# The momentum theory applied to the helicopter rotor

In most, but not all, states of helicopter flight the effect of the rotor may be approxi­mated by replacing it by an ideal actuator disc to which the simple momentum theory applies. More specifically, momentum theory may be used for translational, i. e. forward, sideways or rearwards, flight, climb, slow descent under power and hovering.

9.5.1 The actuator disc in hovering flight

In steady hovering flight the speed of the oncoming stream well ahead of (i. e. above) the disc is zero, while the thrust equals the helicopter weight, ignoring any downward force arising from the downflow from the rotor acting on the fuselage, etc. If the weight is W, the rotor area A, and using the normal notation of the momentum theory, with p as the air density

W = pA V0( Vs – V) = pA V0 Vs (9.44)

since V = 0. Vs is the slipstream velocity and Vq the velocity at the disc.

The general momentum theory shows that

V0=^(VS+V) (Eqn(9.8))

— – К in this case 2

or

Vs = 2V0

 which, substituted in Eqn (9.44), gives W = 2pAVl i. e.

 (9.46) (9.47) (9.48)

 V0 = y/WflpA Defining the effective disc loading, !&, as /de = W/A(7 where и is the relative density of the atmosphere, then W W 1 a 1 2pA Aa 2 p 2po de po being sea-level standard density. Then

 (9.49)

 Vo — ykc/lpo

 = l-pVoVlA = 2pAVl

 (9.50)

 Substituting for Vo from Eqn (9.47) leads to

 ‘ w3/2 jw3

 P = 2pA

 (9.51a)

 2pA

 2pA)

(9.51b)

This is the power that must be supplied to the ideal actuator disc. A real rotor would require a considerably greater power input.