# AXIAL FLIGHT: IMPROVED THEORETICAL ESTIMATES

Although the blade element theory introduced above has allowed the real effect of rotor driving torque arising from profile drag to be included, it has still relied upon the assumption that the inflow is constant along the rotor radius. Likewise, no allowance has been made for three-dimensional flow effects such as tip vortices. Minor modifications to the blade element theory are used to account for these important effects.

2.2.2 Variations in induced velocity

So far, it has been assumed that the induced velocity is constant across the disk. In addition, no description has been given of the precise relationship between blade pitch and induced velocity. In fact flow is induced downwards through the disk as a consequence of the aerofoil’s inclination to the direction of rotation. As such it is often necessary to develop a relationship between blade pitch and induced velocity for any radial station. Combining the momentum and blade element theories introduced earlier leads to:

Therefore:

1 pabc )r [ )r0(r) — (Vc + vi)] = 4p( Vcvi + v2 )nr

or:

where [vih/VT]r represents the ratio of the induced velocity in the hover at radius r to the tip speed, and 0(r) is the pitch at radius r. Figure 2.5 shows the variation of induced velocity for a hovering rotor calculated using Equation (2.9).