Category BASIC HELICOPTER AERODYNAMICS

The vast majority of helicopters are of the single main and tail rotor type. So far, any discussion has focused, primarily, on the main rotor. It provides most of the forces and moments required

to fly successfully and under full control. However, it needs a mechanism for yaw control and this is where the tail rotor contributes. On a pure numbers basis, the main rotor provides control in 5 degrees of freedom:

(c) Vertical motion

(d) Yaw

However, it has to provide this yaw control under very particular circumstances and in areas where the aerodynamic and dynamic forces on the blades are a real problem. It is positioned on a fin which is directly behind the main rotor hub and installation fairings. The tail rotor produces a thrust perpendicular to its disc plane which is reacted by a small lateral tilt of the

main rotor in opposition. It is placed at a vertical location at approximately the same height as the main rotor head in order to minimize any subsequent roll coupling – see Figure 6.13.

Its location at the rear extremity of the airframe can lead to dangers in the event of extreme manoeuvres as exemplified by the rapid deceleration shown in Figure 6.14.

The position of the tail rotor, behind the main rotor hub and top fuselage, results in the forward flight incident flow not being uniform over the tail rotor disc. Figure 6.15 shows a window placed in front of the tail rotor with an indication of the incident mean flow velocity. Efficiency is usually associated with uniformity and the tail rotor is no different. In order to account for the higher flow at the top of the tail rotor disc and the consequent lower flow at the

bottom, a tail rotor rotational direction of backwards at the top would seem advisable. In most conventional helicopters, this is the case; however, there are exceptions. As will be discussed later, the early Westland Lynx helicopter had the opposite rotational direction which caused difficulties in side winds – in other words, there are differing reasons why this rotational direction has merits.

Because of the flight of a helicopter, a tail rotor has to operate in areas not experienced by the main rotor. The tail rotor has to overcome the effects of sideways flight (in either direction), autorotation and high-power climb. These conditions will require a pitch range of the order of 40°, see Byham [9]. This puts the blades at considerable pitch angles, which will introduce kinematic effects which are not normally experienced by the main rotor. One important effect is that of propeller moment.

The loading on a helicopter blade is highly concentrated in the region of the tip, as has been seen (Figure 2.28). It is unlikely that a plain rectangular planform (a typical example is shown in Figure 6.6) is the optimum shape for the task of carrying this load and consequently investigations into tip design are a feature of modern aerodynamic research. Figure 6.8 shows the main rotor blade tip of the Merlin helicopter, which is the BERP planform with anhedral added (note the provision for picketing the blade and the leading edge erosion strip).

Figure 6.9 shows the main rotor blade tip of an Agusta Westland A129 variant. Since resultant velocities in the tip region on the advancing blade are close to Mach 1.0, it is natural to enquire whether sweepback can be incorporated to delay the compressibility drag rise and

thereby reduce the power requirement at a given flight speed or alternatively raise the maximum speed attainable. The answer is not so immediately obvious as in the case of a fixed wing, because a rotor blade tip which at one moment is swept back relative to the resultant airflow, in the next moment lies across the stream. In fact, however, the gain from sweepback outweighs the loss, as is indicated in a typical case by Wilby and Philippe [8] (Figures 6.10a and b): a large reduction in Mach number normal to the leading edge is obtained over the rear half of the disc, including a reduction in maximum Mach number of the cycle (near C = 90°), at the expense of a small increase in the forward sector (C = 130° to 240°). Reductions in power required were confirmed in the case shown.

Shaping the blade tip can also be used to improve the stalling characteristics of the retreating blade. A particular all-round solution devised by Agusta Westland Helicopters is pictured in Figure 6.11. The principal features are:

• approximately 20° sweepback of the outboard 15% of blade span;

• a forward extension of the leading edge in this region, to safeguard dynamic stability;

• a sharply swept outeredge topromote controlled vortex separation and thereby delay the tip stall.

Wind tunnel tests (static conditions) showed this last effect to have been achieved in remarkable degree (Figure 6.12). Subsequently the tip proved highly successful in flight and was used on a version of the Lynx helicopter which captured the world speed record (see Chapter 7).

6.1.1 Rectangular

Performance of a rotor blade is governed, naturally, over its entire length. The tip region, however, plays a very influential role in the blade’s aerodynamic character. For most of its formative years the helicopter rotor blade possessed a uniform blade chord out to the tip giving it a rectangular planform. A typical example is shown in Figure 6.6. The blade usually has a set of weights firmly fixed at the tip end for blade track and balance adjustments to be carried out. These are normally covered by a tip cap which covers these weights and restores an aerodynamically clean shape to the blade tip. For symmetric aerofoils, it is often a surface of revolution about the chordline.

6.3.2 Swept

The first move away from the rectangular blade planform was the inclusion of sweep in the tip region. It was abrupt and the sweep angle was constant. This is aimed at the advancing blade tip where high Mach numbers will be encountered at the higher forward speeds. The rearward

movement of the blade chord at the tip end of the blade will require careful design as the lift centre for the tip and the local centre of gravity will now be behind the blade pitch axis and its shear centre. This will open up the possibility of blade flexing in pitch which can cause aeroelastic problems. An example of this type of tip planform is shown in Figure 6.7.

In the design of rotor blade sections there is an a priori case for following the lead given by fixed-wing aircraft. It could be said, for instance, that the use of supercritical aerofoil sections for postponing the drag-rise Mach number is as valid an objective for the advancing blade of a rotor as for the wing of a high-subsonic transport aircraft. Or again, the use of blade camber to enhance maximum lift may be as valuable for the retreating blade as for a fixed wing approaching stall. Having accepted, say, this latter principle, there remains the problem of adapting it to the helicopter environment: this requires focused research, and substantial progress has been achieved.

The widely ranging conditions of incidence and Mach number experienced by a rotor blade in forward flight are conveniently illustrated by a ‘figure-of-eight’ diagram (sometimes called

Basic Helicopter Aerodynamics, Third Edition. John Seddon and Simon Newman. © 2011 John Wiley & Sons, Ltd. Published 2011 by John Wiley & Sons, Ltd.

0.2 0.4

Boundaries – N АСА 0012

Drag Rise

(c) Stall & Compressibility

Boundaries – PL 9615

Figure 6.1 Figure-of-eight diagrams for a typical blade

sausage plots) (Figure 6.1a) which plots these conditions for a particular station on the blade near the tip (x = 0.91 in the case shown) at a specified value of ц. (In essence, a spanwise section of the blade is taken around a complete revolution and the various aerodynamic quantities recorded.) The hovering condition would be represented by a single point: as ц is increased the figure of eight expands, extending into regions of higher a (or CL) and higher Mach number (M).

Plotting on such a diagram the a-M loci of CL Max and MD (the drag-rise Mach number) for a particular blade section, these being obtained independently, as for example by two­dimensional section tests in a wind tunnel, will give an indication of whether either blade stall or drag divergence will be encountered in the rotor at the particular level of ц. Effectively, the figure-of-eight plot tracks a particular location of a rotor blade around one complete rotation of the rotor. This will highlight instances where the aerofoil section crosses over boundaries such as drag rise and stall. The example in Figure 6.1b relates to a symmetrical, 12% thick, NACA 0012 section. It is seen that the retreating-blade loop passes well into the stalled region and the advancing-blade loop likewise into the drag-rise region.

NACA 0012 was the standard choice for helicopter blade sections over many years. It is symmetric and is expressed mathematically. The function gives the section the following features:

• a parabolic nose shape of given radius;

• a specified thickness/chord ratio with maximum thickness at a specified chordwise location;

• the trailing edge has a given thickness and included angle.

Modern aerofoil sections embodying camber to increase maximum lift have been developed in various series, of which the ‘VR’ Series in the USA and the ‘96’ Series in the UK are examples. Results for a 9615 section – the basis for the Westland Lynx blade – are shown in Figure 6.1c. The figure of eight now lies wholly within the CL Max locus, confirming an improvement in lift performance. Additionally the high-Mach-number drag rise now affects a much reduced portion of the retreating-blade loop, and the advancing-blade loop not at all, so a reduction in power requirement can be expected. It should be borne in mind that any improvement in thrust capability will automatically incur an increase in induced and climb power.

The evidence, though necessary, is not of itself sufficient, however. To ensure acceptability of the cambered section for the helicopter environment, additional aspects of a major character need to be considered. One is the question of section pitching moments. The use of camber introduces a nose-down CM0 (pitching moment at zero lift), which has an adverse effect on loads in the control system. A gain in CL Max must therefore be considered in conjunction with the amount of CM0 produced. One way of controlling the latter is by the use of reflex camber over the rear of a profile. Wilby [1] gives comparative results for a number of section shapes of the ‘96’ Series, tested in a wind tunnel under two-dimensional steady-flow conditions.

A selection of his results appears in Figure 6.2, from which we can see that the more spectacular gains in CL Max (30-40%) tend to be associated with more adverse pitching

moments, especially above a Mach number of approximately 0.75, which would apply on the advancing side of a rotor. Generally, therefore, compromises must be sought through much careful section shaping and testing. Moreover, while aiming to improve blade lift performance for the retreating sector, care must be taken to see that the profile drag is not increased, either at low CL and high Mach number for the advancing sector, or at moderate CL and moderate Mach number for the fore and aft sectors which in a balanced forward flight condition will carry the main thrust load. Figure 6.3 shows the variation of aerofoil sections on the BERP blade fitted to the fifth pre-prototype (PP5) EH101 aircraft. Each section is specifically designed for the particular incidence/Mach number ranges that it will experience.

While static testing of this nature is very useful in a comparative sense, it cannot be relied upon to give an accurate final value of CL Max, because the stall of a rotor blade in action is known to be dynamic in character, owing to the changes in incidence occurring as the blade passes through the retreating sector. Farren [2] recorded, as long ago as 1935, that when an aerofoil is changing incidence, the stalling angle and CL Max may be different from those occurring under static conditions. Carta [3] in 1960 reported oscillation tests on a wing with 0012 section suggesting that this dynamic situation would apply in a helicopter context. Figure 6.4 shows a typical result of Carta’s tests. When the aerofoil was oscillated through 6° on either side of 12° incidence (just above the static stalling angle), with a representative rotor frequency, a hysteresis loop in lift coefficient was obtained, in which the maximum CL reached during incidence increase was about 30% higher than the static level.

Many subsequent researchers, among them Ham [4], McCroskey [5], Johnson and Ham [6] and Beddoes [7], have contributed to the provision of data and the evolution of theoretical treatments on dynamic stall and in the process have revealed the physical nature of the flow, which is of intrinsic interest. As blade incidence increases beyond the static stall point, flow reversals are observed in the upper surface boundary layer – but for a time these are not transmitted to the outside potential flow. Consequently the lift continues to increase with incidence. Eventually, flow separation develops at the leading edge (or it may be behind

a recompression shock close to the leading edge), creating a transverse vortex which begins to travel downstream. As the vortex rolls back along the upper surface into the mid-chord region, lift continues to be generated but a large nose-down pitching moment develops owing to the redistribution of upper surface pressure. The passage of the vortex beyond the trailing edge results in a major breakdown of flow. Finally, when the incidence falls below the static stall angle as the blade approaches the rear of the disc, the flow reattaches at the leading edge and normal linear lift characteristics are re-established. The important consideration is that the aerodynamic influences with increasing incidence are not matched with the following reduction in incidence. This is the basis for the hysteretic effects seen in oscillating aerofoils.

Some further results for the RAE 9647 aerofoil section are shown in Figure 6.5, in this case from blade oscillation tests over four different incidence ranges. As the range is moved up the incidence scale, the hysteresis loop develops in normal force coefficient (representing CL) and the pitching moment ‘break’ comes into play. In practice it is the latter which limits the rotor thrust, by reason of the large fluctuations in pitch control loads and in blade torsional vibrations which are triggered. It is of interest to note that in the results shown, the normal coefficient reached at the point of pitching moment break is about 1.8. Considerably higher values may in fact be attained; however, it is to be noted that this value on the retreating blade is not particularly important in itself, since what matters more is the amount of lift produced by the other blades in the fore and aft sectors, where in a balanced rotor the major contributions to thrust are made.

Initially, one saw a situation on blade section design still capable of further development. The emphasis was placed on improving the lift capability of the retreating blade. As the aspect of fuel economy in helicopter flight gains in importance, the incentive grows to reduce blade profile drag, particularly for the advancing sector. In this area there are probably improvements to be had by following the lead given by fixed-wing aircraft in the use of so-called supercritical wing sections. A further comment putting the incentive into context is made in Chapter 7. What is now under scrutiny is the influence of efficiency. This is a result of the perceived need to conserve energy resources and the aeroelastic behaviour of a blade is now being used to

improve performance and to reduce both fuel consumption and vibratory influences. The initial aeroelastic effects are passive in nature; however, the research direction is now directed towards active aerodynamic techniques such as blade morphing and tip blowing. This allows blade camber to be adjusted in a live environment and to control the nature and strength of the tip vortex.

6.1 Introductory

In this chapter are described some of the trends in aerodynamic design which in the latter part of the twentieth century and the beginning of the twenty-first century are making the helicopter a considerably more efficient flying vehicle than it formerly was. In earlier years the low power – to-weight ratio of piston engines necessitated the use of large rotors to provide the all-important vertical lift capability: both profile drag and parasite drag were unavoidably high in conse­quence and forward speeds were therefore so low as to consign the problems of refining either the lift or drag performance to a low, even zero, priority. With the adoption of gas-turbine engines, and an ever-increasing list of useful and important applications for helicopters, in both military and civil fields of exploitation, forward flight performance has become a more lively issue, even to the point of encouraging comparisons with fixed-wing aircraft in certain specialized contexts (an example is given in Chapter 7). Some improvements in aerodynamics stem essentially and naturally from fixed-wing practice. A stage has now been reached at which these appear to be approaching and, in certain areas, to have arrived at, optimum levels in the helicopter application and therefore a substantial description here is appropriate. Further enhancements, concerned with the fundamental nature of the rotor system, may yet emerge to full development: one such is the use of higher harmonic control, which is described briefly. In the concluding section an account is given of a step-by-step method of defining the aerody­namic design parameters of a new rotor system.

Calculations have been made to illustrate in broad fashion the ways in which parameters discussed in the foregoing analysis vary with one another and particularly with forward speed. For this purpose the following values have been used:

The parasite drag factor is a form of expression in common use, in which f is the ‘equivalent flat plate area’ defined by:

Dp = 2 pV2f (5.88)

Dp being the parasite drag and A the rotor disc area.

Figure 5.21a shows the variation of inflow factor l with advance ratio m at two levels of thrust coefficient. As previously mentioned, l, as defined in Equation 5.39, is relative to the TPP, so is denoted by 1T in the diagram. The variation shows a minimum value at moderate m, inflow being high at low m because the induced velocity is large and high again at high m because of the increased forward tilt of the TPP required to overcome the parasite drag. The lower the thrust coefficient, the more marked the high-m effect.

Figure 5.21b shows the corresponding variation of collective thrust angle в, for CT/s = 0.2. The variations of в and l are similar in character, as might be expected from Equation 5.87.

Combination of Equations 5.39 and 5.87 leads, on elimination of l, to a direct relationship between CT and в which, using the chosen values of aircraft weight ratio and parasite drag factor in the final term, is:

в = з ^ + 2 m^ + m^3 ~2m2^ (5.89)

where B is a slowly decreasing function of m.

Note that when m is zero, B = 1/3 and Vj/Vj0 = 1, so that we have Equation 3.29 as previously derived for the hover. Figure 5.22 shows variations of в with CT for different levels of m. The characteristics at low and high forward speed are significantly different. When m is zero or small the variation is nonlinear, в increasing rapidly at low thrust coefficient owing to the induced flow term (the second expression in the equation) and more slowly at higher CT as the first term becomes dominant. At high m, however, the induced velocity factor Vi/Vi0 is so small that the second term becomes negligible for all CT, so the e/CT relationship is effectively linear. The intercept on the в axis reflects the particular value of m while, more interestingly,

with m and s known the slope is a function only of the lift slope a. This provides an experimental method for determining a in a practical case.

A final illustration (Figure 5.23) shows the flapping coefficients a0, aj and b as functions of m. These have been calculated using Equations 5.84-5.86. The coning angle a0 varies only slightly with m, being essentially determined by the thrust coefficient. It may readily be shown in fact that a0 is approximately equal to (3CTg)/(8sa) which with our chosen numbers has the value 0.105 rad or 6.0°. The longitudinal coefficient aj is approximately linear with forward speed, showing, however, an effect of the increase of l at high speed. The lateral coefficient b1 is also approximately linear, at about one-third the value of a1. In practice b1 at low speeds depends very much on the longitudinal distribution of induced velocity (assumed uniform throughout the calculations) and tends to rise to an early peak as indicated by a modified line in the diagram.

References

1. Glauert, H. (1926) A general theory of the autogiro, R & M, 1111.

2. Newman, S. J., Brown, R., Perry, J. et al. (2001) Comparitive numerical and experimental investigations of the vortex ring phenomenon in rotorcraft. 57th American Helicopter Society Annual National Forum, Washington, DC, 9-11 May, Vol. 2, pp. 1411-1430.

3. Drees, J. and Hendal, W. (1951) Airflow patterns in the neighbourhood of helicopter rotors. J. Aircr. Eng., 23, 107-1112.

4. Perry, F. J. (2000) Vortex ring instability in axial and forward flight – comparisons with test, Private Technical Note 0002/2000, June.

5. Brand, A., Kisor, R., Blyth, R. et al. (2004) V-22 high rate of descent (HROD) test procedures and long record analysis. 60th American Helicopter Society Annual National Forum, Baltimore, MD, June 7-10.

6. Mangler, K. W. and Squire, H. B. (1950) The induced velocity field of a rotor, R & M, 2642.

7. Bennett, J. A.J. (1940) Rotary wing aircraft. Aircr. Eng. Aerosp. Technol., 12,139-146.

8. Stepniewski, W. Z. (1973) Basic aerodynamics and performance of the helicopter. AGARD Lecture Series, 63.

The flapping motion is determined by the condition that the net moment of forces acting on the blade about the flapping hinge is zero. Referring back to Figure 4.12, the forces on an element dr of blade span, of mass m dr, where m is the mass per unit span, are:

• the aerodynamic lift, expressed as an element of thrust dT, acting on a moment arm r;

• a centrifugal force rO2m dr, acting on a moment arm rb;

• an inertial force rfim dr, acting on a moment arm r;

• a blade weight moment, small in comparison with the rest and therefore to be neglected.

These lead to the flapping moment relationship given in Equation 4.3. Writing the aerodynamic or thrust moment for the time being as MT, we have:

•R fR

br2O2m dr + br2m dr = MT

Now MT may be written:

	pR (0UT-UPUT)r dr 0T
1 — pac
(5.77)
MT

so that, in dimensionless form:

1 2 g

uP up x dx

(5.78)

a0

where g is defined by:

and is known as the Lock number. It provides a ratio between the aerodynamic forces and the inertial forces which determine the centrifugal loads. Replacing uT and uP by their definitions in Equations 5.37 and 5.43, and substituting for b and db/dC, the right-hand side of Equation 5.78 develops to:

1

2 g

where fS and fC represent functions in sin ф and cos ф respectively.

Since MT is independent of C, its value can be obtained by integrating only the first part of this expression. Hence:

1 ao = 2g	y( x2 + 2m2
-lx x dx
(5.81)
У ( x3 + — m2x ) —lx2 dx
0
41′ T

This is for an untwisted blade (У = constant У0) or in the usual way for a linearly twisted blade with У taken at three-quarters radius.

Also, because of the independence of MT the terms in sin C and those in cos C are each separately equatable to zero. These two equations yield expressions for the first harmonic coefficients aq and bq, namely:

4

3 m^o 1 + 2 m"

The three equations immediately above represent the classical definitions of flapping coefficients, in which У and 1 have been defined relative to the NFP. Equivalent, though rather more complex, definitions relative to the TPP are given by Johnson (p. 189) or Bramwell (p. 157). Bramwell’s equations, while not completely general, are probably accurate enough for most purposes and are quoted here for ease of reference (they are quoted in the order of calculation):

m

	(5.84) (5.85) (5.86)
12 1 + 2 m

The corresponding relationship for thrust coefficient is:

л . – zi ^t ma1

3 V1 + 2m j 2 2

From the preceding discussion, two reference planes have been quoted, namely the TPP and the NFP. The previous analysis used the NFP; however, the rotor downwash is intimately linked with the rotor disc or the TPP. Bramwell’s equations (5.84)-(5.87) quote the l term in terms of the TPP. To avoid confusion, the subscript T is used for the downflow term (1T). This also links naturally with the definition of l in Equation 5.39. The 1T notation is also used in Figure 5.21.

The elementary torque is:

dQ — dH ■ r

— r(dDcosf + dLsinf)

Again there is a profile drag term, dQ0 say, and an induced term dQi. The former is readily manipulated thus (in coefficient form):

o

— 4s ■ Cdo ■ (1+m2)

The induced term, after a lengthier manipulation, is shown (Bramwell, p. 151) to be:

CQi — ICT-rnCHi (5.63)

giving for the total torque:

Using Equation 5.60 this becomes:

Cq = 4 s • Cd0 • (1Am2) aICt—m^Ha2 s • Cdo • m2

4 2 (5-65)

= 4 s • Cdo • (1+3m^ a1Ct—mCH

Now by Equation 5.39 the inflow factor l is a function of the inclination ar of the TPP, which clearly depends upon the drag not only of the rotor but of the helicopter as a whole. Examining the relationships for trimmed level flight, illustrated in Figure 5.20, we have approximately:

(5.66)

Tar — H A Dp

Dp being the parasite drag of the fuselage, including tail rotor, tailplane and any other attachments.

Thus:

li A mar

^ Dp A m

W

Using this in Equation 5.65, the power coefficient is expressed in the form:

cp = cq = ^iCT + 4s • Cd^1 + 3m2) + m (5.70)

which is seen to be the sum of terms representing the induced or lift-dependent drag, the rotor profile drag and the fuselage parasite drag. The first of these had already been derived (Equation 5.22) when considering the adaptation of momentum theory to forward flight.

In practice both the induced and profile drag power requirements are somewhat higher than are shown in Equation 5.70. An empirical correction factor k for the induced power was suggested in Equation 5.23. For the profile drag power the deficiency of the analytical formula arises from neglect of:

• a spanwise component of drag (Figure 5.19);

• a yawed-wing effect on the profile drag coefficient at azimuth angles significantly away from 90° and 270°;

• the reverse flow region on the retreating side.

The first of these factors is probably the most important. They are conventionally allowed for by replacing the factor 3 in Equation 5.70 by an empirical, larger factor, k say. Studies by Bennett [7] and Stepniewski [8] suggest that an appropriate value is between 4.5 and 4.7. Industrial practice tends to be based on a firm’s own experience: thus a value commonly used by Westland Helicopters is 4.65.

With the empirical corrections embodied, the power equation takes the form:

Cp = kiliCt +ts • Cd0 (1 + km2) + m — Ct (5.71)

4 W

This will be followed up in the chapter on helicopter performance (Chapter 7). In the present chapter we take our analytical study of the rotor aerodynamics two stages further: firstly examining the nature of the flapping coefficients a0, aj and b in terms of У, l and m; and secondly looking at some typical values of collective pitch У, inflow factor l and the flapping coefficients in relation to the forward speed parameter m and the level of thrust coefficient CT.

In hover the in-plane H-force, representing principally the blade profile drag, contributed only to the torque. Here, however, since the resultant velocity at the blade is Or + Vsin C (Equation 5.35), the drag force on the advancing side exceeds the reverse drag force on the retreating side, leaving a net drag force on the blade, positive in the rearward direction.

Seen in azimuth (Figure 5.19) the elementary H-force, reckoned normal to the blade span and resolved in the rearward direction, is:

dH = (dD cos f + dL sin f )sin C (5.56)

which may be written as dH0 plus dHi, where the suffices relate to the profile drag and induced drag terms, respectively. Treating the drag term separately and making the usual approxima­tions, we have:

dH0 = dDsinC

In coefficient form, for N blades, this gives:

1 2

pNUj • c • CD0 • sinC dr dCHc = 2 у

2 p(OR)2 pR2

= s • uT • CD0 • sinC dx = s(x + msinC)2 • CD0 • sinC dx

o

— 2 s ■ Cdo ■ m

Overall then for the in-plane H-force, we have:

ch — 2 sCdo ■ m + CHi

Expressions can be obtained for the induced component CHi in terms of У, l, m and the flapping coefficients a0, аг and b: these are derived in varying forms in the standard textbooks, for example Bramwell (p. 148) and Johnson (p. 177). The relations are somewhat complex and since we shall not require to make further use of them in the present treatment, and moreover in the usual case CHi is small compared with CHo, we can be satisfied with the reduction at Equation 5.60.

Following the derivation of the hover analysis in Chapter 3 we write an elementary thrust coefficient of a single blade at station r as:

and for N blades, introducing the solidity factor s and non-dimensionalising,

dCT = s • uTCL dx

On expressing CL in the linear form (i. e. the blade is unstalled):

from which (5.46) becomes:

dCT = sa(euT~uPuT) dx

For the hover we were able to write uT = x, uP = l: in forward flight, however, uT and uP and in general в also, are functions of azimuth angle C The elementary thrust must therefore be averaged around the azimuth and integrated along the blade. It is convenient to perform the azimuth averaging first and we therefore write the thrust coefficient of the rotor as:

To expand the terms within the inside brackets, we recall from Chapter 4 that the flapping angle b may be expressed in the form:

b = a0—a1 cos C—b1 sin C

from which also we have:

—- = a1 sin C—b1 cos C

dc

For the feathering angle в a similar Fourier expansion (Equation 4.9) can be used: however, there is always one plane, the plane of the swashplate or no-feathering plane (NFP), relative to which there is no cyclic change in в; for our analytical solution therefore this will be used as the reference plane. Thus we have в = в0, constantin azimuth, and following the same procedure as
for hover we shall assume an untwisted blade, giving в0 constant also along the span. Averaging round the azimuth will make use of the following results:

*2p

sinC dC = 0

J0

*2p

cosC dC = 0

J0

*2p

sinCcosC dC = 0 (5.52)

J0

*2p

sin2C dC = p

J0

*2p

cos2C dC = p

J0

= в0 ■ x2 m

while:

all other terms cancelling out after substituting for b and (d^/dC) and integrating. Hence finally,

This is the simplest expression for the lift coefficient of a rotor in forward level flight. The assumptions on which it is based are those assumed for hover in Chapter 3, namely uniform induced velocity across the disc, constant solidity s along the span and zero blade twist. As before, it may be assumed that for a linearly twisted blade, Equation 5.55 can be used if the value of в is taken to be that at three-quarters radius. Also in Equation 5.55 the values of в and 1 are taken relative to the non-feathering plane as reference. Bramwell (p. 157) derives a significantly more complex expression for thrust when referred to disc axes (the TPP) but since the transformation involves the assumption that actual thrust, to the accuracy required, is not altered as between the two reference planes, the change is a purely formal one and Equation 5.55 stands as a working formula.