC +


plix,~,zri I /
c(x.~;il


f t+ ((n) uBx h1 V x )2 k2(y I- 2.,2; 1
' C(S,91 x uB 2 u g n(


(j.1)


where k = 1.






NACA TM 158)


The region of integration S(x,y,z) is that part of the x,y-plane
which lies within the characteristic fore-cone of equation (1.4) from
the point with coordinates x,y,z (fig. 4).
The solution of equation (3.1) will give the velocity potential
arising from the additional motion of the wing if C(x,y) is determined
from the boundary conditions of the problem on the x,y-plane.
Let us introduce the new variable of integration 9 into equa-
tion (3.1) in place of


(x F)2 k~z2cos 8


(3.5)


9 y -


Then equation (j.1) becomes


9 x


- 1 x 2 k2z2 cos


pl(xiyIz~t) =


sin 6 ddj +


(x )2 k2z


u(x 5)
.2 _2



S(x,yIz)


a
.2 2


cos 9j X


c scy


-


(x I)2 k~z2 cos 9 -


u2X a2 u 2 a2 V'-~aa


(3.4)


~~(;C1 iy


f~~~~ t x-[) `2cs-


f. t+ a ,y -






NACA TM: 1585


Let us note that for any point M(x,y,z)
to isolate from the region S(x,ys,z) a region
able of integration has the limits


of space it is possible
S' in which the vari-


x kz 6 5 1 C',


O 4 $


,1 = ( )2-k22 9 y x ) -kz


where C'
remaining
on z or


is a constant satisfying the inequality C' < x
region S S' the limits of integration either
depend on z only in the combination kz2.


-kz. In the
do not depend


Differentiating equation (3.4) with respect to a we find the rela-
tion between C(x,y) and a(x,y) and the normal derivative of the
velocity potential 891 d~z at any point of the x,y-plane


(35)


Corpa~ring equation (3.5) with equation (1.0) we conclude that on
the wing


C(x,y) =1 Al(x~y)


(3.6)


i.e., the function C(x,y) is given.

Therefore, the velocity potential pl may be computed from equa-
tion (3.1) by taking equation (3.6) into account for those points M(x,y,z)
of space for which the region of integration S(x,y,z) does not extend
beyond the limits of the wing.

If the leading and trailing edges of the wing are given by x = Jr(y)
and x= X1(y'), respectively, and if, therefore, JI and X1 satisfy


C(x,y) =R ft +afxiy -- z=0






NACA TM 1383


(3.7)



(3.8)


(where a* is the semi-vertex angle of the characteristic cone) on the
leading and trailing edges of the wing, respectively, then in particular,
equation (5.1) yields the effective solution of the problem of finding
the velocity potential pl everywhere on the wing surface because in
this case the region of integration S does not extend beyond the wing
for any point M(x,y,0) on it (fig. 5).

Also, in particular, equation (5.1) gives a solution of the plane
problems if C and a are considered as functions of one variable -
C = C(x) and a = a(x) and the variables of integration in the region
S are considered to vary between


0 [ x km


12 = Y -1 x ()2 -k2z2


< Y+ 1 (x-g2 k,2 1'I


(39)


where ql and 12 are as defined previously.

Considering f in equation (3.1) a constant
equation (3.5), we obtain the fundamental formula
tial cp0 specified by the basic steady motion of


and taking into account
for the velocity poten-
thre wing


PO(x,y,Z) = -


(3.10)


Formula (3.10) contains, as special cases, the results of Prandt1
(ref. 3), Ackeret (ref. 23), Schlichting (ref. 4) when thne wing surface
is a plane and when the leading edge is a straight line perpendicular to
the free stream.


<1 cot a*
dy


d;il 7)
< cot 2*
dy


S(x,y,1) z0 x )2- 22 kz



























n
d(g 2 k2 2 2z2


14 NACA TM 1383


4. HAIRMONIC OSCILLATIONS OF A WING


1. Let us turn. to the case when the additional motions of the wing
are harmonic oscillations, i.e., on the wing equation (1.6) is given as


acq,1, i a~),Ut + "(xtuy
---=RP. lxye


= R.P. A2(x,y)e


(4.1)


where A2(x,y) defines the amplitude and initial phase of the oscillations.
Using the obvious relation ei9 + e-i8 = 2 cos 6 and equation (5-5), the
basic formla for the velocity potential (3.1) is represented as


a lcosl x-(2 k(- k2_r o2 -kz


(q(x,y,s,t) = epx S bz s~3--


dqde (4.2)


where


u2 a2



u2 a2

Keeping the second inequality of equation (3.9) in mind, let us
compute the inner integral after which we obtain a solution of the prob-
lan for a wing of infinite span


cl(x,z,t) = epx x-kz z=0 e4 I~'O~ A L)2 k2z2 d



where IO is the Bessel function of zero order.

By means of eqaion. (4.5) the velocity potential may be computed at
those points of the x,z-plane for which the interval of integration on
the Oxe-axis does not extend beyond the wing, i.e., at those points of the






NACA TM 1858


x,z-plane not affected by the vertices trailing from the wing because

the function 01is given onl on the wing. In. order to copue the
oz
velocity potential at any point of the x,z-plane by equation (4.3) it is

necessary to determine 1, using eqution. (1.8), everywhere on the

Ox-axis outside the wing.

Let us express, by equation (4.3), the velocity potential 1j for
any points lying on the Ox-axis outside the wing, which, according to
equation (1.8), equals on the Ox-axis everywhere outside the wing


rp (x,t) = R.P. 91(E)ev(x-3)
where

iu


and 1 is the abscissa of the trailing edge. Thn we obtain the integral
equation



3x z=e IOJ -8)d = -k ~le-p 3 1z=0*1 ~ )



bzp

we solved such an integral equation. The inversion of equation (4.5) is


ar,1 z0-PX= dF*(x)+ x- dg.6
dz dx


where F" denotes the right side of equation (4.5)J the know function,
and where Il is the Bessel function of first order.

Therefore, keeping equation (4.6) in mind, we can calculate the
velocity potential at any point of the x,z;-plane by equation (4.3).






NACA TM 138)


The problem considered in this section was solved and explained
in refe~rence 5 from another Ipoint of view.


5. INLUN~CE OF THE TIP EEFECrT


1. To calculate the velocity potential according to equation (3.1)
and also through equation (3.10) or (4.2) for those points M(x,y,z) of
space for which the :region of integration S extends outside the limits
of the wing surface, it is necessary to determine the normal velocity

component everywhere in the region of integration Sfo h

boundry conditions of the problem on the z = O plane.

Let us consider the~ case when the region of integration S inter-
sects the wing surface and the region E3 lying outside the wing and
outside the region of the vortex system from the wing. Region ZE
(fig. 6) is part of the region 12 defined above. That is, let us con-
sider the case when the wing tips the arcs ED and E'D' of the wing
contour act on the point M(x,y,z) or so to speak, the influence of
the "tip effect" and not the influence of the vortex sheet trailing from
the wing surface.

The point E on th leading edge is defined so that condition (3.7)
is fulfilled to its left and violated to its right. The point E' is
similarly defined. The points D and D' are, respectively, the right-
most and leftmost points on the wing contour as shown in figure 6.

Let us conzstruct the integral equation for C(x,y), connected to1

by relation (j.5), in 2 .

Let us select the vel~ocity potential rl at any point N(x,y,0)
lying in C3 by means of equation (3.1), equal to zero everywhere in 2
according to equation (1.12). The region of integration S(x,y,0) is
divided into two parts, as shown in figure 7; the region s (x,y) is
that pat of the wing falling in the Mach fore-cone from N(x,y,0), and
the region o(x,y) is that part of Cj lying in the same fore-cone.
According to equation (3.6) C(x,y) is given La s. In aC(x,y) is
unknown. We therefore arrive at the integral equation which C(x,y)
satisfies in r" .






NACA TM 1385


a(x,y)


C!g,g)K(g,9;x~yyttddqdj = F(x,y,t)


(5.1)


where the kernel is


C~~u + (~l- a( u2 a 2I~ l2y-)j
(x ~2k2- g)2-k2_2


K =(E,?;x,ylt) =


t+ a(C,r) u(x 9) a (x -2 1)- 2 d
Lu2 -a2 u2 -2 a2 -



and the known function



F(x,y;t) = A~,)~~~~~~qj
2~stx,y)

If the characteristic coordinates are introduced


(5.2)


(5.5)


xl = x -x k(y yO)s


Y1 = x x0 + k(y yO),


Z1 = kz


(where x0 an~d yO may be any numbers) then integral equation (5.1) is
simplified and in some cases this integral equation is easily inverted
as will be shown below.


6. SOLUTION OF THE INTEGRAL EQUATION FOR A HARNDNICALLY OSCILL~ATING WING

1. If the additional motions of the wing are harmonic oscillations,
i.e., the condition on the wing is given in the form of (4.1), then
equation (5.1) becomes


C06[ \k-~TY-~dldC = F(x,y) (6.1)


IS 9(E,ll)
o(x,y)






NACA 554 1585


where the function 8(x,y) = ---. -p Laaa hrete nw

function is


F(x,y) =- SSA(S,1)co (x-)2-22d d(.2
(x 5) k2( 2 L
s(x y)


wher A(xy) = e-px.B in s. In order to solve this integral

equation we introduce the characteristic coordinates x1, 1, El with
origin at "'O" by means of the formula


(6.5)


X~l' = x y, = x + kyJ El = km


In thte ne~w coordinates the variables of integration in a will vary
between the limits


xE 1 = ~xl'/ I 1


(6* )


where yl = ~(x) is the equation of the wing tip
the wing contour in the transformed coordinates,
abscissa of E defined in section 5 in these same
Equation (6.1) is transformed to


- the are ED of
and xE is the
coordinates (fig. 8).


cos A xl 1 1 1)S'~;~T
xl 1 1 d(1d = FI(1 xl 1)
(6.5)


'xl_ Il eL5 Tl
x-E il, 1171




















cosh A xl 1 71 il)


NACA TM 1383

where the furnctionn


P(x1 71)
e2


61(xlrYl) =


and where the known function is


S

" xldl)


FL(xlY1y) =-


Al 1791)


d11 dS1





(c.)


s (xlfl)
e2


Al=
sl=0


Let us note that the normal velocity of the perturbed flow


a' /8z1 by


is related to


abz1


For brevity, the index "1" will be left
everywhere from now on.


off the independent variable


2. Let us look for a solution of equation (6.5) ia the form of the
power series


e(x,y;h) =
n=0


92n(xry) h2n


(~.7)


Into both sides of equation (6.5) let us introduce


cos A (x )(y rl) = -(-1)n (x ))ny In 2n


(6.8)






NACA TM 1585


Keeping the absolute convergence of equations (6.7) and (6.8) in
mind, we multiply them term by term with the result


e(S,rl)cos[A J(x e)7- ) ]


h2n -1(_)n-k 1(x 9) (y 4) n-k 82k(Sn)
k=0 2(n k)!


=
nw0


(6.9)


Substituting equations (6.7), (6.8) and (6.9) into equation (6.5)
the latter becomes


n=0


1
n-k-
2dyd


k=n
h2n
k=,0


-1 n-k 6 k ( ) ( )
2(naj k)l~x -i


1
n-
2
a-i dS


Sn+1
S(-1) h2nCG (x-) ( 9)
LG-0 (2n)1


= JlA(5,rl)
s(x,y)


(6.10)


Taking into account the uniform convergence of the
sides of equation (6.10) with respect to the variables
integrate tern by term


series in both
5 and we


n-k-1,
2


a k=n n-k xx
fX 2n"t (-1)
Id=0 k=0 2~(n -k)l S
xE r(~


2k, (e,r) ~x i) (Y 91


n+1 n-
12n (--1(2)./ A(i,) [(x- i) (y- 1)] 2 d t

s(x,y)


=


(6.11)


(y 1)






N~ACA TM 1385


In equation (6.11) equating coefficients in identical powers of h
we obtain the integral equation which the functions 02n(x~y) satisfy


x r


2n~g,9) = Fn(x,Y)


(6.12)


where


n-1 k
F (x,y) = fn(x,y) + f_" f (x,y)
k=0-


(6.15)


where, in its turn,


1


s(x,y)


1
n-k+1 n-k-
fk(x,y) = 2(1) x 'u 2k(ll (x-E(y-1 2y
E


(6.15)


from which the functions fr; are defined for k $ 0 and n > 0. Let
us note that the right side En(x,y) of equation (6.12) depends, for
92n,~ on the coefficients 02k but only for k ,,,.,-.Tee-
fore, if we find 90, 82 84,*** 82(n-1), then F,(x,y) is a knzown
function in the equation which the coefficient 92n in the general term
of series equation (6.7) satisfies. For n=0 the right side in equn-
tion (6.12)


F (x,y); = (x,y) =-
s(x,y)
is a knnown function of x and y.


A(S,rl) d?


(6.,16)


Let us solve equation (6.12) for 92n(x,y).






22 NACA TM 1383


The two dimensional tutegral equation (6.12) is equivalent to the
two homogeneous integral equations


x 92n
at = F,(xy)
x
E


(6.17)


and


rY
I


82n(5,'1)
dsl
JY ?


- B~(Sy)


(6.18)


each of which reduces to an Abel equation.

Using the inversion forrmula of the Abel integral equation and
observing that for any n functions Fn(xEJy) = 0 hence the solution of
equation (6.17) for the function 9 (x,y) is


d5


92n(x~y) = 1


Let us turn to equation (6.18). We denote the parameter 5 by x,
and again using the inversion formula for the Abel equation and kee ing
in mind that according to equation (6.19) the right side 92n[x,3(xl of
equation ( 6.18) for y = #(x) is different from zero, the solution of
equation (6.18) for 62n is


1 a2n x,9(x)~ 1
92n(x'y) +


82n?(x,4)
dB


r


(6.20)


Substituting in equation (6.20) in place of 02n(x,y) its vaue from
equation (6.19) we obtain the solution of equation (6.12) in the following
form:

92n(xIy) % ~,~j ag +
x2 .-~? xE ~


FnSt? S,)
x i)(y 4


x2 JxE JS(x)






NACA TM 1585


Thus, according to equation (6.21), we can evalae successively,
the coefficients 90, 82, 84>"** 82k, etc.

Formula (6.21) shows that all the coefficients (n=0,1,2,...) for

y = f(x), i.e., on the wing tip ED, become infinite as R-1/2 where R
is the distance of the point (x,y) from ED. Therefore, the velocity
of the perturbed stream becomes in~finite as the specified order on the
wing tips, approaching from outside the wing.

It is possible to represent the inversion (6.21) of (6.12) as


drl di


a2
92n (x,y)= ---
lr2 axay


- xE (r(x)


(6.22)


which can be confirmed without difficulty by direct differentiation with
respect to the parameter.

Therefore, the solutions of integral equation 16.5) are construted
in the form of the absolutely convergent series (6.7) for any val~ue of
the parameter h.


02n(x,y) are expanded in the series


The coefficients


8' (x,y;h)) = 2nxy)An
n=0


(6.23)


(x~y)
e in. I:


(fig. 6) lying off the wing to the left, from equations (6.21) or (6.22)
by replacing in the latter the function Jr(x) by e2(x) (where
y = 92(x) is the equation of the are E'D' of the wing contour -
the left wing tip) and interchang~e the role of the coordinates.


We find the function 9'(x,y) = ----~' z=






NACA TM 1585


5. Let
cones from
points El


us consider a
El and EI'
and EI' are


wing of small span. Let the characteristic
intersect the wing as shown in figure 9. The
defined just as are E and El in section 5.


Let us divide the x,y-plane where the medium is perturbed into the
regions SO, S1, S2, .., Sn>*-


The regionn Sn
istic aft-cones from


is the M-sh~aped region lying: within the character-
En and E (or within one of them) and outside


the characteristic aft-cones from En+1 and En+1'. In its turn, we
divide the part of the x,y-plane lying to the right and left of the wing
into the strips al, o2, *.. > n ***"d #1 *
ag', .., respectively. The strip on lies within the characteristic
aft-cone from n. Therefore, an and ag' are the parts of Sn lying
respectively to the right and to the left of the wing.

Let us return to the fundamental formula for the velocity potential,
equation (4.2), which is in the characteristic coordinates


e2


e 2


(x g)(y )- z2


-


In order to comrpute the velocity potential by means of this fornnula
in those parts of the space (or, in particular, on the wing surface) for
which the region of integration S(x,y,z) intersects the region Sn of


the x,ylplane, we must first determine 1 e 2


outside the


wing in the strips a 02'.. U, and arl', @2>* *nl *

respectively.


S(x,y,) z=0






NACA ?TM 1585


(x+y)i
891
Let us denote e 2
az

strips by 9, a(2) ,()


in the a 0'"""

B(n). and in al',


CI


(121 *


by 8', er(2)


"n' .


Let us construct the integral equation for 6(2)~

Let us express the velocity potential at the point N(x,y,0) in
by formula (6.24) which is equal to zero everywhere in the strips l
0r2, an correspondinglyy in al 42 ,.' "n

Let us divide the region of integration into the three parts
S = s + (T + a1'* as shown in figure 10.



The fuinction-- e 2 =A(x(,y) is given in six,y) on the
az (x+y)
wing;. In ol'+(x,y) of ol', the function azi e 2 = '(xgy)
determined by the solution of equation (6.23).


(x+yr)
we denote --e
az


by e(2)(x,y).


Th~en we arive~ a~t


In a(x,y)


the integral equation satisfied by a(2)


e ")(5,1) cos[x/(x a(y- )] d d= F( 2) (X~Y)


SS
~(xy>


(6;.25)





NACA TM 1585


where the limits of integration are bounded by xE = x and
~(5) 6 9 y and the known function F(2) is defined as


F 2) (x,Y) =- ASi(Z,1) co hIcry ~ ids d5 -
s(x,yr)

/ 9(%8)cos dCx ~--~ E)y ) aTn at (6.26)


We look for the solution of integral equation (6.25) in the form of
the power series

9(2)(x,y) =~ 6(2)(,y 2n (6.27)
n=0O 2n
MIoreover, by reasoning similarly to the preceding section we arrive
at an integral equation for the coefficient eg, in the general term
of series (6.27)


eg ) (tn) dq dE


n2)xy)


(6.28)


where


Fn 2(x,y) = Fn(x,y) + f(2k(,)

where, in its tur,


(6.29)


n-k+1
f(2)k(x,y) = (1 (9 x-
n C2(n k) I 2


-1

(6.j0)


Equation (6.28) differs from equation (6.12) o:Ely in the form of the
y(2) fuctiojn on the right side. Taking into account the condition on
nh onaJF~I ouino
the ounary 2)EIy) = O for any n=0, 1, 2,. thesouino
(6.28i) for 6~1 is obtained by using the solution (6j.21) or (6.22) of
(6.12) as a final formula if F,(2) replaces Fn in the latter. The


C)(y rl) n- k-






NACA TM 1583


function Fn(2)(x,y) depends on the coefficient 92k(2) where k=0,
1, 2 ., -1.Therefore, just as in the previous section, if the
92k(2) for k = 0,1,2, .. ., n-1 are already found, then Fn(2) in the
right side of (6.28) is a known quantity. Therefore, the functions
e0(2), 82(2)1 .., 8 2n(2), may be found successively.

Let us note that Fn 2, and therefore the coefficient 92n(2
depends only on the first n + 1 coefficients 80 r, 62' > *, *>2n'
of the series expansion of
(x~y)
e'(x,y) = --- e
coz

in 0J1

Reasoning in the same manner, we may find the values of 6 5),

9 ., 9(N). .. in Ir ag, ..,UN .. (corr-espondingly
8 (', 64, ., 9,(N), in "lr > 2: M7

Therefore, the velocity potential can be computed by equation (6.24)
at every point M(x,y,z) of the space for which the region S(x,y,z)
intersects any number of strips aN or UrN

All the results hold for the case when the wing tips are not given
by one equation y = #(x) but consist of curves given by- the equations
y = ek(x) k = 1, 2, .,m The same observation applies to the
leading edges E'E (or E1E ') of the wing. Therefore, in our problem
the wing contour may be piecewise smooth.

If the frequency of oscillation a> of the wing be put equal to zero
then the coefficients a0, 80 g), **, 80()... coincide with t
values of the derivatives as0/az in the strips als Q2, .., aN,.
respectively, for the steady motion of a wing when the streamline conldi-
tion (1.6) on the wing is given in the form

--= Al(Xry)






NACA SS( 1585


7. INFLUENCE OF THIE VORTEX SYSTEM FRON.THE WING FOR A KARMIONIC.MGWf

OSCILATTIVG WING

1. Let us consider the case wh~en the region of integration S(x,y,s)
in formula (4.2~) for the velocity potential intersects the vortex sheet
I1 as shown in figure 26(a) (see also fig. 11). That is, let us consider
the case when -,he trailing edge of the wing the are DTr of the wing
contour or, so to speak., the vortex sheet, acts on the point M(x,y,z)
of space.

Using condition (1.10) we determine b~l/az in the region R of
thie x,y-plane and shown in fig-ure 11.

Thle region 0I is off the wing within the characteristic aft-cone
from D and outside the characteristic cones from T. Therefore, R
is affected by the vortices trailing from the edge DTof the wing but
not :from Dr'T'. The region :i partially intersects the vortex sheet
C1'


Let us return to the characteristic coordinates xl, y1, El which
we introduced earlier by formula (6.5).

As before, for brevity we omit the subscript 1 from the independent
variables .

Condition (1.10) fulfilled on in the characteristic coordinates
'1
is

319 + ~ u +u 1 (7 .1)
at ax ay

From equation (7.1) it follows that the function

.m x+y
9.m 91l(x',y,,t)e u 2

remains constat everywhere on the vortex sheet along lines parallel to
the direction of the incoming stream, i.e., along vortex lines from the
wing.






NACA TM 1585


Since the velocity potential r1 = 0 everywhere in the x,y-plane
off the wing surface and the vortex sheet, then it may be verified that
ap possesses the specified property everywhere in SZ.

Let us construct the equation for the fiction


6(x,y) 0e


in R.

Let us express cpm at the arbitrary point N(x,y,0) lying in
by using the basic fonrmla for the velocity potential (6.24). We divide
the region of integration S into thrz~ee pars, a~s shown in. figure 12,
into s(x,y), a *(x,y) and cr(x,y). The regions s anda a~re
parts of the wing surface and E defined above, respectively, which
fall within theP characteristic fore-cone from. N(x,y,0). The region a is
the part of R in the same cone. The variables of integration in ar
vary' between xD 5 ( x and XE where x is the abscissa
of D and y = X(x) is the equation of the arc DT of the wing contour.
The expression obtained for cpe is differentiated in a direction parallel
to the velocity vector of the impinging streamn.

Therefore we arrive at the integro-di fferential equation, which 4
satisfies in




x_h~ x co[ x 5)(y ?)

a x 4 (1,1) C5h )( lldtl dS +





2 I~ x "y r cos(A/(x 5)(Y '1)1 r SEPXY 7





NACA TM 1585


where 1 2-a and the known function is
we


A(S,?)Kl(S,8;x,y;X) dp dS -


Qb(x,y) = L.


9(5,4)Kl(t,9;xy;A) di di -


A(S,9)Kl(S,9;x~y;A) d? dS -
y)


al(x,y)

012
s(x,


r" S
al(x~y)


(7.5)


9(S,a)Kl(S,9;x,y;A) da di


cos A) ~- 5)y-i 1)
x E)(y )


wJhere K1(5,rl; x,y;A) =


and the operator


- = a-+b,
aL ax. ay


The function 6 is determined from equation (6.7) of the


preceding section.
2. We will- look for a solution of equation (7.2) in the form of the
power series


4(x,y;A) = 42nB(x,y) h~n


(7.4)


Keeping in mind the absolute convergence of equation (7.4) and using
the expansion (6.8) for the cosine we obtain


6(5,9ih) cos 3/(x 4)(y 8


k=n

k=0


(ln -k
-1) 2k(e,9) (x e)(y rl) n-k
C2(n k)f 1


e-


(7.5)


S -Js(x,y)







NACA TM 1385 51


Substituting equation (7.5), (6.8), and (6;.9) into equation (7.2),
the latter becomes


n-k n-k- -=


k~n


a





ax s
n=0


at~ aS +


lo=n n-k n-k- 1
an (-1)) 1 2(c,l) Cx f)(y r) 2a dyd +


n-k nk 1:
(-1) d2k(S,9) (x 6)(Y 9)] n- dtl at
[2(n k)l 1


k~n
h2(n+1) r


(-1)n+ 2n A(,)( 4( )n d? di +
(2n)!


r 0-
s n=0


-t)(y- 9n-k-





- s9 n-2ad ag +



(x~~~~c )y-9nk


k= n n-k+1
h2n ZZ(-1) 92k(E,?) [(X
k=) [2(n k) I


+l n=0


(2) 1 ,,
n=0 2n.


2 h(n+1)
n-0


s




'1


h2(n+1) A(S,9) (x e) (y


k=n n-k+1
-= I( In- k) :2kt,)


(7.6)


Taking into account the uniform convergence of the series with
respect to 5 and ? in both sides of equation (7.6)J we integrate it
term by term. Then, keeping in mind, the uniform convergence of the3
obtained series with respect to xv and y which is also maintained after




















































(7.8)


32 NACA TM 1583


differentiation, we3 differentiate the specified series term by term with
respect to x andy. After these operations on both sides of the
obtained equation we equate coefficients in identical powers of A. There-
fore we3 arriv at thae in~tegro-differential equation which the coefficients
of equation (7.4) satisfy


IL rx


bx xD


3r




gr
XI)


2}' dg at
/(x )(y ))


2# >9~) =~ dSn(xiy)
L(x 5)(y 'I)


(7.7)


where


(-1)+ a
(2n)I BL


art dS
SA(r,8) +
8 x -6)y -1


Pn(x,y) =


(-1) n-5/2
'----~- AI') ~(r,l) (x 6)(y 1)
C2(n -k) 1


d? dS +


khn n-k+1

[~2(n k) 1
k=0


2d >4) (xr[ i)(y n) n-k-

92k~,9)(x I)y _4 2dTn dS +
1l~


kc=0


LI Ir/
a


n-k
(-1)
I
[2(n k)J E


n-k+1


1=


n-k-






MACA TM 1385


in which the last sum and also the terms in CI are defined for n>O.

Let us note that the right side, en, of equation (7.7) for 92n
contains terms with coefficients 42k but only fork= ,12 ...
n-1.

Let us transform equation (7.7). We integrate by parts with respect
to the first integral on the left side of equation (7.7), the second
by parts with respect to ?, afterward we differentiate with respect to
the parameters x and y, respectively. Equation (7.7) becomes

Ix Y 92n5(E,Tn) + 62n (5 1)
dn 15 = *n*(x,y) (7.9)


where

*~ 1 2(xDa'I)
n ,(x,y) = d'l +




1) at + 4(x,y)
(x ) y X(i ) (

Let us note that the first term in equation (7.10) of the right side
of equation (7.9) becomes infinite for x = x:D.

Let us return to expression (7.8) for egand separate out of it
the terms corresponding to the value k = n in the first sum the compo-
nent


a e~~n~g?
~dr dS = R



We integrate this integral by parts with respect to 5 keeping in
mind that the limits of integration in ol* are xE AR x(D and
e(5) 5. 9 y and that 62n(xE,Y) = 0. Then we differentiate with respect
to x





NACA TM 158.5


JxE 1 r
+.
xD e J~s


2#%>8rT )
y.


1I 6 2n( ~D,1) r
/xxD (xD) ~Y _9


(7.11)


Let us subject the desired function 4 in equation (7.2) to a sup-
plemtentar condition.
Let us assume that at the trailing edges the are DT (or D'T',
respectively) of the wing contour and on the straight line DD)*
(figs. 11 and 12) the intersection of the characteristic aft-cone from D
with the z=0 plane correspondinglyy the line D'Dlx) the velocity of
the perturbed flow, and therefore the function 4, is a continuous fune-
tion, then the conditions are fulfilled


6 [x,x(x)] = A x,i(x)

B~xD,y] = a [xD,Y


(7.12)

(7.15)


These conditions are analogous to the Joukowsky~ condition for flow around
a wing: by an incompressible fluid. From eqluatiojn (7.1-3) follows


1 r'-' d2n(xD ~
F-D~~~ J(x y-


~2n(xDI1


1 j
dTn =
Jx xD J(xD


(7.14)


since X(xD) = 9(xD)*

Substituting equations (7.11) and (7.1 ) in equation (7.10), the
latter becomes


92n /~.:(g)

x 8)y X(S


(7.LS)


.x "
xD


o i


62n 51


xE 1


-, Id


.p i i:;






NACA TM 1583


where


(7.16)


on' n h-R


For n = O, the right side in equation (7.9) is a known function of
x aind y


bOO=
xD


(cx S) y X(5)]


Ij dt -


a 0i9


1


3 T


d? dS
(x )(y )


A(5,11) d? at


014


(7.17)


Let us solve equation (7.9) for d2nx + 92ny*

The two-dimensional integral equation (7.9) is equivalent to two
homogeneous equations


xx 42n 7di = On*(x,y)
D x )


(7.18)


and


9 n(t,9) + 92i


Y


daT = B2n*(S~y)


(7.19)


each of which reduces to an Abel equation. Using the Abel inversion
formula we find the solutions of equations (7.18) and (7.19) as


i d 'X(5)d5





HIACA TM 1585


at

xD


ni '(~y
di


n*"(xD Y
y xD


4s2n (x~y) = -
X


(7.20)


42n@(,Y 7 B2ny(@,Y) =I~+S j()


(7.21)


rY
I
Jx(S)


1
+


Substituting equation (7.20) Into equation (7.21), first replacing
in the latter by x, we obtain the solution of equation (7.9) as


e n* xDX(x)
92nx(x,y) + 42ny(x~y) 2 gx~-Tj


1 x enS* EX(x)
dS +



1 r n{i(x, ?DT>
~x-XD-dx +


%c2 iD Xn5 (x) ( Cx 5)(y q)5( t d r

(7.22)
Integrating equation (7.22) along the straight line parallel to the
free-stream betwJeen the limits of ;N(x,y,0) and N(x,y,0) we find the
formula determining 42n in the gBneral forn of equation (7.4)





















ntl*(xD,rl) aT) dX1
xl x l!- -


MACA TM 1585


x xXx1
dxl
up xlxD xJ y-xXixl


92n(x,Y) = 42n(3T,g) + -
x2


x" x, en( g+C,X(l)a di xl
JXD l 5 I xl + Y x X(xl)


Sx rx1+yr-x
2I VX( xl


ont/ ^l(Sr


1-1 x xl y-x+yx
xx -D'(xl)


an at dx1

(7-23)


y are taken as solu-
the value of 42n(x,9i~:)
edge, then we find


If in equation (7.25) the coordinates
tions of y x +- x y = 0 and 2-1()
is determined from condition (7.12) on the
92n on the vortex sheet.


x and
= O and
trailing


If in the same fornaLa, the coordinates and y are set equal to
jI = xD and y7 = y x + xD and the value of 92n(jT,Y) is determined
from equation (7.15) on the line x = xD, then we find 32n outside the
vortex sheet in the region it affects.

Thus, through equation (7.23), we can compute successively th coef-
ficients 40, 92, *.. 2ns**

Therefore, the solution of equation (7.2) is constructed as the
absolutely convergent series (7.4) for any value of A.

The coefficients 42n' are expanded in the series


J'(x,y;h) = 42~n'(x,y)A2n
n=0,


(7.24)






NACA TM 1585


x~ p+y
The funtion 9' e 2 in (r' ig. 11) may~ be computed through

equation (7.23) if the function X2(x) replaces X(x) in it (where
y = X2(x) is the equation of D'T' of the wing contour) and we inter-
change the role of the coordinates.

3. Let us consider thie general case of the flow over an oscillating
wing by a supersonic stream. Let the characteristic aft-cones from El
and El' and Dl and Dqr intersect the winlg as shown in figure 13.
Then El (correspondingly El'), as shown above, are defined so that to
the left on the leading edge equation (3.7) is satisfied and to the right
it is not. The points D1 and DI' are, respectively, the most right
and left points on, the wing plan form.

Tlhe space of the considered wing plan form as transformed by equa-
tion (5.4) is illustrated. in figure 14.

Let us divide the x,y-plane where the medium is perturbed into a
series of regions: the :regions considered in thle preceding section,
SO, Sl>***>S>* *S and the regions nl, a2' "
Ch, The region Spj is the M~-shaped. region bounded downstream
by the intersection of the chlaracteristic cones from DI and D1' with
the a =- O plane. In the z = 0 plane, these lines are the upper
bounds of the region, of influence of thle trailing vortex sheet.

The region n~ is Ml-shaped lying between the characteristic cones
from Da, Dn', D,+1, Dn+11. We divide, in its turn, the part of the
x,y-:plane lying to thne right and left of the wing, respectively, into
thne ;trips al, a12> n *" defined above and into 81,
82 .. ,6n, .. and into al' a n** *"
defined above and 81" 621 *, ** n', correspondingly. The
strip 6n is that partt of Lh to the right and in' is the corresponding
part of Lh to the left of the wing. It is easy to see that the region a
defined at the beginnling of this section is in bl*

In order to solve completely the problem of the flow over the wing
shown in figures 13 and 14, the der,(~vative a891/az must be determined






N~ACA ~TM 183


in 81J s2, *> ;6n, and in 81' 621 ** 6n >**

( x+y)
Let us denote the function e 2 by4 2) g)

ti(n). .. and B', es(2), s~g(n), in the 817 62> **
Sn, and 811 6 21 s **En,. strips, respectively.

Applying equation (6.24) for the velocity potential we construct cp,
for any point N(x,y,0) in 82*

We: divid the? region of integration S which depends on the form of

the Function ----e 2 into the following: S = s + QE + al*' + S" + a,
as shown in figure 15. This functiLon is given in s. It was determline~d
in e*and o'lt in the~ preceding section by the soluions of equ,
tions (6.7l), (6.23), (6j.27), etc. In e4 i~t is determiined by the solu

tion of equation (7.24). We denote --- e 2 in a by 4(2).

Using the boundary conditions (1.10) and (1.12) we~ arrive at the~ integro-

differential equation which g(2 satisfies and which differs from equa-
tion (7.2) only in the form of the right side. On thet one hand the righ

side depends on the solutions 9, 9(2), (N), 6,I et(2)1..
e'(N) and on the other hand on the solutions 9'. We~ construct 9(2) in
the form of a power series in the parameter h.

Requiring the fulfillment of equations (7.12) and: (7.13) for 6(2)
we obtain for the coefficients 0() 2),... n(2)
an expansion in series of 6 2) of equations of the form (7.9) which
differ from each other in the form of the right sid~e.

The right side in the equation for the coefficient 9;tn(2) in the
general term of the series for 9(2) depends on the first n+1 coef-
ficients of the expansion of 6~i and 6' 1) where i takes all values
less than or equal to Na, and on the first n coefficients 4,2),
92(2), .. 2k(2) (k=0, 1, 2, .., n-1) of the series expansion of





NACA TM 1585


the desired function 9(2). Therefore, it is possible to find succes-
siel te oefiiets4(2)J 2(2) 42(2) using the solu-
tion (7.22) of (7.9) as a final formula if there is put in the latter,,
instead of cI,*~, right sides in the equations of the form of (7.9) for
the respective coefficients of the expansion of 42n(2),

By the same reasoning, values may be found of 4 5), 4 ,...
9(k), .. in 85> 64, kr

Therefore the velocity potential may be computed by equation (6.24)
at any point of the space perturbed by the motion of the wing shown in
figures 15 and 14., In particular, the velocity potential may be eval-
uated at any point of the wing surface.

All the results are valid when the contour of the wing is piecewise
smooth.

If the frequency of the oscillations of the wing, m, be put equal
to zero, then the coefficients 901 902) 900() o
cide, respectively, with the values of a90/8 in bl> 62> *
sk, for steady motion when the streamline condition (1.6) is given
on the wing as ac90/az = Al(x,y).

We apply the proposed method of determining ac1 he for the oscil-
Lating mo~ction, of a wing by constructing an integral equation, to wings
of competely arbitrary plan form. For example, the wing contour mayr not
be camibered bu may have the shape shown in figures 18, 24, etc.

In all cases, the part of the x,y-plane where acq1$z must be deter-
mined should be divided into the corresponding characteristic regions.
Thren successi-vely passing downstream from one region to another, construct
the integral and integ;ro-differential equations using the boundary condi-
tions on the x,y-plane. The solution of these equations for al1/32 or
for functions related to a(1/az is obtained as a series in even powers
of the :parameter A, which defines the frequency of oscillation. The
whole problem of determining the coefficients of the expansion reduces to
a double integral equation in each characteristic region. Each of the
equations after transformation appears to be an equation of the same type
which is solved by means of a double application of the inversion fonrula
for the Abel integral equation. The form of the wing contour is the limits
of integration. The influence on the considered region, of determining






NACA TM 1a38


the desired function in the preceding upstream characteristic region, is
reflected in the form of the function in the right side of the integral
equations.


8. FLOW AROjUND AN OSCIL;LATINGC WING OF NON-ZERO THICKH~ESS


1. Let us consider the motion of a thin wing at a small angle of
attack (fig. 10a).

Let the wing be moving forward in a straight line with the constant
supersonic velocity u. Let an additional small oscillating motion be
superposed on the basic motion of the wing so that the wing surface may
be deformed.

The normal velocity component on the upper surface of the wing will
be considered given by


Png = i~u(x,) + R.P. A2u(x~y)ei*t (8.1)


and on the lower surface by


-on = AOl(x~y) + R.P. A2u(x~y)e ir (8.2)


where Ab~u and AO 1 define the wing surfaces and

A~u = Alu(xIy)eiagl~xty) and A21 = Al3(xly)e ia(x,y) define the ampli-
tude and initial phases of the additional oscillating motion of the wing.
We consider the functions A~,Alu and amu given at each point of the
upper surface and AOZ> All, and al given on the lower surface. The
x,y,z coordinates were defined in section 1.

The velocity potential pp is


9Pp(x,y,zrt) = cF(xryIZ,t) + (Ps(x,y,z,t)


(8,5)






NACA TM 1383


The potential cp is specified by the motion of an oscillating wing
of zero thickness, which creates at each moment an antisymmetric flow with
respect to the x,y-plane (fig. 15b). The potential 9,is specified by
the motion of a. thin oscillating wing with a profile symmetric relative
to the x,y-plane. Therefore the motion proceeds in such a manner that at
each moment the wing surface wiLl be symmetric relative to a designated
plane (fig. 15c). Such a wing creates a symmnetric flow and cp, satisfies


Ps(x~y,-z,t) = q,(x,y,z,t)


(8.4)


Each of the potentials


Sand Wa is represented, in its turn, by


(85)


'P = P0 91 C


(8*6)


Q's ''* +0 1s


where cp0 and 0 correspond to the steady motion of the wing and rq
and cY1s correspond to the additional motion of the wing.

Let us set up the streamline condition using the representation (8.5)
for the velocity potential.

We transfer the boundary conditions on the wing surface parallel to
the Oz axis onto the projection C of the wing on the x,y-plane


Therefore, we obtain the streamline conditions based on equa-
tions (8.1) and (8.2)






NACA IN1 1)S3


==AOu(x,y) + R.P. A2u(x~y)eiwt


(8.7)


and



lz=-

which mut be satisfied on the upper and lower sides of C, respec-
tively.

Using equations (8.5) an (8.6) we establish boudr conditions
for the desired potentials cP0s (1, 'P0s, and 4)1s*

Keeping in mind that on the z=0 plane the normal derivatives of
the potentials 91bs and (1ls are specified by the syrmmetry of thre flow
over the wing satisfying the condition

Os Os1 1
(-~', = Ui 4s (8.9)
z "J=+0 z=- %=+ a =-


We find the boundary conditions for c90s and r1s which rmust be satis-
fied on the upper surface to be






where the functions rO and P2, are related to quantities given on the
wing surface through

A~u(x,y) A02(x'y) A2u(X,Y) A2 (X'y)
TO(x,y) = P2(x,y) =
2 2


(8.11)

on the lower surface of


R.P. r2(xty)e'ist(8.12)


The conditions to be satisfied by
j re

I dI a c = 0


z=+


cP0s and Q1s

1g~s -
az=






44 NACA TM 1838


Since the normal derivative of the potentials c'b and cq specified
by the antisymmetric flow over the wing, on the z=0 plane, satisfy


(8.13)


the boundary conditions which muI~st be satisfied simultaneously on the
upper and lower surfaces of I are


90 AO(xIy)
3z


1=R.P. A2(x~y)eiid
az


(8.14)


where AO and A2are related to quantities given on the wing through


A~u + AOL
AO =
2


A~u + A22
2


(8.15)


The boundary problems for c1(x,y,z,t) and c90(x,y,z) were set up
In section 1 where in the case of a harmonically oscillating wing, equa-
tion (8.14) rather than equation (1.6) should be taken on the wing. The
solution of these boundary problems is contained in the present work.

Let us formulate the boundary problems for c1s and r90s*

I. Find r91s(x, y~z,t) satisfying equation (1.4), condition (1.11) on
the disturbance wave, condition (8.10) on the plane region C and


1~s =
bz


(8.16)


everywhere in the x,;'-plane off


r where the medium is perturbed.


II. Find the function cpg(x,y,z) satisfying equation (1.5), condi-
tion (1.11) on the disturbance wave, condition (8.10) in the plane
region Z, and


Os
=o


(8.17)


everywhere off Z in the x,y-plane where the median is perturbed.


-=~ -- a z-








































s z=+0


NIACA ?TM 1385


Since thre potentials 01s and q are functions which are symmetric
relative to the x,y-plane, it is sufficient to solve the problem for the
upper half-space.

The solution of boundary problem I is given by equation (4.2). By
means of this formula it is possible to comp~ute the velocity poten-
tial 91s everywhere since in the case of symmetric flow over a wing the
derivative ac1s//8z is a given quantity for any point M(x,y,z) of the
space in the region of integration S(x,y,z). To compute c1s at M
according to equation (4.2) the function


SR.P. 02(x,y)eirnt


and integration is over that part of


the wing within the characteristic cone from M.

The solution of boundary problem II as is known refss. 21 and 22), is


is replaced by


= EO(IDxy) and integration is also over the region defined imme-


diately above.

If the ving is vibrating as a rigid body then the functions Auand
A~2 coincide and therefore, to solve the flow problem in this case, it is
sufficient in antisymmetric streams excited by the motion of an oscillating
wing with profile of zero thickness to superpose steady symmetric streams.


must be substituted for q


given by fonrmla (3.10) if the function 9b ap
azz0


r








PART IIS


To apply the integral equations method explained in Part I of the
present work, let us consider the problem of the flow over thin wings of
finite span in steady supersonic flow.

The velocity potential rp0 specified by the steady motion of the
wing may be computed through equation (3.10) at those points M(x141l,zl)
of the space for which the region of integration S(xldli,zl), already
known from Part I, does not extend outside the limits of the wing where

---- is given.

If a90/3zl appears to be unknown at any part of S, then, to use
equation (3.10) in these cases, where it has in the characteristic
coordinates 6t.3) the form


90(x ,yl~Zzl) li Sq d=



(21.1)

and to obtain the effective solution of the problem, it is necessary, first
of all, to find 890/az1 everywhere in S by constructing and solving
an integral equation.


1. INFLUENCE OF THE TIP EFFECT FOR STEADY WINIG MOTION


1. The integral equation (5.1) in the coordinates (6.5) is, for the
steady ving motion


1(xldQdqldi F(xlrYl) (21.2)

a(x'1 1

3The results of Part II, sections 1, 2, and 5 were completed in
April, 1948 at the Math. Inst. of the Acad. of Science, USSR.


NACA TM 1838






NACA TM 158)


where 81 is the value of a99/zl on E3 (fig. 6) and where the
known function is



F~14) -A(Elsil) (21.3)
s xl 7Y1)

The function A given on the wing is

a'P0 up0(x,y) _u xl +71 ;v1 xl (14
A x1,Y1) ; (21.4
azl k kl 2 2

It is easy to see that the velocity of the pertured flow --

normal to the x,y-plane is related to arPO/aZ4 through





The regions of integration in ar are x1E: 'i 51 x l and
~(El) 1 81 6 Y1 where, as before, yl = I#(xl) is the equation of the
wing tip ED in the transformed coordinates and X1E is the abscissa
of E in the same coordinates. The regions of integration for 91 in
s are the same limits x1E r 51 < Xl and ltl 1 91~ ~( 9 ) where
Yl1 = 1xl) is the equation of the leading edge E1E of the wing contour.

Let us note that equation (21.2) may also be obtained fronn equa-
tion (6.5) if the frequency a of the wing oscillation is set equl to
zero in it.

Let us delete the index "1" from the independent variables.

We solve the double integral equation (21.2) with respect to 65, by
means of a repeated application of the inversion formula for Abel's inte-
gral equation.





NACA TM 1838


We write equation (21.2) as


dll +\I) dy dI =


Jyrl


S~


J,"


1
Jx-~


(21.5)
This is an Abel equation with right side identically zero, therefore,
the brace equals zero for ( = x. Hence, equation (21.5) is equivalent to


dT) = -A~x,4) dn
(x), 1/y -?x) 1 (x)


(21.6)


which is also an Abel equation. Noting that the right side of' equa-
tion '21.6) is, generally speaking, different from zero for Y = #(x)
we find the~ solution using the well-known inversion formula for the
Abel equation


19-J(x)


01(x,Y) =1


iy
1 8
(x)/y ~-


i (x)
Aix,.)


(21.7)


I 21.7) for the steady motion of a wing may
(6.22) of equation (6.12) for the vibrating
frequency of oscillation as are both set


Let us note that the solution
be obtained from the solution
wing if the index n and the
equal to zero.


Carrying out the operations specified on the right side of equa-
tion (21.7) we find the solution of equation (21.2) to be


91(x~) = _1 1 -(x)
8L('Y)= f -y /I(x) 9()


A(x,l)' W~x dr
Y .


(21.8)


x 3: d y- = (x)


dn' dq






NACA TM 1383

a90
in a smlar manner, wJe find the value -- 1'(x,y) in Zjr
OZ


x (Y- r


RL'(x,y) = A((Y
Jx c-~Y 2 1(y) (


(21.9)


The functions x = 1(rly) and x = 92(y) are, respectively, the equa-
tions of the arcs ED and E'D' of the wing contour solved for x. The
solutions (21.8) and (21.9) show that the velocity of the perturbed stream,
when the arcs ED and E'D' are approached from off the wing, goes to


1
R 2 where R is the distance of N(x,y,0) from the points
(see fig. 7).


infinity as
ED or E'D'


2. Let us find the velocity potential according to equation (21.1) at
the point M(x,y,z) of space for which the region of integration S
intersects the wing surface T, and the region Ly or Zp'
The region of integration S in equation (21.1) is divided intor three
parts: S = sl + 2 + sO, as shown in figure 16


1 Aay dn d

BO+s2


rr dq at


rp0(x,y,z) =


(21.10)


The limits of region s1 are xE 5 ( xA and t (5)'1 & 22
here xA is the coordinate of the point A which is the intersection
of the characteristic forecone from M with the side edge ED of the
.ring. The equation rl = Y z2 x 5 is the equation of the hyperbola
in which the aforementioned cone intersects the z=0 plane. The limits
of region s2 are xE 5 E & xA and Jrl( *5? l()





x -


NACA TM 1585


Using equation (21.8), let us evaluate the integral over
equation (21.10)


s1 to


S1


drl l


(21.11)


. -


we interchange the order of integration of 1,


,2
t(g)-? Y-x-6
~-c i, (5)


I -1 ~1) Ag4


dq dq' dt


(21.12)


Tlhe result of the inner integration is


2
z
x-5


S2 T
x -


(21.15)


Putting the valu of equation (21.13) into equation (21.12) we obtain


I ~ ~ ~ = d5q)

(x -5)y -4) -z


(21.14)


22
x-6













(x l )y )


NACA TM~ 138)


Equating (21.116) and (21.11) we obtain


s1


"2 y(x 5)(y ?) z2


(21.1~5)


Therefore, to find the velocity potential, on the
tion (21.1), at a point M(x,y,z) projected onto
x,y-plane as shown in figure 16, it is sufficient


basis of equa-
M' (x,y,0) in the
to integrate over


A~~t,9) dn a (21.16)

"0O )y )-z


'P0(x,y,z) = 1
2J


The limits of region sO are ~1(() 1'1 y z2,/x- and
xA < Bweex is the abscissa of the poin-t of intersection of
the Mach forecone from M with the leading edge ErE.


The velocity potential
equation (21.16) by setting
integration to be xA E(


on the wing surface can be calculated fromn
=0in it and considering the region of
x and S (C) 6 1 y because the lines of


intersection of the characteristic forecone from M with the x,y-plane,
in this case, are the lines 5 = x and ? = y.

In order to compute the velocity potential at points of space, or in
particular, on the surface of the wing for which the region of inte-
gration S intersects simultaneously I~ and Z I; that is, at points
of space where there is felt the effect of both side edges ED and ErD',
it is sufficient to integrate equation (21.1) over the region
a = S + S e, the cross-hatched region in figure 17. Hence the integral
over S e in equation (21.1) must be taken with the opposite sign, i.e.,
the plus sign.






52 NACA TM 1383


3. L~et us consider the wing of more general forn shown in figure 18.
Let the forward part of the wing have the break, the are EGG'E1', in the
wig contour which affects the flow just as do the side edges.

Let us show how to compute the velocity potential at all points
M(x,y,z) of the space disturbed by the motion of the wing, which is not
affected by the trailing vortex sheet, in particular, on all points of the
wing surface.

We divide the wing surface into the characteristic regions shown in
figue 18.

If the region of integration S in equation (21.1) intersects
regions 2, 2', 3 and does not intersect 4 then the velocity potential
may be evaluated by using equation (21.161 (see figs. 16 and 1'1).

The simple result which is expressible by equation (21.16) does not
hold in the general case.

If S intersects 4 on the wing, in the curvilinear triangle K'4KL
then according to equation (21.1) ar0/os must first of all be found in
the triangle.

Le us express, by equation (21.1), the velocity potential at any
point of Kr'OlK as equal to zero everywhere outside the wing and the
vortex sheet from the wing, hence in K('O1K. Therefore, we arrive at an
integral equation of the form of (21.2) for the function
91*(x,y) = acFypz in K'01K but with a more complicated known function.

Applying the Abel inversion formula twice, we arrive at the solution
in the following final form:




01*(x,y) = 1 1



1 1 I k (~) 27
--_ dS
n /x- #2 1k)x 5 (21.17)


where = 1x i the equation of EG, y = 1l(x) is the equation of E'1
x = 7(y) of El'O' and x = gl2(Y) of E1IE.





NACA TMr 158)


Substituting equaittion~s (21.17), 2.) n 3 )it qa
tion (21.1) we obtain the formula orthe velocity potential at M which
has the projection M' shown on figure 18, and for which the region S
intersects 4 on the wing and, therefore, the region K'O1K outside the
wing, as


I
s*(x,y,z)


A~g~l)drl dE
A x E(y, 9) z


qq(x,y,z) =


tem g ySlx ) --
--$(c) 1 (x () y *(5) -


drl de +
z22


-- ]
ar2


(x ( )(y-1 2


A(Srl) tsn'i I[Fcl~
hx tl(r 1) 22 [Y2(1)


dE drl


~]l~y -I)[x r(?J]- z~~


(21.18)


where y = Y*(x) and x = rl*(y) are the equations of GG'
contour in terms of x and y, respectively.


of the wing


The region S" is the part of the wing shown cross-hatched in fig-
ure 18. The regions Sl*C and S2* are part of S* and are marked in
the same figure by horizontal stripes. The regions Sl* and S2* are
bounded downstream by lines parallel to the coordinate axes passing
through C and G'. The points G and C' are respectively the points
with the largest x and y coordinate on the arc EGG'El

By combining the results of equations (21.1) and (21.18) there is
found in the form of integrals taken over the wing surface, an effective
expression for the velocity potential at points of space for which S in
equation (21.1) intersects 5 or 6 on the wing and therefore a K'01K(
and and c31 off the wing.






NACA TM 1383


2. FLOW OVER WINGS OF SMALL SEAN



1. Let us assume that the characteristic cones from El and El'
intersect the wing as shown in figure 19. This occurs, for example,
for small span wings.

Let usdivide the x,y-plan~e where the mediumm1 is disturbed into the
regions SO S,.. ., Sn',,

The region Sn is an M-shaped region lying between the cha~racter-
istic cones from En and E'(or in one of them) and En+1
and En+1'. In its turn, we divide the part of the x,yr-plane to the
right and left of the wing into the strips ol, a2, .., ag, ..
and al', 02= ', ., respectively. The strip an lies
between the after cones from E, and En+1. Therefore, an is that
part of S, lyinlg to the right of the wing. The coordinates of E
and E' with their indices are shown in figure 19. The strip a,' is
similarly defined.

Let thle leading edge El'El be given as in part I, section 6, by
the equation y ()and the side edges E1En+1 and El n+1' by
y = Jr(x) and y = 2(!x), respectively, or as x = l(y) and x = JI2 7)
correspondingly.

To compute the velocity potential at M according to equation (21.1)
in that part of' space (ojr, in particular, on the wing surface) the region
of which intersects Sn of the x,y-plane but not Sn+1, we must first
of all determine agO/az off the wing in al, cr2, ,... n
and also in crl', 52 / .. n a**


We construct the integral equation for hp0/a5 in the arbitrary
strip ok.

Let us ex-press a velocity potential which is equal to zero every-
where off the wing an~d outside the region of influence of the vortex
system from the winlg, at U~ ofsithe ok strip (fig. 20) according to
th~e fundamental formula (21.1)






NrACA TM 1383


if by dr( di 0
S(x,y,0)


(22.1)


The limits of integration in 8 are xl < 5 5 x and y1 i tl I *
For convenience in. later writing, we make B a rectangle, which is pos-
sible since the medianm ahead of the wing is ntot distured and abla b
is zero. The region S is shown in figure 20 bouned by the lined ~LN,
rL1, L101 and OpL.


Let us denote aby 32 by


91 21 *>


kJ .


and 9311

2" ''


92 -s Sk, in the respective regions al'


... and al1


2 '>


k **


In confojrmance with this new notation we write equ~ation (22.1_) as


/,x


dyL +


drl +


1
Jx 5


k-1

(22.2)


i=k-2 rYi+1 eir S?
dqt +
i=1 Ci -


r7





56 NACA TM 138)


Applyi~ng the Abel inversion fomumLa twice to equation (22.2) we find
efor k 2


-4(x) A(x,1)/9(x) ?
anl +
2(x) 7 -


Bk(XI1 1


syi+1 ~ ~ ds +1(,)()-4
Fi r


i=1


-92(x) ek-Ir(x,9) Vx)- d
Yk-1 -T

(22.3)


Correspondingly, for







Gk'(x,y)=- fi~


we obtain


[ 2 7 )
F)


di +








k() k-


i=1- ~Xi+ da 2()-


(22.4)






NACA TM 138) jT


where the terms in equations (22.5) and (22.4) containing the summations
are defined only for k 5 .

If 61, 82r 8k-1 and therefore, 8l', 621 *~ ** k-/
are already defined in ol, 21, ., crk-1' then we can compute
8k in ok for any k by means of equation (22.5).

The value of a(Fg/az in al and crl' is determined by solving
equations (21.8) and (21.9).

The value of ac90/az in a2 is found from equation (22.5) by
putting k = 2:


1 1 9(x)
e,(x~y) = [


x gy (x )J 2 x)


A(x,rl) d'l +




R,11) at d'l
(y ?)(x 5)/x J2 1)

(22.5)


We find acyg/az in a2' in the same way


1 12
e,'(xJy) = _xT Y A(5,y) x d5 +


-t~(y) ,9(5)2 () -
1. 1 ~ J, A(ST,) d? dS


(22.6)


Thus, step by step we compute arpg/as in ok*






NACA TM 1583


Usihrg the soluion, of equation (22.i), we now prove the relation


= xl:2s _z a'xt da at



(22.7)


where xl*F and x2" are any numbers satisfying x1 < x2* 2
the coordinate of the point A shown in fig. 21), xl xl


For the proof, we write n in the equivalent form


xA. (xA is
< A*


22
[ x-
t(s)


/(x 8)(y 4) z2


a' = F
JX *


xl ~ 5 2(x g)( -r 4)


ai' (5,1) dl at x2 r2(5 $-L1 ,9) drl di


i=k-2 x 71+1


(22.8)






NACA TM 1383


where 8k in the first of the integrals is replaced by its value
according to equatio3 (22.5).

Then, we obtain


0 =- 1 2+ r~


A(S,g') 4) .q I+ dnl di -


ei I) 1'TTrl


i=l-
1
X i=1


x y

x1 1


I" drl' d5 -


-x2"
JX~L" JYk-l C/x-5


I" d?' dS t-


(


*


x, t )

xl 2


A( ) dq dS


i=k-2

i=1


x2 i+1
x~l 1


ei'(S,5) dq di

/(x S)(y 8) z2


x2 "-2

x2 Ik-1


ek-l'(t,9) dq di

/(:x )y ) 2


(22.9)


where I* denotes the integral (21.15) evaluated before. It is easy to
see that all the terms in the right side of equation (22.9) cancel in
pairs. Hence, equation (22.7) is proved.
It is also clear that the following holds


z2
ya Y2Jx----
71* xl


Zjz=0 J(x g)(y rl) z2


O







































at art


NACA TM 1583


where y and y2" are any ntanbers satisfying Y1 1 JT1 <2 Bg BY is the coordinate of B shown in fig. 21) .

Using equations (22.3) and (22.1+) it is possible to prove equa-
tions (22.11) and (22.12) correspondingly


Tx,*
xt*


drl at
kx S)(y ?) "2


132!zC tan-1


(x 5)(y ?) z2]y ( 1 dC d?


2 x2


5 and satisfies 1|(x1r~ y =Yxl


whe re y* maLy depend on


Jy2:


S*


drl dS


lten


t
2 .'`12
nit
Y1


~2('1)

~ Xi


(22.12)


where x* may depend on TI and satisfies #2 Y 1*) < x*C = x- ;L


The relations (22.10)
from equations (22.7) and
changed in the latter.


and (22.12) may be obtained, respectively,
(22.11) if the role of the coordinates is inter-


,- y* 8
-4(() bz 0


Sag a1,






NACA TM 1385 61


Let us note that the result of a single application of the
Abel inversion formula to equation (22.2) or directly to equation (22.1)
yields






Interchianging the role of the coordinates in equation (22.15) we obtain

jI' I J O (22.14)




It is possible to consider equations (22.15) and (22.14) as rela-
tions fulLfilled long the characteristic lines LII and LN in the
x,y-plane where y and x are, respectively, the coordinates of NT or
N' Lying off the wing and off the region of influence of the trailing
vortex system (fig. 20j). The points NI and Ii' lie to the right and
left of the wing, respectively. These relations can be usefull for compu-
tations.

2. Let us turn to ther fundamental formula (21.1). Using equations (22.7r)
(22.10), (22.11), and (22.12) we obtain, by calculation, the formula
for the velocity potential cpg at M(x(,y,z) for which S intersects
S, for any n > 0






NACA TM 1838


where th functions 01 and R2 are defined as


-1 l(x 5)(y rl) -z2 .B "(C)
01L = tan


1(x S)(y Jr)- z2 xA" 2rl

0L2 = tar
[(x- xA)(y -4) -z2 2]()1


and where the regions S~ and Se are regions of the wing marked on
figure 21. Th region Sl* is the vertically-striped region on the wing
surface. The region S2* is the horizontally- striped region of the wing
surface. It is clear that Sl* and S2* intersect each other and Sqp
on the wing.,

The region S1 lies off the wing and is vertically-striped in
figure 21. This region is the sum of the regions over which are taken the
integrals containing 8k' for k=1, 2, .., n-2 in equation (22.15).

The region S2 lies off the wing and is hori zontally- striped in
the figure. All the integrals are evaluated over it which together con-
tain 8k for 1,2 .. n.

If M is such that S in the basic formula intersects S, falling
La the characteristic cone from En and lying outside the cone from E'
then n must be replaced by nr-1 in the second sum and in the last term
of equation (22.15). If S falls inside the cone from Ene and lies
outside the cone from E, then n-1 must be substituted for n in the
first sun and the penultimate term of equation (22.15).

Let us note that the sums in equation (22.15) are defined for n > 3
and the last two terms in equation (22.15) for ny 3.






NACA TM 138)


If n=1, then the formula for the velocity potential in equa-
tion (22.19) is limited to the first two terms. This result was already
obtained before.

If n=2, the formula in equation (22.15) is limited to the first
four terms, the region of integration is shown in figure 22.

Thus, to evaluate the velocity potential, by equation (22.15), at
a point M(x,y,z) which has the projection Ml'(x,y,0) shown in fig-
ure 21, it is necessary, first of all, to compute Bk for k1 ,5
n-2 by equation (22.5) for k>2 and by equation (21.8) for
k=1 (8k1 correspondingly).

As an example we present the expression for the potential for n=3
in the expanded form


1 AE9d i
eb(x,y, z) =1 JJ (~iaid
2xB so/x )(y 9) 2


A(C,?)drl d5

8 (x- )(y ) 22


1


A(C,?)n2 da dS


Sl*


A(S,ri)na1 1 I

(x 4)(y 4) -2 ,2 82


---- an'd5 drl +






I 1
1 5dd


n 7r2 xl





(y- (5
3 x2 71


(22.16)






64 NACA TM~ 1383


The~ region of integration in the last two integrals over 5 and ?
are, respectively, thle regions S1 and S2, lying off the wing and shown
striped in figure 23.

Formula (22.1)) for the velocity potential contains an n-iterated
integral with the integrand an arbitrary given function on the wing:
#0/lbs = A(x,y).

In the general case, it is not possible to reduce the number of
iterations in the compu~utationur of equationU L (22.15) for arbIitray vinj-
tip shapes since the arbitrary functions y4, 2 and A all contain
the variables of integration. If the functions and $2 are fixed
t~hen th~e wing to be considered has completely determined tips and it
is easy to see that all the integrals in equation (22~.15) are reduced
to double integrals taken over the wing surface with integrands containing
thte arbitrary given function A(x,y) which defines the form of the wing
surface.

Let us turn to the wing of small span which has a break in its
leading edge as shown, for example, in figure 24.

The derivative a00/az may be evaluated in Gi andl up by equa-

tions (21.8) and (22.j). It is impossible to evaluate a O/az in 05
using equation (22.3) anrd, therefore, a surface-integral equation must
again be constructed which will also reduce to two Abel equations but
with more complex right sides than occurred for a3 in figure 19.

Hence, we note that it is impossible to construct one formula which
would determine a670/az for all cases, but a single method of solution
may be shown to depend on the wing plan form.

The formation of the surface-integral equation for 870/az is
explained above, for each characteristic region. Each of these equa-
tions is of the same tyrpe, reducing to two Abel equations with different
right sides in different cases. In particular, the right side of one
of the Abel equations, in some cases, may be identically zero.






NACA TM 138)


5. INFLUENCE OF THE VORTEX SYSTEM FROM THE WING FOR STEADY WING MIDTION


1. To study the influence on the air flow of the trailing vortex
system in steady motion, it is convenient to operate with the acceleration
potential 40 which, in linearized theory, is related to the velocity
potential derivatives in the characteristic coordinates through


0 = u 9x 90 (2).1)


Let us turn to the wing shown in figure 25. Let
M(x,y,0) on the wing surface, which lies between the
cones from D and D'. Therefore the trailing edge


us take a point
characteristic
D]T affects M.


Using equation (21.15) the velocity potential at M~ according to
equation (21.1) is


ssis0


A(S,rl) in at


1 e(S,?) dn dS

82rr~Lr-)Y


90(xly,0) =


(2~.2)


where the regions s = s1 + sO and s2 are shown in figure 2$. The
region s2 belongs to R, considered in section 7 of part I and shown
in figure 11. We denoted the derivative 890/az in R by 9 where
this derivative is an unknown.

We subject tf90/az to an additional condition, analogous to the


Kutta-Joukowsky incompressible-flow condition.
perturbation velocity potential at the trailing
and D'T' of the wing contour (figs. 11 or 25)
specified derivative, is a continuous function.
conditions are fulfilled:


We assume that the
edge the arcs D
- and therefore, the
Then the respective





NACA TM 1585


(25.5)


(23.4)
D~T and


In order to obtain the acceleration potential OO at M on the
wJing surface, we must take the derivative of equation (25.2) in a direc-
tion parallel to the oncoming stream. Before differentiating the double
integral with respect to x and y we integrate by parts in the first
case with respect to 5, in the second with respect to ?.

During these operations, we use equation (25.j) and the relation
(22.15) which is fulfilled along characteristic lines, and which on the
line DD" (fig. 25) is


X~xD) A D
fl(xD)


;XxD) Y I


(23.5)


We keep in mind, moreover, that the limits of
x:D 5 ( xA and X(5E)( '1 fl(5) where xD
XA. xA(y) is the~ abscissa of A, the limits
and t() yand finally the limits of


integration of s1 are
is the abscissa of D and
of sO are xA s2 are xD ( 5 ES xA and


After the specified operations, the results of differentiation are


1 ff- A (S,1)fr(1),) + An(S,1) d
sl+sO


s d'l dS -


P~x(x,y) + 44 (x,y) =


25.6)


AZ 8, I 5


21,


(23.6)


9[x,xj~x)] = A x,X(x)

B[x,:,(x)] = A[x,x2(x)]
where, as above, the function y = XIc(x) is the equation of
y = 2(x)is the equation of D'T' of the wing contour.


dS







NACA TMI 1358

where the arc 2 = RP is shown in figure 25. In order to evaluate the
acceleration potential 00 at M according to equation (2j.6) it is
first of all necessary to determine 63 + 4y in s2'

2, Let us construct the integral equation for 9, + d Let us
express the acceleration potential through equation (21.1) at an arbitrary
point N(x,y,0) outside the wing in n affected by the vortex sheet
trailing from the wing


44(x~yo) 1A(EB
2x d' di
s(xy)x- S)(y- .4)

a(x,y)
(23.7)

for which the limits of integration in a are xD ( 5 3 x and
7(( q y and in s, 5 varies between the same limits but rl
between 01(5) I rl X(S) (fig. 26).

Let us differentiate this expression in the free-stream direction.
Since, according to the condition((1.10) of part I) the velocity poten-
tial (p0 off the wing in the x,y-plane remains constant along lines in
the specified direction, then the left side of equation (2).7) goes to
zero as a result of differentiation an~d therefore we obtain


x xy ag)?






axaII d'l = O 2)8
by ~xD 1 ) /(x S)(y ?)





N~ACA TMI 1383


(25.6) by parts
respect to x.




-.1 xD) A xD


We integrate the first two integrals in equation
with respect to 5, after which we differentiate with
result is


The


X(~)


Sd'l +


x

xD


A(E,tl)
/(x~c' -5 g)y )


1
Ilx xD


x(g)


A(g,q) dly-l dg


x


1 ai


(25-9)


Sy 9 xD')
.I X xD) J rl


x
bxgD


rY
JX(5)


' d7 d5 =


x D


ix 5 yy~YSSr)


rx


1 a
~x 5 a~


(25.10)


Keeping equation (2).)) in mind, which is fulfilled on the characteristic
DD*n we substitue equtions (23.10) and (2J.9) into equation (25.8)
obtaining


d? +a I-


Ex


rY


') y--


A(5,1) d


x~)A(S,rl)
1() y 1


IT ,i)


drl +


(25.11)


-dt
9 -






NACA TM 1383

This equatipn is equivalent to


,X(x)

1i(x)



J1(x)


-


9(x,1)
dr



9(x,l)
Jy 1


A(x,rl)
dll +



A(x,rl) =0
dy O


(23.12)


according to the inversion of the Abel integral equation.
We integrate the last two integrals in equation (23.12) by parts
with respect to tl after which, as above, we differentiate with respect
to the parameter. Using equation (23.3) we arrive at


Bx(x,q) + 4 (x,q)
yt =


rX(x)

1~(x)


Ax(x,rl) + A l(x,9)


-y
X.x)


(23.13)


Let uts aply once again Abel's inversion formula, he~eping in. mind
that the right side of equation (23.13), generally speaking, is different
from zero for y = )((x) we obtain the solution for 4x, +y as

yx(x) F
O,(x,y) + b (x"y) = 1 1 Az. x,9) +



<~T(x) -l d 1l 1~ 1~ d 1l(x) j~)-1x
A 4 x @ X-x) dx 7- x)


(25.14)


A x, 1(x Sl(x)
1 --- ---
7 1(x)j dx






NJACA TMu 1585


Using equation (25.14) we prove


d? de =
s2 6(x,9 d)(y )


d9 dS
(x 4)(y ?)


s1


(x S) y 9 (SE


- [
21


(23.15)


where 21 = HQ. The regions s2 and sl are shown in figure 25.

Substituting equation (2).15) into equation (23.6) we obtain the
forml fo~r the acceleration potential


0g(x~Y)
n = 0Px + by =


A (5,9) + A (5,1)

(Ix i)(y 9)


1 s1

sO


dtl dS -


A E,el1~ i


(25.16)
where L = QP, the direction of the integration is shown by the arrows in
figure 25S.
Thus to evaluate the acceleration potential at M on a wing sur-
face two integrals, the surface integral over sO and the contour inte-
gral ove L of the leading edge are to be comIputed.
Let us turn to equation (25.12) and write it in the form


J,1(x)


1(x)


d 9


7 -


- 1


1 )3 d


be z=0


bz z=0






NACA TM 1583


Interchanging the role of the coordinates in equation (29.17) we!
obtain



a+ 0
ax 1y sz z=0, B (P)b =

(23.18)


where x = el(y) is the equation of E'E of the wing leading edge solved
for x in terms of y.

It is possible to consider equations (23.17) and (23.18) as rela-
tions which hold along characteristic lines in the x,y-plane where the vor-
tex sheet has effect.

Relation (23.17) is fulfilled along characteristic lines parallel
to the 0y;-axis (the line NQ on figure 26); the y-parameter is the
ordinate of a point lying off the wing to the right, in the effective
range of the vortex sheet (point N in fig. 26). Relation (23.18) is
fulfilled along lines parallel to the Ox-axis; the x parameter is the
abscissa of a point lying off the wing to the left.

If the point N is thus located to the right of the vortex line DH
or to the left of D'H', then along characteristic lines the respective
relations (22.13) and (22.14) also hold.

If N is located to the left of DH or to the right of D'H',
respectively, then relations (23.17) and (23.18) hold along characteristic
lines. In this case, equations (22.13) and (22.14) are not fulfilled,

In this section, we wrote down the transformation and obtained
the formula for the acceleration potential in the simplest case of the vor-
tex sheet affecting the flow.

For any other case, the potential 0~ is found in an analogous way.
In each case an integral equation is constructed for 9x + y. All the
integral equations are of the same type but with different right sides La
the different cases, and they are inverted by means of a double application
of the A~bel integral equation inversion formula.

In the following paragraph we present results defining the accelera-
tion potential OO at any point of a wing surface.






NACA TM 1585


3. Let us find the velocity potential 90O(x,y,z) at a point M
lying within the characteristic aft-cone from D and outside the charrac-
-teristic aft-cone from D'. The region of integration S in the funda-
mental formula (21.11 intersects the plane region ii (fig. 11) in this
case.

The projection M4' of M on the x,y-plane is shown in figure 26a.

Starting from condition (1.12) (of part 1) we express the derivative
ifr0/az for 3ny point where the velocity potential equals zero and where,
simultaneously, the effect of the vortex sheet is felt through the same
derivative at points located upstream on the same characteristic line
with the point studied. To do this we reason just as we did to obtain
formula (21.8). We then obtain the desired representation for the
derivative







390 1 1 .x'yD-xD OQ0(x'rl~z)x+yD-D
bz x-n~ _l D+xD(x) b z=0 j-


(25.19)

Using equation (23.19) it is easy to prove










-x2" ,E+YD-xD 80da d5,



(23.20)


by the same methods used in proving equation~ (21.15).






TIACA TM 138)


The limits firgaini equation (25.20), xl*~sto and x2*, are
any numbers satisfying xD xl* xF and xDj x2 X' xF where xF is
the coordinate of the point F shown in figure 2ba. Th~e point F is the
intersection of the vortex line DH, wh~ich~ has the equation
y =x + YD xD, with the characteristic cone fr~om the point with the
coordinates (x,y,z).

In particular, there holds



/J 3r00 dndi_ drl dS



(25.21)

where the regions S1 and. S2 are shown in figure 26a. The region S1
is marked with horizontal and the region S2 with vertical crosslines.

Keeping in miind equation (25.21) we obtain an expression for the
velocity potential at the point M1 defined above


1 / A(5,1)dy dS 1 (,d d

2x0/x-8(y-1 2 2 S' (~x S)(y 1) z

(25.22)

where SO and S' are spown on figure 20a.

Therefore, the region of integration S in equation (23.22) inter-
sects the wing surface only in that part of which lies to the left of
the vortex line DH.

Before evaluating the velocity potential by equation (23.22) it is
necessary to determine a'F0/a2 = in the region S' of R.

We find 4 from the solution (25.14) if the latter is integrated
in a free stream direction between N(x,y) and N(x,y). Hence in order
that the obtained expression correspond to the value of the deriva-
tive ac90/az = in .' to the left of DH, the coordinates r. and
on the vortex sheet should be taken as the solution of the equa-
tions yf 32 yD + xD = 0 and y' = :r(x) and the value of 9(ii,-) is
determined from equation (25.5) at the trailing edge.






NACA T1M 1838


If the E and coordinates are set equal to 2 = xD and
fi = y-x+xD and the value of 4(.3,f) is determined on DHI from the
solution of equation (21.8) then the obtained expression will correspond
to the value of acpg/az in R to the right of DH off the vortex sheet
but in its sphere of influence.


4. PRESSURE DISTRIBUTION ON A WING SURFACE


1. L~et us consider a wing of arbitrary plan form. Let the wing
contour in the characteristic coordinates be given by the following equn-
tions: The leading edge E'E by y= J(x) or x = ~1(y), the side
edges ED and E'D' by y = #(x) and y = 92(x) or x = #(y) and
x = Z(2y), the trailing edges DT' and D'T' by y = Xr(x) and
Y = X2Z(x) or x = ji(y) and x = 22 7)*

Let us find the pressure of the flow on the wing surface.

According to the Bernoulli integral, the pressure difference of the
flow above and below the wing is related to the acceleration potential 90


p(x,y) = pZ(xIY) pu(x,y) = 2peO(xiy) (24.1)

where p is the density of the undisturbed flow.

We divide the wing surface into the ten characteristic regions shown
in figures 27 and 28.


Ltet us express the stream pressure on the wing surface in each
characteristic region by the function A(x,y) which is given on the wing,
defining th shape of the surface.

We denote by M and M with a subscript the ends of line segments
parallel to the coordinate axes and lying in the x,y-plane. It is clear
that these segments are parts of the lines of intersection of the charae-
teristic cones, with vertices in the x,y-plane, and the x,y-plane itself.

Region I is the region where the tip effect is not felt. This part
of the wing lies ahead of the characteristic aft-cones with vertices at
E' and E.






NACA TM 1583


Region II is where the tip effect is felt but not the influence of
the trailing vortex sheet. This region lies between the characteristic
aft-cones from E' and E and D and D'. At M of region II, for
which the lines M1MrI and M2Mq intersect on the wing as shown on
figure 27, the pressure difference is


p(x,y)=up


D(S,?;x,y)d? dS + up


D(S,?;x,y)d? d5 +


-d yS) Y
L1


L


- 1


B 9(y),q;x,y dn


1--- B5 (x; dV'rI~i


(24.2)


where S1 is the region of the wing bounded by
NISand M12Mq S2 is the region bounded by


the lines MI~I,
M1yM M2Mq and


the


arc L = ~ZMk and where


Ag(~,?) + A?(~,?)


A(51T~)
B(S,?;xy) =


D(S,~;xy) =





NACA TM 1583


If the lines M1N and 2M2 do not intersect on the
in figure 28, then the pressure difference is


wing, as shown


d i15~, -


p(x,y) = u
82


D(S,9;x,y)d9 dS 1


u. 1 d r(y) B y)8xyd-




L2


(24.5)


wJhere S1 is bounded by the lines MMI> I1
L = MM .~-


IrZM M~g M2MA 4


and


Arrows in the figures show the direction of integration in the con-
tour integral and the integrals taken over the lines L1 = M5"I and
L2 = M4 2*

In region III, which lies between the characteristic cones from E
anrd the characteristic cones from E', D and D', the pressure differ-
ence is


P(x~Y) = US
81


D(g,9jx,y)drl dS -


B ~(y),?jx,y d'


(24.4)


The pressure difference in region III' is expressed in the same way.


.B ,tlE);~y -


B 8 1E);j 1 -d5


-p 1 a~yd; J)-

L2






NIACA TM 158)


p(x,y) = up D(g, ix~y)dl dC -


B 8,9 (S);xiy~l 1~(5 d5 -


dC~dx 1()~B 8, 2g(x) ;x, y di
L2


up


(24.5)


Region IV lies in the characteristic cones from E and E' and D
and outside the characteristic cone from D'. Region IV' is defined cor-
respondingl. At M(x,y) of region TV, when 1MIM and M2 14 intersect
on the wing, the pressure difference is


p(x,y) =p D(t,?;x,y)dy di
Sl


S2


u B E,t1(S);xIyI 1 d ll() -dS -


up id92(x) j~(, ~
X dx


(24.6;)


For the MI, for which M1M~ and M2Nq do not intersect on the wing,
the pressure difference is expressed by equation (24.5). Similarly, the
pressure difference for region I~V' is


u J D(E,9;x1)dy1 di + ufr(e,l;x~y~dy di +
SL S2


p(x,y) = -


d~l()
d5dS -


L


3 .


B E,9 (5)


(24.7)


;x,y] 1


B y,1xy G
L1






NACA TM 1385


if My@and M2 4 intersect on the wing. If these lines do not inter-
sect on the wing the pressure difference can be expressed by equa-
tion (24.4).

In region V, which lies within the characteristic cones from E, E',
D) and D)' where the influence of the trailing vortex sheet is felt, the
pressure difference is



p(x,y) = ]Dj(i,1;x~yldn di + DI(t,9;x.y~dy dr +
S1 S2

Bp I 8, 1(E);,y V1 dd RE(5 (24.8)



if MME and ML2Mq intersect on the wing, and



p(x,y) = -p SD(i,9;xly)dl dS ,9()xp drl(

51 L (24.9)

if they do not intersect.

In region VI, lying in the characteristic cones from E and D
and ouside the characteristic cones from E' and D' (also in
region VI' ) the pressure difference is expressed by equation (24.9). TPhe
pressure difference for rg~ Lon I has the same form.

Ths, if M, at which the pressure is desired, is in one of the
regions II, IV (IV' correspondingly), or V, as shown in the figures, then
to set up the regions and contours of integration in the pressure formulas
it is necessary to proceed as follows: Draw two lines MM1 and MM2
upstream from M to intersect with the side (or trailing) edges of the
ving. From these points of intersection M1 and Mr2 again draw lines
M1M3 and M2 4q upstream to intersect the leading edge E'E at M3 and


If M4 is in region III or VI (III' or VI' correspondingly) then from
M draw the lines N~y and MM1'i upstream; the line MMimmediately
intersects the leading edge E'E at MgZ; MMI1 intersects the side edge






NACA TM 138) 7


ED in the case of region III or the trailing edge DT' in the case of
region VI. From the point of intersection MIagain draw the line I01 53
to intersect the leading edge E'E.

Let us consider particular cases.

(I) Let the side edges of the wing ED and E'D' be straight
lines parallel to the free stream. In this case



by x


and, therefore, formulas (24.2) and (24.3) are simplified substantially;,
because the Last two terms in them become zero.

A particular wing of this class is the rectangular wing.

(II) Let the ving surface be such that


D(S,1;x,y) O


This holds, firstly, when the wing surface is a plane, i.e., the

function A = -uPO/k is given on the wing, where 80 is the angle of
attack, as a constant.

Secondly, this holds when the wing surface is linear, generally
speaking, uncambered, with generators lying in planes parallel to the
y = x-plane (x~z-plane in the original coordinates), then the derivative
of the function A(x,y) given on the wing satisfies the rela-
tion Ax = A In particular this is a wing with a cylindrical surface
formed in the manner described.

In these cases, only the contour integrals and the integrals over the
line segments L1 and L2 remain in the formulas for the pressure.

(III) The pressure formulas take an especially simple form when
the wing surface is such that the function D(S,rl;x,y) 50 on the wing,
at the same time as the side edges ED and E'D' are straight lines
parallel to the stream (combination of cases I and II). In this case, the
pressure difference above and below the wing in any region can be repre-
sented by


p(x,y) = ,1()xy 1- d di~ (24.10)
L 5






NACA TM 1585


where the~ plus sign, is taken if the lines MIM3 and M2M4 intersect on
the wing and the minus sign if these lines do not intersect on the wing.

B~ence, the pressure on the wing surface is expressed by the curvi-
linear integral. taken over the arc L of the wing leading edge.

(IV) Let t~he wing plan form be such that the points D and E
and E' and D' coincide. In this case, calculation of the pressure
on the wing surface is also simplified because there are no regions II,
III and III' on the wing. In particular, the trapezoidal wing belongs
to this case.

2. The pressure forrmulas show that there can exist a geometrical
locus F*(x,y) = O where the pressure on the wing p(x,y) = 0. Down-
stream of this geometrical locus, the pressure difference p = p2 pu
is negative.


For example, if D(t,?;x,y) 5 0
locus F* = O is found in the region
aeteristic cones with vertices E and
regions II and IV or through TV an
The first case occurs only when K;, th


on the wing then the geometrical
of the wing l\ying inside the char-
E' and passing through either
Id V or or lying entirely in V.
re intersections of' the lines 01K


and 02K parallel to the coordinate axes, appears to be outside the
region of influence of the vortex sheet, as shown in figure 27, for
example.. In all these cases, the points T and T' are on the geo-
metrical locus of F* = 0. The curve F* = O may also be shaped convex
downstream and not as shown on the figures.

Let us write t~he equation for the geometrical locus. Fw = 0.

In region II:


d 1E, d5


F*-(x,y) =


= 0


p Y)
/ 1


2 1 idibiL 1-~~) 2 1 -dt2(x) lx fl 02(x)
dy \I x (y) dx y Y \/ 2(x)


(24.11)






N~ACA TMl 138)


In region IV:


JC1(5)1 -


F*(xy) -


'j;i J


x-e[~= O


(24.12)


In region V:


F*(x,y) = 1 ~(x) Y.(y) = 0


(24.13)


If the side edges of the u~ing are
direction or the wing is such that E
inglyr) coincide, then F*F = O takes a
not changed, blut in regions II aLnd
of equations (24.11) aind (24.12)


lines paraLlel to the free stream
and D (E' and D' correspond-
Ssimplle form. In region V it is
IV, we haive, respectively, in place


(~24.1,4)


and


(24.1_5)


In all cases when the pressure difference on the wing, according to
equations (24.2) to (24.9), is expressed onily by means of curvilinear
integrals taken over L of the wing contour, it is easy to constrct the
zero-pressure curve graphically, keeping in mind that the zero-pressuree
curve in these cases is the geometrical locus of such points M on the
wing surface for which the points M1 and Mg on the wing contour coin-
cide. That is, the are on the leading edge over which the cuirvilinear
integral is taken shrinlks to a point.

We construct the zero-pressure curve as follows: From each pojint
NIO on the leading edge we draw the lines NfyI and 110 I2 parallel. to
the coordinate axes intersecting the side edges ED and E'D' as shown


1


F* = $1 2(x) r(y) = 0



F* = [2(x) ~Y) = O





NACA TM 1585


in figures 29 and 3O, or the trailing edges as shown in figures 31 and 32.
From the~ points of intersection N1 and N2 within the wing again we
draw lines N1N and 2NW" parallel to the coordinate axes. The geo-
metrical locus of NS*, where these lines intersect, is the desired zero-
pressure lin3e.

For exampn~le, for a asymmetric wing, if the side edges ED and E'D'
are parallel to the~ stream, the zero-pressure curve passes through G
and G' and is the line equidistant from the leading edge (fig. SO).
The? points G an are shown on figures 29 to 32. If E and D,
E' and D'", correspondinglyr, coincide and the trailing edges are straight
lines then F*n = O passes through C and 0' and. is the curve obtained
by inverting the leading edge 'Erelative to the center of inver-
sion. O*. The center 0* is the point of intersection of the trailing
edges (fig. 31).

If the wing is asynmmmetric and if the side edges ED and E'D' are
parallel to the free stream then the zero-pressure curve is the reflection
of the curve equidistant to thne leading edge and passing through 0 and
G', relative to the line equidistant from the side edges (fig. 29). If
the points E: and D, and also E' and. D', coincide and the trailing
edges are straight lines making identical angles with the stream then the
geometrical locus F*n = O is the reflection of the curve obtained by an
inversion, with center 0*, of the leading edge and passing through the
points G and G;' relative to the line equidistant from the side edges
(fifg. 32).

3. All the obtained results are generalized to the case when the
leading edge E'E is given not by one equation y = gl(x) but consists of
segments of smooth curves given. by y = $1k(x), where k=1 ,...
with n anyr integer. In such cases the surface and contour inteFr~als in
the formulas for the pressure should be divided into component parts for
the actual evaluations.

The side, ED and E'D', and trailing, DT' and D'T, edges may
also be piecewise smooth.

The same generalization holds for the previous three sections.

LC. All the results are generalized in the case of the asymmetric
flow over a ving which occurs, for example, in the motion of a yawed wing.

Let us consider a wing of arbitrary plan form with an angle of
yaw r as shown in figure 55-






NACA TN 158)


The pressure on the wing can be comp~uted by the same formulas if
the equation of the are EO'EO, in the coordinates transformed to the
origin O, is taken as the function y = fl(x).

The equation of EODO (corre spojnd i ngly EO'DO') is y = 4(x). In
this case EODO acts as the~ wing tip.

Finally~, for the trailing edge, DO 0, we have the equation y = X(x)
(correspondingly for DO'TOI)


). As is known, knowing the acceleration potential or the velocity
potential on the wing surface, we can easily compute the aerodyna-mic
forces on the wing.

In order, we represent the aerodynamic-force foruas using the adi-
ginal coordinate system shown in figures 1 and 2.

The lift P on the wing is


P = 2pS O0(x.y) dx d (24.16;)


where the region of integration in C is defined by Jr0(y) fx X X1 (y)
and yD' J Y 6 YD where x E O-r(y) is the equation of D'E'ED atnd
x = X1(y) is the equation of the trailing edge D'TT'D (figs. 217 and 28).
The limits yDL and yD are respectively the coordinates of D' and D)
of the ving.
Since according to linearized theory B0(x,y) = u by, bx then
integrating (24.16~) over x and keeping in mind that the velocity poten-
tial is zero on D'E'ED fran conditions (1.11) and (1.12) of part I,
the lift is

~P = 2pu W 1 b~ldy
D1.


If the trailing edge is piecevise smaooth~, then in actual computa-
tions the contour integral must be divided into its component parts.






NACA TM 1383


The expression for the moment M~by due to lift relative to the
Oy,-axis is



Mby = 2p jl O (x,y)x dx dy 2.)

The moments relative to the other axes have the same form.

6. The explained theory can be generalized to the case of the flow
over a tail or over a biplane in tandem.

We proceed as follows to obtain formulas to compute the pressure on
the tail taking into account the influence of the wing.

Express P0% + (P0y at M(x,yr) on the tail using the basic formula
(21.1). In the expression for q0x t 0y under the integral sign insert
4x y j on the vortex sheet. The function 9x y $ is found from the
Abel integral equation which is constructed by the method of section 5-

Ijn the case of flow over the tail the different characteristic
regions on the tail must be separated just as was done in figures 27 and
28 for the uniform motion over a wing.

Only in this case, to divide the tail surface into regions, there
must be taken into account, on the one hand, the wing effect and on the
other hand, the tip effect and also the effect of the vortex sheet of
the tail itself.






NACA TM 1J83


APPElfDIX


EXAMPLES


The following examples, solved by N. S. Burrow and M. M. Priluk,
will serve to illustrate the methods explained before.


A. Arrow-Shaped (or Svallowtail) Wing

Let us consider the arrow-shaped (or swallowtail) wing plan form
where the leading edges are formed by the segments AD and AD' and
the trailing edges by the segments DB and D'B as shown in figure 34,
Let the following geometric parameters be given: 61 the angle between
the leading edge and the free-stream direction; 62 the angle between
the trailing edge and the free-stream direction and 2 the wing semispan.


The equations of the wing leading
coordinates with origin at 0 are


edges in the


x,y characteristic


line AD

yl + o a+ 1a (1 cot a" tan 61)xl + 21 cot

line AD '




and the trailing edge equations are


line DB

yl =1 + o "tnB(1 cot aWe tan

line D'B


YL=1 cot ate tan 82 c a n


82)xl + 21 cot a*l




82 x 2 ota


where the angle am* is the semiapex angle of the characteristic cone.

Let us consider the wing for which 81 > a+ and 52 > at; that is,
a wing surface not affected by the trailing vortex sheet.






NA.3CA TM 1383


We will assume that the wing surface is a plane inclined by an
angle B0 to the free-stream direction. Thrfoe tedriaiv -

will be a constant everywhere on both sides of the wing. surface and will
bre given in the form


2 p0tn2 (Al)


In conformance with the method we divide the wing surface into the
three characteristic regions la, Ib, and Ic, with each region having its
own analytic characteristic solution and taking into account the angular
point A of the leading edge (fig. 34). Let us canpute the stream
pressure on the wing surface in each region.

Using the formula ($1.3), we find the pressure in the regions la and Ib,
lying outside the characteristic cone from A, to be

2u2pp0
p 2 u2pp0 tan a* (A2~)
lu2
---1 1
Sa2

This formula shows that the pressure in regions la and Ib, is a constant.

In region Ic, lying inside the characteristic cone from A, we find,
by using the same formula, the pressure to be

2u2FPpO tan 81 2 o *tn5 o l-x
p(x,y) = coi tn 1 b2 tan-lOlt LC4t; + ~ 1-X
cot2a* an2 l n1 + cot a* tan 61 y1 I cot 61


p. tan~-1 1 + cot a* tan 61 y 1 cot 61 -x (Ag)
1 cot a* tan 61 71- cot 81 x



















x + y cot a-n
x '1 cot 81j


2u12pp0 tan 61
x
Jco;t2 tan2 51 -

-1 1I cot a" tan 81 1 cit
1 -tan
n 1 + cot (z" tan 61 y cot


I


NACA TM 1585


In the original coordinate system shotin in figures 34 and ~5, (AS)
becomes


p(x,y) =


61 x + y cojt rz4
a" + x 1 cot 61


2 -1 1 + cot a" tan 81 co 1-
-tan i
x V1 cot a" tan 61 y cot a* +


(A4)


These formulas show that the pressure is constant along each ray from A
in region Ic.


figures 56 and j7, respectively, are the pressures along a
parallel to the y-axtis and along the section A2B2 pa~r-


Shown in
section AlBy


allel to the x-axis.

The lift P of the considered wing is

2u2 02tan 62cta(;lbll tan- 82 2 1 ot a*tan 81 1+
tan 82 ot2 n 61 tan 61 -1 cot a* a 8

2.ta 51 tan82 tn-1cot a" tan s1 + 1

xr tan 81 + tan 62 Vcot a+ tan 61 1


-1 lcot a*" tan 62 1
tan
cot a'A tanz 82 + 1


4 tan5 62
" tan 81 1802 61 ta2 62)


(A5)


The lift coefficient Cz is

480 tan 81 2 1 o an
C, = 1acn~2- tan-1 Icta
z K cot a" tan

co2 tan 1 tan 2 gl n' 1 ~ -4 ~
2 tan 61 tan 62 -1cot a+ tan 51 + 1

16B0 tan2 62 tan-1
rc(tan 61 + tan 62) ct~2 a tn g-1


61 1
81 + 1


cot a* tan 82-1
Vcot a+ tan 82 + 1


(A6)






















































See the work of M. I. Gureyic'h: On the Lift of an Arrow-Sha~ped
Wing in Surpersonic Flow. Prik. Mate. Nekb., Vol. X, No. 4, 1946.


NACA TM 1383


As is well known, the wave drag coefficient C, is related to the lift
coefficient through Cx = POC,.

Let us consider particular cases of (A6). In the limit as 61 y>~
we obtain for the triangular iin7g

C, = 4P0 tan a" (A7)

the well known result for the lift coefficient of a triangle.

Comparing (A6i) and (A7) we conclude that for identical wing speeds
and identical angles of attack the lift coefficient of the arrow-shaped
wing exceeds the lift coefficient of the triangular wing.

In the particular case when 62 = 61; we obtain the infinite span
arrow-shaped wing. In the limit as 52' 1 (A6;) yields


4P0 tanl 81
C,
z cot2 a* tan2 61 1

This result shows that the lift coefficient of an infinite span arrow-
shaped wing equals the lift coefficient of an infinite span slipping
wing with slip angle 6L.

Formula (A6) shows that with increasing 61 and 62, the angles
between the leading and trailing edges and the free stream, respectively,
the wing lift coefficient decreases. The dependence of Cz for an
a~rrow-sh-aped wing on 61 and 62 is shown in figures j8 and 59.


B. Semielliptic Wing

Let us consider the wing plan form which is half an ellipse as shown
in figure 40. Let the semiaxis al and bl of the ellipse be given.
Let us assume that the wing moves, as shown in the figure, in the direc-
tion of the axis of symmetry.







NACA ?TM 1838


The equation of the leading edge, the line D'D, in characteristic
coordinates with origin at 0 is


yl = 1

and the trailing edge equation in these same coordinates is


71 a2 bl2 cot2 aw~xl r 2alb1 cot aX a 2 + bl2 cot2 aX x12
al l cot2 ase

In the original x,y coordinates the trailing edge equation is

b \al2 x2
y = + I (Bl)
a

The plus sign relates to the are CD of the ellipse and the minus sign
to the aeC CD'.

Let us assume that the wing surface is a plane inclined at an angle
"i90
BO to the free-stream direction, therefore the normal derivative -
oz1
as given by (Al).

Let us consider the flow around the semietlipise when the cha~racter-
istic cones from D and D' intersect on the wing surface. In con-
formance with the method we divide the wing surface into the four
regions I, VI, VI', and V. Region I is outside the characteristic cones
from D and D', hence the vortex sheet trailing from the wing exerts
no effect here. Region VI is within the characteristic cone from D
but outside the cone from D'. Conversely, VI' is within the cone
from D' and outside the cone from D. Region V, however, falls within
both the characteristic cones from D and D'.

Using the formulas, we compute the pressure in each region on the
wing surface. The pressure in I is constant everywhere and expressed
by (A2). In VI the pressure distribution in the x,y coordinates is
given by

p = u2 pO tan a" x

1-2 sin-1 cot aX B1y + B2 1 + 2albl cot a* 31 ~12 B2





NACA TM 138)


where


B1 = al2 + bl2 cot2 a+s f1 = x + y cot a"


B2 = al2 b12 cot2 a*

Similarly for region VIr. The pressure distribution in V is

2u2 0g tan a+
p(x,y) = x


-1 cot a? Bly' + B2 1 + 2alb1 cot a? IBI 12



1i- cot at Bly + B2 2 2albl cot a* IBI f22
sin- (B3)


where f2 = x y cot a* and Bl, B2, and fl are as defined in (B2).
Graphs of the pressure distributions along the respective sections AlBI
and A2B2 parallel to the y-axils are given in figures 41 and 42 and
along the corresponding segments A3B and A4B4 parallel to the
x-axis are shown in figures 45 an~d 44. Spanwise section lines A BL
and A2B2 a~re shown in figure 45; whereas chordwise section lines 3B3
and A4B4 are shown in figure 4O.

If the seTmiaxis of the ellipse are given in a special way; namely,
if there exists between the semiaxes the relation al = bl cot a", then
forml (~B2) for the pressure distribution in region VI simplifies,
becoming


p(x,y) = u2 Otan a* 2_ sin-1 .,, 2Tx ctx~

(B4)

This corresponds to the case where the characteristic cones with apexes
at D) and D' intersect the wingstrailing edge on the axis of symmetry
of the wing at; the point C; consequently the region V on the wing now
vanishes.














































al2 b cot2 a

cot a* l+ bl2 cot2 a,


NACA TM 1585


In the general case for the flow around a sermielliptical wing, it
may be shown that on the surface of the wing in region V, there exists
a certain curve along which the pressure difference beaten the upper
and lower surfaces of the wing reduces to zero. Downstream, fran this
curve on the surface of the wing the pressure difference becomes nega-
tive. We find the equation for this line of zero pressure by equating
the right side of (BS) to zero.



l2+b2cot 2 ,i t 1 2 -bl cot 2 2 4al2bl12 cot2 ,*x]2


cot2 a*2 4al1bl2 cojt4 + 16a bl32 cot6 a* y2


= 4al~bl2 cot2 a' al2 bl2 cot2 dx2 (al2 + b 2 cot2 a*)


After obvious transformations, we
in the following final form


represent the desired geometric locus


y2
+ -1
b22


(BS)


where


2alb1 cot G*
a =
2 a2 + bl cot2 ~A*


b2 -


(B6')


al82 bl2






NACA TM 1585


These results show. that the3 zero-pressure line is the are of an ellipse
with semiaxes a2 ndb2 related through (B6) to the semiaxes al
and b1 of the are of the ellipse which is the wing trailing edge. The
directions of the semiaxes a2 and b2 coincide with those of the semi-
axes al and bl. In order that the zero-pressure line should not pass
through the wing surface, the elliptical are forming the trailing edge
of the wing should not have a real point of intersection with (BS), which
determines the zero-pressure line. Comparing (Bl) and (B5) we obtain the
following result. In order that the zero-pressure line, of a plane wing
of se~mielliptic plan formn moving at the supersonic speed u, should not
pass through the~ wig surface, it is necessary and sufficient that the
geometric parameters of the wing satisfy the condition




al 5 35 b cot a* (BT)





Constructed in figure 46 is an isometric view of the pressure on a
semielliptic wing in the general case when (B7) is not fulfilled and
there exist the regions I, VI, VI', V on the wing.


C. Hexagonal Wing

Let us consider the wing of hexagonal plan forml shown in figure 47.
Let the leading edges be the lines 081, and 081', the side edges ElD
and ED'parallel to the free stream, and the trailing edges DB and
D'B3. In characteristic-coordinate space, the wing has plan fornn as shown
in figue 48.,

Let us assign the following geometric parameters: a --the angle
the leading edge makes with the free stream; 7 --the angle the trailing
edge makes with the free stream; 2 semispan and h chard.
Let us consider that wing for which a > a*, 7 > a*. The first
:inequality me~ans that the wing surface extends outside of the character-
istic cone from 0. The second inequality means that the wing surface
is outside the sphere of influence of the trailing vortex sheet.






NACA TM 15385


The equaEtions of the lines forming the uing contours are: the
line~ 0El


y = x ta~n a


or in characteristic coordinates


7T1 = E x~l


where


1 cot 0.* tan a
m-
1 + cet a.* tg g


b > a,*; the line OE1'


here m < O, since


y = x tan a and yl = mxl


the line E1D


y = 1 and yl = xl + 2 cot a+2


the line E1'D'


y = I and yl = xl 2 cot a'c3


the line DB

y =- xtan 7 + h tan r and yg= xl+n





tan .1


NACA TM 1385


and finally D)'B


y = x tan r h tan r and yl mlxl + n2


:the re


2h cot a* tan r
nl =
1 + cot a" tan r


1 + co"V a" tan 7
1 cot a" tan~ 7


2h cot a* tan r
n2 =
1 cot a* to' 7


In conformance with the method we divide the wing surface into the
13 characteristic regions shown in figure 48.

Assuming that thie surface of the wing is a plane, we give the stream-
line condition in the form (Al) and we compute the pressure in each
characteristic region. We produce below the results of computing the
pressure on th wing surface as formulas already transformed back to the
original coordinate system.

Th pressure in la and Ib is constant and expressed by (A2).
In le the pressure is

2upp 1 x-cta* y
p(x,y) = c\rmco t2LP + tan-1 __ .
MV ITcota* 2 I- nm x + cot a* y


-1 -cot a* yl
tan- r-n f x- m o X(1
Yx +- cot a*


Hence it follows that the pressure is constant along each ray starting
from 0 in Ic. In IIIa


2u2po (1 m)
p x,y) =
n (- m cot a* .


2 cot a* (1 Y)


(C2)


(m 1)(x +- cot a" y) + 22 cot a*














~


2m cot a*(Y 1)
Y(1 m)(x + cot OL* Y) +- 2ml cot a*t


2u 2pp0(m ) i a-1 1 x-co *y
p(x,y)=tn -
Jrf\fm cot a* JC x + cot a y


2 cot a*(l y)


(1 m)(x + cot af y) + 2E cot a2*


x cot a* y
tan-1 (-IE
x + cot a* y


(03)


In IIIc


2u2 O(1 m)
It {-cTE cot ar*


-1 2m cot a"(y 2)
ttan
(1 )(x + cot a*" y) + 2mZ cot a*


(c4)


In IIa.


tan-1


2
2u pPO0 m)
p(x,y) =
r J-m cot a*


tan- 1 (1 m)(x cot ar* y) 21 cot at 1
r ~2 cot cr*(l + y)


(CS)


NACA TM 138)


In IIIb


-


ta-1 i





I


.I


S(1 m)(x +- cot a* y) 21 cot a*


-1 1(1 m)(x + cot at y) 21 cot a+
tan-
2 cot a+(l + y)



tn1 x-cot a* y
tan-~- mx + cojt a* y


(1 m)(x cot a*y)~ + 2 cot (z*


-1 2 cot a*(l y)
tan' ;
(1 m)(x + cot a* y) + 2m? cot a*


96


In It


NACA TM 1585


2u
p~~y=2u pBO(1 m) -1n~ 1 Ix cot 2* y
n {1~ E cot a* (- mYx + cot a2* y


2 cot a*C(2 y)


tan-1


(C6)


In IIe


2
2u FpB(m 1) 1
p(x,y) =I tan-
i 1J--mcot a*


1


1
-'


2 cot a"(2 + y)


1 1 x" cot a* y
tan"
(7 m Vx + cot a* y


x cot a" y
x + cot a* y


+ tani II m


(C7)


Formulas for the pressure distribution on the wing surface in
regions IIIa', IIIb', IIIc', and IIa' may be obtained from (C2), (C3),
and (CSf), :respectively, if coordinates appropriate to the specific
regions are chosen,

The formulas for the pressure show that there is a zero-pressure
line on the wing surface, downstream of which the pressure difference
below and above the wing becomes negative. This line is formed of the
two segments KN and K'the equations of which are


(C4 );







NACA TM 1J83


y = x tan 8 22 tan aA tan a y = x tan 6 + 23 tan aiw tan~ a. (CS)


and which are parallel to the~ leading edges E10 and E1'O.

The zero-pressure line mayr easily be constructed graphically.

Graphical representations of the respective pressure distributions
in the sections A1Bl, A2B2, A~B A4B6, and AjB5 parallel to the
y-axis are given in figures 49, jO, 51, 52, and 53.

An isometric pressure surface is shown in figure Sk for the
hexagonal plane wing.


Translated by Morris D. Friedman






NACA TM 1583


REFERENCES


1. Sedov, L. I.: Theory of Plane Motion of an Ideal Fluid, 1939,
(Russian)

2. Kochin, N. E.: On the Steady Oscillations of a Wing of Circular Plan
Form. P.M.M., vol. VI, no. 4, 1942. NACA translation.

3. Prandtl, L.: Theorie des Flugzeugtragfluigels in Zusammnendruickbaren
Medium. Lufifahrtforschung, no. 10, vol. 13, 1936.

4. Ackeret, J.: Gasdynamik. Handbuch der Physik, vol. VII.

5. Krasilshchikova, E. A.: Disturbed Motion of Air for a Vibrating Wing
Moving at Supersonic Speeds. P.M.M., vol. XI, 1947. Also D.A.N.,
vol. LVI, no. 6, 1947. Brown translation.

6. Krasilshchikova, E. A.: Tip Effect on a Vibrating Wing at Supersonic
Speeds. D.A.N., vol, LVIII, no. 5, 1947.

7, Krasilshchikova, E. A.: Effect of the Vortex Sheet on the Steady
Motion of a Wing at Supersonic Speeds. D.A.N., vol. LVIII, no. 6,


8. Krasilshchikova, E. A.: Tip Effect on a Wing Moving at Supersonic
Speed. D.A.N., vol. LVIII, no. 4, 1947.

9. Krasilshchikova, E. A.: On the Theory of the Uhsteady Motion of a
Compressible Fluid. D.A.N., vol. LXXII, no. 1, 1950.

10. Falkovich, S. V.: On the Lift of a Finite Span Wing in a Supersanic
Flow. P.M.M., vol. XI, no. 1, 1947.

11. Gurevich, M. I.: On the Lift on an Arrow-Shaped Wing in a Supersonic
Flow. P.M.M., vol. X, no, 2, 1947.

12. Curevich, M. I.: Remarks on the Flow Over Triangular Wings in Super-
sonic Flow. P.M..M., vol. XI, no. 2, 1947.

13. Karpovich, E. A., and Frankl, F. I.: Drag of an Arrow-Shaped Wing at
Supersonic Speeds. P.M.M., vol. XI, no. 4, 1947. Brown translation.

14. Frankly, F. I., and Karpovich, E. A.: Gas Dynamics of Thin Bodies.
1948. Translation published by Interscience Publ., N. Y. 1954.




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