Blade mean lift coefficient

Blade mean lift coefficient

Characteristics of a rotor obviously depend on the lift coefficient at which the blades are operating and it is useful to have a simple approximate indication of this. The blade mean lift coefficient provides such an indication. As the name implies the mean lift is that which, applied uniformly along the blade span, would give the same total thrust as the actual blade. Writing the mean lift coefficient as L we have, from Equation (3.12),

from which


The parameter CT/s is thus of fundamental importance and this explains the preference some workers have for using it as the definition of thrust coefficient (see Equation (3.18)) instead of CT. Expanding the definition gives

Ideal twist

Ideal twist

The relation in Equation (3.33) contains one particular case when l is indeed constant, namely if qr is constant along the span, that is qt being the pitch angle at the tip. This non-linear twist is not physically realizable near the root but the case is of interest because, as momentum theory shows, uniform induced velocity corresponds to minimum induced power. The analogy with elliptic loading for a fixed-wing aircraft is again recalled. The twist in Equation (3.34) is known as ideal twist. Inserting in Equation (3.21) gives

Non-uniform inflow A questionable assumption which has been made so far is that the induced velocity is uniform across the blade span. The effect of nonuniformity can be allowed for by using differential forms of the appropriate equations in the combination of blade element theory and momentum theory. Equation (3.21) in the blade element theory is replaced by

 Non-uniform inflow

A questionable assumption which has been made so far is that the induced velocity is uniform across the blade span. The effect of nonuniformity can be allowed for by using differential forms of the appropriate equations in the combination of blade element theory and momentum theory. Equation (3.21) in the blade element theory is replaced by  which expresses the element of thrust on an annulus of the disc at radius r. The corresponding equation from momentum theory, again using the hover case for simplicity, is the replacement of Equation (2.2), namely.

Rotor in Vertical Flight: Blade Element Theory

Rotor in Vertical Flight: Blade Element Theory

Basic method

Blade element theory is basically the application of the standard process of aerofoil theory to the rotating blade.A typical aerodynamic strip is shown in Fig. 3.1 and the appropriate notation for a typical strip is shown in Fig. 3.2. Although in reality flexible, the blade is assumed throughout to be rigid, justification for this lying in the fact that at normal rotation speeds the outward centrifugal force is the largest force acting on a blade and in effect is sufficient to hold the blade in rigid form. In vertical flight, including hover, the main complication is the need to integrate the elementary forces along the blade span. Offsetting this, useful simplification occurs because the blade incidence and induced flow angles are normally small enough to allow small-angle approximations to be made. Fig. 3.3 is a plan view of the rotor disc, viewed from above. Blade rotation is anticlockwise (the normal system in Western-world countries) with angular velocity W.The blade radius is R,the tip speed therefore being WR, alternatively written as Vt. An elementary blade section is taken at radius y, of chord length c and spanwise width dy. Forces on the blade section are shown in Fig. 3.4. The flow seen by the section has velocity components Wy in the disc plane and (vi + Vc) perpendicular to it. The resultant of these is The blade pitch angle, determined by the pilot’s collective control setting (see Chapter 4), is q. The angle between the flow direction and the plane of rotation, known as the inflow angle is f, given by 

Wake analysis methods

 Wake analysis methods

By analysis of his carefully conducted series of smoke-injection tests, Landgrebe5 reduced the results to formulae giving the radial and axial coordinates of a tip vortex in terms of azimuth angle, with corresponding formulae for the inner sheet. From these established vortex positions the induced velocities at the rotor plane may be calculated. The method belongs in a general category of prescribed-wake analysis, as do earlier analyses by Prandtl, Goldstein and Theodorsen, descriptions of which are given by Bramwell. These earlier forms treated either a uniform vortex sheet as pictured in Fig. 2.11 or the tip vortex in isolation, and so for practical application are effectively superseded by Landgrebe’s method. More recently, considerable emphasis has been placed on free-wake analysis, in which modern numerical methods are used to perform iterative calculations between the induced velocity distribution and the wake geometry, both being allowed to vary until mutual consistency is achieved. This form of analysis has been described for example by Clark and Leiper6. Generally the computing requirements are very heavy, so considerable research effort also goes into devising simplified free-wake models which will reduce the computing load. Calculations for a rotor involve adding together calculations for the separate blades. Generally this is satisfactory up to a depth of wake corresponding to at least two rotor revolutions. A factor which helps this situation is the effect on the tip vortex of the upwash ahead of the succeeding blade – analogous to the upwash ahead of a fixed wing. The closer the spacing between blades, the stronger is this effect from a succeeding blade on the tip vortex of the blade ahead of it; thus it is observed that when the number of blades is large, the tip vortex remains approximately in the plane of the rotor until the succeeding blade arrives, when it is convected downwards. In the ‘far’ wake, that is beyond a depth corresponding to two rotor revolutions, it is sufficient to represent the vorticity in simplified fashion; for example free-wake calculations can be simplified by using a succession of vortex rings, the spacing of which is determined by the number of blades and the mean local induced velocity. Eventually in practice both the tip vortices and the inner sheets from different blades interact and the ultimate wake moves downward in a confused manner. 

Complexity of real wake

Complexity of real wake

The actuator disc concept, taken together with blade-element theory, serves well for the purposes of helicopter performance calculation. When, however, blade loading distributions or vibration characteristics are required for stressing purposes it is necessary to take into account the real nature of flow in the rotor wake. This means abandoning the disc concept and recognizing that the rotor consists of a number of discrete lifting blades, carrying vorticity corresponding to the local lift at all points along the span.Corresponding to this bound vorticity a vortex system must exist in the wake (Helmholtz’s theorem) in which the strength of wake vortices is governed by the rate of change of circulation along the blade span. If for the sake of argument this rate could be made constant, the wake for a single rotor blade in hover would consist of a vortex sheet of constant spanwise strength, descending in a helical pattern at constant velocity, as illustrated in Fig. 2.11. The situation is analogous to that of elliptic loading with a fixed wing, for which the induced drag (and hence the induced power) is a minimum. This ideal distribution of lift, however,is not realizable for the rotor blade, because of the steadily increasing velocity from root to tip. The most noticeable feature of the rotor-blade wake in practice is the existence of a strong vortex emanating from the blade tip, where because the velocity is highest the rate of change of lift is greatest. In hover the tip vortex descends below the rotor in a helical path. It can be visualized in a wind tunnel using smoke injection (Fig. 2.12) or other means and is often observable in open flight under conditions of high load and high humidity. An important feature which can be seen in Fig. 2.12 is that on leaving the blade the tip vortex initially moves inwards towards the axis of rotation and stays close under the disc plane: in consequence the next tip to come round receives an upwash, increasing its effective incidence and thereby intensifying the tip vortex strength. Figure 2.13 due to J.P. Jones2 shows a calculated spanwise loading for a Wessex helicopter blade in hover and indicates the tip vortex position on successive passes. The kink in loading distribution at 80% span results from this tip vortex pattern, particularly from the position of the immediately preceding blade. The concentration of the tip vortex can be reduced by design changes such as twisting the tip nose-down, reducing the blade tip,area or special shaping of the planform, but it must be borne in mind that the blade does its best lifting in the tip region where the velocity is high. Since blade loading increases from the root to near the tip (Fig. 2.13), the wake may be expected to contain some inner vorticity in addition to the tip vortex.This might appear as a form of helical sheet akin to that of the illustration in Fig. 2.11, though generally not of uniform strength. Definitive experimental studies by Gray3, Landgrebe4 and their associates have shown this to be the case. Thus the total wake comprises essentially the strong tip vortex and an inner vortex sheet, normally of opposite sign. The situation as established by Gray and Landgrebe is pictured, in a diagram which has become standard, by Bramwell (p. 117) and other authors. Figure 2.14 is a modified version of this diagram, intended to indicate vortex lines making up the inner sheet,emanating from the bound vorticity on the inner part of the blade. The Gray/Landgrebe studies show clearly the contraction of the wake immediately below the rotor disc. Other features which have been observed are that the inner sheet moves downward faster than the tip vortex and that the outer part of the sheet moves faster than the inner part, so the sheet becomes increasingly inclined to the rotor plane.

Summary remarks on momentum theory

 Summary remarks on momentum theory

The place of momentum theory is that it gives a broad understanding of the functioning of the rotor and provides basic relationships for the induced velocity created and the power required in producing a thrust to support the helicopter. The actuator disc concept, upon which the theory is based, is most obviously fitted to flight conditions at right angles to its plane, that is to say the hover and vertical flight states we have discussed. Nevertheless further reference to the theory will be made when discussing forward flight (Chapter 5). Momentum theory brings out the importance of disc loading as a gross parameter: it cannot however look into the detail of how the thrust is produced by the rotating blades and what design criteria are to be applied to them. For such information we need additionally a blade element theory, corresponding to aerofoil theory in fixed-wing aerodynamics: to this we shall turn in Chapter

Complete induced-velocity curve

 Complete induced-velocity curve

It is of interest to know how the induced velocity varies through all the phases of axial flight. For the vortex-ring and turbulent-wake states, where momentum theory fails, information has been obtained from measurements in flight, supported by wind tunnel tests (Gustafson (1945), Gessow (1948), Brotherhood (1949), Castles and Gray (1951) and others). Obviously the making of flight tests (measuring essentially the rate of descent and control angles) is both difficult and hazardous, especially where the vortex-ring state is prominent, and not surprisingly the results show some variation: nevertheless the main trend has been ascertained and what is effectively a universal induced-velocity curve can be defined. This is shown in Fig. 2.10, using the simple momentum-theory results of Equations (2.10) and (2.13) in the regions to which they apply. We see that moving from hover into descent the induced velocity increases more rapidly than momentum theory would indicate. The value rises, in the vortex-ring state, to about twice the hover value, then falls steeply to about the hover value at entry to the windmill-brake state.

Vertical descent

 In vertical descent the nature of flow through the rotor undergoes significant changes. The stream velocity Vc is now negative while the induced velocity vi remains positive as the rotor continues to maintain lift. Initially small recirculating regions develop around the blade tips, as shown in Fig. 2.5. Becoming evident when Vc reaches a level about half vi, an interaction takes place between the upward flow around the disc and the downward flow through it, resulting in the formation of a vortex ring encircling the rim of the disc, doughnut fashion.The situation is illustrated in Fig.2.6.As this vortex-ring state develops the flow becomes very unsteady and the rotor exhibits high levels of vibration. It appears that the ring vortex builds up strength and periodically breaks away from the disc, spilling haphazardly into the flow and causing fluctuations in lift and also in helicopter pitch

and roll. Flight in the developed vortex-ring state, which reaches its worst condition when the descent rate is about three quarters of the hover induced velocity, is unpleasant and potentially dangerous. Because of the dissipation of energy in the unsteady flow, simple momentum theory cannot be applied. As the descent rate approaches the level of the induced velocity, a modified state is observed in which, corresponding to the near equality,there is little or no net flow through the disc.Now the flow is characterized by vortices shed into the wake in the manner of the flow around a solid bluff body. In this turbulent-wake state (Fig. 2.7) flight is still rough but less so than in the vortex-ring state. Simple momentum theory is again not applicable, since energy is dissipated in the eddies of the wake. At large descent rates, when Vc is numerically greater than about 2vi, the flow is everywhere upwards relative to the rotor, producing a windmill-brake state, in which power is transferred from the air to the rotor. With a flow pattern as in Fig. 2.8, simple momentum theory gives a reasonable approximation: thus with Vc negative and vi positive the thrust is:

Momentum theory for hover

The helicopter rotor produces an upward thrust by driving a column of air downwards through the rotor plane. A relationship between the thrust produced and the velocity communicated to the air can be obtained by the application of Newtonian mechanics – the laws of conservation of mass,

momentum and energy – to the overall process. This approach is commonly referred to as the momentum theory for helicopters.It corresponds essentially to the theory set out by Glauert1 for aircraft propellers, based on earlier work by Rankine and Froude for marine propellers. The rotor is conceived as an ‘actuator disc’, across which there is a sudden increase of pressure, uniformly spread. In hover the column of air passing through the disc is a clearly defined streamtube above and below the disc: outside this streamtube the air is undisturbed. No rotation is imparted to the flow.

The situation is illustrated in Figs 2.1a–2.1c. As air is sucked into the disc from above, the pressure falls. An increase of pressure Dp occurs at the disc, after which the pressure falls again in the outflow, eventually arriving back at the initial or atmospheric level p•. Velocity in the streamtube increases from zero at ‘upstream infinity’ to a value vi at the disc and continues to increase as pressure falls in the outflow, reaching a value v• at ‘downstream infinity’. Continuity of mass flow in the streamtube requires that the velocity is continuous through the disc. Energy conservation, in the form of Bernoulli’s equation, can be applied separately to the flows before and after the disc. Using the assumption of incompressible flow, we have in the inflow: