Region between Hover and Windmill State

In the region —2<Vc/vh<0 momentum theory is, strictly speaking, invalid because the flow can take on tw’o possible directions and a well-defined slipstream ceases to exist; this means that a control volume cannot be defined that encompasses only the physical limits of the rotor disk. However, the velocity curve can still be defined empirically, albeit only approximately, on the basis of flight tests or other experiments with rotors. Unfortunately descending flight accentuates interactions of the tip vortices with other blades and so the flow becomes rather unsteady and turbulent, and experimental measurements of rotor thrust and power are difficult to make. Also, the average induced velocity cannot be measured directly. Instead, it is obtained indirectly from the measured rotor power and thrust – see Gessow (1948, 1954), Brotherhood (1949), and Washizu et al. (1966b).

The measured rotor power can be written in the assumed form as

Pmeas = T(VC + V{) + P0, (2.92)

where P0 is the profile power and where Df is recognized as only an averaged value of the induced velocity through the disk. Using the result that Ph = Tvh we get

Ve T" Vi ______ Pjneas Po _________ ^meas Pq ___________ (^meas Po)+J2pA /n

vh ~ ~Ph ” T*fT/2pA ~ ‘ KL’yS)

Therefore, in addition to the measured rotor power Pmeas, to obtain an estimate for the averaged induced velocity ratio it is necessary to know the rotor profile power. As shown previously by means of Eq. 2.42, one simple estimate for the profile power coefficient of a rotor with rectangular blades is CPq = crC^/8, where is the mean (average) drag coefficient of the airfoil sections comprising the rotor and a is the rotor solidity.

Because of the high levels of turbulence near the rotor in this operating state, the derived measurements of the average induced velocity contain a relatively large amount of scatter. The smooth curve fit shown in Fig. 2.18 is taken from Gessow (1954) and is a composite
of flight tests and wind tunnel measurements made by Lock et al. (1926), Brotherhood (1949), and Drees & Hendal (1951). Note, however, that the curve follows the nonphysical branch of the induced velocity curve derived on the basis of momentum theory up to about Vcfvh ~ —1.5, after which it drops off precipitously and joins the branch of the curve defined on the basis of the momentum theory result in a descent. The higher values of measured power in hover and at low rates of descent are a result of higher induced power losses, which, as explained previously, are not predicted directly by the simple momentum theory.

Because the nature of the induced velocity curve is not analytically predictable in the range —2 < Vc/vh < 0, the experimental estimates can be used to find a “best-fit” approx­imation for vt at any rate of descent. Various authors, including Young (1978) and Johnson (1980), suggest a linear approximation to the measured curve. Following Young (1978), one approximation is

where к is the measured induced power factor in hover. A better approximation to the measured curve is

A continuous approximation to the measured induced velocity curve is the quartic

with k = —1.125, &2 = —1.372, k3 = —1.718, and = —0.655, which is valid for the full range —2 < Vc/vh < 0.