Weighted Solidities of Tapered Blades

The equivalent solidities for rotors with taper or other planform variation can now be computed. One must be cautious, however, in that the rotor planforms must not be too radically different. This is because the definition of weighted solidity, although based on the blade planform, has a hidden assumption in that the Q distribution over the blade is assumed to be nominally constant. If this is not the case, then the definition breaks down. Consider a linearly tapered blade that can be described by o(r) = oq + or, where oq and oj are constants. The thrust weighted solidity is given by

= cr0 + – cri = solidity at 3/4 (75%) blade radius.

The corresponding power-torque weighted solidity is given by

4 / err[17] dr = 4 / (сто + <Уг)гъ dr

Jo Jo

= ao + – ai = solidity at 4/5 (80%) blade radius.

Similar results can be obtained for blades with taper extending only over part of the span (see Question 3.12). In this case, the integral for the equivalent weighted solidity must be split into two parts, where the inboard portion of the blade has a constant chord cr from the root to a point r = rі (i. e., where a = ar) and then a linear taper from Г] to the tip at r = 1 where the chord is ct (i. e., a = ^^£77) (r — 1) + a,). Then for this blade the equivalent thrust weighted solidity is given by

(3.157)

3.4.6 Mean Lift Coefficient

Another parameter that is useful in rotor analyses is the mean lift coefficient, Cl. The mean lift coefficient3 is defined to give the same thrust coefficient as

Ct = f ar2ci dr

when the entire blade is assumed to be operating at Q = Cl – Thus,

CT = і f crr2CL dr = }-Cl f err2 dr = aCL 2 Jo 2 Jо o

or simply that the mean lift coefficient is

Therefore, the quantity Cl/Ci0 can be viewed as a mean AoA of the blades, a. Because the values of blade loading coefficient, Cj jcs, for a contemporary helicopter rotor will vary between 0.08 and 0.12, it is found that Cl is the range 0.5 to 0.8, and so a for hovering flight will vary from about 5 to 8 degrees. Remember that, like the weighted solidity factors, the mean lift coefficient is derived on the basis of uniform Q. In practice, however, for any rotor an estimate of the mean lift coefficient remains a good overall indicator of the average working state of the blades.

Based on the simple corrected momentum theory developed in Chapter 3, notice that the figure of merit can be written in terms of Cl as

which simply confirms what is known all along, in that the use of airfoils with a high average lift-to-drag ratio is required for good hover performance (see also Question 3.7).