Zhukovskii's Theorem

a theorem dealing with the lifting force that acts on a body located in a plane-parallel flow of a fluid or gas. According to the theorem, the lifting force acting on a body in a stream of fluid or gas is determined by the vortices associated with the body around which the flow occurs (attached vortices, generated because of the viscosity of fluid). The existence of such vortices results in a flow around the airfoil with a nonzero circulation. The theorem was formulated by N. E. Zhukovskii in 1904.

If a stable, plane-parallel potential flow of an incompressible fluid encounters an infinitely long cylinder perpendicular to its generating lines, then that portion of cylinder that has a unit length along the generating line is acted upon by the lifting force Y. which is equal to the product of the density of the medium ρ and the flow velocity v at infinity and the circulation Γ of the velocity within any closed path enveloping the cylinder past which the flow occurs, Y = ρvΓ. The direction of the lifting force is obtained from the direction of the velocity vector at infinity by rotating the vector 90° against the direction of circulation.

Zhukovskii’s theorem is also valid for a subsonic flow of a compressible fluid or gas around a profile. For sonic and supersonic velocities of flows past a body, the theorem cannot be proved in a general form.

The modern airfoil and propeller theories are based on Zhukovskii’s theorem. The lifting force of a wing with finite span, the thrust of a propeller, and the pressures acting on turbine blades and compressor blades can be determined with the aid of Zhukovskii’s theorem.