Momentum theory

Short description: Mathematical model in fluid dynamics

An actuator disk accelerating a fluid flow from right to left

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]

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:

[math]\displaystyle{ P = \sqrt{\frac{T^3}{2 \rho A}} }[/math]

where:

* T is the thrust
* [math]\displaystyle{ \rho }[/math] 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. https://hdl.handle.net/2027/mdp.39015022700325?urlappend=;seq=47;ownerid=13510798885881399-61.
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. https://babel.hathitrust.org/cgi/pt?id=hvd.hnhtv1&seq=424.

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