An assumption has been made so far, which is that the induced velocity is uniform across the rotor disc. This is an idealized situation, which in fixed-wing terms refers to the elliptically loaded wing which, itself, generates a uniform downwash. The effect of non-uniformity can be introduced by dividing the rotor disc into elements of concentric annuli, where we can treat each annulus as a lifting element and apply BET and momentum theory as before. By restricting the analysis to a generic annulus, over which the downwash can be considered constant, the restriction of uniformity of downwash can be removed. We restrict the analysis to hover and use
instead of l.
If we have an annulus of radius r and width dr, the thrust produced by this annulus can be expressed using BET, giving:
This thrust is expressed using momentum theory as:
dT = p • 2nr dr • Vi • 2Vi = 4ppr drV-2 = 4ppr dr(QR)2 • l2
The inflow distribution may now be calculated as a function of x and the thrust evaluated from Equation 3.26.
As a numerical example let us consider the case of a blade having linear twist, from a collective pitch setting of 12° to 6° at the tip (the root cutout can be ignored for this purpose). The rotor solidity (s) is 0.08 and the lift curve slope value (a) is 5.7. The value of the pitch angle at 75% radius is then O75% = 7.5°.