Historical Background

As a historical note, the roots of conformal mapping as applied to airfoil design date back to the early days of the NACA. Eastman Jacobs thought that Theodorsen's method for airfoil analysis (Theodorsen 1931) could be used as the basis of the design method by inverting the analysis process. With some contributions made by Theodorsen and Garrick (1933), the approach proved possible and eventually lead to the development of a series of low-drag laminar-flow airfoils, namely the NACA 6-series airfoils (Jacobs 1939) that are still applied today. Jacobs's design method, although crude by present standards, was the first of what have come to be known as inverse airfoil design methods as described.

Surprisingly, the inverse approach of Jacobs was initially received by considerable skepticism among the group of theoreticians at Langley, and, even after it proved successful, it was never fully appreciated. Ira H. Abbott who worked with Jacobs at the time and who later took part in compiling and publishing the airfoil work done at the NACA (Abbott, Von Doenhoff and Stivers 1945; Abbott and Von Doenhoff 1959) said (Abbott 1980):

We were told that even the statement of the problem was mathematical nonsense with the implication that it was our ignorance that encourages us.
In the mind of E.I. Garrick, this view had hardly changed when he later wrote (Garrick 1952):
In the opinion of the writer several difficulties arise or exist in defining this problem to satisfy both the mathematician and the aerodynamicist. For one thing, attempts have not been successful in making precise statements of the problem in regard to uniqueness, closure, proper trailing edge, leading edge contours, avoidance of grotesque nonstreamline figures most likely to be subject to separated flow, or of no physical significance as figures eight (or worse). For another, the prescription of pressure distributions with respect to a reference chord leads to nonuniqueness; and prescription with regard to normals to the boundary surface leads to undefiniteness, since the physical boundaries are being sought. Another difficulty is the fact that our insight and knowledge of flow behavior are not developed to the point that an exactly defined desirable pressure distribution can be specified.
Predictably, these attitudes did not foster continued growth in airfoil research at Langley as high-speed aerodynamics came into the forefront in the 1940s and 1950s. Interestingly, Barger (1974, 1975a,b) in a series of NASA reports extended the inverse method of Jacobs, but it was never received with much enthusiasm. Other more powerful methods had come into favor.

These more powerful methods were based on conformal mapping (like Jacobs's method) and had their origin in Europe, notably Mangler (1938) in Germany and Lighthill (1945) in Britain. It is not clear if Jacobs's work had any direct impact in these developments; chances are it did not since Jacobs never fully published his approach and only few details exist (Theodorsen and Garrick 1933; Theodorsen 1944).

These new methods of Mangler and Lighthill showed clearly for the first time that the velocity distribution specified around the airfoil could not be entirely arbitrary. Specifically, they showed that the velocity distribution had to satisfy three important integral constraints: one in order to guarantee compatibility with the freestream velocity and two in order to ensure closure of the airfoil profile. While these theories did much to dispell doubts about the theoretical soundness of the inverse approach, practical application was hampered severely by the lengthy calculations involved in obtaining the final airfoil shape; it was said that a skilled mathematician could perform the calculations in approximately 20 hours. Thus, most of the early work done following the theory of Mangler and Lighthill was focused primarily on improving the numerical solution, both its speed and accuracy (Peebles 1947; Glauert 1947; Timman 1951; Peebles and Parkin 1956). Starting in the 1960s emphasis had shifted towards practical application through the use of the computer (Nonweiler 1968; Ingen 1969; Arlinger 1970; Strand 1974; Polito 1974).

By the 1970s, the inverse approach had matured into a very powerful tool for design, but it was not without shortcomings - shortcomings that still exist today. The all-important integral constraints are expressed in terms of the velocity distribution around the airfoil not as a function of arc length but as a function of the angular coordinate around the circle from which the airfoil is mapped. In other words, the desired velocity distribution can only be specified indirectly as a function of the circle angular coordinate. Iterative techniques, however, were introduced by Arlinger (1970) and James in 1970, as discussed by Liebeck (1990), so that the desired velocity distribution could be specified from the outset, subject of course to the integral constraints.

From the need to satisfy the integral constraints arises a different problem. Since there are three integral constraints, it is necessary to introduce into the specified velocity distribution three, free parameters in order to satisfy them. Many successful ways have been devised to do this. A difficulty occurs when the values determined for these free parameters lead to unrealistic velocity distributions which in turn correspond to unrealistic airfoils, for instance, crossed airfoils or figure of eights as referred to by Garrick (1952). Essentially, all practical inverse methods employ some kind of iterative technique to overcome this difficulty.

Finally, the last shortcoming pertains not to the application of the method but to the theory itself. Methods based on the theory of Mangler and Lighthill may be regarded as single-point inverse airfoil design methods; that is, the desired velocity distribution is prescribed at a single angle of attack. The fact is that many airfoils must operate over a range - not at a single point. Whether or not an airfoil designed by a single-point method satisfies multipoint design requirements must be determined through post-design analysis at the operating conditions of interest. Consequently, the single-point design methods tend to be very tedious if multipoint design requirements are imposed. Although design by this single-point method has lead to many successful airfoils, a theory that has the explicit capability of handling multipoint design requirements from the outset is favored.

While efforts were underway in the 1950s to improve the numerical techniques of the single-point design methods, Eppler published his theory for multipoint design (Eppler 1957). Since this time, the method has been improved and made readily available as a computer program (Eppler and Somers 1980a,b; Eppler 1990). To this day the program enjoys increasingly widespread use. The Eppler method allows the airfoil to be divided into a desired number of segments along each of which the velocity distribution is prescribed together with the design angle of attack at which the velocity is to be achieved. In this way, multipoint design requirements can be satisfied during the actual design effort, not iteratively through post-design analysis. Despite the versatility of the method as a practical design tool, the actual theory has received very little attention; notable exceptions are Miley (1974), Ormsbee and Maughmer (1984) and Selig and Maughmer (1991). Miley (1974) applied the Eppler method to the design of low Reynolds number airfoils, and Ormsbee and Maughmer (1984) derived necessary conditions and integral constraints for the attainment of finite trailing-edge pressure gradients. Additional contributions were made by Selig and Maughmer (1991). More recently the general approach was applied to airfoils in cascade (Selig 1992) and airfoils with slot suction (Selig and Saeed 1995) for application to advanced transports (and here too). These last two links point to Prof Ilan Kroo's WWW site.

References in TeX format can be found here.