momentum theory

{{Short description|Mathematical model in fluid dynamics}}

File:Actuator disk.svg

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),{{Cite journal |last=Rankine |first=W.J. Macquorn |date=6 April 1865 |title=On the Mechanical Principals of the Action of Propellers |url=https://hdl.handle.net/2027/mdp.39015022700325?urlappend=%3Bseq=47%3Bownerid=13510798885881399-61 |journal=Transactions of the Royal Institution of Naval Architects |volume=6 |pages=13 |hdl=2027/mdp.39015022700325?urlappend=%3Bseq=47 |via=Hathi Trust}} Alfred George Greenhill (1888) and Robert Edmund Froude (1889).{{Cite journal |last=Froude |first=Robert |date=12 April 1889 |title=On the Part Played in Propulsion by Differences in Fluid Pressure |url=https://babel.hathitrust.org/cgi/pt?id=hvd.hnhtv1&seq=424 |journal=Transactions of the Royal Institution of Naval Architects |volume=30 |pages=390 |via=Hathi Trust}}

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:

:$P\; =\; \backslash sqrt\{\backslash frac\{T^3\}\{2\; \backslash rho\; A\}\}$

where:

• T is the thrust
• $\backslash rho$ 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.