Some examples of the applications of the transonic and supersonic area rules to the prediction of wave drag

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Material Information

Title:
Some examples of the applications of the transonic and supersonic area rules to the prediction of wave drag
Series Title:
NACA RM
Physical Description:
46 p. : ill. ; 28 cm.
Language:
English
Creator:
Nelson, Robert L
Welsh, Clement J
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics, Transonic   ( lcsh )
Aerodynamics, Supersonic   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: The experimental wave drags of bodies and wing-body combinations over a wide range of Mach numbers are compared with the computed drags utilizing a 24-term Fourier series application of the supersonic area rule and with the results of equivalent-body tests.
Bibliography:
Includes bibliographic references (p. 26-27).
Statement of Responsibility:
by Robert L. Nelson and Clement J. Welsh.
General Note:
"Report date January 18, 1954."
General Note:
"Classification changed to unclassified Authority: NASA Technical Publications Announcement No. 8 Effective Date: July 22, 1959 WHL."--stamped on cover

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003809079
oclc - 133092791
System ID:
AA00006400:00001

Full Text
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NACA RM L56D11 CONFIDENTIAL

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


RESEARCH 1 NEMORAIDUl4


SOME EXAMPLES OF THE APPLICATIONS OF THE TRANSONIC

AND SUPERSONIC AREA RULES TO THE

PREDICTION OF WAVE DRAG

By Robert L. Nelson and Clement J. Welsh


SUMMARY


The experimental wave drags of bodies and wing-body combinations
over a wide range of Mach numbers are compared with the computed drags
utilizing a 24-term Fourier series application of the supersonic area
rule and with the results of equivalent-body tests.

The results indicate that the equivalent-body technique provides a
good method for predicting the wave drag of certain wing-body combina-
tions at and below a Mach number of 1. At Mach numbers greater than 1,
the equivalent-body wave drags can be misleading. The wave drags com-
puted using the supersonic area rule are shown to be in best agreement
with the experimental results for configurations employing the thinnest
wings. The wave drags for the bodies of evolution presented in this
report are predicted to a greater degree of accuracy by using the frontal
projections of oblique areas than by using normal areas. A rapid method
of computing wing area distributions and area-distribution slopes is given
in an appendix.


INTRODUCTION


The area rule, first advanced by Whitcomb in reference 1, has con-
siderably altered the methods for predicting wave drag of wing-body com-
binations. Studies leading to the discovery of the area rule showed that
interference drag between wing and body components could be very large.
Therefore, estimation of drag by component buildup without somehow evalu-
ating the interference drag could give misleading answers. However, in
consequence of the transonic area rule, a valuable tool was made avail-
able to the designers in assessing the transonic drag. This was the
equivalent-body concept, which states that at transonic speeds the pres-
sure drag of the airplane is the same as that for a body of revolution


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2 CONFIDENTIAL NACA RM L56D1


having the same longitudinal distribution of cross-sectional area. As a
result, the drag of the configuration is obtained by either estimating
or experimentally determining the equivalent-body drag. Experimental
checks for airplane configurations presented in reference 2 generally
support this concept in the transonic speed range.

The supersonic area rule, given by Jones in reference 5, provided a
powerful method for calculating the wave drag at supersonic speeds. In
references 4 and 5 the mechanics of the drag calculations were discussed
together with a number of comparisons of calculated and experimental
drags generally at low supersonic speeds. Jones pointed out in refer-
ence 3 that the method could be expected to give good results for thin
wings mounted on vertically symmetrical bodies. Later, Lomax in refer-
ence 6 gave the complete linearized theory expressions for the drag.
The added terms in Lomax's result represented the limitation pointed out
by Jones.

The purpose of the present paper is to provide a better feel for
the range of applicability of both the transonic and supersonic area
rules. For the transonic area rule, this is done by making additional
comparisons between equivalent-body and wing-body experiments. For the
supersonic area rule, comparisons are made of calculated and experimental
results for both body and wing-body combinations over a wider range of Mach
numbers than heretofore made. The supersonic-area-rule calculations were
made by using a 24-term Fourier series expression for the slope of the area
distribution.


SYMBOLS


A frontal projection of the area cut by a Mach plane or wing
aspect ratio


an = dx r A sin no do
an =J \dx 2 q

CD drag coefficient, -
qS

Cp pressure coefficient

c wing local chord

cO root chord of particular pointed wing tip

cr wing root chord


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Ct wing tip chord

D drag

d maximum body diameter, 2rm

f resultant pressure force

f(v) wing-thickness-distribution function, z
max

G(K,v0) wing-area-distribution function

H(K,Vo) wing-area-distribution slope function


K m(l + cos ) for the left-wing panel; ml cos e for
tan A / tan A /
the right-wing panel

1 length of configuration

lt total length of area distribution

l/d body fineness ratio

M Mach number

m= A (1 + tan A
4 (1 -

n integer

q dynamic pressure

r local body radius

rm maximum body radius

S reference area

Sb body frontal area

Se wing exposed area

S, wing total plan-form area


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So
s



t/c

x,y,z

x0

xt

xl

max

P = yM2 1




0 = cosl(2

A






S=
c

V0

Vr


wing semispan

semispan of particular pointed-tip wing

2Zmax
wing thickness ratio, 2z

Cartesian coordinates

point of intersection of Mach plane with the x-axis

x-coordinate measured from wing leading edge

dummy variable

local maximum wing ordinate


x
It

wing

wing


Mach


1)

leading-edge sweepback angle

taper ratio, -t
cr

angle, sin-1
R


value of v at root of particular pointed-tip wing

value of v at root of actual wing


6 angle between z-axis and line of intersection of Mach plane
with the y,z plane


REVIEW OF THE BASIC THEORY


From reference 6, the equation for the wave drag of any system of
bodies or wings and bodies can be written as:

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D 21 d t It d d2A(x,e) t df(x,9) d2A(xi )
q 42 0 0 l x2 2q dx dX-2


S-df(xl ) log(x Xl) (1)
2q dxl J

The equation is subject to the usual limitations of the linearized theory.

Before discussing the terms in the drag equation, it is well to review
the definition of Mach planes. The physical significance of equation (1)
is understood if the configuration is cut by Mach planes. Mach planes are
easily visualized by considering a Mach cone originating at a point on the
x-axis which is alined with the remote relative wind. A Mach plane is
simply a plane tangent to the Mach cone and at an angle of roll, 8 about
the x-axis measured from the y-axis. By moving the vertex of the Mach
cone along the x-axis, a series of parallel Mach planes will cut the con-
figuration for a fixed roll angle 0.

In the drag equation the term A(x,8) represents the frontal pro-
jection of the oblique area cut by a particular Mach plane, whereas f(x,9)
represents the net force normal to the stream direction on this section in
the 8 direction. These relationships are illustrated in figure 1 for
angle of roll 8 of the Mach plane of 0 and 900. By neglecting the
term _2 df(x,) the equation reduces to the supersonic-area-rule formula
given by Jones in reference 5. Evaluation of f(x,8) requires the pres-
sure distribution on the configuration which when integrated over the
configuration gives the drag directly. As a result, large values of
Sdf x impose a limitation on the supersonic area rule, even within

the framework of the linearized theory.

It is not the purpose of the present paper to evaluate the drag of
configurations including the effect of the pressure term but to evaluate
the drag of configurations using the supersonic-area-rule formula of
Jones. The influence of the pressure term was evaluated for one simple
case.

Equation (1) can be written in coefficient form as

2 1
CD = f CD(8)de (2)
C I=B%


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where


c(e) =-2 t St Lt dA d df og(x xl)dx dx (5)
O) 1qdx
tD 2n/ox0 V 2
The quantity CD(e) is most readily determined by solving the inte-
gral for CD(6) through a Fourier sine series expression for d fol-
lowing the method of reference 7 if


= cos- l)


an J dsin no d

then CD(() can be written as

CD(8) = Ena2

For the computations of this paper, only 24 terms were used in the
Fourier sine series expression for dA-. Thus,
dx
n=24
CD(e) -i__ 2
CD() = 4 nan2
n=l


BODY DRAG RESULTS


For bodies of revolution, the calculation of the drag is simplified
to some extent because the area distributions are identical for all roll
angles. However, except for high-fineness-ratio bodies, it is not pos-
sible to assume that the frontal projection of the oblique area cut by
the Mach plane is the same as the normal area. Figure 2 shows an example
of this for two parabolic bodies of revolution having different fineness
ratios and shapes. The area-distribution curve slopes were calculated
from the expression

dA = 2 fu (x xo)dx
,dx J 2R2(x) (x xo)2


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The derivation of this expression is given in appendix A. It has also
been assumed for the calculations (and all succeeding body calculations)
that a cylinder can be added at the base of the body without altering
the drag. If this were not done, the solution would require the flow to
fill the area behind the base which would exceed the limitations of the
linearized theory. Figure 2 shows large changes in the peak slope over
the afterbody of the fineness-ratio-6.04 configuration; these changes
would lead to a significant drag variation with Mach number.

The evaluation of the slope of the oblique area distributions is
extremely difficult except for simple bodies. There naturally arises
the question as to whether this is worth while if the pressure term is
ignored.

As derived in appendix B, the local force acting on the oblique
area of a body of revolution is

f dCp
q dx

The only assumption made in derivation of f is that -d is
q dx
constant over the oblique area. This is a reasonable assumption except
for bodies having discontinuities, and high local slopes. Then,


iA f dA 2A dCp
dx 2 q dx 2 dx

Thus, the error in the drag introduced by ignoring the pressure term is
dCp
dependent on the pressure gradient ---.
dx

It would be expected that the drag for a conical nose with an
attached shock wave over which the pressure is constant at zero angle of
attack would be least affected by the pressure term. (The pressure term
takes on a value only near the juncture with the cylinder; however, the
pressure term was not evaluated in -his region.) Figure 5 presents a
comparison of the drag of various cones calculated with the supersonic-
area-rule formula with the exact theory drag of reference 8. The lowest
Mach number of the comparison corresponds to the lowest Mach number for
entirely supersonic flow on the cone as calculated with the exact theory.
The highest Mach number of the comparison was arbitrarily taken as that
at which the slope of the Mach line equaled one-half the slope of the
nose. The agreement between the two theories is remarkable, within
5 percent except for a few points.


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A better comparison may be made by plotting CD(I/rm)2 against
Prm/l, the quantity which defines the frontal projection of the oblique
area distribution. This has been done in figure 4 to give drag in dimen-
sionless or collapsed form. The drags from exact theory (50 half angle
cone was chosen as representative), slender body theory (ref. 9), and
supersonic-area-rule theory are shown. The comparison shows the great
improvement of the area-rule theory over the slender-body theory at
values of Prm/l greater than 0.2, and the good agreement of the area-
rule result with the exact theory to prmjl of about 0.7. At higher
values of prm/1 the area-rule theory is in error, possibly first because
the pressure term is neglected but finally, near prm/1 = 1, because the
assumptions of the linearized theory are violated. At prm/1 = 1, the
Mach line lies on the cone surface, which corresponds to the realm of
hypersonic flows. (See ref. 10.)

For a body with curvature, for example, a nose of parabolic profile,
the pressure over the nose is variable, and the influence of the pressure
term may be significant. Figure 5 presents the drag for noses of parabolic
profile in collapsed form. Here the supersonic-area-rule theory is an
improvement over slender-body theory but in only partial agreement with
the more exact second-order theory of reference 11. Inclusion of the
pressure term, evaluated by using second-order pressure distributions,
however, does give agreement with some of the second-order-theory results.
Since the second-order-theory drags do not collapse into one curve, agree-
ment should be expected only with those points for which the pressure dis-
tribution used in evaluating the pressure term apply. However, this was
not the case. For example, the pressure distribution used for the pres-
sure term calculation at prm/l = 0.5 corresponds to the flagged symbol.
For the parabolic noses, both the area-rule theory and the area-rule
theory plus the pressure correction cannot be expected to apply near and
above Prm/l = 0.5, where the slope of the Mach line equals the slope of
the nose tip.

Figure 6 presents a comparison of the pressure drag from supersonic-
area-rule theory with experiment and slender-body theory for a family of
parabolic bodies of revolution. The experimental drags were taken from
references 12 and 13. In determining the experimental pressure drags,
the friction drag was assumed turbulent and evaluated by using the sub-
sonic drag level and the results of reference 14 for the effects of Mach
number and Reynolds number; the fin pressure drags were assumed identical
and taken from reference 15; and the base drags were small and were sub-
tracted when available. The slender-body-theory drags were calculated
using the curves of reference 9.


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As would be expected, the comparisons show the increasing ability of
both the area-rule theory and slender-body theory to predict the drag as
the body fineness ratio is increased. In most cases, the area-rule theory
offers a significant improvement over slender-body theory. The area-rule
theory and slender-body theory are in agreement near M = 1, since at this
Mach number the supersonic-area-rule theory reduces to slender-body theory.

From these nose and complete-body comparisons that have been made,
the following conclusion can be drawn. The area-rule drag of bodies can
be predicted to a greater degree of accuracy by using the frontal projec-
tion of oblique areas at a given Mach number than by using normal areas,
if, at the Mach number under consideration, the limitations of the line-
arized theory are not exceeded. This is illustrated by the comparison
between the drag at a given Mach number and the drag near M = 1 especially
for the low-fineness-ratio bodies. It is not to be inferred from the above
statement that the supersonic-area-rule method is recommended for evalu-
ating the drags of bodies of revolution. However, when the drags of wing-
body combinations for which the body area distribution is needed are deter-
mined, the oblique area distribution should be used if the body is of low
fineness ratio or has low-fineness-ratio components.


CALCULATION OF WING-BODY DRAG


The difficulty in computing the wave drag of wing-body configurations
can be considerably reduced if the configuration meets the following con-
ditions: first, the body is of sufficiently high fineness ratio so that
the change in body-area distribution with Mach number is small, and sec-
ond, the wing is thin. These conditions imply also that the pressure
term is negligible. Some feel for the body fineness ratios necessary for
the above condition to be met can be obtained from the preceding section
on bodies of revolution. The assumption of a thin wing allows the Mach
plane intersecting the wing obliquely to be replaced by a plane perpen-
dicular to the wing chord plane intersecting the wing plane along the
same line as the Mach plane. Note that, at zero roll angle, the Mach
plane is normal to the wing chord plane but is not normal to the wing
chord plane at any other roll angle for a Mach number other than M = 1.
Also the angle between the Mach plane and the normal to the wing chord
plane is greatest and equal to tan-'l at a roll angle of 900.

Appendix C presents a simple analytical method for evaluating wing
area distributions and area-distribution-curve slopes. The curves neces-
sary for evaluating these quantities (figs. 16 and 17) are applicable only
to 65A series airfoils, but similar curves can be made up for other air-
foil sections.


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In order to get an idea of the applicability of the thin-wing assump-
tion, a calculation has been made of the true area-distribution-curve-
slope variation for 600 delta wing having an NACA 65A006 airfoil section
for a roll angle of 900 and a Mach number of 1.414. In order to simplify
the calculation, the wing was approximated by a sufficient number of
linear-slope elements to define the airfoil section adequately. With
this approximation the Mach plane intersection with the wing surface was
made up of straight lines. The expression for the frontal projection of
the oblique area was then easily evaluated and differentiated to obtain
the slope. The results of the calculation are presented in figure 7.
Although the slopes for the upper and lower half wings are significantly
different, the total slope agrees almost exactly with the slope obtained
by using the thin-wing assumption. On the basis of this result, it is
felt that the thin-wing solution should be adequate for wings of present-
day interest.

For the wing-body combinations of this paper, an additional simpli-
fication was allowed in the supersonic-area-rule wave-drag calculations.
Since the tail fins mounted on the models were thin and relatively small
(see ref. 16), their drags were subtracted as tares. Then, since the
bodies for all cases were of high fineness ratio (and identical), the
body-area-distribution-curve slopes were considered independent of Mach
number, and the changes in the area-distribution-slope curves with Mach
number and roll angle were due entirely to the wings. As derived in
appendix C, the area distribution for a given wing (m fixed) is depen-
0 cos 0
dent only on the value of ta Thus, the area-distribution-slope
tan A
curves for the wing-body configuration are dependent only on the value
B cos 8
of Cos Then, from equation (2) and because of the symmetry of the
tan A
configuration,


CD(M) 2 / CD(9)d9


In order to obtain the wave drag of the configuration, a plot of CD
against B is required. This can be computed if a plot of CD against
B cos B B
Ctos is given, since the angle 8 is known for fixed values of
tan A tan A
and C s. The configuration drag is simply the average drag between
tan A
S= 0 and

For the wing-body calculations of this paper the bodies were identi-
cal. The body-area-distribution-slope curves are shown in figure 2(b).
The curve for M = 1.414 was chosen as representative for the Mach number

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NACA RM L56D11


range of interest. The wing area distributions and area-distribution-
curve slopes were obtained by using methods similar to that given in
appendix C. In addition, a limiting value of P cos = 0.8 was set for
tan A
configurations having blunt leading-edge airfoils. (Above t cs A = 1,
the Mach line lies behind the wing leading edge, and the linear theory is
no longer valid for blunt airfoils,)

An example of the wave drag calculation for the most extreme configu-
ration investigated (600 delta wing, IACA 65A006 airfoil) is presented in
figures 8, 9, and 10. Figure 8 shows nondimensional plots of dA against
dx
0 cos d
for various values of Figure 9 shows the effect on CD of
tan A
the number of terms in the series solution. Except at P Cos 8 = 0 and 0.8
tan A
convergence was apparently obtained within 24 terms. Figure 10 shows the
variation of the area distribution drag with P0 s the variation of
tan A '
area distribution drag with roll angle, and the variation of the config-
uration drag with P
tan A*


WING-BODY DRAG COMPARISONS


Figure 11 presents some wave-drag comparisons for wing-body combina-
tions. The experimental wing-body results were taken from references 16
to 19. The wing-body wave drags were obtained in the following manner.
The friction drags were assumed to be turbulent and were estimated by
using the results of reference 14. Base drags and fin pressure drags were
subtracted using the results of reference 16. The equivalent-body drags
for a Mach number of 1 were obtained experimentally by using the helium-
gun technique described in reference 2. These models had four scaled
tail fins. The friction drag was assumed to be the subsonic drag level
corrected at higher speeds for Reynolds number and Mach number by using
the results of reference 14. Base-drag rise and fin-drag rise were not
evaluated for the equivalent-body models. These quantities, however,
should be small in the Mach number range where comparison is valid. The
supersonic-area-rule-theory drags were evaluated by using the method of
the preceding section. No attempt was made to evaluate the drag with
the pressure term included. The drag coefficients presented in figure 11
are based on total wing area.

The inability of the supersonic-area-rule theory to predict the drag
near M = 1 is evident for nearly all cases. However, the agreement at
the higher Mach numbers between the theoretical drags and the experimental


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12 CONFIDE TRIAL NACA RM L56Dll


wing-body drags is excellent and within the accuracy of evaluating the
experimental wave drag, except for three configurations. Two of these
configurations (figs. 11(c) and 11(g)) had 6-percent-thick wings which
were the thickest wings investigated. The third configuration (fig. 11(h))
had a 4-- percent-thick airfoil but with fairly steep wedge components.
2
For these configurations, a significant effect of the neglected pressure
term may be possible. As a result, the drags calculated for configura-
tions having wings of these thicknesses and sections should be viewed
with caution.

The comparisons in figure 11 show that the equivalent-body drags
give a good approximation to the experimental wing-body drags up to a
Mach number of 1, except for the two configurations having 6-percent-
thick wings (figs. 11(c) and 11(g)). This result is in agreement with
reference 2 which shows the validity of the transonic-area rule decreases
with increasing wing-thickness ratio. At Mach numbers above 1, the agree-
ment is variable but tends to be consistent with the flatness of the cor-
responding theoretical curve. That is, as the theoretical drag variation
with Mach number becomes smaller, the equivalent body gives a better
approximation of the supersonic-drag level. This would be expected, since
a flat theoretical curve indicates that the variation in area-distribution
drag with roll angle or Mach number is small. Then the drag for the Mach
number 1 or roll angle 900 area distribution (corresponding to the equiv-
alent body) is representative of the configuration drag.

Figure 12 shows the comparison between experimental-configuration
drag and equivalent-body drag for an airplane configuration. The com-
parison shows an extreme example, compared with the relatively good results
of reference 2, of the inability of the equivalent body to predict the
supersonic-drag level. The equivalent-body drag is approximately 40 per-
cent low in spite of the low aspect ratio of the configuration. Apparently,
the configuration tail surfaces cause the area distribution to change mark-
edly at low supersonic speeds. Below M = 1, the equivalent body gives a
fair representation of the configuration drag. The drag of the configu-
ration minus the tail surfaces could probably be calculated to the degree
of accuracy shown in figure 11. The influence of the tail surfaces, how-
ever, may be difficult to evaluate. If the horizontal tail supports a
load when the configuration is at zero lift, the influence of the pres-
sure term may be significant. Although no supersonic-area-rule drag cal-
culations were made for this airplane, reference 20 indicates that gen-
erally good predictions of complete-airplane drag can be made.1



1Subsequent bo the preparation of this paper, NACA RM A56107 has
been prepared at the Ames Laboratory and presents supersonic-area-rule
calculations for a configuration similar to the one shown in figure 12
but with small differences in area distribution in addition to the
absence of a canopy.


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EVALUATION OF CuMI.POOPEIIr AND IrJPT1FEFETICE DRAGS


As was shown in the preceding section, the supersonic area rule can
be a useful tool in evaluating the supersonic drag of a wing-body con-
figuration. In order to assess the efficiency of the combination as a
whole, however, the effects of the combination on the component drags
and the interference drag between the components must be known. The
supersonic area rule provides a valuable method for evaluating these
effects.

The supersonic-area-rule equation can, of course, be used in evalu-
ating the drags of individual wing and body components. This was done
for a number of bodies in a previous section of this paper. The same can
be done for isolated wings. An example of this is shown in figure 15
where the drag of delta wings having 65A series sections is plotted in
collapsed form. The area-rule result is compared with a result obtained
by the method of Beane (ref. 21). The two methods are just two forms of
the same linearized wing theory. The agreement between the two methods
is good.

An example of the effect of the wing-body configuration on wing drag
is presented in figure 14. The calculation is for the configuration having
the closest agreement between the theoretical and experimental drags
(fig. 11(b)). In this figure, the drag of the exposed-wing panels based
on total and exposed wing areas is compared with the isolated wing drag.
Separation of the wing panels gives approximately a 10-percent reduction
in wing drag coefficient at Mach numbers above 1.5. As the Mach number
approaches 1, this favorable effect disappears. This would be expected,
for at M = 1 the area distributions of the exposed wing panels and the
isolated wing would be identical if the body were cylindrical.

Figure 15 shows an evaluation of the interference drag for the same
configuration. The sum of the calculated body and wing drags is compared
with the calculated configuration drag. The curves show a favorable inter-
ference effect at Mach numbers below M = 1.3. At Mach numbers above 1.5,
interference drag is, for all practical purposes, zero. Thus for this
configuration, at Mach numbers greater than 1.3, the only beneficial effect
of combining the wing with the body comes from the separation of the wing
panels.


CONCLUSIONS


An investigation has been made of abilities of the equivalent-body
technique and a 24-term Fourier series application of the supersonic-
area-rule method to predict wave drag at transonic and supersonic speeds.
From the theoretical and experimental comparisons made, the following
conclusions can be drawn:


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1. The area-rule drag of the bodies of revolution presented in this
report are predicted to a greater degree of accuracy by using the frontal
projection of oblique areas at a given Mach number than by using normal
areas.

2. The supersonic wave drag of slender-wing-body configurations can
be predicted with the supersonic-area-rule formula. For the wing-body
configurations investigated, the best agreement was obtained for the con-
figurations employing the thinnest wings.

5. The equivalent body technique provides a good method for predicting
the wave drag of certain wing-body combinations at and below Mach number 1.
At Mach numbers above 1, the equivalent body wave drags can be misleading.


Langley Aeronautical laboratory,
National Advisory Committee for Aeronautics,
Langley Field, Va., April 6, 1956.


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APPENDIX A


AREA DISTRIBUTION SLOPE FOR BODIES OF REVOLUTION

CUT BY OBLIQUE MACH PLANES


The area distributions are identical for all roll angles.
simplicity a roll angle of 900 will be used in the derivation.


For


The frontal projection of the oblique area cut by the Mach plane (see
sketch (a)) is given by the equation:


A = 2 f uy dz
z1

From the equation for the Mach plane, z is related to x by


Z = X

and

dz =1 dx


The equation relating y and x is given by


y = R2(x) 2


R2() (x-xo) 1
P2 P


2R2x) (x xo)2


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Then,


A =X 2R2(x (x 0)2 dx


Differentiating the expression for A gives


(A2)


dA 2 Fxu
dx0 2 Jx


(x XO)

/2R2(x) (x xo)2


where x, and xu are the roots of x = x0 PR(x) and x = x0 + pR(x),
respectively.


CONFIDENTIAL


(A3)


CONFIDENTIAL






NACA RM L56D11


APPENDIX B


THE NET FORCE ACTING ON THE OBLIQUE AREA OF A BODY OF

REVOLUTION AT ZERO ANGLE OF ATTACK


For a body of revolution at zero angle of attack, the net force is
independent of roll angle. The derivation will be made for a roll angle
of 90.


xu


r = R(x)


x = x0 + Pz


NdSY

dy
dy


The net force in the e direction (the z-direction for a roll angle of
900) (see sketch (b)) can be written as


= f sin a dS
f C
C


Since dS = dy2 + dz2 and


ay
sin a =
dy2 + dz2


= Cp dy

The pressure coefficient at zero lift is a function of x only.
The equation for y in terms of x is given by equation (1) of appen-
dix A as follows:


y pR2 (x) (x 2
y P


CONFIDENTIAL


CONFIDENTIAL





NACA RM L56Dll


and
02R(x) (x O)
dy = dx
p pR2(x) (x xo)2

Then, the net force can be written as


SCpO C
0


) upper surface


Cp2R(x) (x- xO]

2R,]2(x) (x xo)2


+ i(2 Cp
0


dx + f x
Px,


Slower surface


C 02R(x) (x x0
Sdxdx
O2R2(x) (x x)2


f 2f xu
x 1


Cp[P2R(x). (x x0j

JpR2(x) (x xo)2


Integrating the expression by parts gives


f 2- Xu
q 0' x
X22


dCp x) (x x0 2 dx
d RP2R2(x)_ (x d xO)


dOn
If dc is essentially constant between the limits of integration,
dx


f= 2 C xu
q TdxJ
xl


22(x) (x x)2 dx


Then, from equation for the frontal projection of the oblique area
in appendix A (eq. (A2))

= C
q dx


CONFIDENTIAL


= -2 /X0


CONFIDENTIAL






NACA RM L56Dll


APPENDIX C


METHOD FOR DETERMINING WING-AREA DISTRIBUTION AND

AREA-DISTRIBUTION-CURVE SLOPE


This method assumes that the wing is thin and that the oblique Mach
plane can be replaced by a plane perpendicular to the wing chord plane.
The method also assumes the wing has straight leading and trailing edges
and constant thickness ratio.

The method is developed first for pointed-tip wings. Then, correc-
tions are made for curved-wing-body junctures and finite wing tips. In
addition, the right- and left-hand wing panels are considered separately.

Pointed-tip wings.- Consider the right-wing panel shown in the fol-
lowing sketch:


Zmax


Sx' = x0 + y(p cos e tan A)


\0


(c)
CONFIDENTIAL


CONFIDENTIAL






NACA RM L56D11


The frontal projection of the area of one wing panel cut by the Mach
plane is given by

A = 2 z dy

and z can be written as
c

z_ Zax z I_ t f(1 )
c c zmax 2 c

t
Then, for L constant,


A = s c0 L ) dT


The value of q and v are related by the intersection line of the Mach
plane and the wing chord plane for the right-wing panel given by the
equation

x' = x- + y(p cos 0 tan A)

and for the left-wing panel by the equation

x' = x0 + y(p cos 6 + tan A)


Then,


v = [0 O + ( cos e tanA)


With


cO


v = i +


-CO(P cos e tan A)q


V0 V
VO V
(tan A cos 9). v
CO

sO
m = tan A
CONFIDENTIAL
CONFIDENTIAL


and


Let


CONFIDENTIAL






NACA RM L56D11


In terms of tapered wing geometry, m is given by


m = A 1 + tan A
41 -


Then,


T =


V V
0t
ml t P Cos -
tan A /


K= m(


for the left-wing panel and


K = ml


for the right-wing panel.


+ cos o
tan A



p cos
tan A /


Then,


SV0 -V
K-v

dT K- v
dH K v0
dv (K v)2


K v0
K-v


c (K vo)2 dv
dq = -^-- dv
do- (K V)

The equation for the area can then be written as


t-S (K
A = -S0C0 K


- 2 f upper

lower


f(v)3 dv = s0c0 G (K, 0
(K- v)3


CONFIDENTIAL


CONFIDENTIAL





NACA RM L56D11


The slope of the area-distribution curve is obtained by differentiating
the expression for A


dA t upper f() d +
dvo O dv +
O v0lower (K v)


K vO)2

(K- ~lower)


dvlower
do


f (lower)


(K v0)2

(K *upper)5


dvupper
dv


f(Vupper)


dA sct H(Ko)
d-o = sOcO v)


Curves of G(K,vo)
airfoil and are given in


to 2.4 and


and H(K,v0) have been made up for
figures 16 and 17 for values of K


v0 from 0 to 1. In evaluating / f(v) dv,
(K v)


assumed to vary linearly between airfoil ordinate stations.
gives a plot of f(v) for this assumption.


a 65A series
from 0

f(v) was

Figure 18


For K and vo
by the expressions


greater than 1, G(K,V0)


and H(K,v0)


G(K,vo) = G(K,1)


For K and vg
the expressions


H(K,vo) =

less than


H(K,1) (K )
(K 1)


0, G(K,vO)


G(K,v0) = G(K,0) (K v0)2
K2


H(K,Vo) = H(k,O) (K K0)
K


and H(K,vo) are given by


CONFIDENTIAL


are given


(K v)2

(K )2


CONFIDENTIAL






NACA RM L56D11


Correction for curved-wing-body juncture.- The following sketch
shows a pointed-tip wing mounted on a curved body.


Line of intersection of Mach
plane and wing chord plane


The areas and slopes will be
(cr, s, and 7). The areas
pointed-wing tip.


(d)
referred to
and slopes,


the actual wing geometry
however, will be for the exposed


In sketch (d) consider one point of intersection of the wing panel
with the body. The area of the wing cut by the Mach plane through this
point is determined only by the product of SOC0 of the exposed wing
through the point and the value of v0 for the exposed wing at the point
of intersection. As the point of intersection changes, sO, cO, and vO
change and account for the intersection line. Expressed in terms of the
actual wing-body characteristics, SoC0 is given by


S00 = c r ( -

The quantity r is related to v0 by the expression

v (1 A)m ]
s
1 (1 x)

The area of the exposed wing panel cut by the Mach plane can be written
as


CONFIDENT AL


CONFIDENTIAL


I






NACA RM L56D11


Crs t- 2
A = --- 1 (1 )r G(K,v0)


The area is calculated for given value of vO. The center-line value of
v is given by

Vr = v[l (1- )s] + K(l A)r


The slope
is given by


is obtained by differentiating the expression for


dA
dvr


A and


1 (1 )F+ (K vo)(1 -


If d: O,
dvO


dA
dvr


= 1 (1 A)H(Kvo)
1. -A A-


Correction for finite wing tip.- In order to correct the pointed-
tip wing panel and slopes for the finite wing tip, the areas and slopes
outboard of the wing tip are subtracted.


Intersection line of Mach plane and wing


cr


- -


s -- sO


From sketch (e):


SoCO = ers


N2
1 A


CONFIDENTIAL


CONFIDENTIAL






NACA RM L56D11 CONFIDENTIAL 25


Then the areas and slopes are given by

Crs 2 2
Atip = 1- G(K,vo)


t

dvr i A

The center-line value of v is given by


Vr = + K(l -)


CONFIDENTIAL






NACA RM L56D11


REFERENCES


1. Whitcomb, Richard T.: A Study of the Zero-Lift Drag-Rise Character-
istics of Wing-Body Combinations Near the Speed of Sound. NACA
RM L52H08, 1952.

2. Hall, James Rudyard: Comparison of Free-Flight Measurements of the
Zero-Lift Drag Rise of Six Airplane Configurations and Their
Equivalent Bodies of Revolution at Transonic Speeds. NACA
RM L55J21a, 1954.

5. Jones, Robert T.: Theory of Wing-Body Drag at Supersonic Speeds.
NACA RM A55H18a, 1955.

4. Holdaway, George H.: Comparison of Theoretical and Experimental Zero-
Lift Drag-Rise Characteristics of Wing-Body-Tail Combinations Near
the Speed of Sound. NACA RM A55H17, 1955.

5. Alksne, Alberta: A Comparison of Two Methods for Computing the Wave
Drag of Wing-Body Combinations. NACA RM A55AO6a, 1955.

6. Lomax, Harvard: The Wave Drag of Arbitrary Configurations in Line-
arized Flow As Determined by Areas and Forces in Oblique Planes.
NACA RM A55A18, 1955.

7. Sears, William R.: On Projectiles of Minimum Wave Drag. Quarterly
Appl. Math., vol. IV, no. L, Jan. 1947, pp. 561-566.

8. Staff of the Computing Section, Center of Analysis (Under Direction
of Zdenek Kopal): Tables of Supersonic Flow Around Cones. Tech.
Rep. No. 1, M.I.T., 1947.

9. Fraenkel, L. E.: The Theoretical Wave Drag of Some Bodies of Revolu-
tion. Rep. No. Aero. 2420, British R.A.E., May 1951.

10. Van Dyke, Milton D.: Application of Hypersonic Small-Disturbance
Theory. Jour. Aero. Sci., vol. 21, no. 5, Mar. 1954, pp. 179-186.

11. Van Dyke, Milton D.: Practical Calculation of Second-Order Super-
sonic Flow Past Nonlifting Bodies of Revolution. NACA TN 2744,
1952.

12. Hart, Roger G., and Katz, Ellis R.: Flight Investigations at High-
Subsonic, Transonic, and Supersonic Speeds To Determine Zero-Lift
Drag of Fin-Stabilized Bodies of Revolution Having Fineness Ratios
of 12.5, 8.91, and 6.04 and Varying Positions of Maximum Diameter.
NACA RM L9I50, 1949.

CONFIDENTIAL


CONFIDENTIAL






NACA RM L)6D11


13. Wallskog, Harvey A., and Hart, Roger G.: Investigation of the Drag
of Blunt-Nosed Bodies of Revolution in Free Flight at Mach Numbers
From 0.6 to 2.3. NACA RM L55Dl4a, 1955.

14. Chapman, Dean R., and Kester, Robert H.: Turbulent Boundary-Layer
and Skin-Friction Measurements in Axial Flow Along Cylinders at
Mach Numbers Between 0.5 and 3.6. NACA TN 5097, 1954.

15. Stevens, Joseph E., and Purser, Paul E.: Flight Measurements of the
Transonic Drag of Models of Several Isolated External Stores and
Nacelles. NACA RM L54L07, 1955.

16. Morrow, John D., and Nelson, Robert L.: Large-Scale Flight Measure-
ments of Zero-Lift Drag of 10 Wing-Body Configurations at Mach
Numbers From 0.8 to 1.6. NACA RM L52D18a, 1953.

17. Wallskog, Harvey A., and Morrow, John D.: Large-Scale Flight Measure-
ments of Zero-Lift Drag and Low-Lift Longitudinal Characteristics of
a Diamond-Wing-Body Combination at Mach Numbers From 0.725 to 1.54.
NACA RM L55C17, 1953.

18. Welsh, Clement J., Wallskog, Harvey A., and Sandahl, Carl A.: Effects
of Some Leading-Edge Modifications, Section and Plan-Form Variations,
and Vertical Position on Low-Lift Wing Drag at Transonic and Super-
sonic Speeds. NACA RM L54KOl, 1955.

19. Sandahl, Carl A., and Stoney, William E.: Effect of Some Section
Modifications and Protuberances on the Zero-Lift Drag of Delta
Wings at Transonic and Supersonic Speeds. NACA RM L53L24a, 1954.

20. Holdaway, George H., and Mersman, William A.: Application of Tchebichef
Form of Harmonic Analysis to the Calculation of Zero-Lift Wave Drag
of Wing-Body-Tail Combinations. NACA RM A55J28, 1956.

21. Beane, Beverly: The Characteristics of Supersonic Wings Having
Biconvex Sections. Jour. Aero. Sci., vol. 18, no. 1, Jan. 1951,
pp. 7-20.


CONFIDENT IAL


CONFIDENTIAL






IIACA RM L56Dll


(a) e = 90.


(b) e = 00.


Figure 1.- The areas and pressures which influence the drag of configura-
tions at supersonic speeds.


CONFIDENTIAL


CONFIDENTIAL






NACA RM L56D1l



2 -



0.


,e/x/


CON FIDENTIAL


.6
x/I


(a) Low-fineness-ratio body; 1 = 6.04.
d


/ ,4
l^A.r/


H-.4z*

M


0 -



-2-
0


(b) High-fineness-ratio body; = 10.
d

Figure 2.- The effect of Mach number on the area-distribution-curve slope
of bodies of revolution.


CONFIDENTIAL





rIACA RN L56Dll


Cb (Area


50
7.5
/0
/2.5
/5
/75
20


M
/./5-4.68
. 07-3.94
. 12-3.01
A /6-2.47
/.22-2.12
/ 28-1.89
/33-/. 70


/Lie of perifet agreement '


0 ,2 .3 .4 .5
C, /Exact theory)


Figure 3.- Comparison of the drag of cones calculated with the area rule
with the drag from exact theory (ref. 8).


CONFIDENTIAL


CONFIDENTIAL






NACA RM L56D11


4


E-a/ theory /( = 5/
Area-rule theory
S/endaer-hody fthry


2-


.2 .4 r, .6 .


Figure 4.- The drag of cones in collapsed form.


SS/ender-body theory
Area-rule /heory
Area-rukl /theory pressure erm
0 Second-order /thry

0 .2 .4 .8 LO


Figure 5.- The drag of parabolic noses in collapsed form.


CONFIDENTIAL


CONFIDENTIAL







CONFIDENTIAL


NACA RM L56D11


"0
i
s
u

u








a





:s 5
r
F
? I

s


P


0



r
c








Sci






rda
.0


o0
a,4


o o






0r
LO
(d $ [,















0 w

cd-


*9 '
49


CONFIDENTIAL


lit t Ii
r1




I ,


I 1I
'1 "


N
'0'9-



N

N





'0hr
^


ll

I ,


I I I I
'0 '


'* '
4.


N I


4-o






NACA RM L56D11


_______ _______
/


.6 Z /a = 6.04



.4



.2- --Area-r/e hevy
-Slender-o4dy theory



.8 1.0 /2 /.4 ,
M


2/ = 8.91


Co, .2-


.8 /.0 .2 1.4 .

.4


2/d = 12.





.a /V, 0 1.2 1.4 L


(b) nose = 0.2.
1

Figure 6.- Concluded.




CONFIDENTIAL


CONFIDENTIAL







CONFIDENTIAL


%at


NACA RM L56Dll


o



0*
0 *

*r4
0


I .:4





0 00

.l 4. I


f1-



o I
H .-4-
rd ii



l-0
or4






















rd
4-3
o wI



*H .0
0 aH

Cd r *H

u 0


D I

i 0


CONFIDENTIAL






NACA RM L56D11


24d


/Scos a
tan .A-

0--
,2 -7
.4 --
.6
.8


Figure 8.- Example of the area-distribution-curve slope for a wing-body
configuration for various values of P cos 9/tan A. 600 delta wing;
NACA 65A006 airfoil section.


CONFIDENTIAL


CONFIDENTIAL







36 CONFIDENTIAL NACA RM L56D11










U mo
izi

cno
(u


or-i












0 1
NO








0 0
ao






z bo
W 0O








w




ON
;q







1Q I 0


CONFIDENTIAL







NACA RM L56Dll


0 .2 .4 geos&a .8


(a) Area distribution drag.

8
taeA
-cffA





i i i i i i


60 75 90


(b) Area distribution drag against roll angle.


.4 O/.tanA


.8 /.0


(c) Configuration drag.

Figure 10.- An example of the calculation of the configuration drag from
the drag of the area distributions at various values of p cos 9/tan A.
600 delta wing; NACA 65A006 airfoil section.


CONFIDENT LAL


.03



.02

Ca

.0/



0


0 /5 30 45
*, dae


IVl l l i


CONFIDENTIAL






NACA RM L56D11


.0" --- Experiment
CD -- /rr-Arvr* hery

.8 .9 .0


I


/./ ..2


3 .4 /.5 /.6


(a) Basic body; model 1 (ref. 16); Sb = 0.0305.
Sw


- -Arwa tuA. thoojy


Co .o0/


N


.9 1.0 i /M
Ml


/.3 /4 .S .6


(b) Model 4 (ref. 16); 600 delta wing; NACA 65A005 airfoil section;

Sb
--= 0.0305.
Sw

.03
-- A------ fir /entr
-- Equval/ent mody

I




0 l I I I I
S 9 1.0 /./ .2 /3 /.4 .S 1.6

(c) Model 5 (ref. 16); 600 delta wing; IJACA 65AO06 airfoil section;

Sb
--= 0.005.
Sw


Root ailo/l

Cot -


o


-i A~p,iu~m'm1por
L -- Eq..ao/e'nf booy

I r-u Ih~r


/3 /.4 /.5 .6


(d) Model 6 (ref. 19); 600 delta wing. Thickness ratio varies from 0.05

at root to 0.06 at 0.9 semispan. -= 0.0305.
Sw

Figure 11.- Comparison of the calculated drag with experiment and
equivalent-body test results for wing-body combinations.


COIIFIDEITIAL


.8 9 /.0 /./ /.
M


CONFIDENTIAL






NACA RM L56D11

.03


.02 -


CONFIDENTIAL



Arir-rule thMr-y
- E4uiYa/eLt b.vy


o/ I-


SJ/1


.9 .O /./ /.2
M


I I I


/3 /.4 /5 /.6


(e) Model of reference 17; A = 2.51; Ac/2 = 0; NACA 65A005 airfoil section;

b = 0.0606.
Sw


Aa,--- erime n'
- Area-rule /~kuw-
- -- Equval/,t kody


/
I I -- I


.9 /0 /./ /.2
M


I I


/3 /4 /.5 /6


(f) Model C-3 (ref. 18); A = 3; A = 0.2; Ac/4 = 450; NACA 65A003 air-
Sb
foil section; = 0.0606.
Sw



section; I mey 6.
.02 quivaent body ---- ---













Sb
section; 1-= on.066.

Figure 11.- Continued.


COIIFIDEINTIAL


. .. I m m m I )






NACA RM L56D11


Ex, 'er/.4ent
Area-rule AIOry
- EqwDva/eff body


.9 /. 0 / /.2
M


/.4 /5 /.6


(h) Model 2 (ref. 16); A = 3.04; A = 0.594; A5c/4 =


00; = 0.045;
c


b = 0.0606.
Sw

Figure 11.- Concluded.


Exp-ermenl

Equid/ae'n Co'y


-------


.9 /.0 /./ /2
114


1.3 /.4 1/.


Figure 12.-


Comparison of equivalent-body drag and configuration drag
for an airplane configuration.


COIIFIDENTIAL


4irhil


CONFIDENTIAL







NACA RM L56D11


/6



14 -


-Area-rule theory
-Beane theory


.3 .4
/9 ianE


.5 .6


.7 .
.7 .8


Figure 15.-


Comparison of the drag of delta wings calculated with two
versions of the linearized theory.


C CFIDENTIAL


Co
ft/iftan#
9 -



6



4


CONFIDENTIAL






NACA RM L56Dll


.0/6



.0/2



(a .008



.004


/./ 1/. /3 /. /S /16 / 7
/W


Figure 14.- Effect of wing-panel separation on wing drag.
wing; NACA 65A005 airfoil section.


600 delta


/. /2 /.3 / 4 /.S /.6 /.7
/n


Figure 15.- Comparison of the sum of component drags with the configura-
tion drag. 600 delta wing; NACA 65A003 airfoil section.


COIFI DETIAL


-Based on4, iC1j
Baswd onSu -


.01/2


Co .o00


CONFIDENTIAL






NACA RM L56D11


0 .2 .4 .4 .8 A.O /.2 /16 / 20 22 2A


Figure 16.- Area distribution parameter G(K,VO) for 65A series airfoil.


CONFIDENTIAL


CONFIDENTIAL






NACA RM L56D11


6 9 O '0 #1 j.6 /8 20 22
N


(a) v0 from 0 to 0.45.

Figure 17.- Area-distribution-slope parameter H(K, v) for 65A series
airfoil.


COINFIDEITIAL


CONFIDENTIAL






NACA RM L56D11


(b) v0 from 0.5 to 1.0.

Figure 17.- Concluded.


COIIFIDENTIAL


CONFIDENTIAL






NACA RM L56D11


0 A i/old orda/le sfr/ons-a


0 .2 .6 .8


/.0


Figure 18.- Approximation of 65A series airfoil for the calculation of
G(K,VO) and H(K,VO).





d!


CONFIDENTIAL


NACA Langley Field. Va.


CONFIDENTIAL















































































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