Algorithm for Calculating Coordinates of Cambered Naca Airfoils at Specified Chord Locations

The equations for the NACA 4-digit and 4-digitmodified sections are in algebraic form and easily incorporated into various geometrical procedures that define a vehicle and any necessary flow field grid. The 6-series and 6A-series airfoils are more complex because they are developed by conformal mapping procedures. Even though there are computer programs available (refs 1-3) that can produce a large table of points on the surface of the airfoil, there is a frequently expressed desire for an algorithm that will calculate the upper and lower ordinates and slopes of a cambered airfoil at a specified chord location that requires no interpolation on the part of the user. The purpose of this paper is to present such an algorithm and describe subroutines that may be used for these calculations. A public domain computer program incorporating these procedures has been written and may be downloaded from the author’s web site. This program is modular, allowing its internal procedures to be used in other programs. INTRODUCTION In spite of the advanced airfoils available to the modern designer, the NACA series of airfoils continue to be of interest. Computer programs were written in 1974-1975 (Ref 1-2) and updated in 1996 (Ref. 3) that enable a user to produce a table of surface coordinates of an airfoil with thickness and camber from the NACA families. These original programs have been included in the collections of public domain aeronautical software distributed by the author (Ref 4). Users of these programs identified two areas where the programs did not totally satisfy their needs. Firstly, many users wanted to specify the chord position of the airfoil and compute the upper and lower ordinates of the surface directly above and below this chord point. The output from the programs of refs 1-3 displaces the upper and lower xcoordinates because the thickness is applied perpendicular to the mean line. Secondly, many users wanted subroutines that performed these calculations that could be incorporated in a larger analysis or design program. The monolithic structure of the programs of Refs 1-3 discouraged any such use. In order to satisfy these user requirements, the present algorithm was developed. THICKNESS OF 6AND 6A-SERIES AIRFOILS 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, ] named and that were chosen to satisfy a prescribed velocity distribution about the airfoil. The definition of the and 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 and 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