Induced Velocity and Rotor Power

It has been shown previously using Eq. 2.7 that momentum theory can be used to relate the rotor thrust to the induced velocity at the rotor disk by using the equation

T = mw — m(2vi) = 2(pAvi)Vi = 2pAvf. (2.14)

Rearranging this equation and solving for the induced velocity at the plane of the rotor disk gives

= = Ш5)

The ratio T/ A is known as the disk loading, which is an extremely important parameter in helicopter analysis. Notice that vh == щ is used to represent the induced velocity in hover. This value will be used later as a reference when the axial climb and descending flight conditions are considered.

The power required to hover (or the time rate-of-work done by the rotor on the fluid per unit thrust) is given by

nr t3/2

P = Ты = Tvh = ТІ————— = ———– . (2.16)

V 2pa rr

This power, called ideal power, is entirely induced in nature because the contribution of viscous effects have not been considered in the present level of analysis. Alternatively, we can write

P = Tvi = 2 rhvf = 2(pAvi)vf = 2pAv.

From this equation it is noted that the power required to hover will increase with the cube of the induced velocity (or inflow) at the disk. Obviously, to make a rotor hover at a given thrust with minimum induced power, the induced velocity at the disk must be small. Therefore, the mass flow through the disk must be large and this consequently requires a large rotor disk area. This is a fundamental design feature of all helicopters.