This is the user sandbox of Ssscienccce. A user sandbox is a subpage of the user's user page. It serves as a testing spot and page development space for the user and is not an encyclopedia article. Create or edit your own sandbox here.
Finished writing a draft article? Are you ready to request review of it by an experienced editor for possible inclusion in Wikipedia? Submit your draft for review!
Inviscid, frictionless fluid
Actuator disc (uniform flow)
Air far above the rotor is at ambient pressure pa and zero velocity, far downstream it's at wake velocity Vw and at ambient pressure pa again (we assume inviscid and frictionless fluid, so the wake and the surrounding air don't mix). Directly under the rotor, the velocity is Vd and the pressure is pd.
The thrust delivered is equal to the change in momentum with time, so (with dotted m meaning the mass flow rate):
(why Vw and not Vd? Because at Vd there's an additional pressure component)
The power delivered by the rotor is
But at the location of the propeller disc, the power is equal to the work done by the thrust:
So
, the velocity under the rotor is half of the ultimate wake velocity.