Kutta–Joukowski theorem

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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 generated by a right cylinder to the speed of the cylinder through the fluid, the density of the fluid, and the circulation. The circulation is the line integral of the velocity of the fluid, in a closed loop enclosing the cylinder. It can be understood as the total amount of "spinning" of the fluid around the cylinder. In descriptions of the Kutta–Joukowski theorem the right cylinder is usually confined to a circular cylinder or an airfoil.

The theorem refers to two-dimensional flow around a cylinder (or a cylinder of infinite span) and determines the lift generated by one unit of span. When the circulation $Γ$ is known, the lift $l$ per unit span of the cylinder can be calculated using the following equation: ^[1]

$l = \rho V\Gamma\,$

where $ρ$ is the fluid density, $V$ is the speed of the cylinder through the fluid, and $Γ$ is the circulation.

Kuethe and Schetzer state the Kutta–Joukowski theorem as follows:^[2]

The force per unit length acting on a right cylinder of any cross section whatever is equal to $ρ V Γ$ and acts perpendicular to V.

Formal proof of the theorem is to be found in standard texts.^[3] However as a plausibility argument, consider a thin airfoil of chord $c$ and infinite span, moving through air of density $ρ$ . Let the airfoil be inclined to the oncoming flow to produce an air speed $V$ on one side of the airfoil, and an air speed $V + Δ V$ on the other side. The circulation is then

$\Gamma = (V+ \Delta V)c-(V)c = \Delta Vc\,$

The difference in pressure $Δ P$ between the two sides of the airfoil can be found by applying Bernoulli's equation:

$\frac {\rho}{2}(V)^2 + (P + \Delta P) = \frac {\rho}{2}(V + \Delta V)^2 + P\,$

$\frac {\rho}{2}(V)^2 + \Delta P = \frac {\rho}{2}(V^2 + 2 * V * \Delta V + \Delta V^2)\,$

$\Delta P = \rho * V * \Delta V\,$ (ignoring $\frac{\rho}{2}\Delta V^2\,$ )

so the lift force per unit span is

$l = \Delta P * c = \rho * V * \Delta V * c =\rho V\Gamma\,$

A differential version of this theorem applies on each element of the plate and is the basis of thin-airfoil theory.

[edit] References

Batchelor, G. K. (1967) An Introduction to Fluid Dynamics, Cambridge University Press
Clancy, L.J. (1975), Aerodynamics, Pitman Publishing Limited, London ISBN 0 273 01120 0
A.M. Kuethe and J.D. Schetzer (1959), Foundations of Aerodynamics, John Wiley & Sons, Inc., New York ISBN 0 471 50952 3

[edit] Notes

^ Clancy, L.J., Aerodynamics, Section 4.5
^ A.M. Kuethe and J.D. Schetzer, Foundations of Aerodynamics, Section 4.9 (2^nd edition)
^ Batchelor, G. K., An Introduction to Fluid Dynamics, p 406