Kutta–Joukowski theorem: Wikis

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Encyclopedia

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 defined as the line integral, around a closed loop enclosing the cylinder or airfoil, of the component of the velocity of the fluid tangent to the loop.[1] The magnitude and direction of the fluid velocity change along the path.

The flow of air in response to the presence of the airfoil can be treated as the superposition of a translational flow and a rotational flow. It is, however, incorrect to think that there is a vortex like a tornado encircling the cylinder or the wing of an airplane in flight. It is the integral's path that encircles the cylinder, not a vortex of air. (In descriptions of the Kutta–Joukowski theorem the airfoil is usually considered to be a circular cylinder or some other Joukowski 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 $\Gamma_\infty\,$ is known, the lift $L\,$ per unit span of the cylinder can be calculated using the following equation: [2]

$L = \rho_\infty V_\infty\Gamma_\infty,\,$   (1)

where $\rho_\infty\,$ and $V_\infty\,$ are the fluid density and the fluid velocity far upstream of the cylinder, and $\Gamma_\infty\,$ is the circulation defined as the line integral,

$\Gamma_\infty = \oint_{C_\infty} V\cos\theta\; ds\,$

around a path $C_\infty\,$ (in the complex plane) far from and enclosing the cylinder or airfoil. This path must be in a region of potential flow and not in the boundary layer of the cylinder. The $V\cos\theta\,$ is the component of the local fluid velocity in the direction of and tangent to the curve $C_\infty\,$ and $ds\,$ is an infinitesimal length on the curve, $C_\infty\,$. Equation (1) is a form of the Kutta–Joukowski theorem.

Kuethe and Schetzer state the Kutta–Joukowski theorem as follows:[3]

The force per unit length acting on a right cylinder of any cross section whatever is equal to $\rho_\infty V_\infty \Gamma_\infty$, and is perpendicular to the direction of $V_\infty.$

Derivation

Two derivations are presented below. The first is a heuristic argument, based on physical insight. The second is a formal and technical one, requiring basic vector analysis and complex analysis.

Heuristic argument

For a rather heuristic 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 + v on the other side. The circulation is then

$\Gamma = (V+ v)c-(V)c = v c.\,$

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

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

so the lift force per unit span is

$L = c \Delta P = \rho V v c =\rho V\Gamma.\,$

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

Notes

1. ^ Anderson, J.D. Jr., Introduction to Flight, Section 5.19, McGraw-Hill, NY (3rd ed. 1989.)
2. ^ Clancy, L.J., Aerodynamics, Section 4.5
3. ^ A.M. Kuethe and J.D. Schetzer, Foundations of Aerodynamics, Section 4.9 (2nd ed.)
4. ^ Batchelor, G. K., An Introduction to Fluid Dynamics, p 406

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