# How were complex analysis methods, like the Joukowsky transform, used in early aircraft design?

The Joukowsky transform is a conformal mapping of a disk to an airfoil shape. The wiki page says that "it was historically used to understand some principles of airfoil design". That's kind of a vague statement and I'd like to know more about precisely how it was used. I know that early work on theoretical fluid mechanics before digital computers were available used potential theory and other techniques from complex analysis. Was complex analysis integral to the design and engineering of aircraft before computers, or was it a theoretical and conceptual tool used to help understand fluid mechanics but not used directly in the design of aircraft?

I'm asking this because I occasionally encounter people rolling their eyes at learning about imaginary numbers, and I like being able to tell them that an important historical application of complex numbers was the defeat of the Nazi menace in the skies. I'd like to make sure I'm being truthful there.

"Theoretical and conceptual tool" is probably closer to what happened than "used directly", but how "integral" it was is probably in the eye of the beholder. Joukowsky's 1906 model was an influential early part of gradually developing and intertwined theoretical and experimental work with feedback going in both directions, see Highlights From The History of Airfoil Development by Jones.

Joukowsky's airfoil itself was too idealized for direct use, but, together with abundant experimental data, it became the basis of more refined models that were used in design more directly, such as Munk's thin airfoil model of 1921 elaborated by Glauert. The thin airfoil model manipulates a Fourier series ansatz for the vortex distribution rather than uses complex analysis, but...

"The vortex distribution function... is Glauert's approximation and is based on Joukowski transformation results ($$A_0$$ term) which mainly covers the effect of angle of attack, plus a Fourier series variation ($$A_n$$ terms) to account for camber. It automatically obeys the Kutta condition with zero vorticity at the trailing edge. It is based on a mapped angular position $$θ$$ rather than an exact surface location $$s$$ to allow for ease of integration." [Auld-Srinivas, "2-D Thin Aerofoil Theory. ]

Here is from Jones:

"In 1902, W. M. Kutta calculated the lift of a thin, cambered plate at zero angle of attack and obtained a substantial lift force without drag in a frictionless fluid. In 1906, a theory of airfoils having rounded leading edges and varying angles of attack was developed by N. E. Joukowski... Beginning in 1921, NACA began collecting airfoil data from laboratories around the world and presenting them in a uniform notation... During this period, the design of airfoils was largely intuitive and based on the experience gained in testing numerous variations. While the experimenters were thus occupied, attempts were being made to extend the theory of Kutta and Joukowski to cover a greater variety of shapes. Thus, R. von Mises, a well-known aerodynamicist, was able to extend Joukowski's theory to cover an infinite series of profiles, all obtained by transformation of a circle. Also, H. Glauert, and T. von Karman obtained generalizations of the Joukowski airfoils.

A new direction in experimental technique appeared when Max Munk of NACA Langley proposed the variable density wind tunnel (see NACA TN 60, 1921)... Soon after the variable density tunnel, Munk developed his "thin airfoil theory." Although less accurate, this theory was a great simplification of earlier theories and permitted the relations between shape, pressure, and lift to be seen more clearly than before. With thin airfoil theory it was relatively easy to design airfoils that had a fixed or even a stable center of pressure. A systematic series of these, known as the M sections, were tested in the variable density tunnel at Langley in the 1920s. Of these, the M-6 and the M-12 found application; for example, on the Waco Taperwing. I used the M-12 on a small racing airplane (Pobjoy Phantom, Fig. 7) which flew in the 1930 National Air Races. Another important result of the thin airfoil theory, brought out by Glauert, is the magnifying effect of a plain trailing-edge flap when used as a control surface."

A more up-to-date reference that's fairly comprehensive for the first half of the 20th century would be

Robinson, A.; Laurmann, J. A. Wing theory. Cambridge, at the University Press, 1956. ix+569 pp.

This is described as follows in Math Reviews:

This is an admirable compendium of the mathematical theories of the aerodynamics of aerofoils and wings. Almost all the important results are referred to, even though there can be only a brief reference to literature in connection with the more difficult topics. It begins with a remarkably economical account of the fundamentals of hydrodynamics, including all the principal results concerning irrotational and rotational inviscid flow, the elements of viscous flow in so far as they are needed for a simple account of boundary layer theory, and some account of turbulent flow and wakes. The second chapter gives the two-dimensional aerofoil theory for incompressible flow. After the standard complex variable theory, the different methods of calculating pressure distribution for arbitrary aerofoils, or designing aerofoils for given pressure distribution, are set out. The limitations of the Kutta-Joukowski condition, and the preferability of using empirical values of lift to fix circulation, are clearly stated. In addition, biplanes and cascades of aerofoils are treated, and the chapter ends with an account of the Squire & Young method of calculating profile drag.

The review provides further details; see there (if you have access).

This is a very interesting question but, nevertheless, one could view this from a position of experience, a perspective in fluid mechanics, and an overview of aircraft history, reaching, thereby, a completely different answer. Given this view, the emphasized question within your post may be answered as follows: complex analysis was integral to the design and engineering of aircraft before computers, and it was used as a theoretical and conceptual tool to help understand fluid mechanics and applications in aircraft design, particularly in regard to airfoils and potential-flow theory. The upshot was that the application of conformal mapping and analysis was almost exclusively limited to, and used in, airfoil analysis and design. Since about 1975, the process has been extensively developed for computer application.

One may safely say that Joukowski and his colleagues eschewed cumbersome calculations, and with the method he developed, the tools used for analysis were those available to any grade-schooler, namely, compass, ruler and reflective-metal straight edge. The process used in the Joukowski transformation is known as conformal mapping. Given an elementary understanding of the transformation process, understanding the mathematics becomes relatively straightforward. Tools used, other than a pencil and ledger-tabulating paper, would be a slide rule and planimeter, and possibly a Comptometer, or as later available, a Marchant-style calculator. Of course, one would need a set of mathematical tables for various trigonometric functions and logarithms. The numerical methods would be based on simple differences and interpolations which, in many cases, were used for differentiation or integration. The methods similarly used in graphical and mechanical calculation were well known.

The use of conformal mapping in relation to elementary problems in fluid flow was hardly new. In 1738 Jacob Bernoulli first expressed the momentum and energy relations in fluid flowing from one location to another. Then in 1753 Euler proposed potential flow as a means for understanding irrotational fluid flow. More than 150 years later in 1904, Kutta would observe that the flow around an airfoil could not be irrotational if the flowing fluid was to smoothly leave the trailing edge of the airfoil. Independently, in 1906, Joukowsky would make a similar observation and work through a solution using conformal mapping and potential-flow analysis. The method provides a means for determining the velocity of the fluid flow around an airfoil. The fluid is treated as inviscid (that is, without viscosity), and Kutta's observations are included by superimposing a rotational component on the flow solution.

Extended computational methods in applications of the Joukowski transformation and airfoil analysis were developed during the period after WW-I. The basic process of hands-on-paper analysis and wind-tunnel verification was undertaken by the National Advisory Committee for Aeronautics. This was done for a large number of already known airfoils, and an extensive series of easily developed wing sections that could be applied in general aviation and more advanced aircraft. Two areas of particular interest were in drag reduction and in propeller design. Applications were in understanding fluid mechanics and the consequences in perfection of aircraft construction methods for the most efficient use of power in flight.

Wind-tunnel testing, however, presented a series of problems, the principal of which was expense. Further, the facilities for the wind tunnel were very large and required significant power and a number of personnel to operate. No short-cuts could be undertaken in building models used in wind-tunnel testing of wing sections. In fact, comments are made within the reports of the NACA regarding the stringent standards to which models needed to be constructed, and the need to develop an adequate theory for the on-paper evaluation of wing sections based upon mathematical methods and insights. Theodorsen, along with Garrick, proposed such a mathematical approach in the 1930s. Although the mathematical methods may have seemed somewhat obscure, they were, in general, relatively simple and fundamentally could be accomplished with little more than hand tabulation and analysis of results from graphical and mechanical computation. Greater precision in results could be obtained with relatively direct mathematical methods that were readily available and known. More detailed hand computation was involved, and could be expensive. Even so, along the approach taken in the mathematical analysis was the realization that the Joukowski method was easily used where the resulting transform-circle used in the method was altered in its shape to conform to that of the airfoil of interest. In fact, Jones and McWilliams show precisely how this is done and the kind of results that could be obtained.

The objective in using the Joukowski method, and variations, was to determine the pressure distribution of the flow along the surface of the airfoil contour. Using conformal mapping, the pressure distribution could be derived directly for an airfoil of interest, or for an airfoil inversely derived from the pressure distribution. Of particular interest was the velocity distribution of the flow over the contour of the airfoil. This velocity distribution could be derived directly from the pressure distribution.

One aspect in understanding the fluid mechanics of the flow in the near-vicinity of the surface related to the maintenance of laminar flow and an understanding of turbulent-flow separation. This was especially important in reducing drag. Consequently methods were advanced that allowed the velocity distribution and shape of the airfoil to be adjusted by changes in the shape of the transform-circle used in the conformal mapping.

For the use of conformal mapping in airfoil design per se, this refined approach was initially used in the design of the series 6 wing section for the P-51. The design and construction of this aircraft was in essence a tour de force accomplished by the NACA and North American Aviation, that resulted in a completed prototype from lofting of plans to prototype assembly and initial flight testing, within 90 days. The newly-designed laminar-flow wing section had favorable aerodynamic characteristics that made the aircraft easy to fly and control over the range of flight conditions needed for bomber escort duty and aerial combat, especially in the European theater.

Methods used by the NACA engineers in the design and hands-on-paper analysis of the series 6 airfoils, were not overly complex and wonderful. Every Joukowski airfoil may be developed in a process using three circles, namely, the basis circle (centered on the y-axis and known principally as the unit circle), the transform circle, and a resulting invert circle. The basis circle simply establishes the relationship of the line of centers used in the transformation. If this line is inclined, the transformation result is a cambered airfoil. If the basis circle is transformed onto itself, the result is the camber line of the airfoil. However, if the center of the basis circle is at the origin and that of the transform circle offset adjacently on the abscissa, the resulting airfoil is uncambered and symmetrical. The approach taken by the NACA in developing the series 6 wing sections was similar, namely, to simply distort the shape of the transform circle and map it back onto the basis circle to obtain a symmetrical airfoil. This process is termed mapping the near circle onto the unit circle. When this is done, the functional relationship regarding the angular distortion and difference in shape of the near circle relative to the unit circle, can be used in a relatively simple expression defining the ratio of the fluid velocity at the airfoil surface, to that of the free-stream velocity some distance away. In this way, desirable characteristics of the velocity distribution could be derived for enhancement of laminar flow.

After the end of the Second World War, for this type of airfoil design work there were no electronic computers readily available for perhaps more than 20 years. In the United States, the NACA laminar wing sections were used on many general aviation aircraft, and ultimately used in the perfected design of the Sisu-1A, a high performance sailplane. However, after 1950 the methods used by the two German engineers are of particular interest as they are relevant to the posted question. Both of these engineers were interested in the optimized design of laminar airfoils for high-performance soaring flight, and for applications in general aviation aircraft. They were well aware of the methods used by the NACA in developing the series 6 airfoils. The interests of Richard Eppler were in the understanding how to control turbulent separation and optimize the airfoil design for minimum drag. Eppler was also interested in calculating the drag of an airfoil based upon an adequate understanding of turbulence and flow separation. His work, although relatively obscure in the post-war period for nearly 20 years, became very highly regarded and well known in applications for high performance sailplanes and aircraft. Although Eppler was well published, his work did not step into the forefront until after 1975. At about this time, with the help of associations in the United States (particularly with NASA), Eppler transferred his methods to digital computation. His principal work in airfoil design, nevertheless, was based on the mainstay of conformal mapping derived from modifications to the Joukowski transformation, and consequent desired results to the airfoil shape in order to optimize the velocity distribution and airfoil design. Although a practicing aeronautical engineer in the early portion of his career, Eppler ultimately joined the faculty of the University of Stuttgart.

F X Wortmann, a contemporary of Eppler, became much more well known and in the forefront of airfoil design in the post-war period from the early 1950s and into the 1980s. His interests were also in turbulent separation and optimization of laminar flow in airfoil design. From the very beginning of Wortmann’s work, he understood the dynamics of boundary-layer separation and the need to appropriately shape the airfoil to control the velocity profile and boundary-layer separation. Wortmann became director of the Institute for Aerodynamics and Gas Dynamics at the University of Stuttgart. His most well-known work was in the development of low-drag airfoils for high-performance sailplanes. An apparent fundamental aspect of Wortmann’s airfoil-design work was in the modification of the Joukowski profile for conformance of the profile velocity distribution to meet specified objectives regarding laminar flow and turbulent separation.

Is the Joukowski transformation still used today? The answer is yes. Aspects of airfoil design, particularly for low-speed, high-performance aircraft, are reliant on the use of the method, and variations of it, to achieve desired results.

• It may be worth citing some sources for all this. Feb 22 at 9:50