Special Case, a = 0
If the disk AoA is zero (a = 0), an exact analytical solution for X can be determined. This is a physically unrealistic situation because the rotor must always be tilted slightly forward to produce a propulsive force. However, this special solution serves to illustrate the basic form of the induced part of the inflow through the rotor disk in forward flight and also provides a check case for the numerical solution (considered next). With a — 0, the induced velocity ratio in forward flight is
Squaring both sides of the above equation and rearranging gives
xf + ^x2-x4h = 0.
Dividing by XAh gives
which is a quadratic in (Xi/Xh)2. This quadratic has the solution
This result is shown in nondimensional form in Fig. 2.24. Notice that the induced velocity decreases quite quickly with increasing forward flight velocity. The asymptotic
approximation is obtained by letting д A, so that in high-speed forward flight Eq. 2.127 gives
ki kfo Cj
– —– > — or that = — (Glauert’s high-speed approximation), (2.131)
A/j Д 2д
which is denoted by the broken line in Fig. 2.24. For most purposes, this approximation is satisfactory for д/А/, > 2, which in practice is the case for д > 0.1.