Momentum theory

In fluid dynamics, momentum theory or disk actuator theory is a theory describing a mathematical model of an ideal actuator disk, such as a propeller or helicopter rotor, by W.J.M. Rankine (1865),^[1] Alfred George Greenhill (1888) and Robert Edmund Froude (1889).^[2]

An actuator disk accelerating a fluid flow from right to left

The rotor is modeled as an infinitely thin disc, inducing a constant velocity along the axis of rotation. The basic state of a helicopter is hovering. This disc creates a flow around the rotor. Under certain mathematical premises of the fluid, there can be extracted a mathematical connection between power, radius of the rotor, torque and induced velocity. Friction is not included.

For a stationary open rotor with no outer duct, such as a helicopter in hover, the power required to produce a given thrust is:

${\displaystyle P={\sqrt {\frac {T^{3}}{2\rho A}}}}$

where:

• T is the thrust
• ${\displaystyle \rho }$ is the density of air (or other medium)
• A is the area of the rotor disc
• P is power

A device which converts the translational energy of the fluid into rotational energy of the axis or vice versa is called a Rankine disk actuator. The real life implementations of such devices include marine and aviation propellers, windmills, helicopter rotors, centrifugal pumps, wind turbines, turbochargers and chemical agitators.

See also

• Blade element theory
• Circulation (fluid dynamics)
• Disk loading
• Kutta–Joukowski theorem

 

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References

1. Rankine, W.J. Macquorn (6 April 1865). "On the Mechanical Principals of the Action of Propellers". Transactions of the Royal Institution of Naval Architects. 6: 13 – via Hathi Trust.
2. Froude, Robert (12 April 1889). "On the Part Played in Propulsion by Differences in Fluid Pressure". Transactions of the Royal Institution of Naval Architects. 30: 390 – via Hathi Trust.