Joukowski foil

The Joukowski airfoils are a family of two-dimensional airfoils developed by the Russian mathematician N.E. Joukowski and for which he solved analytically the flow field. Nikolai Egorovich Joukowski (1847-1921) did some extensive research in aerodynamics. His name is also spelled Zhukovsky or Zhukoskii. In 1906, he published two papers in which he gave a mathematical expression for the lift on an airfoil. Today the theorem is known as the Kutta-Joukowski theorem. Martin Wilhelm Kutta (1867-1944) was a German engineer and mathematician, known also for his contribution to the numerical solution of differential equations: i.e., the Runge-Kutta method. The Kutta-Joukowski equation was presented in his 1902 dissertation.
Considering the flow past a Joukowski airfoil, the incomplete flow net, seen below, illustrates the streamlines in orange, including the boundary of the foil, and the stagnation point at the leading edge. At the trailing edge, there is a discontinuity in velocity in the general cases. The Joukowski hypothesis states that the circulation around an airfoil always adjusts itself so that velocity is finite at the trailing edge, a hypothesis well substantiated experimentally.
The Joukowski airfoil shape was commonly used in aircraft design during the first half of the 20th century. However, speed and manoeuvrability requirements imply not only good lift characteristics but also low drag and favourable pressure distributions over a broad range of angle of attack. In the particular case of water flows, separation may be associated with cavitation. Further three-dimensional phenomena are not considered, although they are important in real-fluid flows.
Since the mid-20th century, profiles other than the Joukowski profiles were devised: e.g., the NACA profiles (Chanson 2014, pp. 171-172).
The Joukowski airfoil and Kutta-Joukowski theorem are essential concepts in applied hydrodynamics and aerodynamics. These are discussed in the relevant RUclips video movies in the same Playlist at:
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References
JOUKOWSKI, N.E. (1890). “Modification of Kirchoff’s Method for Determining the Two Dimensional Motion of a Fluid at a Prescribed Constant Velocity on a Given Streamline.’’ Proc. Math. Symp., Moscow, Russia, Vol. XV.
ROUSE, H. (1946). "Elementary Mechanics of Fluids." John Wiley & Sons, New York, USA, 376 pages.
STREETER, V.L. (1948). "Fluid Dynamics." McGraw-Hill Publications in Aeronautical Science, New York, USA.
VALLENTINE, H.R. (1969). "Applied Hydrodynamics." Butterworths, London, UK, SI edition.
CHANSON, H. (2009). "Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows." CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages (ISBN: 978-0-415-49271-3).
CHANSON, H. (2014). "Applied Hydrodynamics: An Introduction." CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 448 pages & 21 video movies (ISBN 978-1-138-00093-3).