REPRESENTING AIRFOIL DATA WITH EQUATIONS
Airfoil lift and drag characteristics as a function of angle of attack and Mach number are often used in a computer as a bivalue table in conjunction with some table lookup scheme. There are occasions, however, when it is desirable to convert the airfoil data to equation form for use in simple hand analysis, in a small computer with limited storage capacity, or even in a large computer for quick comparisons of rotor performance using different airfoils. The simplest analytical expressions based on ignoring the effects of stall and compressibility are:(Note that the drag equations are unsymmetrical with respect to angle of attack and will give unrealistically low values of the drag coefficient for negative angles of attack. It is suggested that, if the negative angle of attack region is of importance in a specific analysis, the drag equations be rewritten in terms of only even powers of a. A cambered airfoil does not have zero lift at zero angle of attack. This can be accounted for by writing the equations in terms of a — aLO or by redefining the angle of attack such that it is zero at zero lift.)
A procedure for writing equations for the lift and drag coefficients for cases in which compressibility and stall cannot be ignored is given next.
For purposes of illustration, the lift and drag data synthesized from whirl tower tests of a rotor with the NACA 0012 airfoil as published in reference 6.66 will be used. These data are shown as solid lines in Figure 6.43. Both the lift and the drag coefficients of the NACA 0012 airfoil change characteristics at a Mach number of about 0.725, where compressibility effects are first evident. Because of this, two sets of equations must be written for the two separate Mach number regimes.
Lift Coefficient below M — .725
Theoretically, the slope of the lift curve should follow the PrandtlGlauert relationship:
Source: Carpenter, “Lift and ProfileDrag Characteristics of an NACA 0012 Airfoil Section as Derived from Measured HelicopterRotor Hovering Performance,” NACA TN 4357, 1958.
a0
a = —,
Figure 6.44 shows, however, that the slope is somewhat lower, being better fit by the equation:
The angle of attack at which the lift coefficient first shows the effects of stall will be defined as dL and is a function of Mach number. For the NACA 0012, dL is essentially linear with Mach number, as shown in Figure 6.44. An approximate equation is:
a£= 15 – 1Ш
Above dL, the lift coefficient can be represented by:
c’*>4 = «a – K,(a – aL)*‘
The exponent, K2, at any Mach number is obtained by plotting the difference between ad and the measured lift coefficient versus (a — a£) on loglog paper. The slope of the straight line faired through the points is the value of the
exponent to use. For the NACA 0012, this procedure, as illustrated in Figure 6.45, gives:
AT 
K2 
0.2 
1.95 
0.5 
1.57 
0.7 
1.38 
These points lie nearly on a straight line, whose equation—favoring the highest Mach number points—is:
K2 = 2.05 – 0.95 M
The coefficient, Kb at any Mach number is found by evaluating the equation for ct at the angle of attack corresponding to the maximum lift coefficient:
aac, – о
for the NACA 0012 airfoil:
M 
*i 
0.2 
.0233 
0.5 
.0257 
0.7 
.0497 
When K{ is plotted against Mach number, as in the top portion of Figure 6.46, it is seen to be a constant plus some power of M. Again using the loglog paper technique, as in the lower portion of Figure 6.4(5, the exponent can be evaluated and an approximate equation can be generated:
fCj = 0.0233 + 0.342 M715
Thus the lift coefficient below 0.725 Mach number and above stall is:
Ci < 0.725, a > aL = І 0Л – 0.0Ш a
– (0.0233 + 0.342ЛТ 15)(a – 15 + 16М)(2 05°9Ш)
Lift Coefficient above 0.12b Mach Number
The slope of the lift curve of the NACA 0012 breaks at 0.725 Mach number. The slope above this value is shown in Figure 6.44 to be nearly a. straight line, represented by:
a = 0.677 – 0.744 M
In order to satisfy the experimental lift characteristics by equations, the following constants were evaluated by the same methods used at the lower Mach numbers:
aL= 3.4
K, = 0.575 – 0.144(M – 0.725)044 K2 = 2.05 – 0.95 M
Thus:
c, = (0.677 – 0.744Л1)а
vH>.725,a>3.4 4 I
– [0.0575 – 0.144(M – 0.725)044] [a – 3.4](2 05“095Ar)
Mach Number
FIGURE 6.46 Evaluation of Equation for К^
Figure 6.43 shows the correlation of measured lift coefficient and the generated lift coefficient.
Drag Coefficient below Drag Divergence
At the lowest test Mach number, the incompressible drag coefficient can be represented by a power series of the form:
Cj. = Cj + Cj + Cj Q,2 + . . . +Cj d”
"incomp a0 al d2 an
The coefficients can be evaluated by selecting n control points and solving a set of n simultaneous equations. Before selecting the control points, the angle, aD, at which the individual drag curves break away from the incompressible curve should be established. For the 0012 airfoil, this angle, as shown in Figure 6.44, is approximately:
aD= 17 23.4M
(Note that the line represented by this equation goes through the drag divergence Mach number of 0.725 at a = 0.) The control points should include a = 0 and aD for the lowest test Mach number and n — 2 other points in between. For the 0012 a satisfactory fit was obtained with the 0.1 Mach number test data using a fiveterm series evaluated at 0, 2, 6, 10, and 14.7 degrees. This gives (with a in degrees):
cd. = 0.0081 + (350a + 396a2 – 63.3a} + 3.66a4) x кг6
“mcomp ‘ ‘
(Warning: too many terms in the series may introduce large fluctuations of the curve between the control points).
For Mach numbers above 0.1, the drag coefficient breaks away from the incompressible value as a exceeds aD. The curves have the characteristics represented by the equation:
^=4co„p + K>(aa’>)’;4
where Kb and KA are evaluated in the same manner in which Kx and K2 were evaluated in the equation for the lift coefficient. For the NACA 0012 airfoil:
Al 
*3 
*4 
0.3 
0.00071 
2.60 
0.5 
0.00063 
2.48 
0.7 
0.00064 
2.57 
Using average values gives:
Kb = 0.00066 K4 = 2.54
Thus:
Cd =cd. + 0.00066 (a – 17 + 234АГ)2 54
яЛ{<.725,а>а£) “incomp ‘ /
Drag Coefficient above Drag Divergence
For Mach numbers above drag divergence, another term must be added to account for the drag increment at zero angle of attack. The equation becomes:
= Чкошр + Къ(а " аоҐ4 + K>(M ~ Mdd)K6 For the NACA 0012, the coefficients and exponents that give the best fit are:
aD = o *, = 0.00035 K4 = 2.54 *5 = 21 *6=3.2
Thus:
cdu = cd + 0.00035a234 + 21(M – 0.725)3 2
aM>. 725 “incomp ‘ ‘
The correlation of the drag coefficient generated by this procedure with the test data is shown in Figure 6.43.
Equations Suited for Forward Flight Analysis
The foregoing procedure is adequate for hover performance methods and has been used to prepare the hover charts at the end of Chapter 1. For forward flight, however, the equations should be modified somewhat. Since negative angles of attack are possible on the advancing tip, the equation for the incompressible drag coefficient should be written in terms of even powers of a so that it is symmetrical about a = 0. For the NACA 0012, the fourterm series on a in degrees has been found to give satisfactory representation:
cd = 0.0081 + (65.8a2 – 0.226a4 + 0.0046a6) x 10~6
irtcomp v 7
A second modification accounts for the fact that some of the inboard elements on the retreating side will be subjected to angles of attack up to 360°, although at low Mach numbers. Figure 6.47 shows measured NACA 0012 lift and drag coefficients from 0° to 180° taken from reference 6.15. Equations for the lift coefficient an be written by dividing the angle of attack range up into segments:
Lift Coefficient Ci = generated coefficient Ci = 1.15 sin 2a ci = 0.7
o = o. i (a – 180°)
Ci = 0.7
0 = 1.15 sin 2a
Cj = generated coefficient
Similarly, the drag coefficient is represented by:
Moment Characteristics of a Cambered Airfoil
whirl tower and twodimensional wind tunnel tests. The whirl tower method has the advantage that threedimensional effects are accounted for, but these tests are relatively expensive. Twodimensional wind tunnel tests are relatively inexpensive but do not include any threedimensional effects. The airfoil data that are readily available at this time and are considered suitable for rotor analysis are summarized in Table 6.1. Much of the data is plotted in a consistent manner in reference 6.57.
TABLE 6.1 Sources of TwoDimensional Airfoil Data

Chord, In. 
Tb’/ Apparatus 
Date 
Ref 
7 
Boeing Tunnel 
1965 
6.57 
16 
Langley Whirl Tower 
1958 
6.66 
16 
UAC Wind Tunnel 
I960 
6.10 
1.5 
UARL 4′ Rotor Rig 
1972 
6.67 
2.7 
UARL 9′ Rotor Rig 
1961 
6.68 
21 
Bell Rotor in 40 x 80 Tunnel 
1965 
6.69 
16 
NSRDC Tunnel 
1977 
6.24 
24 
Boeing Tunnel 
1968 
6.23 
3, 4, 14 
Langley 6 x 28 Tunnel 
1980 
6.70 
10 
NPL Tunnel 
1968 
6.22,6.57 
20 
Langley 6 x 28 Tunnel 
1977 
6.71 
11 
Langley Whirl Tower 
1958 
6.72 
6 
Ames Tunnel 
1945 
6.17,6.57 
6 
Ames Tunnel 
1945 
6.17,6.57 
7 
Boeing Tunnel 
1965 
6.57 
6 
Boeing Tunnel 
1965 
6.57 
8 
Onera Tunnel 
1965 
6.57 
24 
Boeing Tunnel 
1968 
6.23,6.57 
24 
Boeing Tunnel 
1968 
6.23 
5 
ARA Tunnel 
1971 
6.57 
4 
Langley 6 x 19 Tunnel 
1971 
6.71 
1.5 
UARL 4′ Rotor Rig 
1972 
6.67 
6 
Ames Tunnel 
1945 
6.17,6.57 
7 
Boeing Tunnel 
1971 
6.57 
8 to 24 
Lockheed Tunnel 
1965 
6.46 
13 
Langley Whirl Tower 
1956 
6.73 
6 
Ames Tunnel 
1952 
6.47 
6 
Boeing Tunnel 
1969 
6.57 
6 
Boeing Tunnel 
1969 
6.57 
6 
Boeing Tunnel 
1969 
6.57 
64A(4.5)12 
—4 to 14 
.4 to.96 
64A612 
—6 to 13 
.4 to.90 
64 A516 
—4 to 14 
.4 to.80 
64006, 8, 10, 12 64A006406 
2 to 31 
.3 to.9 
64AO 10, 4101 
2 to 28 
.3 to.92 
65213 65006 64009 f 
—4 to 16 
.3 to.9 
64210 } 64215 ‘ 
2 tO 13 
.1 to.47 
64 AO 10 (Mod) 65 215 1 
—2 to 14 
.3 to.9 
66215 1 
0 to 16 
.3 to.85 
FX69H098 
—4 to 16 
.3 to.78 
FX69H098 
—4 to 13 
.35 to.90 
NPL 9615 NPL 9626 
—2 to 13 
.3 to.85 
NPL 9627 t 
0 to 13 
.3 to.75 
NPL 9660 
2 to 13 
.3 to.85 
NACACAMBRE 
—2 to 16 
.3 to.9 
VR7 
10 to 20 
.3 to.92 
VR7.1 
10 to 20 
.2 to.71 
VR8 
10 to 20 
.2 to.95 
NLR1(722362) 
2 to 11 
.35 to.85 
NLR1 
10 to 20 
.2 to.9 
SC1095 
—4 to 8 
.3 to 1.1 
SC1095R8 
—4 to 18 
.35 to.9 
DBLN518 
5 to 17 
.3 to.8 
BHC540 
—3 to 12 
.35 to.89 
SC1095 
—4 to 18 
.35 to.88 
RC10(N)1 
4 to 13 
.33 to.87 
RL(1)10 
—4 to 14 
.35 to.89 
RC(l)10MODl 
ft 
ft 
RC(1)10MOD2 
tf 
ft 
RC(3)08, 10, 12 
ft 
ft 
Al 
—2 to 14 
.2 to.84 
6 
Boeing Tunnel 
1969 
6.57 
6 
Boeing Tunnel 
1969 
6.57 
6 
Boeing Tunnel 
1969 
6.57 
6 and 11 
Langley Tunnel 
1953 
6.48 
6 
Ames Tunnel 
1954 
6.13 
16 
NSRDL Tunnel 
1977 
6.24 
12 to 36 
Langley Tunnel (Lift Only) 
1952 
6.12 
6 
Ames Tunnel 
1956 
6.8 
6 
Ames Tunnel 
1945 
6.17 
18 
UAC Tunnel 
1973 
6.53,6.57 
20 
Langley 6 x 28 Tunnel 
1977 
6.71 
10 
NPL Tunnel 
1968 
6.22 
10 
NPL Tunnel 
1969 
6.74 
10 
NPL Tunnel 
1973 
6.57 
8 
Onera Tunnel 
— 
6.57 
29 
Boeing Tunnel 
1971 
6.57 
29 
Boeing Tunnel 
1971 
6.57 
29 
Boeing Tunnel 
1971 
6.57 
4 
Langley 6 x 19 Tunnel 
1977 
6.71 
25 
Boeing Tunnel 
1977 
6.75 
16 
NSRDL Tunnel 
1977 
6.24 
3, 4, 9, 15 
Langley 6 x 28 Tunnel 
1980 
6.70 
16 
NSRDL Tunnel 
1977 
6.24 
20 
Langley 6 x 28 Tunnel 
1977 
6.71 
3, 4, 14 
Langley 6 x 28 Tunnel 
1980 
6.70 
15, 25 
Langley 6 x 28 Tunnel 
1981 
6.76 
24 
Langley 6 x 28 Tunnel 
1981 
6.77 
ft 
ff 
ft 
6.77 
ft 
ff 
ft 
6.77 
tf 
ff 
1982 
6.78 
Ames 2×2 Tunnel 
1980 
6.79 
The existence of several sets of published data for the NACA 0012 airfoil presents an opportunity for a comparison of test variables. Figure 6.51 shows the significant lift and drag characteristics of four sets of data as a function of angle of attack and Mach number. The comparison shows that the maximum lift coefficient and the drag coefficient are both strongly influenced by the chord of the test airfoil. This is a Reynolds number effect, which has been demonstrated in other tests.
Line 
Source 
Eff. Chord 
RN at 600 ft/sec 
Ref. 
Whirl Tower 
16" 
5.05 x 106 
6.66 

— 
Model Whirl Tower 
1.5" 
.47 x 106 
6.67 
— 
2D Tunnel 
9" 
2.84 x 106 
6.66 
— 
2D Tunnel 
20" 
6.30 x 1Q6 
6.80 
FIGURE 6.51 Comparison of NACA 0012 Airfoil Data 
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