On slender-body theory at transonic speeds

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Title:
On slender-body theory at transonic speeds
Series Title:
NACA RM
Physical Description:
12 p. : ; 28 cm.
Language:
English
Creator:
Harder, Keith C
Klunker, E. B
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

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

Notes

Abstract:
Abstract: The basic ideas of the slender-body approximation have been applied to the nonlinear transonic-flow equation for the velocity potential in order to obtain some of the essential features of slender-body theory at transonic speeds. The results of the investigation are presented from a unified point of view which demonstrates the similarity of slender-body solutions in the various Mach number ranges. The transonic area rule and some conditions concerning its validity follow from the analysis.
Bibliography:
Includes bibliographic references (p. 12).
Statement of Responsibility:
by Keith C. Harder and E.B. Klunker.
General Note:
"Report date January 18, 1954."
General Note:
"Classification changed to unclassified Authority: J.W. Crowley Date: 9-7-55 Change No. 3075."--stamped on cover

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003808938
oclc - 132692991
sobekcm - AA00006153_00001
System ID:
AA00006153:00001

Full Text


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ON SLENDER-BODY THEORY AT TRANSONIC SPEEDS
By Keith C. Harder and E. B. Klunker










Langley Aeronautical Laboratory
Langley Field, Va.
C.: '.OMM TO .W.
SEARCH MEMORANDUM


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March 24 1954

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NACA RM L54A29a


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


RESEARCH MEMORANDUM


ON SLENDER-BODY THEORY AT TRANSONIC SPEEDS

By Keith C. Harder and E. B. Klunker


SUMMARY


The basic ideas of the slender-body approximation have been applied
to the nonlinear transonic-flow equation for the velocity potential in
order to obtain some of the essential features of slender-body theory
at transonic speeds. The results of the investigation are presented
from a unified point of view which demonstrates the similarity of
slender-body solutions in the various Mach number ranges. The primary
difference between the results in the different flow regimes is repre-
sented by a certain function which is dependent upon the body area
distribution and the stream Mach number. The transonic area rule and
some conditions concerning its validity follow from the analysis.


INTRODUCTION


Slender-body theory originated with Munk's paper (ref. 1) in 1924
in which the forces on slender airships were calculated for low-speed
flight. In 1958 Tsien (ref. 2) pointed out that Munk's airship theory
also applied to the flow past inclined, pointed bodies at supersonic
speeds. The subject gained new importance in 1946 with the appearance
of Jones's paper (ref. 5) in which it was shown that the basic ideas of
the slender-body approximation could be used to calculate the forces on
slender lifting wings at both subsonic and supersonic speeds provided
that proper account was taken of trailing-vortex sheets. Since Jones's
paper, the subject has received wide treatment in the literature. In
an important paper in 1949, Ward (ref. 4) developed a general unifying
theory for the flow past smooth, slender, pointed bodies at supersonic
speeds which contains as special cases the lifting planar wings of Jones
and the slender nonlifting bodies treated by Von Karman (ref. 5). The
corresponding problem at subsonic speeds has been examined by Adams and
Sears (ref. 6) who also extended the slender-body concepts to shapes
which are "not so slender." Lighthill (ref. 7) has given a method for
calculating the flow past bodies with discontinuities in slope. Keune
(ref. 8) has developed solutions for slender wings with thickness and
various lifting configurations have been treated by Heaslet, Spreiter,
Lomax, Ribner, and others refss. 9 to 14).
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The slender-body theory presented in references 2 to 14 has been
based upon the linearized equation for the velocity potential. In the
present paper, the basic ideas of the slender-body approximation are
applied to the nonlinear transonic equation for the velocity potential
in order to obtain some of the essential features of slender-body theory
at transonic speeds. The attempt has been made to present the results
from a unified point of view which demonstrates the similarity of the
slender-body solutions in the various Mach number ranges.

The authors wish to acknowledge the invaluable aid and advice of
Dr. Adolf Busemann of the Langley Laboratory during the writing of this
paper.


SLENDER-BODY APPROXIMATION

Slender-body theory deals with that class of shapes whose length
is large compared with any lateral dimension. For such shapes at both
subsonic and supersonic speeds, the flow in planes normal to the stream
direction can be approximated by solutions of Laplace's equation. The
justification is that for very slender wings or bodies the variation
of the geometrical properties in the stream direction is small and,
consequently, the rate of change of the longitudinal component of the
velocity in the stream direction is also small. The various slender-
body solutions have all been developed on the basis of the linearized
potential equation. However, a similar development can be made on the
basis of the nonlinear transonic equation.

The simplest differential equation for the disturbance potential 0
which is generally valid at transonic speeds (ref. 15, for example) is


M2 (7 )M2x xx + Orr + r + 0 (1)
r r2

where x, r, and a are cylindrical coordinates, M is the stream
Mach number, and 7 is the ratio of specific heats at constant pressure
and constant volume. With the introduction of the dimensionless
coordinates and n by x = Z| and r = br, and of the dimension-
2
less potential 0 by 0 = -b2 0(,,), the transonic potential-flow
equation becomes


( M2- (7 + 1)M2 + + =0 (2)
\zI~J~


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NACA RM L54A29a


where Z is a characteristic length and b is a characteristic width
such as the largest lateral dimension of the configuration. For suffi-
ciently small values of the width parameter b/1, it appears that the
terms involving derivatives in the stream direction can be neglected
to obtain the result that the flow satisfies Laplace's equation


1rr + r + = 0)
r r2


in the cross-flow plane. Equation (5) represents the slender-body
approximation to equation (1). Some conditions will be determined
subsequently which are necessary in order for solutions of equation (5)
to be approximate solutions of equation (1).

The boundary conditions for the flow about a body in a uniform
stream are the vanishing of the disturbance velocities at infinity and


= dn(1 + Ox) dn
on dx dx

on the body where n is in the direction normal to the body contour in
the cross-flow plane. For flows satisfying Laplace's equation in the
cross-flow plane, the surface boundary condition can be integrated
(ref. 4, for example) to give


S dt = S'(x) (4)


where dt is an element length in the direction of the tangent to any
contour C in the cross-flow plane enclosing the body, S(x) is the
cross-sectional area distribution of the body, and the prime denotes
differentiation with respect to the indicated argument.

The slender-body solution of equation (1) can be represented by a
solution of equation (5) plus a function of integration. Since equa-
tion (5) is independent of Mach number, the form of the solution is
identical with the known slender-body solutions for subsonic and super-
sonic flow. The slender-body analyses of references 4 and 6 have
established that the solution can be represented by a distribution of
sources and higher order singularities on the axis; an equivalent form
is given by a distribution of sources in the region of the cross-flow-
plane interior to the surface boundary. The slender-body solution is
then expressed as


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where cp denotes the solution of Laplace's equation in the cross-flow
plane with i appearing as a parameter introduced by the geometry of
the cross section at |. The function cp, being independent of the
stream Mach number, can be evaluated for incompressible flow past the
shape under consideration.

A necessary condition for equation (6) to be an approximate solu-
tion of equation (1) is that the terms neglected in equation (1) be
small compared with those retained. By assuming for the moment that
log p is the only singular term in 0, the ratio of the term
I M2 (7 + l)M2x ]xx, which is neglected in the slender-body
approximation, to any of the remaining terms in equation (1) is of the
order


()2 1 M2) log + 0(1] + 2 (log2 )+ (log + 0(1)


where O( ) denotes order of and the nonsingular terms are denoted
by 0(1). This ratio can be made smaller than any prescribed value E
by restricting the solution to the interior of a cylinder of some
radius, say R/Z. For given values of M and e, the radius of this
cylinder increases with decreasing b/1 and the ratio R/b approaches
infinity as b/Z approaches zero. Moreover, for given values of E
and R/Z, larger values of b/Z are permitted as M approaches 1.

From equation (6) it is apparent that S'(x/Z) and its deriva-
tives must be finite in order to satisfy the requirement of small dis-
turbances. Moreover, an additional restriction on the asymmetry of
the body is sometimes required (ref. 4), particularly for lifting con-
figurations namely, the radius of curvature of the body boundary in
the cross-flow plane must be O(b) where the boundary is convex outward.
The restrictions on body shape imposed by the function g(x/Z) will be
considered subsequently after the nature of this function is established.

The function g(x/1) is determined from considerations involving
the complete transonic differential equation (eq. 1) and, consequently,
is dependent upon the stream Mach number. Ex: at r= < shows that the contribution of the asymmetric term of
the source distribution to the potential can be made smaller than any
prescribed value by making b/Ro sufficiently small -

To this order of approximation, then, the flow field external to Ro
is axisymmetric. Moreover, there is an axisymmetric flow which approxi-
mately matches the pressure and flow direction of the slender-body


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NACA RM L54A29a


solution at r = Ro. The potential 0o for this associated axisymmetric
flow satisfies equation (1) and is expressed symbolically as


So/2 ) log + go(x/z,r/ (7)



where the characteristic width b and length I are taken the same as
for the asymmetric shape. In equation (7), so(x/1) is the area dis-
tribution of the associated body of revolution which has the same outer
flow (r Ro) as the asymmetric shape. If this axisymmetric flow
satisfies the slender-body conditions, then go is independent of r
for r < R. In order for the radial derivatives to match at r = Ro,
So(x/l) must be equal to s(x/1); that is, the associated body of
revolution must have the same axial distribution of cross-sectional
area as the asymmetric shape. In order for the pressure distributions
to match at r = Ro, go(x/2) must be the same as g(x/1). Thus,
g(x/Z) is the same function as that for a body of revolution having
the same axial distribution of cross-sectional area.

In the preceding discussion the region of validity of the slender-
body approximation to (o was tacitly assumed to be at least as large
as that for 0. This condition is certainly true since the singular
terms in the two solutions are the same.

In the slender-body approximation the term 2 (7 + l)Vx.x,
is required to be small compared with any of the other terms in the
transonic differential equation. If this condition is to be satisfied
in the neighborhood of weak shock waves where g'(x/1) would be required
to have a jump proportional to the pressure rise and g"(x/Z) would be
infinite, the quantities (which were included in the terms denoted
as 0(1))


g"(x/Z)[- M2 (7 + l)M o] (8)

and



g"s a -


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NACA RM L54A29a


must be bounded there. Thus, at shock waves the coefficients of g"(x/I)
in expressions (8) and (9) must vanish. In the first of these expres-
sions, which is axisymmetric, the coefficient of g"(x/Z) vanishes for
a local Mach number of 1. Moreover, the average value of the local
Mach number ahead and behind a weak normal shock wave is 1. With this
interpretation of shock waves, then, the coefficient of g"(x/1) vanishes
and the solution should represent the flow in the neighborhood of weak
normal shocks. Also, since g(x/1) is the same function for both a
slender configuration and its associated body of revolution, their
shock-wave systems would have the same strength and location. In order
for expression (9) to be bounded in the neighborhood of a shock wave,
the double integral must vanish. Since this double integral gives rise
to asymmetric shapes and is zero for chordwise locations where the body
is axisymmetric, the additional restriction is obtained that the body
cross section must be circular in the vicinity of shock waves.

The slender-body solutions in the various Mach number ranges are
similar in that they are all represented by equation (6) although the
function g(x/Z) differs for the various speed ranges. Analytic
expressions for g(x/1) have been obtained for supersonic and subsonic
flows by considering general solutions of the complete linearized equa-
tion satisfying the boundary condition of vanishirn disturbance veloc-
ities at infinity. These solutions were then expanded in the neighbor-
hood of the body to evaluate g(x/z). Ward (ref. 4) has determined
this function for supersonic flows as


g(x/1) = 1 log M2 1 x/l s( log -
4 0 1

The corresponding result at subsonic speeds was obtained by Adams and
Sears (ref. 6) as


g(x/) (x/1) log s"(l)log 1)d. +



1 1 X
2- s"(l)log(1 )dJ
L/ Ixl



where the body extends from x = 0 to x = 1. Although an analytic
expression for g(x/l) at transonic speeds is not known, it has been
established that the stream Mach number enters the solution only through
g(x/1) and that the only geometric property of the body influencing


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8 CONFIDENTIAL NACA RM L54A29a


this function is the area distribution. The transonic similarity rule
for bodies of revolution (ref. 15 or 16) shows that g(x/1) can be
expressed in the form


g(x/z1 = ( ) log + 1)N + f(x/Z;K)


where the similarity parameter is


1 M2
K =
2
( + 1)M2(




AERODYNAMIC FORCES


Since the slender-body solutions are all represented by equation (6),
formal expressions for the aerodynamic forces can be determined which are
valid throughout the Mach number range. Consequently, many of the
essential features of slender-body theory at transonic speeds can be
obtained without resorting to detailed calculations.


Lift

The most significant difference between the slender-body solutions
at subsonic, transonic, and supersonic speeds is that the function g(x/Z)
differs in these various speed ranges. However, the term in the pressure
arising from the function g(x/Z) makes only a uniform contribution to
the pressure at any value of x and, therefore, cannot affect the lift
distribution or the lift. Thus, within the slender-body approximation,
the lift distribution depends only upon the function q9 and, conse-
quently, is independent of the stream Mach number. Several investigators
(Robinson and Young (ref. 17) and Heaslet, Lomax, and Spreiter (ref. 9),
for example) have previously noted that the linearized slender-body
theory gave consistent results, even at a Mach number of 1, for planar
systems.

According to slender-body theory, the lift distribution can be
obtained completely from solutions of Laplace's equation in the cross-
flow plane. Since this equation is linear, the lift is proportional
to the angle of attack even at transonic speeds. Ward has obtained
an especially simple form for the drag due to lift in which


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NACA RM L54A29a


1
DL = aL

where a is the angle of attack measured from zero lift and L is
the lift.


Drag

By computing the momentum change of the fluid passing through a
cylinder enclosing the body, the drag is determined as


D = (b2 2 s'(E)g'()dt + f d dt (10)
q q O c' 6n (


where the body extends from 0 = 0 to 1 = 1, C' denotes the contour
of the body at the stern which in the case of wings or wing-body com-
binations includes the trailing-vortex wake, q is the stream dynamic
pressure, and Db is the base drag. Equation (10) is valid throughout
the Mach number range provided the appropriate forms of the func-
tion g(x/1) are employed. The line integral is zero for nonlifting
configurations if the body is closed or if the body ends in a cylindrical
section whose elements are parallel to the stream. The effect of Mach
number (excluding the variation of base drag with Mach number) is con-
tained in the term involving g(x/1).

When the subsonic form of g(x/Z) is used in equation (10), the
correct result is obtained that the drag of nonlifting configurations
is zero. By using the supersonic form of g(x/L), the drag varies with
Mach number like s,'(l)]2log(M2 1). For pointed bodies, or for
bodies which end in a cylindrical section, the supersonic slender-body
theory indicates that the drag is independent of Mach number. For
bodies which do not satisfy these conditions, the supersonic result
indicates that the drag approaches infinity as the Mach number
approaches 1. These results from linear theory cannot be considered
satisfactory at transonic speeds since they give a discontinuity in
the drag as the Mach number is increased through 1; whereas experimental
data show that the drag starts to increase rapidly at a subsonic Mach
number and varies smoothly through 1. However, the few known solutions
of the nonlinear transonic-flow equation are in good agreement with
experiment in this regard. Consequently, the drag rise of slender shapes
should be correctly approximated by equation (10) when the transonic
form of g(x/1) is employed in the drag equation.


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Transonic Area Rule

The body shape enters into the function g(x/Z) only as a function
of the cross-sectional area distribution throughout the Mach number
range. This property of the slender-body solutions leads to an important
result even though the analytic expression for g(x/2) is not known at
transonic speeds. Examination of equation (10) shows that the body
cross-sectional shape enters into the slender-body drag expression only
through the contour integral evaluated at the stern of the configura-
tion. For a fixed geometry at the base, then, the drag of nonlifting
configurations depends only on the axial distribution of the body cross-
sectional area and is independent of the cross-sectional shape. Thus,
within the slender-body approximation, the drag of a nonlifting configu-
ration is the same as that of the associated body of revolution having
the same streamwise distribution of cross-sectional area provided the
base geometry is fixed. It is in this sense that an equivalent body of
revolution is associated with a wing-body combination. This result,
often referred to as the area rule, is especially significant at
transonic speeds where larger values of the width parameter b/Z are
permitted than in other speed ranres.

The property of the dependence of the drag upon the distribution
of cross-sectional area has previously been obtained by Ward (ref. 4)
and Graham (ref. 18) for supersonic flow and has been observed experi-
mentally by Whitcomb (ref. 19, for example) at transonic speeds. The
importance of this result was first noted by Whitcomb who demonstrated
that the area rule could be used as a basis for the design of low-drag
wing-body combinations at transonic speeds.

From the preceding development, the transonic area rule is subject
to the restrictions of slender-body theory with the additional condition
that the base geometry be fixed. These restrictions imply that modifi-
caLions to the equivalent body of revolution must be performed in the
region between shock waves. However, the seriousness of violating this
condition is not well understood at the present time. For example,
schlieren photographs seem to indicate that, even in cases where the
presence of the wing affects the strength of the shock, the average
strength of the shock may be close to that for the equivalent body of
revolution.


Langley Aeronautical Laboratory,
National Advisory Committee for Aeronautics,
Lr~.ilcy Field, Va., January 18, 1954.


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REFERENCES


1. Munk, Max M.: The Aerodynamic Forces on Airship Hulls. NACA
Rep. 184, 1924.

2. Tsien, Hsue-Shen: Supersonic Flow Over An Inclined Body of Revolu-
tion. Jour. Aero. Sci., vol. 5, no. 12, Oct. 1958, pp. 480-485.

5. Jones, Robert T.: Properties of Low-Aspect-Ratio Pointed Wings at
Speeds Below and Above the Speed of Sound. NACA Rep. 835, 1946.
(Supersedes NACA TN 1052.)

4. Ward, G. N.: Supersonic Flow Past Slender Pointed Bodies. Quarterly
Jour. Mech. and Appl. Math., vol. II, pt. 1, Mar. 1949, pp. 75-97.

5. Von Ka'rman, Th.: The Problem of Resistance in Compressible Fluids.
Atti del V Convegno della Fondazione Alessandro Volta, R. Accad.
d'Italia, vol. XIV, 1956, pp. 5-59.

6. Adams, Mac C., and Sears, W. R.: Slender-Body Theory Review and
Extension. Jour. Aero. Sci., vol. 20, no. 2, Feb. 1953, pp. 85-98.

7. Lighthill, M. J.: Supersonic Flow Past Slender Bodies of Revolution
the Slope of Whose Meridian Section Is Discontinuous. Quarterly
Jour. Mech. and Appl. Math., vol. I, pt. 1, Mar. 1948, pp. 90-102.

8. Keune, Friedrich: Low Aspect Ratio Wings With Small Thickness at
Zero Lift in Subsonic and Supersonic Flow. KTH-Aero TN 21, Roy.
Inst. of Tech., Div. of Aero., Stockholm, Sweden, 1952.

9. Heaslet, Max A., Lomax, Harvard, and Spreiter, John R.: Linearized
Compressible-Flow Theory for Sonic Flight Speeds. NACA Rep. 956,
1950. (Supersedes NACA TN 1824.)

10. Spreiter, John R.: Aerodynamic Properties of Slender Wing-Body
Combinations at Subsonic, Transonic- and Supersonic Speeds. NACA
TN 1662, 1948.

11. Lomax, Harvard, and Heaslet, Max A.: Linearized Lifting-Surface
Theory for Swept-Back Wings With Slender Plan Forms. NACA TN 1992,
1949.

12. Spreiter, John R.: Aerodynamic Properties of Cruciform-Wing and Body
Combinations at Subsonic, Transonic, and Supersonic Speeds. NACA
TN 1897, 1949.


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NACA RM L54A29a


15. Heaslet, Max A., and Lomax, Harvard: The Calculation of Pressure
on Slender Airplanes in Subsonic and Supersonic Flow. NACA
TN 2900, 1955.

14. Ribner, Herbert S.: The Stability Derivatives of Low-Aspect-Ratio
Triangular Wings at Subsonic and Supersonic Speeds. NACA TN 1425,
1947.

15. Busemann, Adolf: Application of Transonic Similarity. NACA TN 2687,
1952.

16. Oswatitsch, K., and Berndt, S. B.: Aerodynamic Similarity at
Axisymmetric Transonic Flow Around Slender Bodies. KTH-Aero
TN 15, Roy. Inst. of Tech., Div. of Aero., Stockholm, Sweden,
1950.

17. Robinson, A., and Young, A. D.: Note on the Application of the
Linearized Theory for Compressible Flow to Transonic Speeds.
Rep. No. 2, College of Aeronautics, Cranfield, Aero. Res. Council
(British), Jan. 1947.

18. Graham, Ernest W.: Pressure and Drag on Smooth Slender Bodies in
Linearized Flow. Rep. No. SM-13417, Douglas Aircraft Co., Inc.,
Apr. 20, 1949.

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























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DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE UBRARY
P.O. BOX 117011
GAINESVILLE, FL 32611-7011 USA


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