Numerical technique for analysis of wave drag at lifting conditions for supersonic aircraf

Harris, R. V., Jr.

English

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27. A NUMERICAL TECHNIQUE FORANALYSIS
OF WAVE DRAG AT LIFTING CONDITIONS
By Roy V. Harris, Jr.
NASA Langley Research Center
SUMMARY
A numerical technique for an application of the far-field linear theory
is presented. This technique, when adapted to the high-speed electronic dig-
ital computer, provides a practical means for analyzing the total wave drag
of an airplane at lifting conditions. An indication of the accuracy of the
method at lifting conditions is shoe by comparing measured pressure signa-
tures at various azimuth angles about an airplane model with those predicted
by theory. Calculations are made to illustrate some effects of configura-
tion variables on the total wave drag of an airplane at lifting conditions.
INTRODUCTION
The recent development of numerical techniques for estimating and
optimizing the lift-drag characteristics of supersonic crui'se vehicles has
led to significant improvements In aerodynamic performance at supersonic
speeds (ref. 1). Most of these techniques have resulted from applications
of existing theories to the high-speed electronic digital computer. As a
result, analytical approaches which once were considered too complex for
practical application can now be used to conduct aerodynamic trade studies
in time to affect the preliminary design of airplanes.
Two approaches to the analytical drag buildup of an airplane at super-
sonic speeds are illustrated in figure 1. On the left of the figure is an
illustrative drag polar for a supersonic-cruise vehicle with the cruise lift
coefficient indicated by the solid symbol. For purposes of analysis, the
drag is usually considered to be composed of skin-friction drag, zero-lift
wave drag (or wave drag due to volume), and drag due to lift. This approach
to the drag calculation was used in paper no. 26 by Harry W. Carlson and
F. Edward McLean. It should be noted, however, that the drag due to lift
consists of part vortex drag and part _ve drag due to lift, and that the
approach utilized by Carlson and McLean neglects any interference effects
between the wave drag due to lift and that due to volume. A more funda-
mental approach, as illustrated by the bar graph on the right, is to con-
sider the drag to be composed of its three basic elements, friction drag,
total wave drag (including that due to lift), and vortex drag. Some aspects
of the techniques for calculating skin-friction drag are discussed in paper
no. 30. by John B. Peterson, Jr., and William J. Monta and in paper no. 31 by
K. R. Czarnecki. The calculation of vortex drag at supersonic speeds for
the condition of 100 percent leading-edge suction is the same as the
399
https://ntrs.nasa.gov/search.jsp?R=19750065533 2020-03-20T00:52:18+00:00Z
calcu?_tio_ of induced drag at subsonic speeds and depends only on the span-
wise load distribution (ref. 2). For the more realistic condition in which
little or no leading-edge suction is achieved at supersonic speeds, the problem
is more complex and no suitable numerical technique presently exists. A numer-
ical technique has been developed for the analysis of airplane wave drag at
lifting conditions, and the purpose of this paper is to present this new appli-
cation of the far-field linear theory which, when adapted to the high-speed
electronic digital computer, provides a practical meansfor analyzing the total
wave drag of an airplane at lifting conditions.
SYMBOLS
A
CD
CD,wave
CL
D
Z
L
14
P
q
C
x_y_z
X,Y,Z
equivalent-body area due to volume
total-drag coefficient
wave-drag coefficient
lift coefficient
wave drag
componentof section lift along intercept of airplane and the Mach
cutting plane, taken in direction of e
length of equivalent bgdy
Math number
reference static pressure
incremental pressure due to flow field of airplane or model
8
d,vnamiepressure
total equiva!ent-bodo_ area
coordinates along X, Y, and Z
axis system of airplane or model
axes
azimuth annie referred to control cylinder, as shownin figure 2
Ma_:l _ngle
Primes are used to indicate derivatives with respect to x.
4O0
2DISCUSSION
Calculation of Wave Drag at Lifting Conditions
The far-field linear-theory approach to supersonic flow is illustr_"
figure 2. Consider an airplane in a steady supersonic stream and the cylino_%--
cal control volume suggested by Hayes in reference 2. Define the azimuth
angle e to be such that 8 is zero to the side of the airplane_ with nega-
tive values below the airplane and positive values above it. Consider next a
point on the surface of the control volume below the airplane and between the
forward and rearward trailing Mach cones. The upstream Mach forecone from this
point represents a surface of coincident signals in that all disturbances which
lie in this plane arrive at the point simultaneously. It is possible, there-
fore, to determine a linear distribution of singularities that produce the same
pressure disturbances as the airplane at the surface of the control volume.
If the dimensions of the control volume are allowed to become infinitely large_
the surfaces of coincident signals become planes in the vicinity of the air-
plane, and the linear source-sink distribution can be related to an equivalent-
body area distribution (shown in fig. 2), which is determined by the intercepts
of the planes of coincident signals (referred to as the "Mach cutting planes")
and the airplane. The dotted lines shown in the area-distribution sketches
indicate the equivalent-bodF area distribution due to volume and the solid
lines indicate the total equivalent-body area distribution including the effects
of llft.
The mathematical basis for the concept of equivalent area due to lift can
be determined from the following equation for airplane wave drag (refs. 2
and 3) that was derived by use of the far-field linear-theory approach:
This equation shows that the ratio of the wave drag to the free-stream dynamic
pressure is a function of the second&erivative of the equivalent-body area
distribution due to volume A(x,e) and is also a function of a term that is
proportional to the first derivative of the longitudinal distribution of lift
_(x,e), as determined by the Mach cutting planes. If the term S(x,e) is so
defined that
s"(x,e) = A"(x,e) -2-_ Z'(x,e)
then
j_o xS(x,e) = A(x,e) - 2-_ Z(x,e)dx
401
It can thus be seen that S(x,e) is equal to the equivalent-body area dis-
tribution due to volume minus a term which has units of area and is a function
of the longitudinal lift distribution as determined by the Mach cutting planes.
This term is defined as the equivalent-bod_ area due to lift. As illustrated
in figure 2, the equlvalent-body area due to lift varies from positive values
below the s_rplane to negative values above the airplane. To the side of the
airplane; the area due to lift becomeszero since there is no componentof
] _ft -i.._ this direction. It should be noted that the pressures that occur
_ the airplane cause the sonic boom along the ground track. Also, the
equivalent-body area distribution corresponding to e = -90 ° is used in the
calculation of sonic-boom signatures. This subject is treated subsequently in
paper no. 29 by McLean, Carlson, and Hunton.
An illustration of the procedure that has been programed for determining
the equivalent-body area distributions of an airplane at lifting conditions is
presented in figure 3. The left side of the figure illustrates the procedure
for determining the area due to volume and the right side of the figure illus-
trates the procedure used to determine the area due to lift. For calculation
of the equivalent-body area due to volume, a mathematical representation of
the airplane in terms of the x 3 y, and z coordinates is used as input to
the computer. The computer then solves for the normal projection of the area
intercepted by the Mach cutting planes and thus defines the equivalent-body
area due to volume A(x,e). For calculation of the equivalent-bo_y area due
to lift, the wing camber surface in terms of x, y, and z coordinates is
used as input to the computer, and the lifting-surface pressure distribution
is determined by the method which was described in paper no. 26 by Carlson and
McLean. The computer then solves for the intercept of the Mach cutting planes .
and the wing camber surface and integrates the lifting pressures along this
line to determine the component of force normal to the free stream and in the
e direction. This component of force Z(x,e) is then used to define the
equivalent-body area due to lift. Finally, the computer sums the area due to
volume and the area due to lift and solves for the total wave drag.
Direct comparisons between measured and computed wave drag at lifting con-
ditions cannot be made because of the difficulty in determining the supersonic
vortex drag. Numerous correlations at zero lift, however, have shown good
agreement with the theory (ref. 4). An indication of the accuracy of the
equivalent-body concept at lifting conditions can be made by comparing the
measured pressure signatures at various azimuth angles about an airplane model
with those predicted from its equivalent bodies. Such a comparison is made in
figure 4. These data were taken at a Mach number of 1.4 and a lift coefficient
of O.lO for the configuration shown in the sketch. The upper three plots show
the equivalent-body area distributions for the azimuth angle of -90 ° (which is
directly below the model)3 the azimuth angle of 0o (which is to the side)_ and
the azimuth angle of 90 ° (which is above the model). The dashed lines indicate
the area due to volume and the solid lines indicate the total equivalent-body
area including the effects of lift. The three corresponding lower plots show
a comparison between the pressure signatures predicted by theory and those
measured in the wind-tunnel tests. The classic far-field type of pressure sig-
nature was not achieved in these tests because of the dimensional restraints
imposed by the wind tunnel. The near-field effects have been included,
4O2
however, in the theoretical predictions (ref. _). The good agreement between
the measuredand predicted pressure signatures at the various azimuth angles
indicates that good agreement for the wave drag at lift should be expected.
Calculations of the wave drag at several lifting conditions have been
madefor the configuration shownin figure 4 at a Machnumber of 2.7, and some
of the results are shownin figure _. The upper plots showthe equivalent-body
area distributions for lift coefficients of 0 and 0.08. The lower plots show
the corresponding ratio of the equivalent-body wave drag to dynamic pressure
as a function of the azimuth angle. The dashed lines indicate the integrated
average and, hence, the total wave drag of the airplane. Consider, first, the
results at zero lift for which the entire equivalent-body area is due to vol-
ume and, therefore# only positive areas exist. This portion of the figure is
slmplifiedby showing only selected area distributions for positive values of
the azimuth angle. Comparing the shape of each equivalent-body area distribu-
tion with its corresponding drag contribution in the lower plot indicates that
the equivalent bodies having the lowest fineness ratios and the steepest slopes
make the largest contribution to the wave drag. For example, the equivalent
body for an azimuth of 90 ° has the largest maximum area and the shortest length
and, therefore, the largest contribution to the wave drag of the airplane. At
a lift coefficient of 0.08, the area due to lift significantly alters the shape
of the equlvalent-body area distributions. The equivalent body for an azimuth
of 90 ° , for example, although having the same length as the zero-lift condi-
tion, now has a much larger area change occurring over that length and there-
fore makes a much larger contribution to the airplane wave drag. Thus, a
direct relationship can be established between the wave drag of an airplane
at any given lift coefficient and its equivalent-body area distributions.
Configuration Effects on Wave Drag at Lifting Conditions
A comparison of the wave drag of two similar configurations at a Mach
number of 2.0 and a lift coefficient of O.lO is shown in figure 6. Both
wlng-body combinations were symmetrical, employed the same body, and had
wings of equal area, equal span, and equal thickness ratio. One configura-
tion, however, had a wing of delta planform, whereas the other had a wing of
arrow planform. Comparing the equivalent-body area distributions at azimuth
angles of -90 °, 0°, and 90 ° indicates that the effect of spreading the lift
over a greater length with the arrow wing is to reduce the rate of area growth
of the equivalent bodies. The effect on the airplane wave drag is shown in
the lower plots, where the total wave drag of the arrow-wing configuration is
about30 percent less than that of the delta-wing configuration.
Up to this point, all configurations that have been considered would be
expected to have little or no interference between wave drag due to lift and
wave drag due to volume. An example for which interference effects do exist
between wave drag due to lift and that due to volume is shown in figure 7.
Consider the arrow-wing--body configuration shown in figure 6, again at a Mach
number of 2.0 and a lift coefficient of O.lO, but now with a high wing position
as shown at the left in figure 7 and with a low wing position as shown at the
right. The equlvalent-body area distribution for e = 0° (which is to the
403
side of the airplane) is unaffected by changes in the vertical position of the
wing and, thus, is the same for both configurations. The Mach cutting planes
intercept the high-wing configuration in such a manner that the area due to lift
is shifted forward to the center line for the azimuth angle of 90o and is
shifted rearward for the azimuth angle of -90o . For the low-wing configuration,
the reverse is true. The area due to lift is shifted rearward for an azimuth
angle of 90 ° and forward for an azimuth angle of -90 ° . The effects of inter-
ference between lift and volume on the total wave drag can be seen in figure 8.
The bar graph shows a comparison of total wave-drag coefficient based on wing
area for the low-wing_ midwing, and high-wing configurations. For this example,
the low-wing arrangement results in a 7.5 percent higher wave drag than the
high-wing arrangement.
CONCLUDING REMARKS
A numerical technique has been presented for an application of the far-
field linear theory. This technique, when adapted to the high-speed electronic
digital computer, provides a practical means for analyzing the total wave drag
of an airplane at lifting conditions. Numerous correlations at zero lift have
sNown good agreement with the theory. An indication of the accuracy of the
method at lifting conditions was shown by comparing measured pressure signa-
tures at various azimuth angles about an airplane model with those predicted by
theory. The good agreement between the measured and predicted pressure signa-
tures indicates that good agreement for the wave drag at lifting conditions
should be expected. Results from a limited number of calculations showed some
effects of differences in configuration on the total wave drag of airplanes at
lifting conditions.
REFERENCES
1. Robins, A. Warner; Morris, Odell A.; and Harris, Roy V., Jr.:
Research Results in the Aerodynamics of Supersonic Vehicles.
No. 65-717, Am. Inst. Aeron. Astronaut., Nov. 1965.
Recent
Paper
2. Hayes, Wallace D.: Linearized Supersonic Flow. Rept. No. AL-222_ North
American Aviation, Inc., June 18, 1947.
3. Lomax, Harvard: The Wave Drag of Arbitrary Configuratioi,s in Linearized
Flow as Determined by Areas and Forces in Oblique Planes. NACA
RMA55AI8, 1955.
4. Harris, Roy V., Jr.: An Analysis and Correlation of Aircraft Wave Drag.
NASA TMX-947, 1964.
5. Niddleton, Wilbur D.; and Carlson, Harry W.: A Numerical Method for Cal-
culating Near-Field Sonic-Boom Pressure Signatures. NASA TN D-3082, 1965.
4o4
L¸,
TWO APPROACHES TO ANALYTICAL DRAG BUILDUP
CD
VORTEX AND
WAVE DUE
TO LIFT WAVE
0
CL
ZERO-LIFT
WAVE
FRICTION
VORTEX
FRICTION
FigureI
ILLUSTRATION OF FAR-FIELD LINEAR-THEORY APPROACH
Figure2
4O5
f _,%
ILLUSTRATION OF MACHINE COMPUTING PROCEDURE
AREA DUE TO VOLUME=A(x,8)
MATHEMATICAL MODEL
Z OF_
.....
AREA DUE TO LIFT= "_]Q Z(x,O)dx
WING CAMBER
SURFACE
_'_ e,_, _ ,/
Figure 3
EXPERIMENTAL VERIFICATION OF EQUIVALENT-BODY CONCEPT
M = 1.4_ CL= O.IO
O EXPERIMENT
THEORY
X X X
Figure4
//
RELATIONSHIP BETWEEN WAVE DRAG AND EQUIVALENT-BODY AREA
M= 2.7
e, DEG
_o
X
q WAVE DRAG
-90 0 +90
e, DEG
_ _ _ c_=o.oe e, DEG
:_ -90
,,,'---+45
+90
X
WAVE DRAG
-90 0 +90
e, DEG
Figure 5
S 0
O(e__.!
q
-90
CONFIGURATION EFFECTS ON WAVE DRAG AT UFT
I
//_ e, DEG
1 / I _-90
!
+,o
M= 2.0; CL=0.10
C
X
i I J
0 +90
e, DEG
, I
e, DEG
/_,, .,o
, {/-o
i
+90
X
I I
-9O 0 +90
8, DEG
Figure6
z_07
SO
INTERFERENCE EFFECTS BETWEEN LIFT AND VOLUME
AREA DISTRIBUTIONS AT M= 2.0 AND CL:O.t 0
HIGH WING
! I e,DEG
' I /---
I +90
X
_LOW WING
= li e,DEGI
I j
J
----...-_,,_,-- o
+90
Figure 7
INTERFERENCE EFFECTS BETWEEN LIFT AND VOLUME
WAVE DRAG AT M=2.0 AND CL=O.IO
CD, wove
.006 -
.004 -
.°°it-
WING POSITION
LOW
7-7-/"2: '
f////,
f////,
///,.
,//,
MID
_'7777 HIGH
_1111
,-,-..-,,.4
lJJJJ
_'/J//
Figure8
408


A numerical technique for analysis of wave drag at lifting conditions

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