Information about Kutta Joukowski Theorem
The Kutta-Joukowski Theorem is a fundamental theorem of aerodynamics. 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. The theorem relates the lift from an airfoil to the free stream velocity of the fluid and the circulation. The circulation is the line integral of the velocity of the air, in a closed loop around the boundary of an airfoil. It can be understood as the total amount of "spinning" (or vorticity) of air around the airfoil. When the circulation is known, the airfoil lift can be calculated using the following equation:
where ρ is the air density, V is the free-stream airspeed, and Γ is the circulation.
Formal proof of the theorem are to be found in standard texts (see eg ref. 1, p 406). However as a plausibility argument, consider the flow round a thin airfoil, length L, in a flow of undisturbed velocity V. Let the plate be slightly distorted to produce an increase v in the flow on the upside of the airfoil, and a decrease in velocity v on the downside. The circulation is then (V+v)L-(V-v)L= 2vL. Applying Bernoulli's theorem, the difference in pressure p between the top and bottom side of the plate at any point is 2.ρVv, so the lift force is l = pL= 2.ρVv.L=ρVG
A differential version of this theorem applies on each element of the plate and is the basis of thin-airfoil theory.
- l = ρVG
where ρ is the air density, V is the free-stream airspeed, and Γ is the circulation.
Formal proof of the theorem are to be found in standard texts (see eg ref. 1, p 406). However as a plausibility argument, consider the flow round a thin airfoil, length L, in a flow of undisturbed velocity V. Let the plate be slightly distorted to produce an increase v in the flow on the upside of the airfoil, and a decrease in velocity v on the downside. The circulation is then (V+v)L-(V-v)L= 2vL. Applying Bernoulli's theorem, the difference in pressure p between the top and bottom side of the plate at any point is 2.ρVv, so the lift force is l = pL= 2.ρVv.L=ρVG
A differential version of this theorem applies on each element of the plate and is the basis of thin-airfoil theory.
Reference
Batchelor, G. K. (1967) An Introduction to Fluid Dynamics, Cambridge University Press Martin Wilhelm Kutta (November 3, 1867 – December 25, 1944) was a German mathematician.
Kutta was born in Pitschen, Upper Silesia (today Byczyna, Poland). He attended the University of Breslau from 1885 to 1890, and continued his studies in Munich until 1894, where he
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Kutta was born in Pitschen, Upper Silesia (today Byczyna, Poland). He attended the University of Breslau from 1885 to 1890, and continued his studies in Munich until 1894, where he
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Nikolai Yegorovich Zhukovsky (Russian: Николай Егорович Жуковский) (January 17 [O.S.
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airfoil (in American English, or aerofoil in British English) is the shape of a wing or blade (of a propeller, rotor or turbine) or sail as seen in cross-section.
An airfoil shaped body moved through a fluid produces a force perpendicular to the motion called lift.
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An airfoil shaped body moved through a fluid produces a force perpendicular to the motion called lift.
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