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Kutta-Zhukovsky theorem

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Kutta-Zhukovsky 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 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\, is known, the lift L\, per unit span (or L'\,) of the cylinder can be calculated using the following equation:[2]

1

 

 

 

 

(})

where \rho_\infty\, and V_\infty\, are the fluid density and the fluid velocity far upstream of the cylinder, and \Gamma\, is the (anticlockwise positive) circulation defined as the line integral,

\Gamma= \oint_{C} V \cdot d\mathbf{s}=\oint_{C} V\cos\theta\; ds\,

around a closed contour C enclosing the cylinder or airfoil and followed in the positive (anticlockwise) direction. This path must be in a region of potential flow and not in the boundary layer of the cylinder. The integrand V\cos\theta\, is the component of the local fluid velocity in the direction tangent to the curve C\, and ds\, is an infinitesimal length on the curve, C\,. 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 whatsoever is equal to -\rho_\infty V_\infty \Gamma, 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 \rho. 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 = Vc-(V+ v)c = -v c.\,

The difference in pressure \Delta 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 + 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.

Formal derivation

Extension

The lift predicted by Kutta Joukowski theorem within the framework of inviscid flow theory is quite accurate even for real viscous flow, provided the flow is steady and unseparated.[5]

For an impulsively started flow such as obtained by suddenly accelerating an airfoil or setting an angle of attack, there is a vortex sheet continuously shed at the trailing edge and the lift force is unsteady. This is known as the Wagner problem [6] for which the initial lift is one half of the final lift given by the Kutta Joukowski formula.[7]

When a source is present outside of the body, a force correction due to this source can be expressed as the product of the strength of outside source and the induced velocity at this source by all the causes except this source. This is known as the Lagally theorem.[8]

For free vortices and other bodies outside of the body, a generalized Lagally theorem holds,[9] with which the forces are expressed as the products of strength of inner singularities (image vortices, sources and doublets inside each body) and the induced velocity at these singularities by all causes except those inside this body. The contribution due to each inner singularity sums up to give the total force. The motion of outside singularities also contributes to forces, and the force component due to this contribution is proportional to the speed of the singularity.

For two-dimensional inviscid flow, the classical Kutta Joukowski theorem predicts a zero drag. When, however, there is vortex outside of the body, there is a vortex induced drag, in a form similar to the induced lift.

When in addition to multiple free vortices and multiple bodies, there are bound vortices and vortex production on the body surface, the generalized Lagally theorem still holds, but a force due to vortex production exists. This vortex production force is proportional to the vortex production rate and the distance between the vortex pair in production. With this approach, an explicit and algebraic force formula, taking into account of all causes (inner singularities, outside vortices and bodies, motion of all singularities and bodies, and vortex production) holds individually for each body,[10] with the role of other bodies represented by additional singularities. Hence a force decomposition according to bodies is possible.

For general three dimensional, viscous and unsteady flow, force formulas are expressed in integral forms. The volume integration of certain flow quantities, such as vorticity moments, is related to forces. Various forms of integral approach are now available for unbounded domain[7][11][12] and for artificially truncated domain.[13]

See also

Notes

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
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