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R. & M. No. 331ff
MINISTRY
OF
AVIATION
AERONAUTICAL RESEARCH-COUNCIL
REPORTS AND MEMORANDA
F u r t h e r E x p e r i m e n t a l I n v e s t i g a t i o n s of t h e
Characteristics of-Cambered Gothic W i n g s
.... - at Mach N u m b e r s from 0. 4 to 2.o
By L. C. SQUIRE, P h . D .
LoNDoN:
HER MAJESTY'S STATIONERY OFFICE
I963
FIFTEEN.
SHILLINGS
NET'
Further Experimental Investigations of the
Characteristics of Cambered Gothic Wings
at Mach Numbers from o. 4 to z-o
By L. C. SQUIRE, P h . D .
COMMUNICATED BY THE DEPUTY CONTROLLER AIRCRAFT (RESEARCH AND DEVELOPMENT),
IVIINISTRY
OF
AVIATION
Reports and Memoranda No. 33zo *
December, z96z
Summa~.
The wind-tunnel tests on cambered gothic wings reported in Reports and Memoranda No. 3211 have been
extended to include the effects of changes in design lift coefficient and of changes in spanwise camber without
changes in camber incidence distribution.
It was found that the camber was successful in that the flow was attached over the whole wing at the design
lift. Also at the design lift the lift-dependent drag was close to the predicted values. However, the lift-dependent
drag of the uncambered wing was also close to this value so that the benefit of camber on lift/drag ratio was
very small. At subsonic speeds the cambered wings were less stable than the uncambered wing; also the changes
of stability with incidence and Mach number were greater, particularly near M = 1- 0.
Changes in spanwise camber, without changes in incidence distribution, do not alter the force characteristics
near the design lift, but do alter the off-design characteristics.
1. Introduction.
In connection with the design of slender wings for high lift over drag, an experimental investigation
of slender-wing models with curved leading edges and various types of camber is being made in the
3 ft Tunnel at the Royal Aircraft Establishment, Bedford. The results of tests at supersonic speeds on
the first two wings in this programme were presented in Ref. 1. Both of these wings were of gothic
planform, aspect ratio 0.75, with the same thickness distribution. One wing was uncamber~d and
the other was cambered by Weber's method 2 to have completely attached flow and low drag-due-tolift at the design lift coefficient {(CL) a = 0.1}. Tests, both at supersonic speeds ~ and at low speeds a,
on this first cambered wing showed that the large droop of the wing near the leading edge retarded
the development of leading-edge separation at off-design conditions.
In the present report tests at supersonic speeds on two more cambered gothic wings are described.
On both these wings the amount of leading-edge droop was decreased by reducing the design lift
coefficient to 0.05; the two designs were obtained by integrating the camber incidence distribution
with two different initial conditions (see Section 2.1 for details).
T h e report also includes results o f tests on all four wings at subsonic and transonic speeds.
Previously issued as R.A.E. Tech. Note No. Aero. 2803--A.R.C. 23,724.
2. Details of Tests.
2.1. Description of Models.
All the cambered wings tested in the present programme were designed by a method described
by Weber 2. One feature of this design m e t h o d is of particular relevance in the present programme.
T h e camber design is based on 'slender, thin wing' theory which yields, for a given load distribution,
a formula for the local incidence distribution, a/ax{z(x, y)}. This incidence distribution must then
be integrated with respect to x to obtain the camber surface. T h e integration introduces an arbitrary
function of y (the spanwise co-ordinate); this function Weber fixed by making the wing trailing edge
straight, i.e., by putting z = constant at x = c0. For structural and aerodynamic reasons it may be
desirable to modify this condition, and the fourth wing in the present series is designed to find the
effect of a change in this function of y (see Fig. 4 and below).
Full details of the four wings are given in Table 1 and in Figs. 1 to 4, where the wings are
designated by the numbers 1 to 4. Wing 1 is the uncambered wing and wing 2 the cambered wing
of Ref. 1. Wing 3 has the same type of camber incidence distribution as wing 2, but the a m o u n t
of camber (and hence of leading-edge droop) has been decreased by reducing the design Cr. to 0" 05.
Wing 4 has the same camber incidence distribution as wing 3, but differs in actual shape as the camber
surface was obtained by making the Wing straight (i.e., z = constant) at x = 0.8c 0 instead of at
x = co as on wing 3.
Wing 1 was made of steel throughout, b u t the three cambered wings were made of glasscloth
and Araldite formed onto a metal core. In all models a small circular body of 1.35 inches diameter
was used at the rear of the model to shield the balance and sting support (Fig. 1).
2.2. Range of Tests.
T h e tests were made in the transonic and supersonic test sections of the 3 ft T u n n e l at R.A.E.,
Bedford. T h e range of force tests, consisting of measurements ~f lift, drag and pitching moment,
is given in the following table:
:
Mach numbers
Incidence Range
(1 ° steps)
Wings
Test Section
3
Supersonic
1.42, 1-61, 1.82, 2.0
- 5 ° to +12 °
4
Supersonic
1.42, 1-61, 1.82, 2.0
--5 ° to +10 °
1 and 2
Transonic
0.4, 0.7, 0.8, 0-85, 0-9, 0.94,
0.98, 1.02, 1.25, 1- 30
Transonic
0.4, 0.7, 0.9, 0.94, 0-98, 1.02,
1.25, 1.30
- 2 ° to +13 °
4
Transonic
0.4, 0.7, 0.9, 0-94, 0.98
- 2 ° to +10 °
3 and 4
Nominal supersonic test section
with unshaped (flat) wall
0'4
- 2 ° to +17 °
-
-
5° to + 10°
All tests were made at a Reynolds n u m b e r of 2 x 106 based on aerodynamic mean chord.
2
The force tests on wings 1 and 2 at supersonic speeds 1 with free transition showed that extensive
regions of laminar flow occurred on the wings at low incidence. It was also found that turbulent
flow over the whole wing could be obtained with bands of carborundum along the leading edge;
these bands of carborundum did n.ot change the lift or pitching moment, but did increase the drag.
For all the present force tests therefore, transition was fixed with bands of carborundum along the
full length of the leading edge; the bands were approximately 0.5 in. wide normal to the leading
edge and started 0. 125 in. inboard of the edge. Two grades of carborundum were used, the grain
size being 0.007 in. for M = 1.42 and above, and 0.003 in. for lower speeds. Oil-flow tests
suggested that these bands had in fact fixed transition throughout the incidence range at all Mach
numbers.
During initial tests on wings 1 and 2 in the transonic test section, large fluctuating stresses occurred
in the balance% Near M = 0.80 these reached a peak value which was considered dangerous;
hence no measurements were made on wings 3 and 4 between M = 0-70 and 0.90.
In addition to force tests, oil-flow investigations were made at various Mach numbers and
incidences; details are given in the following sections. Early tests with and without roughness
bands showed no significant changes in vortex position or shape. Thus, since the main region of
interest in the flow development occurs near the leading edge, the main oil-flow tests were made
without roughness bands.
2.3. Reduction and Accuracy of Results.
Force results have been reduced to the usual coefficient form; on all four wings the reference
areas and chords are based on the common platform. Pitching-moment coefficients are given about
the quarter-chord point of the mean aerodynamic chord. The drag has been corrected to a base
pressure on the minimum body equal to free-stream static pressure.
No corrections have been applied for wind-tunnel interference o1" for angularity of the tunnel
flow. The former correction is zero at Mach numbers above M = 1.3 since the reflection of the
bow wave strikes the model support well downstream o f the model base. Below this Mach number
the interference effects are probably small, except near ~/I = 1.0, where the measured speed may
have been somewhat in error. Also, at M = 1.25 and 1.30 measurements of base pressure suggested
that flow at the model base was still affected by wall interference. Hence the drag results at these
Mach numbers were considered not reliable and have not been presented.
Apart from tunnel interference it is estimated that the accuracy of the results is as follows:
cL + 0.003
C~ + 0"0005
CD + 0"0004 at CL = 0 )
+ 0 . 0 0 1 a t C5 = 0.3t
M I > 1.42
c¢ _+ 0.05 ° from measurement, together with a possible error of ~ 0.1 °
from flow angularity.
The errors in drag measurement may be slightly greater at subsonic speeds owing to the balance
vibration mentioned in last section; also all the results at M = 0-4 are subject to larger errors than
those listed above owing to the low level of loading at this Mach number.
~ The vibrations which were responsible for these fluctuating stresses occurred when the natural frequency
of the model system (i.e., model, balance and sting) corresponded with a frequency of main disturbances in
the tunnel air stream: they had no other aerodynamic significance.
3
(86513)
A*
3. Presentation and Discussion of Results.
Full results for wings 1 and 2 at supersonic speeds are given in Ref. 1. Force results for wings 3
and 4 at supersonic speeds and for all four wings at subsonic and transonic speeds are given in the
present report. Graphical presentation has been adopted (see Figs. 5 to 8 and 12 to 20).
In discussing the results some cross-reference to the earlier report is obviously necessary. In order
to keep this reference to a m i n i m u m the discussion has been divided into four parts. In the first
part, results for wing 3 at supersonic speeds are discussed, and the behaviour of this wing is
compared with that already described for wings 1 and 2 in Ref. 1. T h e other three sections then deal
with topics which are less related to the subject matter of Ref. 1.
3.1. Flow and Force Development on Wings 1 and 3 at SupersoJ~ic Speeds.
T h e variations of C L with % and of C,~t, C D and LID with C L for wings 1 and 3 are compared
in Figs. 5 to 8. Oil-flow photographs for wing 3 at a series of incidences at M = 1.61 and M = 2.0
are given in Figs. 9a, 9b and 10. Comparable photographs for wing 1 (though not at identical
incidences) are shown in Figs. 11a and 1lb.
Before discussing the results it should be recalled that in the investigation 1 of the flow development
on wing 2 at supersonic speeds it was found that the flow remained attached at.the leading edge,
and over the whole wing, for an incidence range on both sides of the design incidence; also the
lift-curve slopes were linear t h r o u g h o u t the test range. At the same Mach numbers the flow separated
from the leading edge of wing 1 at very low incidence and the separated sheet rolled into a vortex
which produced a non-linear lift contribution; the size of this contribution decreased with increase
of Mach number.
T h e results plotted in Figs. 5 to 8 show that apart from a displacement of the curves the force
results for wing 3 are similar to those of wing 1, and again the non-linearity of the lift and m o m e n t
curves decreases with increase in Mach number.
T h e oil-flow photographs (Figs. 9a and 9b) at M = 1.61 for wing 3 show that separations do
occur on this wing at small incidences away from the design point (C L = 0.05), but there is still a
small incidence range in which the flow is attached, for example at c~ = 3.1 ° (C L = 0 . 0 7 0 ) and
o~ = 2.0 ° (C L = 0.045) the flow was attached on both surfaces of the wing*. ( T h e actual flow
patterns at ~ = 2 ° are almost identical to those at ~ = 3- 1 ° and so are not presented.) Above these
incidences the lower-surface oil pattern remains similar to that at ~ = 3.1 ° but a separation occurs
on the upper surface. T h e vortex associated with this separation is quite small but can be seen in the
photograph for ~ = 4" 2 ° (C L = 0.097) over the outer half of the leading edge where the attachment
and secondary separation lines are clearly visible ~. At higher incidences (~ = 6.3 ° and 8.4 °) separation
starts nearer the apex and the attachment find secondary separation lines move inboard. It should
be noted that at ~ = 6.3 °, i.e., approximately 4 ° above the design pointl the shape and position of
* In the interpretation of these photographs it should be noted that the regions of unmoved oil which exist
at most incidences represent regions of attached laminar flow. For example on the lower surface at c~ = 3" 1°
the oil has formed streamlines near the leading edge due to the high shear in the laminar attached flow there.
Farther inboard the oil is unmoved except for two regions at the rear of the wing where the well defined oil
lines indicate that transition has occurred. It should be noted that roughness which was used to fix transition
in the force tests was removed for the visualisation tests in order to obtain details of the flow in the immediate
vicinity of the edge. Thus the transition changes do not occur in the force tests.
the vortex is similar to that on the plane wing at a = 4 ° (Fig. 1 lb). At incidences below ~ = 2 °
separation occurs along most of the leading edge, and a vortex lies along the lower surface (see
photograph for a = 0).
At M = 2.0 the photographs (Fig. 10) of oil flow do not show a single large vortex, even at
= 8°; instead at this incidence the oil flow could be interpreted as showing a series of small
vortices running back over the wing. However, the lift-curve slope for this Mach number begins
to increase at incidences above about 6 ° incidence, the rate of increase being less than at lower
Mach numbers. Thus it would appear that although a single vortex has not formed at 8 ° incidence,
a change in the flow has nevertheless taken place which causes an increase in lift-curve slope.
On wing 1 there appears to be little change in the surface flow pattern as the Mach number is
increased from 1.6 to 2.0; the only change being that the vortex starts nearer the wing apex at the
lower Mach number. T h u s the decrease in non-linear lift on this wing with increase in Mach
number is probably associated with changes in the vortex strength, which might not produce large
changes in the surface flow pattern, rather than with the disappearance of the vortex as on wing 3.
3.2. Effects of Camber-Design Lift Coeffcient.
3.2.1. Lift and pitching m o m e n t . - - T h e variation of C L with ~ and of C m with C z for wings
1 and 3 is compared in Figs. 5 and 6 for Mach numbers between M = 1.42 and M = 2-0. Similar
curves for wings 1, 2 and 3 between M = 0.4 and M = 1.3 are given in Figs. 12 and 13, 15 and
16, and 18 and 19.
These figures show that in addition to giving a positive no-lift angle and a non-zero pitching
moment at zero lift, camber also causes a displacement to higher lift coefficient of the m i n i m u m slope
of the lift and moment curves. This minimum slope occurs at, or near, the condition of flow
attachment and the non-linear behaviour of the curves away from this condition is associated
with the growth of leading-edge separations. In order to study this non-linearity in more detail
results at Mach numbers of 0.7, 0.9, 1.02, 1.61 and 2-0 have been replotted in the form of
(C z - Ore) against ( ~ - ~ ) , and (C m - Cm) against (C z - Oz) {Figs. 21 and 22}, where e, 0 z and O,,
are values of % C z and C m corresponding to the m i n i m u m lift-curve slope.
Fig. 21 shows that the lift development about e is similar for all three wings although there are
some small differences between the wings. For example, at supersonic speeds the lift-curve slope
of wing 2 at a = m is larger than that of the other wings, also the lift of wing 2 is linear throughout
the test range, and at the highest values of (c~- m) wing 3 has the greatest lift. Fig. 21 also includes
linear lift-curve slopes as given by slender-wing theory (C z = ~v/2 Ao~) and, at supersonic speeds,
by not-so-slender-wing theory 5,~. Also included are curves of C L = ~/2 Ao~ + 4 ~ ; 4~ 2 being the
non-linear lift increment derived by Smith 7. At M = 0.7, C r = ~r/2 A s + 4a 2 slightly overestimates
the totallift of wing 1 whereas at M = 1.02 it provides a slight underestimate. However, at M = 1- 02
the initial lift-curve slope is higher than 7c/2A so that 4~ ~ is still a fair approximation to the nonlinear lift increment. At M = 1.6 the lift is again in fair agreement with Smith's estimate; however,
a study of the curves shows that this agreement is fortuitous since the initial lift-curve slope is much
greater than ~r/2 A, and is in fact in excellent agreement with the not-so-slender value.
T h e curves of C,~ - 0,~ against Cr - Oz show an almost complete collapse at supersonic speeds
(Fig. 22), but at subsonic and transonic speeds the range of C L about 0 L in which the aerodynamic
centre remains at a constant position before moving back, increases w i t h camber-design lift
coefficient. This difference in behaviour is presumably associated with the changes in leading-edge
separation discussed in Section 3.1. The effect of this increase in range of constant aerodynamiccentre position is clearly illustrated in Figs. 23 and 24, where the variations of centre=of-pressure
position with CL at fixed Mach number, and of centre-of-pressure position with Mach number for
fixed CL are compared for the three wings '~. It should be noted, however, that the large forward
shift in centre-of-pressure position which occurs on both cambered wings near M = 1.0 is associated
both with a forward shift of aerodynamic centre (see Fig. 22) and with a decrease in C r, in this speed
range. Reasons for the forward shift in aerodynamic centre near M = 1.0 have not been found.
The fact that the effect only occurs on the cambered wings rules out wind-tunnel interference as
the main cause, although the actual magnitude of the forward shift may be influenced by tunnelconstraint effects.
3.2.2. Drag.--Drag polars for wings 1 , 2 and 3 at transonic and subsonic speeds are
presented in Figs. 14, 17 and 20. Similar curves for wings 1 and 3 at supersonic speeds are shown
in Fig. 7. All results are for wings with fixed transition. Comparative drags are plotted for fixed
Mach number in Fig. 25 and for fixed CL in Fig. 26.
From Fig. 25 it can be seen that at negative and low positive lift coefficient, the drag of wing 1 is
less than that of the cambered wings, but that with increase in C L the drag of wing 1 becomes greater
than that of the cambered wings. The drag of wing 3 {(CL),Z = 0"05} is always less than that of
wing 2 {(CL) d = 0" 10}. The large increase in zero-lift drag of wing 2 as compared with wings I and 3
is mainly due to the extensive flow separations which occur on the lower surface of wing 2 at lift
coefficients below 0.10.
The drag results have also been analysed in terms of a lift-dependent drag factor t {~rd(CD - C' D o)/
CL2}. The variations of this factor with C L for the three wings are compared in Figs. 27a to d at Mach
numbers of 0.4, 0.7, 0.9, 0.98, 1.42, 1.61, 1.82 and 2.0. These figures also include the theoretical
values of this factor as given, at supersonic speeds, by not-so=slender theory 2, 5. It will be seen that
the induced-drag factor of wing 3 is equal to, or less than, the theoretical value, although the theoretical
value only applies near C L = 0.05 where the flow is attached. The experimental factor for wing 2
is always higher than the theoretical value at the design point (CL = 0.1), but it drops below
the design value at lift coefficients above 0.2. The differences in shape of the curves at low values
of C L for the various wings are associated with the relative positions, and shapes, of the drag polars
as shown in Fig. 25.
In Fig. 28 the zero-lift drag of the plane wing is compared with theoretical estimates. The
skin-friction drag was calculated by a strip method 1 based on a fiat-plate turbulent boundary layer.
The wave drag at supersonic speeds was calculated by slender-body theory; two values are given
in Fig. 28; the upper corresponds to the wave drag of the wing alone, i.e., ignoring the small body at
the rear of wing, while the lower curve includes the effect of this body (with zero base drag). When
allowance is made for form drag it will be seen that agreement between the estimated and measured
drags is good throughout the speed range.
:~ Note that, for convenience of presentation, different vertical scales are used in Figs. 23 and 24.
]" C'D 0 is equal to the zero-lift drag of the uncambered wing together with an increment to allow for the
greater wetted areas of the cambered wings. For wing 2 the increment was 0. 0006, and for wing 3, 0.0003.
6
3.3. Effect of Change hz Wing Shape.
In this section the results on wings 3 and 4 are compared. It' will be recalled that these two wings
have the same local incidence distribution, but that this incidence distribution has been integrated
to produce two wingshapes (see Section 2.1 and Fig. 4). On wing 3 the trailing edge is straight and
all spanwise sections have drooped leading edges; on wing 4 spanwise sections forward of 0.8 of
the root chord have drooped edges, but aft of this point the tips turn up.
The variations of C L with a, C,~ with C L and C1) with Cj; for the two wings are presented in
Figs. 29, 30 and 31. Figs. 29 and 30 show- that near the design condition (C L = 0.05), where the
flow is attached, the lift and pitching moments of the two wings are similar, although wing 4 has
slightly less lift; this lower lift corresponds to an increase of about 0.15 ° in the zero-lift angle of wing 4
as compared with wing 3. Away from the design point the lift develops less rapidly and the increase
in stability with increase in C L is less on wing 4 than on wing 3. Oil-flow patterns on wing 4 did not,
however, show any significant differences to those on wing 3 (Figs. 9a and b). In spite of this, the
vortex may be weaker on wing 4, or the wing shape may be less efficient in converting the low
pressures associated with the vortex into lift.
The differences in the drag results are, in general, consistent with the lower non-linear lift of
wing 4; that is wing 4 produces a given lift at a higher incidence than wing 3, and so has greater
drag due to lift. Near the design lift coefficient the drags of the two wings are identical at supersonic
speeds, but wing 4 has a slightly higher drag at subsonic speeds; it is thought that this increase
in drag is due to slight differences in the transition fixing on the two wings.
These results show that large changes in wing shape, without changes in local incidence
distribution, do not produce significant effects on the overall forces when the flow is attached.
However, in the present case the wing with the straight trailing edge develops more non-linear lift.
3.4. Low-Speed, High-Incidence, Results for Wings 3 and 4.
A comparison of the results for wings 1 and 2 at M = 0"4 with Keating's low-speed results s for
equivalent wings gave excellent agreement. It was thus decided to dispense with low-speed models
of wings 3 and 4, and to extend the tests at M = 0.4 on these two wings up to 17 °. These tests
could not be made in the transonic test section due to limitations on the incidence range and so they
were made in the supersonic test section with a flat top wall. The results are presented in Figs. 32,
33 and 34; for comparison these figures also include results from the transonic test section up to
10 ° incidence.
The purpose of these tests at high incidence was twofold. (i) To check whether any undesirable
features (for example, pitch-up) occurred at these high incidences. (ii) To study the non-linear
lift at high incidence, since Keating found that, for given incidence, the wing with camber designed
for CL = 0.1 (wing 2) had much less lift than the plane wing.
The results show that the--forces develop smoothly with increase in incidence throughout the
test range. They also show a rapid increase in non-linear lift, particularly for wing 3. At 15 ° incidence
the lift coefficients for wings 3 and 4 are 0.465 and 0.425 respectively; Keating's values at this
incidence were 0-50 (wing 1) and 0.37 (wing 2). Thus at these high incidences wing 3 produces
nearly as much lift as wing 1 because the increased strength of the non-linear lift compensates for
its positive no-lift angle (0-6 °) and for the smaller range of incidence where there is positive
non-linear lift.
7
(86513)
A**
4. Conchtsions.
The tests on cambered gothic wings reported in Ref. 1 have been extended to include the
investigation of changes in design lift coefficient, and of changes in wing shape without changes in
the local incidence distribution. The results show that:
(1) The camber design is successful in that the flow is attached over the whole wing at the design
incidence, and for a limited range on either side of it. The incidence range over which the flow is
attached on the cambered wings appears to increase with increasing supersonic Mach number,
whereas on the uncambered wing the flow separates from the leading edge at a small incidence for
all Mach numbers.
(2) At any given incidence the cambered wings give less lift than the uncambered wing because
of the positive no-lift angle and because positive non-linear lift does not commence until a higher
incidence.
(3) At subsonic and transonic speeds the rate of growth of non-linear lift is similar on all wings,
but at Mach numbers above M = 1.4 the cambered wing with a design lift coefficient of 0.05,
and a straight trailing edge, appears to develop more non-linear lift than the uncambered wing,
whereas the wing with (Cz)cz = 0" 1 develops no non-linear lift.
(4) Camber causes a forward moment of the wing centre-of-pressure position at subsonic and
transonic speeds; this shift is most marked near M = 1.0. At supersonic speeds the effect of camber
on centre-of-pressure position is small.
(5) Both the camber shapes designed for C L = 0.05 have slightly lower drags than the plane
wing at positive CL, but the drag of the camber designed for C z = 0" 10 is greater than that of the
uncambered wing at lift coefficients below about 0" 1 at subsonic speeds and 0.15 at supersonic
speeds.
(6) Changes in spanwise camber, without changes in the camber incidence distribution, do not
alter the force characteristics when the flow is attached, but with separated flow the wing with the
straight trailing edge develops the most non-linear lift.
L I S T OF SYMBOLS
A
Aspect ratio
c(y)
Local chord
CO
Root chord
Mean aerodynamic chord
f
--sT
/fi:
T
Lift coefficient = Lift/qS
c~
( c~)o
Cm
CL~ Cm
M
Drag coefficient = Drag/qS
Drag coefficient of plane wing at zero lift
Pitching-moment coefficient = Pitching moment/qS~ (referred to quarter-chord
point of the mean aerodynamic chord)
Lift and pitching-moment coefficients at minimum lift-curve slope (see Section
3.2.1)
Free-stream Mach number
q
Free-stream dynamic pressure
S
Wing area
s(x)
s~,
s(x)
(X
Cross-sectional area distribution
Semi-span at trailing edge
Equation of leading edge (local semi-span)
Wing incidence: incidence of uncambered centre section of cambered wings
Incidence at minimum lift-curve slope
REFERENCES
No.
1
Author
L . C . Squire
2 J. Weber
Title, etc.
..
An experimental investigation at supersonic speeds of the
characteristics of two gothic wings, one plane and one cambered.
A.R.C.R. & M. 3211. May, 1959.
. . . . . .
Design of warped slender wings with the attachment line along
the leading edge.
Unpublished M.o.A. Report.
3
R . F . A . Keating . . . . . .
Low-speed wind-tunnel tests on sharp-edged gothic wings of
aspect ratio 3/4.
A.R.C.C.P. 576. May, 1960.
4
E . C . Maskell
Flow separations in three dimensions.
A.R.C. 18,063. November, 1955.
5
MacC. Adams and W. R. Sears ..
Slender-body theory--review and extension.
fi Ae. ScL Vol. 20. No. 2. February, 1953.
6
L . C . Squire
..
Some applications of 'not-so-slender' wing theory to wings with
curved leading edges.
A.R.C. R & M 3278. July, 1960.
7
J . H . B . Smith
. . . . . .
. . . . . .
....
A theory of the separated flow from the curved leading edge of a
slender wing.
A.R.C.R. & M. 3116. November, 1957.
10
TABLE 1
Details of Models
Dimensions (all models)
Planform
Centre-line chord (Co)
20 inches
Span (2st)
10 inches
Aerodynamic mean chord (~)
15 inches
Area (S)
133.3 square inches
Distance of ~ aft of apex
8.75 inches
Volume ( = 0. 009c0a)
72 cubic inches
Thickness distribution (without sting fairing)
Area distribution
S(x) = 100.a t ~ )
1-
( )ll
1
I
+
square inches.
Centre-line semi-thickness
z _ ~.~.
--
__
U
x
.[Z{~
x
x
--
__
x 2
-~
,
•
C0
Camber distribution (Ref. 2)
On all the cambered wings the local incidence distribution is given by
3z
-~ = C (constant) for 0 < I*?l "<< ~/0(x)
= c
-([.I-.o)"
] forVo(*)-< IvI < i,
i - (i + 2,o.) co~--nT,
o 7 ~ o g ( 1 - ,o~)
where
0.8
= y/s(x)
~0(x) -
2 -
(~/eo)"
The constant C is equal to 0. 0956 for wing 2 and 0. 0478 for wings 3 and 4.
11
i~
0
s
S C A L E IN [NCHE,5
i
I
1
I
[
2.
3
4
i
.5
(z_h)
×
WING U N C A M B E R E D
IN T H I S T R I A N G U L A R
REGION
,
\
\
REGION OF
NG
bO
13)
7 :l ',,:
II
./
ST
~
.....
,
~OOT
CENTRE-LINE
SECTION
,5ECTI 0 N
ON 0 " 4 SEMI-SPAN
PLANFOP, M
S C A L E IN I N C H E S
CHORD
FIe. 1. Details of plane wing (wing 1).
1
0
I
I
I
Z
I
3
FIG. 2.
I
4-
I
5
Details of the camber design.
51DE- VIEW
x
co
=0"2
WlNG ~ (UNCAMBERED)
I~
x
u
co
X
=0.4
WING3 { ( C L ) & =O,05~"
Eo = 0 ' 4
=0"(~
C o
X
Co
= 0'8
.t WINC 4
x
~=0"8
~ ~ ~xxxx Wl~q 3
WINq ATRAILING g.O C,B.
TRAILING EDqE
FIo. 3.
Details of wing cross-sections: wings 1, 2 and 3.
FIC. 4.
WING 3
Details of wing cross-sections: wings 3 and 4.
CL
//"
ZERO Cm
FOR
M:2-O
ZERO
CL
lVl = Z'O
.
"
/
t /
/
M=1"82
//
M :1',51
1.82,
~N
r
l,/ol
I '42
M =l '4:
i e
S f"
-0"04
Crn
-0"08
WING 3
-Oq2
-.4"
-2
(3
2
4
6
8
I0 ~..o
"FIG. 5. Variation of C L with c~: wings 1 and 3.
IZ
-0"2
-0, I
o
o-I
0.2
0.3 CL
0-4
Fro. 6. Variation of Cn~ with CL: wings 1 and 3.
6'0
L
WING 3
o.oe
wine i
~
j
I
,~1
2"0
1'8Z
"5'"
0.04
ZERQ
FOR M=2"O
.%.
¢J't
ZERO Co
"-.'~. ~°~oo
4"0
j ~N=Z.O
- ..-<..,
WIN~ 3 "~
I "82
/
WiNG | /
~,~
M=2.0
I'~1
\
p~
1"82
1"42
1'42-
-0-2.
-0-1
Fie. 7.
0
O-I
0'2
Variation of C D with
1 and 3.
0"3
CL
0.4
CL: wings
-O-Z
-0.I
0
0"[
O'Z
0'3
C L O°L~-
FIG. 8. Variationof LID with Cr: wings
1 andS3.
(% = 3 . 1 ° :
Upper
surface
CL
= 0.070
(% " 3 . i ° :
Lower
surface
CL
= 0.070
(% " O:
FIG. 9a.
Lower
surface
CL
- -0.010
Oil-flow photographs. Wing 3. M = 1.61.
~6
a " 8.4°:
Upper surface
CL • 0.223
ix - 6.30:
upper s u r x a c e
a = 4.2°:
Upper s v r f a c e C L - 0 . 0 9 7
FIG. 9b.
C L - 0.155
Oil-flow p h o t o g r a p h s . W { n g
3. M
=
1.61.
17
(~513)
B*
(% = 8 . 4 ° :
Upper
surface
CL
= 0.193
(% = 4 . 2 0 :
Upper
surface
CL
= 0.082
(% = 0 ° :
FIG. 10.
Lower
surface
CL
" -0.015
Oil-flow photographs. Wing 3. M = 2.0.
18
M
=
:~. =
FIG. l l a .
1.61
2. 3
Oil-flow photographs. Wing 1. o~ - 2 °.
19
14 = 1 . 6 1
M
"2.0
FIG. 11b. Oil-flowphotographs. W i n g | . ~ - 4-°.
20
u-~,o
Cm
r
ZERO
0'IZffRO CL
FOR
M= I " 3 0
/ J
J
t
i/
j/
Cm
FOR
'
J
M=I'80
M=I'2~
M=I'OZ
..-
° "~--
~
~
~"
M=I,O
bO
i/
/
7"
/ /
M=0.94
f-"
/~
j
.j-
1~
11"/" .,o
;9
tf" / o~O
,a~'gc
./5/
,i"
j-
M=0'94
M=0,85
/
M~o.o
~
M=0-40
~
M=0"80
.i-"
M=O.~O
-
,.-I
.fi
,,
-4
-7.
FIG. 12.
E1
0
2
4
C~
8
eL °
Variation of C L with e~: Wing 1.
lO
- 0 - ::'
-Oq
FIo. 13.
O
o-I
0.2:
Variation of C m with
0.~
CL
CL: wing
0.4
1.
v-c
H =1,3C
A<
0-1
O-OB
Co
Z-ERO C L
F O R M=I'BO
0'0~
I-'1= bE~
0.04
M = b02.
./
//
0"02
ZERO
CO
FOR
2/~ //"i
M=(
M= I-O~
%
/•o.8!
M:0"98
hO
hJ
M:O'g4
~
M=O,5~
/
M=O'50
H=O'85
M:O'90
M:0"85
t / V~0~o
%
M=O'~O
//
//
//
/
M= O , 7 0
M:O-80
NI = O.~l-O
M:O:70
,%
M:O'40
-O,Z
-O'1
Fm. 14.
0
0.1
0.2
0-8
CL
o'4.
Variation of Cz) with CL: wing 1.
-4
/
/k/-
/
FIG. 15.
/
,J"
f
-?-
/-
~ >--~" I.:o~
0
2.
4
6
~
0~o
Variation of C z with c~: wing 2.
I0
o-...
Cm
0-04-
0-08
CO
ZERO C m
/:OR
M: 1.30
0.06
O'O~
.o*
M = I'O'~
• M = I ' O ~-
ZERO ~ , .
FOR
M= 1-02
M= 0 4 8
M:
/
~ ""~--~~
Co
/~
"FL
O'g,
~'-'~ M~ 0-94
b~
H= 0-9C
M=0.86
~'~"-~ -m-~_m._~._eM: 0. a 5
M:O'80
Lg.,~
,.~._,~...,~... "-~ M= O.BO
H:O'70
",.
M=0-80
M=0.70
M: C)'40
M:o.~o
B T=I=0 , 4 0
f*
~..%
yO
~
[°
M=0"40
-0"2
-0'1
F~. 16.
0
0.~
0.2
0,3
C~. 0"4-
Variation of C., w i t h CL: w i n g 2.
-o.a
-o.~
Fro. 17.
o
o.,
. o-a cL 0-3
0.4
Variation of C D with CL: wing 2.
-2
o
FIG. 18.
2
4
6
~
I0
Variation of C z with o~: wing 3.
24
12
c4._Q
14
Cm
--0.04
-.-
--O.tO
CO
ZERO Cm
FORM=,.3C
,
~
.
~
--~'08
--0-0~
--0
•
/•:0-9•
04
CD FOR
i
=
"
f4:-l.OZ
~.
M=0'98
f
bO
---<.
/
//
" /M=0"9(
/
~M:0.70
M:O-90
M= 0,90 - - - //M=0.40
M=0"70
M=0.40
M=0"40
-o.I
0
FIG. 19.
o.,
°-2
1
0.3
0-4 cL
?s
Variation of C m with CL: wing 3.
J
~4:0.70
.M: 0
'40
-Oq
0
FIG. 20.
0'I
0.2
0"3
0.4- CL 0'5
Variation of C D with CL: wing 3.
I
0.4,
e WING I : UNCAMBERED
13 WINCIZ:(CL)d :O-I
A WING 3: (CL)d :0'05. O-~,
,...~
I
J
I
(c,- E:L)
0.2
I
Cm-Em
Q 'PIING t
B WINC 2
0 -02
,',-WINC 8
-%
YM>o'TU
Z
o.I
\\
ZERO M: 0-70
ZERO M : o - g o
\
m& j i
....
ZERO M = 0 " 9 0
"~ ~,,o.
90
ZERO M= I-O2
tO
Ox
ZERO M : I" 0 2
~1:1"61
\
ZER~ M : 1,61
ZERO M = 1-61
//
\
/
ZERO M= 2-OO
%
Z"
ZERO M : 2 . 0 0
I
I
THEORY;SLENDER WINC
THEORY: SLENDER WING
--
~Q/~/
-~>
THEORY:NOT,SO-SLENDER- -
T-HEORY-- SLENDER WlklC +4#,2 - -
THEORY : NOT-80-SLENDER WING
-4
Fzo. 21.
-z
o
z
4
~,(~ _~.)o 8
io
Variation of (CL-CL)with (c~-~): wings 1, 2
and 3.
-0"3
-0-2
-O.I
o
O-I
0"2
0"5
0"4
FIG. 22. Variation of (Cm-C,~,) with (CL-CL):
wings 1, 2 and 3.
-o-|
O-I
O.2
0"3
CL
0%
4 0 % ¢o
M= 0'40
flu
WJNCI
-o,,
O,I
0.~
0.2-
CL
40~/o Co
M=O'~4
Z
soO/
2
o
Z
J. . ° . -
<--<-<l
o
P
40%
{1-
WlNC
55%
I
co
o
407,
WING
...........
55~ CO
I
U
50% ~
o%~
-0-1
0
0"|
0-2.
0.3
CL
0%
40% Ca
I
--0.|
0
O.I
0,2-
0"~,
CL
~ 0 ~ Ca
M= 0'70
~o~
- , ° - . . . . . . ~. . . . . .
E
,
407,
55%Co
4o~
Q:
.......
THEORETICAL
~J
WING
WING
u
so%
P O S I T I O N AT M : I - 0
2 : 3G~ ~ AT CL=O'I
3 : $6~/o ~ A T Cm : 0 " 0 5
~ o %'
"tO
-0-1,
o-I
0-2
I
o'/5
o-3
CL
O.I
4 0 % Co
0.2.
0-~,
CL
4 0 ~ CO
I
M : O-,B0
\\
M= I'OZ
x
so;
z
o
~ J
v-
~
"~._
--.._.._~
o 40%
o~
55%¢o
0
-O.I
0-1
0"2
0"3
o~ 40;
Q.
0:
i ......
0
i''"
.
" - . . ....
5 5 ~ CO
!
CL
4 0 % Co
I
O~
-O-I
" 0
" (O.I
M = 0-90
M=P30
0"2.
I
0.3,
C'L
140~
Co
~,O
~ J
I-
-.~
,-; . . . . . . . , . . . . . . . . .
.,.-°
55% co
o
o.
155~oCQ
5 ~O~o'r...... [---,
5oo~
Flos. 23a and b.
Variation of centre-of-pressure position with C n at constant Mach number: wings 1, 2 and 3.
/
"0'1
Ct
o.r
0.2.
0-3,
EL.
4ON c~
M=1-42
•WINCI ~,
o
1.-"
~'~f
.WINC 2
•WINC |
o
55% Co
_._..J
d
~o~
0
-O.I
0"I
O'g
0'3
Cu
4 0 %c,,
1112.
z
M=l'bl
~o2,
o
E
gl
0
1,
S5 % c~
,
/
~2
u
o
~
=':"2T. . . .
~o~
-0.1
0
O'l
O'Z
I
0"3
C.
4.0 % Co
M = 1'SZ
-z
o
SO%'
I-
o
s5% ¢o
.4o%
/
¢.
.... o
.U
sog,
-o.l'
0
O,I
M=2"O
30%
O'Z
C,.~,
~
40°~
Co
i
55%c~
o_
'6
SO%
"'~x
FIG. 23c. Variation of centre-of-pressure
position wlth C L at constant Mach number:
wings 1, 2 and 3.
28
0.5
~.o
I~-'--"
r-s
WIMC z
:
48~
o
I'0
o.s
!
1"5
I jWING
. WING 8
'
//
2-0
-'
48:
C L : O.i !
,,
t
WINC
o
r
"" . . . . . . . . "
I
o
o0=
I
I
f
d s~%
ds~
I
~o%
7
0-5
. ..o
I
1"5
1"0
M
......
~(~
2"0
O'S
~'5
I-0
M
2,0
CL : 0"8
CL=0'2
WINe 2
u 487
I-
2-0
3
.~'~.~WINC 2
CL = 0 . 0 5
M
.WING 3
WINC 3
~-5~
sz%
WING [
ec
(J
i1:
-'~,.~%--.
5(0%
O'S
I'0
WINq I
'°51
o.. o. ....
1"5
2,0
0.5
I'0
I'S
2,0
FIG. 24. Variationof centre-of-pressure with Mach number, position at fixed CL: wings 1, 2 and 3.
29
0"07
O.Oto
[
CL: 0"3 /
-T-. . . . . . .
CL ~
"'
CL= O-I
:~---
CD
0"0~ ' ............ ~........... I . . . . . . . . . " ~ ' ~
//
CD
0 '06
I /':
0"0.
WING t
~
0 "0:
0'04
.:-..z. --~.
- F~
\
.
""
0-0~
M:O'70
I
I
NING 3
C~
-0'2
-0"1
0
O'l
0'2
.- .--:~ . - /
WINC 3
0'0;
f
1
0'0
0'3 CL 0-4
u-uo
0.6
0'8
bO
1'2
1'4
1"6
1"8
P(o
1,8
M
2-0
CD
OO
,&
o.03
::::W
I
CD
CL= 0
WINe 7 -
0.0Z
X
N
"~
-o-2
x
-o-I
FIG. 25.
0.02 -
"".-,.0.02
0
M : I'~,I
o-I
0-2
o.3 C L 0.4
Comparison of drag polars of
wings 1,2 and 3.
.
/
o.o, L - - : ~ - - . - . I . - 7 ~ ;
. . . . . . . r - -- i ......
O-~
Fro. 26.
0-8
sK,.,-~R,c~-,o.
1.0
I'Z
_
1",4-
M 20
Variation with Mach number of the drag at
fixed CI: wings 1, 2 and 3.
i.o
~R
L /WIN~
A (CO-C~o)
TrA (co- Cdo)
CkZ
1
CLZ
-~
l'Z
/WIN(:; 2
1.6
2
)~__~ ~w,~q,
~
WING I
I "2
-o
f
--0"8
/
WING 3
~/INC 3
--0
--0'4
-0-!
O. I
0.2
0.5
"4
-o-I
CL 0 ' 4
0
O.I
0'2
0"3
CL 0 . 4
M=0"90
M = 0,40
o
-a.
A(co-Cdo)
w A Cc~-Cgo)
EL2
CL z
1"2
-O'l
1-2
/
/
--0'8
--0"8
--
--0
0"4
0
\
1'6
l.e.
0-1
0-2
M=0"70
FIGS. 27a and b.
0-3
C
0'4
-0"I
'4
0
0"I
O'Z
0"3
CL
0'4
IVi = 0 ' 9 8
Variation with CL of the lift-dependent drag factor: wings 1, 2 and 3.
a
Z-O
~
/
1"6
~
WING 2
)VING I
~/W]NG
1"6
/
WING I.
WING ~.
WING
CL2
I.~'
1,2
THEORETICAL
VALUE AT
DESIGN C.L
O-8
-O.I
2-o
O
0.1
THEORETICAL,
VAL J~
AT DESIGN CL
H:I'4Z
0'4
t'~
0"8
O.P-
M = 1.81~
0.4-
O'
0.~ CL 0 - 4
-0.1
20[
/
o
G-I
0-2
0-3 C L 0.4-
\
1"6
F ^ (~,,-c,..)
~ A(Co-C~o)
%2
cd
I,~
I-2
Y
0.6
THEORETICAL
VALUE AT
DE~.IGN C L
0'4-
0.8
THEORETICAL VALIE
AT DESIGN CL.
M= 1.61
0
-0.1
O
o.I
O-P_
H = Z.-OC
o.4
O,.~ cl..
o,4
O
- O .l
O
O-t
0-2
O-3
cL
o-4
FIGS. 27c and d. Variation with Cr, of the lift-dependent drag factor: wings 1, 2 and 3.
<3.0ZO
Coo
I
L
1
J
t
~'LAT-PLATE SKIt,I-FRICTION PLUS WINC ALONE, WAVE DRAG
~'LAT-PLATE SKIN- FRICTION PLUS WINC AND BODY WAVE DRAG"'~"-..
O "015
;®
- ~ .-........
O.OlO
,000
0
FLAT-PLATE. 5KIN-FRICTION
0.00!
0
O'Z
0"4
FIG.
28.
0'6
0"8
I'0
M
1"2
[.4
1"~
Comparison of zero-lift drag o£ wing 1 with theory.
33
I'8
2.0
CL.
0".%
I
I
I
A WING 3 : STRAIGHT A T
TRAILING EDGE
--0.08
' V WING4:
Cm
%1
--0.04
ZERO
CL FOR
zero C.m
M=2-0
M: 2'0
M :I'SZ
M= 1"82
k~"~,i
M: I , 4 Z
v'a'%a~"
/
/
/4
M =I'(M
M:1.4Z
w
STRAIGHT A T
0 . 8 C,~
,
- -
]
/
,,~M:O-9.
%,.°,-8z
-
.4
,
/,,~vM:O.9(
M:O -98
M= 0"94 ~'~,v~,~
M:0.94
/ f
M:OqO
/-
%~""~
M : 0'.70
v t~'V'~a
~ ' ~ ' ~
~.~
-z
FIo. 29.
0
2
/x
WING ~ : $TRAICHT TRAILING EOCE
~7
WINC4 :STRAIGHT AT O ' S c a
4
~
6
lO olC°
Variation of CL with c~: wings 3 and 4.
~"~M
M : 0.40 v~,~.~
M:0.40
-,4-
M:O'90
"--~
IZ
-0.[
0
o-I
0-2
0.3
0.70
:0"4 0
0 . 4 CL. 0 . 5
FIG. 30. Variation of C~,zwith Oc: wings 3 and 4.
....
I,
/l/.,a,
-oo0
Co
/
-o.o~
"/
,.o~°
~o,,
i
b., .<
..,.,,,..,._
~,.o.,
I/
M.o.94
I
A
/
/
/ I / ;.,;Lo~°
!/~'
yl<'¢o,o
/I/
~/
,/
./
,,/I
t
,.
" >"P 5/,/'
M:o.9o
~'vz,...y~..~.,..~...~W-T
H=O.erc .
~/
"-..-~"iI / " i
,
i
/
I
M=Z.O0
.
.
.~
.
A WING 3 :STRAIGHT AT TRAILING EDGE
~WING
-0.1
0
Fro. 31.
0.1
~- :STI~.AIGHT AT O ' 8 C o
0.2
O.3
C u 0.4-
0.2
Variation of Cz) with Cr,: wings
3 and 4.
FIG. 32.
Variation of Cz with o~: M = 0.4j wings 3 and 4.
to
\
+O.OI
ZERO
/
--O,Ig
\
--O.14.
Cm
Co
O
WIN6~
~
--O.i~_
--
-O,OI
--0"~0
%
-0-O2
\\
\
\
C~
\
-O*03
\=
--0"0(
/
.\
--O'Ot;
--o.o,~
/
X
AWING 3:TRANSON;C TEST SECTION
~
~WING 3:SOL~D TEST SECTION
~7 WINca 4:TRANSON;C TEST SECTION
~'WING ~-:SOLID TEST .SECTION
, -O.OS
-0"06
ZERO WtNG 4
\
-0'07
-O'I
FtG, 33.
o.i
0-2
0.3
o.+
/
o.s c c o.~
Variation o f C m w i t h O1;: M = 0 . 4 :
w i n g s 3 and 4.
////
/
,:t~ '
f
-D'|
zERO ~ t N G 3
O
Fla. 34.
o.1
O.R
~WINC-. %: TRANSONICTEST S~CTION
AYWING, %: S O L I D TEST SECTION
R7 WIN& 4 : TRtkNSONle-, "fl
SEC'fION.
~7'WIN6. 4; SOLID TESTSECTION
Cl.
Variation of C;D w i t h C L: M = 0 . 4 :
w i n g s 3 and 4.
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Between Nos.
Between Nos.
Between Nos.
Between Nos.
Between Nos.
Between Nos.
Between Nos.
Between Nos.
2351-2449
z451-z549
255i-z649
z651-2749
275 I-z849
285 I;~949
295 I-3o49
3o5 i-3 I49
R. & M.
R. & M.
R.&M.
R.&M.
R.&M.
R.&M.
R.&M.
R.&M.
No.
No,
No.
No.
No.
No.
No.
No.
245o
255o
z65o
2750
2850
z95o
3o5o
315o
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zs. 6d. (post 3d.)
zs. 6d. (post 3d.)
zs. 6d. (post 3d.)
3s. (post 3d.)
3s. 6d. (post 3d.)
3s. 6d. (post 3d.)
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