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,
~
~
ioo""~
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/~/
iJl
x
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,~~
+
o
.2
-t
.-:l~ ~t
.002
-.2
C,
.4
.8
.6
LO
FIG. 51. Comparison of calculated and experimentally measured polars for N.ACA 67,1-215 airfoil.
.014
.010
I
t
.012 '-
+~
~.008
~..........
~
.........
,
J
/ /'
¥
!~ "'--
...~
...... ~+~- ... .-:. ~
---
-~
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+ Experimedol
.004
measurement -
.oO?:s
Theoretical
Or--'"
I---
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colcu/olion
-.6
-.4
-.2
o
.2
Cz.
.4
.6
.8
1.0
/2
1.4
Comparison of calculated and experimentally measured polars for NACA ~015 airfoil at R = 5.9 X 10'.
FIG. 52.
where 8 = sum of momentum thicknesses on upper and lower surfaces In the usual case, the pressure at the trailing edge is not the same as free-stream static pressure, and some means must be found for finding the effective momentum thickness at a point far downstream in the wake
THE EFFECTS OF VISCOSITY
109
where the pressure has returned to the free-stream static value. Squire and Y oung'P derived the required relation by setting the value of T in Eq. (5.11) equal to zero in the wake and finding an empirical relation between Hand q/qo- The resulting expression for the profile drag is
(8)
ctl=2- (U~ - (Jl+fl)/2 c t 11 t
(5.27)
where the subscript t designates conditions at the trailing edge. The value of H is the value used in calculating the boundary-layer thickness. The agreement to be expected between experimental and calculated profile drag coefficients is indicated by Figs. 51 and 52. The calculated 14 --...r- r- drag coefficients presented in these .... 8 """' --. .............. ~ figures were obtained by a method" r-----. ........... A r---... r-- fundamentally the same as that pre- ~ 12 r-- senteel here. ¢ 1.0 5.14. Effect of Mach Number on Skin Friction. Solutions for the velocity distribution through the 080 1 4 2 3 5 laminar boundary layer in corn Mach number pressihle flow and for the correspond- FIG. 5~1. Skin-Iriction eoeffir-ienrs, ( ...4) No ing skin friction have been obtained heat transferred to wall. (Ill wall tern perature one-quarter of Iree-streurn temperature, by von Karman and Tsien1ft for the case of the flat plate. The results of the skin-friction calculations are pre sented in Fig. 53 for the case of no heat transfer to the plate and for the case of a plate whose absolute temperature is one-quarter of that of the free stream. These results show a moderate decrease in laminar skin friction with increasing Mach number. Although these results indicate that the laminar skin friction increases with beat transfer from the fluid to the plate, this increase is small even for the extremely 10"" plate temper atures for which these results were obtained. For the turbulent boundar)' layer, Theodorsen and Regier!" found ex perimentally that the skin friction is independent of the Mach number (at least up to a value of 1.69). The experimental data are presented in Fig. 54. These data were obtained by measuring the torques (moments) required to rotate smooth disks in atmospheres of air and Freon 12 (CCI2F2) at various pressures. Data obtained by Keenan and X eumannf for the skin friction in pipes appear to confirm Theodorsen's conclusion for the fully developed turbulent flow. Analytical studies by Leeg6% on the effect of Mach number and heat transfer on the lower critical Reynolds number for the case of the flat plate indicate that increasing th~ Mach number has a destabilizing in fluence on laminar flow when there is no heat transfer to the plate. Heat
---
THEORY OF WING SECTIONS
110
transfer to the plate has a stabilizing effect, while the opposite is the case for heat transfer from the plate to the fluid, This effect becomes stronger I
-1.2
I I I Moch P1essute (in.HgobsJ tvmber Q24toQ62 r- 1.8 fo.JO Air .4010 .96 Freon /2 30 t- 14 .5J 10/.42 FI1!OII 12 to.77101.69 Fr«Jfl 12 5
I
I
Symbol 0
-1.4
04
x D
~ IIlloo....
~ ~ ~ ........ 100.... """"
4.8
5.0
52
~
KDrrnDns -Ys
r--. ........
"""
~~ 4 ~ KcrnD1s /ominar-f/oW"'/ i" ~ -2.2 fomvlo, C",=.J.871r~ ~,
-2·~.6
-
futtJu/enI-f/oN formub, /"" Lm CM=O.I46R
-~
"'"",,
-2.0
[,I
I
60s
5.4
5.6
a -"'1fI
M=O.53 ~=l69 ~
~ ~ ~.
~
5.8
6.0
6.2
6.4
....:..; ~ ....... + it-+
6.6
6.8
-ZO
LDgI()R FIG. M. Moment coefficient for disks as function of Reynolds number for several values of Mach number with air and Freon 12 as mediums, Maximum Mach number, 1.69.
as the Mach number is increased, and, at moderate supersonic speeds, comparatively small amounts of heat transfer to the plate stabilize the laminar layer to very high values of the Reynolds number.
CHAPTER 6 FAMILIES OF WING SECTIONS 6.1. Symbols. Ps resultant pressure coefficient R radius of curvature of surface of modified NAC.A four-digit series symmetrical sections at the point of maximum thickness IT velocity of the free stream a coefficient a mean-line designation; fraction of the chord from leading edge over which loading is uniform at the ideal angle of attack
e chord
ci. section design lift coefficient
c.e/. section moment coefficient about the quarter-chord point
d coefficient k1 constant m maximum ordinate of the mean line in fraction of the chord p ehordwise position of 111 r leading-edge radius in fraction of the chord r, leading-edge radius corresponding to thickness ratio t maximum thickness of section in fraction of chord v local velocity over the surface of a symmetrical section at zero lift !:Av increment01local velocity over the surface of a wing section associated 'lith camber Av.. increment of local velocity over the surface of a wing section associated with angle of attack % abscissa. of point on the surfaee of a symmetrical section or a chord line XL abscissa of point on the lower surface of tl. wing section Xu abseisaa of point on the upper surface of a wing section XC abscissa. of point on the mean line YL ordinate of point on the lower surface of a wing section Yu ordinate of point on the upper surface of a wing section Yc ordinate of point on the mean line y, ordinate of point on the surface of a symmetrical section Qi design angle of attack 8 tan-I (dyc/dzc ) T trailing-cdge angle
6.2. Int:"oduction. Until recently the development of wing sections has been almost entirely empirical. Very early tests indicated the desirability of a rounded leading edge and of a sharp trailing edge. The demand for improved wings for early airplanes and the lack of any generally accepted wing theory led to tests of large numbers of wings with shapes gradually improving as the result of experience. The Effie! and early RAF series were outstanding examples of this approach to the problem. 111
THEORY OF WING SECTIONS
112
The gradual development of wing theory tended to isolate the wing section problem from the effects of plan form and led to a more systematic experimental approach. The tests made at Gottingen during the First World War contributed much to the development of modem types of wing sections. Up to about the Second World War, most wing sections in common use were derived from more or less direct extensions of the 'york at Gottingen. During this period, many families of wing sections were tested in the laboratories of various countries, but the 'York of the NACA was outstanding. The NACA investigations were further systematized by separation of the effects of camber and thickness distribution, and the experi y ./0
°u(xy,Yy)
e -./0
'I p!!_.LJ_-_~-=---
~(XLIYLJ 'Rodius fhrough end of chord (mean line slopeOf a5 % chord)
Xy=X-Jf sin9 XL=X"'t
sinS
Yu=~
+11 cos9
JL 7C -JI cos e
Sample calculationA lor derivation 01 the NACA 86.3-818 airfoil (0 -= 1.0)
-
tan'
- ---1----1
sin"
FIG. 56. Method of combining mean lines and basic-thickness forms.
mental work was performed at higher Reynolds numbers than were generally obtained elsewhere. The wing sections now in common use are either N .~CA sections or have been strongly influenced by the NACA investigations. For this reason, and because the NACA sections form consistent families. detailed attention will be given only to modem NACA wing sections. 8.3. Method of Combining Mean Lines and Thickness Distributions. The cambered wing sections of all NACA families of wing sections con sidered here are obtained by combining a mean line and a thickness distribution. The process for combining a mean line and a thickness distribution to obtain the desired cambered wing section is illustrated in Fig. 55. The leading and trailing edges are defined as the forward and
FAMILIES OF WING SECTIONS
113
rearward extremities, respectively, of the mean line. The chord line is defined as the straight line connecting the leading and trailing edges. Ordinates of the cambered lying sections are obtained by laying off the thickness distributions perpendicular to the mean lines. The abscissas, ordinates, and slopes of the mean line are designated as Xc, Ye, and tan 8, respectively. If Xu and Yu represent, respectively, the abscissa and ordi nate of a typical point of the upper surface of the wing section and y, is the ordinate of the symmetrical thickness distribution at ehordwise position %, the upper-surface coordinates are given by the following relations: Xv
= :r -
y, sin 9 }
yv = Yc + y, cos "
(6.1)
The corresponding expressions for the lower-surface coordinates are XL = X + Yt sin 8 } YL = Ye - Yt cos 8
(6.1)
The center for the leading-edge radius is found by drawing a line through the end of the chord at the leading edge with a slope equal to the slope of
the mean line at that point and laying off a distance from the leading edge along this line equal to t.he leading-edge radius. This method of con struction causes the cambered wing sections to project slightly forward of the leading-edge point. Beeause the slope at the leading edge is theoreti cally infinite for the mean lines having a theoretically finite load at the leading edge, the slope of the radius through the end of the chord for such mean lines is usually taken as t.he slope of the mean line at x/c equals 0.005. This procedure is ~ustified by the manner in which the slope
increases to the theoretically infinite value as x/c approaches o. The slope increases slowly until very small values of x/c are reached. Large values of the slope are thus limited to values x/c very close to 0 and may be neglected in practical wing-section design.
or
The data required to construct some cambered wing sections are pre sented in Appendixes I and II, and ordinates for a number of cambered sections are presented in Appendix lll. 6.4:. NACA Four-digit Vmg Sections. a. Thickness Distributions. "'hen the N ACA four-digit wing sections were derived.P it was found that the thickness distributions of efficient wing sections such as the Gottingen 398 and the Clark Y were nearly the same when their camber was removed (mean line straightened) and they were reduced to the same maximum thickness. The thickness distribution for the NAC~4\. four-digit sections was selected to correspond closely to that for these wing sections and is \' given by the following equation: ±'Yt,=_t- (O.29G90·~-O.12600x-O.35160x2+0.28430r-O.l0150zC)(6.2) 0.20
THEORY OF WING SECTIONS
114
where t == maximum thickness expressed as a fraction of the chord The leading-edge radius is r, = 1.1019t2
(6.3)
It will be noted from Eqs. (6.2) and (6.3) that the ordinate at any point is directly proportional to the thickness ratio and that the leading-edge radius varies 88 the square of the thickness ratio. Ordinates for thickness ratios of 6, 9, 12, 15, 18, 21, and 24 per cent are given in Appendix I. b. Mean Line«. In order to study systematically the effect of variation of the amount of camber and the shape of the mean line, the shape of the mean lines was expressed analytically as two parabolic arcs tangent at the position of maximum mean-line ordinate. The equations" defining the mean lines were taken to be m
Yt: :;: p;. (2px - x 2 )
forward of maximum ordinate
and
(6.4) Yc
=
m
(1 _ p)t [(1 - 2p)
+ 2px -
r]
aft of maximum ordinate
where m = maximum ordinate of mean line expressed as fraction of chord p == chordwise position of maximum ordinate It will be noted that the ordinates at all points on the mean line vary directly with the maximum ordinate. Data defining the geometry of mean lines with the maximum ordinate equal to 6 per cent of the chord are presented in Appendix II for chord,vise positions of the maximum ordinate of 20, 30, 40, 50, 60, and 70 per cent of the chord.. c.. Numbering System. The numbering system for NACA wing sections of the four-digit series is based on the section geometry. The first integer indicates the maximum value' of the mean-line ordinate Yc in per cent of the chord. The second integer indicates the distance from the leading edge to the location of the maximum camber in tenths of the chord. The last two integers indicate the section thickness in per cent of the chord.. Thus the NACA 2415 wing section has 2 per cent camber at 0.4 of the chord from the leading edge and is 15 per cent thick. The first two integers taken together define the mean line, for example, the NACA 24 mean line. Symmetrical sections are designated by zeros for the first two integers, as in the case of the NACA 0015 wing section, and are the thickness distributions for the family. d. Approximate Theoretical Characteristic8. Values of (V/V)2, which is equivalent to the low-speed pressure distribution, and values of v/V are presented in Appendix I for the NACA 0006, 0009, 0012, 0015, 0018, 0021, and 0024 wing sections at zero angle of attack. These values were cal culated by the method of Sec. 3.6. Values of the velocity increments AIl.IV induced by changing angle of sttaok are also presented for an
FAMILIES OF WING SECTIONS
115
additional Iift coefficient of approximately unity. Values of the velocity ratio v/V for intermediate thickness ratios may be obtained approximately by_linear scaling of the velocity Increments obtained from the tabulated values of v/V for the nearest thickness ratio; thus
(!!.V.)" = [(~) 1
II
1J ~ + 1 t1
(6.5)
Values of the velocity increment ratio ~vc/V may be obtained for inter mediate thicknesses by interpolation. The design lift coefficient. Cl i , and the corresponding design angle of attack oa, the moment coefficient Cm.lf) the resultant pressure coefficient P R , and the velocity ratio ~/V for the NAC.L~ &2, 63, 64, 65, 66, and 67 mean lines are presented in Appendix II. These values were calculated by the method of Sec. 3.6. The tabulated values for each mean line may be" assumed to vary linearly with the maximum ordinate Ye; and data for similar mean lines with different amounts of camber, within the usual range, may be obtained simply by scaling the tabulated values. Data for the NACA 22 mean line may thus be obtained simply by multiplying the data for the NACA 62 mean line by the ratio 2:6, and for the X.~CA 44 mean line by multiplying the data for the N.t\CA 64 mean line by the ratio 4:6. Approximate theoretical pressure distributions may be obtained for cambered lying sections from the tabulated data for the thickness forms and mean lines by the method presented in Sec. 4.5. 6.6. NACA Five-digit Wlng Sections. 4. Thickne&8 Distributions. The thickness distributions for the NAC.A~ five-digit wing sections are the same as for the NACA four-digit sections (see Sec. 6.4a). b. Mean. Lines. The results of tests of the N.&-\.CA four-digit series wing sections indicated that the maximum lift coefficient increased as the posi tion of maximum camber was shifted either forward or a~t of approximately the mid-chord position. The rearward positions of maximum camber were not of much interest because of large pitching-moment coefficients. Be cause the type of mean line used for the NACA four-digit sections was not suitable for extreme forward positions of the maximum camber, a new series of mean lines was developed, and the resulting sections are the NACA five-digit series. . The mean lines are defined 46 by two equations derived so as to produce shapes having progressively decreasing curvatures from the leading edge aft. The curvature decreases to zero at a point slightly aft of the position of maximum camber and remains zero from this point to the trailing edge. The equations for the mean line are Yc = ~kl[x3 - 3mx2 + m2(3 - m)x] from x = 0 to z = ml y.; = %k1m3(l - x) from x = m to oX = c = l~
(6 6) .
THEORY OF WING SECTIONS
116
The values of m were determined to give five positions p of maximum camber, namely, 0.000, O.IOc, O.ISe. O.2Oc, and O.25c. Values of k1 were initially calculated to give a design lift coefficient of 0.3. The resulting values of p, m, and k1 are given in the following table. Mean-line designation
Position of camber p
""
kl
210
0.05 0.10 0.15 0.20 0.25
0.0580 0.1260 0.2025 0.2900 0.3910
361.4 51.64 15.957 6.643
220
230 240
250
3.230
This series of mean lines was later extended" to other design lift coefficients by scaling the ordinates of the mean lines. Data for the mean lines tabulated in the foregoing table are presented in Appendix II. c. Numbering System. The numbering system for wing sections of the N..\ CA five-digit series is based on a combination of theoretical aerodynamic characteristics and geometric characteristics. The first integer indicates the amount of camber in terms of the relative magnitude of the design lift coefficient; the design lift coefficient in tenths is thus three-halves of the first integer. The second and third integers together indicate the distance from the leading edge to the location of the maximum camber; this distance in per cent of the chord is one-half the number represented by these .-integers. The last two integers indicate the section thickness in per cent of the chord. The NACA 23012 wing section thus has a design lift coefficient of O.3 J has its maximum camber at 15 per cent of the chord: and has a thickness ratio of 12 per cent. d. Approximate Theoretical Characteristics. The theoretical aerody namic characteristics of the N ACA five-digit series wing sections may be obtained by the same method as that previously described (Sec. 6.4airfoil
/ ~ ~~ ---:
-
~)
--fbJgh
--5rnodh I J I Symbols with fJogs correspond 10 sinllbfed splil fbp deflected 60-
.4
I
00
4
8
I
I /2
I
I
I
16
Airfoillhick~pen;:enI
20
24
of chotrI
(a) N ACA four- and five-digit series. 58. Variation of muimum section lilt coefficient with airfoil thickness ratio and camber for .veral NACA airfoil sections with and without simulated split ftaps and standard rough ness. B, 6 X loa. FIG.
lift-curve slope increases with increase of thickness ratio and with forward movement of the position of minimum pressure on the basic thickness form. The effect of thickness ratio is comparatively small for the NAC.\ 66-series sections. The thick-wing-section theory (Chap. 3) shows that the slope of the lift curve should increase with increasing thickness ratio in .he absence of viscous effects. For wing sections with arbitrary modifica tions of shape near the trailing edge, the lift-curve slope appears to decrease with increasing trailing-edge angle. Some KACA6-series wing sections show jogs in the lift curve at the end
.
EXPERl}.{ENTAL CHARACTERISTICS OF WING SECTIONS
133
of the low-drag range, especially at lo,v Reynolds numbers. This jog be comes more pronounced with increase of camber or thickness ratio and with rearward movement of the position of minimum pressure on the basic thickness form. This jog decreases rapidly in severity with increasing Reynolds number, becomes merely a change of lift-curve slope, and is practically nonexistent at a Reynolds number of 9 million for most lying I
-
...i
"6
-
-
_~o.'
2.8 -
.1,.
-'" 0
i , '"'\
-. ..."
~--
~ L-...... ~
'\ £ //
"'0 . 2 -
-
o
I °Il
r--
~'i
~ ,...:3 ... )\, -- :~ ~, ~-
~):;J'
\ Y ~ M ~:~ r-, 7"
Cl 1
~
k:-: -_ .I' ~ r-, \.. 0.'J,. ,"F='=" - e ~" ~ I ,. / V~·V -&.-o ~ ...... JII 1/ -- :i'~ .4~ _J V,.)
~
~
~"O ~~
.... L:..-
~
~
ft
.2
~
"00
~
......,s:: o
o
:
C&l
~
1.2 l - - ~
...
6
.2~
e e L
L.. iIl.............
~
~--
90.6-
e
~~
~ .~
..0-
----
... ~,,~
..... ~
,'.
~-
~
·f ~-\~ ~;;
Plain
a1rt'o11
-6
~
\ \\ ~ '"' \\ ~'( -vo.6-4 C~l
8IDOotb
\
\.- -6
~
-0 .2 00
I
ltouab
I--SJlDbola with 1"l-p correepood ~o 8!'-w..ated- ap11t 1'1ap :Serlect~d 600
o o
A1rt'oll W1tb ep11t. ~J.ap
~ 8 12 16 A1rtoll th1ckDe... perceD t or (6) NAC.A 63-series. }"'IG. 58. (Continued)
20
Cbor4
sections that would he considered for practical application. This jog may he a consideration in the selection of wing sections for small low-speed airplanes. An analysis of the flow conditions leading to this jog is pre sen ted ill reference 134. The values of the Iift-curve slopes presented are for steady conditions and do not necessarily correspond to the slopes obtained in transient con ditions when the boundary layer has insufficient time to develop fully at each lift coefficient. Some experimental resultglOG indicate that variations of the steady value of the lift-curve slope do not result in similar variations of the gust loading,
THEORY OF WING SECTIONS
134
c. Mazimum Lift. The variation of maximum lift coefficient with thickness ratio at a Reynolds number of 6 million is shown in Fig, 58 for a considerable number of NACA wing sections. The sections for which data are presented in this figure have a range of thickness ratios from 6 to 24 per cent and cambers up to 4 per cent of the chord. From the data for the NACA four- and five-digit wing sections (Fig. 58a), it appears that the
'.2
I °Il
2.8 Ie E
~
..
0
....
~~ -.",,~
.~",.
2.0
S 0
_~
1.6
....
...... 0
••
i
-
°Il
o~
v
f'
'70.6- ..../ ~ 06=3
-
g
~,;~ ('., ~ ~ ::::- ~
... ..J~•
~
...~
:,
or,
, ~ o>~ .,.,~ -1 ~ , " ~ ,JI
,p
.......i
~
.4- -- -, ~ 0 .~ v ~ .2 "a .1' t l O - ~~.~ lS' ~.,~ h: :.II
..• 2.4
0
--
~
.. 0.6
.4- -"'"
01
"Y-
~
- ~'V .~ ,," 00---, :~ ... 1~,-
iI_~ or~
-,
--.. ..J!\
-,\_0lil
'\1:
-G
~
~
.)00 .).--
----
I
-
ir:
I
~
-- ...... ~--
Cl
~
'-!\
Smooth
4\ ~
~
..--
~
(d)
NACA
65- aerles. A
.012
~.#Q
A
.008
~
Rough
1·~--
.o~ ~oo7h
'r
.()-~ ).-~
.....
v
,)--
__ ...4 ~.,~
'~
v
-
8 12 4 16 Alrrol1 thickness. percent or (e) NACn 66- series.
FIG.6S. (Corduded)
c:> 0 0.2
~
A
V
o
--A
20
Cho~a
l
0.4 0.6
THEORY OF WING SECTIONS
154
ratio. The effect of thickness ratio is shown in Fig. 69 from which the center of the low-drag range is seen to shift to higher lift coefficients with increasing thickness ratio. This shift is partly explained by the increase of lift coefficient above the design lift coefficient for the mean line obtained when the velocity increments caused by the mean line are combined with
.032 G
a
.028
e A
IfAOA 641 - Ja,12 RACA '42-~5 MCA 6~_~
.ADA 611:4-1;21
:'.~
..
.t .p
I
o
.020
I Z.016
...a
If tJ
W v IPI/
~
~
8
co
.012
~
.1
~
.008
'i:1
~
~~
""'i ~
.~
o
-1.'
~V
~V
~ -.
~
...... ~ ~
-1.2
-.8
.
~ rJy
AI.~
--
_.... 0 .... .8 8ectlOil 11ft coetftclent.O&
1.2
1.6
69. Drag characteristics of some NACA 64-series airfoil sections of various thicknesses, cambered to a deei&n lift coefficient of 0.4. R, 9 X 10'.
FIG.
the velocity distribution for the thickness form according to the first approximation method of Sec. 4.5. At the end of the low-drag range, the drag increases rapidly with increase of the lift coefficient. For symmetrical and low-cambered wing sections for which the lift coefficient at the upper end of the low-drag range is moderate, this high rate of increase does not continue (Fig. 71). For highly cambered sections for which the lift at the upper end of the low-drag range is already high, the drag coefficient shows a continued rapid in
crease. Comparison of data for wing sections cambered with a uniform-load
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
2
..
..,;
8
V
~~
o
i
.:s := '-t
i•
Ir--B
I
-.•
~
1 ' - - 1---
V
1J
.... ... ~
na.
~e
I
0
.
I I I I I I I I Vppel'lJa1t of low-4rq . . . . ~J1 1JIdt of l_-4ras
I
-
155
~-
~
_.....
J
~
...
II
-
....~
"':"
1-· o
2
o
(a)
8
~ 20 2JJ
111,.148 . .bel'• •
12
Variation of upper and lower limits of low-drag range with Reynolds number,
(
J
•x J.o' 2.0 .0
e
.028
I I
II
~.o
•
61.'
~
" 25·0 ~
'5.0
,
>
.~
~
J.,
',-
e~ ~
.008
~ .,...
.-. e
-1.2
FIG.70
III
.
~
a; rw:...
-.8
-.4
0
---
~r.;)
-.~
-
.1,.
.8
"o~l_ 1U~ coer~~ol.at.
o~
1.2
L'
(b) Section drag characteristics at various Reynolds numbers. Variation of low-drag range with Reynolds number for the NAC_4 65(421)-420 airfoil.
THEORY OF WING SECTIONS
156
mean line with data for sections cambered to carry the load farther forward shows that the uniform-load mean line is favorable for obtaining low-drag coefficients at high lift coefficients (Fig. 72). Data for many of the wing sections given in Appendix IV show large reductions of drag with increasing Reynolds number at high lift coefficients.
• 028 4~
.024
IA
tcJ
0
-
.p
cG)
is/
.020
V if?
.....
...o '-t
~
CD
.016
0 0
tQ
~
~
s: 0
.012
..... .p 0
C)
It)
.008
l~
HACA 6~-4l8"
\ ~r\ ~
~.jl~
.~
I
I
I
I
I CA
:U
~. "t:>: ~~ ~. ~ .~
.oa,.
.",,:
I
65,-218
RACA
I
'"\
\
\
I
6 53-018,\
i\
.~
d~J ':> ~ ~
1\
I~ ~
~
~~
~
1.
L
-
~
p~
~.~~
.7
J
r 1M
.... Li. ~
....
~\
RACA 6~-618JlACA
).\
65.,-8J.8- W
0 ...........- ......- ............- ......- ...----......- .....----.....- .....- ............ o 1.2 .8 1.6 -1.2 -.8
-.4
Sectlon
~lrt
coefficient.
CI
FIG.71. Drag characteristics of some NACA. 65-seriea airfoil sections of 18 per cent thickness
with various amounts of camber. R, G X 10'.
This scale effect is too large to be accounted for by the normal variation of skin friction and appears to be associated with the effect of Reynolds number on the onset of turbulent flow following laminar separation near the leading edge.1M A comparison of the drag characteristics of the XACA 23012 and of three NACA 6-series wing sections is presented in Fig. 73. The drag for the N.~CA 6-series sections is substantially lower than for the NACA 23012 section in the range of lift coefficients corresponding to high-speed flight, and this margin may usually be maintained through the range of lift
EXPERIl.{ENTAL CHARACTERISTICS OF rVING SECTIONS
157
coefficients useful for cruising by suitable choice of camber. The NACA 6-series sections show the higher maximum values of the lift-drag ratio. At high values of the lift coefficient, however, the earlier NACA sections .generally have lower drag coefficients than the N ACA 6-series sections. c. Effect oj Surface Irregularities Oil Drag Characteristics. Numerous measurements of the effects of surface irregularities on the characteristics of wings have shown that the condition of the surface is one of the most important variables affecting the drag. Although a large part of the drag increment associated with surface roughness results from a forward move ment of transition, substantial drag increments result from surface rough ness in the region of turbulent flow." It is accordingly important to maintain smooth surfaces even when extensive laminar flow cannot be expected. The possible gains resulting from smooth surfaces are greater, however, for wing sections such as the XA(~A G-series than for sections where the extent of laminar flow is limited by a forward position of mini mum pressure, No accurate method of specifying the surface condition necessary for extensive laminar flow at high Reynolds numbers has been developed, although some general conclusions have been reached, It may be presumed that, for n given Reynolds number and chordwise location, the size of the permissible roughness will vary direct ly with the chord of the lying section. It is known, at one extreme, that the surfaces do not have to be polished or optically smooth. Such polishing or waxing hus shown no improvement in tesu,.3 in the Kf\CA two-dimensional low-turbulence tunnels when ap plied to satisfactorily sanded surfaces. Polishing or waxing a surface that is not aerodynamically smooth ,,;U, of course, result in improvement, and such finishes may be of considerable practical value be cause deterioration of the finish may be easily seen and possibly postponed. Large models having chord lengths of 5 to 8 feet tested in the NACA two-dimensional low-turbulence tunnels are usually finished by sanding in the ehordwise direction with Xo, 320 carborundum paper when an aerodynamically smooth surface is desired," Experience has shown the resulting finish to be satisfactory' at flight values of the Reynolds number. Any rougher surfuee texture should be considered as a possible source of transition, although slightly rougher surfaces have appeared to produce satisfactory results in some cases.. l.;oftin6S 8ho\\'00 that small protuberances extending above the general surface level of an otherwise satisfactory surface are more likely to cause transition Ulan are small depressions. Dust particles, for example, are more effective than small scratches in producing transition if the material at the edges of the scratches is not forced above the general surface level. Dust particles adhering to the oil left on wing surfaces by fingerprints may be expected to cause transition at high Reynolds numbers.
THEORY OF WING SECTIONS
158
Transition spreads from an individual disturbance with an included angle of about 15 degrees. H • 42 A few scattered specks, especially near the 2.8
..
I I
2.0
I
o
e
DCA 6~-418
D DCA ~-
t----
r---
1.2
65,-418, a
---.. =0.5 - r-.
~~~ ..........
-~--
~.
rJ) ~
~.-
I
r:'
'r
II
.8
r
; l~ ~
o
I
o
-rwlr
--4
sa •o
.....
-.2
J •
- .8
7 1
~...I
-
--
~
-
~
" (.;1
'-t ~
S o
-.~
~
I
= -.4 -16
-8 8tQ~lOD
0
a
aDS1. or attack.
~b (10.
deg
24
FIo.72. Comparison of the aerodynamic characteristics of the NACA 65,-418 and NAC_~ 66,-418, CI ==.0.5 airfoils. B, 9 X 10'.
leading edge, will cause the flow to be largely turbulent. This fact makes necessary an extremely thorough inspection if low drags are to be realized. Specks sufficiently large to cause premature transition can be felt by hand. The inspection procedure used in the N ACA two-dimensional low-turbulence
EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS
159
tunnels is to .feel the entire surface by hand, after which the surface is thoroughly wiped with a dry cloth. 0J2
028
o td
0- •
G24
...•
020
. 1i ...o
•
.~ ~.....'r---.....-_.8 I
.2
or-......._
1.0
.. _ + - - - t - _ t - - - t -......._+---I
rI
'-4 '-4
8 0
.ou :: . a :
.b a0
012
0
If /J
008 AftL "T
.
o· 0
....-•..
-
0
4
1;
a.o. position
0
""8
....
..1
Q
-.2
JUOA-'5,~8
-II IIICA 653-418,
0
I -.,
011
=0.5
-.060
.267
-·Qq1
I I
•
-1.6
•
zlo .2'5
-1.2-
-.8
-.J..
0
.k
"etlon 11ft coeff1cient. FIG. 72. (Ccmcluded)
7/0
I •J
1.2
1.6
z.0
c&
It has been noticed that transition caused by individual sharp pro tuberances, in contrast to waves, tends to occur at the protuberance. Transition caused by surface waviness appears to move gradually up stream toward the wave as the Reynolds number or wave size i~ increased. The height of a small cylindrical protuberance necessary to cause
THEORY OF WING SECTIONS
160
transition when located at 5 per cent of the chord with its axis normal to the surface" is shown in Fig. 74. These data were obtained at rather low 2.8
I 1
T
II~
e
DCA
EI lIACA .1IACA
""""-
I I I I
I
I
I
I
-l&J.~
~~
I--
.
I
Jl
x
106
6
6 lIACA 65~~5 9 DCA " 215)~,
o .: 1. 6
I
!012 9 9 roupn... 9
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~. Q~
...
I
9
F
nJ. L.
~~
e:
....•u
.......
I
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,1. 2
.p
.......
"
4"1
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g
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ro
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.Art
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0 ~
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-
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\,.I
.......
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~
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f\
<JsV
j3 .~
rv
7
-1 .2 ~
e
•
!
-.4
.-1. .6
-2J,.
-16
seotion
-8
8D&1~
0
of attack.
8
l'
"0' dec
24
FIG. 7~. Comparison of the aerodynarnie characteristics of SOD1C N.ACA airfoils from tests in the Langley two-dimensional low-turbulence pressure tunnel.
values of the Reynolds number and show a large decrease of the allowable height with increase of Reynolds number. Analysis'S of these data showed that the height of the protuberance that caused transition depended on the shape of the protuberance and on the Reynolds number based on the height of the protuberance and the local
EXPERIMENTAL CHARACTERISTICS OF It''lNG SECTIONS
161
velocity at the top of the protuberance. This Reynolds number is plotted against the fineness ratio of the protuberance in Fig, 75 for protuberances located at various chordwise positions on two wing sections. •0,2
I I I
t ·02" i•
9 Ie 10' ..... .9 ~"' r-,
$-ur;
-415
~ IlACA
o •
"
6 ~ ,~
StaDel rougbne•• • DCA. '52-415 " DCA 66(215)~
Iu
1
It
~
G DCA B DCA
.;: • ... . .--
, ,I
1
9 -_.~
\ \
.
Ol~
.I 012
~
'" ~
008
,
~
f\
',-""
..... ' -v"f..:J
~
~
.
o
.....
l~" f\Vp ) ~W
,
J~ ~v
I
j /)1
/ I/~ V
J~ /
_llt V
~I
I'oT
V
1_ 1"1 ~
~
V
0
o
.....
..;.
:-or
o
...-
....:. -
...~ .2 ...o Ii CI.
1
·m
0
I
I
....
W
...
-
v
v
--..
I
-.~
-.~o
.2
D ~.o x 1 4 .0 x 10 " 9.0 x 10
I
...
..c. poaIt}On 7./c 7. c
o 9.0 " l~
. .,
-
- ...
R
.
~
~
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-
.1
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r
:1
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to
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.26
-.
.265
I
I
2
-.105
I
I
1
I -1.2
-.8
-.If.
0
-fa.
sectIon 11ft coefficient. FIG.
.8
c~
1.2
1.6
2. o
73. (Cuncluded)
The effect of Reynolds number on permissible surface roughness' is also indicated in Fig. 76, in which a sharp increase of drag at a Reynolds number of approximately 20 million occurs for the model painted with camouflage lacquer. Experiments with models finished with camouflage paint" in
162
THEORY OF WING SECTIONS
dicate that it is possible to obtain a matte surface without causing pre mature transition but that it is extremely difficult to obtain such a surface sufficiently free from specks without sanding. ~
.050
i .* ~:i I .
"', ',,
~i·*
S'" ",I
....
~:.Q20
~> ""
..,Ito...
I
.2 section 11ft coe1'£lclent.
Fro. 89. Flight measurements of transition on a NACA
............
~
c~
f)()"~rics wing
within and outside
the slipstream.
similar conclusion is indicated by X ..\.C..\ flightdata/ (Fig. 89), which show transition as far back as 20 per cent of the chord in the slipstream. Even fewer data are available on the effects of vibration on transition. Tests in the X . .\ CA 8-foot high-speed tunnel 43 showed negligible effects, but the range of frequencies tested may not have been sufficient.ly wide to represent conditions encountered on airplanes. Some K .~(~.~ Hight data" showed small but consistent rearward movements of transition outside the slipstream when the propellers were feathered. This effect was noticed even when the propeller on the opposite side of the two-engined airplane was feathered, and it was accordingly attributed to vibration or noise. In
THEORY OF WING SECTIONS
174
some cases, increases of drag have been noted in wind tunnels when the model or its supports vibrated. A tentative conclusion may be drawn from the meager data that vibration may cause small forward movements of transition on airplane wings, The skin friction associated with a turbulent boundary layer on a smooth surface decreases with increase of Reynolds number, as shown in Chap. 5. This favorable scale effect is not obtained at high Reynolds
,
,
.
e; .IS~ o ..,
~
.s
'\
....
~
~
.. .5
~pVtcl
Cl mas maximum section lift coefficient
~Imax increment of maximum section lift coefficient
em section pitching-moment coefficient, m/~p lltc2
Cm~/, section pitching-moment coefficient about the quarter-chord point
c.1 section pitching-moment coefficient about the quarter-chord point with flap neutral ~ section pitching-moment coefficient about the quarter-chord point with flap deflected ~c". incremental section pitching-moment coefficient about the quarter-chord point Cal section normal-force coefficient with flap neutral Cnt section normal-force coefficient with flap deflected ~a incremental additional normal-force coefficient associated with flap deflection Ca" incremental basic normal-force coefficient associated with flap deflection C-n/a incremental flap normal-force coefficient At:. incremental section normal-force coefficient (cp)J center of pressure of the load on a flap measured from the leading edge of the flap k section flap hinge-moment coefficient (positive in direction of positive flap deflection) 188
HIGH-LIFT DElT/CES
l, m n/ a q r
v ~I'a
x y a
ao &to "Yael' "Yb&
8 8/ a
a/_
~cl p T
Tn. T..
189
lift force acting on the flap per unit span
section pitching moment
incremental flap normal force per unit span
dynamic pressure, ~p VI
leading-edge radius
thickness of wing section local velocity over the surface of a symmetrical section at zero lift increment of local velocity over the surface of a wing section associated with angle of attaek
distance parallel to chord
distance normal to chord
angle of attack
section angle of attack increment. of section angle of zero lift ratios of the flap normal force to the section normal force for the incremental additional and the incremental basic normal forces flap deflection
deflection of fore flap or vane
deflection of main flap of a double-slotted flap configuration
increment of flap deflection
mass density of air turbulence factor, effective Reynolds number /tcst Reynolds number
see Eq. (8.8)
8.2. Introduction. The auxiliary devices discussed in this chapter are essentially movable elements that per mit the pilot to change the geometry and aerodynamic characteristics of the wing sections to control the motion of the airplane or to improve the per formance in some desired -manner. The desire to imp ove performance by increasing the wing loading while maintaining acceptable landing and take-off speeds led to the developmen t of retrae able devices to improve the maximum lift coefficients of wings without changing the characteristics for the cruising and high-speed flight conditions. Some typical high-lift de vices are illustrated in Fig. 95. The aerodynamic characteristics of some typical high-lift devices are presented and discussed in the following sec tions. Primary emphasis is given to the capabilities and relative merits
SPlIT FLAP
C----~:::--=
~
EXTERNAl AIRFOIL RAP
c:
~
DOUBLE· SLOTTED FlAP"
/
~-----:::-::--==LEADING EDGE SlAT FIG.
95. Typical high-liit devices.
THEORY OF WING SECTIONS
190
of the various devices. Numerous references to the literature are given to provide design information.
•8 I
I
I
I
I
.,
~
/~
1
.,.f ~
0 CD Cf-t
ft-t
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as
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CD
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~
/
.~
s::CD
.L
~" ri.7
,-..
~
10-'"
r\ I
BxperlaenU.l
~ ~
~
C
~!J
j~
/rfJ !/ r 0
(a)
.8
range f'rom 0° to 10°.
I I I I I I
.,
4fI#'
~~
-:lheoretlcal
l ~",
-'~
AI ~ ~
,/
/~
.4
IAl
,
"L I
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y
~
'Iii'
\ I
BxperSmental -
-
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itf
/
til
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.JIll'
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,.
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s::
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r-, ,-~
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I
'1heoret1cal
/~ ~~
,
J~
~
o 0V
.1
.2
.,
.0
•
flap-chord ratio, E (b)
6
range from 0 0 to 200
•
Flo.96. Variation of section flap effectiveness with flap chord ratio for true-airfoll-eontour flaps without exposed overhang balance on a number of airfoil sections; gaps sealed; Cz -= o.
8.3. Plain Flaps. Plain trailing-edge flaps are formed by hinging the rearmost part of-the wing section about a point within the contour. Down ward deflections of the trailing edge are called "positive-flap deflections." Deflection of a plain flap with no gap effectively changes the camber of the wing section, and some of the resulting changes of the aerodynamic ehsrae
Sym hoi
-
0
+ X 0 -
o - V
Air-flow characteristics
Type of flap
Basic airfoil
NACA 0009 NACA0015 NACA 23012 NACA 66(2x15)-009
NACA 66-009
NACA-
-
63,4-4(17.8) approx,
~
66 (2xl5)-216, a = 0.6
-
- ~
- b. - ~
17 $)
- JJ - -0 - b
S>
- ~
- -~
.....
~
o
IO "V'll
,..... ~
~ 20 16-- r---
.4
t' .-...,
~
~ ~.
~
.
.....
-
-.
~ l...d.. /2
.8 /.6 Seclion lifl txJeffici~Ct FIG.
I
~
~
..........
-
fd8g.)
~ ~ ......
~ ~ .....
r--c:... ~ ~
-.4
-
.. ---
~
lIo.... ....
65~ ~
-~8
..D..
A
r
5~ rd:::: r---..
-
0
'j'U
1,35~l---... I--
~
-. In
126. (Condtule4)
.... 2.0
-
--::: ~
1 r-i l:.,,;R
-
2.4
,
... ~f>I
2.8
_d
3.2
.,
HIGH-LIFT DEVICES
223
I I ',,..A) N.A.CA. 65., -/18 A!A.aA./4/0~ )
(~
4
FIG.
I l. --N.A.C.A.64-series (f
_
8
20
/2
/6
SecIiaIlhickness lf1fio, ~ lpercenf 127. Effect of thickness ratio and type of wing section on maximum lift with double.
slotted 1laps.
Ftpdud
line
~---------
. T 1 ' 5 c - - - - - - - - -....
(o)AIRFOL WITHRAP
XI
c:: Airfo7chord line~
,,;
r
12·
8~~
FIG.
128. Typical airfoil and flap configuration.
THEORY OF WING SECTIONS
224 3.2
I
I
I
I
Double slottedflop wiIh fore tip .~ r---...... ......~
01t75c
2.8
J
.>---
2.4
~--
r----... ~
~_J.
---,---
~) ----4)
~
J...
~ 2.0
i
Plain wing section
fA
~ 1.6
~
.~
/1
~
..............
11
:~
~
...............
J
1.2 ~--
p-' ......
)-_1
-
."'4
.8
--Smooth ---- Leading edge rough .4
':'2
.3
.4
.5
.6
Position ofmimil1lU71 pteSSIIe, % maximum section lift coefficient with position of
.7
129. Variation of minimum pressure for 80rne NACA tHeries wing aectiODS of 10 per cent tbiclmees and a design lift coefficient of 0.2 .. R, 6 X 10'.. FIG.
FIXED AUXILIARY WING SECTION (AXED SLAT)
~< ~
./ ~
.2
~ PDI
V
?~ V
~
~
/
~
'" V
~
~
V
7 ./
."".,.,.,...,
~~
.c~2 P,
/'
---r-, ~
'-.
~
~
-
~ -..............
v~
r-----.. ~
r----.-
~~
L4
1.8
26
3.4
4.2
M, FlO.
145. Pressure, density, and Mach-number relations for a normal shock.
low Mach numbers. These conditions are approximated for thin wing sections with small amounts of camber at low lift coefficients at speeds well below the speed of sound. The Glauert-Prandtl relation is Cl c
at;
ci,
a;
1 vI-M2
-=-=-==~
(9.30)
where the subscripts c and i denote, respectively, the compressible and incompressible eases. In deriving this formula, Glauert considered all the velocities over the wing section to be increased by the same factor. Equa tion (9.30) may therefore be applied equally well to the moment coefficient. The Glauert-Prandtl rule agrees remarkably well with experimental data, considering the assumptions made in its derivation. A comparison of this rule as applied to the slope of the lift curve for three wing sections of 6, 9, and 12 per cent thickness is given" in Fig. 146. KapianM obtained a first-step improvement of the Glauert-Prandtl rule
EFFECTS OF COMPRESSIBILITY AT ~UBSONIC SPEEDS
257
for the lift of an elliptical cylinder and extended it to arbitrary symmetrical profiles, The Kaplan rule is
~~ = I' + 1 ~ t [1'(1' - 1) + ~('Y + 1)(1'2 - 1YJ where
Il
(9.31)
1 vl-M2
= -_""'/=== ;
t = thickness ratio
.24.-----..-----or----or----..,.--....--....
.201-----t-------t------+------1,........--+----1
.'61-----+-----+------I----I-oI-~---&.--_t
dCt
da, .12....------4-----+-
~-~~-+-_+__-t----_t
.081-----'-------1-----+------;------1
.2
.6
.4
.8
·ID
N FlO. 146. Lift-eurve slope variation with Mach number for the NACA 0012-03. 0009-63. and 0006-63 airfoils.
Equation (9.31) approaches the Glauert-Prandtl rule as the thickness ratio approaches zero. It will also be noted that the Kaplan rule shows a "peak dependence on the ratio of the specific heats 1'. The Kaplan rule for wing sections 5, 10, 15, and 20 per cent thick in air together with the Glauert Prandtl rule are plotted in Fig. 147. b. Effect oj j{ach number on tile Pressure CoejJicicnt. The Glauert Prandtl rule supplies a first approximation to the variation of pressure coefficient with Mach number. C~l
Cp ,
1
=
vI - ]\{2
THBORY OF WING SECTIONS
258
Numerous attempts have been made to obtain more accurate expressions for the variation of pressure with Mach number, notably by Chaplygin,15 Temple and Yarwood, 119 and von Karman and Tsien. 129,14O Garrick and Kaplan" succeeded in unifying these results 88 approximate solutions of
It',l 0.20
~
J5
2.5
rule
./0 1.05
I
I I ~Kaplon
J
I I
1.Pra1dt1-G!aJerf I I
Ir' III
rule
JI
2.0
flI '1 / J
1.5
~
1·°0
-----ae
~ 0.4
0.6
0.8
1.0
M1
FIG. 147. Ratio of lifts for compressible and incompressible fluids as function of stream Mach number.
the general problem and presented two other approximations. No simple general solution of the problem is known. The I{arman-Tsien relation is widely used in the United States. Ex perimental evidence appears to indicate that this relation is as applicable as any of the solutions. This relation is CPM 1
CPt = viI - M2 + [M2/(I + viI - M2)J(C p J 2) (9.32)
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
259
A plot of the pressure coefficient as a function of the free stream Mach num ber AI is given in Fig. 148 for a number of values of the incompressible
/.0
.9
.8
1 .~
i· 7 ~
t~
~
~
.6
~
u~
.5 .4 .3
.2
.
~
./
°0
./
.4 .5 .6 .7 Free-stream Mach 1III11ber,Mo
148. Variation of local preesure coefficient and local Mach number with free stream Mach number according to KArm8.n-Tsien.
FIG.
pressure coefficient. Also shown in Fig. 148 are curves of the local Mach number ML. A special use for relations such as Eq. (9.32) is the prediction of the stream Mach number at which the velocity of sound is reached locally
THEORY OF WING 8ECTIONS
260
over the wing section. Jacobs" applied the Glauert-Prandtl rule to this problem. It is now customary in the United States to use the Karman 3.5
./
.4
,,5
.6
.7
Free -stream MochnumberlMo FIG.
.8
.9
148. (Concluded)
Tsien relation [Eq. (9.32)]. This problem may be solved from Fig. 148 by following the curve for the highest incompressible pressure coefficient on the wing section to the line ML -= 1 and reading the corresponding stream
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
261
Mach number. Figure 148 may be used conveniently when the maximum local pressure coefficient is known experimentally at some subcritical Mach number. If the incompressible pressure coefficient is known theoretically, it is more convenient to use Fig. 149, which gives the corresponding criti cal Mach number directly. 1.0
.9
.8
1
\
\
I
\
II
~
-,
r-,
1 0.691 the pressure rise on the surface below that FIG. 153. Schlieren photograph of separated flow for rear portion corresponding to a normal shock. of NACA 23015 airfoil. XACA The first factor is the pressure gradient rectangular biJ!h-slx-t'd wind through the boundary layer previously dis tunnel. Airfoil ehord, [; inehes. Angle of ut tar-k, (. dt~~rN'~' cussed. The importance of this factor may he inferred from Figs. 152 and 153. The lines in the schlieren photographs correspond to density gradients. Dark Jines are 8ho''''11 in the vicinity of 81:
()
~.4-
~ ~
.s-: M Q810
t1
~ .2
15
~
.708
./
.~
Col)
§
0 240 200
/60
/20
80
40
0
40
80
/20
160
200
240
Distance across KOte.l perceri chord FIG. 154. '''''ake shape and total pressure defect as influenced by Mach number, 0012 airfoil, a = 0 degrees; NACA rectangular high-speed wind tunnel.
~ACl~.
the shock making a small angle with the surface. Although these density gradients may be partly associated with temperature gradients, the sharp ness and intensity of these lines are thought to indicate the presence of appreciable pressure gradients. The second factor is the possible reaction of the boundary layer on the shock. The thickening or separation of the boundary' layer tends to pro duce oblique shocks, Such shocks would produce some pressure rise ahead
THEORY OF WING SECTIONS
268
of the normal shock and would accordingly reduce the intensity of the normal shock, especially close to the surface. The extent to which the normal shock is locally softened in this manner is uncertain, but Figs. 152 and 153 suggest that this mechanism may be of considerable importance. Investigations made by the NACA, by Liepman,· and by Ackeret4 and his associates emphasize the complexity of the interaction of the shock and the boundary layer. These investigations indicate important differences between the shock phenomena for laminar and for turbulent boundary layers. Although exceptions have been noted, laminar boundary layers .4
CD
em -·~I
.2
.3
.4 .5.6 .8 Mach numbtJr,M FIG. 155.. Force and moment coefficient variation with Mach number. NACA 23015 airfoil. a == 0 degrees; NACA rectangular high-speed wind tunnel.
tend to produce multiple shocks of lower intensity, or U lambda-type" shocks, while those associated with turbulent boundary layers tend to resemble intense normal shocks more closely. The understanding of these phenomena awaits further theoretical and experimental investigations. The typical changes of forces experienced at supercritical speeds by a wing section of moderate thickness are shown!" in Fig. 155. The Mach number for force break is seen to be appreciably higher than the critical Mach number. At" Mach numbers higher than that for force break, the lift coefficient decreases, the drag coefficient increases, and the moment coefficient usually increases negatively. 9.6. Experimental Wing Characteristics at High Speeds. The existing three-dimensional lying theory, which is the basis for the concept of section characteristics, is not valid for supercritical speeds where part of the flow is supersonic. Consequently section data cannot be applied quantitatively to the prediction of the characteristics of wings at such speeds. Moreover, present trends for wings designed for efficient operation at supercritical speeds are toward low aspect ratios and large amounts of sweep. As
EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS
269
pointed out previously, section data are of doubtful quantitative signif icance for such plan forms at any speed. Under these circumstances, the detailed consideration of section characteristics as design data is not warranted. Nevertheless, the characteristics of wings at supercritical speeds de pend to some extent on the lying sections employed. The variation of the wing characteristics with wing section appears to be qualitatively similar to the variation of the wing-section characteristics- if the lying plan form is such as to permit any semblance to two-dimensional flow, Consequently, the purpose of this section is to present typical data showing qualitatively the major effects of variation of the wing profile at high subsonic Mach numbers. a. Lift Characteristic«; One of the first and most severe effects of com pressibility encountered in flight was the tendency of many airplanes to "tuck under" in high-speed dives. This U tucking under" tendency con sisted of a large negative shift in the angle of trim tbgether with a large increase of stability that resisted the efforts of the pilot to trim at the desired positive lift coefficients required for recovery from the dive. The resulting large elevator motions required to recover from the dive, or even to prevent the dive from becoming steeper, corresponded to excessive con trol forces, thus leading to the impression that the "~t.ick became frozen." These changes in longitudinal stability limited the maximum safe flying speed for many airplanes. These effects are intimately associated with tile changes in lift charac teristics of wing sections at Mach numbers above that for force break. The relatively thick cambered wing sections commonly used during the Second ,Vorld ar experienced a positive shift of the angle of zero lift and a reduction of lift-curve slope. Although ID3ny effects associated with other characteristics of the airplane are present, the change in angle of zero lift directly affects the angle of trim and the reduction of lift-curve slope directly increases the longitudinal stability. The shift in the angle of zero lift is associated with the camber of the wing section. ..~ symmetrical section shows, of course, no change in the angle of zero lift 116 (Fig. 156). The rather large positive shift in the angle of zero lift shown!" in Fig. 157 for the X.~(~.:\ 2409-34 section is typical for cambered sections without reflex. In this case, the change amounts to about 2 degrees for a change of Mach number from 0.80 to only 0.83. The effect of increasing the camber is showu'" by Fig. 158. Doubling the amount of camber causes a similar shift to occur at a lower Mach number. The effects of thickness on the lift characteristics of"symmetrical sec tions are shown" in Figs. 159 to 161 for the N..~CA 0006-34, 0008-34, and 0012-34 sections. The 6 per cent thick section shows a large increase of lift coefficient at a given angle of attack with increasing Mach number at
"r
THEORY OF WING SECTIONS
270
values below about 0.85. At Mach numbers above the force break, the lift coefficient decreases rapidly, but it never goes below the low-speed values, at least at Mach numbers up to 0.95. The 12 per cent thick section, however, fails to show much increase of lift coefficient at subcritical Mach numbers, but it shows a large decrease of lift coefficient at Mach numbers just above the force break. At a Mach number of 0.85, the lift coefficient,
c::
----~ -====
I/.
4 ,
~
....-:"'~ ~
~~
~
~~
~~
, t
.06
,,
\
-.
,
I
.... ... f~-~ f'... .. ~---~
---- , ,, -.2
I
,
,,
f--
--
o
~
:~
,
,,
,,
,
,,
,/
l,/
,
l
"
M= ~
.40-.60-.70--.80 ---- +-.83 ---,-- t
r
V
V
~
_......... :,...-' ~ ,~......... ~
-- --.2
I I
.- ---
- r-
~
i'-- t - - .
-
~ """"'-
~~
~
1.0
.8
(J
y
x
II
(V/V)2
»rv
0 0.978 1.179 1.490 2;035
0 0.750 0.885 1.020 1.129
0 0.866 0.941 1.010 1.063
5.0 7.5 10 15 20
2.810 3.394 3.871 4.620 5.173
1.204 1.240 1.264 1.296 1.320
1.097 1.114 1.124 1.139 1.149
0.685 0.559 0.482 0.388 0.328
25 30 35 40 45
5.576 5.844 5.978 5..981 5.798
1.338 1.351 1.362 1.372 1.335
1.156 1.162 1.167 1.171 1.156
0.281 0.247 0.221 0.199 0.177
50 55 60
5.480 5.056 4.548 3.974 3.350
1.289 1.243 1.195 1.144 1.091
1.136 1.115 1.093 1.070 1.044
0.158 0.138 0.122 0.103 0.088
2.695 2.029 1.382 0.786 0.288
1.037 0.981 0.928 0.874 0.825
1.018 0.990 0.963 0.935 0.908
0.074 0.063 0.052 0.045 0.028
0
0.775
0.880
(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5
65 70 75 80
85 90 95 100
L.E. radius: 1.040 per ce~t c NACA 041-Oi2 Buic Thickness Form
I
~Vtl./V
2.379 1.663 1.508 1.271 0.943
II I
0' .--
~
~"
APPENDIX I
351
.22 LtlWt!r ."(Dce
.2
.4
.6
xjc
.8
1.0
I
X
Y
1---_
(v/l'F
I
tJ/V -----I-----!-----:
(per cent c) (per cent c)
~o/l' I
o
o
o
o
0.5 0.75
1.208 1.456 1.842 2.528
0.670 0.762 0.896 1.113
0.819 0.873 0.947 IJJ55
1.231
1.109
1.284
15
3.504 4.240 4.842 5.785
20
6.480
1.323 1.375 1.410
1.133 1.150 1.172 1.187
0.670 0.559 0.482 0..389 0.326
25
6.985 7.319 7.482 7.473 7.224
1.454 1.470 1.485 1.426
1.198 1.206 1.213 1.218 1.195
0.285 0.250 0.225 0.202 0.179
1.25
2.5 5.0 7.5 10
30
35 40
45 50
70
6.810 6.266 5.620 4.895 4.113
75 80
3.296 2.472
55 60 65
85
1.677
90 95
0.950 0.346
1.434
1.365 1.300
1.233 1.167 1.101
1.033 0.967 0.902 0.841 0.785
1.168 1.140 1.110 1.080 1.049
1.016 0.983 0.950 0.917 0.886
1.939 1.476 1.354 1.188 0.916
i
I
I
I i I i
I "
0.158 0.135 0.121 0.105 0.090 0.078 0.065 0.054 0.041 0.031
100 o 0.730 o -----~-----_...:..-----_I~~._I_ -----1 L.E. radius: 1.590 per cent c
NACA 64 r015 Basic Thickness Form
THEORY OF WING SECTIONS
352
()
.4
y x (per cent c) (per cent c)
0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20
25 30 35
40 45 50 55
60 65
70 75 80 85
90
95 100
0 1.428 1.720 2.177 3.005 4.186 5.076 5.803 6.942 7.782
8.391 8.789 8.979 8.952 8.630
(V/V)2 0
0.54.6 0.705 0.862 1.079
I
I
1.244 1.327 1.380
1.450 1.497 1.535 1.562 1.585
vii'
1.518
J1fJ./l'
0.739 0.840 0.920 1.039
1.646 1.360 1.269 1.128 0.904
1.115 1.152 1.175 1.204 1.224
0.669 0.558 0.486 0.391 0.331
1.239 1.259 1.265 1.232
0.288 0.255 0.228 0.200 0.177
0
1.250
1.600
1.0
.8
.6
Z/c
8.114 7.445 6.658 5.782 4.842
1.354 1.272 1.190 1.109
1.198 1.164 1.128 1.091 1.053
0.154 0.134 0.117 0.102 0.088
3.866 2.888 1.951 1.101 0.400
1.028 0.952 0.879 0.812 0.747
1.014 0.976 0.937 0.901 0.864
0.074 0.063 0.051 0.039 0.027
0
0.695
0.834
0
1.436
i
L.E. radius: 2.208 per cent c
NAC.A 64r018 Basic Thickness Form
APPENDIX I
353
(~)z
.4
-I.
1.0
.8
.6
%,C
I I
(per ::nt C) (per
0 0.5 0.75 1.25
2.5 5.0 7.5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
~nt C)
I
()
j
1.646
I
\
{V!V)2_
3.485
0 0.462 C.603 0.759 1.010
4.871 5.915 6.769 8.108 9.095
1.248 1.358 1.431 1.527 1.593
9.807 10.269 10.481 10.431 10.030
1.654
I
I
1.985
!
2.517
I
t
I
1.681 1.712 1.709 1.607
9.404 8.607 7.678 6.649 5.549 4.416 3.287 2.213 1.245 0.449 0
1.117 1.165 1.196 1.236 1.281 1.297 1.308 1.307 1.268 1.228
0.932 0.851 0.778 0.711
1.010 0.965 0.923 0.882 0.844
0.653
0.808
1.020
..
1.458 ;
1.274 1.203
1.084 0.878 I i
1.262
1.186 1.143 1.099 1.055
1.112
I I
0.680 0.776 0.871 1.005
~./V
_ _-
0
1.507 1.307
!
----
1.406 1.209
I
v/lf"
I
I
I
1
I
I
i
!
1I
II
I
I
I
I
L.E. radius: 2.884 per cent c
NACA 64.-021 Basic Thickness Form
0.665 0.557 0.486 0.395 0.335 0.293 0.259 0.232 0.202 0.178 0.155 0.134 0.116 0.099 0.084 0.071 0.059 0.047 0.036 0.022
0
THEORY OF WING SECTIONS
354 0
l2
(vt
..-I P:-rJf
II
/ "I =.02 Upper surface
H--
~02 Lowersurface
...............
.8
...........
~
NACA 64AOO6
.--~
o
.2
y x (per cent c) (per cent c)
.4
lO
.6
x/c
(V/V)2
v/V
t.Vo/V
0 0.485 0.585 0.739 1.016
0 1.019 1.046 1.076 1.106
0 1.009 1.023
4.688 2.101 1.798
1.037 1.052
0.980
5.0 7.5 10 15 20
1.399 1.684 1.919 2.283 2.557
1.118 1.126 1.132 1.141 1.149
1.057 1.061 1.064 1.072
0.694 0.564 0.482 0.382 0.321
25
2.757 2.896 2.977 2.999 2.945
1.154 1.158 1.162 1.165 1.156
1.074 1.076 1.078 1.079 1.075
0.278 0.246 0.219 0.197 0.177
70
2.825 2.653 2.438 2.188 1.907
1.142 1.125 1.107 1.087 1.066
1.069 1.061 1.052 1.043 1.032
0.159 0.143 0.126 0.112 0.099
75 80 85 90 95
1.602 1.285 0.967 0.649 0.331
1.043 1.018 0.992 0.964 0.935
1.021 1.009 0.996 0.982 0.967
0.087 0.074 0.061 0.047 0.033
100
0.013
0
0
0
0 0.5 0.75 1.25 2.5
30 35
40 45 50 55 60 65
1.068
L.E. radius; 0.246 per cent c T.E. radius: 0.014 per cent c NACA 64A.OO6 Basic Thickness Form
1.422
355
APPENDIX I / Cl
= .046 Upper surface
/
~ l2
(vt K J
I
~r--
~
r---. ~ ...
'.tJ4.6 Lower sarfoce
.............
.8
~
NACA 64,4008
.4 ~
,........
-.-.,
r- ..........
o
.2
.6
.8
LO
xle x
y
(per cent c) (per cent c)
0 0.5 0.75 1.25
0 0.646 0.778 0.983
2.5
1.353
(V/V)2
v/V
lw./V
0 0.947 1.005 1.068 1.122
0 0.973 1.033 1.059
3.546 1.972 1.697 1.352 0.971 0.692 0.564 0.481 0.382 0.323
1.002
5.0
1.863
7.5
2.245 2.559 3.047 3.414
1.151 1.165 1.176 1.191 1.201
1.073 1..079 1.084 1.091 1.096
3.681 3.866 3.972 3.998 3.921
1.2\>9 1.217 1.221 1.225 1.211
1.100 1.103 1.105 1.107 1.100
3.757 3.524·
1.191 1.167 1.141 1.113 1.084
1.091 1.080 1.068 1.055 1.041
0.158 0.141 0.125 0.111 0.098
1.026
1.010 0.993 0.975 0.956
0.084 0.072 0.059 0.045 0.032
0
0
10 15 20
25 30 35 40 45 50
55 60 65
3.234
70
2.897 2.521
75
2.117
80 85
1.698
90 95
1.278 0.858 0.438
1.053 1.020 0.987 0.951 0.914
100
0.018
0
L.E. radius: 0.439 percent c T.E. radius: 0.020 per cent c NACA 64AOOS Basic Thickness Form
I I
I
0.279 0.247 0.221 0.198 0.177
THEORY OF WING SECTIONS
356
L6 /C,
0,
= .08 Upper surface
Ii
12
r t. ~ '-::=:
t;(
r
-
~loo...-
~
"""'iiillIII
~
~~
~
t', .08 Lower 6urface
~
NACA 64AOIO
A
--
~-
r---- I--....
o
~
.6
.4
t - - - ..... ~
....-
.B
LO
x/c %
11
(V/V)2
vtV
&J./V
0 0.804 0.969 1.225 1.688
0 0.868 0.952 1.042 1.130
0 0.932 0.976 1.021 1.063
2.868 1.845 1.603 1.300 0.957
2.327 2.905 3.199 3.813 4.272
1.178 1.201 1.217 1.238 1.254
1.085 1.096 1.103 1.113 1.120
0.280 0.248 0.221 0.199 0.177
(per cent c) (per cent c)
0 0.5 0.75 1.25 2.5
5.0 7.5 10 15 20
I I
II
0.688 0.562 0.480 0.382 0.324
25
4.606
1.266
30
4.&17
25
4.968 4.995 4.894
1.275 1.282 1.268
1.125 1.129 1.132 1.135 1.126
4.684 4.388 4.021 3.597 3.127
1.240 1.208 1.174 1.139 1.102
1.114 1.099 1.084 1.067 1.050
0.158 0.140 0.124 0.109 0.096
1.063
95
2.623 2.103 1.582 1.062 0.541
0.981 0.938 0.893
1.031 1.011 0.990 0.969 0.945
0.083 0.070 0.058 0.044 0.031
100
0.021
0
0
40
45 50 55 60 65
70 75 80
85 90
1.288
1.023
L.E. radius: 0.687 per cent c T .E. radius: O'()?-3 per cent c
I
II
0
APPENDIX I
l6
,
0
#
357
/ c1 =./3 Upper surFQ(Y
- ~........ ~ ~~~ ~~~ ~
1.2.
L
{~
(vt V .8
~
........
~
"J~ lower surFace
~~
~
I
~
NACA 64,AO/2
~
-r--- !---
I'"""""""
f'-.-. ~
o
.2
.4
.6
x/c z
I
r--. .-- ........
L.- ~
LO
,._ i
l7/ v
y
(V/V)2
I
0 0.792 0.893
1.25
0 0.961 1.158 1.464
1.006
2.5
I
2.018
1.127
l.Oti2
0.941
5.0 7.5 10 15
1.201
1.000
1.235
1.111
1.257
1.121
20
2.788 3.364 3.839 4.580 5.132
25 30
5.534 5.809
35 40
5.965 5.993
45
5.863
50 55 60
5.605
(per cent c) (per cent c) 0
0.5 0.75
65
70 75
80 85
5.244 4.801 4.289 3.721
I
1.135
1.144
1.324
1.151 1.156 1.160 1.164 1.152
0.281 0.249 0.221 0.199 0.177
1.135 1.118
1.099 1.079 1.057
0.157 0.139 0.123 0.108 0.094
0.873
1.035 1.011 O.98i 0.962 0.934
0.080 0.068 0.056 0.042 0.029
0
0
0
1.34fi
1.289 1.250 1.207 1.164 1.118
0.974
1.263
0.925
0.644
100
0.025
I.U71 1.023
i
1.254
1.308
3.118 2.500 1.882
90 95
!
2.408 1.720 1.515
1.2..~
1.354 1.326
I
0
0.890 0.945 1.003
0.681 0.560 0.478 0.383 0.325
1.336
I
i
~l'CI/V
L.E. radius: 0.994 per cent c "f.E. radius: 0.028 per eent c
NAC.A.. 64 1A012 Basic Thickness Form
THEORY OF U'ING SECTIONS
358
,
L6
/c, = .2/ Upper surface
.-- r----- ~~ ~ ~
,0
l2
/
~
If J K,
(vt '/ .8
~~
~ ~
~
I
~~
~
".2/ lower surface
r-,
NACA 64-2 A O/5
I
.4
- r--- r---
~
r '-- r--- -
o
.2
- .6
~~
.4
-------
,......
~
LO
.8
x/c x 11 (per cent c) (per cent c)
(VJV)2
vJV
.:1t·a / l'
0.5 0.75 1.25 2.5
0 1.193 1.436 1.815 2.508
0 0.678 0.789 0.936 1.110
0 0.823 0.888 0.967 1.054
1.956 1.552 1.404 1.189 0.912
5.0 7.5
3.477 4.202
1.226 1.280
4.799 5.732 6.423
1.314 1.390
1.107 1.131 1.146 1.166 1.179
0.671 0.552 0.478 0.384 0.326
40 45
6.926 1.270 7.463 7.481 7.313
1.413 1.430 1.445 1.458 1.414
1.189 1.196 1.202 1.207 1.189
0.283 0.249 0.222 0.201 0.177
50 55 60 65 70
6.978 6.517 5.956 5.311 4.600
1.364 1.311 1.255 1.198 1.139
1.168 1.145 1.120 1.095 1.067
0.156 0.137 0.121 0.106 0.091
75 80
1.079 1.020 0.961 0.901 0.843
1.039 1.010 0.980 0.949 0.918
0.078 0.065 0.053 0.041
95
3.847 3.084 2.321 1.558 0.795
100
0.032
0
0
0
0
10
15 20 25
30 35
85 90
.1.360
LE. radius: 1.561 per cent c T.E. radius: 0.037 per cent c NACA 64tA015 Basic Thickness Fonn
0.027
APPENDIX I ~,Cls.OJ
.I.e 10
359 I
I .............
~r:L
r--:Ol Lower""'or:e
.8
I
I
Lt;per surfoce
r----.. r-----...
-.......
r-.
I NACA 65-006
.4
--
~
D
.4
x
y
5.0 7.5 10
15 20 25 30 35 40 45
50 55 60
65 70
75 80 85
90 95
100
/.0
.8
(V/V)2
v/V
~Va/V
0 0.476 0.574 0.717 0.956
0 1.044 1.055 1.063 1.081
0 1.022 1.027 1.031 1.040
4.815 2.110 1.780 1.390 0.965
1.310 1.589 1.824 2.197 2.482
1.100 1.112 1.120 1.134 1.143
1.049 1.055 1.058 1.065 1.069
0.695 0.560 0.474 0.381 0.322
2.697 2.852 2.952 2.998 2.983
1.149 1.155 1.159 1.163 1.166
1.072 1.075 1.077 1.078 1.080
0.281 0.247 0.220 0..198 0.178
2.900 2.741 2.518 2.246 1.935
1.165 1.145 1.124 1.100 1.073
1.079 1.070 1.060 1.049 1.036
0.160 0.144 0.128 0.114 0.100
1.594 1.233 0.865 0.510 0.195
1.044 1.013 0.981 0.944 0.902
1.022 1.006 0.990 0.972 0.950
0.086 0.074 0.060 0.046 0.031
0
0.858
0.926
0
(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5
.6
L.E. radius: 0.240 per cent
c
NACA 65-006 Basic Thickness Form
f'HEORY OF WING SECTIONS
360
~ 1
,
1/' -.001 lJp~r sur'a~
0 1.2
~
I I """'ilIlIiiii
~IH LtItIWr ~1ICtI
~
"'" r-,
.1 NACA 85-008
·4 ~
~
-
r---........
.....
~
a
.4
x 11 (per cent c) (per cent c)
Z/c
1.0
.8
.11
(V/V)2
V/V
Avo/V
0 0.627 0.756 0.945 1.267
0 0.978 1.010 1.043 1.086
0 0.989 1.005 1.021 1.042
3.695 2.010 1.696 1.340 0.956
5.0 7.5 10 15 20
1.745 2.118 2.432 2.931 3.312
1.125 1.145 1.158 1.178 1.192
1.061 1.070 1.076 1.085 1.092
0.689 0.560 0.477 0.382 0.323
25 30 35 40 45
3.599 3.805 3.938 3.998 3.974
1.203 1.210 1.217 1.222 1.226
1.097 1.100 1.103 1.105 1.107
0.281 0.248 0.221 0.199 0.178
50 55 60 65 70
3.857 3.638 3.337 2.971 2.553
1.222 1.193 1.163 1.130 1.094
1.105 1.092 1.078 1.063 1.046
0.160 0.145 0.128 0.113 0.098
75 80 85
2.096 1.617 1.131 0.664 0.252
1.055 1.014 0.971
0.873
1.027 1.007 0.985 0.961 0.934
0.084 0.072 0.059 0.044 0.031
0
0.817
0.904
0
0 0.5 0.75 1.25 2.5
90
95 100
(i923
L.E. radius: 0.434 per cent c
NACA 65-008 Basic Thickness Form
APPENDIX I I.G
I /e I -.IM
1.2
361 r I I tP/HIr ",'act1
f
C\ ,...-r
~
~~
I'-
~.D6
I -Low.- .ur'ac.
f'
V
~
.8 /MeA
ts5-009
~
r----
I000o......
o
L---
--
r--- I....
-
~
1.0
.8
.2
x y (per cent c) (per cent c)
"' r-,
(vlV)2
vjV
I1v.IV
0 0.700 0.845 1.058 1.421
0 0.945 0.985 1.037 1.089
0 0.972 0.992 1.018 1.044
3.270 1.962 1.655 1.315 0.950
5.0 7.5 10 15 20
1.961 2.383 2.736 3.299 3.727
1.134 1.159 1.177 1.200 1.216
1.065 1.077 1.085 1.095 1.103
0.687 0.660 0.477 0.382 0..323
25 30 35 40
4.050 4.282 4.431 4.496 4.469
1.229 1.238 1.246 1.252 1.258
1.109 1.113 1.116 1.119 1.122
0.280 0.248 0.220 0.198 0.178
4.336 4.086 3.743 3.328 2.856
1.250 1.220 1.185 1.145 1.103
1.118 1.105 1.089 1.070 1.050
0.160 0.144 0.128 0.111 0.097
2.342 1.805 1.260 0.738 0.280
1.059 1.013 0.963 0.912 0.856
1.029 1.006
0.981 0.955 0.925
0.084 0.071 0.059 0.044 0.030
0
0.797
0.893
0 0.5 0.75 1.25 2.5
45
50 55 60 65 70 75
80 85 90
ss 100
L.E. radius: 0.552 per cent c
NACA 65-009 Basic Thickness Form
I
0
THEORY OF WING SECTIONS
362 1.4
1.2
.8
I
-- -- (/
)\
r
t
--::: :::-
~ \08
I
J
I
/ ', •.08 IJpptIr 6Ur'ace
,
~
~
LDWf!r
Sur~DC4P
~
""'--,
NACA 65-010
,--.............
- r--
.z
(J
y
x
.4
--
~
.s
I"'--- ~~
.8
/.
(V/V)2
v/V
l:.VA/V
2.5
0 0.772 0.932 1.169 1.574
0 0.911 0.960 1.025 1.085
0 0.954 0.980 1.012 1.042
2.967 1.911 1.614 1.292 0.932
5.0 7.5 10 15 20
2.177 2.647 3.040 3.666 4.143
1.143 1.177 1.197 1.224 1.242
1.069 1.085 1.094 1.106 1.114
0.679 0.558 0.480 0.383 0.321
25
4.503 4.760 4.924 4.996 4.963
1.257 1.268 1.277 1.284 1.290
1.121 1.126 1.130 1.133 1.136
0.280 0.248 0.222 0.199 0.179
70
4.812 4.530 4.146 3.682 3.156
1.284 1.244 1.202 1.158 1.112
1.133 1.115 1.096 1.076 1.055
0.160 0.141 0..126 0.110 0.097
75 80 85 90 95
2.584 1.987 1.385 0.810 0.306
1.062 1.011 0.958 0.903 0.844
1.031 1.005 0.979 0.950 0.919
0.082 0.070 0.058 0.045 0.030
0
0.781
0.884
0
(per cent c) (per cent c) 0 0.5 0.75 1.25
30 35 40 45 50 55 60 65
100
L.E. radius: 0.687 per cent c NACA 66-010 Basic Thickness Form
APPENDIX I
It---+--t-----i~
363
NACA 6~-OI2
.41----+---+-~f--__+_-_+_____f-__+_-_+___t-"'"""1
o
.2"
x
y
.4
1.0
.8
.6
X/C
(t,/lr)~
v/1T
&toll"
0 0.923 1.109 1.387 1.875
0 0.848 0.935 1.000 1.082
0 0.921 0.967 ].000 1.040
2.444 1.776 1.465 1.200 0.931
5.0 7.5 10 15 20
2.606 3.172 3.647 4.402 4.975
1.162 1.201 1.232 1.268 1.295
25 30 35 40
5.406
1.316
5.;16
1.332
5.912 5.997 5.949
1.343 1.350 1.357
(per cent c) (per cent c)
0 0.5 0.75 1.25 2.5
45 50 55
iI
I
1.078 1.096 1.110 1.126 1.1.38
1.147 1.154 1.159 1.162 1.165
0.282 0.251 0.223 0.204 0.188
1.159 1.138 1.115
60 65
70
3.743
1.295 1.243 1.188 1.134
75 80 85 90 95
3.059 2.345 1.630 0.947 0.356
1.073 1.010 0.949 0.884 0.819
1.036 1.005 0.974 0.940 0.905
0
0.748
0.865
100
I
0.702 0.568 0.480 0.389 0.326
5.757 5.412 4.943 4.381
1.343
1
1.065
0.169 0.145 0.127 0.111 0.094 0.074 0.062 0.049 0.038 0.025
1.090
L.E. radius: 1.000 per cent c
NACA. 65 1-012 Basic Thickness Form
I
0
THEORY OF WING SECTIONS
364
c, •. 22 ~ surl'oce
o
.2
x
'Y
8
6
1.0
(V/V)2
V/V
Av./l
0 1.124 1.356 1.702 2.324
0 0.654 0.817 0.939 1.063
0 0.809 0.904 0.969 1.031
2.038 1.729 1.390 1.156 0.920
5.0 7.5 10 15 20
3.245 3.959 4.555 5.504 6.223
1.184 1.241 1.281 1.336 1.374
1.088 1.114 1.132 1.156 1.172
0.682 0.563 0.487 0.393 0.334
25
6.764 7.152 7.396 7.498 7.427
1.397 1.4:18 1.438 1.452 1.464
1.182 1.191 1.199 1.205 1.210
0.290 0.255 0.227 0.203 0.184
7.168 6.720 6.118 5.403 4.600
1.433 1.369 1.297 1.228 1.151
1.197 1.170 1.139 1.108 1.073
0.160 0.143 0.127 0.109 0.096
3.744 2.858 1.977 1.144 0.428
1.077 1.002 0.924 0.846 0.773
1.038 1.001 0.961 0.920 0.879
0.078 0.068 0.052 0.038 0.026
0
0.697
0.835
0
(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5
30 35
40 45
50 55 60
65 70 75 80 85
90 95
100
L.E. radius: 1.505 per cent c X ACA. 65r015 Basic Thickness Form
Y
APPENDIX I
(J
.4
x
JI
(per cent c) (per cent c) 0 0.5 0.75 1.25 2.5
0
1.337 1.608 2.014 2.751
365
.s
z/c
14
(v/V)~
v/1T
j.r:./V
0 0.625 0.702 0.817 1..020
0 0.i91 0.838 0.. 904 1..010
1.746 1.437 1.302
1.123 0.858
5.0 7.5 10 15 20
3..866
4.733 5.457 6.606 7.476
1.192 1.275 1.329 1.402 1.452
1..092 1.129 1.153 1.184 1.205
0.650 0.542 0.474
25
1.488 1.515 1.539 1.. 561 1.578
1.220
45
8.129 8.595 8.886 8.999 8.901
0.285 0.251 0.225 0.203 0.182
50
8..568
55
1.526 1.440 1.353 1.262
1.235 1.200
65 70
8.008 7.267 6.395 5.426
75
4.396
80
30 35 40
60
85 90
95
100
1.170
0.. 385 0.327
1.231 1:241 1.249 1.256
0.157 0.137 0.118 0.104
1.163 1.123 1.082
3.338 2.295 1.319 0.490
1.076 0.985 0.896 0.813 0.730
1.037 0.992
0
0.657
0.811
0.087 0.074 0.062 0.050 0.039 0.026
0.947 0.902 0.854
L.E. radius: 1.96 per cent c
NACA 65r 0 18 Basic Thickness Form
I
0
THEORY OF ,"lING SECTIONS
366 2.0
1.2
.8
»:
'I I ...--- IV ~
x
I
----
[\ /~ ~
~ r--"\
-: or / v / / \.J/4 I IV
a
I
I
\ . / 1 :1 -.-I/. ~ svr'oce
I 1.6
I
I
r:
--"-
661 -0 15
r--- to---
1----
t---
o
~
.2
I
.4
z c) (per cent Y c) (per cent
i
X/C
.6
----
~
r--- "--~
~
1.0
.8
I
("'IF):
to/I'
0 0.760 0.840 0.929 1.055
0
2.139
0.872 0.916 0.9641.027
1.652
j
..1"./1'
0 0.5 0.75 1.25 2.5
0 1.122 1.343 1.675 2.235
~.O
7.5 10 15
3.100 3.781 4.358 5.286
20
~.995
1.288 1.317
1.078 1.099 1.114 1.134 1.148
25
6.543 6.956 7.250 7.430 7.495
1.340 1.356 1.370 1.380 1.391
1.158 1.164 1.170 1.175 1.179
7..450 7.283 6.959 6.372 5.576
1.401 1.411 1.420 1.367 1.260
1.184 1.188 1.192 1.169 1.122
4.632 3.598 2.530 1..489 0.566
1.156 1.053 0.949 0.841 0.744
1.075 0.974 0.920 0.863
0.080 0.065 0.051 0.039 0.025
0.639
0.799
0
30 35 40
45 50
55 60
65 70 75 80 85
90 95
100
1.163 1.208
1.242
0 ~~---_
... -_._......-- ..... _-
1.026
L.E. radius: 1.435 per cent c
N ACA 66::-015 Basic Thickness Form
j I
I I I
I
!
t
I
i
1.431 1.172 0.895
0.663 0.047 0.473 0.381 0.322 0.280 0.248 0.222 0..200 0.180
0.163 0.146 0.131 0.113 0.096
7.'HBORY OF WINO .SECTIONS
378
% 11 (per oeot c) (per cent c)
(fJ!V)I
fJ/V
MJ./v
0 0.806 0.857
1.773 1.456 1.312 1.121
0 0.5 0.15 1.25
0 1.323 1.571 1.952
2.5
2.646
1.0 7.5 10
3.690 4:.613 5.210
16
20
8.333 7.las
1.393
1.134: 1.182 1.1SO
25 30 3& 40 4&
7.848 8.348 8.101 8.918 8.998
1.423
1.193
1.4:45
1.202
1.484 1.481
1.210 1.217 1.223
SO IS 60
8.942
8.733 8.323
65 70
7.580 6.197
1.S09 1.522 1.534 1.438 1.302
75 80
5.4:51
8S
90 95 100
0 0.650
0.735 0..850 1.005 1.114 1.234 1.285 1.350
1.496
0.897 1.002 1.074
1.111
1.228 1.234
1.238 LI99 1.141
0..858
0.649 0.645 0.472 0.381 0.323 0.282 0.250 0.223 0.201 0.181 0.163 0.147 0.131 0.114 0.095
0.848
1.172 1..046 0.922 0.803 0.692
0.896 0.832
0.077 0.061 0.048 0.037 0..022
0
0.S87
0.766
0
4.206 2.934 1.714
1.083 1.022 0.9S0
L.E. radius: 1.955 per cent c
NACA 66r018 Basic Thickness Form
APPENDIX I
.4 z JI (per cent c) (per cent c) 0
0
0.5 0.75 1.25 2.5
1.525 1.804
5.0 7.5 10 15 20 25
30 35 40 45
SO 55 60 65
70
75
2.240 3.045
4.269
Z/c
(V/V)J
0 0.580 0.635 0.755 0.952
0 0.~1
0.797 0.869 0.976
&1.1 V 1.547 1.314 1.218 1.054
0.828
6.052 7.369 8.376
1.148 1.185 1.208
9.153 9.738 10.1M 10.407 10.500
1.499 1.528 1.551 1.574 1.594
1.224 1.236 1.245 1.255 1.263
.0.251 0.224 0.202 0.183
1.611 1.629 1.648
1.269 1.276 1.284
10.434 10.186 9.692 8.793 7.610 6.251 4.796 3.324
1.069 1.116
0.283
1.508
1.228
1.335
1.155
0.165 0.148 0.132 0.114 0.093
1.084
0.073
1.015 0.944 0.873 0.805
0.058 0.046 0.034 0.020
0.734
0
90
1.924
0.717 0
0.539
100
_IV
0.635 0.542 0.472 0.381 0.324
95
85
1.0
.•
1.143 1.2-&6 1.318 1.405 1.459
5.233
1.176 1.031 0.891 0.763 0.648
80
379
L.E. radius: 2.550 per cent c NACA 66r021 Basic Thickness Form
THEORY OF WING
380
~ECTIONS
2.D I
/e, • .J2 1.6
1.2
(v)' 8
(/ e
t.
/ ~
~
I---~
t=-
~
-
~
~
~
,~
/MeA
t->
I
svrFac»
i"'"-./2 LorNtN'" .,ftl~
r/ I
.4
o
~
I Uppttr
----- -- .z
x
'!J
(per cent c) (per cent c)
-,
67,1-015
r---~
., x/c' (V/V)2
-----.. r--..... ~
~
/.0
.8
6
»rv
~G/V I
0 0.5 0.75 1.25
2.5 5.0 7.5 10 15 ~ 25 30
35 40 45 50
55 60
65 70 75 80 85
90 95
100
I
0 1.167 1.394 1.764 2.395
0 0.650 0.970 1.059
0 0.806 0.985 1.029
1.140
1.068
2.042 1.560 1.370 1.152 0.906
3.245 3.900 4.433 5.283 5.940
1.209
1.100 1.113 1.122 1.134 1.142
0.667 0.548 0.470 0.370 0.312
6.404 6.854 7.155 7.359 7.475
1.318 1.330 1.341 1.851 1.360
1.148 1.153 1.158 1.162 1.166
0.276 0.248 0.221 0.201 0.180
7.497 7.421 7.231 6.905 6.402
1.368 1.375 1.381 1.388 1.390
1.170 1.173 1.175 1.178 1.179
0.160 0.142 0.124 0.111 0.108
5.621 4.540 3.327 2.021 0.788
1.321 1.176 1.018 0.864 0.712
1.149 1.084 0.930 0.844
0.094 0..071 0.060 0..045 0.025
0
0.570
0.755
0
1.239 1.259 1.285 1.304
1.009
L.E.. radius: 1.523 per cent c NACA 67,1-015 Basic Thickness Form
II
APPENDIX I 2.0
381
r--....,.-...,...-.....--r--....... .......
-....-~
~l
............_ ....
I aurtlce =.22 Upper
,r-oIo---+--I-L--- ......K~
•8
~
II '(
.. . . .
I .IJ--;---t----ir--f--+---+--+--J----+_-I
----I--
'-........r-- r----I-_.... _ _I---,- .2
.4
xlc. 6
~
.8
1.0
j
%
11
i
(per cent c) (per cent r) i - - - - - -.._ - ~_._.---
(I'll')"! --..
-"--~"---
~t~CI/l!
1,'/1"
- _.._
..
---
---
()
0
()
0.5 0.75
1.199
1.25
1.801 2.462
0.660 0.799 0.942 1.100
0 0.812 0.894 0.971 1.049
2.028 1.680 1.560 1.325 0.990
6.384
1.201 1.259 1.295 1.339 1.369
1.096 1.122 1.138 1.156 1.170
0.695 0.551 0.465 0.383 0.324
6.898 7.253 7.454 7.494 7.316
1.390 1.409 1.42"& 1.435 1.391
1.179 1.187 1.193 1.198 1.179
0.283 0.252 0.224 0.199 0.176
7.003 6.584 6.064 5.449 4.738
1.348 1.306 1.265 1.221 1.178
1.161 1.143 1.125 1.105 1.085
0.156 0.138 0.122 0.108
3.921 3.020 2.086 1.193 0.443
1.115 1.027 0.938 0.852 0.774
1.056 1.013 0.969 0.923 0.880
0
0.703
0.838
1.435
2.5 5.0 7.5 10 15 20 25 30 35 40
3.419 4.143
4.743
5.684
I I I
45 50
55 60 65 70 75 80 85 90
95
100
I
i
! i
I t
I
I I
---_._-
I
I
0.093 0.079 0.065 0.052
O.MO
I
I
L.E. radius: 1.544 per cent. c
NACA 747A015 Basic Thickness Form
0.028 0.018
APPENDIX II MEAN LINES CONTENTS
NACA Designation
62
63 . . . . .
Page
383
384
64
65
66
385
386
387
388
389
67
210 .
220 .
390
230 .
240 250 .
a91 .
392
393
394:
4==0
4
== 0.1
4
== 0.2
a == 0.3 a -= 0.4 a::s= 0.5 4 == 0.6
a == 0.7
a == 0.8 . . . . . . ....
a := 0.8 (modified for use with N ACA 6A-eeries sections)
a -= 0.9 a -= 1.0 .
382
395
396
397
398
399
400
401
4
l
r
v
J
.... ~~
,j
1/
iJ II'
-.4
• rc
~J
'K
il:l
fl 1~1I\
~~
"I
~
iJ
~
-.8
'\II~J
I.J~
11
. . . rv low.
e
_
1-7)
,
~1
-ce
Ir4
."
l \.
, \..
-1.6
.-.5
-2.IJ. -~
-18
-8
0
8
/6
Section ongIe of Q/~ . . deg NACA 0009 Wing Section
24
455
APPENDIX IV .IJ3,
.2 ~~
..-
~
..•
--.
-,
I
\
'\
.e
.8
.IJ
.4
.D3I
/
" -_.
"'.
1.0 .--
*
.024
II
~
\ \
CI· oeo .......
.§
~ .016
.....
•e
,
In
iQ~
~.OI2
~
t
~
,..
\
\
\.
•
"\
\. ~,
\
en
~
-
..
...
II ~
( lfS V jV / ¥
d
,.,. t>
...... ~ r-~
l.)
J
I~ ~ V
........ ..Mol!
....
u..
1-
-
.004
o
:1 J
1/
'- I\,
"
~ .008
J
~
'l\. I"\kl ~ ""~ ~....,~
~
I
7
01
I\. ~
&
I
D
z-
~
~
I
t
~
...
'"
..
I"lIo.. !-
--"""
..... I
""""
t f
I
t
I
i
, f
2
l
1
G.c. ppsifion
R->--,. *~~I o 30-'0_ .250 0 I I a 6.01 I .2500 I I
:J
c 9.0~
250 .005 6 6.0 ~ Standard roughness . O.zOe simotatea split flop deflected 60· v 6.0
4
I
5.
StJ"do).d'
6.0
p
Iial
I
I I I I ·1.2
-.8
I ..4
I
I ~ks
rout}
J I I I I I I I I I t I I I 0
.4
Section liff coefficient, e,
NACA 0009 ·,\ing Section (Conlinued)
.8
1.2
/4
THEORY OF WING SECTIONS
456
2.4
2.0
1.8 I
I.. 1.2
,
0.1
..
~ I
0
'C.I
..:
:I
1 ...
-0.1
5 J
J
··i
IA··
l"
+. ;ilJ ..•!
i·· I
~
0.4
! ! ···To.
j
l···· . } .. -r :
1":."~
0.'
f-t~~~~-. . ..' .. ..
-
i-:
~.
J.
···f··
I
I
i:
...
o
I..
_ ••••
-'1--1---;
.
.
.
1- . l I;"':
'j'" ...
~I
1'.
+. 10: t
~ _i,~
j""~'+
.~.-\-..
••!
-0.8
'1"+ .·r··· I
'='0.3
-0.4
;"~~~ _:
..·t·.
':'.'I":If-..'1--"
:
R
., • •'..saN
--
... ::!~:
~
I
.......
~
J
,
-\j
I
I
\
J .......- +-W"Wo-+--l n ~ .OI6.......- +--+-.....,..-+--+-........ --t--+-+--t~-+-+-4--+--+-+-I!-+-+--++-l....-+
...
a
I
J
J
,, , ,
(
IP
B'.OI2.......-+-+-+---.......-+-....-.+--+-"lIlil-+--t-+-+...-+--+-+--t-+-tI~+-+-+-~~...-. Y/~ ...... ~
~
s """
_"
,~
r1
v
\.)
J: .008t-+-+-+-+-+-P-+~~a.-+-+-+-......-+-+-+-.-+-+~~..,......-t-+-+--+-+......
-./ t--+-+--+-+-t--t--+-0f-4-t--+-+-4~-+-+-4~-+-+-lJ-+"-'+--I-"""'-+-o1-~""""'"
i
~
,'" .i-.2t--t--+-t-t--t-it-+-~""""'~I--+-~~~~"""""'--I-~--I--t--+--+-..f---f.-+--I
~
0
~3
:0.10 .:r:$~;;
o~
...... -. 1--+--+--+-1-+............
~
9.0 4
~
~ -.4
-.8
.~~~~~~~~~~~
.24 7
.OO7~~-+-~I--+..-.+--+··
I Staodord rOUtJhneSs O.2OC simulafed· spk.f flop defkH:fed 60
6.0 I
~ ~Z -1.2
.~
·St;'~ ~~t-+-t-~I-+-+-o+-f
-.4 0 .4 Section lift coeffldenf, c1
NACA 1410 Wing Section (Continued)
.8
1.2
1.6
THEORY OF WING SECTIONS
472 3.6
rr
-
-
3.2
-
r- ~
2.4
LY
r\
1..1
l
I\.,
II' .J t'1"
2.0
rJ
,.,
1(/
:L
dJf
1.6
'(I J.~
~
n
#I
Jf
I Jr
~
IllS
II I r
,,..Id J
.N.
11 tfi
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APPENDIX III
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APPENDIX IV
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APPENDIX IV
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1.6
THEORY OF WING SECTIONS
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APPENDIX IV
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24
APPENDIX IV
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1.2
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568 3.6
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APPENDIX IV
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24
APPENDIX IV
581
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1.2
1.5
THEORY OF WING SECTIONS
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1.2
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THEORY OF WING ljECTION8
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8
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THEORY OF WING SECTIONS 3.6
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APPENDIX IlT .2
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THEORY OF WING SECTIONS
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24
631
APPENDIX IV
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APPENDIX IV
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APPENDIX IV
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THEORY OF WING SECTIONS
636 36
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APPENDIX IV .1)16
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THEORY OF WING SECTIONS
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32
APPENDIX IV .2
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APPENDIX IV
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THEORY OF WING SECTIONS
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APPENDIX IV
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THEORY OF WI.LVG SECTIONS
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APPEA·DIX IV'
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24
APPENDIX IV
687
INDEX
A Ackeret, J., 268
Adiabatic law, 249
Aerodynamic center, experimental, 182
theoretical, 69
wing, 5, 19
Aerodynamic characteristics, experimen tal, 449
Airfoils (see Wing sections)
Allen, H. Julian, 65, 164, 194, 202, 236
Anderson, Raymond F., 9
Angle of attack, 4
effective, 22, 25
ideal, 70
induced, 20
wing, 8,11
Angle of zero lift, experimental, 128
theoretical, 66, 69
approximate methods, 72
variation with flap deflection, 192
wing, 16
Aspect ratio, 6
at high speeds, 296
B
Bamber, Millard J., 234
Bernoulli's equation, 35, 251
Birnbaum, Walter, f~
Blasius, H., 87, 96
Boshar, John, 20
Bound~layer, 80
compressibility effects, 109
concept, 81
crossftows, 28
ht.minar,85
Blasius solution, 87
characteristics, 85
separation, 83, 86, 93
skin friction, 87
thickness, 89, 92
transition, 105, 157
velocity distribution, 87
Boundary layer, momentum relation, 90
slip, 83
thickness, displacement, 89
momentum, 92
turbulent, 95
pipe flow, 95
separation, 83, 103
skin friction, 97, 174
thickness, 102
velocity distribution, 97, 100
Boundary-layer control, 231
c Cahill, Jones F., 206, 221
Camber, 65
application to thickness forms, 75, 112
design of mean lines, 73
effect of, on aerodynamic center, 182
on angle of zero lift, 128
on lift curve, 132
on maximum lift, 136
on minimum drag, 148
on pitching moment, 179
on profile drag, 151, 178
induced, 28
shapes and theoreti~l characteristics,
382
Chaplygin, Sergei, 258
Chord, 113
mean aerodynamic, 21
Circular cylinder in uniform stream, 41,
49
with circulation, 44, 49
Circulation, 37
lift, 45
thin wing sections, 65
vortex, 43
Complex number, 47
Complex variable, 47
Compressibility effects, 247
boundary layer, 109
drag characteristics, 281
first order, 256
689
7'REORY OF lVING SECTIONS
690
Compressibility effects, flap effectiveness,
281
lift characteristics, 269
pitching-moment characteristics, 287
pressure coefficient, 257
wing characteristics, 268
Conformal mapping, 47, 49
Conformal transformations, 49
Continuity, 32, 38, 249
D Design of wing sections, 62
Designation of wing sections, III
German designations of NACA sec-
tioDS,118
KACA four-digit sections, 114
XACA five-digit sections, 116
X.ACA modified four- and five-digit
sections, 116
XACA L-series sections, 119
X A.CA 6-series sections, 120
K.-\CA. 7-series sections, 122
Doublet, 41, 49
Downwash, 9, 20
Drag characteristics, 148
calculation, 107
compressibility etIects, 281
minimum, 148
roughness effects, 157, 175
section induced, 25
variation with lift, 149
wing induced, 16, 27
E Experimental techniques, 125
Flow around wing sections, 83
at high speeds, 261
Fluid, perfect, 32
Fullmer, Felicien F., 231
G
Garrick, I. E., 63, 258
Glauert, H., 9, 65, 67, 192
Glauert-Prandtl rule, 256
Goldstein, Sidney, 63
Gruschwitz, E., 103
H
Harris, Thomas A., 218, 227
High-lift devices, 188
I
Irrotational motion, 37
J Jacobs, Eastman N., 194, 260
JQDes, Robert T., 296, 299
Jones edge-velocity correction, 11, 25
l~
Kaplan, Carl, 256, 258
Kaplan rule, 257
}{&nnaD integral relation, 90
}{arman-Tsien relation, 258
Keenan, Joseph H., 109
}\:night, Montgomery, 234
Koster, H., 230
Krueger, W., 230
Kutta-Joukowsky condition, 52
F
L
Families of wing sections (see Wing sec
tions)
Fischel, Jack, 218
Flaps, 188
external airfoil, 215
leading-edge,7Zl
plain trailing-edge, 190
split trailing-edge, 197
slotted trailing-edge, 203
Laminar flow (see Boundary layer)
Lanchester-Prandtl wing theory, 7
Leading edge, 113
Lees, Lester, 109
Lemme, H. G., 230
Liepman, n. \V., 268
Lift characteristics, compressibility effects,
269
design lift, 70
691
INDEX
Lift characteristics, experimental, 128
ideal lift, 70
maximum lift, 134
theoretical, 53
wing, 3, 8
maximum, 19
Lift-curve slope, effective, 11
experimental, 129
theoretical, 53, 69, 256
wing, 11
Limiting speed, 254
Loads, chordwise, 53, 60, 73, 75
flaps, 236
plain trailing-edge, 192, 236
slotted trailing-edge, 215, 221
split trailing-edge, 202, 236
span\\?ise,9, 19, 25
Loftin, Laurence K., Jr., 157
Lowry, John G., 227
M Mach line, 248
Mach number, 81, 248
critical, 259, 261
(See also Compressibility effects)
Mean lines, 382
(See also Camber)
Millikan, C. B., 93
Moment (see Pitching-moment charac
teristics)
Motion, equations of, 32, 85
irrotational, 37
Munk, Max ~I., 23, 65, 72
N
Naiman, Irven, 56, 74
Navier-Stokes equations, 85
Keely, Robert II., 20
Neumann, Ernest P., 109
Xikuradse, J., 99
Normal shock, 254
Noyes, Richard W., 304
o Ordinates of cambered wing sections, 406
p Pankhurst, R. C., 72
Pinkerton, Robert M., 61
Pitching-moment characteristics, 179
compressibility effects, 287
theoretical, 66, 69
approximate, 72
wing, 4, 16, 27
Platt, Robert C., 215, 226
Pohlhausen, K., 93, 106
Prandtl, L., 9, 20, 85, 96
Prandtl-Glauert rule, 256
Pressure coefficient, 42
Pressure distribution, theoretical, 53
approximate, 75
empirically modified, 60
Propeller slipstream, effect on drag, 170
Purser, Paul E., 218
Q Quinn, John H., Jr., 234, 236
R
References, 300
Regier, Arthur, 109
Reynolds number, 81
(See also Scale effect)
Reynolds, 0., 95
Rogallo, Francis l\{., 202
s Sanders, Robert, 225
Scale effect, 81
aerodynamic center, 182
lift-curve shape, 133
lift-curve slope, 129
maximum lift, 137
minimum drag, 148
profile drag, 151
Schlichting, H., 105
Schrenk, Oskar, 233
Schubauer, G. n., 105
Sections (see \Ving sections)
Sherman, Albert, 20
Shock, normal, 254·
on wing sections, 263
Shortal, Joseph A., 227
THEORY OF WING SECTIONS
692
Sinks, 39
Sivells, James Co, 20
Skin friction (Bee Boundary layer)
Skramstad, H. x., 105
Slats, leading-edge, 225
Slots, 2:1.7
Sound, velocity of, 249
Source, 39, 49
Speed, limiting, 254
Squire, H. s., 101, 109
Stack, John, 272
Stagnation points, 45
Stream function, 36
Stream tube, compressible flow in, 249, 252
Streamlines, 36
Superposition, 39, 75
Surface condition, 143, 157
effect of, on drag, 157, 175
on lift, 143
on transition, 157
standard leading-edge roughness, 143
SWL-ep, 296
T Taper ratio, 11
Tani, Itiro, 20
Techniques, experimental, 125
Temple, G., 258
Tetervin, Neal A., 102, 103
Theodorsen, Theodore, 54, 63, 65, 70, 109
Thickness distributions, 309
effect of, on aerodynamic center, 182
on angle of zero lift, 129
on lift-curve shape, 133
on lift-curve slope, 132
on maximum lift, 134
on minimum drag, 148
on pitching moment, 179
on profile drag, 148, 178
Thickness forms and theoretical data; 309
Thickness ratio, effect of, on aerodynamic
center, 182
on angle of zero lift, 129
on lift-curve shape, 133
on lift-curve slope, 53, 132
on maximum. lift, 134
on minimum drag, 148, 178
on pitching moment, 179
on profile drag, 151
(See also Wing sections)
Trailing edge, angle, 117
effect of, on aerodynamic center, 182
on lift-curve slope, 132
definition, 113
Transformation of circle in to wing sec tion, 50
Transient conditions, 133, 143
Tsien, H. S., 109, 258
Turbulent flow (Bee Boundary layer)
Twist, 11
u firich, A., 105
U nconservative sections, 175
Uniform stream, 39, 49
v
Velocity, limiting, 254
Velocity potential, 38
Vibration, effect on drag, 173
Viscosity (see Boundary layer)
von Kdrmdn, To, 82, 90, 93, 109, 175, 258
Vortex, 42, 49
system on wings, 9
Vorticity, 37
w Weick, Fred E., 225-227
Wenzinger, Carl J., 194, 202, 215
Wing sections, combinations of NACA
sections, 123
concept, 8
experimental characteristics, 124, 449
families, 111
N ACA four-digit, 113
NACA five-digit, 115
N ACA modified four- and five-digit,
116
NACA T-series, 118
NACA 6-scri~,s, 119
NACA 7-series, 122
ordinates, 406
unconservativc sections, 175
Wings, 1
aerodynamic center, 5, 19
angle of attack, 8, 11
angle of zero lift, 16
applicability of section data, 28
693
INDEX Wings, aspect ratio, 6
drag, 3, 16, 27
induced, 8, 16, 27
forces, 2
lift, maximum, 19
lift-curve slope, 11
'lift distribution, 9, 25
pitching moment, 16, 27
Wragg, C. A., 215
y Yarwood, J., 258
Young, A. D., 101, 109
z Zalovick, John A., 168
\. '-v
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(continued on back flap)
