Hovering Performance

The helicopter rotor in hover or in vertical climb is relatively easy to analyze in comparison with its other states of operation. Neither the blade forces nor the blade pitch varies with azimuth position. In addition, a trailing vortex pattern is established underneath the rotor, which makes possible the application of propeller vortex theory.

The principles developed in Chapter 4 can be applied directly to the heli­copter in hover. Hence from Eq. (4-10) the ideal power required by a hovering rotor would be

Pt = Tw

J3/2

~ у/Щ’

In conformity with past practice, thrust and power coefficients for a helicopter rotor are defined by

(5-3)

In terms of these coefficients, the thrust and ideal power are related by

The power, according to Eq. (5-1) or (5-4), is the least possible power with which the thrust of the rotor can be attained. This ideal power is low for two reasons. First, the momentum theory ignores any profile drag of the blades and, second, the actuator-disk concept is optimistic because of the tip losses incurred by a physical rotor with a finite number of blades.

A hovering rotor certainly performs a useful function, but, regardless, it accomplishes no useful work, so that its efficiency is always zero. Therefore the performance of hovering rotors is sometimes evaluated on the basis of the figure of merit M. This parameter is defined as

Pi

P’ where Pi = ideal power according to Eq. (5-1),

P — actual total power required by the rotor.

If P is written as Pt + AP, then M can be written as

ACP, for a given rotor, does not depend appreciably on the thrust co­efficient. Thus the figure of merit of a given rotor will approach unity as the thrust coefficient increases, provided the rotor does not stall or encounter compressibility. Therefore it is important when comparing two different rotors by use of the figure of merit to compare them at equal thrust co­efficients ; otherwise the comparison can be misleading.

Consider a family of rotors, all of which produce the same thrust and have the same tip speed and solidity. Solidity, as defined in Chapter 4, is the ratio of total blade area to disk area and for a rectangular blade is equal to

В being the number of blades and c, the constant chord. As the radius of the rotor is increased, the induced power for the constant thrust will de­crease. However, the profile power will increase with increasing radius. Thus there will be some optimum radius for which the required power will be a minimum. This radius can be found approximately as shown in Ref. 2 by assuming that the profile drag coefficient Cd is a constant. For a rectangular blade the total power in hover is given approximately by

p _ T312 , P°Cd0V3rnR2

– t + C>R1-

To find the optimum R, dP/dR is equated to zero. dP

M

or, multiplying by R,

= 2 C2R2.

Thus for the optimum radius the induced power is equal to twice the profile power, and the figure of merit for the optimum physical rotor (which must have some profile drag) is

M = f.

The foregoing analysis is possibly not too realistic. In defining the optimum rotor for a given application, we must consider other restrictions such as structural and controllability requirements. The root stresses on a blade of uniform cross section result primarily from the centrifugal forces. Since these forces vary directly with the cross-sectional area of the blade, the root stresses depend primarily on the rotor tip speed and do not vary with the chord (assuming that the thickness-to-chord ratio is a constant). For controllability we usually specify an average lift coefficient for the rotor. Because, approximately,

T = В J ±p(cor)2cC, dr,

if C, and c are assumed constant, an average С, = C, can be calculated as

Thus, in comparing one rotor with another for a given application, we should probably hold CL constant, which means that a will vary in accord­ance with the above.

In terms of CL, the total power can be written

p _ T3/2 3TVT cd

Hence for the same CL the profile power is not a function of radius. From the above, from a purely aerodynamic standpoint, the best rotor is one with large radius and low tip speed. Practically, other considerations that must be taken into account include transmission weight as the tip speed is reduced and blade weight and blade clearance problems as the blade radius

increases.

It is very enlightening to express the figure of merit in terms of CL.

or

Thus the figure of merit depends on the section drag-to-lift ratio and on the ratio of tip velocity to the downwash velocity.

To allow a margin below the stall, most rotors are designed for a CL of approximately 0.5. At this value of CL the drag-lift ratio is approximately

0. 021. Typical values of the ratio of tip velocity to downwash velocity range from approximately 17 to 23. Hence a typical figure of merit would be approximately 0.76.

In several respects the figure of merit is useless. As an academic exercise, it is interesting, but in a practical application we must consider the power; for example, for a constant T, VT, and CL, increasing the disk loading improves the figure of merit but increases the total power.

The fact that one rotor has a higher figure of merit than another is not sufficient to indicate its relative superiority but might mean simply that the first rotor is acting at a higher thrust coefficient so that its induced power is high.