It is proposed in this section of the report to present, briefly, a summary of the results of the existing thin-airfoil theory (based on the section mean line) as applied to the prediction of certain section characteristics. Such a summary is desirable because at present the results must be obtained from several different sources which give them in a form not easily applied. Three characteristics are considered; namely, (1) the angle of zero lift aL0, (2) the pitching-moment coefficient Cmc/4 and (3) the "ideal" angle of attack aI, or the corresponding lift coefficient CLI, that is, values corresponding to the unique condition for which the theory gives a finite velocity at the nose of the airfoil. (See reference 16.)
Expressions for lift and moment coefficients may be written as follows if the angles are measured in radians:
If the leading end of the mean line is chosen as the origin of coordinates and the trailing end is taken on the x axis at x =1, then the parameters aL0, aI and (beta) are given by the following integrals
where
and y is the ordinate of the mean line at a given abscissa x. The integrals (4) and (6) may be shown to be identical with the corresponding integrals given by Glauert (reference 15) and by Munk (reference 17), and integral (5) is given by Theodorsen (reference 16).
The evaluation of these integrals for the N.A.C.A. airfoil sections given in this report was accomplished analytically. The values of arO (changed from radians to degrees), Cmc/4 and CLI, so computed, are given in tables III, IV, and VII, respectively, in the main body of the report. This method of evaluation, however, cannot be applied to many of the commonly used sections because they do not have analytically defined mean lines; hence, an approximate method must be used. A graphical determination gives good results and for convenience the values of the three functions, (7), (8), and (9), at several values of x, are given in the following table:
In general, some difficulty would be expected with the graphical method because the values of the above functions tend to infinity at the leading and trailing edges. Actually, because the ordinates of the mean-line extremities are zero, the integrand may approach zero, and does at the leading edge for the integral (4), and at the leading and trailing edges for the integral (6). Difficulty, however, is encountered at the trailing edge for the integral (4) and at the leading and trailing edges for the integral (5). In order to avoid this difficulty, integral (4) is evaluated graphically from x= 0 to x=0.95, and the increment contributed by the portion from x=0.95 to x=1 is determined analytically. Likewise, integral (5) is evaluated graphically from x = 0.05 to x= 0.95 and analytically for the extremities. The analytical determination of the increments is accomplished by assuming the mean line near the ends to be of the form
Evaluating the integrals gives
where y'0 and y'l are the mean-line slopes at the leading and trailing edges, respectively.
Table of Contents | Summary | Introduction | Description of Airfoils | Apparatus and Methods | Results | Discussion | Supplementary Airfoils | Conclusions | Appendix | References