## Kutta - Joukowski

**Kutta-Joukowski Lift Theorem for a Cylinder**

Lift per unit length of a cylinder acts perpendicular to the velocity (V) and is given by: L = ρVG (Lbs/Ft)

Where: ρ = Fluid Density (slugs/Cu Ft) G = Vortex Strength (Sq Ft/sec) (G=2.Π.b.Vr) V = Flow Velocity (Ft/sec) Vr = rotational speed (Ft/sec) (Vr=2.Π.b.s) b = radius of cylinder s = revolutions/sec pi = 3.14159 Two early aerodynamicists determined the magnitude of the lift force, Kutta in Germany and Joukowski in Russia. The lift equation for a rotating cylinder bears their names. The equation states that the lift L per unit length along the cylinder is directly proportional to the velocity V of the flow, the density ρ of the flow, and the strength of the vortex G that is established by the rotation.

**L = ρ * V * G**

The equation gives lift-per-unit length because the flow is two-dimensional. (Obviously, the longer the cylinder, the greater the lift) Determining the vortex strength G takes a little more math. The vortex strength equals the rotational speed Vr times the circumference of the cylinder. If b is the radius of the cylinder,

**G = 2.0 * b* pi * Vr**

Where pi =3.14159. The rotational speed Vr is equal to the circumference of the cylinder times the spin s of the cylinder.

**Vr = 2.0 * b* pi * s**

**WARNING: **Be particularly aware of the simplifying assumptions that have gone into this analysis. This type of flow field is called an ideal flow field. It is produced by superimposing the flow field from an ideal vortex centered in the cylinder with a uniform free stream flow. There is no viscosity in this model (no boundary layer on the cylinder) even though this is the real origin of the circulating flow! In reality, the flow around a rotating cylinder is very complex, depending on the ratio of rotational speed, free stream speed, angle of incidence of the free stream flow, viscosity of the fluid, and size of the cylinder, the flow off the rear of the cylinder can separate and become unsteady. But, the simplified model does give the first order effects; it gives an initial good prediction of the force on the cylinder when compared to experiments.

**In Conclusion: **All that is necessary to create lift is to turn a flow of air. We are familiar with the lift generated by an airplane wing or a curving baseball. But a simple rotating cylinder will also create lift. In fact, because the flow field associated with a rotating cylinder is two-dimensional, it is much easier to understand the basic physics of this problem than the more complex three-dimensional aspects of a curveball. However, the details of how a rotating cylinder creates lift are still pretty complex. Next to any surface, the molecules of the fluid will stick to the surface. This thin layer of molecules will entrain or pull the surrounding flow in the direction that the surface moves. If we put a cylinder that is rotating about the longitudinal axis (a line perpendicular to the circular cross section) into a fluid, it would eventually create a spinning, vortex-like flow around the cylinder. If we then set the fluid in motion, the uniform velocity flow field can be added to the vortex flow.