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The Kutta–Joukowski theorem is a fundamental theorem in aerodynamics used for the calculation of lift of an airfoil (and any two-dimensional body including circular cylinders) translating in a uniform fluid at a constant speed so large that the flow seen in the body-fixed frame is steady and unseparated.
When an airfoil is moving with an angle of attack, the starting vortex has been cast off and the Kutta condition has become established, there is a finite circulation of the air around the airfoil. The airfoil is generating lift, and the magnitude of the lift is given by the Kutta–Joukowski theorem. [5]: § 4.5
Lifting line theory supposes wings that are long and thin with negligible fuselage, akin to a thin bar (the eponymous "lifting line") of span 2s driven through the fluid. . From the Kutta–Joukowski theorem, the lift L(y) on a 2-dimensional segment of the wing at distance y from the fuselage is proportional to the circulation Γ(y) about the bar a
Calculating the lift per unit span using Kutta–Joukowski requires a known value for the circulation. In particular, if the Kutta condition is met, in which the rear stagnation point moves to the airfoil trailing edge and attaches there for the duration of flight, the lift can be calculated theoretically through the conformal mapping method.
This is known as the Kutta–Joukowski theorem. [6] This equation applies around airfoils, where the circulation is generated by airfoil action; and around spinning objects experiencing the Magnus effect where the circulation is induced mechanically. In airfoil action, the magnitude of the circulation is determined by the Kutta condition. [6]
The horseshoe vortex model is unrealistic in that it implies uniform circulation (and hence, according to the Kutta–Joukowski theorem, uniform lift) at all sections on the wingspan. In a more realistic model, the lifting-line theory , the vortex strength varies along the wingspan, and the loss in vortex strength is shed as a vortex sheet all ...
Example of a Joukowsky transform. The circle above is transformed into the Joukowsky airfoil below. In applied mathematics, the Joukowsky transform (sometimes transliterated Joukovsky, Joukowski or Zhukovsky) is a conformal map historically used to understand some principles of airfoil design.
Martin Wilhelm Kutta (German:; 3 November 1867 – 25 December 1944) was a German mathematician. In 1901, he co-developed the Runge–Kutta method, used to solve ordinary differential equations numerically. He is also remembered for the Zhukovsky–Kutta aerofoil, the Kutta–Zhukovsky theorem and the Kutta condition in aerodynamics.