The results of this investigation are here discussed and analyzed to indicate the variation of the aerodynamic characteristics with variations in thickness and in mean line form. For the analysis of the effect of thickness, test data from consecutive tests of airfoils having different thicknesses and the same mean-line form are used. The analysis of the effect of the mean-line form is made with respect to consecutive tests of airfoils of the same thickness (12 percent of the chord) and related mean-line forms. The results are com-pared, where possible, with the results predicted by thin-airfoil theory, a summary of which is presented in the appendix.
LIFT
Lift curve.—In the usual working range of an airfoil section the lift coefficient may be expressed as a linear function of the angle of attack
where aO is the slope of the lift curve for the wing of infinite aspect ratio and aL0 is the angle of attack at zero lift.
The variation of the lift-curve slope with thickness is shown in figure 81. The points on the figure represent the deduced slopes as measured in the angular range of low profile drag. These results confirm previous results (reference 1) in that they show the lift-curve slope to decrease with increasing thickness. The camber has very little effect on the slope, as indicated in figure 82, although a rearward movement of the position of the camber tends to decrease the slope slightly. Table II gives the numerical values of the slope in convenient form for noting the general trends with respect to variations in thickness and in camber. It will be noted that all values of the slope lie below the approximate theoretical value for thin wings, 2,r per radian; the measured values lie between 95 and 81 percent, approximately, of the theoretical.
The angle of zero lift is best analyzed by means of a comparison with that predicted by the theory. Thin airfoil theory states that the angle of zero lift is proportional to the camber if the camber is varied, as with these related airfoils, by scaling the ordinates of a given mean-line without altering the
camber position. The theory also predicts an increased negative angle as the position of the camber moves back along the chord. The experimental values are compared with the theoretical values in figures 83 and 84. The experimental values lie between 100 and 75 percent, approximately, of the theoretical values, the departure becoming greater with a rearward movement of the position of the camber and with increased thickness (above 9 to 12 percent of the chord). Numerical values of the angle of zero lift are given in table III.
Maximum lift.—The variation of the maximum lift coefficient with thickness is shown in figure 85. It will be noted that the highest values are obtained with moderately thick sections (9 to 12 percent of the chord thick, except for the symmetrical sections for which the highest values are obtained with somewhat thicker sections). The variation with camber, shown in figure 86, confirms the expected increase in maximum lift with camber. The gain is small, however, for the normal positions of the camber, but becomes larger as the camber moves either rearward or forward. It will be seen by reference to figure 85 that the camber becomes less effective as the thickness is increased. This reduced effectiveness of the camber is in agreement with a conclusion reached in reference 13 that for airfoils having a thickness ratio of approximately 20 percent of the chord, camber is of questionable value. Numerical values of the maximum lift coefficient are given in table IV.
Air-flow discontinuities.—These and other windtunnel tests indicate that at the attitude of maximum lift the air forces on certain airfoils exhibit sudden changes which in many instances result in a serious loss of lift. The probable cause of these airflow discontinuities is discussed briefly in reference 13. The stability or instability of the air flow at maximum lift may be judged by the character of the lift-curve peaks indicated for the various airfoils. The curves are classified into three general types as noted in table IV, but the degree of stability is difficult to judge. It may be generally concluded that improved stability may be obtained by (1) having a small leading-edge radius, which causes an early breakdown of the flow with a consequent low value of the maximum lift, (2) increasing the thickness (beyond the normal thickness ratios), or (3) increasing the camber (for airfoils having normal camber positions; i.e., 0.3c to 0.5c).
MOMENT
Thin-airfoil theory separates the air forces acting on any airfoil into two parts: First, the forces that pro-duce a couple but no lift (they are dependent only on the shape of the mean line); second, the forces that produce the lift only, the resultant of which acts at a fixed point. We then have in the working range an expression for the total moment taken about any point
CM=CM0+nCL
where CM0 is the moment coefficient at zero lift and nCL is the additional moment due to lift.
As with the angle of zero lift, the theory states that the moment at zero lift is proportional to the camber and predicts an increase in the magnitude of the moment as the camber moves back along the chord. Figures 87 and 88 show the values of the moment coefficient as affected by variations of camber and thickness compared with the theoretical values. Referring to figure 87, the plotted data indicate that the moment coefficients are nearly proportional to the camber. It will also be noted that the curves representing the ratios of the experimental coefficients to the camber are nearly parallel to the equivalent curve representing the theoretical ratios except that the curves tend to diverge for positions of the camber well back. Figure 88 shows that the experimental values lie between 87 and 64 percent, approximately, of the theoretical. Numerical values of the moment coefficient at zero lift are given in table V.
If the resultant of the lift forces acted exactly through the quarter-chord point, as predicted by the theory of thin airfoils, there would be no additional moment due to the lift when the moments are taken about this point. The curves of Cmc/4 against CL, however, show a slope in the working range which indicates that the axis of constant moment is displaced somewhat from the quarter-chord point. The factor n represents the amount of this displacement as obtained from the deduced slopes of the moment curves in the normal working range. The variation of this displacement with thickness and with camber is shown in figures 89 and 90. Table VI gives the numerical values. Beyond the stall all the airfoils show a sharp increase in the magnitude of the pitching moment. The suddenness of this increase follows the degree of stability at the stall as indicated by the type of the lift-curve peak.
The total drag of an airfoil is considered as made up of the induced drag and the profile drag. Considering the profile drag as the minimum value plus an additional drag dependent upon the attitude of the airfoil, we have in coefficient form
The induced-drag coefficient CD2, which is computed by means of the formula given in reference 8, is considered to be independent of the airfoil section. The variation of the profile-drag coefficient with the shape variables of the airfoil section is analyzed with respect to the variations of the two components of the profile drag.
Minimum profile drag.—The variation of the minimum profile-drag coefficient with thickness for the symmetrical sections is shown in figure 91. The cambered sections show the same general variation with thickness but, to avoid confusion, the results are not plotted. The variation of the minimum profile-drag coefficient with the profile thickness may be expressed by the empirical relation
where t is the thickness ratio and k (which is approximately constant for sections having the same mean line) represents the increase in CD0min, above that computed for the symmetrical section of corresponding thickness. The variation of CD0min with camber is indicated by the variation of k as shown in figure 92.
The effect of camber is small except for the highly cambered sections having the maximum camber well back. Numerical values of CD0min are given in table VII.
Additional profile drag.—The additional profile drag, which is dependent upon the attitude of the airfoil, has previously been expressed as a function of the lift (reference 4) by the equation
where CLopi may be called the optimum lift coefficient; that is, the lift coefficient corresponding to the minimum profile-drag coefficient. This equation holds reasonably well for the normally shaped airfoils at values of the lift coefficient below unity.
A convenient practical method of allowing for the increased values of CD0 at moderately high values of the lift coefficient is to include the additional profile drag with the induced drag, as suggested in reference 2. For the symmetrical airfoils of moderate thickness the term to be added to the induced-drag coefficient was given as 0.0062 CL2. The relative importance of this term may be better appreciated by considering that it represents 11.7 percent of the induced drag of an elliptical airfoil of aspect ratio 6. The same method may also be applied to other airfoils if the value of the optimum lift is not too large.
Andrews (reference 14), using the part of these data published in references 2, 4, and 5, suggests for the additional profile drag the form
This function is represented in figure 93 as the curve determined from the results for the symmetrical airfoils and for the airfoils having a camber of 2 percent of the chord. As the camber is increased, the dispersion of the plotted points from the curve becomes greater. In general the points above the curve correspond to thick sections and sections in which the maximum camber is well back. The departure from the curve becomes greater with increased thickness and with a rearward movement of the maximum-camber position. The points well below the curve correspond to the thin airfoils.
Because the additional profile drag is not a simple function of the lift, and also because the results as presented in figure 93 are difficult to follow, generalized curves for the relation
are given in figure 94. These curves are given to represent more accurately the additional profile drag for the normally shaped sections.
Optimum lift.—The optimum lift, as defined above, is the value of the lift corresponding to the minimum profile drag. As the determination of this value of the lift is largely dependent upon the fairing of the profile-drag curves, special curves were faired for this purpose on enlarged-scale plots corresponding to certain related airfoils grouped together. The values of the optimum lift coefficients obtained in this manner are give in table VIII. It may be noted by reference to this table that the optimum lift coefficient increases with camber and for the highly cambered sections a definite increase accompanies a forward movement of the camber.
More important than these variations, however, is the variation with thickness. The rapid decrease in the optimum lift with increased thickness indicates that it is not primarily dependent upon the shape of the mean line. Nevertheless it is interesting to compare the optimum lift coefficients with the values included in table VIII representing the theoretical lift coefficients at the "ideal" angle of attack for the mean line; i.e., the angle of attack for which the thin-airfoil theory gives a finite velocity at the nose. (See the appendix.)
The general efficiency of an airfoil cannot be expressed by means of a single number. The ratio of the maximum lift to the minimum profile drag is, however, of some value as the measure of the efficiency of an airfoil section. The variation of this ratio with thickness is shown in figure 95. The curves of this figure indicate that the highest values of the ratio are given by the sections between 9 and 12 percent of the chord thick. The variation with camber, shown in figure 96, is less important. An increase in the camber above 2 percent of the chord and a rearward move-ment of the camber (for the highly cambered sections) tend to decrease the value of CLmax/CD0min. The numerical values of the ratio are given in table IX.
Table of Contents | Summary | Introduction | Description of Airfoils | Apparatus and Methods | Results | Discussion | Supplementary Airfoils | Conclusions | Appendix | References