HN

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LECTURE NOTES

ON

HELICOPTER AEROMECHANICS

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Rotorcraft Aeromechanics

2 Pre-requisite:

Aerodynamics, Flight Mechanics and Engineering Mechanics

Course objectives:

 Perpetuate, cultivate and advance the understanding of a still unusual and very capable aircraft: the helicopter

 Learning the first principles of helicopter flight

 Using the knowledge framework from the previous years to understand the multidisciplinary aspects of helicopter (aerodynamics, structural dynamics, performance, Aeroelasticity, and optimisation)

 To understand basic and advanced concepts related to aerodynamic loads, vehicle performance, basic rotor dynamics, and control of helicopters and tilt-rotor aircraft (i.e. VTOL aircraft).

 To develop the students' understanding of helicopter aerodynamics.

 To develop the students' understanding of momentum theory and blade element theory in estimating helicopter performance.

 To develop the students' understanding of helicopter performance problems and how to estimate the performance of an example helicopter.

Program outcome:

 Understand the characteristics of helicopter rotor flow fields for all phases of helicopter flight, hover, vertical flight, forward flight, autorotation, etc.

 Understand theoretical and empirical techniques used to analyse and predict the aerodynamic characteristics of helicopters in hover, vertical flight and forward flight.

 Understand helicopter main rotor and tail rotor aerodynamic design considerations including airfoil selection and rotor configuration trade-offs.

 Understand momentum theory uses and limitations and apply it to the estimation of rotor performance of a selected helicopter.

 Understand and apply rotor blade element theory to estimate rotor performance and trim conditions for a selected helicopter.

 Understand and apply combined momentum and blade element methods to analyse helicopter rotors.

 Understand and apply empirical corrections to both momentum and blade element techniques.

 Understand and apply component drag build-up techniques to the estimation of the total drag of a selected helicopter.

 Understand and apply performance estimation techniques to hover, vertical climb, autorotation, range and endurance.

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3 References:

1. J. Gordon Leishman, “Principles of Helicopter Aerodynamics”, Cambridge University Press, 2000

2. Prouty, R.W., “Helicopter Performance, Stability and Control”, R.E. Krieger Pub. Co. Florida, 1990

3. Seddon, J., “Basic Helicopter Aerodynamics”, B.S.P. Professional Books, 1990

4. Johnson, W., “Helicopter Theory”, Princeton University Press, New Jersey, 1980

5. Mil, M. et al, “Helicopters – Calculation and Design: Vol. I Aerodynamics”, “NASA TTF-494, 1967.

6. Mil, M. et al, “Helicopters – Calculation and Design: Vol. II Vibrations and

Dynamics”, “NASA TTF-519, 1968.

7. Gessow, A. and Meyers, G.C., “Aerodynamics of the Helicopter”, Dover Publication

8. Bramwell, A.R.S., “Helicopter Dynamics”, Edward Arnold Pub., London, 1976

9. Stepniewski, W.Z. and and Keys, C.N, “Rotary Wing Aerodynamics, Vol. I and

II”, Dover Publications, 1984.

10. Padfield, G.D., “Helicopter Flight Dynamics: The Theory and Application of

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4 Pitch Angle (

θ

):

The blade pitch angle (θ) is the angle between the plane perpendicular to the rotor shaft and the chord line of a reference station on the blade. For a hovering helicopter, angle of incidence (i) is different from θ. As the rotor blade rotates, a downward velocity ( iv ) is

induced. The resultant velocity V_R is a combination of this induced velocity

 

v and the _i

linear velocity (Ωr) in the plane of rotation at a distance r from the hub. The angle between induced velocity

 

v and the linear velocity (Ωr) is defined as inflow angle φ and the angle _i

of incidence (i) is reduced from θ by the inflow angle φ.

Azimuth Angle ():

The helicopter rotor blade moves through 360 degree azimuth. The azimuth position is measured positively in the direction of rotation from its downstream position.

Collective change of Pitch

Within limits of stall, lift coefficient increase with increase in . If the pitch of all the blades is increased simultaneously, the overall lift and hence thrust increases. Therefore, changing the thrust to values more than or less than weight will cause the helicopter to climb or descend. The means of achieving this change of pitch of all blades simultaneously is called “collective” pitch change.

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5 Cyclic change of pitch

With cyclic pitch lever, the pilot can increase the blade pitch at one azimuth position (A) and decreases it at a diagonally opposite position (B). As a result all the blades coming to position A steadily have increasing pitch values those receding from A and going to B have steadily decreasing values. This causes increased angle of attack at position A and decreased angle of attack at position B. this cyclic variation of pitch along azimuth position is called “cyclic” pitch change.

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6 Rotor Hinges

The development of the autogyro and, later, the helicopter owes much to the introduction of hinges about which the blades are free to move. The use of hinges was first suggested by Renard in 1904 but the first successful practical application of hinges was due to Juan de la Cierva in the early 1920s. There are three hinges in the so-called fully articulated rotor:

i. Flapping hinge ii. Drag or lag hinge iii. Feathering hinge

Flapping hinge

The flapping hinge solves the problem of rolling moment when the helicopter is in forward flight. In hover, pitch is maintained the same throughout the azimuth position. However, when the rotor moves forward horizontally at a velocity V, the advancing blade (at = 90

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degree) is at a velocity V + r and the retreating blade (at =270 degree) is at V-r. Thus, if the pitch is same, the advancing blade gives higher lift than the retreating blade. This production of unequal lifts on either side of the helicopter would result in undesirable rolling moment and excessive alternating air blades on the blade. One way of correcting this is by setting the pitch on the advancing side lower and the retreating side higher by use of some sort of lateral control.

Flapping Hinge with offset „a‟

As the blade advances and develops more lift, it begins to flap upward. This then introduces a downward vertical component of velocity in relation to the blade which reduces its angle of incidence and hence the lift of the advancing blade. As it retreats, the opposite is true, for a downward flapping of the blade produces an increased lift. The changes in speed in advancing and retreating blades are compensated by opposing changes in angle of incidence (and lift) and net rolling moment about flapping hinge becomes zero.

•

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8 Retreating Side (flapping down increases angle of incidence)

It is to be noted that this flapping motion is caused automatically by unequal velocities only (i.e. without any control force by pilot) and it is referred to as aerodynamic flapping.

Drag (Lag) Hinge

The next important hinge is the drag hinge. In addition to the flapping hinge, a hinge is essential to cater for the lead-lag motion of the blade; this is the drag hinge. The blade is hinged about a vertical axis near the center of rotation so that is free to oscillate or “lead and lag” in the plane of rotation. This flexibility makes the net moment about drag hinge zero. Both the flapping and drag hinge (in a so-called fully articulated rotor).

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9 Feathering Hinge

Pitch of the blades can be increased or decreased by the pilot simultaneously or differentially (collective and cyclic pitch change) by the use of feathering hinge.

Types of rotors

Three fundamental types of rotors have been developed so far:

Rigid rotor

In these rotors, the blades are connected rigidly to the shaft. Such rotors do not have either flapping or drag hinge. Usually, such rotors are two-bladed.

See-saw (or teetering) rotor

Rotors in which blades are rigidly interconnected to a hub but the hub is free to tilt with

respect to shaft. These rotors are two bladed. The blades are mounted as a single unit on a “see-saw” or “teetering” hinge. No drag hinges are fitted and therefore lead-lag motion is not permitted. However, bending moments my still be reduced by under-stinging the rotor. The principle of see-saw rotor is similar to that fully articulated rotor (having both flapping and drag hinges) except that blades are rigidly connected to each other. The “see-saw” hinge is like the flapping hinge located on the axis of rotation and because of rigid interconnection between two blades, when the advancing blade, flaps up, the opposite (retreating) blade flaps down.

Fully articulated rotor

Rotors in which blades are attached to the hub by hinges, free to flap up and down also swing back and forth (lead and lag) in the plane of rotation. Such rotors may have two, four or more blades, such rotors usually have drag dampers which present excessive motion about the lag hinge.

General Expression to determine the location of K

th

blade

Azimuth of Kth blade is given by

( ) N- Number of blades

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10 Rotor Provides

 Lift (thrust)

 Horizontal Propulsive Force

 Forces needed for control

Hover:

Helicopters Designed to be operationally efficient there.

For forward flight the rotor is tilted forward

 Blades Flap

 Vortices create vibration and noise

 Higher forward speed: Transonic flow

o Increased drag and noise (swept blades help)

o Retreating side, High AOA, Dynamic stall, Vibration

 Forward flight speeds are limited

High Speed Forward Flight Limitations

• As the forward speed increases, advancing side experiences shock effects, retreating side stalls. This limits thrust available.

• Vibrations go up, because of the increased dynamic pressure, and increased harmonic content.

• Shock Noise goes up.

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11 Power plant Limitations

• Helicopters use turbo shaft engines. • Power available is the principal factor.

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An adequate power plant is important for carrying out the missions.

Momentum Theory of Rotors (Actuator Disk Theory)

• Developed for marine propellers by Rankine (1865), Froude (1885). • Extended to include swirl in the slipstream by Betz (1920)

• This theory can predict performance in hover, and climb. • First approximation, no details of shape

Applications:

Propellers, Rotor and Ducted Fans

Assumptions:

• Rotor is modeled as an actuator disk (infinitely thin disk of area A which offers no resistance to air passing through it) which adds momentum and energy to the flow. • Flow is incompressible (compressibility corrections can be made).

• Far Upstream and Far Downstream the pressure is freestream static pressure • Flow is steady, inviscid (no drag and no momentum diffusion), and irrotational. • Flow is purely one-dimensional

• Thrust loading and Inflow velocity are uniform over the rotor disk (equivalent to assuming infinite number of blades).

• There is no swirl in the wake (no rotational effect) • Low Disk Loading

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Consider an actuator disk (thin circular disk with radius of rotor radius and has infinite number of blades. It is permeable to the air flow and supports the pressure difference on the top and bottom surface of the disk) of area A and total thrust T. It is assumed that the loading is distributed uniformly over the disk. Let be the induced velocity at the rotor disk and w be the wake-induced velocity infinitely far downstream. A well-defined smooth slipstream is assumed with and w uniform over the slip-stream cross section. The rotational energy in the wake due to the rotor torque is neglected.

Mass flow through the rotor

̇ (1)

By conservation of mass the mass flux is constant all along the wake. Mass conservation ( ) (2) √ (3) √ Actual Value 0.78

Thrust from Momentum equation is given by, considering stations (0) – (3)

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Applying Bernoulli‟s equation from station (0) – (1)

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Bernoulli‟s equation cannot apply between 1-2 stations

Below the disk, between stations (2) & (3), the application of Bernoulli‟s equation gives

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( )

Because the jump in pressure is assumed to be uniform across the disk, this pressure jump must be equal to the disk loading, T/A, that is

From equation (5) & (6)

( ) (7)

Therefore rotor disk loading is equal to dynamic slipstream pressure.

. / ( ) (8)

. / ( ) (9)

Therefore, the static pressure is reduced by ( ⁄ )( ⁄ ) above the rotor disk and increased by ( ⁄ )( ⁄ ) below the disk.

( ) (10)

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̇ ( ) (12) √ √( ) (13) Power required to hover:

√

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̇ ( ) (15)

Power to hover reduces when induced velocity is small and mass flow rate through the disk is large (large rotor disk area)

√ √ (16)

where

Pressure Variation Plot in Axial Direction:

According to momentum theory, the velocity deficit in the far wake is twice the velocity deficit at the rotor disk.

Momentum theory gives an expression for velocity deficit at the rotor disk.

It also gives an expression for maximum power produced by a rotor of specified dimensions. Actual power produced will be lower, because momentum theory neglected many sources of losses- viscous effects, tip losses, swirl, non-uniform flows, etc.

Preliminary Remarks

Momentum theory gives rapid, back-of-the-envelope estimates of Power. This approach is sufficient to size a rotor (i.e. select the disk area) for a given power plant (engine), and a given gross weight. This approach is not adequate for designing the rotor.

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15 Limitations of the Momentum Theory:

The analysis made by the simple momentum theory is idealised because it neglects profile drag losses, non-uniformity of induced flow (including the energy losses due to spilling of the air about blade tips, commonly known as tip losses) and slipstream rotation losses. Thus an actual rotor would require more power to hover with a given load than an “ideal” rotor (i.e., a rotor having zero profile drag and uniform inflow) and therefore would be less efficient. The order of magnitude of the rotor losses not considered by simple momentum theory, expressed as a percentage of the total power required is as follows:

Profile drag losses: 30% Non uniform inflow: 6% Slipstream rotation: 0.2% Tip losses: 3%

Lastly, it does not provide any information as to how the rotor blades should be designed for a given thrust.

Hover Power Losses

Momentum theory gives the induced power loss of an ideal rotor in hover, √ . A real rotor has the power losses as well, in particular the profile power loss due to the drag of the blades in a viscous fluid. There is also an induced power loss due to the non-uniform inflow of a real, non-optimum rotor design. The distribution of the power losses of the rotor in hover is approximately as follows:

Induced Power – 60% Profile Power – 30%

Non-uniform Inflow – 5% to 7% Swirl in the wake – less than 1% Tip losses – 2% to 4%

The main rotor absorbs most of the helicopter power, but there are other losses as well. The engine and transmission absorbs 4% to 5% of the total power with turbine engines. The tail rotor absorbs about 7% to 9% of the total helicopter power, and there is an additional loss of about 2% due to aerodynamic interference.

Disk Loading and Power Loading

T/A or DL = Disk Loading T/P or P/L = Power Loading At hover T = W

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√

( ) (17)

For a single rotor helicopter in hover, the rotor thrust, T, is equal to the weight of the helicopter, W; the disk loading is sometimes written as W/A. Disk loading is measured in pounds per square foot. The power loading is defined as T/P, which is denoted by PL, Power loading is measured in pounds per horsepower or newton per kilowatt. Power required to hover is given by . This means that the ideal power loading will be inversely proportional to the induced velocity at the disk. The ratio T/P decreases quickly with disk loading. Therefore, vertical lift aircraft that have a low effective disk loading will require relatively low power per unit of thrust produced (i.e. they will have high ideal power loading) and will tend to be more efficient.

Disk loading for helicopters are usually in the range of 100 – 500 N/m^2 and the corresponding inflow velocity is in the range of 6.4 – 14.3 m/s (at sea level density condition). ⁄ N/m^2 Rotor 100 – 500 Propeller 2500 Ducted fan 2500 - 5000 Jet 50000

The higher the disk loading, the higher the induced velocity, and the higher the power. For helicopters, disk loading is between 5 and 10 lb/ft2 (24 to 48 kg/m2).Tilt-rotor vehicles tend to have a disk loading of 20 to 40 lbf/ft2. They are less efficient in hover. VTOL aircraft have very small fans, and have very high disk loading (500 lb/ft2).

Power Loading

For a given gross weight, that is, a high power loading with a large value of T/P is required. Power loading is the ratio of the thrust produced to the power required to hover, that is,

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(18) ( ) (19)

This quantity should be as close as possible to the ideal value for best hovering efficiency. Because

( ) ( )

Maximizing the power loading requires a low tip speed ( ).

On the basis of simple momentum theory considerations, the ratio P/T is given by

√ √ √

( ) (20)

To maximize the power loading (that is minimize the ratio P/T) the disk loading should be low (i.e. the disk area should be large for a given gross weight to give a low induced velocity and the tip speed should be low).

When using the modified momentum theory, the ratio P/T is given by √ : √ ; ( √ ₎ √ √ √ √

This means that the best rotor efficiency (maximum power loading) is obtained when the disk loading is minimum and the figure of merit is a maximum.

Pure helicopters have a power loading between 6 to 10 lb/HP. Tilt-rotors have lower power loading – 2 to 6 lb/HP. VTOL vehicles have the lowest power loading – less than 2 lb/HP.

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18 Induced Inflow Ratio

 Induced Velocity (Hover)

The induced inflow velocity, , at the rotor disk can be written as

(Non-dimensional quantity) = Angular speed; R = Rotor radius;

Thrust and Power Co-efficient

In helicopter analysis the rotor thrust co-efficient is formally defined as

Where the reference area is the rotor disk area A and the reference speed is the blade tip speed. All velocity components are non-dimensionalized by tip speed so the inflow ratio is related to the thrust co-efficient in hover by

√ √ ( ) √

This is based on the 1-D flow assumption made in the preceding analysis, which means that this value of inflow is assumed to be distributed uniformly over the disk.

If Thrust coefficient goes up, Inflow ratio goes up and if inflow goes up, power goes up. As the goes up Thrust coefficient decreases and tip speed increases beyond the critical value. The rotor power coefficient is defined as

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So that based on momentum theory the power coefficient for the hovering rotor is

( ) ( ( ) ) . /

√

Again this is calculated on the basis of uniform inflow and no viscous losses, so is called the ideal power coefficient. The corresponding rotor shaft torque coefficient is defined as

In non-US countries an extra ½ is used in the denominator for

Comparison on Theory and Measured Rotor Performance

The Comparison of measured and theoretical results is shown in the above figure. In terms of coefficients it is apparent that the ideal power according to the simple momentum theory can be written as

√

Notice that the momentum theory under predicts the actual power required, but the predicted trend that is essentially correct. These differences between the momentum theory and experiment occur because viscous effects (i.e. non-ideal effects) have been totally neglected so far.

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In hovering flight the induced power predicted by the simple momentum theory can be approximately described by an empirical modification to the momentum result in

√

Where k is called an induced power factor. This factor is derived from physical effects such as non-uniform inflow, tip losses, wake swirl, less than wake contraction, finite number of blades and so on.

Induced power factor =

= Profile power; = Induced Power

∫

Where is the number of blades and D is the drag force per unit span at a section on the blade at a distance y from the rotational axis. The drag force can be expressed conventionally as

( )

Where c is the blade chord. If the section profile drag coefficient, , is assumed to be constant (= ) and independent of Reynolds number and Mach number and the blade is not tapered in planform (i.e. a rectangular blade), then the profile power integrates out to be

( ) ( ) ()

: Solidity ratio, which is the ratio of blade area to rotor disk area. Typical values of solidity ratio for a helicopter range between 0.05 and 0.12.

Armed with these estimates of the induced and profile power losses, it is possible to recalculate the rotor power requirements by using the modified momentum theory result that

√

Figure of Merit

The figure of merit is a measure of rotor hovering efficiency, defined as the ratio of the minimum possible power required to hover to the actual power required to hover. The figure of merit is equivalent to a static thrust efficiency and defined as the ratio of the ideal power

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required to hover to the actual power required, that is,

Ideal power is the power required to lift the weight of the helicopter (i.e. minimum power without any loss). Ideal power is nothing but induced power.

_√ √

As thrust coefficient increases FM reaches a maximum or drops off slightly. This is because of the higher profile drag coefficients (> ) obtained at higher rotor thrust and higher blade section AoA. increases as increases; behaviour depends on airfoil stall characteristics. In practice, FM values between 0.7 and 0.8 represent a good hovering performance for a helicopter rotor.

Because a helicopter spends considerable portions of time in hover, designers attempt to optimize the rotor for hover (FM~0.8). A rotor with a lower figure of merit (FM~0.6) is not necessarily a bad rotor. It has simply been optimized for other conditions (e.g. high speed forward flight).

Using the modified form of the momentum theory with the non-ideal approximation for the power, the figure of merit can be written as

√ ⁄ ⁄√ ⁄

Notice that at low operating thrusts the figure of merit is small. This is because the profile drag term in the denominator is large compared to the numerator. As the value of increases, however, the importance of the profile power term decreases relative to the

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induced term and FM increases. This continues until the induced power dominates the profile term and the figure of merit will begin to approach a value of 1/k. In practice, however, the profile drag contribution decreases this value. In practice, at higher values of rotor thrust the profile drag (and power) increases quickly as the blade begin to stall, which will again cause a reduction in FM.

High solidity (lot of blades, wide-chord, large blade area) leads to higher Power consumption, and lower figure of merit. Figure of merit can be improved with the use of low drag airfoils. If the solidity ratio ( ) value is small FM value goes up.

We know that

√

If we need high figure of merit, value should be small. But in the equation (A), value goes down value goes up and reaches stall condition. So optimum value of should be decided to compromise both the equations.

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23 Induced Tip Loss

The formation of trailed vortex at the tip of each blade produces a high local inflow over the tip region and effectively reduces the lifting capability.

Tip vortex reduces lift tip loss B: tip loss factor

BR = =effective radius (< R)

( ) Adding inner root cut-out

Prandtl (vortex theory) showed that when accounting for the tip loss, the effective blade radius, , is given by ( ) √ ( ) √

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( √ ) √ √

B is shown to decrease with decreasing (less blade-to-blade interference) and also with increasing (spacing of vortex sheet below the rotor). In practice, values of B for helicopter rotors are found to range from about 0.95 to 0.98, depending on the number of blades.

Gessow & Myers (1952) suggest an empirical tip-loss factor based on blade geometry alone where

c: tip chord

Sissingh (1939) has proposed the alternative geometric expression ( )

Where is the root chord of the main blade and is the blade tip ratio (ratio of the tip chord to the root chord).

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25 Blade Element Theory

Two primary limitations of the momentum theory are that it provides no information as to how the rotor blades should be designed, so as to produce a given thrust. Also, profile drag losses are ignored. The blade element theory provides means for removing these limitations. The blade element theory, which was put in practical form by Drzewiecki, is based on the assumption that element of a propeller or rotor can be considered as an aerofoil segment that follows a helical path. Lift and drag are then calculated from the resultant velocity acting on aerofoil, each element being considered independent of the adjoining element. The thrust and torque of the propeller or rotor are then obtained by integrating the individual contribution of each element along radius.

It is a “strip” theory. The blade is divided into a number of strips, of width dr. The aerodynamic force lift generated by that strip, and the power consumed by that strip, are computed using 2-D airfoil aerodynamics. The contributions from all the strips from all the blades are summed up to get total thrust, and total power.

The resultant local flow velocity at any blade element at a radial distance y from the rotational axis has an out-of-plane component (perpendicular to the rotor) as a result of climb and induced inflow

An in-plane component (parallel to the rotor) because of blade rotation

The resultant velocity at the blade element is, therefore √

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The relative inflow angle (or induced angle of attack) at the blade element will be

₍ ₎

Thus, if the pitch angle at the blade element is , then the aerodynamic or effective AoA is ( ) ₍ )

For low axial velocities and a radial position sufficiently outboard from the axis of rotation

( ) ( )

Aerodynamic section of the rotor blades starts from 20 to 25% of the rotor. The elemental lift and drag, associated with a segment of width dr is

For a constant chord blade, and assuming a constant drag coefficient, the profile power coefficient can be evaluated as

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For an accurate calculation of the profile power loss, the variation of the drag coefficient with the angle of attack and Mach number should be included. Consider a profile drag polar of the form

By properly choosing the constants , and the variation of drag with lift for a given airfoil can be well represented for angle of attack below stall. This representation for drag coefficient was used by Bailey (1941) and his numerical example is frequently found in helicopter calculations.

The lift dL and drag dD act perpendicular and parallel to the resultant flow velocity respectively.

The quantity c is the local blade chord.

When is small, is large and is small ( ) ,( ) - ∫ ,( ) ( ) - ( ) ∫ ,( ) ( ) - ( )

These forces can be resolved perpendicular and parallel to the rotor disk plane giving

Therefore the contributions to the thrust, torque and power of the rotor are : Number of blades comprising the rotor

For hover and axial flight independent of azimuth angle ( ) ( )

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( )

( ) For helicopters the following simplifying assumptions can be made:

√

This is valid approximation except near the blade root, but the aerodynamic forces are small here anyway.

The induced angle is small, so that

The drag is at least one order of magnitude less than lift, so that the contribution ( ) is negligible.

Applying these simplifications to the preceding equations results in ( ) ( ) Non-dimensionalization: ̅ ( ) ( ) ( ) The inflow ratio can be written as

( ) . / ̅ ( ) . / ( )( ) ( ) . / . / ( ) ̅ ̅ For a rectangular blade (c = constant)

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̅ ̅ Similarly ( ) ( ) ( )( ) ( ) ( ) ̅ ̅ ( ) ̅ ̅ For a rectangular blade the thrust coefficient is

∫ ̅ ̅

Torque / Power coefficient:

∫ ( ) ̅ ̅

∫ ( ̅ ̅ ) ̅

Since

Drawbacks of Blade Element Theory

It does not handle tip losses.

Solution: Numerically integrate thrust from the cutout to BR, where B is the tip loss factor. Integrate torque from cut-out all the way to the tip.

It assumes that the induced velocity v is uniform. It does not account for swirl losses.

The Predicted power is sometimes empirically corrected for these losses.

√

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31 Integrated Rotor Thrust and Power

To evaluate and it is necessary to predict the span wise variation in the inflow as well as the sectional aerodynamic force coefficients .

If 2-D aerodynamics are assumed

( ) ( ) Where Re and M are the local Reynolds number and Mach number

Thrust Approximations

Based on steady linearized aerodynamics, the local blade lift coefficient can be written as ( ) ( )

of the airfoil section For an incompressible flow

per radian

( ) For symmetric airfoil

∫ ̅ ̅ ∫ ( ) ̅ ̅ ⁄

∫ ( ̅ ̅) ̅

Untwisted Blades, Uniform Inflow

Zero twist Uniform Inflow ∫ ( ̅ ̅) ̅ 6 ̅ ̅ 7

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[ ]

√

< √ =

This can be solved iteratively to find for a given √ √ is almost equal to 6

⁄ gives the average angle that required during hovering condition or angle need to be provided during hover. ⁄ is also known as blade loading or mean pitch angle required for operation.

When the weight increases, pitch angle also increases. After certain angle, helicopter cannot lift-off, because blade reaches stall angle.

Linearly Twisted Blades, Uniform Inflow

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( ) ̅

where is the blade twist rate per radius of the rotor.

Using this variation in ( ), we get

∫ ,( ) ̅ ̅- ̅ 6 ̅ ̅ ̅ 7 [ ] Using at 0.75 radius ( ) ( ) ( ̅ ) ∫ *,( ( ̅ ) ) ̅ - ̅+ ̅ ∫ *,( ̅ ̅ ̅ )- ̅+ ̅ [ ] [ ]

Ideal Twist

This twist distribution while not physically realizable at the root is of interest because it will be found to give uniform inflow with the constant chord blades.

̅

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vary inversely with non-dimensional radial distance ( ̅)

So

[ ]

Rotor Torque

Another important quantity associated with rotor behaviour is the torque, or the moment needed to overcome the drag and keep the rotor turning at a certain RPM in steady state conditions. One can express the elemental torque

Total torque

∫ ( )

Elemental Torque due to all the blades

[ ( ) ] In non-dimensional form ( ) ∫ [ ( ̅ ̅) ̅ ] ̅ ∫ 0 ( ̅ ̅)1 ̅ ∫ ̅ ̅ ∫ ∫ ̅ ̅ ∫ , ̅ ̅- ̅

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∫ ∫ ∫ ̅ ̅ If the climb velocity is constant

∫

In non-dimensional form the climb power will be the product of velocity and the total weight of the helicopter.

Induced velocity in climbing condition is different from induced velocity in hovering condition.

For hovering flight becomes

∫ ∫ ̅ ̅

In this case at each section is assumed to be constant. If different aerofoils used in the rotor blade then will become the function of span and Mach number.

∫ [ ( ) ] ∫ 8 6 ( ) 7 9 6 ( ) 7 [( ) ] ( ) Combining the above equations

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[( ) ] [( ) ]

The first term is usually called the induced torque (because it is due to the induced drag) and the second term is called the profile torque.

The power coefficient is defined as

( )

Next it is interesting to connect the expressions which have been derived with momentum theory in hovering flight. Recall for this case power is given by

( ) ( ) Since for hover

So

Ideal torque coefficient in absence of friction is identical from both blade element theory and momentum theory. In hovering flight √ √ < √ = From which

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( ) √ √ ( ) ( ) ( ) ( )

This is a quadratic equation for , when the value of collective pitch is given. Thus

√ Where = Profile torque coefficient

Ideal torque coefficient,

√

Another useful quantity often used in helicopter engineering is the Rotor Figure of Merit (FM) which is defined as

√

( ) So that the ideal rotor figure of merit is

√ ( )

For an actual rotor the Figure of Merit indicates the magnitude of the losses due to non-uniformity of flow, tip loss, and profile drag for a particular rotor. For a good rotor FM 0.75 where “good” implies a well-designed rotor.

In the equations used above has appeared a number of times, a good approximation for is given by

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38

For a reasonable angle of attack this normally yields a value of 0.012

In hover case

√

k = Empirical factor which represents additional losses (+15%)

In this case we are integrating the equations from zero to one directly (i.e. along the full span of the blade) but near the tip the above equation will not look like this. In real rotors, at the tip there will be lift drop. But the equation gives the lift per section.

Lifting-line theory is not strictly valid near wing tips. When the chord at the tip is finite, blade element theory gives a non-zero lift all the way out to the end of the blade. In fact, however, the blade loading drops to zero at the tip over a finite distance because of three-dimensional flow effects. Since the dynamic pressure is proportional to .

Beyond 0.5 the lift value will be large. In the actual rotor near ̅ , lift drops and then to zero value at the tip exactly. At the tip, lift becomes zero bu there will be always drag value and non-uniform inflow. Analysis at the tip region is very complicated and it cannot be done by Momentum Theory.

Tip-Loss Factor

Effective Blade

B = Prandtl tip-loss factor; B = 0.95 – 0.98

∫ ̅ ̅ ∫ ( ̅ ̅) ̅ For Untwisted blades and uniform inflow

[ ]

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39

∫ ( ) ̅ ̅ 6( ) ̅ 7 ( )

Because B is between 0.95 and 0.98, we find a 6-10% reduction in rotor thrust resulting from tip-loss effects for a given blade-pitch setting under the stated assumptions.

The correct interpretation is to consider that for the same thrust the induced inflow will be increased to a value

√

√ ( ) √

(or ) is increased by a factor _{compared to the case with no assumed tip losses. For} untwisted blades and uniform inflow with tip losses alone, the thrust becomes

[

] For ideal twist and uniform inflow, the thrust now becomes

[ ]

Because of tip-loss effects, a real rotor will always have a higher overall average induced velocity compared to that given by momentum theory and so the induced power will also be increased relative to the simple momentum result.

Tip loss constitutes an additional source of non-uniform inflow, and would normally be factored into the value of k. Using the BET, the induced power can be written as

∫ ∫ ̅ ̅

Using untwisted rectangular blades and uniform inflow assumptions, then with a tip loss the total power can be approximated by

[

]

Where the induced power factor k includes the effects of both tip loss and non-uniform inflow over the remainder of the blade. In general we can write that

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40 Combined Blade Element-Momentum Theory

A combination of momentum theory and blade element theory has been developed later. Identical equations may be derived by means of vortex theory, but it is believed that the combination of momentum and blade element theory has greater physical significance and can be easily grasped. The combined theory can be applied for performance analysis of helicopter in hover, vertical climb and forward flight.

In the previous sections we dealt with momentum theory which gave estimates for power and induced velocity in terms of the thrust. We also studied blade element theory, which allowed us to incorporate features such as number of blades, airfoil section drag and lift characteristics, taper and twist, etc. The latter, however, assumed that the flow through the rotor disk is uniform as in the momentum theory.

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The annulus is at a distance y from the rotational axis, and has a width dy.

Equating differential thrust over annular area from BET and Momentum Theory. Assume that the annular area is not affected by either the flow inside or outside that.

On the basis of simple one-dimensional momentum theory, the incremental thrust on the rotor annulus as the product of the mass flow rate through the annulus and twice the induced velocity at that section. In this case the mass flow rate over the annulus of the disk is

̇ ( ) ( ) so that the incremental thrust on the annulus is

( ) ( ) This has also been known as the Froude-Finsterwalder equation.

( )( ) ( ) ( ) ( ) ( ) ( ) . / . / . / ̅ ̅

Therefore, the incremental thrust coefficient on the annulus can be written as ̅ ̅ ( ) ̅ ̅

This assumes no swirl in the wake. Consider first the hovering state where . The incremental thrust and power contributions of the annulus are given by

̅ ̅ ̅ ̅ The total thrust coefficient of the rotor is

∫ ∫ ̅ ̅

And the corresponding induced power coefficient is

∫ ̅ ̅

From the BET the incremental thrust produced on an annulus of the disk is ( ̅ ̅) ̅

Equating the incremental thrust coefficients from the momentum and blade element theories we find that

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( ̅ ̅) ( ) ̅ Which gives ̅ . / ̅ This quadratic equation in has the solution

( ̅ ) √(

)

̅ (

)

In hovering flight condition, the above equation simplifies to

( ̅)

:√

̅ ;

The numerical implementation of combined BEM theory is identical to classical blade element theory. The only difference is the inflow is no longer uniform. It is computed using the formula given earlier, reproduced below:

( ̅ ) √(

)

̅ (

)

Average Lift Coefficient

Let us assume that every section of the entire rotor is operating at an optimum lift coefficient and the rotor is untapered.

̅ ∫ ( ) ̅ ̅ ( ) ̅ ̅ ̅

Rotor will stall if average lift coefficient exceeds 1.2, or so. Thus, in practice, CT/s is limited to 0.2 or so.

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With ideal twist the performance of the rotor can now be recalculated. If

̅ ∫ ( ) ̅ ̅ ( ) ( ) ( ) :√ ; √

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44

√

Alternate Derivation

The elemental thrust of the blade elements contained in the annular ring based on the blade element theory is given by

( ) [ ( ) ]

For the same annular ring shown, the elemental thrust based on momentum theory is given by , ( )- ( )

Equating this two expressions yields

( ) [ ( )

] , ( )- ( ) Which is a quadratic equation for ( )

Rewrite the equation in non-dimensional form ( ̅) ̅ 6 ( ̅) ( ̅) ̅ 7 ̅ , ( ̅)- ( ̅) ( ̅) ̅ 6 ( ̅) ( ̅) ̅ 7 , ( ̅)- ( ̅) ( ̅) ̅ 6 ( ̅) ( ̅) ̅ 7 , ( ̅)- ( ̅) ( ̅) ̅ 6 ( ̅) ( ̅) ̅ 7 , ( ̅)- ( ̅) ( ̅) ( ̅), ( ̅) - ̅ [ ( ̅) ̅̅̅] ( ̅) ( ̅) 6 ( ̅) 7 ̅[ ( ̅) ̅̅̅] ( ) ( √ , ̅ )

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45

or axial flight. Once the induced velocity is known, the inflow angle at the blade element can be determined from

̅

It is a completely general expression which allows one to determine the inflow velocity for any blade planform and pitch distribution. A number of special forms of this equation are quite useful.

CASE (A) when chord is constant and the blade twist is ideal (inversely proportional to( ̅),

one obtains ( ̅) constant over the disk

( ̅) ( ) ( √ , )

CASE (B) Another useful relation is obtained for the case of hover (V=0) and constant

chord. Assuming that the inflow velocity at 0.75 R is representative of a uniformly distributed inflow for one has

( ̅) ( ) ( √ , ) ( ̅) ( ) :√ ; ( ̅) ( ) :√ ;

which is an approximate relation for uniform inflow frequently used in aero-elastic calculations.

Optimum Rotor for Hover

Here we are interested in the optimum rotor for hover including real fluid effects. We are seeking for max (L/D)

Still want = constant over the disk.

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( ) 6 ( ) ( ) 7 ( ) 6 ( ) ( ) 7 We want ( ) Let ( ) ( ) Therefore ( ) ( ) ̅ . / ( ̅) ̅√

Ideal Rotor vs Optimum Rotor

• Ideal rotor has a non-linear twist: ⁄ ̅

• This rotor will, according to the BEM theory, have a uniform inflow, and the lowest induced power possible.

• The rotor blade will have very high local pitch angles q near the root, which may cause the rotor to stall.

• Ideally twisted rotor is also hard to manufacture.

• For these reasons, helicopter designers strive for optimum rotors that minimize total power, and maximize figure of merit.

• This is done by a combination of twist, and taper, and the use of low drag airfoil sections.

Optimum Rotor

• We try to minimize total power (Induced power + Profile Power) for a given T. • In other words, an optimum rotor has the maximum figure of merit.

• From earlier work, figure of merit is maximized if

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• All the sections of the rotor will operate at the angle of attack where this value of

coefficient of lift and drag are produced.

• We will call this lift coefficient the optimum lift coefficient.

All radial stations will operate at an optimum value at which

is maximum Once angle of attack is selected, we find from

.

/ √

This determines how the blade must be twisted. Variation of chord for the optimum rotor is given by

( ) Note that is constant (the optimum value). It follows that

( ) 6

7

Such planforms and twist distributions are hard to manufacture, and are optimum only at one thrust setting. Manufacturers therefore use a combination of linear twist, and linear variation in chord (constant taper ratio) to achieve optimum performance.

Accounting for Tip Losses

We have already accounted for two sources of performance loss-non-uniform inflow, and blade viscous drag. We can account for compressibility wave drag effects and associated losses, during the table look-up of drag coefficient. Two more sources of loss in performance are tip losses, and swirl. An elegant theory is available for tip losses from Prandtl.

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the tip, and unity in the interior.

( ₎

When there are infinite number of blades, F approaches unity, there is no tip loss. ( )

All we need to do is multiply the lift due to inflow by F (incorporation of Tip Loss model in BEM)

Thrust generated by the annulus

( ) From BET ( ) ( ) . / Resulting Inflow (Hover) will be

( )

<√

=

Rotors Hovering in Ground Effect

Consider a rotor hovering in close proximity to the ground. Recall that when the rotor hovers far from the ground one can obtain the inflow from momentum theory. Because the ground must be a streamline to the flow, the rotor slipstream tends to rapidly expand as it approaches the surface. This alters the slipstream velocity, the induced velocity in the plane of rotor and therefore the rotor thrust and power. Similar effects are obtained both in hover and forward flight, but the effects are strongest in the hovering state. When the hovering rotor is operating in ground effect, the rotor thrust is found to be increased for a given power. When the rotor is near the ground

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49

In this case one can expect the inflow for an equal amount of thrust. One can develop a fairly simple analytical model for this case by using an analytic image effect, which is schematically shown below

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requirement is lower. At a constant power setting descent tends to increase thrust.

Out of ground effect

√ In ground effect √

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51

√ √ ( ) ( ) And thus √

For forward flight the effect of V (forward flight velocity) has a beneficial effect as show in the figure

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52 Axial Climb & Descent

The rotor is climbing at velocity V and therefore the flow is downward through the rotor disk. It is assumed that the induced velocities and w at the rotor disk and in the far wake respectively.

The mass flux is

̇ ( ) Momentum conservation gives

Thrust = Change in Momentum ̇( ) ̇ ̇ From energy conservation

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( ) ̇( ) ̇ ̇( ) ( ) ̇ ( ) ̇ ( ) ̇( ) ( ) In hover

In hover, climb or descent, rotor supports the same weight. So T will be the same ( )

( ) ( )

This is a simple quadratic relationship of non-dimensional inflow w.r.t inflow at hover condition.

The solution of the above quadratic equation is

: √( ) ; ⁄

The positive root of the equation should be used, since the induced velocity is positive. If we use the negative root, induced velocity becomes negative, Inflow variation becomes

√( )

Net flow at rotor disk

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54

√( ) √( ) √( )

From the above expressions, we can say that inflow decreases as the helicopter rise up or climb up. Mass flow rate increases as the velocity changes.

Power Needed in Climb and Hover

It is convenient to non-dimensionalize these graphs, so that universal behavior of a variety of rotors can be studied.

( )

< √( )

=

Descent condition

Thrust is always up and weight is down. The rotor to support the weight it has to push the air down. In this case is downward and V is upward. As the rotor pushes the air down, the farfield downstream also moves downward.

The mass flux is

̇ ( ) Momentum conservation gives

Thrust = Change in Momentum

Climb Velocity, V Power

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̇( ) ̇ ̇

Thrust acting on the rotor disk is upward, but on the fluid is opposite in direction. From energy conservation

( ) ( ) ̇( ) ̇ ̇( ) ( ) ̇ ( ) ̇ ( ) ̇( ) ( )

Momentum Theory gives incorrect Estimates of Power in Descent. ( )

< √( )

=

In hover

In hover, climb or descent, rotor supports the same weight. So T will be the same ( )

V/v

_h

(V+v)/v

_h Climb Descent

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56

( ) ( )

This is a simple quadratic relationship of non-dimensional inflow w.r.t inflow at hover condition.

The solution of the above quadratic equation is

: √( ) ; ⁄

√( ) The roots are positive in both the cases. When V is less than

√(

)

becomes imaginary value. This is valid when V is more than .

is positive in both the roots When V is much larger than the term

√(

)

Gives positive value, otherwise this becomes imaginary value. At rotor disk, √( ) Climb (V<0) √( )

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57

√( )

Descent (V>0) (V positive upward)

√( )

If we consider V<0 for descending condition Helicopter Up (Positive) and Down (Negative)

√( ) √( ) √( )

Assume that these expressions are valid everywhere.

Inflow diagram for vertical flight is expressed in non-dimensional form. Continuous lines are valid. In dotted line region, flow is highly mixed so it is difficult to calculate inflow value. Dotted lines are extrapolated values. When , there is no flow through the disk. The lines divide the plane into four regions, where the rotor operating condition is named the normal working state (climb and hover), vortex ring state, turbulent wake state and windmill brake state. For climb it was assumed that the air is moving downward throughout the flow field ( ). The flow through the disk and in the wake is downward while the flow outside the slipstream is upward. The climb solution may be expected to be valid for small rate of descent, however, where at least near the rotor the flow is all downward.

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field ( ). For the upper branch of the descent solution, however, , so the flow is downward in the far wake while it is upward everywhere else, including outside the wake slip-stream.

In the vortex ring and turbulent wake states the flow outside the slipstream is upward while the flow inside the far wake is nominally downward. Because such a flow state is not possible, there is no valid momentum theory solution for the moderate rates of descent between V = 0 and .

The line corresponds to ideal autorotation, P=0, and is in the center of the range where momentum theory is not valid.

At hover V=0, becomes 1. So the curve starts from 1 in climb. is a constant quantity for a given rotor. It is a fixed value in climb condition. As the velocity increases the below term decreases and reaches zero asymptotically at very high velocity value.

√( )

Expression given below is valid both in descent and climb condition

√( )

In descent, velocity is negative so the result from the above expression keeps on increases. The climb curve will reach technically.

Asymptotically the curve will reach the line at 45 degree. If we extrapolate to descent, the curve will be like as shown in graph.

If Or Then √( )

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becomes imaginary value. We cannot have the root less than -2.

In descent

√( ) In this case both roots are valid, because

√(

) Is smaller than

When V is less than or

< 2

√(

) becomes imaginary value.

For each value of

,

we will have two roots one above the and one below it. The line is the ideal autorotation case here; for points the above ideal autorotation the rotor is absorbing power and for points below it is producing power for the helicopter. The universal inflow curve crosses the ideal autorotation line at about ⁄ (the scatter extends over roughly ⁄ to -1.8. Real autorotation occurs at a high rate of descent, in the turbulent wake state. In the turbulent wake state the induced velocity curve can be approximated fairly well by a straight line on the ( ) ⁄ vs ( ⁄ ) plane. Joining the ideal autorotation intercept ( ⁄ ) and the windmill brake state boundary (( ) ⁄ ) gives

( )

(-2,-1) is the point where the root for the descent condition starts. The above equation represents the straight line equation.

If we substitute X as -1.6 to -1.8

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( ) We know that ( )

Autorotation is the state of rotor operation with no net power requirement. The power to produce the thrust and turn the rotor is supplied either by auxillary propulsion or by descent of the helicopter. A component of the aircraft forward velocity directed upward through the

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rotor disk supplies the power to the rotor, in the autorotative descent of the helicopter, the source of power is the decrease of the gravitational potential energy. More directly, the descent velocity upward through the disk supplies the power to the rotor. The net rotor power is zero for vertical descent in autorotation. The decrease in potential energy (TV) balances the induced (T ) and profile losses of the rotor. Neglecting the profile losses gives ideal autorotation

( )

The velocity with which the helicopter descends will not be at ( ) . It will be slightly at increased descent velocity. It will not be at ( ) . It will be more than this value.

Typically the descent velocity is in the range of 15 to 20 m/s in autorotation. ( ) ( ) ( ) ( ) ( ) √ In autorotation, shaft power = 0

√ 4 √ 5 4 √ 5

If ⁄ is less or small, then autorotative descent velocity also becomes small.

The rotor in descent operates in a number of stages, depending on how fast the vertical descent is in comparison to hover induced velocity. Four different flow states have been

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identified with descending flight.

(a) Normal thrusting state , ( ) ( ) ( )- (b) Vortex ring state , ( ) ( ) ( )- (c) Turbulent Wake state , ( ) ( )-

(d) Windmill brake state , ( ) ( ) ( )-

Assume that the induced velocity is constant and uniform over the entire disk. Velocity of the flow inside the slipstream is same in the direction everywhere (downward). The value of induced velocity vary depend on flight condition. This condition is defined as Normal Working state (Flow everywhere inside and outside the slipstream is same in the direction).

Normal thrusting state:

The normal working state includes climb and hover. For climb, the velocity throughout the flow field is downward with both V and positive. From mass conservation it follows that the wake contracts downstream of the rotor. A wake model with a definite slipstream is valid for this flow state and momentum theory gives a good estimate of the performance. There will also be entrainment of air into the slipstream below the rotor and some recirculation near the disk, particularly for hover. Although such phenomena are not included in the momentum theory model, their effect on the induced power is secondary.

Hover (V=0) is the limit of the normal working state, still momentum theory models the flow well in the vicinity of the rotor disk and hence gives a good performance estimate even though hover is nominally a limiting case

Vortex ring state:

When the rotor starts to descend, a definite slipstream ceases to exist because the flows inside and outside the slipstream in the far wake want to be in the opposite directions. Therefore, from hover to the windmill brake state the flow has large recirculation and high turbulence. Sometimes this entire region is called the vortex ring state.

Descent condition is split into three regions, depending on the kind of flow. Vortex ring state begins when helicopter just started descending from the hovering condition. In descent

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condition, the velocity of the flow ( ) is smaller than the inflow. is larger than V. So it is directed downward( ) .

V = Farfield Upstream; = Farfield downstream velocity; = Velocity at the disk

During the descent, flow inside the slipstream will not be in one direction. It becomes mixed flow. As the rotor descends down and the flow goes up, rotor generates vortex flow from bottom to top. When the strength of the vortex becomes large enough, it detaches from the rotor (Vortex Ring State).

V = Negative because flow is coming from down to up.

= Positive because downward (Flow at the rotor disk is down) = Positive because downward

In vortex ring state condition, the flow is downward and in turbulent wake state condition, the flow is upward. There is a condition at which the flow at the rotor disk is zero (i.e. no flow at the disk)

We know already that, the power is split into three terms.

( )

When , Power (P) becomes zero, but still the rotor is supporting the weight. Power supplied to the rotor disc to generate thrust decreases as the descent velocity increases and becomes zero at one particular condition.

The vortex ring state is defined by P = T (V+ ) > 0, so that the power extracted from the air stream is less that the induced power.

The region with P = T (V+ ) < 0, is called the turbulent wake state. Partial power descents occur in the vortex ring state. Equilibrium autorotation will usually occur in the turbulent

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wake state. At small rates of descent, recirculation near the disk and unsteady, turbulent flow above it begin to develop. The change in flow state for small rates of climb or descent is gradual; the momentum theory solution remains valid for some way into the vortex ring state.

Turbulent Wake State

In the turbulent wake state condition, the velocity (V+ ) is in the downward direction and (V+ ) is in the upward direction, so there is a circulation of flow inside the slipstream (wake region). In this condition, vibration will not be as high as in condition (vortex ring state). When the descent velocity increases more, V becomes negative. At the rotor disk, inflow is down, is less than the velocity V (net value is upward in direction). Farfield downstream velocity is positive in this case also.

When the rotor had no profile power losses, power-off descent would be in this condition, since P = T (V+ ) = 0 for it. While nominally there is no flow through the disk, actually there is considerable recirculation and turbulence. The flow state is similar to that of a circular plate of the same area (no flow through the disk, a turbulent wake above it). The flow still has a high level of turbulence, but since the velocity at the disk is upward there is much less recirculation through the rotor. The flow pattern above the disk in the turbulent wake state is very similar to the turbulent wake of a bluff body.