On the range of applicability of the transonic area rule


Material Information

On the range of applicability of the transonic area rule
Series Title:
Physical Description:
21 p. : ill. ; 28 cm.
Spreiter, John R
Ames Research Center
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


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


Abstract: Some insight into the range of applicability of the transonic area rule has been gained by comparison with the appropriate similarity rule of transonic flow theory and with experimental data for a large family of rectangular wings having NACA 63AXXX profiles.
Includes bibliographic references (p. 21).
Statement of Responsibility:
by John R. Spreiter.
General Note:
"Report date June 28, 1954."
General Note:
"Classification changed to unclassified Authority: J.W. Crowley Date: 9-7-55 Change No. 3076."--stamped on cover

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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 - 003808573
oclc - 130052568
sobekcm - AA00006162_00001
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Full Text
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By John R. Spreiter

es Aeronautical Laboratory
Moffett Field, Calif.




GCAiB go 3o076

)ATE: 9-7-55

i niormatlon affecting the N al Defense ofat the Unted Staes within the meaning
Tite 18, U.S.C., S.es. 79 and 794, the ramneaislo or revelation of which In any
ed person s prohiited ki law.


August 23, 1954


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By John R. Spreiter


Some insight into the range of applicability of the transonic area
rule has been gained by comparison with the appropriate similarity rule
of transonic flow theory and with available experimental data for a large
family of rectangular wings having NACA 63AXXX profiles. In spite of the
small number of geometric variables available for such a family, the
range is sufficient that cases both compatible and incompatible with the
area rule are included.


A great deal of effort is presently being expended in correlating
the zero-lift drag rise of wing-body combinations on the basis of their
streamwise distribution of cross-section area. This work is based on the
discovery and generalization announced by Whitcomb in reference 1 that
"near the speed of sound, the zero-lift drag rise of thin low-aspect-ratio
wing-body combinations is primarily dependent on the axial distribution
of cross-sectional area normal to the air stream." It is further conjec-
tured in reference 1 that this concept, known as the transonic area rule,
is valid for wings with moderate twist and camber. Since an accurate pre-
diction of drag is of vital importance to the designer, and since the use
of such a simple rule is appealing, it is a matter of great and immediate
concern to investigate the applicability of the transonic area rule to the
widest possible variety of shapes of aerodynamic interest.

The experimental data contained in reference 1 and many subsequent
papers have shown that this simple rule is often remarkably successful for
a wide variety of shapes ranging in complexity from simple bodies of revo-
lution to models of complete airplanes. Furthermore, important reductions
in the transonic drag of wing-body combinations have been realized by
indenting the body so that the axial distribution of cross-section area
corresponds to that of smooth bodies of revolution having low drag at



supersonic speeds. On the other hand, it is important not to overlook
the fact that there are a number of test results on equivalent bodies for
which the correlation of drag rise by the area rule is unsatisfactory.
Inasmuch as the models tested are generally of complex geometry, and only
the original model and an equivalent body of revolution are tested, it is
difficult to ascertain whether these discrepancies are attributable to
viscous phenomena or to the fact that the drag rise may depend on other
geometric parameters than the axial distribution of cross-section area.

It is the purpose of this note to examine in further detail the
applicability of the area rule. Despite the fact that most of the emphasis
in the tests relative to the area rule has been on wing-body combinations,
there exists such a scarcity of experimental data of a sufficiently sys-
tematic type that the present discussion will be confined to rectangular
wings without bodies. Transonic drag data are available from bump tests
in the Ames 16-foot high-speed wind tunnel for a large family of rectangu-
lar wings having NACA 63AXXX sections, aspect ratios varying from 0.5 to
6.0, thickness ratios from 2 to 10 percent, and both symmetrical and cam-
bered profiles. These results are reported in references 2, 3, and 4 and
have been studied by McDevitt refss. 3 and 5) who showed in a convincing
manner that the experimental data can be correlated successfully by means
of the transonic similarity rules. These same results will be used herein
to evaluate one phase of the transonic area rule. The relationship between
the two rules is naturally of interest and will also be explored.

Since the transonic area rule is considered to apply equally to all
low-aspect-ratio wing-body combinations, detailed examination of such a
limited class of aerodynamic shapes as a family of rectangular wings is
not without value inasmuch as limitations revealed in special applications
must appear as a limitation in the general case. The restricted range of
the investigation is compensated somewhat by the fact that the geometric
simplicity increases the chances of understanding the underlying causes.
Although the results can only be said to apply with surety to the specific
cases investigated, the method of approach is not restricted and may be
applied similarly to other cases as more data become available and as
understanding increases. In this way, the present discussion may be con-
sidered more as suggestive than definitive.


A aspect ratio

A [(7 + 1) Mo2T1A

b wing span




CDow D ow



CMg ['2 (Y + )1/ C
EL 7T2/3 CL

CZi ideal lift coefficient

c wing chord

D drag

Do drag at zero lift

Dow wave drag at zero lift

o Do Do
cc 2 qc2)
qC'jMO (q M-ref

f function of indicated variables

g, dimensionless function describing thickness distribution

gg dimensionless function describing camber distribution

H c

h maximum camber of wing

h -t

L lift

Mo Mach number

Mref reference Mach number less than the critical

q dynamic pressure, U2




R Reynolds number

Rref reference Reynolds number corresponding to Mref

S area

Sc cross-section area

Sm maximum cross-section area

Sp plan-form area

s dimensionless area distribution function

t maximum thickness

Uo free-stream velocity

x,y Cartesian coordinates in plane of wing where x extends in
the direction of the free-stream velocity

Y dimensionless function describing plan form

Z ordinates of wing profiles

Zu ordinates of upper surface of wing profiles

Zz ordinates of lower surface of wing profiles

a angle of attack


7 ratio of specific heats, for air 7 = 1.4

o Mo2 1
o [(7 + 1) Mo T]

Po density





The forces on a body moving through an infinite fluid are dependent
on a number of parameters equal to that necessary to describe the problem.
Thus, it is well known from dimensional considerations that the drag D
of a body can be expressed as the product of a pressure, say the dynamic
pressure q = -- Uo2, a characteristic area S, and some function of Mach
number Mo, Reynolds number R, the geometry of the body, and gas proper-
ties such as the ratio of the specific heats 7, etc. This can be written
symbolically as follows:

D = qSf(Mo, R, geometry, gas properties) (l)

(Note that S does not necessarily refer to the plan-form area, but,
rather, to any combination of geometric lengths having the dimensions of
area.) The geometry of a body can be described by a number of dimension-
less parameters, the number of which depends on the complexity of the
shape. If these considerations are applied to the present family of rec-
tangular wings having NACA 63AXXX profiles, the description is particularly
simple since the wing plan form is determined by specifying the aspect
ratio A, the profile by the thickness ratio T = t/c and the camber ratio
H = h/c, and the inclination of the wing by the angle of attack a. In
this way, equation (1) can be rewritten in the following more explicit
D = f(Mo, R, A, T, H, a, gas properties) (2)
If, as in nearly all problems of aerodynamic practice, all measurements
are made in the same fluid, air, the gas properties can be represented by
merely a set of constants and therefore disappear as parameters in equa-
tion (2) leaving only

D = f(Mo, R, A, T, H, a) (3)

If attention is confined to the drag at zero lift Do, as is the case with
the transonic area rule, further simplification results because a is
then a dependent parameter and may be eliminated even though it is not a
constant for variously cambered wings. Then
DO = f(Mo, R, A, T, H) (4)

The proof of the foregoing step is as follows: The relationship for lift
analogous to equation (3) is

CL = L- = f(Mo, R, A, T, H, a) (5)




where f is, of course, a different function of the indicated variables.
This can be solved for a

a = f(Mo, R, A, T, H, CL) (6)

and substituted in its place in equation (3) to produce

S= f(Mo, R, A, T, H, CL) (7)

In this form, it is clear that the restriction to zero lift fixes the
value of the last parameter, thereby leaving Do/qS dependent only on the
remaining parameters as in equation (4).

Although dimensional considerations such as the foregoing display the
parameters or factors that influence the drag, they do not provide any
information on the nature of the functional relations involved. Thus, it
may be that some of the parameters are more important than others, or that
a parameter is of importance over a certain range of values but of negli-
gible importance in another range, or that a parameter is important or not
depending on the value of another parameter, etc. Examples are that
Do/qS depends on Mo at transonic and supersonic speeds but not at sub-
sonic speeds, Do/qS depends on A for small A but not for large A
(if S refers to the plan-form area), etc. Since there exists a great
body of literature pertinent to the nature of these dependencies which is
probably quite familiar to the readers, no further amplification appears
necessary here.


Theoretical Considerations

The customary verbal statement of the transonic area rule is given
in the INTRODUCTION. It should be noted that the term "drag rise" does
not refer to the increase in drag force Do, but to the increase in the
ratio Do/q between a subcritical reference Mach number Mref arid a
transonic Mach number Mo. If Sc(x) represents the streamwise or axial
distribution of cross-section area, the transonic area rule states that
the variations A(Do/q) with Mach number Mo are the same for all low-
aspect-ratio wing-body combinations having the same Sc(x). Note that
the rule in this form relates a very wide class of wings, bodies, etc.,
but that all related configurations must have the same longitudinal
length c. This restriction can be removed easily by recasting the state-
ment in dimensionless form using c as the characteristic length. Thus,
we can say that the variation of A(D/qc2) with Mach number Mo is the
same for all low-aspect-ratio wing-body combinations having the same




Sc(x/c)/c2. From a slightly different point of view, the area rule states
that, near the speed of sound, A(D/qc2) is a function of Mo and
Sc(x/c)/c2, or

A ) = o fLMo, Sc(x/c)j

The axial distribution of cross-section area Sc(x/c) can be written
in the form

Sc(x/c) = Sms(x/c) (

where Sm is the maximum cross-section area and s(x/c) is an area-
distribution function. Application of the transonic area rule to the drag
results of the present family of rectangular wings is facilitated partic-
ularly by the fact that s(x/c) is the same for all wings. As a result,
the area distribution Sc(x/c) of any wing in this family is specified by
stating the value of the maximum cross-section area Sm, so that

A(Do/qc2) = f(Mo, Sm/c2) (10)

Since the chord c and thickness t of each wing are constant across the
span b, the maximum cross-section area is equal to the product of the
span and thickness

Sm = bt (11)


Sm bt
C2= AT (12)
c c c

whence equation (10) can be rewritten for this family of airfoils as

AD) = f(Mo, AT) (13)

The remarkable simplicity of these statements is emphasized by com-
parison with the functional relation revealed by dimensional consideration
alone. Thus, if the unspecified area S is replaced by c equation (4)

A( ) = A 2) = f(Mo R, A, T H) f(Mref, Rref, T, H) (14)

where Rref refers to the value of Reynolds number associated with the
subcritical reference Mach number Mref, and the symbol f refers in each
case to the appropriate function of the indicated variables. In the cus-
tomary discussion of drag-rise data, Mref and Rref are constants and no



longer appear as parameters in equation (14). Further simplification
occurs in most cases because the wind-tunnel or flight test technique
determines a specific relation between the Mach and Reynolds numbers. As
a result, either Mo or R can be removed as parameters since the value
of either is determined by that of the other. Since the present problem
is more closely connected with effects of compressibility than of vis-
cosity, it is appropriate to retain Mo as the significant parameter. In
this way, equation (14) reduces to

S ) = f(Mo, A, T, H) (15)

Comparison of equations (13) and (15) highlights the fact that the
area rule affirms, in dimensionless terms, that the drag-rise parameter
A(Do/qc2) for the present family of wings depends on Mach number and the
product of aspect ratio and thickness ratio AT (or the maximum cross-
section area parameter Sm/c2) but is independent of camber ratio H, and
aspect ratio A or thickness ratio T taken separately.

Comparison with Experiment

Application to wings having identical area distributions.- The
applicability of the transonic area rule to the present family of rectangu-
lar wings can be examined in several ways. Perhaps the most obvious way
is to actually compare the variation of A(Do/qc2) with Mo for two or more
wings having identical area distributions. For the present family of
wings, this means comparing wings having the same S/c2 or AT. The tran-
sonic area rule predicts that the variation of A(Do/qc2) with Mo should
be the same for all such wings.

An example of such a
The experimental data are

comparison is shown in sketches (a) and (b).
from references 2 and 3. Both wings have
SA=2, r=.08
-0"o o A=4, r=.04

Sketch (a) Sketch (b)
Sketch (a) Sketch (b)



symmetrical sections (H = 0) and AT = 0.16 but one has an aspect ratio of
2 and thickness ratio of 0.08; whereas, the other has an aspect ratio of
4 and thickness ratio of 0.04. The first sketch shows the total drag and
the second the drag rise determined by subtracting the value of Do/qc2
at Mo = 0.6. Although the two curves in sketch (b) are not identical as
predicted by the area rule, they are closely related. Innumerable reasons
could be advanced to explain the differences between the two curves; per-
haps there are viscous effects which may significantly affect the drag
rise, perhaps the measurements are not sufficiently accurate or the flow
field sufficiently uniform, or perhaps the aspect ratio or thickness ratio
is too large, etc. In any case, this particular comparison would probably
be scored in favor of the transonic area rule.

Another comparison, this time among three wings of aspect ratio 2,
thickness ratio 0.06, but different amounts of camber is shown in sketches
(c) and (d). The amount of camber is specified by the ideal lift coef-
ficient Cz. in accordance with the NACA scheme for airfoil section
designation. Again, the basic data
are presented in the first sketch and
the drag rise in the second. In this A=2,r=.06 % i
case, however, the variation of .10 4
A(Do/qC2) with MO is definitely not
the same for the three wings and the o8s
use of the area rule could lead to .
serious error. At a Mach number of
unity, where the area rule is sup- o6
posed to be most accurate, the drag
rise of the wing with greatest camber .04
is nearly twice that of the uncambered
wing. .o0

Wings having similar area dis- _
tribution.- Although a certain num- .6 .7 .8 .9 i0 u 1
ber of additional comparisons of the
type described in the preceding sec- Sketch (c)
tion can be made using the data of
references 2 through 5, the number is
definitely limited because the test
program was not designed to preserve .06 4--
a single value for AT for all wings.
It is furthermore not practical to .04 .
carry out extensive programs of such 4
a type because it necessitates the
testing of very thin wings of high
aspect ratio and thick wings of low
aspect ratio. As mentioned in the ~ 7 .9
derivation of equation (13), however, M,
all members of the present family of Sketch (d)
wings have similar distributions of



cross-section area (a single area-distribution function s(x/c)), and the
transonic area rule can be extended to include such cases by introducing
Sm/c2 or AT as a second parameter. In exchange for being able to cor-
relate the drag rise of bodies having not only identical, but also similar
area distributions, we incur the complications of a dependence on two
parameters rather than only one. Thus, whereas the curves representing
the variation of A(Do/qc2) with Mo for all wings having identical
Sm/c2 or AT should coincide to form a single line, those for a family of
wings having similar area distributions should form a family of lines.
Simplicity can be regained, however, by restricting attention to a single
Mach number and ascertaining the variation of A(Do/qc2) with Sm/c2 or AT.
Since the customary statement of the area rule restricts attention to
near-sonic speeds, the most appropriate single Mach number to select for
such a comparison is unity.

Sketch (e) shows the variation of A(Do/qc2) with Sm/c2 for all the
uncambered wings of references 2 and 3. It can be seen that all these
results fall near to a single curved line for wings of all aspect ratios
up to 3 but that those for wings of aspect ratios 4 and 6 depart from this
line in a systematic manner.

The same results are replotted in sketch (f) versus the square of
Sm/c2 rather than the first power. It can be seen that the curved line

0 .1 .2 .3 .4
Sm/c' Ar
Sketch (e)

.5 .6

(Sm/c'', (A '
Sketch (f)




of sketch (e) for wings having aspect ratios less than 3 is now a straight
line, indicating that the sonic drag rise for these wings is proportional
to the square of the maximum cross-section area. Such a dependence is
consistent with the formulas of linearized compressible flow theory for
the wave drag of nonlifting slender wings and bodies at supersonic speeds.

These plots can be interpreted as showing that the sonic drag rise
of the uncambered members of the present family of wings depends on the
cross-section area in accordance with the area rule, provided the aspect
ratio is about 3 or less. On the other hand, the results for wings of
higher aspect ratio can only be interpreted as indicating that some parame-
ter other than the cross-section area must be involved. Inasmuch as the
geometry of wings of the present family is described completely by the two
parameters, aspect ratio and thickness ratio, it is clear that these
parameters must assume importance in some form other than their simple
product AT for wings of larger aspect ratio. One could seek the new
relation empirically, but the transonic similarity rules provide a theo-
retical basis for proceeding. Some properties of transonic similarity
rules are reviewed in the following section, although the reader is refer-
red to the original references for further details.


Statement of Rule

Transonic similarity rules are derived from the nonlinear equation
of inviscid flow theory and are known for thin wings (e.g., ref. 6) and
slender bodies of revolution (ref. 7), but not for wing-body combinations
with pointed noses. In contrast to the transonic area rule which relates
the zero-lift drag rise of families of bodies having identical or similar
axial distributions of cross-section area, the transonic similarity rules
relate the aerodynamic properties of much more highly restricted families
of bodies. Even for wings alone, the restrictions imposed on the members
of a single family are much more severe than for the area rule, since all
members of a single family must have affinely related plan forms, affinely
related thickness distributions, and affinely related camber distributions.
To be more explicit: if the plan form is given by

= Y(x/c) (16)




y as indicated in sketch (g), it is required that
Y(x/c) be a single function for all wings of a
given family. Furthermore, if the ordinates of
Y= A() the wing surface are given by

Z t 1 (x
-c c -2 c' b


Sketch (g)

where the plus sign is associated with the ordinates of the upper surface
Zu and the minus sign with those of the lower surface Z2, it is required
that gl(x/c, y/b) and g2(x/c, y/b) be single functions for all wings of
a given family. The first of these restrictions requires that related
plan forms be obtainable one from another by a differential lengthening of
lateral and longitudinal dimensions. Thus rectangular plan forms consti-
tute one family, triangular plan forms with straight trailing edges
another, etc. Examples of related plan forms are illustrated in
sketch (h). The relationship expressed in equation (17) requires that the
thickness distribution and camber
I => EL z _] => / variations must be the same for all
wings of a particular family. The
magnitudes of the maximum thickness
ratio t/c and camber thickness
S> ratio h/c, as well as the angle of
attack a, may be different for vari-
Sketch (h) ous members of a single family.

From the present point of view, one of the most significant proper-
ties of the family of wings described above is that all members have the
same dimensionless area-distribution function s(x/c). This can be shown
as follows:

S(x/c) =

(Zu ZZ)dy =

tg1 X) d,


= bt Y/b

b (b)

= btg(x/c)

= Sms(x/c)




t f e


where R represents the integral of the preceding expression and is pro-
portional to s(x/c), since Sm is proportional to bt for a family of
wings. Because of this fact, a family of wings suitable for correlation
by the transonic similarity rules is also part of a family of bodies suit-
able for correlation by the area rule.

For wings of such a family, the similarity rules of inviscid, small-
disturbance, transonic flow theory provide that the wave drag Dw be
given by
D Sw S TM/s [( M 22/ [(+1)Mo71/3 A;!, h
DW [Mo=(T+I) ] / M [( (+l)Mo2T]2/s,

= qS [Mo2(7+1)]1/ f(o, A, h, I ) (19)


T t/c

Mo2 1
A0 [(7+l)Mo2Tr]2/

A [(,y+DMo2]1/3 A

and Sp is the plan-form area of the wing. Equation (19) is not in the
proper form for comparison with the area rule since the latter is con-
cerned with conditions at zero lift and, hence, imposes an indirect
requirement on F. We can proceed toward the desired form, however, by
introducing the transonic similarity rule for the lift coefficient
CL = L/qSp

CL = [M2(7+1)]i/3 f( o a) (20)

and defining the reduced lift coefficient
X [_Mo2(y+l))1) 3
L = -- CL = f(' i, F1, a) (21)

Now since CL is a function of the same four variables that appear in
equation (19), a can be replaced with CL in the latter equation, whence

Dw = qSP [Mo,(7)]1/ f(Et A, C L) (22)




The condition of zero lift eliminates the last parameter, leaving only
Dow = qS [Mo2(7+)]/s3 f(So' )

= qSp [Mo (7+1)+1/)3 ( o2

[(Y+)Mo2T]1/s A, h/t} (23)

Application to Rectangular Wings and Comparison
with Transonic Area Rule

The restrictions introduced in the derivation of the transonic simi-
larity rule for zero-lift wave drag are such that they permit the direct
application of equation (23) to the present family of rectangular wings
having NACA 63AXXX profiles. It is natural to compare this functional
relation with the corresponding relations of equations (10) and (13) given
by the transonic area rule.

A = f(Mo, Smc2) = f(Mo, AT) (24)

At first glance, the two sets of relationships appear to bear only slight
resemblance. It can be seen upon closer examination, however, that some
of the apparent differences are superficial and of little or no signifi-
cance. For instance, equation (23) is concerned with wave drag Dow
whereas equation (24) is concerned with drag rise. It is evident, how-
ever, that the two rules are actually concerned with the same quantity,
since the drag rise can be considered to be an approximation for the wave
drag under the assumption that the friction drag coefficient is independent
of Mach number.

Another point of apparent lack of resemblance is that equation (23)
does not show an explicit relationship between wave drag and maximum
cross-section area, since the latter is not a function of SpT5/s nor of
o, A, or E taken separately. This does not mean that the transonic
similarity rule is incompatible with the area rule because there are sev-
eral possibilities for making such a dependence visible. Two permissible
procedures are to multiply the right side of equation (23) by either to
or A. The first procedure is of no help for the discussion of conditions
at Mo = 1, since the quantity Mo2 1 appears in two places and an inde-
terminate form ensues. The second is perfectly acceptable, however, and
produces the following relationships:




Dow = qSpAT2 f(o, A, h)

= qc 2f((, A, h) (25)
(bt 2

Since the maximum cross-section area Sm of the present family of rec-
tangular wings is equal to the product bt, equation (25) can be rewritten
as follows:

DowIc2 2 f(o, A, h)

qc2 2 f
c 7 [(7+l)Mo.T]2/'

[(7+l)Mo2eT]1/ A, h/t} (26)

Equation (24) can likewise be rewritten by multiplying f by the square
of Sm/c2,

( = f(M, Sm/c2) = f(Mo, AT) (27)

This appears to be the closest that the two rules can be brought
together without introducing additional simplifications. Both rules are
now concerned with essentially the same quantity, Dow/qc2 and A(Do/qc2).
Each rule states that this quantity is proportional to the square of
Sm/c2 times some unknown function of certain specified parameters. The
two rules disagree completely, however, as to the nature of the parameters.
The transonic similarity rule specifies three parameters to, A, and h,
whereas the area rule specifies only two, Mo and AT. Since neither set
can be transformed into the other, it is apparent that the only way in
which both rules can be universally correct is for the function f to be
actually a constant and not to depend on the value of any of the five
parameters. It is obvious that such is not the case, however, since it
requires, for instance, that the wave drag be independent of Mach number.

There is a way out of this apparent impasse, however, if the range
of validity of one or both of the rules is restricted sufficiently that
f is independent of the remaining variables. Comparison of the two rules
shows that the drag depends in both cases on the Mach number of the stream
and the geometry of the wing and, in the case of the transonic similarity
rule, on the ratio of the specific heats 7. Since nearly all problems of
aerodynamic interest are concerned solely with air, however, 7 is a con-
stant and need not be retained as a parameter. The resulting simplified
equation can be written in full in either of the following forms:




B2 2 f M2 1)2/ (MoT)1/s A, h/t
qc2 f (MoT)2 j

= (AT)2 f o27) (M2 T)1/ A, h/t (28)

With regard to further simplifications, it should be noted that the
derivation of the transonic similarity rule requires that the thickness
and camber be small with respect to the chord, but does not restrict the
Mach number or aspect ratio. The transonic area rule, on the other hand,
restricts itself to Mach numbers near the speed of sound and to wings
having low aspect ratios.

The functional relation given in equation (27) representing the area
rule simplifies if attention is fixed on any given Mach number, since Mo
appears as an isolated parameter, thus

A ( C2 f(AT) (29)

On the other hand, the only Mach number at which the corresponding rela-
tion given by the transonic similarity rule simplifies, irrespective of
the thickness ratio, is Mo = 1. The resulting expression is

Do =( f(A1/s3, h/t) (30)
qc2 \c/

This expression simplifies further for wings having symmetrical sections,
since h/t is zero. For such cases, the transonic similarity rule for
wave drag at MO = 1 reduces to

D 2 = S f(AT1/) (31)
qc2 c2/

The foregoing analysis has developed certain equations relating to
the drag rise or wave drag of a family of wings having relatively simple
geometry. Despite the great restrictions imposed by the selection of such
a family of bodies, the analysis has disclosed a number of significant
points which complement those discussed in connection with the experimental
data presented in sketches (a) through (f). First of all, it has been
shown that the only way in which the wave drag at Mo = 1 can depend on
the axial distribution of cross-section area (defined in the present fam-
ily of wings by the value of the maximum cross-section area Sm) and still




be compatible with the transonic similarity rule for drag is for Dow/qc2
to be proportional to the square of Sm/c2. This is precisely the depend-
ence disclosed by the experimental data for low-aspect-ratio wings having
symmetrical sections and shown in sketch (f). The transonic similarity
rule states that the wave drag may depend on the camber, whereas the area
rule states that it does not. The experimental data in sketch (d) show
that camber has a significant effect on drag.

It is evident, both from a priori considerations and from the experi-
mental results shown in sketch (f), that some change must occur in the
relation between wave drag and maximum cross-section area as the aspect
ratio becomes very large. The only possibility permitted by the transonic
area rule is that A(Do/qc2) varies with Sm/c2 in some other manner than
as the square. It can be seen from sketch (e), however, that the data for
wings of the present family cannot be correlated on this basis if the
aspect ratio is greater than about 3. The transonic similarity rule for
the zero-lift wave drag of uncambered wings at Mo = 1 provides a differ-
ent dependence by stating that Dow/qc2 is equal to the square of Sm/c2
times some function of ATI/3. This statement is compatible with the
area rule if Dow/qc2 is independent of AT1/3 for small values of the
latter. If wave-drag data for wings of larger aspect ratio can be cor-
related successfully by considering the value of this quantity, a theoret-
ical basis for a limit to the range of applicability of the transonic area
rule has been found.

Comparison with Experiment

The foregoing discussion has served to focus attention on the fact
that the parameter AT /3, already familiar from prior papers on transonic
flow (e.g., ref. 3), may be of importance in defining the limit of appli-
cability of the transonic area rule as applied to a family of affinely
related wings. The functional relation of equation (31) suggests that if
Doc2/qSm2 is plotted as a function of AT1/3, all the data for the
uncambered wings should fall on a single curve. Moreover, the values of
Dow2/qSm2 should be independent of AT1"3 over whatever range the area
rule applies. Although this method of plotting is conceptually simple,
it imposes severe requirements on the accuracy of the experimental deter-
mination of wave drag because any errors are magnified as a result of
dividing by the square of the maximum cross-section area. For results
such as the present in which the wave drag is not actually measured, but
is inferred from measurements of total drag by subtracting an estimated
friction drag, greatest difficulties are experienced if the wings are thin
and of low aspect ratio so that the wave drag is only a small fraction of
the total drag. Consequently, data points for which the wave drag is less
than half the total drag are omitted in the plot of A(Doc2/qSm2) versus




3.o ATS/3 shown in sketch (i). Even with
this precaution, the data evidences
2s-- considerable scatter for thin, low-
-- .-a-a aspect-ratio wings. (The symbols refer
20o A42 to the same wings as in sketch (e).)

_____ __ The principal points of interest
Z A 6 in this plot are threefold. First,
,o /_ except for the smallest values of
h=o ATI3 where the scatter is too large
5 ______ __ to provide any positive conclusions,
the points determine essentially a
__single line, indicating that the sonic
0 4 .8 2 .6 20 24 28 wave-drag characteristics of the uncam-
bered wings of the present family can
Sketch (i) be correlated successfully by the tran-
sonic similarity rule (this has been
shown previously by McDevitt, ref. 3); second, A(Doc2/qSm2) is, at best,
independent of AT1/3 for values of the latter up to about unity, indi-
cating that the drag in this range varied in accordance with the area rule
as well; third, A(Doc2/qSm2) varies appreciably with AT1/3 at larger
values of the latter, indicating that the range of applicability of the
area rule was exceeded.

Before leaving the discussion of the correlation of drag data at
Mo = 1 by use of the transonic similarity rule, it is worthwhile to call
attention to another method of plotting which illustrates the same factors
although in a slightly different manner. This procedure, which is
described more completely in references 3, 5, and 6, is based on equa-
tion (23) rather than equation (26) and consists, for the zero-lift drag
of a family of uncambered wings in an air stream with Mo = 1, of plotting
the variation of CDow/T5/s with AT1/3 where CDow represents Dow/qSp.
3 Sketch (j) shows the data of sketch (i)
Sreplotted in this manner. (Once again,
30o the lack of wave-drag data requires the
\-A4 A'6 substitution of drag-rise information
2.5- and the introduction of the drag-rise
/ coefficient ACDo, defined as equal to
2.0 R- A(Do/qSp).)
ti / /' -A=2 h=O
Z5 h= The curve formed by the data
o + points on this type of plot is perhaps
somewhat simpler than that of sketch
/ a-- (i) since it is asymptotic to straight
lines at both large and small AT1/3.
o ____ At small AT1/3, the points define a
0 4 .8 2 Ar6 20 24 2S straight line passing through the
origin. Such a line is in accordance
Sketch (j)



with the area rule. At values of AT1/3 larger than about unity, how-
ever, the line determined by the data points departs from this initial
trend and turns toward the horizontal. This trend is contrary to the
area rule but consistent with the fact that the results for wings of high
aspect ratio must tend toward those for wings of infinite aspect ratio.
It cannot be emphasized too much that the critical value of unity for
AT1/3 is determined solely on the basis of data for a very special family
of rectangular wings having symmetrical profiles. Other families of wings
would be represented by different curves on such a plot.

It is interesting to consider for a moment the nature of this limit
and to compare it with the verbal restriction of the area rule to low-
aspect-ratio wings. Although the one requires that AT"1/ be small, and
the other that A be small, these two statements are in better agreement,
insofar as engineering applications are concerned, than might appear at
first glance. This agreement results from the fact that other consider-
tions, such as designing for structural strength or for the avoidance of
excessive drag, tend to preserve a rather narrow range for the values of
T likely to be met in practice. The effectiveness of T as a parameter
is further diminished by the fact that only its cube root is involved.
Consequently, the restriction to small AT /3 represents a limitation
primarily on the aspect ratio and only secondarily on the thickness ratio.

It is also interesting to exam-
ine the shapes of equivalent bodies of
revolution having the same axial dis-
tribution of cross-section area as
wings lying on either side of the
limit. Accordingly, sketch (k) has
been prepared showing the shapes of
bodies of revolution equivalent to
one of the wings having AT1/3 much
less than unity, to the two wings
having AT1/3 nearest unity, and to
the wing having the largest AT1/3 of
any tested. It can be seen that the
equivalent bodies are blunt and stubby
rather than pointed and slender. Thus,
although it has been shown that the
drag-rise characteristics of the var-

A=/, r=.04, Ar=i342

A=3, r .04, Ar =/026

A=6, r /0, Ar 2784

Sketch (k)

ious members of the present family of uncambered rectangular wings are
related to one another in the manner predicted by the transonic area rule,
provided AT1/3 is less than about unity, it would appear that the drag-
rise characteristics of the equivalent bodies of revolution might be con-
siderably different.

It is
drag rise.
mum camber

shown in sketch (d) that camber has an effect on the zero-lift
The transonic similarity rule suggests that the ratio of maxi-
to maximum thickness h/t might be an appropriate parameter to



A=2, r=./O, AriT.928


use in addition to AT1/3 to correlate the sonic zero-lift wave-drag
characteristics of a family of cambered wings. Accordingly, sketch (z)
has been prepared showing the influence
40 444 of h/t for wings of various ATI/3.
3.5 t63A406) For reasons of simplicity, only three
_5 values for h/t, namely, 0, 0.222, and
j3OC / I -- 0.444, are included on this plot. The
/' important effects of camber are readily
25 / evident from this graph. It is appar-
h / h222 ent that the area rule is not applica-
( -20 26 ble to wings having different camber-
0 /0 thickness ratios, even if the values of
5 -h/t=0 AT1/3 are sufficiently small to permit
Successful correlation of the wave drag
/0o of uncambered wings. On the other hand,
AM I/ it is permissible in the theory and
.5 seems to be indicated by the experimen-
tal data that the area rule is applica-
0 4O 8/ /2 6 20 24 2.8 ble to families of cambered wings pro-
Ard viding h/t is maintained constant.
This result is recognized in sketch (1)
by the fact that A(CD /T5/3) for the
Sketch (Z) wings of constant h/t is approximately
proportional to AT1/3.


The range of applicability of the transonic area rule has been inves-
tigated by comparison with the appropriate similarity rule of transonic
flow theory and with available experimental data for a large family of
rectangular wings having NACA 63AXXX profiles. These wings are of affinely
related geometry and are hence immediately amenable to analysis by the
transonic similarity rules. On the other hand, the axial distributions of
cross-section area are not identical, in most cases, but merely similar.
(The ratio of the local cross section to the maximum cross section is a
given function.) It is shown, however, how the transonic area rule can
also be used to correlate the sonic drag-rise data for such a family of

It is found that the sonic zero-lift drag-rise data for the present
family of wings can be successfully correlated on the basis of the area
rule, provided the wing profiles are symmetrical and the product of the
aspect ratio and the cube root of the thickness ratio is less than about
unity. Within this range, the sonic drag rise varied as the square of the
maximum cross-section area, all wings having equal chords. It is demon-
strated that this is the only dependence of drag on maximum cross-section
area for a family of wings like the present that is compatible with both
the area rule and the transonic similarity rule.



It was found that the addition of camber greatly increased the sonic
drag rise and that the application of the transonic area rule to a family
of vings, some of which are cambered and others not, could lead to serious
error. On the other hand, it is indicated by the transonic similarity
rules and the experimental data that the area rule is applicable to fami-
lies of cambered wings, provided the camber distribution, as well as the
area distribution, are similar and that the ratio of the maximum ordinates
of the camber and thickness distribution is maintained constant.

Ames Aeronautical Laboratory
National Advisory Committee for Aeronautics
Moffett Field, Calif., June 28, 1954


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. Nelson, Warren H., and McDevitt, John B.: The Transonic Characteris-
tics of 17 Rectangular, Symmetrical Wing Models of Varying Aspect
Ratio and Thickness. NACA RM A51A12, 1951.

3. McDevitt, John B.: A Correlation by Means of the Transonic Similarity
Rules of the Experimentally Determined Characteristics of 22 Rec-
tangular Wings of Symmetrical Profile. NACA RM A51L17b, 1952.

4. Nelson, Warren H., and Krumm, Walter J.: The Transonic Characteris-
tics of 38 Cambered Rectangular Wings of Varying Aspect Ratio and
Thickness as Determined by the Transonic-Bump Technique. NACA
RM A52D11, 1952.

5. McDevitt, John B.: A Correlation by Means of Transonic Similarity
Rules of the Experimentally Determined Characteristics of 18 Cam-
bered Wings of Rectangular Plan Form. NACA RM A53G31, 1953.

6. Spreiter, John R.: On the Application of Transonic Similarity Rules
to Wings of Finite Span. NACA Rep. 1153, 1953.

7. Oswatitsch, K., and Berndt, S. B.: Aerodynamic Similarity at Axisym-
metric Transonic Flow Around Slender Bodies. Kungl. Tekniska
HRgskolan, Stockholm. Institutionen for Flygteknik. Tech. Note 15,


NACA-Langley 8-23-54 325


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