A theoretical investigation of the drag of generalized aircraft configurations in supersonic flow

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Title:
A theoretical investigation of the drag of generalized aircraft configurations in supersonic flow
Series Title:
NACA TM
Physical Description:
iv, 108 p. : ill ; 27 cm.
Language:
English
Creator:
Graham, E. W
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Drag (Aerodynamics)   ( lcsh )
Aerodynamics, Supersonic   ( lcsh )
Airplanes -- Design and construction   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
It seems possible that, in supersonic flight, unconventional arrangements of wings and bodies may offer advantages in the form of drag reduction. It is the purpose of this report to consider the methods for determining the pressure drag for such unconventional configurations, and to consider a few of the possibilities for drag reduction in highly idealized aircraft. The idealized aircraft are defined by distributions of lift and volume in three-dimensional space, and Hayes' method of drag evaluation is the fundamental tool employed. Other methods of drag evaluation are considered also wherever they appear to offer simplifications. The basic singularities such as sources, dipoles, lifting elements, and volume elements are discussed, and some of the useful interrelations between these elements are presented. Hayes' method of drag evaluation is derived in detail starting with the general momentum theorem. In going from planar systems to spatial systems, certain new problems arise and are discussed.
Bibliography:
Includes bibliographic references (p. 107-108).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by E.W. Graham ... et al..
General Note:
"Report date January 1957."

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003874442
oclc - 156939359
System ID:
AA00009195:00001


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1 O r 1 0 f I .. 1 5 y / 1 f-

TABLE OF CONTENTS


I. SUMMARY . . . 1


II. INTRODUCTION . . 2


III. SINGULARITIES USED IN THE LINEARIZEDD" DESCRIPTION
OF THE FLOW ABOUT AIRCRAFT . . 5

A. BASIC SINGULARITIES . . 5

B. SOME EQUIVALENT SINGULARITY DISTRIBUTIONS 12


IV. THE EVALUATION OF DRAG . . 25

A. THE "CLOSE" AND THE "DISTANT" VIEWPOINTS 25

B. GENERAL MOMENTUM THEOREM FOR THE EVALUATION OF DRAG 26

C. HAYES' METHOD FOR DRAG EVALUATION . 355

D. LEADING EDGE SUCTION . . 49

E. DISCONTINUITIES IN LOADINGS . 50

F. THE USE OF SLENDER BODY THEORY WITH THE
DISTANT VIEWPOINT . . 52

G. THE DEPENDENCE OF DRAG COEFFICIENT ON MACH NUMEBEF 55

H. SUPERPOSITION PROCEDURES AND INTERFERENCE DRAG 55

I. ORTHOGONAL DISTRIBUTIONS AND DRAG REDUCTION
PROCEDURES . . 5$6

J. THE PHYSICAL SIGNIFICANCE OF INTERFERENCE DRAG 56

K. INTERFERENCE AMONG LIFT, THICKNESS AND SIDEFORCE
DISTRIBUTIONS . . 57

L. REDUCTION OF DRAG DUE TO LIFT BY ADDITION OF A
THICKNESS DISTRIBUTION . . 59








V. THE CRITERIA FOR DETERMINING OPTIMUM DISTRIBUTIONS
OF LIFT OR VOLUME ELEMENTS ALONE . 60

A. COMBINED FLOW FIELD CONCEPT . 60

B. COMBINED FLOW FIELD CRITERION FOR IDENTIFYING
OPTIMUM LIFT DISTRIBUTIONS . 60

C. COMBINED FLOW FIELD CRITERION FOR IDENTIFYING
OPTIJMUM VOLUME DISTRIBUTIONS . 6i

D. UII-FOPJR DOWNWASH CRITERION FOR MITND-IU4 VORTEX DRAG 61

E. ELLIPTICAL LOADING CRITERION FOR MINIMUM
WAVE DRAG DUE TO LIFT . . 61

F. "ELLIPTICAL LOADING CUBED" CRITERION FOR MfI14UM
..!AVE DRAG DUE TO A FIXED TOTAL VOLUME . 62

3. COMPATIBILITY OF MINIMUM WAVE PLUS VORTEX DRAG
WITH MINM-UM WAVE OR MINflIDTUlI VORTEX DRAG 62

H. ORTHOGONAL LOADDIG CRITERIA . 65

APPENDIX V. DISTRIBUTION OF LIFT III A TRANSVERSE
PLANE FOE MINIMUM VORTEX DRAG . 64


VI. THE OPTIMUM DISTRIBUTION OF LIFTING ELEMENTS ALONE 68

A. THE OPTDIMUMll DISTRIBUTION OF LIIT' THROUGH
A SPHERICAL SPACE . . 68

B. THE OPTEIMUM DISTRIBUTION OF LIFT THROUGH
All ELLIPSOIDAL SPACE . . 70

C. THE OPTIMUM DISTRIBUTION OF LIFT THROUGH
A DOUBLE MACH CONE SPACE . 72

APPENDIX VI. DERIVATION OF OPTD-IUMI DISTRIBUTION
OF LIFT THROUGH A SPHERICAL SPACE . 75


VII. THE OPTDIMUM DISTRIBUTION OF VOLUME ELEMENTS ALONE 78

A. THE SINGULARITY REPRESENTING AN ELEMENT OF VOLUME 78

B. THE DISTRIBUTIONi OF VOLUME ELEMENTS . 79


* ii










C. THE DRAG OF VOLUME DISTRIBUTIONS ON A STREAMWISE
LINE AND THE SEARS-HAACK BODY . .

D. THE SEAFS-HAACK BODY AS AN OPT.IMUM
VOLUME DISTRIBUTIONI IN SPACE . .

E. RING '.JirIG AND CENTRAL BODY OF REVOLUTI01[ COMBINATION
HAVING THE SAME DRAG AS A SEARS-HAACK BODY .

F. OPTIMUM THICKNESS DISTRIBUTION FOR A PLANAR WING
OF ELLIPTICAL PLAfJFORM . .


VIII. UNIQUENESS PROBLEMS FOR OPTIMUM DISTRIBUTIONS IN SPACE .

A. THE IION-UNIQUENESS OF OPTE-I.UM DISTRIBUTIONS
IN SPACE "ZERO LOADINjGS" . .

B. UNIIUEJhESS OF THE DISTANT FLOW FIELD PRODUCED
BY AN OPT.fL-1 FAMILY . .

C. -ENIQUJEirESS OF THE ENTIRE "EXTERNAL" FLOW FIELD
PRODUCED BY AN OPTIMUM FAMILY . .

D. EXISTENCE OF SYMMETRICAL OPTIMUM DISTRIBUTIONS
IN SYMIETRICAL SPACES I. . .


IX. IliVESTIGATION OF SEPARABILITY OF LIFT, THICKNESS,
AND SIDEFORFCE PROBLEMS . .

A. THE SEPARABILITY OF OPTIR1MU DISTFJRIBUTIONS
PROVIDING LIFT AND VOLUblE . .

B. THE HON-INTERFERENCE OF SOURCES WITH OPTIMUM
DISTRIBUTIONS OF LIFTING ELEMENTS IN A SPHERICAL
SPACE . . .

C. THE JONi-IUJTERFERENCE OF SIDEFORCE ELEMrENTS WITH
OPTRIM1 DISTRIBUlTIONS OF LIFTING ELEMENTS IN A
SPHERICAL SPACE . . .

D. IlITERFERENCE PROBLEMS IN CERTAIN SPACES BOUNDED
BY MACH ENVELOPES . .


. 94


. 94




* 97




. 98


iii







E. THE I.'ITEPFEREIICE BETWEEN LIFT AI1D SIDEFORCE ELEMENTS
AND AN OPTII.IUT i DISTRIBUTION OF VOLIJUlE EL1EMETS 102

F. THE RING WING AND CENTRAL BODY OF REVOLUTION
HAVING ZERO DRAG . . .. 105


X. RESULTS Arm COnCLUSIONS . . 106


XI. REFEEENrCES . . .. 107







NATIONAL ADVISORY COMMITTEE FOB AERONAUTICS

TECHrICAL M1EMORAUDUM 1421



A THEORETICAL INVESTIGATION OF THE DRAG OF GENERALIZED

AIRCRAFT CONFIGURATIONS IN SUPERSONIC FLOW*

By E. W. Graham, P. A. Lagerstrom, R. M. Licher,
and B. J. Beane


CHAPTER I. SUMMARY


It seems possible that, in supersonic flight, unconventional arrange-
ments of wings and bodies may offer advantages in the form of drag reduc-
tion. It is the purpose of this report to consider the methods for deter-
mining the pressure drag for such unconventional configurations, and to
consider a few of the possibilities for drag reduction in highly idealized
aircraft.

The idealized aircraft are defined by distributions of lift and
volume in three-dimensional space, and Hayes' method of drag evaluation,
which is well adapted to such problems, is the fundamental tool employed.
Other methods of drag evaluation are considered also wherever they appear
to offer simplifications.

The basic singularities such as sources, dipoles, lifting elements
and volume elements are discussed, and some of the useful inter-relations
between these elements are presented. Hayes' method of drag evaluation
is derived in detail starting with the general momentum theorem.

In going from planar systems to spatial systems certain new problems
arise. For example, interference between lift and thickness distributions
generally appears, and such effects are used to explain the difference
between the non-zero wave drag of Sears-Haack bodies and the zero wave
drag of Ferrari's ring wing plus central body.

Another new feature of the spatial systems is that optimum configu-
rations generally are not unique, there being an infinite family of lift
or thickness distributions producing the same minimum drag. However it
is shown that all members of an optimum family produce the same flow
field in a certain region external to the singularity distribution.

Other results of this study indicate that certain spatial distri-
butions may produce materially less wave drag and vortex drag than com-
parable planar systems. It is not at all certain that such advantages
can be realized in practical aircraft designs, but further investigation
seems to be warranted.

uneditedd by the HIACA (the Corrmittee takes no responsibility for the
correctness of the author's statements).



correctness of the author's statements).






NACA T!4 1421


CHAPTER II. INTRODUCTION


The primary purpose of this report is to consider the problems
involved in exploring a broader class of aircraft configurations than
is ordinarily studied for supersonic flight. It is necessary to deter-
mine whether any unconventional arrangements cf dings and bodies offer
sufficient aerodynamic advantages in the form of drag reduction to merit
more detailed study. As a first step in this direction attention is
directed to optimum configurations, even though they are highly idealized
in form and do not necessarily represent practical aircraft.

In the preliminary exploration of such ccnfigurations it is not
necessary to know their detailed shapes. It is sufficient to define the
aircraft as a distribution of lift and volume in space, without knowing
the camber and twist of the wing surfaces supporting the lift distri-
bution, and knowing only approximately the shapes of the bodies con-
taining the volume.

Hayes' method of drag evaluation is well adapted to this type of
analysis and is one of the primary tools used. However other methods
and points of view are employed wherever they appear to offer further
understanding of the problems.

The properties of sources, dipoles, etc., are reviewed, and a sin-
gularity corresponding to an element of volume is introduced. Some
useful relations between three-dimensional distributions of different
types of singularities are developed and later applied. Also Hayes'
method for drag evaluation is developed in detail.

Since this report is exploratory in nature the investigations made
are frequently incomplete and somewhat isolated from each other. Some
of the material of Ref. 2 and most of the material of Ref. 5 are included
in this report for convenience. The latter has also been published in
The Aeronautical Quarterly, [lay 1955, under the title, "The Drag of
Non-Planar Thickness Distributions in Supersonic Flow." Permission to
reproduce this material has been granted by The Royal Aeronautical
Society.


tical
Society.






NACA TM 1421


CHAPTER III. SINGULARITIES UTILIZED IN THE LINEARIZEDD"


DESCRIPTION OF THE FLOW ABOUT AIRCRAFT


A. BASIC SINGULARITIES

The Source


For incompressible, non-viscous fluids the
flow is the Laplace equation,


)2 +2
;x2 ,y2


2
+ ) = _
3Z2


where 0 is the perturbation velocity potential.
exhibits spherical symmetry, is the source,


equation governing the


(5a-l)


A basic solution, which


S(x 4)2 + (y )2 + (z 2


(5a-2)


This solution can be interpreted as representing the emanation of unit
volume of fluid per unit of time from the point t, 1, t. Because of
the linearity of Eq. (j5a-l), other solutions of it can be built up by
a superposition of sources through the use of certain limiting proce-
dures; such resulting solutions are the horseshoe vortex, doublet, line
vortex, etc. Much is known about these solutions and with them the
flow over wings and bodies can be described mathematically.

In supersonic flow the governing differential equation is the
linearized potential equation,


2 62o
- x2
8x2


2
,= 0
6z2


(3a-3)


where x is the coordinate in the stream direction and 0 = M2 1. Equa-
tion (5a-5) can also be considered as the two-dimensional wave equation
where the x coordinate is thought of as the "time" variable.

If the y and z coordinates in the Laplace equation (Eq. 5a-l) are
multiplied by io, then that equation is transformed into the wave equa-
tion; a similar transformation of the source potential from Eq. (3a-2)
results in





[JACA TM 1421


(5a-4)


(x )2 P2 [ n)2+ ( )2]


which can easily be shown to be a solution of Eq. (5a-3). Equation (5a-4)

is real inside the forward and rear Mach cones, (x 0)2> |2 y T)2 +

(z )2] and imaginary elsewhere; however, due to the nature of super-
sonic flow only the solution in the rear Mach cone is used to represent
a source. Since half of the real solution is discarded, the constant


RE6S/ON 0c /i'ZUEA/CE OF SUPERSON/C SOURCE


associated with the incompressible 2 must be doubled to represent a unit
supersonic source. Thus the supersonic source at 1, T1, ( has the potential


(5a-5)


Elsewhere


where the x axis is in the free stream direction. It can be shown that
Eq. (Ba-5) represents unit volume flow from the point E, T, t; however,
care must be taken in the proof because of the singularities on the Mach
cone (cf. Ref. 4). In the proof given by Robinson(k) he made use of the
concept of the finite part of an infinite integral, an idea originally


Os~


x- t 'P (y -i)2+ (z-_ )2


S






NACA TM 1421


introduced by Hadamard"5. As in incompressible flow, other solutions
of Eq. (5a-5) can be built up by superposition of the basic source solu-
tions; some of the solutions can also be obtained, as was the source,
by analogy with the incompressible solutions.

Before i-cine on to other solutions let us examine the supersonic
source in more detail. Since the velocities are infinite on the Mach
cone from a finite source, care must be taken in using such sources to
describe real flows. It is instructive to examine the isolated source
in terms of the limit of a finite line of sources in the free stream
direction as the length tends to zero while the total strength remains
constant. Under the assumptions of slender body theory, if the line of
sources extends from x = 0 to x = x with strength Kx, it represents a
cone of semi-vertex angle K/2U with a semi-infinite cylindrical after-
body (Fig. 5a-la). The velocities are constant along conical surfaces


///N/TE AT REAR MACH
VELOCITIES1 CONE, ZERO AT FORWA RD
MACH CONE


SOURCE A
STRE THO


pvX0


U
-X


Fig. 5a-la: Cone-cylinder and
source distribution repre-
senting it


Fig. 5a-lb: Perturbation flow
lines in x-z plane for super-
sonic source, M = F2


VELOCiTrIES INFINITE
AT MACH CONE b






IACA TM 1421


from the origin; but on the Mach cone from x = xo the velocities become
infinite due to the discontinuity in source strength. The total inte-
grated source strength C is equal to 1/2 Kxo2. If xo is allowed to
approach zero while C remains constant then in the limit a concentrated
source of strength C is obtained. The flow pattern in the xz plane for
the source at M = f2 is shown in Fig. 5a-lb. (See also Ref. 6.) For a
source of finite strength the velocities are infinite on the Mach cone.


The Three-Dimensional Doublet

The three-dimensional doublet (or dipole) is a second basic solution
of the wave equation; it is obtained by allowing a source and sink of
equal strength to approach one another while the product of source strength
and distance between source and sink remains constant (and equal to unity
for a unit doublet). The axis of the doublet is defined here as the
vector extending from the center of the sink to the center of the source;
positive values are taken to be those along the positive directions of
the coordinate system. For a doublet with its axis vertical, the above
method of derivation is equivalent to taking the negative partial deriva-
tive of the source potential in the z direction; that is,



S 2_-- 2z-- x 3r (5a-6)
2 (x2 22

where r2 = y2 + z2. Equation (5a-6) represents a positive doublet at
the origin, i.e., one with the source above the sink.


The Horseshoe Vortex

In supersonic theory, as well as in subsonic, the flow around a
wing of finite span can be described by certain solutions of the wave
equation called horseshoe vortices. In the subsonic case this singularity
is derived by integrating in the streamwise direction a semi-infinite line
of negative doublets with axes vertical. The supersonic horseshoe vortex
can be derived in the same way as the subsonic one if only the finite
part of the integral, as defined by Hadamard) is taken as the solution.
This solution can also be obtained without the use of the Hadamard finite
part if a streamwise line of sources is differentiated in the vertical
direction; thus,





NACA TM4 1421 7


0 px-pr 1 x-pr
HSV = Finite part {xr O d} x- O3 d9
f. 0I 6z 0

_____xz x > pr (5a-7)
2itr2 x2 -22

The flow pattern for a horseshoe vortex in planes normal to the
free stream axis is shown in Fig. 5a-2. Far behind the bound vortex,


Fig. 5a-2: Flow pattern for supersonic horseshoe vortex






8 NACA TM 1421


the flow near the x axis is similar to the flow far downstream around
a subsonic horseshoe vortex and it is this part which gives rise to the
"vortex" drag. The drag associated with the flow near the Mach cone
is called "wave" drag. Equation (5a-7) represents a supersonic horseshoe
vortex of unit strength, i.e., unit circulation around the bound vortex.
Since a force pUr is associated with a bound vortex of strength r, we
shall, for convenience, discuss unit lifting elegients which have as their
velocity potentials


= xz
21tpUr2 fx2 -22


x prr


(5a-8)


Similarly, the potential for a unit side force element is


= 1 0 f o
SF = U px-1r
pU byJO


S dE = xy
2itpUr2 x2 P2r2


The force associated with Eq. (3a-9) is directed in the positive y direc-
tion; a force in any direction normal to the flow direction may be repre-
sented by a combination of lift and side force elements.

In the light of the discussion of the horseshoe vortex, the three-
dimensional doublet (Eq. 3a-6) may be given added significance as a lift
transfer element or element of moment. That is, the doublet potential
can be formed by subtracting the potential for a horseshoe vortex at x = Ax
from one at x = 0 (Fig. 5a-5) and applying the proper limiting processes
(equivalent to differentiating the horseshoe vortex potential); in this
process the trailing vortices from the negative or rear element are can-
celed out by those from the positive one, and the remaining part forms
the doublet or lift transfer element.


U
i-t>


z z


Fij. 5a-5:


Formation of doublet or lift transfer element from
horseshoe vortices


x > pr


(5a-9)






HACA TM 14i21


The horseshoe vortex consists of a bound vortex of infinitesimal
length plus two free vortices trailing back to infinity. Since the
vortex drag and the lift associated with a finite wing can be evaluated
by considering the flow velocities far behind the wing, it is useful to
consider the trailing vortices as they appear in the Trefftz plane far
downstream from the bound vortex. The Trefft-j plane flow represents a
two-dimensional doublet or dipole and its potential is obtained by
letting x->- in Eq. (5a-7). Thus


(5a-10)


02D z + 2
2n y2 + z2)


It should be noted that the potential for this doublet is independent
of Mach number, and thus the vortex drag calculations for a given lift
distribution are the same for supersonic and incompressible flows. The
flow pattern about the doublet will be similar to the planar flow inside
the dashed circle in Fig. 5a-2.


The Volume Element

Another useful solution is the doublet with its axis in the stream-
wise direction; it has as a potential


d v
OV 60S


S-x

21t(x2 02r2)3/2


x ? or


(5a-ll)


Equation (5a-Ul) can be shown tc represent the potential for a unit of
volume equal to 1/U (see Chapter VII) at the origin. A distribution of
volume elements along the x axis with strength f(x) has as a potential


rx-pr
= 0f
Jo0


f(t )v dt


f (U x-3r
24( (x- *)2 p~r220


1- fx-r
2 A-o


The first term in Eq. (5a-12) is infinite; but if only the finite part
of the integral is considered (as defined by Hadamard(5) ), then Eq. (5a-12)


(x )2 -p2r2


(5a-12)






NJACA TI- 1421


represents the potential for a source distribution of strength f'(0).
Thus a body can be built up from a series of volume elements as well as
from a series of sources and sinks.

The Closed Vortex Line
Equation (5a-11) can be considered not only as a volume element but
also as a closed vortex line of circulation strength 1/ 2 in the yz plane
(Fig. 3a-4a). The line carries a constant intensity of forces directed
inward so that the total vector force is zero. The negative of Eq. (3a-11U)
would represent an element with the forces directed outward from it. The
potential for the closed vortex line can also be obtained by applying the
standard limiting process to an element composed of two pairs of horseshoe
vortices of strength 1/P 2, one with its axis in the negative z and the
other in the negative y direction (Fig. 5a-4b); when added together the
trailing vortices cancel leaving the closed vortex line.


r-

4,S.


+


C


(a)


Fi,. 5a-4: Formation of closed vortex line from horseshoe vortices


Two-Dimensional Singularities


In subsonic flow two-dimensional sources, obtained by integrating
a line of three-dimensional sources in the lateral direction, have proven
useful in many problems; so also has the infinite bound vortex obtained
by a lateral integration of horseshoe vortices. The same types of solu-
tions can be derived for supersonic flow and these provide more insight
into the nature of the supersonic solutions. The two-dimensional source
potential is






NACA TMt 1421


+ 1 x2_2z2

28 J- -x2p2z2 S = 20 x > M ( 1
02sf12



All of the disturbance created by the two-dimensional source is
concentrated on the Mach planes from it, thus creating a potential jump
across these planes. The two-dimensional vortex potential is


+1 x2 2Z2 +1 z> 0
2V 0 &V dT = 2 x > pZ (3a-14)
1 -x2-p_23z2 -2 z < 0


Again all of the disturbance is concentrated on the Mach planes. There
is a potential jump across the Mach planes and also across the z = 0
plane, the latter due to the discontinuity in the past history of the
fluid particles above and below the plane.






NACA T1 1421


B. SOME EQUIVALENT SINGULARITY DISTRIBUTIOIUS

Statement of the Problem

The first section of this chapter reviewed the basic singularities
which represent elements of lift, side force and volume in linearized
supersonic flow theory. It was noted that these singularities may all
be obtained from the source singularity with the aid of the simple proc-
esses of integration and differentiation. The fact that the basic
singularities are so related will be shown to imply that certain dis-
tributions of singularities are equivalent, i.e. they produce the same
flow field, at least outside of a finite region. In the present section
an equivalence theorem will be proved regarding constant strength dis-
tributions of sources, lifting and side force elements and vortex sheets.
Such a theorem will later prove useful in the study of interference
between distributions (Ch. IX B,C,F). Note that if the distribution A
is part of a larger distribution (A,B) and if A is replaced by an equiva-
lent distribution Al then the drag of (A1,B) is the same as that of (A,B).
This follows from the fact that the substitution of Al for A does not
change the flow field at infinity (Ch. IV).

The distributions to be studied will be located on a cubic shell
which has two faces perpendicular to the free stream direction. One
face of the cube will be covered by sources of constant strength and
the opposite face by sinks of constant strength. The remaining four
faces will be encircled by vortex lines of constant strength. Two cases
may then be distinguished: (A) The source distribution is on a face
parallel to the free stream; (B) The source distribution is on a face
normal to the free stream. These two cases are illustrated in Fig. 5b-l.
The source, sink and vortex distributions are uniform and of constant
intensity as indicated. The vortex lines are continuous around the cube
with the circulation directed so as to induce in the interior of the
cube downwash velocities in case A and upstream velocities in case B.


J Z SOURCES VORTEX LINES
(IsrENGTrH ) (cCIRCULATION 4)


-^ .- -^-y x ^ A-C

MOTEX
-iNES 6i '-f S/- SOU--ES 57 (--/ )
(ciRCUIAr/oN -&) (S TREPNGTH-w) (STREfNGTH

CASE A CASE B


Fig; 5b-1






NACA TM 1421


We shall now prove the following theorem.


Theorem

In both cases A and B the perturbation velocities are zero every-
where outside the cubic shell. Inside the shell the downwash is constant
in case A (w = -k) and the pressure is constant in case B (u = -k).

This theorem implies in particular that the source sink distribution,
say in case A, is equivalent to the negative of the vortex-line distribu-
tion in case A, in the sense that the associated flow fields are identical
outside the cube. Note that in case A the vortex distribution on the
front and rear faces gives rise to a lifting force, whereas the vorticity
on the side faces produces no force. In case B the vorticity on the top
and bottom produces lift and the vorticity on the side faces produces
side force.

The theorem will first be proved by a geometrical argument and then
an alternative proof by analytical methods will be outlined.


Geometrical Proof of Theorem

Consider first case A. We shall construct a geometrical configu-
ration which corresponds to the distribution of singularities indicated
in Fig. 5b-1. This construction will proceed in several steps by succes-
sively cutting down configurations of infinite extent. The vortex dis-
tribution on the front face is equivalent to a distribution of lifting
elements of constant strength.

To begin with we shall assume the whole infinite plane containing
the front face to be covered by lifting elements. This may be physically
realized by a cascade of doubly infinite (two-dimensional) wings of con-
stant angle of attack a and such that the vertical distance between two
neighboring wings is equal to the wing chord divided by M2 1,
(Fig. 5b-2). In the limiting case of zero chord length the plane x = -x
is then covered by vortex lines with the circulation (of strength k)
oriented as in Fig. 5b-1. The value of the constant k is then k = 2aU.
The lift per unit area in the plane x = -3 is then 2apU2. Since the
wings are spaced so as not to interfere with each other but still influence
every point downstream of the cascade, the flow field at any point P with
x > -3 may be described as follows (Fig. 5b-2). The point P receives a
unit of downwash (-w = cU) from the wings A and B each. It also receives
a positive unit of pressure (-u = aU /M2 from A and a negative unit


m A and a negative unit






NACA TM 1421


^P

)7


---P- x


Fig. 5b-2


of pressure from B.


The net pressure (referred to p.) received at P is


then zero and the net downwash is -w = 2aP = k.


The cascade may now be terminated from above by a wedge of opening
angle 2a located in the plane z = T with its exterior surface parallel
to the free stream direction (Fig. 5b-5). Actually this wedge corre-
sponds to a source distribution of constant source strength k = 2aU.
If the cascade is removed for z > T the flow field is zero there since
the wedge isolates this region from the rest of the cascade and since
the exterior surface of the wedge is at zero angle of attack. For z < T
the flow field is unaffected by the introduction of the wedge. To see
this consider a point P with z < T (Fig. 5b-5). The wing at B acts as
before to produce a downwash of -adU at P. Only the point C on the wedge


A, C 2C -

A2-' >P
/



Fig. 5b-5

affects the point P and this point C is already in the downwash field -al
of wing A2. Thus the wedge turns the flow downward only by an angle a


A4I






MACA TM 1421


so that the total downwash at P is again -2aU. Conditions at P are the
same as in the infinite cascade.

Similarly the cascade may be terminated from below at z = -Z by
placing a wedge there of opening angle -2a.. This corresponds to a dis-
tribution of sinks of strength -k.

The cascade may then be cut down to finite width by placing planes
of zero thickness parallel to the plane at y = *y and removing the part
of the wings for |y| > Y. Since no sidewash is present the flow field
is undisturbed by the introduction of these planes. Thus for x > -x,
|y < Y, |z < Y the downwash is -w = 2UJ = k and the pressure is zero.
Outside this region all perturbation quantities are zero.

Finally one may restrict the flow field to the inside of a cube by
taking the negative of the above configuration and placing it at x = x.

Thus the resulting flow field has constant downwash and zero pres-
sure inside the cube -x3 < x < x, -y < y < y, -z < z < z. Outside thiis
cube the perturbation velocity is zero. Thus the front face is a cascade
of lifting wings at an angle of attack a, which bends the flow down. The
rear face is a cascade of wings of angle of attack -a which straighten
the flow out again. The top and bottom faces consist of wedges whose
inside surfaces follow the direction of the flow which has been bent by
the cascade. These outside surfaces are parallel to the free stream.
(Note that for the wedge of negative angle the "interior" top surface is
directed downward at an angle 2am and the "exterior" bottom surface is
parallel to the flow.) Finally the side faces are planes that carry no
forces. For each such plane the downwash is -w = 2aU on the inside and
w = 0 on the outside. These planes are then surfaces of constant vortic-
ity. However, the vorticity vector is parallel to the free stream and
hence no force results.

Thus a geometric configuration (using a wedge of negative opening
angle) corresponding to case A has been constructed and the theorem has
been proved for this case.

The corresponding construction for case B will only be indicated.
The source distribution on the front face is obtained by placing wedges
there (Fig. 5b-4) of half-angle a = pk/2U.


A4


N, x
0.1
\k I N f
^ [-^


Fig. 5b-4






NACA TM 1421


At a point P then the downwash is zero and the pressure is given by
-u = 2aUJ/M2 1 = k.

By inserting planes of zero thickness at z = z, y = y and removing
the wedges outside these planes the infinite configuration is cut down
to a configuration with a finite cross section. Outside these planes the
flow is undisturbed. Inside these planes -u maintains its value
2aU/M2 1 = k. These planes are then pressure discontinuities and
hence carry lift and side force respectively. They are also vortex
sheets.

Finally the configuration may be terminated by placing its negative
at x = +3.

A geometric configuration (again using wedges of negative opening
angles) corresponding to case B has thus been constructed and the theorem
has been proved for case B.


Analytical Proof of Theorem (Outline)

Case A. Source Distribution Face Parallel to Free Stream

Consider a cube with sources of strength k on the top and -k on the
bottom, and with lifting elements of strength pUk on the front face and
-pUk on the front face and -pUk on the rear face. On the side faces of
the cube there are no forces associated with the vortex lines parallel
to the flow direction; these are the trailing vortex system of the ele-
ments on the rear face.

In computing the potential due to the singularities on the cubic
shell, various regions of the flow field are considered separately. For
the region ahead of the foremost Mach waves from the cube no disturbance
is possible in supersonic flow. Behind the cube, if the forward Mach
cone from a point includes all of the shell, the potential at that point
may be found simply by integrating the total effect of the singularities
covering the shell. The potential due to individual unit source elements,
lifting elements, and side force elements are given in Eqs. (5a-5), (5a-8),
and (3a-9). The strengths of the distributions considered in this case
are indicated in Fig. 5b-5.


n Fig. 5b-5.





NACA TMn 1421


/ fy'


SOURC~s (i4)
(-i~9,
44
-~


Pos/;r/v EI/ /A____V-/j
IFT/ ;- --^ 7( ,-S)
ELEMENTS f/('-- /

(-x,- y,- z) //(x,-y,-2)
Ls/wHsp.4


Fig. 5b-5

The potential for the entire shell is then

k (x y) (z- zo)dyo dzo





"-^^ly2-y y)2 + (2]^-)2j ?^ ^ ^2
2A (x+E) _
[y _7 ) +2(Z -zo) 2] X-x ) 2 [(y-yo) + (z-zo) 2

(X z -zo)dyo dzo
0 )2x -( 3 )
-y yo 2+ (-z 2 (x- ) (Y_ Y" 702+ (Z zo) 2]


k x y do do




f rJ dxo dyo (3b-1)
-(x -x2 2 YYo )2+ (z+.)2


This, after evaluating the integrals, equals zero.







NACA TM 1421


A third region of the flow field contains points slightlyr behind
and far to the! side of the3 cube, where forward N~ach cones from the points
(1)
include part, but not all, of the cube. For this region, Hayes' method
can be used to sho that the? potential again is zero. Th~is method is
described in, Ch. IVC. It requires that the distance from the cube to any
point P whre the potential is to be compulted must be Large compared to
thne dimecnsions of the cube~. In addition, P must lie near the Mach cones
emnanatingi from the singuilarities on the cube. P is then a point at some
angle 8 (measured from the horizontal plane) on a distant cylindrical
control surface surrounding the cubic she~ll. An "equivalent lineal dis-
tribution" of singularities is formd by finding the singularity strength
intercepted from the cube by a set of parallel_ planes originating at
angle 6 on the control cylinder and inclined at the Mach angle to the
free streak direction. The sinigularities intercepted by a given Mach
plan are lumped together at the intersection of the Mlach plane and the
axis of the cylinder, such that the total strength of the equivalent
distribution is equal to the total. strength of the original distribution.
After determining the strength (h) of the equivalent lineal distribution
which re-presents the cubic shell for a fixed 6, the effect of' all those
s-ingularities which influence the flow field att P can be summed. Hayes
writes the expression for h as


h = +f gz sin 9 gy cos 6 (jb-2)


where f is the source strength. (per unit length), gzI p the circulation
strength (per unit length) of the lifting elements, and g 0ll the circu-
lation strengh of the side force elements.

Figure 5b-6 indicates the notation to be used in describing the
geomretry of the intersections of the Mach planes with planes containing
thie x,y,z axes. The Mlach plane is inclined to the axis of the control
cylinder at the Mach angle CL = sin-l(1/M) and it is tangtent to a cross-
section of the cylinder at angle 6.* The trace of the Masch plane in a
horizontal (x-y) plane~ is inclined to a normal to the flow direction at
angle 6, where tan 6 = cot 4 cos 9 = 8 cos 8. Th trace of the M~ach
plane in a vertical (x-z) plane f'onns an angle a with a line parallel
to the z-axis, where tan a = p sin 8.






1IACA TM 1421


TrACE OF A ACH PLANE
/N THE X-Z PLAVE


p I




/^>
TRACE OF AA CH /
PLA'E PV THE
X-y PLANE /


Fig. 5b-6


With this brief description of Hayes' procedure in mind, an equiva-
lent lineal distribution of singularities is now to be computed for the
specific case of the cubic Shell described previously. Figure 5b-7a
shows the intersections of two parallel Mach planes with the shell; the
Hach planes are assumed to be separated by an infinitesimal distance.
The case illustrated shows only three faces of the cube intersected by
the M-ach planes since the procedure would be the same if four faces were
affected. In order to better define the geometry and notation, Fig. 5b-7b
shows the cubic shell as though it were cut along the corner edges and
flattened out in one plane.


Tj






NACA TM 1421


/9


Fig. b-7a b LIe
di'
FRONT SIDE



(5/NVKS)
BOT TOM

Fig. 5b-7b



The net singularity strength cut out by these Mach planes must be
l-mnped along a length dx of the axis. The total source strength is the
product of the strength per unit area (k) and the area intercepted from
the top surface of the cube by the Mach planes:


x_ xli xl
f = k x1 dn, = k dx = -k 1 dx
cos( = tan 8 cos e
[ 2;


(5b-5a)


(The negative sign is inserted because 9 is in the second quadrant for
the example shown, but f is positive.) The total lifting element strength
is pUk multiplied by the area intercepted from the front face:

Zi
I = pUk 1- dn5 = pUkzi dz tan 6
p cos(n 67)


Ata dx x, dx
= pUk t a 1= -pUk 2 1 6 cos
tan20o p2sin 8 cos 9


(5b-3b)






NACA TM 1421


Again, a negative sign is inserted because cos 9 is negative while I
should be positive. There are no forces on the side faces. In computing
the strength of the equivalent lineal distribution from Eq. (5b-2) it
must be remembered that gp, gz from that formula are circulation strengths
multiplied by 0; i.e.,


g = PS
z pu y =pu

Then

h = +f g. sin 9 gy cos 9


-kx1 dx -pUkx Cl dx
=-1- -Li- -- -- sin 9 =0 (5b-4)
P cos 9 pU p2sin 0 cos


That is, the net singularity strength is zero. This is true for all
angles 8, and similar calculations show that it is also true for every
station x along the cylinder axis. Therefore, the velocity potential
is zero at all distant points for which Hayes' method is applicable.

There remains to find the velocity potential in the neighborhood
of the shell. The cube may be subdivided into smaller cubic shells,
each similar to the original. Singularities on interfaces of adjoining
shells then cancel so the net singularity distribution is unchanged.
Those shells which lie behind and outside the forward Mach cone from
any point cannot influence the velocity potential at that point. It
was shown earlier that those shells which lie completely inside the for-
ward Mach cone from the point also do not influence the potential there.
Therefore, only those shells lying along the forward Mach cone need be
considered. However, these may be further subdivided into cubic shells
of elementary proportions so that the distance from the point to any
one of the shells is very large compared to the dimensions of that shell.
Then the analysis based on Hayes' procedure shows that these shells do
not contribute to the velocity potential at the point either. This indi-
cates that the velocity potential is zero everywhere outside the cubic
shell.

To find the velocity perturbations inside the shell, again consider
it divided into smaller shells. None of these except the one containing
the point P can influence the potential at P according to the preceding
analysis. Therefore, all of the small shells located more than a dis-
tance E ahead of P can be removed without affecting the potential at P.
The forward Mach cone from P then intersects only the front face of the






NACA ITM 1421


remaining part of the original cube, so that, effectively, P is aware
only of an infinite distribution of lifting elements. Since this result
is independent of the location of P inside the original cubic shell, the
downwash inside the shell must be constant.


Case B. Source Distribution Face formal to the Free Stream

Consider now a cube with lifting elements of strength pUk on the
top face and -pUk on the bottom'face, with side force elements of
strength pUk, -pUk on the side faces, and with sources of strength 02k
on the front face and -p2k on the rear face.

2


eLmeEvrsf-pU-k)


Fig. 5b-8




First, the potential ahead of the foremost Mach waves of the cube
is, of course, zero. At a downstream point whose forward Mach cone
includes all of the sources and lifting elements the potential is





NACA m 1421


k (z- (x-x)dx, dyo


E- -(Y Y-Yo)2+(z )2(x -x) 2 2 [( Yo)2+(z-T)2]



(x + -, r_(y x-xo)dxo d0z+





-x (y(y yy))2
_y y) + zo_ x-z)2o 0 (y )2 + (Z zo)



(Y + x xo) dxo, dzo,
3E~y +' ( )2 +( z _z) 2
x -z )2+(z)o 2(xxo)2_ 2(y+y) 2+ z zo)2]



2k dyy d 4 zo
-y xo)2

.x -X), z d & I

SO (5b-5)
-( + -- [ + 3 (z -zJ 12


Carrying out the integration, it is found that

S= -2k- for < J < y and-z < z < z


Elsewhere


( = 0






NACA TM 1421


By Hayes' procedure, when forward Mach cones from distant points
include only part of the singularities, the potential at those points
is the same as would be contributed by a lineal distribution whose
strength, h, can be computed in the manner described previously. For
Mach planes intersecting the cube in the same location illustrated for
another case in Fig. 3b-7, one finds that


X1 dx xli dx Xdxl
f = -k S = -pk 1S, 1 = -pUk x (5b-6)
sin 6 cos 0 p sin e' cos 6


and so

Xh- dx px dx ex d
h = -k -1 -pUk 1 cos -LpUk sin 6=0
sin 9 cos 9 pU p sin / pU cos

(5b-7)


Thus, the potential due to the cubic shell is zero at all distant points
of the flow field which lie near the Mach cone of the shell.

In the neighborhood of the cube, the same arguments used for the
first cubic shell show that the perturbation velocities for this case are
zero there also. Therefore, the perturbation velocities are proved to
be zero in every region of the flow field external to the cubic shell.

To find the potential at a point inside the shell, the shell is sub-
divided as before into smaller shells, each similar to the original. The
analysis just completed shows that the velocity perturbations at P cannot
be influenced by any of these shells except the one containing P. There-
fore, all of the small shells located more than a distance e ahead of P
can be removed. The net singularity, strength intersected by, the forward
'Tach cone frrom P then includes only sources on the front face of the
remaining group of cubes. Effectively, then, conditions at P are the
same as behind an infinite distribution of sources of constant intensity.
This result is independent of the location of P inside the cubic shell,
so the pressure must be constant inside the shell and the potential is
of the form 0 = ex.






NJACA m 1421


CHAPTER IV. THE EVALUATION OF DRAG


A. THE "CLOSE" AND THE "DISTANT" VIEWPOINTS

The non-viscous drag for a wing and body moving at supersonic speeds
may be obtained from two different points of view(1-), using linearized
theory. First, the drag can be evaluated by integrating the local pres-
sure times frontal area over the wing and body surfaces. Second, the
drag can be evaluated from momentum or energy considerations involving
the flow field at a great distance from the aircraft. These two pro-
cedures are actually variations of the same basic method.

In the latter case part of the drag due to lift is associated with
the production of kinetic energy in the trailing vortex system, and is
called "vortex drag." This drag is Identical with that produced by the
same spanwise lift distribution in an incompressible flow, (frequently
called "induced drag").

The remainder of the drag due to lift and all of the drag due to
thickness is associated with the production of energy near the surface
of a downstream Mach cone whose vertex is in or near the aircraft. This
is called wave drag, and the associated energy is half kinetic and half
potentially).

The wave drag plus the vortex drag is equal to the drag evaluated
at the wing and body surfaces by the first method. (It may be necessary
to retain nonlinear terms in the expression for pressure coefficient to
get this agreement.)

The momentum theorem is utilized in both of the above methods but
different "control surfaces" are used. In the first case the control
surface is close to the aircraft surface, but in the second case the
control surface is a distant one. For example Hayes(1) uses a circular
cylinder with axis passing through the aircraft and parallel to the free
stream direction. The radius of the cylinder is chosen to be very large
compared to the aircraft dimensions since this simplifies the calculations.

The wave drag. of the aircraft is then evaluated from the rate at
which momentum (in the free stream direction) is carried across the sur-
face of the cylinder. (If the control surface had been chosen as a sur-
face containing streamlines instead of a perfect cylinder, then the wave
drag would have appeared as pressure on the streamline surface.)

The cylindrical control surface is closed far downstream by a plane
normal to the flow direction. The vortex drag is then determined, as in
incompressible flow, by the rate at which the kinetic energy of the
trailing vortex system passes through this plane, or alternatively
through momentum and pressure considerations.


ure considerations.






NACA ITM 1421


B. GENERAL MOMENTUM THEOREM FOR EVALUATION OF DRAG

In the present section a momentum integral for the drag, as given
by linearized theory, will be derived (Eqs. 4b-53,53). The drag will be
given as an integral over an arbitrary control surface enclosing the
solid. The integrand is a quadratic expression in the velocity compo-
nents as given by linearized theory.

First a more general momentum integral will be considered. Consider
a control surface S enclosing a solid (Fig. 4b-l). A surface element
on S of area dS will be represented by its outward normal din where the
length of dn is equal to the area of the surface element. Thus dn = (dS)n
if n is the outward normal of unit length. Let the hydrodynamical stress


U
------ 0-


Fig. 4b-l


tensor be denoted by a, and
region outside S. Then


f = a dn = Force exerted


let I be the region inside S and II the


by II on I across surface element


(4b-l)


If a system of coordinates xl, X2, x3 is chosen dn may be repre-
sented by its three components (dn)i and a by a 35x3 matrix oaij. The
above equation may then be written


5
fi = (a dn)i => .ij(dn) j

j=l


(4b-2)


where (a dn)i is the ith component of the force.






NACA TM 1421


For a non-viscous fluid the only hydrodynamical force is the pres-
sure p and the stress tensor is


a = -pl = -(Pij) (4b-5)

where I is the identity tensor whose matrix is the Kronecker delta bij.
In this case the force across the element is


f = -p(I dn) = -p dh (4b-4)

or

fi = -p(dn)i


The bydrodynamical momentum equation states that the stress tensor
is balanced by flow-of-momentum tensor. (This is actually a restatement
of Newton's law that force = (mass) times (acceleration).) To define
the flow-of-momentum tensor we first introduce the concept of a dyadic
_> I>
product of two vectors. Let a and b be two vectors with components (ai)
and bi). The dyadic product is then the tensor whose ij component
(a b is aib i.e.


a b = ai bj (4b-5)


Note that if c is any vector then


( b) = ((aibq)c = (aibjCj) = ( ) (b-6)


where b c is the ordinary dot product.

The flow of momentum tensor is then the dyadic product of pq
(momentum per unit volume) and q velocity:


Flow-of-momentum tensor = pq q )


(4b-7)






NACA TM 1421


Its physical interpretation may be seen by applying this tensor to the
normal dn

> = P->
(pq q)dn = p(q dn) =kMomentum transported through dS per unit time.

(4b-8)


The basic momentum equation for stationary flow for a surface Sl
which does not enclose a body is then



S(pq q)dn = f a dn (4b-9)
S1 JS1

This is analogous to the law of conservation of mass which states that


Spq dn = 0 (4b-10)
S1


Consider now the composite surface consisting of the surface S in
Fig. 4b-l and the body surface E. Let dn denote normals on E which point
outwards with respect to the body (i.e. into region I). From the defini-
tion of the stress tensor a


F = Total force exerted by fluid on body = a dn (kb-ll)


Since the flow through E is zero one obtains by applying Eq. (4b-9) to
the composite surface 81 = S + E


S(p )dq. =Lf an- adn (4b-12)
S S fE


The minus sign in the last term is due to the convention that on the
surface E the quantity dn denotes the inward normal with respect to the
region I. Comparing Eqs. (4b-11) and (4b-12) one obtains






NACA Tm 1421


F = (pq )d + a dn
S S


(4b-15)


This is the fundamental momentum formula which gives the total hydro-

dynamical force on the solid as an integral over a control surface

enclosing the solid.

Note that in Eq. (4b-ll), the force is given by an integral of the
stresses on the body surface. This is the "close" point of view for
evaluating the force. Eq. (4b-15) shows how the same force may be evalu-
ated from the distant point of view.

A slight modification of Eq. (4b-15) will be needed later. Denote
the flow quantities at infinity as follows


q, p, p, a at infinity = U, po, p0, O0, respectively


(4b-l4)


The difference between a flow quantity and its value at infinity will
be denoted by a "prime." Thus


q =q U, p' =p p, po = p p0, a' = a o


From the continuity equation (Eq. 4b-10) it follows that


(kb-l5)


(4b-l6a)


/ (pU q)dn = 0
JS


Furthermore, since a. = Constant


aCT dn = 0


Subtracting Eqs. (4b-16a, b) from Eq. (4b-15) one obtains


F = (pq' q)dn + a' Cn
S B


(4b-l6b)


(4b-17a)


U < pq dn =
S






IJACA TI-1 1421


where, for a non-viscous fluid,



0-' dn = p' dn (4b-17b)




This is the fundamental momentLum formula in terms of perturbation

quantities. iHote that the latter are not assumed to be small.

The drag is the component of F in the free stream direction. We
shall take this direction as the x-direction and use the following
notation.


L' = i, q = (uvw), q = (u',v',w') (4b-18)


where


u = U + u', v' = V, w' = w


From Eq. (4b-17a) then follows the fundamental momentum formula for drag:


Drag = F i = pu'q in + i f a' din (4b-19)


The momentum integrals may be further simplified for special choices
of the control surface S, in particular by letting S recede to infinity.
However, we shall first derive an approximate form of the drag formula,
valid within linearized theory. In the following section this linearized
formula will then be specialized to a special infinitely distant control
surface (method of Hayes(l)).


Inviscid Second-Order Drag

It will be shown below that for a thin or slender body the largest
contribution to the drag may be evaluated by an integral of a quadratic
expression of the linearized perturbation velocities. It is usually
stated that the drag is of second order. However, it should be remembered
that the values of the perturbation velocities are computed from first-
order (linearized) theory. The result is a formula for drag according
to first-order theory. The tdrm "second-order drag" refers to the fact






[ACA TM 1421


the integrand is quadratic in u', v' and w' and hence of second order
if u', v' and w' are themselves of first order. Furthermore, the second-
order correction to u', v' and w' will contribute nothing to the secorid-
order expression for drag. The final formula is given by Eqs. (4b-55,34)
and the reader interested only in the final result may skip the deriva-
tion now presented below.

We shall first assume non-viscous flow, so that the stress tensor
is given by Eq. (4b-5). Furthermore, we shall assume that the solid
is characterized by a parameter E, which is small, e.g. the fineness or
thickness ratio. We shall f-urthermore assume that the flow quantities
may be expressed by power series in e:


u = U + EUl + Eu2 + (4b-20)


v = V1 + E2v2 + .


w = Ew + E2p +2 .


p = p + Ep + Ep + .


0 = po + epl + E 2 P2 + .


Such an expansion is valid at a distance from the body. It should
be remembered, however, that in slender body theory, terms involving
log e are of importance very near the body.

The coefficients of e are the first order terms and are given by
linearized theory. The coefficients of E2 are the second order terms,
etc. The lowest order term in the expression for the drag will now be
found using the isentropic pressure-density relation and Bernoulli's
law.

From isentropy it follows that density is a function of pressure
alone. One defines



(p) = a2
S)constant entropy-


where a is the isentropic speed of sound. Then

p = p+ p p + (4b-21)
ao






HACA TM 1421


from which then follows

P1
p^
1 a 2


Bernoulli's law may be written


d(u2 + v2 + w2) + dP =
2


where


p
P = p
T^o


(u' + U)2 + 2 + w2 + 2P = U2


Using Eq. (4b-21) P may be expanded to second order


1 p '
- O 02o
p0^ \Poao


(4b-24)


P 2poa)2
PO 2poao 2


Expanding the terms in Eq. (4b-25) to second order one obtains


S2 u+ 2 + v+ 2 + wl2 1
EUui + 2 + E + p
o


+ E2P2 ---- = 0
-2poa3


Collecting the terms of order e one finds the linearized Bernoulli's
law


PI + PoulU = 0


(4b-22)


P dp'
p +
0 a02


(4b-25)


(4b-25)






NACA TM 1421


Comparing with Eq. (kb-22) one sees that


Uu1
P = -PO 2
O*


The terms of order ield the following expression for p2
The terms of order yield, the following ex'pressioni for p2


ul2 + v2 l+ w2 p0ulM2
Po0Uu2 + Po 22 + + 2 2 = 0


(4b-27)


where Eq. (4b-25) has been used and


M=U
ao

In the momentum formula, Eq. (4b-17), the stress and momentum flow
tensors may be combined to form a tensor A


A = -pq' q p'I

Using Eqs. (4b-20, 25, 26) one may evaluate All


All = -pu'(U + u') p'


= -E (PoulU + p E 2(pou2U + plu1U + pu2 + p2


p p ul2 12 + w2
= -2P Uu2 M2u1 + U12 Uu2 +Vi2 1



Finally,


A E11 = oP [M2 lu2 + V12 + W2


Ul2M2
2)


(kb-28)


Simrnilarly


A12 = -pu'v' u -C2PoulV1


(kb-26)


(4b-29)







NACA Thi 1421


A13 = -pu'w' I -2 2pou1w (4b-50)


Since only the first row (All, A12, A15) enters in the drag computation
we have proved the following:

1. The dominant term in the drag formula is of second order in E

2. The integrand in the drag formula is, to second order, a second
degree polynomial in the first order velocity perturbations. The velocity
perturbations of second order, or pressure and density perturbations of
second order, do not t ter into this expression.

Thus while drag is of second order, it may be computed on the basis
of first order theory linearizedd theory). On the cther hand, one may
easily check from the above expressions that in general the lift has a
first order term. Furthermore to compute lift to second order one needs
to know u2, that is, u to second order.

In the remainder of this report we shall only be concerned with the
drag as given by linearized theory. It is then convenient to introduce
a change of notation. We shall let u, v, w stand fcr the linearized
velocity perturbation; in other words

Eul, EVJ, Ew1 are replaced by u, v, w (4b-1l)

Furthermore a velocity potential 0 will be introduced such that


Grad = u, v, w (4b-52)


The above results may then be summarized as follows. The drag to
second order is given by the formula


D = Al dn (4b-55)
S


where S = Control surface enclosing the body






NACA TM 1421


A = (All, A12, A1)


A = +Po 0u2 + v2 + W2) p2 = M2 1

Ali + Pou
(4b-34)
A12 = -POuv


A5 = -Pouw

and u, v and w are the components of the perturbation velocity given by
linearized theory.


C. HAYES METHOD FOR DRAG EVALUATION

The method developed by Hayes in Ref. 1 consists in applying the
drag formula Eq. (4b-55) to a special control surface, a truncated cir-
cular cylinder, surrounding the body and in considering the limiting
case when the control surface recedes to infinity. The general momentum
integral for the drag then assumes a simplified form. (This results in
certain simplifications in the integrand.) Furthermore, if the body is
represented by singularities (sources, lifting elements, etc.) as dis-
cussed in Ch. III, the velocities at large distances may be represented
very simply in terms of the strength of the singularities. As a result
the drag may also be represented as an integral over the singularities
(distribution of source strength, etc.). This result of Hayes' gener-
alizes a previous result by von Kirmain(7) for a body of revolution.

First a somewhat detailed demonstration of the method of Hayes will
be given for the case of a lineal source distribution. This part may
be skipped by a reader not interested in mathematical details. Then the
results of Hayes and related results will be stated in intuitive terms
for general three-dimensional distribution of sources, lifting elements
and side-force elements. Detailed proofs will not be given. However,
the results may be proved by methods closely analogous to the method
exhibited for the case of a lineal source distribution.


Hayes Method for Lineal Source Distribution

We shall consider a distribution of sources along the x-axis between
x = 0C and x = L. The corresponding solid is then a body of revolution.
The source strength will be denoted by f. It will be assumed that


f(0) = 0, f(L) = 0


(4c-1)






NACA TM 1421


These assumptions lead to certain restrictions on the body shape.
Let the radius of the body be r(x). The cross sectional area S(x) is
then nr2(x). Since f(x) = U S'(x), f(0) means that r(0) r'(0) = 0.
This is fulfilled if r ~ xn, n > 1/2 near the origin. In particular,
f(0) is equal to zero if the body starts in a point with finite slope,
i.e. r ~ x near x = 0. The analogous condition at x = L insures f(L) = 0.
In addition, f(L) = 0 if the body ends smoothly in a cylinder with con-
stant radius, i.e. if S(x) = Constant for x l L and S'(x) is continuous
and hence zero at x = L. It will be indicated in the proof below why
the restrictions on f are necessary.


Expression for Velocities


The potential due to the source distribution is then


l x-pr
^xQ1r) = 2 f


(4c-2)


f(E)d-
(x t)2 02r2


where r2 = y2 + z2. For x pr L the upper limit may be replaced by L.

Using the condition f(0) = 0 one finds by partial integration of
Eq. (4c-2) and differentiation that the perturbation velocities are


x-Pr
,, 1 ^-
x -
21 0


r 2~tr x-fpr


f'()dE
(x t)2 p2r2


S(x g)f(g)d
x 2r2
V(x -) r


In Eqs. (4c-3), the upper limit is replaced by L for x pr >j L.

We shall introduce the notation


= x )

Then t = 1 on the downstream Mach cone from x = | and 0 < t <. 1 inside
this Mach cone. For x "r > L one may also write the velocity components
as


(4c-5a)



(4c-5b)






NACA TM 1421


x 1= L f() [(x- )2 P2r2] -3/2(x S)dt

L 2(/ p5/2

L f()(x ) -21 t 2 -3/2d



= pL r 1 -r 2 d
21t fr ^xs2?? p1d(
-12t~ L -/


0 f L _2 t &
=IJFLf(W)X E) _tti -


(4c-5a)


(k4c-5b)


Hayes' Control Surface


Following Hayes we now introduce the control surface shown in
4c-l. It consists of a circular cylinder of radius rl, truncated
front disc x = Constant < 0 and a rear disc x = xl > L. The drag


FORWlARD
o/SC -








-y


-REAR DISC


Fig. 4c-1


Fig.
by a





NACA TM 1421


integral (Eq. 4b-55) will be evaluated for this control surface as rl
and x, tend to infinity. The ratio between xl and rl will be determined
later in such a way that the contribution of the rear disc to the drag
will vanish in the limit.

Contribution of Rear Disc

According to Eqs. (4b-55,54) the contribution of the rear disc to
the drag is


D = rl (p2 x2 + r 2)2r dr
0


(4c-6)


The velocity components may be evaluated as follows. Write f(g)
as a difference of two positive functions


f(0) = f+(0) f_(g),


f+(S), f-(6) o 0


(4c-7)


Then by the mean value theorem and Eqs. (4c-5a,b)


L
0


f+(S)dt


L
0


2x 3)2(l tt5 2)


2mx ) (l t 5/2
2% 2 21 -te22)


where 0 : EY, |2 L. A similar expression is valid for .r Note that
in Eq. (4c-8) the continuous source distribution is replaced by a posi-
tive source at t and a sink at g2. However, 93 and t2 depend on x
and r.

As is easily seen


0 x L

tL i


i = 2, 5


(4c-8)


(4c -9)


f-f()dS


1 : 1 + tt 1 + tt2 < 2






NACA Tm 1421


Hence, replacing E and E2 by L increases the absolute magnitude of
both terms in Eq. (4c-8). Hence, on the rear disc x = xl, 0 r rl,


X2 < 2-2A (4c-10)
(xl L) 1 tL)

where A is independent of x, and rl, and

2r, rp2x2 dr A fro1 x, r L 1 20 dr (4c-il)
O (x1 L)20 xi L ( 2)5 xl L


If one puts y = 1- tL 2, then dy = -2tL dtL = -t^ 2p dr/(x L).
Hence

Sr_20ar A 12 dy A I I (4c-12)
o (xl L)2 y1 y5 2(x/ -L)217A

where yl = 1 (1 e )2 = 2E1 E12 and el is explained in Fig. 4c-1.
Equation (4c-12) may be written

rB rpx2dr< B C (4c-15)
o (xi L) 212 r122

where C is independent of ri and x1 for rl and x1 sufficiently large;
e is explained in Fig. 4c-1. The fact that x/(x1 L)- l, e/e--1l
as r- :. has been used above. It is then clear that if e is constant
or if E = r", a < 1, then the integrand in Eq. (4c-15) tends to zero

as r-c. A similar estimate may be shown for fyr2dx. A comparison

with Eq. (4c-6) shows that:






NACA TM 1421


Contribution of rear disc to drag is zero even if E decreases

as r increases. However, E should decrease more slowly than r-.

Since the distance BC is of the order er it follows that this

distance becomes infinite in the limit.


Contribution of Cylindrical Part

Since the contribution of the forward disc to the drag integral is
identically zero it follows then in the limit rl, x1--m the entire drag
contribution comes from the cylindrical part, provided E varies as pre-
scribed above. Thus


D = Limit D2 (4c-14)

where D2, the contribution of the cylindrical part, is



D2 = -po2irlI x+pr1E x r dx' (4c-15)
o3r,


Note that the radial component Ai. of the vector A1 in the drag formula
Eq. (kb-4) is -pdx r.) In the above equation 1/1 E has been replaced
by 1 + e which may be done without loss of generality.

To evaluate D2, we write Eq. (4c-5a,b) in the following form


1 x_,L r1) 1
x 2n 021 1 V l + 20rl

(4c-16)
S= x',L f'(f2) xf + or 2
2Ko x' g2 + 2pr, Pr1


where x' = x pr,.






fNACA TM 1421 41


The upper limit is x' for x' L and L for x' L. Hence


p o erl x1'L xl'L 1iF) 20pr,
2 = 0 0 2 x' x' 2 x' + 2prl


x' + prl t2
r -- dl 2 dx' (4c-17)
x' 2 + 2pr, Pr1 j



Limiting Case for Infinitely Distant Control Surface

We shall now evaluate D2 as rl--. The three ratios within the
second bracket all tend to unity as rl-:o? and may hence be neglected
in the limit. Note that this approximation implies

1 1 1 ,1 1 (4c-18a)
(x 12 p2r2 x o) pr f2pr x' 1 20r

Furthermore, applying the same approximation to Eq. (4c-16) one obtains
that

Or ~ -0Ox (4c-18b)

r and x both vanish as l/2f3rl. Their ratio, however, is given by the
above equation. The corresponding relation with or replaced by Oy is
exact for two-dimensional flow. Thus the flow is approximately two-
dimensional at large distances near the Mach cone from the leading edge
(E small, i.e. tt almost unity for 0 S L).

Hence

D = Limit D2 I (4c-19a)
41r

where

prl x',L x',L f'f l)f'( 2) d d2dx'
I = /$ d/ dx' \j_
J0 JQ J ^x 1 ^x, 2






NACA TM 1421


The domain of integration is a region in x', l, t2 space whose cross-
section for x' = Constant is the square 0 1 & x', 0 E2 :! x' for
0 5 x' L and the square 0 < l, E2 L for L x' E per,. Let I,
be the integral where 0 y L and 12 the integral over the domain
L y per,. Since the integrand is symmetric in 1 and t2, half its
value is obtained by integrating only over the triangle ABC in the k1,
2 plane (i.e. l < t2) as shown in Fig. 4c-2. In evaluating I over




B C






^______________ V
A X/


Fig. 4c-2


its domain (a truncated triangular cylinder with base at x = L and vertex
at x' = 0) we shall first integrate along a line parallel to the y-axis.
For tl' S2 fixed this line is inside the pyramid only when ki, x' < L.
Sl may vary inside the triangle between 0 and E2, and for t2 any value
between 0 and L may be chosen. Hence Il may be written


pL n2 PL f gl 2a
II = 2 2L ft N fIj2 dX1 dt1 dk2 (4c-20)
0 0 S2 F xt ^ ~2


The integral 12 is (domain is triangular cylinder)


0Er 0 L t2 f'(1 2f'S
l2 = 2 a/ / l 2--did dx' (4c-21)
"L J0 0 ^x' El ^x' (2






NACA TM 1421


Interchanging the order as above one obtains



I = Il+12 = 2 L t2 ft 2 rlf'_x) x_ --log C dt, d2
fo (o L f t t t2 id
(4c-22)

Here C is a constant and it has been introduced under the assumption that

fL
f f'(E)dt = 0, which, since f(0) = 0 means f(L) = 0
J0


(cf. Eq. 4c-1). (lcute that the limit of integration for t1 may be
replaced by L if the factor 2 is omitted.)

Now


,ppe r l C- + 21p e r l p ,2)
Er log C = [0x' E) E2 +( 1) (x!;-2)] t2



= -log(t2- 91) + log2r-1-E2+2 (-1 1)(w -Q (4.-25)
C

Hence if one chooses C = 43Erl, the second term will tend to log 1 = 0
as r7-4-. Note that for this it is essential that er1 as ri--)>
(cf. pg.TV16). In other words the simplicity of the proof depends on
the fact that e--0 as rl->- (cf. Eqs. 4c-l8). On the other hand e
may not tend to zero so fast that erl remains bounded. In this case the
above proof would be invalidated. Actually a drag contribution would
come from the rear disc in that case.

By combining the Eqs. 4c-19a, 22, 25 one obtains the final drag
formula


D = pOfLf 2 1 Qd)a d9 42 (c-24)







NACA TM 1421


This is von Karman's drag formula for a lineal source distribution

such that f(D) = f(L) = 0. It has been derived above by the method of

Hayes. This derivation has the advantage that it may be extended

immediately to cases of a more general distribution of singularities.

Such generalizations will now be discussed.


General Three-Dimensional Source Distributions

We shall now consider a more general case of a spatial distribution
of sources. It will still be assumed that no lifting or side force ele-
ments are present. The source strength will be denoted by f(x,y,z).
It will be assumed that f = 0 outside a certain finite region V. A
special case is a planar distribution, say in the plane z = 0 in which
case f(x,y,z) = f2(x,y)8(z). Another special case is the lineal distri-
bution on the x-axis which was discussed above. In this case
f(x,y,z) = fl(x)S(r). It will be shown below how in a certain sense the
drag evaluation for the general three-dimensional case may be reduced
to a consideration of certain equivalent lineal distributions. In the
course of this discussion certain restrictions on f(x,y,z) will be made
in addition to the requirement that it vanish outside a finite region.

Consider a line in the streamwise direction passing through V. The
position of the line which will be taken as the x-axis is actually arbi-
trary, but for practical purposes it will be assumed that it is "well
centered." This is, of course, a somewhat vague requirement. However,
if for example f has rotational symmetry, the x-axis will be its axis
of symmetry. On the x-axis choose as origin a point, 0, whose downstream
Mach cone contains V. For convenience choose this point as far down-
stream as possible. Also choose the point, L, for convenience as far
upstream as possible, whose upstream Mach cone contains V. An equivalent
requirement is that the downstream Mach cone from L is contained within
the downstream Mach cones from every point in V. Let the value of x at
point L be L. Thus the downstream Mach cone from x = 0 and the upstream
Mach cone from x = L touch but do not penetrate V. We now introduce a
control surface and define e and el as in the lineal case (cf. Fig. 4c-1).
This is shown in Fig. 4c-5. It will be assumed that rl and x, tend to
infinity as described in discussion of the lineal case.






IJACA TM 1421 45















4X














Fig. 4c-5: Hayes control surface in three-dimensional space


That is as x and r1 tend to infinity e and el will tend to zero. In
that sense the line AC will come arbitrarily near the Mach cone from the
origin. On the other hand c and E Will tend to zero slower than 1/r1
so that the line AC becomes infinitely long as rl-m.

By the same methods that were used in the lineal case, it may be
easily seen that the contribution of the rear disc, x = xl, becomes zero
in the limit. All the drag thus comes from a portion on the cylindrical
surface arbitrarily near the Mach cone from the origin and is hence pure
wave drag.

To evaluate the drag contribution from the cylindrical surface we
introduce cylindrical coordinates x, r, 9 where


x = r cos 6, z = r sin


(4c-25)






NACA TM 1421


Let the drag contribution of a strip on the cylinder between 9 = Bo
and 9 = o00 + AO be AD. We define


dD = Drag contribution per unit angle = lim r as AO -;0 (4c-26a)
To9 AO

Then

r2jx
D = Total drag = d d (4c-26b)
J 0 dO

Consider now a fixed meridian plane 6 = 09, and a point P = (x,,rlo)
on the cylinder between A and C (Fig. 4c-5). The potential O(P) depends
on the contribution from all sources inside the upstream Mach cone from P.
The contribution from a source at Q = (t,rjt) is proportional to the
source strength f(Q) and inversely proportional to the hyperbolic dis-
tance rh(P,Q) between P and Q where


rh2 = (xo )2 p2 (r1cos 00o TI)2 + (rl sin 9 t)2] (4c-27)


This hyperbolic distance is constant on hyperboloids of revolution with
r = rl, 0 = o00 as axis. Consider now the sources between two such hyper-
boloids which intersect the x-axis at x = t and x = + d. To evaluate
the contribution to O(P) of these sources one may transfer their total
source strength to the axis. In this way the distribution in V is
replaced by an equivalent lineal distribution i.e. by an equivalent body
of revolution. So far this lineal distribution depends on xo and r, as
well as o00.

Consider now, still for fixed 0 = o00, the limit as rl-->co. Then
the hyperboloids may be replaced by Mach planes which intersect the
meridian plane 0 = o00 orthogonally along Mach lines. Note that for this
it is necessary that as rl--w> any point between A and C comes arbi-
trarily near the downstream Mach cone from the origin in the sense
described above. The source strength between two such neighboring planes






NACA T4 1421


e0=q


X=o


Fig. k4c-4:


Evaluation of O(P)


may then be transferred to the x-axis as above. However, in this limiting
case the resulting equivalent body of revolution depends at most on 0o.
It becomes independent of r1 and x,,. The corresponding lineal source
distribution will be denoted by f(x;9o). A consequence of the independ-
ence of x. and ri is that f(x;9) may be used for computing 0, and as
well as 0 at P. In general it may not be used for computing e9. Clearly
00 is zero for a lineal distribution, whereas the 00 resulting from the
original volume distribution is not. On the other hand 9 is not needed
for drag evaluation on the cylindrical surface.

Since or and x may be computed from the equivalent body of revolu-
tion for fixed 6 it follows that dD/d9 may be computed in exactly the
same way as the drag of a body of revolution was computed. The result
will differ from Eq. (4c-24) only by a factor 2it. Hence we have proved
the following: The drag D of a volume distribution of sources of
strength f(t,q,() is given by the formulas


)(= L.






48 MACA TM 1421


21r
D = de (4c-28a
Jo de



dL = L t2 1 ; 2;)og (2 1 2 (4c-28b



f(S;9)dt = f f f(Q)dQ (4c-28c
v(6,e)

where V(g,9o) is the region contained between two Mach planes perpen-
dicular to 9 = 0o and intersecting the x-axis at x = t and x = g + de.

This result was obtained by Hayes in Ref. 1. It is thus seen how
Hayes' derivation of von Karman's drag formula for bodies of revolution
admits an easy generalization to the general three-dimensional case.

This proof obviously presupposes the following requirement on the
strength distribution f(Q) in addition to the requirement that it vanish
outside a finite volume: f(Q) must be such that for each 9 f(x;9) sat-
isfies the same requirements as f(x) in the lineal case. In particular
for each 9: f(0;9) = f(L;9) = 0 and f(x;9) must be differentiable with
respect to x.

If f(Q) has rotational symmetry, i.e. depends on r and x only then
it may obviously be replaced by one equivalent lineal distribution,
independent of 9, for computing the distant flow field and the drag.
In the special case when f(Q) is lineal to begin with, Eqs. (4c-28)
reduce to the previously established formula (Eq. 4c-24).


Extension to Include Lift and Side Force Elements

For simplicity only sources have been considered in the preceding
development. However lift and side force elements can be included and
were included by Hayes in his original report. We will not go into the
details here, but merely indicate the final results, since the funda-
mental ideas of the method have been illustrated in the discussion of
source distributions.

Following Hayes we define a function h such that

h = f g ggsin 9 + gy cos 0) (4c-29






NACA Tm 1421


where


f = f(t;9) = Source strength


pUgz/P = Z(-;9) = Lifting element strength


pUgy/P = s(;9) = Side force strength


The term (g sin a + g cos e) is proportional to the component of force
in the direction 0, and is the only component contributing to the wave
drag in the Hayes calculation. Equation (4c-28b), as extended to include
lift and side force elements, is






pL fL
TO) e 2 OL, (h' gl;0)h' E2;O)10g (2 1 1d~ dE2



= L L h' (El;9)h'(2;9)log J2 d11 d2 (4c-50)



where h(t;9) is the equivalent lineal distribution (for a given station 0)
of the original spatial distribution of singularities.

This equation makes it possible to determine the wave drag of an
arbitrary spatial system containing thickness and carrying both lift
and side forces. In order to determine the total pressure drag of the
system it is necessary to evaluate the vortex drag produced by the lift
and side force. In Hayes method the vortex drag appears as a momentum
outflow through and a pressure on the end of the cylindrical control
surface. It can be evaluated by calculating this momentum and pressure
or by determining the kinetic energy associated with the vortex system
in the Trefftz plane. Since this is identical with the induced drag
problem of incompressible flow, we will not discuss it further.


D. LEADING EDGE SUCTION

The evaluation of the drag of a lifting wing of zero thickness by
integrating local pressure times frontal area over the wing surface is
not theoretically-complete until leading edge suction is accounted for.






50 NACA TM 1421


This means that the infinite negative pressures acting on subsonic
leading edges should be included. In practical applications this leading
edge suction is sometimes discarded since in many cases only a fraction
of the theoretical value is actually realized.

However, from the distant viewpoint, leading edge suction cannot
be isolated. This is true because there is no point-to-point corre-
spondence between the close and the distant control surfaces. At the
distant control surface the velocity field created by the wing leading
edges is merged with the fields created by other areas on the wing and
body.

From the distant point of view leading edge suction is automatically
assumed to be fully effective, and therefore it must be so assumed from
the close viewpoint to get correspondence in the drag values.


E. DISCO'TIIJUITIES IN LOADINGS

For a planar wing, vortex drag is dependent only on the spanwise
lift distribution. A discontinuity in the ordinates of this lift dis-
tribution produces a concentrated vortex of finite strength and infinite
energy, which corresponds to infinite drag.

Wave drag is similarly affected by discontinuities in loadings.
For example, consider a distribution of sources on a streamwise line.
If there is a discontinuity in source strength, then the drag evaluated
on the distant control surface is infinite.

To prove this, assume a source distribution with a discontinuity
at the point x = X (see sketch). The velocity potential at a point (x,r)
downstream of the rearward Mach cone from Y may be written


I f xr f(t)d 1 -E f ( ()d r
= +
2 0 V(x- _)2 p2r2 2x O (x _)2 2r2

x-r f(dE(4e-1)

3i r(K- E p2r:J


f2 (5)






NACA TM 1421 51


The u-component of velocity at the point (x,r) is found by differentiating
Eq. (4e-l) axially. (In order to avoid indeterminant forms in the differ-
entiation, the equation is first transformed by means of the relation
= x f3r cosh u.) This process gives the result (assuming f(0) = 0):


U j f 1'()d + x-Pr f2()d Af (x )
ax 21t o(x g)2 _p2r2 J (x E)2 p2r2 (x -X)2 2r2

(4e-2)
where Af(x) = fl) f 2).

At the distant control surface it previously was shown (Ch. IV-C)
that one need consider only conditions very near the Mach cones from the
source distribution. Introducing the approximations used in Hayes'
method (i.e., (x t)/pr t 1), Eq. (4e-2) can be expressed

Tf I( f e-5)4
2r 2pr 0o + X fx' ( ) x' (x [

where x' = x pr and x' O pr. Since the radius of the control sur-
face is large compared to the length of the source distribution, the Mach
cones originating at the sources are essentially plane waves when they
intersect the control surface, so that the radial component of velocity
(at the control surface) is (Eq. k4c-18b)


v = pu (ke-4)

The drag, being equal to the transport of horizontal momentum across
the control surface, is proportional to the product of u and v integrated
axially along the control surface. From Eqs. (4e-5) and (4e-4) it is
readily seen that the drag includes a term of the form


dxt
J^ (x' )


The integral is non-convergent. An infinite drag contribution therefore
results from a discontinuity in the strength of the source distribution.







NACA TM 1421


F. THE USE OF SLENiDER BODY THEORY WITH THE DISTANT VIEWPOINT

If slender body theory is applied, then the source strength is
assumed proportional to the rate of change of cross-sectional area,
dS/dx, for a corresponding body of revolution. This means that infinite
drag will be predicted (by the distant procedure) for all bodies of
revolution having discontinuities in dS/dx. Such a prediction is, of
course, incorrect, and the error is caused by the application of slender
body theord to bodies which are not sufficiently smooth.

The use of slender body theory requires that smoothness should be
maintained at the nose and tail of the body and therefore dS/dx should
be zero at these locations. In order that dS/dx should be zero at the
nose or tail of a closed body of revolution it is necessary that the
variation of body radius, R, with distance, d, from the nose or tail
(1 (/2)+kc
should be of the form R ~ d(2)+k where k > 0. This does not elimi-
nate blunt noses or tails entirely, but excludes "excessive" bluntness.
(Note that the Sears-Haack optimum shape is blunt.)

The linearized theory requirement that all velocity perturbations
be small theoretically excludes all bluntness, but this is unimportant
if very small regions of the flow field are affected.

Bodies which begin or end in cylinders also may satisfy the smooth-
ness requirements.

For a body to be sufficiently smooth to permit the use of slender
body theory, it is necessary to restrict the "short" wave length fluctu-
ations in the plot of cross-sectional area versus length. The word
"short" cannot be defined exactly here, but should probably apply to
all wave lengths less than the body diameter times F2 1)

Figure 4f-l illustrates the effect of wave length on the accuracy
of the slender body theory. The drag for an infinitely long corrugated
cylinder according to strict linear theory was found by von Karman (7).
Slender body theory is in good agreement with these results only where
the reduced wave lengths are large compared to the cylinder radius. At
the other extreme two-dimensional theory is approached.

It should be remembered that when the distant viewpoint is used
the drag of a singularity distribution is evaluated. The body shape
corresponding to the singularities may, be determined either by "exact"
linear theory or approximated by slender body theory. For example in
Fig. kf-2 a specific source distribution is considered, and is inter-
preted as a "bump" on a cylinder by "exact" linear theory and by the
slender body approximation. For this ratio of wave length to cylinder






NACA TM 1421


COMPARISON OF THEORETICAL CALCULATIONS
FOR DRAG OF CORRUGATED CYLINDER



(A)PRATIO OF THE DRAG COMPUTED BY SLENDER BODY THEORY
TO THE DRAG COA4PU/TED BY L /NEAR THEORY
(B.) RATIO OF THE DRAG COMPUTED BY TWO-D/MENSIONAL
L/NEAR THEORY (ASS(UM/AlNG TWO-D/MENS/IONAL FLOW/N
EACH MERIDIAN PLANE) TO THE DRAG COMPUTED
BY T/REE-D/MENS/ONAL LINEAR THEORY













c 40
'K d
S16 -.


I I -o- --
Z\ PROFILE OF CYLI/NDER
(Q: K ( CCROSS-SEC7IONSA RE CIRCULAR)
Cr 12 ----










4 IN_________ __ __ __
^4~ --_----- --_-

k i

RATIO OF REDUCED WAVE Z ENGTH


TO CYL /IDER DIAMETER,,
d


(FOR LINEAR THEORY DRAG OF CORRUGATED CYLINDER, SEE PEF:7)


Fig. if-1






54 NACA TM 1421



Qt





(0


~ n.
Zt N






RI r kII












c~co
41.




Ell
cz^ i ^$

M4P? ^^
0^ fe |^ ?


~0
0






NACA TM 1421


diameter the bump shapes and locations are quite different. It is of
interest however that the net volumes contained in the bumps are iden-
tical. This has been proved by Lagerstrom and Bleviss and generalized
by Bleviss in Ref. 22. (This suggests that "volume elements" may retain
their significance even when slender body theory does not apply.)


G. THE DEPENDETJCE OF DRAG COEFFICIENT ON MACH RJMLBER

Hayes(1) has pointed out that, for a distribution of singularities
on a single streamwise line, the drag, evaluated from the distant view-
point, is independent of Mlach nuriber. If the singularities are sources,
and slender body theory is applied, this indicates that the drag of a
given body of revolution is independent of Mach number. However the
application of slender body theory in conjunction with the distant view-
point requires that dS/dx = 0 at the tail of the body.

Hayes' result is.therefore consistent with a fact previously deter-
mined, that the drag coefficient of a slender body satisfying the
"closure" condition (dS/dx = 0 at the tail) is independent of Mach
number.

If the singularities are not confined to a single streamwise line,
then the distant viewpoint gives a drag coefficient which varies with
Mach number. This can be seen from the fact that the projection of the
singularity distribution onto a single streamwise line varies with the
inclination of the Mach planes used for the projection.


H. SUPERPOSITION P1RO'CEDURES AND INTEPFERENCE DRAG

In all the developments discussed in this report the linearized.
supersonic flow equation is used. This means that one flow field and
the lift (or volume) distribution which causes it can be superimposed
on a second flow field with its corresponding lift (or volume) distri-
bution. If the individual flow fields satisfy the linearized flow
equation, then their sum does also.

For example, let a pressure field, pl, correspond to a downwash
field, a.,, and a second pressure field, p2, ?orrespond to a second down-
wash field, p2, then the pressure field pl + p2 corresponds to the down-
wash field al + a2.

However, the drag of the sum of the two fields is not in general
the sum of the drags of the individual fields. For example, the drag

of the first field would be D1 = J 1 dS, where the integration extends






NACA TM 1421


over the wing and body surfaces, and similarly the drag of the second

field is D2 = -Na2 dS. However, the drag of the combination is


D1+2 = fP1 + P2 g + a2)dS. The terms involving cross products give


the interference drag, Di = 1(pia2 + p2al)dS.



I. CiPTHOGlIAL DISTRIBUTIONS A1D DRAG REDUCTION PROCEDURES

If the interference drag is zero then the two distributions are
said to be orthogonal. The use of orthogonal distributions for reducing
drag has been studied in Refs. 8, 9, 10, and 11.

For example consider two types of lift distributions which are
orthogonal and assume that each one carries a net lift. It has been
shown (see for example Ref. 9) that some combination of the two will
carry a given total lift with less drag than would be produced if either
one of the individual types of distribution carried all of the lift.

On the other hand, any given (non-optimum) lift distribution can
be improved by adding the proper amount of a non-orthogonal type of dis-
tribution which carries zero net lift. The improvement is obtained by
utilizing negative interference drag. This can be seen as follows. The
total drag of the combination is the sum of the individual drags plus
the interference drag. The interference drag can always be made negative
by proper choice of the sign of the distribution that carries zero net
lift. Also, since the strength of the zero lift distribution enters
linearly into the interference drag, but enters quadratically into its
individual drag, the magnitude can be so chosen that the interference
drag dominates. Thus the total drag of the combination can be made less
than the drag of the given (non-optimum) lift distribution.


J. THE PHYSICAL SIGNIFICANJCE OF INTERFERENCE DRAG


It has been stated that the interference drag, Di, is pl2 + P2ca, dSE

where the subscripts designate the two flow fields which have been super-
imposed, and the integration is to be carried over all surfaces. Assume
that both flow fields are produced by thickness distributions. Then the
a values are the body surface inclinations which correspond to dS/dx,

the rate of change of cros-Ee-ctional area for the body. The fpila2 dS






NACA Ti 1421


gives the drag produced by the pressure field of the first body acting

on the cross-sectional area distribution of the second. The term p2al dS

has a similar interpretation.

Assume that both flow fields are produced by lift distributions.

Then pla2 dS is the drag created by the downwash field of the second

distribution acting on the lifting elements of the first distribution.
(The surface which supports the lift corresponding to p1 must be inclined
further because of the downwash due to p2)

Let the first field be produced by a lift distribution and the second

by a thickness distribution (a body). Then plax2 dS is the drag produced

by the downwash field of the thickness distribution acting on the lift
elements plus the drag caused by the pressure field of the lift distri-
bution acting on the cross-sectional area distribution of the body. The

f p2l, dS gives no contribution to the drag in this case.


Assume that the first field is produced by a lift distribution and

the second by a side force distribution. The p1a,2 dS is drag corre-

sponding to the downwash field of the side force distribution acting on

the lift elements, while the p2a1 dS is produced by the sidewash field

of the lift elements acting on the side force distribution.


K. INITERFEREICE AMOUG LIFT, THICKIJESSI, AID SIDE FORCE DISTRIBUTIONS

For planar distributions of lift and thickness (the lift being normal
to the plane) there are no interference drag terms, and the two problems
can be studied independently. However, for spatial distributions, inter-
ference generally exists. This has been discussed by Hayes, and the
physical meaning of the interference irag has been discussed in the
preceding sections.







NACA TI 1421


Suppose that a source and a
lifting element are located as shown
in Fig. 4k-l, the direction of flow
being perpendicular to the page.
Then the component of the lift which
lies in the line connecting the two
singularities causes all of the
interference. If the lift element
were located on the y-axis (corre-
sponding to a planar wing problem)
there would be no interference.

For lift and side force ele-
ments, as shown in Fig. 4k-2, there
is interference between the force
components which lie in the line
connecting the singularities, and
also interference between the com-
ponents normal to the connecting
line.

If the side force element lies
either on the y-axis or on the z-axis
(as shown in Fig. 4k-5a and b), then
there is no interference. This can
also be seen from symmetry considera-
tions, which show that the lift ele-
ment produces no sidewash at the
side force element and similarly
the side force element produces no
downwash at the lift element.


SOURCE-,


Fig. 4k-1


L IF r
LEMEN/T*

_ _


S/DE FORCE
EL E44EN 7';a

oS
Fl. aT.


Fig. 4k-2


Fig. 4k-3a


Fig. 4k-5b


E LEMENr


--- P-1W y






NACA TL 1421


L. REDUCTION OF DRAG DUE TO LIFT BY ADDITION OF A THICKNTESS DISTRIBUTION

Consider the two-dimensional system sketched in Fig. 41-1. The
cross-hatched area is a thickness distribution lying partly in the pres-
sure field of a flat-plate wing. The relative geometry of the thickness
distribution and the lifting surface are indicated in the figure. Also,
the pressure distributions, relative to the two-dimensional pressure 2amq/p,
are shown in parentheses.


Fig. 42-1


As long as the pressure field of the thickness distribution does
not intersect the flat-plate, the lift of the system is the same as for
the flat-plate by itself. On the other hand, the interference between
the pressure field of the flat-plate and the thickness distribution pro-
duces a negative drag contribution, so that the total drag of the system
(omitting friction) is 12-1/2 percent less than the drag of the flat-
plate alone. Thus, the total lift in this case is unaffected by intro-
duction of the thickness distribution and a drag reduction is obtained.

This example is related to the Busemann biplane. The result obtained
illustrates the fact that, in the general case (non-planar systems),
sources and lifting elements have an interference drag.


terference drag.







NACA T1 1421


CHAPTER V. THE CRITERIA FOR DETERJIIJG OPTII.MUM DISTRIBUTIONS

OF LIFT OR VOLUME ELEIIETTS ALOIE


A. THE "C'MBlJED FLOW FIELD" CONCEPT

The idea of the "combined flow field" was introduced by Munk'(12)

and extended by R. T. Jones(15,14). Consider a distribution of lifting
elements in a free stream of given velocity. A certain downwash velocity
and pressure are produced at each point in the field. If the direction
of the free stream is now reversed without moving the lift elements or
altering the directions and magnitude of these lift contributions, then
in general different downwash velocities and pressures are produced at
each point in the field.

One-half the sum of the downwash velocities produced at a given
point in the forward and reverse flows is called the downwash velocity
of the combined flow field at that point. A similar definition applies
to sidewash velocity. One-half the difference of the pressures in the
forward and reverse flows is called the pressure in the combined flow
field. These definitions follow from the super-position of the
perturbation velocity fields for forward and reverse flow. It should
be remembered that in the flow reversal the lift distribution (not the
wing geometry) is fixed.

The same ideas may be applied if other singularities such as sources,
side force elements and volume elements are considered. When sources are
used the signs must be reversed when the flow direction is reversed. A
source in forward flow becomes a sink in reverse flow.


B. COMBINED FLOW FIELD CRITEP.IOII FOR IDEfTIFYIN1G
OPTIMUM LIFT DISTRIBUTIONS

A necessary and sufficient condition for ndminimum wave plus vortex
drag was given by T. Jones (15) in connection with planar systems. The
condition is that the downwash in the combined flow field shall be con-
stant at all points of the platform. This result depends on the fact
that a pair of lifting elements has the same drag in forward arid reverse
flow, which is also true when the lifting.elements are not in the same
horizontal plane. Hence the above criterion can be extended immediately
to lift distributions in space by requiring constant downwash (in the
combined flow field) throughout the space.






NACA TM 1421


C. THE COMBIrNED FLOW FIELD CRITERION FOR
IDENTIFYING OPTDIUl VOLUME DISTRIBUTIONS

A necessary and sufficient condition for minimum wave drag due to
thickness was given by R. T. Jones(13) in connection with planar systems.
If total volume is fixed then the optimum distribution of volume gives
a pressure gradient in the combined flow field which is constant over
the platform.

As in the case of lifting elements this criterion can be extended
to cover thickness distributions in space. It is then necessary for the
pressure gradient in the combined flow field to be constant throughout
the space.


D. UNIFORM DOWJWASH CRITERION FOR [1-iIIJIiT1 VORTEX DRAG

A necessary and sufficient condition for vortex drag alone to be
a minimum is -hat the downwash velocity throughout the wake of the win:
system shall be constant in the Trefftz plane. (The wake cross-section
is the projection of the wing system on the Trefftz plane.) This condi-
tion was given by Munk(15).

If the wake of the wing system has an elliptical cross-section then
a constant intensity of lift over the cross-section satisfies the above
condition and gives the minimum possible vortex drag. (See Appendix V-l).
In particular when the cross-section of the wing wake degenerates into
a horizontal line, (corresponding to a planar.wing) the familiar require-
ment of elliptic spanwise load distribution is obtained.


E. ELLIPTICAL LOADING CRITERION FOR I1IUTUNIRl WAVE DRAG DUE TO LIFT

In special cases ellipti' loadings identify minimum drag configura-
tions, as has been shown by Jones(1 Let the space containing the
lifting elements be cut by a series of parallel planes each inclined at
the Mach angle to the flow axis. Consider all the lift intensity cut by
any one plane to be located at the intersection of the plane with the
flow axis. If the resulting load distribution on the axis is elliptical,
and if this is true for all possible sets of parallel planes (inclined
at the Mach angle)., then the wave drag is a minimum.


In Hayes(1) procedure for calculating drag (see Ch. IV) this con-
dition corresponds to obtaining the minimum possible drag contribution
at every angular position on the cylindrical control surface.







NACA TM 1421


Such minima cannot be attained in general since the condition is
sufficient but not necessary. However if they are attained and if the
vortex drag is also a minimum then the more general criterion (constant
downwash in the combined flow field) is satisfied.


F. THE "ELLIPTICAL LOADING CUBED" CRITERION FOR
MIIJDLRJl WAVE DRAG DUE TO A FIXED TOTAL VOLUME


Sears(16) and Haack(17) in determining optimum shapes for bodies
of revolution in supersonic flow have also determined sufficient condi-
tions for identifying optimum distributions of volume elements within
a prescribed space.

We consider a distribution of volume elements within a prescribed
space and ask how these elements should be arranged in order that they
should cause the least wave drag while providing a fixed total volume.
If the equivalent body of revolution for a given angular position 01
on the distant control surface (see Ch. IV) conforms to the Sears-Haack
:.ptinium shape then the wave drag contribution at 01 is a minimum. There-
fore if the equivalent bodies of revolution for all values of 9 are
optimum shapes the total wave drag is a minimum.

The density of the lineal distribution of volume elements repre-
senting the Sears-Haack optimum shape corresponds to the cube of an
elliptical distribution over the length of the line. Hence if all the
equivalent lineal distributions have this form an optimum is ensured.

Such minima cannot be attained in general since the "Elliptical
Loading Cubed" criterion is a sufficient, but not a necessary condition
for minimum drag. When such minima are attained the more general cri-
terion (constant pressure gradient in the combined flow field) is also
satisfied.


G. COMPATABILITY OF 1IIIIflUM WAVE PLUS VOPTEX DRAG
WITH 11IlIDi-lA WAVE OR M.1IrFnIJ1 VORTEX DRAG

It is possible for minimum wave plus vortex drag to be obtained
when neither the wave nor the vortex drag is individually a minimum.

For example consider that the "space" within which lifting elements
may be distributed is the platform shown in the figure. For the vortex
drag to be a minimum it is necessary to maintain an elliptic spanwise
loading over b. This requires a finite load on "a" which in turn pro-
duces infinite wave drag if the chord for "a" goes to zero. However






NACA TM 1i+21


b






the minimum drag due to lift for the platform is certainly finite (load
the end pieces only and consider them as isolated wings) hence minimum
vortex drag is not consistent with minimum total drag in this case.

On the other hand, for a planar wing of elliptical platform minimum
wave drag and minimum vortex drag are obtained with the same (constant
intensity) lift distribution.


H. ORTHOGONAL LOADING CRITERIA

Optimum distributions can be identified also through orthogonality
considerations(8,9). The optimum distribution of lifting elements in a
space is orthogonal to every distribution carrying zero net lift and is
not orthogonal to any other distributions.

A similar statement can be made for the optimum distribution of
volume elements alone (assuming for the moment that negative local vol-
umes are not excluded). However if lifting (and side force) elements
are introduced in addition to volume elements, then the criterion must
be modified. For example the rotationally symmnietric wing plus central
body having zero wave drag is orthogonal to all singularity distributions
although it contains a net volume.*

The criteria discussed in preceding sections of this chapter have
not been thoroughly investigated for cases involving lift and volume
elements simultaneously. However, some material on interference between
lift and volume distributions is given in Ch. IX.


See p. 105 ff. Since the wave drag is zero the disturbances on a
distant control cylinder are identically zero. Hence its interference
with any other singularity distribution is zero.






NACA iTM 1421


APPENDIX V


DISTRIBUTION OF LIFT IN A TRANSVERSE

PLANfE FOR I4IIMT-UM VORTEX DRAG


As stated by Mank's Stagger Theorem(15), the vortex drag of a spa-
tial wing system is not changed if all lift and side force elements in
the system are projected onto a single plane normal to the flight direc-
tion (see Fig. A'-l). Furthermore, if there are no side force elements,




DO/SrTQ/87T/ON OF y
Z IFTIN SPACE
PROJECT/ON OFLIFT
ONrO )Ye PLANE
U ^ / ____


Fig. A5-1


then Munk's criterion for minimum vortex drag is that in the Trefftz
plane, the downwash in the wake must be constant. (The wake cross-
section is defined as the projection of the wing system on the Trefftz
plane.) Assume that the downwash field associated with the optimum lift
distribution is w = -wo and that a uniform field w = +wo is superimposed
on the original field in the Trefftz plane; then the resulting two-
dimnensional flow pattern is equivalent to a uniform flow around a solid
body. Munk gives the expression for the lift distribution in the trans-
verse plane in terms of the velocity potential of this new flow for






NACA TM 1421


certain bodies symmetrical with respect to the x-z plane; for example,
if 0 is the two-dimensional potential flow around an elliptic cylinder,
then


zopt = 2pU"()
Sdzboundary


dvortexmin


where I and d are
transverse plane.
potential is(18)


Fig. A5-2
the lift and drag intensities per unit area in the
For an ellipse oriented as in Fig. A5-2, the


= wo(a + b)cosh@ go)sin I


where


y + iz = a2 b2 cosh(S + in)


The curve t = to corresponds to the boundary of the lift distribution
in the transverse plane. From the above equations one obtains


I = 2pU (
T1 dz _t
$=So


2pUwo(a + b)
b


so that the lift intensity in the transverse plane must be constant to
obtain minimum vortex drag. With S = nab, the drag is


-(w,\ L2
Dvortexmin L gS(+ a/b)


where L is the total lift generated. Thus to obtain minimum vortex drag
for a spatial distribution of lift whose Trefftz plane projection is an
ellipse with one axis vertical, the lift should be distributed so as to
give a constant intensity when projected on the Trefftz plane.


\u) Iopt






NACA TM 1421


This proof can be extended to cases in which the projected lift
distribution covers a rolled ellipse, as shown in Fig. A5-5. If only
lift (and no sideforce) elements are
allowed, Munk's criterion of con-
L stant downwash still holds, but the
lack of symmetry precludes use of the
formulas given above. However, the
(P optimum lift distribution can be
determined by a superposition of two
--- / symmetrical optimum distributions,
( / as shown in Fig. A5-4. L1 and L2





Fig. A5-5






-4 -W --W
1w = W, -f W

(a) (b) (c)


Fig. A5-4



are constant intensity lift distributions over the elliptic areas which
produce constant downwashes wl and w2 over those areas. Because the
governing equation is the Laplace equation, which is linear, the lift
distributions L1 and L2 and the flow fields they produce can be super-
imposed. If L1 = L cos 0 and L2 = L sin 0 and Fig. A--4c is rotated
through the angle 0, then Fig. A5-4c corresponds to Fig. A5-3. There
is a uniform downwash w corresponding to the uniform lift L. Thus
Munk's criterion is satisfied and the drag is a minimum. It can be
shown by symmetry that the total interference drag between the lift
distributions L and L2 is zero so that the drag of L is obtained
simply by adding the drags of L1 and L2; that is






NACA TM 1421


L 2 L2 2
I'ortexin 4qS(l + a/b) + qS(l + b/a)


L2(a sin2o + b cos2)
4qS(a + b)


It should be noted that for this optimum rolled ellipse case there is
also a uniform sidewash generated. If a distribution of side force
elements were available, it would be possible to utilize the uniform
sidewash to reduce the vortex drag below the value given above.






68 NACA TM 1421


CHAPTER VI. THE OPTIMUM DISTRIBUTION OF LIFTING ELEMENTS ALONE


A. THE OPTIMUM DISTRIBUTION OF LIFT THROUGH A SPERICAL SPACE

Consider a sphere of radius "R" with its center at the origin, and
let a total lift "L" be distributed through the sphere with local inten-

sity "Z." If I = L r being the radial distance from the
2RR2 2 r72
origin, then elliptic loadings are obtained when the sphere is cut by
any set of parallel planes (see Appendix VI for derivation). The fact
that elliptic loadings are produced when the planes are inclined at the
Mach angle (to the free stream direction) insures that the wave drag is
a minimum (Ch. V). The cross-section of the wake is circular, and if
the lift intensity is projected onto a plane normal to the free stream
direction it can be shown that the lift is uniformly distributed over
this circular cross section. This insures that the vortex drag is also
a minimum (Ch. V).
L
The lift distribution 7 = then gives the minimum pos-
I(P2R 2 2
sible wave and vortex drag. By Hayes' procedure it can be found that
LB22
the minimum wave drag is Dmin wave = 2 ; the minimum vortex
2nq(2R)dM2

_L2
drag(l) is Dmin vortex and the minimum total drag is
2nrq(2R)2


1>-- L' 212 l|
Dmin I LI.L
2 vq(.2R)2 1.2

The largest planar wing of circular planform contained in the sphere

has a minimum drag(14) which is greater by the ratio 21 This is
2M2 1
a factor of 1.885 at M = {2. However, the drag comparison is, of course,
not complete without consideration of the viscous drag (and thickness
drag. For the spatial lift distribution described above, the required
wing area is infinite and so, then, is the viscous drag. But the same
minimum of wave and vortex drag can be achieved with a number of wing
systems having finite wing area. For example, consider the infinite set
of cascades enclosed in a spherical space as shown in Fig. 6a-1. At







NACA TM 1421


O rT/tsw/M FOR MIN/IM/UM
voqrec ORAG(COnsrA^r
IN rTENSITY)


Oor/mA4UM 1Erui/ALENT LINEAL DISTRABUTrioN
PORA4IN/Mf/O-I1 -VAVE DRAG (Et-t LPrT/C)


Fig. 6a-l: Cross-sectional view of an optimum set of finite area
lifting surfaces in a spherical space

M = (2 this set of cascades covers the region adequately so that the
equivalent linear distribution will be continuous. Determining the lift
distributions for the cascades is essentially a stepwise process in that
the vortex drag criterion is satisfied over part of the space and then
the wave drag criterion over part, alternating back and forth until both
conditions are satisfied everywhere. In this example rotational symmetry
is assumed and the center cascade is used to satisfy the vortex drag
p
requirements; thus, the outer region L- < r P. of this cascade must
V-2






NACA TM 1421


carry a constant intensity of lift. The cascades of radius R/VF are
used to give the equivalent linear distribution the required elliptic
shape for R/V2$< t < R2. The next step is to evaluate the distribution
over another section of the center cascade to give constant lift inten-
sity when elements are summed up in the free stream direction, then
satisfy the wave drag criterion with the next cascade, etc. This proc-
ess is continued working inward to the center of the space; although
an infinite number of cascades are required the total wing area is
finite. Each of the small cascades has a radius 1/W2 times the radius
of the next larger one and the total wing area is S = 2.172itR2 (Ch. VI B).
It should be noted that this is not necessarily the minimum wing area
that could be used, so the distribution obtained is an optimum one with
respect to wave and vortex drag only and not with respect to friction
drag.


B. THE OPTIMUM DISTRIBUTION OF LIFT THROUGH AN ELLIPSOIDAL SPACE

The spherical space with its optimum lift distribution can be
changed into an ellipsoidal space with a corresponding lift distribution
by a scale transformation of one of the cartesian coordinates. This
transformation transforms planes into planes so that elliptical loadings
are preserved for the ellipsoid and minimum wave drag is obtained.

Also a constant intensity of lift over the wake cross-section is
maintained for the ellipsoid so that the vortex drag is also a minimum.

Although the optimum lift distribution for an ellipsoid is obtain-
able from the spherical case, the value of the minimum drag is not nec-
essarily the same. For an ellipsoid formed by revolving an ellipse of
semi-major axis B and semi-minor axis R about the free stream (major)
axis, the optimum distribution of lift is

L
Zopt = xB2 112
tA2R2B 1 (x/B)2 (y/R)2 (z/R)2


The wave drag, computed by Hayes' method, is


Dmin wave = rq 2L2[/ + ]
8 TtqR 2 [( B /1R,)2 + 02]






1ACA TM 1421


and the vortex drag is also a minimum,

l2
Dmin vortex L
8itqR2


so that the total drag is



Dmin = L2 P2 +1
81tqR2 (B/R)2 +21



For B = R the results reduce to the spherical case.

Several limiting cases can be examined; in one an ellipsoid is
collapsed into a horizontal planar wing of elliptic planfornm carrying
constant pressure. Optimum cases of this type were first discussed by
R. T. Jones(1'). Another limiting case which gives minimum drag occurs
when an ellipsoid is collapsed into a plane normal to the flow direc-
tion (B/R->0). Then the wing system can be interpreted as a uniformly
loaded airfoil cascade (of zero chord and gap) within the elliptical
cross-section. The entire cascade can be analyzed as a two-dimensional
system. If the chord is chosen to be p times the gap then the airfoils
in the cascade are non-interfering but the lift distribution is suffi-
ciently continuous (Fig. 6b-l). In other words, when the cascade is
cut by planes inclined at the Mach angle, the resulting load distribu-
tions used in Hayes' method will be continuous. The total wing area is
then p times the area of the ellipse.





/4 /X/
U_

7'\ \* \



/A/ rRFm-A/VVG HON-/NTrRFER?/NG MOv-,NTrERFeR1NG
A /.FO/LS- A/RFO/LS AIRFO/LS W1/THSU/FFICENL7Yf
CONrI/N/OUS L/FrTD/SrR/BU TION


Examples of airfoil spacing in cascades


Fig. ob-1:






NACA T14 1421


A third limiting case is the slender body obtained when B/R-4->;


then


Dmin L + A
2nq(2R)


-2L2 2 = Dvortex + Dwave
2nq(2B)


The wave drag portion is the samnie as that obtained by Jones for a planar
slender ding while the vortex drag for the spatial distribution is
one-half that obtained by Jones for the planar distribution.


C. -THE OPTEIUM DISTPIBUTIO[ OF LIFT THROUGH A "DOUBLE RACH CONE"


Consider
(Fig. bc-1).
and chord) is


a space consisting of two Mach cones placed base to base
If a uniformly loaded cascade of airfoils (with zero gap
placed at the maximum cross-section of this space then


*RADIUS = R


u


Fig. 6r-1: Double 11ach cone space
with optimnuit --ascade


elliptic loadings will be obtained when the space is cut by planes
inclined at the I-lach angle. This airfoil cascade consequently produces
the minimum possible wave drag fo.r wing systems contained within the
space and carrying a specified lift. The uniform distribution of load
over the circular cross-section insures minimum vortex drag also, so
the lift distribution is an optimum for the double Mach cone.

The value of the minimum wave drag (obtained by Hayes' method) is

1 L2
Dwave = r :- and the vortex drag has the same magnitude in this
anql2e.
case.






NACA T1M 1421


r2
The wave plus vortex drag is then D = L This is equal to
nq(2R)
the minimum vortex drag alone for a planar wing of span 2R. If the air-
foil cascade is compared to the largest planar wing of diamond planform
which can be contained within the double Mach cone, the minimum wave
plus vortex drag of the diamond planform is approximately 1.52 times
(2)
greater than for the cascade(2)

Again it must be emphasized that the drag comparison is not com-
plete without the inclusion of viscous drag and thickness drag for the
wing system.

Since the circular cascade is an optimum arrangement, it satisfies
Jones' criterion (Ch. V). This can be checked as follows: By two-
dimensional analysis the downwash, e, in the aft Mach cone is 2a where a
is the angle of attack of each airfoil (Fig. 6c-2). Since the downwash
is zero in the fore Mach cone, the downwash velocity in the combined
field is constant and equal to dU throughout the double Mach cone.

STREAML/NE
/_5 _____ V ~EXPANSION WAVe



4R +A C CONE O CASCA DE

F~ReA R MAdACHCONE Oc* CASCADE


\T--CO4fPRESS/ONV WAVE

Fig. 6c-2: Two-dimensional analysis of downwash in rear Mach cone
of an optimumr cascade






NACA TM 1421


Far behind the cascade in the wake of the wing system c = a.; this
can be shown by equating lift to rate of change of vertical momentum.
The individual wings of the cascade are non-interfering and, in the
limit as gap and chord go to zero, have two-dimensional wing character-
istics. The wing area for a sufficiently continuous lift distribution
(Ch. VIB) is equal to the cascade cross-sectional area A times 3. Con-
sequently L = CLqS = (1Ix/p)q(pA). By Munk's criterion (see Ch. V and
Rnf. 12) the downwash in the Trefftz plane over the area behind the
cascade is constant; thus, the vertical momentum of the fluid in the
downwash region behind the cascade is (pAU) (cU). The vertical momen-
tum of the surrounding fluid can be evaluated from the known "virtual
mass" of a solid circular cylinder of cross-sectional area A moving
downward in the fluid; this latter momentum is equal to that of the
downwash region itself. Thus, by the momentum theorem, L = 2pAU(EU)
and equating the two expressions for L gives E = a.

The airfoil cascade is not the only distribution of lift in the
double Mach cone which has minimum wave drag. A true lineal distribu-
tion of lift distributed as an elliptic loading along the axis of the
double Mach cone will produce the same minimum value of wave drag. So
also will a lift distribution of constant intensity throughout the entire
double Mach cone. However, the latter two cases will not give the mini-
mum value of vortex drag; in fact, the true lineal distribution will
have infinite vortex drag.


ortex drag.






NACA o4 1421


APPEfDIX VI


DERIVATION OF OPTD.IUMI DISTRIBUTIOIl OF LIFT THROUGH A SPHERICAL SPACE


A sufficient condition for minimum drag is that each equivalent
lineal distribution of lift should be elliptic (Ch. V). For the spherical
space these equivalent lineal distributions will be the same at all angu-
lar stations if the optimum lift distribution is rotationally symmetric.
For simplicity, examine the problem from the angular position 9 (on the
control surface) equal to 900; then the Mach planes will be parallel to
the y axis. The notation to be used is illustrated in Fig. A6-1; cylin-
drical coordinates (t,S,O) and the radial coordinate r will be used.

If the spatial lift distribution is 2(r) = 1( 2 + S2) then the equiva-
lent lineal distribution along the axis will be


r R2t 2 r2t/ \ t 2
F() = / 2 (2 + S 2 Sd dS = 2n t
D 3=0 Jo=0 0


9.-z


U


S1 (2 + S2dS


7^1 -2~ R2


71 If


K>


Fig. A6-1


However,


F() opt = K1-- (/R)2






76 I1ACA TM- 1421


where K depends on the total lift of the sphere. Introducing the radial
coordinate r, the integral equation to be solved is


K 1l (R/B)2 = 2it R rl(r)dr



The solution to this equation, found by differentiation with respect
to t, is

K
I(r) = -- --
21rR 2 r2


The total lift of the sphere is


L = 2 R K l (/R)2 d K = nRK
L=22


so that the distribution of lift for minimum wave drag is


1(r) = L
IC2R2 :R r2

For application of Hayes' method, the equivalent lineal distribution
along the x axis is needed. A plane t = C' intersects the x axis at
x = -Mt'; since the distribution is spread out over a larger distance
along the x axis, its maximum intensity will be less; thus,


F(x)opt = 1 (x/MR)2 1 (x/MR)2


Hayes defines two functions such that for the lifting case (Ch. IV)


F pUgz


-2|3L sin u8 /_,
h = -g. sin =,-2L sin x/ME)
npULRM






NACA TN 1421


The expression for the wave drag contribution at each angular station 9
is, from Eq. (4c-50),


dD = Pffh'(x2)h(xl)lnx2 x l dx dx2


and the total wave drag is

p2i
Dwave f2 dO


The integration for dD/d9 has been carried out by Sears(16) in terms of
a Fourier series expansion of an arbitrary function h. For the wave
drag optimum the distribution h is elliptic and only the first term in
the series for h appears. (Note the similarity to the vortex drag opti-

mumrs in Incompressible flow.) If h = C01 (x/MR)2 then dD/d9 = pC2 /16.
Substituting in the equations above leads to the final result,

02L2
Dwave =- o- o
8-nqR -m






78 NACA TM 1421


CHAPTER VII. THE OPTfIUM DISTRIBUTION OF VOLUME ELEMENTS ALONE*


A. THE SINrGULARITY REPRESENTING AN ELEMENT OF VOLUME

The investigation of lift distributions is simplified by the use of
a singularity which represents an element of lift. This singularity is
the elementary horseshoe vortex. The intensity of lift corresponds to
the strength of the singularity and the location of the lift force is
identical with that of the bound vortex. The study of volume (or
thickness) distributions is similarly simplified by identifying the sin-
gularity which corresponds to an element of volume.

Consider a source and sink of equal strength and located on the same
streamwise line. In each unit of time a certain quantity of fluid is
introduced into the flow pattern by the source and the same quantity is
removed by the sink. The volume occupied by the fluid flowing from source
to sink depends on the strength of the source and sink and the distance
between them, and also depends on the velocity and density of the fluid
flowing from source to sink. However, if the volume is to be considered
a linear function of the strength of the singularities, then the mean
value of density times velocity must be unaffected by the perturbation
velocities created by the source and sink. This means that in a line-
arized treatment of the problem the fluid flowing from source to sink
may be considered to have free stream density and velocity.

Let m = Mass of fluid introduced per unit time

d = Distance between source and sink

p = Free stream density

Uo = Free stream velocity

Then the volume occupied by the fluid is

vol = md poU)

Since the volume is proportional to md, doubling the intensity of
source and sink and halving the distance between them should produce a
shorter, but thicker volume of the same magnitude. This suggests pro-
ceeding to the limiting case (as in incompressible flow) where the source


The contents of this chapter have appeared in the paper "The Drag of
Non-Planar Thickness Distributions in Supersonic Flow," published in
the Aeronautical quarterly, Vol. VI, May 1955.






NACA TM 1421


and sink are combined in a dipole with axis in the free stream direction.
This singularity should represent an element of volume, although the
fineness ratio of the element is zero.

The potential for a unit source at (E,0) in supersonic flow is


OS = 24x -12
2n (x -)2 -r2

where = VM 1; x and S are coordinates in the streamwise direction
and r is radial distance from the x axis.

Differentiating with respect to x gives


Sx =


(x )

2g (x _)2 2r2]5/2


where is the potential for the unit dipole or an element of volume
equal to 1 /U0.


B. THE DISTRIBUTION OF VOLUME ELEMENTS

For a distribution of volume elements along the t axis with inten-
sity f(), starting at t = 0, the potential is


= 1fx-pr
1 0


f(t)(x 2)dt

[(x )2 2r2


Integration by parts gives


f(-) x-Or
2n (x t)2 p2r2IO


1 lfx-Or
2_ 0


f'(t)dt
(x E)2 p2r2


The first term in the expression for the potential is infinite,
and apparently corresponds to the "roughness" of the body, which is an
assembly of blunt elements (see illustration).


IU


= O






NACA TM 1421


The smoothly faired body (indicated by dash lines) is all that we
are concerned with, and this creates the finite part of the potential.
This finite part is also the potential for a source distribution of
intensity equal to +f'(|). This source distribution can be used to con-
struct a body of revolution extending from -Z/2 to +1/2.

The shape of the body of revolution created by the singularity dis-
tribution may be obtained approximately by slender body theory or more
accurately by "exact" linear theory. In the first case the volume is
+1/2
/ I`d which agrees exactly with the sum of the volume elements.
-1/2 U0
An example of the second case is shown in Fig. 4f-2 where a singularity
distribution on the axis is interpreted first by slender body theory
then by "exact" linear theory as a "bump" on a cylinder. The bump shapes
and locations are quite different but the volumes are identical. This
has been proved by Lagerstrom and Bleviss and generalized by Bleviss in
Ref. 22.

A planar distribution of volume elements may be interpreted by
("exact") linear theory as a thin planar wing. The volume contained in
this wing is exactly equal to the sum of the volume elements.

The concept of the volume element is not necessary for the study of
smooth slender bodies of revolution and planar wings, since these con-
figurations are relatively simple. However the use of the volume element
does help to clarify problems involving more general spatial distributions
of thickness.

The points to be emphasized are that fixing the sum of the volume
elements fixes the total volume, and fixing the distribution of volume
elements determines the drag. It is therefore possible to study the
drag of a distribution of volume elements without calculating the exact
shape of the corresponding body. This is analogous to the fact that the
drag of a distribution of lifting elements can be studied without calcu-
lating the twist and camber of the corresponding wing surfaces.


C. THE DRAG OF VOLUTE DISTRIBUTIONS Oil A
STREAlW-MISE LIIE AJ1D THE SEARS-HAACK EODY

A body of revolution may be constructed from a distribution of vol-
ume elements along a streamwise line, or from the equivalent distribu-
tion of sources. The body constructed from volume elements is an
"infinitely rough" body and has infinite drag. However, discarding the
infinite part of the potential leaves a "smooth" body (with finite drag)
which is equivalent in every respect to the body created by a source
distribution.






NACA '1 1421


If f(x) is the intensity of the volume element distribution for a
body of revolution of length "1" then the drag is given by(16)


p+1/2 +1/2
D = /2 41/2


f" (x1)f" (x2) In xl x2 dxl dx2


To maintain constant total volume according to linearized theory
+ 1/2
it is necessary that f-/2 f(x)dx = Constant. The body shape giving

minimum drag for a given length and volume has been determined by Sears(16)
and Haack(17) independently. The corresponding f(x) (which is propor-
tional to the cross-sectional area) is given by


opt (x) = 1 -


+1/2
-5/2 / f(x)dx .
2x 32 1 1/2 f(x)dx 8Uo volume
I 5n 1/2 L


- ) /2


Thus the optimum distribution of volume elements along the axis
corresponds to the cube of an elliptical distribution. (For lifting
elements the optimum distribution is elliptical.)

The value of the minimum drag is


Dmin = V2-


2 -2
8q volume
~ 2) (/2)5


D. THE SEARS-HAACK BODY AS AN OPTIMUM
VOLUME DISTRIBUTION IDJ SPACE

If the volume elements are not confined to a single streamwise line,
then the drag contributions at different angles, 9, on Hayes' cylindrical
control surface are not necessarily the same. For any one angle, 9, the
drag is given by


-.+1/2 +1/2
= 2- /
Sir2 _1/2 -1/2


f" (xl),9 f" t(x,9) In xl x2l dxl dx2






NACA TM 1421


Here f(x,9) is determined by the use of "Mach planes" for the
angle 9. All the volume elements intercepted by any one "Mach plane"
are transferred (in the plane) to the streamwise axis. The resulting
distribution along the axis is f(x,9). The problem of finding the mini-
mum drag contribution at the one angle 9 is then similar to the Sears-
Haack problem. If f(x,9) corresponds to the cube of an elliptical dis-
tribution for every 9, then the total drag is a minimum, and the drag
contribution at each 8 is a minimum and corresponds to that of an equiv-
alent Sears-Haack body.


It is not always possible to simultaneously mini
contributions at all angles 8. However if we consider
distribution of thickness
within a space which has
rotational symmetry about
a streanwise axis, then
it may be possible that
all the equivalent bodies
are Sears-Haack bodies
having the same length.
For example, consider that
a double Mach cone bounds
the space within which ----
thickness is to be distrib-
uted. The Sears-Haack
body placed on the axis
is an optimum for this
space. It has the same
drag contribution at SEARS-HAACK BOD
every angle on the cylin- BYDOUBLE AIACH
drical control si-rface,
and of course, the
"equivalent" body of
revolution for any angle 9 is identical with the real
a "ring" wing (which carried no radial forces) plus a
revolution can be designed
to have exactly the same
drag as the Sears-Haack
body. The equivalent -
bodies of revolution are
all identical with the- -
Sears-Haack body. This
is discussed in the next
section. (For the case --
in whic-h radial forces
are carried on the ring
wing see Ch. IX.) RING WINVG PLUS C


nize the drag
r the optimum


Y BOUNDED
CONE SPACE



body. However,
central body of


EA'7RAL


BODY IA V/NG SAME ORAG
AS SEARS-1VAACK BODY






NACA um 1421


E. RINIG WING AND CENTRAL BODY OF REVOLUTION
COMBINATIOIJ HAVING THE SAME DRAG AS A SEAJRS-HAACK BODY

Consider a ring-wing plus a central body of revolution contained
within the space bounded by a double Mach cone. Because of the rota-
tional symmetry of this particular system, the equivalent body of revo-
lution is independent of the angle 9 on the cylindrical control surface.
In this case, if the local radial force on the wing is everywhere zero,
the drag of the equivalent body of revolution is, according to Hayes'
formula, identical to the drag of the original system. Thus, a ring-
wing (which carries no radial force) plus a central body of revolution
will have exactly the same drag as a Sears-Haack body if the equivalent
body of revolution is a Sears-Haack body.

To design such a system, we may select any smooth, slender profile
for the ring-wing and compute the cross-sectional areas cut from this
wing by a set of parallel Mach planes. These areas must then be sub-
tracted from the cross-sectional areas which would be cut from a central
Sears-Haack body by the corresponding Mach planes. The resulting area
difference defines the area distribution (in the Mach planes) of the
correct central body. (This area must be projected normal to the flow
direction to obtain the cross-sectional area of the central body defined
in the usual way.) This body, together with the ring-wing originally
selected, is an optimum distribution of thickness within the double Mach
cone space.

As an example, consider a ring-wing with thickness distribution
corresponding to a bi-parabolic arc profile. The camber necessary for
zero local radial force need not be determined, since it does not affect
the shape of the central body. Assume that the wing is six percent thick
and located half-way between the axis and the apex of the'space. If the
central body of revolution is designed so that the equivalent Sears-Haack
body is of fineness ratio 5, the resulting shape of the central body of
revolution is as shown in Fig. 7e-l.


as shown in Fig. 7e-1.







84 NACA TM4 1421





RING W/V/ NG, /C = 0.06









EQUIVALENr BODY OF
REVOLUTION

CENTER BODY OF
REVOLUrTION








Fig. 7e-l: Cross-sectional view of ring-wing and central body
(an optimal distribution of thickness within the double Mach
cone space)

F. OPTDTUJ THICKITESS DISTRIBUTION FOB A
PLAIUAR WIJG OF ELLIPTICAL PLAIFORV

It is desired to find the optimum thickness distribution for a
planar wing of elliptic planform and given volume; this problem was
first solved by R. T. Jones(14). A geometrically simpler problem, which
will be examined first, is to find the optimum thickness distribution
for a circular wing of given volume. The method of Hayes(l) in which
the drag is evaluated by summing increments of drag at each angular sta-
tion around a cylindrical control surface far away from the body, will
be used. For the total drag to be a minimum, the increment of drag at
each angular station should also be a minimum.

If the thickness distribution of the circular platform is rota-
tionally symmetric, then the equivalent bodies at each angular station
will have the same shape (although different "fineness ratios") due to
symmetry. If t(r) is the thickness distribution to be optimized for a
given volume V, then






NACA TM 1421


R
V = 2t t(r)r dr
0


(7f-1)


where R is the radius of the circular wing and r, 0 are polar coordirnates
from the wing center (Fig. 7f-l). The area cut out at each point along
the t axis by planes normal
to that axis is


+ t
s(0 = / r^ t


.R
=2
J


t(r)r dr
r2 2


(7f-2)


The equivalent lineal distribution along the x axis is


R
S(x) = 2 cos p
X COS 11


rt(r)dr
r2 x2cos2p


+R sec .

-R sec c


S(x)dx = V


For minimum drag, this distribution should be (Ch. VF)


S(x) a 1 (x/B sec )2 /2

Thus the integral equation to be solved for t(r) is


K I (x/B sec


)2] = 2 cos B t(r)r dr
Scos I r2 2cos2
fJ xCos V yr x Cos j.


U


with


(7f -5)


(7f-4)


(7f-5)






86 NACA TM 1421


where K is a constant dependent upon the given wing volume. By a suit-
able transformation of coordinates, Eq. (7f-5) may be written in the form

K (2 (7f-6)d)
R sec p o .


where

= 1 (x/R sec 2)

a = 1 (r/R)2


Eq. (7f-6) is called Abel's equation and its solution is well known,
c.f., Ref. 19. The solution to Eq. (7f-6) is


t(r) = [K (r/R)2]
4R cos uL 1

and substitution of this in Eq. (7f-l) determines K; then


t(r) = 2V 1 (r/R)2] (7f-7)


Equation (7f-7) thus gives the distribution cf thicKness which will
result in minimum drag for the circular planform wing of given volume.

To apply the circular planform solution to the original problem of
finding the optimum thickness for an urnyawed elliptic planform, make
the following change of coordinates:


y X I w
i-b (7f-8)

-yY/. x
yAY// ^
U -00 A 4 & --


Fig. 7f-2






NACA TMi 1421 87


The circular wing is then transformed into an elliptic wing whose equa-
tion is


(x)2 + ()2 1
a b

It can be verified that the thickness distribution


2V XY2 /\
t = 2V1 ( (7f-9)
rtab[ ( a ( b2]

obtained from Eq. (7f-7) through the transformation Eq. (7f-8) is the
optimum for this more general case; that is, the equivalent linear dis-
tribution for Eq. (7f-9) with a set of Mach planes inclined at the
angle as shown in Fig. 7f-2 is

13/2
s(x) = 1 (2 (7f-10)


where


S= a2 + b2tan2.

Since Eq. Tf-10) represents a Sears-Haack body, the thickness given by
Eq. (7f-') is optimum for the unyawed elliptic wing.

Determination of the total drag in this optimum case involves an
integration of the drag increments from these Sears-Haack bodies as
seen at each angular reference station. If the reference station is
at an angle 6 from the horizontal, then the Mach planes cut the elliptic
planforir at an angle p. defined as (Ch. IVC).


tan p = M2 1 cos 9 (7f-ll)


and the total draF is


~de
D 21c dO
lo d9





88 NACA TM 1421


The increment of drag at each reference station is (Ch. IVC)



f S"(x)S"(6)iZnx fIdx d

(7M-12)
4q2
\214


and the total drag for the optimum thickness distribution Eq. (7f-9' is



SM2 1 + 2a2

Dopt = 4ab )/2 (7f-15)
+2 a2
Di- 1n+i-
bD

Defining


t = to 1


(x)2
a


(Y J2


D = CDqnab


then


M2 -1 + a- 2

CDopt (2 +
M2 1+ 5/2
Ti l + b2

This result a,-reec with that given by Jones( 14.


(7f-14)






NACA TM1 1421


CHAPTER VIII. UIJIQUEifESS PROBLEMS FOR UPTIRr-1 DISThIBUTIONS IJ SPACE


A. THE IOfJ-UNIiUEirESS OF OCPTMlM DISTRIBUTIONS IN
SPACE "ZERO LOADINGS"

In the subsonic flow of a perfect fluid the only drag caused by a
lifting wing is vortex drag. The minimum possible vortex drag for a
planar ding is obtained when the spanwise lift distribution is elliptical.
According to [unk's stagger theorem(15) the chordwise location of the
lifting elements is unimportant, so there are infinitely many distribu-
tions of lift over a given planform which produce the minimum drag.

In supersonic flow lift causes both vortex drag and wave drag. The
chordwise location of lifting elements is still unimportant in deter-
mining vortex drag, but does affect the wave drag. For this reason the
optimum lift distribution for a planar wing is generally unique in super-
sonic flow. However, spatial lift distributions offer more freedom in
the arrangement of lifting elements and the optimum distributions in
space are not generally unique even in supersonic flow.

For example, the minimum wave drag due to lift in a double Mach
cone space can be attained with each of three different simple lift dis-
tributions. (See VI-C.) The first is a constant intensity over the
circular disc located at the maximum cross-section of the space. The
second is an elliptical intensity concentrated on the axis of the double
11ach cone. The third is a constant intensity throughout the entire
double Mach cone. If the first two distributions are superimposed, one
carrying a unit of positive lift and the other a unit of negative lift,
the result is a net lift equal to zero. Also, the net strength of the
lifting elements intercepted by any cutting plane inclined at the Mach
angle is zero. This means that the combined distribution has zero wave
drag. Furthermore, there are no disturbances whatsoever produced on
the distant control surface near the Mach cone and no wave drag inter-
ference can exist with any other loading. If another such combined dis-
tribution with opposite sign is placed on the same streamwise line with
the first one, then, by Munk's stagger theorem, the vortex drag is zero
also. This is one example of a "zero loading" (see illustration), and
many others can be constructed.






NACA TM 1421


low1
-- ^ / / K /--








A "ZERO LOADING "PLACED WI/I//N AN ELLIPSO/DAL SPACE

Such a "zero loading" placed within any space alters neither the
lift nor the drag of the original lift distribution. For this reason
optimum lift distributions in three dimensions are never unique (unless
the space degenerates into a surface).

Similar arguments can be applied to optimum distributions of volume.
For an example of non-uniquieness in such cases see Ch. VII.


B. UNIQUENESS OF THE DISTANT FLOW FIELD
PRODUCED BY AN OPTDU4JM FAMILY

It has been shown that optimum lift or volume distributions in
space are not generally unique, since a group of optimum distributions
can be obtained from one given optimum distribution by superposition of
"zero loadings." Each member of the group produces the same (minimum)
value of drag for a given total lift or volume.

From the method of construction of this group (by the use of "zero
loadings") it follows that each member produces the same velocity per-
turbation field in the Trefftz plane and on the distant control surface
near the Mach cone. It can also be shown that there are no optimum dis-
tributions outside this group, since all possible optimum distributions
are indistinguishable from the "distant" viewpoint.

Assume that flopt(EYTI,) and f2pt(t,T1,) are members of the opti-
mum family not included in the original group (whose members were related
through "zero loadings"). Assume also that f lopt and f2pt do not pro-
duce identical perturbation velocity fields far from the singularity







NACA TM 1421


distribution. Then the drag of fl equals the drag of f2opt (or

Dlopt = D2pt) by definition of the optimum family. Also f2opt may be
set equal to flpt + Lf, where 2f carries zero net lift (or volume),
but has a velocity, perturbation field which is not identically zero far
from the singularities.

The distribution Lf is orthogonal to (does not interfere with) flopt
-opt,
This follows because an,' given lift or volume distribution can be improved
through combining it with a distribution having zero net lift or volume
if there is interference drag. However flt by definition, cannot be
improved, and must, therefore, be orthogonal to Af.

Since If is orthogonal to flo, D2,,pt = DIopt + DAf, but we also
lopt' -opt lopt I
know that D2 = DI and, therefore, Dp must equal zero. Here we
dopt opt
can obtain a contradiction since both the vortex drag and the wave drag
depend on the squares of velocity perturbations (in the Trefftz plane
and far out on the Mach cone) and the drag contribution from each portion
of the control su-rface is non-negative. If t6f produces any disturbances
far from the lifting system it must have positive drag, and so6f must
produce identically zero disturbances to have zero drag.

The above contradiction shows that all the members of the optimum
family are indistinguishable from the distant viewpoint.

If drag is computed from the "close" viewpoint the above argument
cannot be made. Drag contributions then appear as the product of local
pressure times angle of attack on the wing surfaces, and these quantities
are not necessarily non-negative at every point on the surface.


C. UNIIULENESS OF THE ENTIRE "EXTlERIJAL" FLOW FTELD
PRODUCED BY AN OPTEfUA FAMILY

It has been shown that any two members of an optimum family produce
identical velocity perturbations on the distant control surface.

If fl (t,,t) and f2 (t 1 are two members of an optimum
-opt 2opt
family, then flopt f2pt must produce identically zero velocity per-
-'opt 'opt
turbations on the distant control surface, and the drag will be zero.

Let "S" designate the space within which the singularity distribu-
tion flopt f2pt exists, and let "E" represent the external flow field







NACA Ti-1 1421


consisting of points whose aft Mach cones
do not intersect "S." Assume that at
some point in the external field "E" the C
resultant velocity vector is inclined to \
the free stream direction. Then an ele-
mentary wing can be inserted at that
point with the angle of attack adjusted
to give negative drag on the wing. Since F
the singularities in "S" are outside the
aft Mach cones of all points in "E," the
net drag change produced by the elementary -
wing is negative. However, f f2
'opt opt c
is a singularity distribution causing zero -
drag, so flopt f2opt plus the elementary
wing is a system having negative drag, although it is an isolated system
inserted in a uniform flow field. However, the drag of this system eval-
uated on a distant control surface comes from a summation of positive
quantities and cannot be negative. This contradiction shows that the
external flow field "E" produced by flopt f2pt must consist of velocity
opt opt
vectors aligned with the free stream direction. These vectors must also
have the magnitude of the free stream velocity; hence, the external flow
field is completely undisturbed, and it can be concluded that all members
of the optimum family produce the same flow pattern in the external
field "E."

It is of interest that a similar proof cannot be made for subsonic
flows. In such cases there is no external region where an elementary
airfoil can be inserted without producing interference effects at the
original singularities.


D. E:ISTETICE OF SYI.-1ETRIC AL OPTI-TUI DISTRIBUTIONS
IN SYMMETRICAL SPACES

It can be shown that, if the boundary of a space has a horizontal
plane of syrrmmetry, then there is one member of the family of optimum
lift distribution within the space which is symmetrical about the plane.
The proof is as follows:

Let iopt(x,y,z) represent an optimum lift distribution in the space.
The distribution iopt(x,y,-z) has the same drag and lift (the drag of







NACA Ti 1421


the individual lifting elements
is unaltered by the change of
position, and the interference
drag of any element pair is
unaltered also).

Since topt(x,y,-z) has the
same lift and drag as opt(x,y,z))
it is also a member of the optimum
family. All members of the opti-
mum family produce the same exter-
nal flow field, and any distribu-
tion producing that field is an
optimum. The distribution,

lopt(xy,-z) + -lopt(x,y,z) pro-

duces the same external flow field
as lopt(x,y,z). It is, therefore, an optimum, and since it is also
symmetrical about the horizontal plane the proof is completed.

Similar proofs can be developed for cases where lift, thickness,
and side force elements are present. Also certain other planes of
symmetry can be used.






NACA 24 1421


CHAPTER IX. IlfVESTIGATIOrJ OF SEPARABILITY OF LIFT,

THICKNESS ANJJD SIDEFORCE PROBLEMS*


A. THE SEPARABILITY OF OPTE-MU
DISTRIBUTIOUIS PROVIDING LIFT AND VOLUME

Separability Questions

For the purpose of drag evaluation a complete aircraft is repre-
sented by a distribution of lift elements, volume elements and possibly
sideforce elements in space. A certain net lift must be provided to
support the weight and a net volume must be provided to house payload,
fuel, structure, etc. The drag should then be made as small as possible
with the net lift and volume equal to the prescribed values.

Several questions arise. Can we first study the problem of how
best to provide the required lift (with no net volume), then determine
the best way to provide the required volume (with no net lift), and
finally by superposition obtain the optimum distributions of singulari-
ties for simultaneously providing the net lift and volume 7 If this
procedure is possible will the drag of the combination be the sum of
the drags of the two superimposed distributions? Does the optimum way
of providing the lift with no net volume require only lifting elements
or are volume and sideforce elements necessary? Similarly does the
optimum way of providing the volume with no net lift require singulari-
ties other than volume elements?

For horizontal planar systems the answers to these questions are
comparatively simple. The lift and volume problems can be studied sepa-
rately and the optimum singularity distributions superimposed. The drag
of the combination is the sum of the drags of the individual distribu-
tions. Finally, the optimum way of providing the lift requires only
lifting elements and the optimum way of providing volume requires only
volume elements.

All of the above results follow from the fact that in horizontal
planar systems there is no interference drag among lift, sideforce, and
volume elements. However this is not true in general for non-planar
systems, and consequently the above problems must be re-investigated
for these more general configurations.


* Portions of this chapter have appeared in the paper "The Drag of
Non-Planar Thickness Distributions in Supersonic Flow," published in
the Aeronautical Quarterly, Vol. VI, -lay 1955.




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