Category Helicopter Test and Evaluation

For low rates of descent, that is when the vertical velocity is less than the induced velocity (f < 0.5), the rotor is said to be operating in the vortex-ring state. As the rotor descends with power in this state, it is effectively pumping air from beneath the rotor by the action of the vortex ring located in-plane with the rotor. At high power settings, it is possible to pump so much air from beneath the rotor that the rate of descent increases thereby exacerbating the problem. The pilot may attempt to arrest the rate of descent by application of increased collective pitch but this tends to increase the rate of descent still further. If on the other hand the pilot lowers the collective lever the thrust is reduced and the helicopter also increases its rate of descent. This dangerous phenomenon is often called power settling. It can cause helicopters to develop significant rates of descent when operating below maximum gross weight even when full torque is applied.

Care must be taken to avoid entry into vortex ring particularly when hovering without external references at high altitude and when making steep approaches. Operation of the rotor in a state of fully developed vortex ring can be prevented by avoiding vertical descents at rates between 70% and 150% of vih (1700 to 3600 ft/min for the example helicopter). However vertical descents at modest speeds may be required operationally in these circumstances: the onset of significant vibration; a sudden uncommanded increase in the rate of descent; or a tendency for the helicopter to gyrate in pitch and roll, are taken as indicators of the incipient stages of power settling or the vortex ring condition and recovery action is initiated.

Other causes of vortex ring are the application of a high collective setting in a flare manoeuvre or operating in an upflow (equal to vi) close to a cliff. It should be noted that yaw manoeuvres can establish a vortex-ring state in the tail rotor.

Momentum theory and simple blade element theory are based on constant energy flow above and below the rotor. They both also assume that the slipstream has a finite velocity either side of the rotor. When these conditions are not satisfied these theories can no longer be used to make theoretically accurate predictions. Provided the rotor ascends, continuous flow through the rotor exists and use of the momentum theory is valid. The hover therefore represents the lowest vertical climb speed for which momentum theory is sound. Consider a vertical descent at low rates of descent: the rotor is now working to impart a downwards velocity on the fluid whilst the general flow along the streamtube is upwards. Clearly smooth flow is not possible in this situation and thus a vortex will form in the region of the rotor. A limiting condition arises when the rate of descent matches the induced velocity necessary to generate the thrust. Here the flow stagnates in the disk and fully developed vortex flow has been established. Such a situation leads to a rapid descent with uncontrolled pitching and rolling motion being caused by a violent flapping motion of the rotor blades. At rates of descent above this limiting value a steady smooth flow upward through the rotor is established and the momentum theory can be applied. It is possible to determine the value of VD ( — Vc) equating to fully developed vortex ring flight. It is often surmised that provided VD exceeds the induced velocity required for hovering flight then the rotor is extracting sufficient energy as it descends through the atmosphere to support the weight of the helicopter. Thus momentum theory:

• applies in vertical ascents and in the hover (when Vc p 0);

• is invalid in the vortex-ring and turbulent windmill-brake states (when 0 < VD < 2vih);

• applies in the windmill-brake state (when VD > 2vih).

The variation of induced velocity through all phases of axial flight can be found using a combination of momentum theory for the vertical ascent and windmill-brake states and empirical relationships for the vortex-ring and turbulent windmill-brake states. An expression developed by Glauert [2.3] has found favour:

3f 2

(1 — f) 2

where:

2

U = Vc±v,

Plotting the variation of 1/f with 1/F, see Fig. 2.7, is a convenient method of portraying the different flow states. As has already been mentioned momentum theory cannot be applied for rates of descents less than twice the induced velocity. It can be shown that this condition equates to a value off < 0.25. The hover condition relates to a value of F equal to zero. These two conditions form boundaries inside which non-momentum flow takes place.

Proximity to the ground results in a decrease in the induced power to produce a given thrust. The presence of the ground is effectively the same as that of having a mirror image rotor below ground. The slipstream of the image rotor induces an upward velocity at the real rotor so that the inflow velocity will be reduced and the incidence of the real rotor increased at constant pitch or feather angle. Hence, the lift developed by the real rotor is increased and the induced power required to maintain a hover is reduced.

Figure 2.6 shows the general trend in the form of a graph of the ratio of the induced power required in and out of ground effect against the height of the rotor above ground (Z), non-dimensionalized by the rotor diameter, D. We can see that ground effect decreases with increasing height, effectively disappearing at about one rotor diameter from the ground. Empirical relationships exist which describe the variation of power, in-ground-effect, with hover height [2.4]. One example is:

2.2.3 Tip loss and non-uniform flow

A characteristic of the actuator disk theory is that the linear theory of lift is maintained right out to the edge of the disk. In reality, because the rotor consists of a finite

number of blades, some air is able to escape outwards, from the streamtube, by the action of the tip vortices. The total induced flow is thus less than the actuator disk theory would prescribe. This deficiency is called tip loss and is shown by a rapid reduction in lift over the last few per cent of the span near the tip. It is common practice to account for this discrepancy by assuming that beyond a certain outboard station the blade sections develop drag but no lift. Thus the thrust integral becomes:

1 fBR

CT = 2 sal (6r2 – ^r)dr

Jo

This increases the hover power to:

C = 1 C 3/2+ — C or C =^— C3/2 + – C

C = BV2 C 3 + 8 Cd° or Cp = V2 C 3 + 8 Cd°

In 1927, Prandtl and Betz approximated the tip loss factor, B, to:

Goldstein and Lock showed satisfactory correlation with this approximation for lightly loaded rotors [2.9]. The value of к can be increased slightly to account for the losses incurred due to the non-uniform nature of the swirling flow from a real rotor. Typical values for к vary between 1.13 [2.4] and 1.15 [2.10].

The slipstream from the rotor exerts a downward force on the helicopter fuselage, which is in addition to any vertical drag associated with axial flight. This means that in the hover the main rotor must generate sufficient thrust to support not only the weight of the helicopter but balance this download. The rotor wake contracts from the diameter of the rotor to its ultimate wake size in about a quarter of rotor radius

THRUST COEFFICIENT FUNCTION (CT15 = (T/pAVT2)15)

Fig. 2.4 Typical hover performance results.

[2.6]. Even so it is often assumed that the fuselage is immersed in the ultimate wake and receives the full effect of the downwash. Simple estimates of download use a projected area and assume a drag coefficient of 0.3 [2.6]. It is often convenient to express the vertical drag as a percentage of gross weight (GW) and typical values range between 1% and 4% of GW.

Although the blade element theory introduced above has allowed the real effect of rotor driving torque arising from profile drag to be included, it has still relied upon the assumption that the inflow is constant along the rotor radius. Likewise, no allowance has been made for three-dimensional flow effects such as tip vortices. Minor modifications to the blade element theory are used to account for these important effects.

2.2.2 Variations in induced velocity

So far, it has been assumed that the induced velocity is constant across the disk. In addition, no description has been given of the precise relationship between blade pitch and induced velocity. In fact flow is induced downwards through the disk as a consequence of the aerofoil’s inclination to the direction of rotation. As such it is often necessary to develop a relationship between blade pitch and induced velocity for any radial station. Combining the momentum and blade element theories introduced earlier leads to:

Therefore:

1 pabc )r [ )r0(r) — (Vc + vi)] = 4p( Vcvi + v2 )nr

or:

where [vih/VT]r represents the ratio of the induced velocity in the hover at radius r to the tip speed, and 0(r) is the pitch at radius r. Figure 2.5 shows the variation of induced velocity for a hovering rotor calculated using Equation (2.9).

Before examining how the realism of the theories introduced above might be improved, it is necessary to discuss the concept of non-dimensional coefficients. These coefficients are analogous to the lift and drag coefficients that are a common feature of aerodynam­ics. Rather than lift and drag, in the case of rotorcraft, coefficients of thrust (CT), torque (Cq) and power (CP) are used. These coefficients are defined as:

CT =

CQ =

Note that power coefficient is numerically equal to torque coefficient (though, of course, they are different physically). Now converting Equation (2.7) into coefficient terms gives:

ct=2 abf u 9o+4 9i-1 *

CT = 2 as (3 90 +191 -1 *

where s is defined as the solidity of the disk that is the ratio of total blade area to disk area. Hence for a rotor with rectangular blades s = bcR I A. The relationship for thrust

coefficient can be simplified by altering the definition of blade pitch from the blade root to a position at three-quarter radius (60.75), since:

fixed and the rotor profile drag coefficient is constant then the power required to drive the rotor will vary as )3 so that CP will remain unchanged. This simple rule forms the basis of hover performance testing and is illustrated in Fig. 2.4.

Most modern blades feature a degree of negative twist, decreasing the pitch angle towards the tip, as means of optimizing the blade loading distribution. This can be expressed using the following equation:

r

0(r) = 0O = r

where 60 is the collective pitch applied at the hub; 91 is the total twist (normally negative) and rIR factors the total twist according to radial position. Now:

T = 2 pabcVJ R (I +1 -*

Simple momentum theory treats the rotor as an actuator disk through which a uniform flow passes. Unfortunately this theory tells us little about the flow around the individual blades that make up the rotor system. Momentum theory cannot, therefore, be used to predict the magnitude of any losses associated with realistic flow around rotor blades. Blade element theory overcomes some of the restrictions inherent in momentum theory as it is based upon the idea that the rotor blades function as high aspect ratio wings constrained to rotate around a central mast as the rotor system advances through the air. As before our study of blade element theory begins with purely axial flight.

2.2.1 Elemental forces, thrust, torque and power

Consider a rotor consisting of b blades climbing at speed Vc. The blades are each of length R, and turn at a rotational speed of ) rad/s. If we now examine the forces generated on a small element, Sr, of a blade located at r from the hub we can gain insight into how a complete rotor system generates thrust and drag. Figure 2.2 shows a blade element and Fig. 2.3 depicts the forces acting on such an element. Each element of the blade is assumed to develop the full aerodynamic forces and moments as it would in two-dimensional flow at the same conditions that occur at its radial station. No allowance is made at this stage for finite span or blade wake effects.

Firstly, we must determine the total flow through the disk. As we have seen for a rotor in a climb this is composed of the climb velocity, Vc, and the induced velocity v;. The resultant velocity at a blade element therefore has a vertical component, Vc + v;, and a horizontal component )r. From Fig. 2.3 it can be seen that the resultant velocity, V is given by:

V = V(Vc + v; )2 + )2r2

This is commonly approximated to V = )r. Justification for this approximation can be seen by considering the example helicopter in a rapid vertical climb (2000 ft/min or

10.2 m/s). Using Equation (2.4), vih = 12.3 m/s and later work will show that under these conditions vi = 0.7vih = 8.6 m/s. If the blade tip is considered then V = )R = 227.5 m/s whereas including the vertical component gives V = 228.3 m/s. Thus in this case
the standard approximation underestimates the total velocity by only 0.3%. Now the blade section incidence, a, will depend on its radial position, r, and from Fig. 2.3 for any spanwise position r:

a(r) = 0(r) — ф(г)

Using small angle approximations:

a(r) = 0(r) — = 0(r) — I(r)

This local incidence will lead to elemental lift and drag:

8L =1 p V1SCL = 2 p )2r2c(r)aa(r)8r

SD =1 p V 2SCd = 2 p )2r2c(r) CD(r)8r

which, from Fig. 2.3, can be combined to give elemental thrust and torque. Assuming the inflow angle is small and the lift/drag ratio is large leads to:

8T = b8L

8Q = b^8L + SD)r

The elementary power, 8P, required to produce the elementary thrust can be found from the elementary torque:

8P = )8Q = b Q^8L + 8D)r = )г(Ф8Т + b8D) = (Vc + v, )8T + )rb8D If this elemental equation is integrated along the rotor blade:

P = (Vc + v, )T + 2 p Q3b |c(r)r3CD dr (2.5)

Equation (2.5) is similar to the result obtained by adopting momentum theory except that we now have an extra term in the expression that represents the profile power required to keep the rotor turning against the torque produced by the profile drag. The total rotor thrust, torque and power can be obtained by integrating analytically the expressions for 8T, 8Q and 8P over the span of the blade. For a rectangular blade at constant pitch, where CD is now the mean profile drag coefficient for whole rotor, then:

T=1 pabV R (5—I)

Q = T ^ + 8 pbcVT R2Cd> (2.6)

P = T(Vc — v,) + 8 pbcVTrcd,

The total change in energy per unit mass (*E) along the streamtube is given by:

*E = 2(VC + vi )vi

As power can be defined as massflow multiplied by change in energy: P = m*E = pA(VC + vi) x 2(VC + vi)vi

P = T(VC + vi) = TVC + Tvi (2.2)

So the power required to drive a climbing rotor can be seen to come from two sources: the power required to generate the rate of climb (the useful power = TVC) and the power required to generate the thrust (the induced power = Tvi). The power calculated using Equation (2.2) represents an ideal minimum value because this simple theory neglects all forms of losses.

It has been seen that power estimations require a knowledge of thrust, rate of climb and induced velocity. Whilst the thrust can be related to the weight of the helicopter

and the rate of climb is an easily specified variable the induced velocity is more difficult to determine. However, Equation (2.1) can be re-written as:

v2 + v;vc-^-7 = 0 2pA

Only the positive root of this quadratic has any meaning in this case, so:

Vc Vc2 T

V = 2 + J 4 + 2pA

Note that Equation (2.3) can be used to predict the induced velocity in the hover (Vc = 0):

(2.4)

The Froude theory postulates that the flow above and below a climbing rotor can be considered as constant energy with energy being imparted only by the actuator disk. This energy is added to the airflow in the form of an increase in static pressure. The theory then draws some conclusions about the streamtube that the disk influences. Far above the rotor, the velocity of flow in the streamtube is equal to the freestream that is dependent on the rate of climb of the rotor itself. As the rotor draws air through its disk the velocity just above the disk is greater than the freestream and as a consequence of the continuity equation the streamtube will have contracted; also by virtue of Bernouilli’s relationship the pressure just above the rotor will be less than ambient. Immediately below the disk the pressure will be greater than ambient because of the energy added by the rotor however, the velocity and streamtube area will be the same as just above the rotor. Far below the disk in what is termed the ultimate wake, the flow will achieve a pressure equal to ambient but the velocity will exceed the freestream again because of the energy imparted by the rotor. The continuity equation and Bernoulli relationship require that the cross-sectional area of the ultimate wake be less than the disk area, see Fig. 2.1. Adoption of the concept of an actuator disk leads to a very simple relationship for the thrust developed by a rotor:

T = A(P2 — p)
where A is the disk area.

The pressure difference (P2 — Pr) generated by the disk can be related to the vertical velocity by considering the changes in pressure and velocity occurring in the streamtube. Consider a helicopter climbing vertically at speed Vc and assume that the

Vr + kv,

Fig. 2.1 Flow through the actuator disk.

acceleration of flow caused by the action of the rotor results in an increase in the flow velocity of vi (Fig. 2.1). Likewise, assume that the continued acceleration of flow below the rotor leads to a total velocity rise of kvi at the ultimate wake. Now Bernoulli states that:

P„ + 2 pvc2 = p + 2 P(VC + vi )2 P2+2 p( Vc + vi )2 = Pm+2 p( Vc + kvi )2

Thus:

T = A(P2 – Pt) = 2 pA(2VC + kvi)kvi

This relationship requires a value for k before it can be used to estimate the thrust required for a given rate of climb. If the momentum change caused by the disk is considered, an alternative expression for thrust can be developed. Recalling that force is equal to rate of change of momentum or massflow multiplied by a change in velocity gives [2.2]:

T = pA(VC + vi )kvi

Hence k= 2 and:

T = 2pA(VC + vi )vi (2.1)

This fundamental equation predicts that the induced velocity at the rotor disk is equal to half the total increase in flow velocity required to match the thrust requirement of the rotor. The maximum increase in velocity occurs at the ultimate wake, which is usually taken as one rotor diameter downstream of the disk. This momentum disk model can be applied to any working state of the rotor in which a continuous streamtube is formed.