Historical Note: Kutta, Joukowski, and the Circulation Theory of Lift

Frederick W. Lanchester (1868-1946), an English engineer, automobile manufacturer, and self-styled aerodynamicist, was the first to connect the idea of circulation with lift. His thoughts were originally set forth in a presentation given before the Birmingham Natural History and Philosophical Society in 1894 and later contained in a paper

submitted to the Physical Society, which turned it down. Finally, in 1907 and 1908, he published two books, entitled Aerodynamics and Aerodonetics, where his thoughts on circulation and lift were described in detail. His books were later translated into German in 1909 and French in 1914. Unfortunately, Lanchester’s style of writing was difficult to read and understand; this is partly responsible for the general lack of interest shown by British scientists in Lanchester’s work. Consequently, little positive benefit was derived from Lanchester’s writings. (See Section 5.7 for a more detailed portrait of Lanchester and his work.)

Quite independently, and with total lack of knowledge of Lanchester’s thinking, M. Wilhelm Kutta (1867-1944) developed the idea that lift and circulation are related. Kutta was bom in Pitschen, Germany, in 1867 and obtained a Ph. D. in mathematics from the University of Munich in 1902. After serving as professor of mathematics at several German technical schools and universities, he finally settled at the Tech – nische Hochschule in Stuttgart in 1911 until his retirement in 1935. Kutta’s interest in aerodynamics was initiated by the successful glider flights of Otto Lilienthal in Berlin during the period 1890-1896 (see chapter 1 of Reference 2). Kutta attempted theoretically to calculate the lift on the curved wing surfaces used by Lilienthal. In the process, he surmised from experimental data that the flow left the trailing edge of a sharp-edged body smoothly and that this condition fixed the circulation around the body (the Kutta condition, described in Section 4.5). At the same time, he was con­vinced that circulation and lift were connected. Kutta was reluctant to publish these ideas, but after the strong insistence of his teacher, S. Finsterwalder, he wrote a paper entitled “Auftriebskrafte in Stromenden Flussigkecten” (Lift in Flowing Fluids). This was actually a short note abstracted from his longer graduation paper in 1902, but it represents the first time in history where the concepts of the Kutta condition as well as the connection of circulation with lift were officially published. Finsterwalder clearly repeated the ideas of his student in a lecture given on September 6, 1909, in which he stated:

On the upper surface the circulatory motion increases the translatory one, therefore

there is high velocity and consequently low pressure, while on the lower surface the

two movements are opposite, therefore there is low velocity with high pressure, with

the result of a thrust upward.

However, in his 1902 note, Kutta did not give the precise quantitative relation between circulation and lift. This was left to Nikolai Y. Joukowski (Zhukouski). Joukowski was bom in Orekhovo in central Russia on January 5, 1847. The son of an engineer, he became an excellent student of mathematics and physics, grad­uating with a Ph. D. in applied mathematics from Moscow University in 1882. He subsequently held a joint appointment as a professor of mechanics at Moscow Uni­versity and the Moscow Higher Technical School. It was at this latter institution that Joukowski built in 1902 the first wind tunnel in Russia. Joukowski was deeply interested in aeronautics, and he combined a rare gift for both experimental and theoretical work in the field. He expanded his wind tunnel into a major aerodynam­ics laboratory in Moscow. Indeed, during World War I, his laboratory was used as a school to train military pilots in the principles of aerodynamics and flight. When he died in 1921, Joukowski was by far the most noted aerodynamicist in Russia.

Much of Joukowski’s fame was derived from a paper published in 1906, wherein he gives, for the first time in history, the relation L’ = Vx Г—the Kutta-Joukowski

theorem. In Joukowski’s own words:

If an irrotational two-dimensional fluid current, having at infinity the velocity Vx surrounds any closed contour on which the circulation of velocity is Г, the force of the aerodynamic pressure acts on this contour in a direction perpendicular to the velocity and has the value

L’^p^V^Y

The direction of this force is found by causing to rotate through a right angle the vector Voc around its origin in an inverse direction to that of the circulation.

Joukowski was unaware of Kutta’s 1902 note and developed his ideas on circu­lation and lift independently. However, in recognition of Kutta’s contribution, the equation given above has propagated through the twentieth century as the “Kutta – Joukowski theorem.”

Hence, by 1906—just 3 years after the first successful flight of the Wright brothers—the circulation theory of lift was in place, ready to aid aerodynamics in the design and understanding of lifting surfaces. In particular, this principle formed the cornerstone of the thin airfoil theory described in Sections 4.7 and 4.8. Thin airfoil theory was developed by Max Munk, a colleague of Prandtl in Germany, during the first few years after World War I. However, the very existence of thin airfoil theory, as well as its amazingly good results, rests upon the foundation laid by Lanchester, Kutta, and Joukowski a decade earlier.

4.15 Summary

Return to the road map given in Figure 4.2. Make certain that you feel comfortable with the material represented by each box on the road map and that you understand the flow of ideas from one box to another. If you are uncertain about one or more aspects, review the pertinent sections before progressing further.

Some important results from this chapter are itemized below:

A vortex sheet can be used to synthesize the inviscid, incompressible flow over an airfoil. If the distance along the sheet is given by. v and the strength of the sheet per unit length is y(s), then the velocity potential induced at point (x, y) by a vortex sheet that extends from point a to point h is
ф(х, y) = j 9y(s) ds	[4.3]
The circulation associated with this vortex sheet is
r=f y(s)ds	[4.4]
Across the vortex sheet, there is a tangential velocity discontinuity, where
у = u, ~ ІІ2	[4.8]

The Kutta condition is an observation that for a lifting airfoil of given shape at a given angle of attack, nature adopts that particular value of circulation around the airfoil which results in the flow leaving smoothly at the trailing edge. If the trailing-edge angle is finite, then the trailing edge is a stagnation point. If the trailing edge is cusped, then the velocities leaving the top and bottom surfaces at the trailing edge are finite and equal in magnitude and direction. In either case, у (ТЕ) = 0 [4.10]

Thin airfoil theory is predicated on the replacement of the airfoil by the mean camber line. A vortex sheet is placed along the chord line, and its strength adjusted such that, in conjunction with the uniform freestream, the camber line becomes a streamline of the flow while at the same time satisfying the Kutta condition. The strength of such a vortex sheet is obtained from the fundamental equation of thin airfoil theory:

1 fc Y(S)dS „ / dz 2л J0 x – f °° v dx

[4.18]

Results of thin airfoil theory: Symmetric airfoil 1. Сі = 2л a. 2. Lift slope = dci/da = 2л. 3. The center of pressure and the aerodynamic center are both at the quarter-chord point. 4. Cm, с/4 — C/n, ac — Cambered airfoil 1. сі = 2л 1 г dz a—– —— (cos вд – l)d90 [4.57] _ л Jo dx _ 2. Lift slope = dci/da = 2л. 3. The aerodynamic center is at the quarter-chord point. 4. The center of pressure varies with the lift coefficient.

The vortex panel method is an important numerical technique for the solution of the inviscid, incompressible flow over bodies of arbitrary shape, thickness, and angle of attack. For panels of constant strength, the governing equations are

(i = 1, 2, …, n)

Lee cos Pi

and

Yi = ~Yi-1

which is one way of expressing the Kutta condition for the panels immediately above and below the trailing edge.

Problems

Consider the data for the NACA 2412 airfoil given in Figure 4.5. Calculate the lift and moment about the quarter chord (per unit span) for this airfoil when the angle of attack is 4° and the freestream is at standard sea level conditions with a velocity of 50 ft/s. The chord of the airfoil is 2 ft.

Consider an NACA 2412 airfoil with a 2-m chord in an airstream with a velocity of 50 m/s at standard sea level conditions. If the lift per unit span is 1353 N, what is the angle of attack?

Starting with the definition of circulation, derive Kelvin’s circulation theorem, Equation (4.11).

Starting with Equation (4.35), derive Equation (4.36).

Consider a thin, symmetric airfoil at 1.5° angle of attack. From the results of thin airfoil theory, calculate the lift coefficient and the moment coefficient about the leading edge.

The NACA 4412 airfoil has a mean camber line given by

Using thin airfoil theory, calculate (a) aL=o (b) ci when a = 3°

For the airfoil given in Problem 4.6, calculate стхц and лср/г when a = 3°.

Compare the results of Problems 4.6 and 4.7 with experimental data for the NACA 4412 airfoil, and note the percentage difference between theory and experiment. (Hint: A good source of experimental airfoil data is Reference 11.)

Starting with Equations (4.35) and (4.43), derive Equation (4.62).

For the NACA 2412 airfoil, the lift coefficient and moment coefficient about the quarter-chord at —6° angle of attack are —0.39 and —0.045, respectively. At 4° angle of attack, these coefficients are 0.65 and —0.037, respectively. Calculate the location of the aerodynamic center.