Non-uniform inflow

Although Glauert [2.11] provides a simple method of calculating the mean induced velocity for a rotor in horizontal flight he, along with others who have studied lifting

rotors [2.15 and 2.16], appreciated that the induced velocity over a rotor is far from uniform. Consequently they proposed additional formulae designed to generate an upwash ahead of the rotor disk and a steadily increasing inflow along the rotor diameter similar to that seen along the chord of an aerofoil. The formula proposed by Glauert is sufficiently accurate for our purposes:

Note that if К is set to a value greater than 1 (typically 1.2 [2.17]) an upwash results and a linear fore/aft variation in induced velocity is predicted. Lateral variations, although present in reality, have been ignored at this level of simplification. The term vi0 represents the mean induced velocity found using Equation (2.12). Before determining the effect of non-uniform induced velocity on the induced power required in forward flight, it is necessary to consider the skew angle of the rotor wake. This skew angle (%) is measured from the vertical and indicates the direction taken by the ultimate wake from the rotor disk. Once again several formulae exist to predict % [2.6, 2.15 and 2.18]. Although some include wake contraction, the simplest assume a cylindrical wake. The following relationship from Prouty [2.6] is most often quoted:

1 = Vo

tan % V

Note that since a cylindrical wake is assumed, the induced velocity at the rotor hub is used rather than the ultimate wake value of 2vi0. Now from Equation (2.12):

1 = гг ИГ

tan % J 2 + V 4 + V4

Figure 2.15 shows a typical variation in skew angle with horizontal speed. As with tip losses in axial flight it can be shown [2.17] that the extra induced power required by non-uniform inflow can be taken into account by applying a factor к to the estimate of induced power based on a uniform inflow. Therefore, for small disk tilt:

Pi = кРі0 = KTvi = KWvi

Estimates for к vary from 1.17 [2.17] to 1.2 [2.10].