Kutta-Joukowski theorem

The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics, that can be used for the calculation of the lift of an airfoil, or of any two-dimensional bodies including circular cylinders, translating in a uniform fluid at a constant speed large enough so that the flow seen in the body-fixed frame is steady and unseparated. The theorem relates the lift generated by an airfoil to the speed of the airfoil through the fluid, the density of the fluid, and the circulation. The circulation is defined as the line integral, around a closed loop enclosing the airfoil, of the component of the velocity of the fluid tangent to the loop. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky (or Joukowski) who first developed its key ideas in the early 20th century. Kutta–Joukowski theorem is an inviscid theory which for pressure and lift is however a good approximation to real viscous flow for typical aerodynamic applications.

Kutta–Joukowski theorem relates lift to circulation much like the Magnus effect relates side force (called Magnus force) to rotation. However, circulation here is not induced by rotation of the airfoil but by some intrinsic mechanism described below. Due to this circulation, the flow of air in response to the presence of the airfoil can be treated as the superposition of a translational flow and a rotating flow. This rotating flow is induced by joint effect of camber, angle of attack and sharp trailing edge of the airfoil and should not be confused with a vortex like a tornado encircling the cylinder or the wing of an airplane in flight. Seen from a distance large enough to the airfoil, this rotating flow may be regarded as induced by a line vortex (with the rotating line perpendicular to the twodimensional plane). In the derivation of the Kutta–Joukowski theorem the airfoil is usually mapped into a circular cylinder. This theorem is proved in many text books only for circular cylinder and Joukowski airfoil, but it holds true for general airfoils.

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