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Thickness Function for NACA 6-Series Airfoils

The NACA 6-series and 6A-series airfoils are defined by means of conformal transformations. These relate the flow over an airfoil to that of a near-circle and that to a circle. The basic reference is Theodorsen, NACA 411.

The NACA 6-series airfoils are calculated by a nonlinear mapping of a unit circle by a four-step algorithm that uses a pair of functions defined on [0,pi] named psi and epsilon that were chosen to satisfy a prescribed velocity distribution about the airfoil. The definition of the psi and epsilon functions is described in refs 7-8. Each of the five members of the 6-series family and the three members of the 6A-series family has its own psi and epsilon functions. These functions are multiplied by a scale factor to produce airfoils of various thickness to chord ratios. The mapping is shown in figure 1. A given value of the scale factor is used to multiply both basic parameters giving new values of the psi and epsilon functions that will be used in the mapping. A given value of scale factor will produce a certain thickness to chord ratio of the airfoil in the normalized physical plane. It is not known in advance just what thickness will result from a given value of the scale factor. The algorithms of references 1 and 3 use an iterative procedure to determine the scaling factor required to achieve an airfoil of a given thickness.

The algorithm used in the present method is based upon a study of the scaling factor required to achieve a given thickness. Calculations were made of the thickness resulting from a given value of scale factor for each of the eight airfoil families. The dependency is somewhat nonlinear but easily fitted as a polynomial with four coefficients. The fitting is done on the data as if scale factor c is a function of t/x.

c=K1(t/c) + K2(t/c)2 + K3(t/c)3 + K4(t/c)4

and the K-values for each family are given by:

63 8.1827700 1.3776209 -0.0928517 7.5942563
64 4.6535511 1.0380630 -1.5041794 4.7882784
65 6.5718716 0.4937629 0.7319794 1.9491474
66 6.7581414 0.1925377 0.8128826 0.852090
67 6.6272890 0.0989966 0.9675977 0.9053758
63A 8.1845925 1.0492569 1.3115094 4.4515579
64A 8.2125018 0.7685596 1.4922345 3.6130133
65A 8.2514822 0.4656936 1.5013018 2.0908904

Now, for a specified family and thickness, the thickness distribution may be determined without iteration. From the thickness, the scale factor is computed from the polynomial function shown above. Then, the scale factor is used to multiply the basic values of the psi and epsilon functions for this airfoil family. These scaled psi and epsilon functions are used in mapping the z-plane to the z'-plane shown in Figure 1. The Joukowski function zeta = z' + 1/z' then maps the z'-plane into the zeta-plane and these results are normalized so that the leading edge is at x=0 and the trailing edge is at x=1.

Public Domain Aeronautical Software (PDAS)