Effect of Taper

Constant chord blades are easy to design and to manufacture, but tapered blades can be made to be more efficient aerodynamically. A special combination of taper and twist can produce not only the uniform induced velocity that is the special characteristic of blades with ideal twist, but can also make the local angle of attack constant, thus giving the opportunity to operate each blade element at the airfoil’s most efficient angle of attack where c]/2/cd is a maximum.

If a constant angle of attack, and therefore a constant lift coefficient, is to be maintained along the blade, then the increment of thrust on the annulus of Figure

1.18 must be:

AT = b ^ (f2r)V, cAr

but, from the momentum theory,

or, with some algebraic manipulation,

If both v1 and ct are to be constants, then the quantity cr must be a constant. This can be accomplished by defining the local chord, c, such that:

This type of taper produces a rotor that looks like the one in Figure 1.25—one that is impractical to build but is interesting in being theoretically the most optimal hovering rotor that can be designed. If the lift coefficient is to be kept constant, then the local angle of attack must be constant where:

In order that both a and vl are constants, then the blade must be twisted so that:

This special twist distribution is shown in Figure 1.26, where it is compared with the ideal twist for a constant-chord blade. A series of calculations for the ideally twisted, constant-chord blade discussed in the previous paragraph shows that this rotor has a maximum Figure of Merit of 0.85 at an average lift coefficient of 0.94.

FIGURE 1.25 Rotor with Ideal Taper

If the angle of attack had been a constant, the Figure of Merit would have been the value from the equation for maximum Figure of Merit:

Thus ideal taper can increase the maximum hover performance about 4% over a blade with constant chord and ideal twist. Since the ideally tapered blade is impractical to build, it is sometimes approximated with a blade that has either a linear taper or a constant chord out to some radius station and a linear taper from that point. Such a rotor, when properly twisted, can achieve a portion of the benefit of an ideally tapered and twisted blade. Even without twist, taper is aerodynamically beneficial. A study of the equation for the nonuniform induced velocity distribution will show that a constant induced velocity can be achieved with a constant pitch if the chord is inversely proportional to the blade station, as well as for a constant chord with the pitch inversely proportional to the blade station. Reference 1.10 reports a comparison of two model rotors, each with untwisted blades, which were identical except that one was untapered and one had a taper ratio of 2 to 1. As can be seen in Figure 1.27, the rotor with the tapered blades had about a 10% performance advantage at low and moderate thrust values but suffered earlier stall. Reference 1.10 attributes the earlier stall to a larger amount of tip vortex interference, but it is also possible that the lower tip Reynolds number resulted in a significantly lower maximum lift coefficient.