Helicopters / Filipe Szolnoky CunhaSlide 1Momentum Theory in Hover

•We saw that the helicopter’s rotor

provides three basic functions:

•Generation of Lift

•Generation of propulsive force for forward

flight

•Generates forces to control attitude and

position

Momentum Theory

Helicopters / Filipe Szolnoky CunhaSlide 2Momentum Theory in Hover

• The helicopter must be able to operate in a variety of flow regimes:– Hover

– Climb

– Descend

– Forward flight

– Backward flight

– Any flight regime that is a combination of the above

Momentum Theory

Helicopters / Filipe Szolnoky CunhaSlide 3Momentum Theory in Hover

• The main goal of the helicopter is it’s ability to

HOVER

• Hover is also the simplest of the flight regimes, so

it should be the easiest to model

• Although it’s the simplest flight regime it is still

complicated enough.

Momentum Theory

Helicopters / Filipe Szolnoky CunhaSlide 4Momentum Theory in Hover

• Let’s simplify our first approach and develop asimple method capable of predicting the rotorthrust and power

Momentum Theory

• First developed by Rankine (1895) for marinepropellers and developed further and generalizedby several other authors

Momentum Theory

Helicopters / Filipe Szolnoky CunhaSlide 5Momentum Theory in Hover

Assumptions

• Conditions in hover:

– No forward speed

– No vertical speed

– The flow field is axisymetrical

– There is a wake boundary with the flow outside this

boundary being quiescent

– The flow velocities inside this boundary can be quite

high

Helicopters / Filipe Szolnoky CunhaSlide 6Momentum Theory in Hover

• Momentum theory concerns itself with the global

balance of mass, momentum, and energy.

• It does not concern itself with details of the flow

around the blades.

• It gives a good representation of what is

happening from a view far away from the rotor.

• This theory makes a number of simplifying

assumptions.

Assumptions

Helicopters / Filipe Szolnoky CunhaSlide 7Momentum Theory in Hover

• Rotor is modeled as an actuator disk which adds

momentum and energy to the flow.

• Flow is incompressible.

• Flow is steady, inviscid, irrotational.

• Flow is one-dimensional, and uniform through the

rotor disk, and in the far wake.

• There is no swirl in the wake.

Assumptions

Helicopters / Filipe Szolnoky CunhaSlide 8Momentum Theory in Hover

Representation and notation

Helicopters / Filipe Szolnoky CunhaSlide 9Momentum Theory in Hover

Conservation of Mass

– Air inflow trough control surface 0:

– There is no inflow/outflow through the side boundaries:

– Airflow trough control surface ∞

Helicopters / Filipe Szolnoky CunhaSlide 10Momentum Theory in Hover

Conservation of Mass through the rotor

disk

– Air inflow trough the rotor disk control surface 1:

– Air inflow trough the rotor disk control surface 2:

– Since the two surfaces (A1=A2=A) are equal:

– There is no velocity jump across the rotor disk. vi is the induced velocity at the rotor disk.

Helicopters / Filipe Szolnoky CunhaSlide 11Momentum Theory in Hover

Hover conditions

• In hover Vc→0:

– The velocity at station 0 is 0

– The velocity at the rotor is the induced velocity at the

rotor vi

– The velocity at the far field is the induced velocity at

the far field w

Helicopters / Filipe Szolnoky CunhaSlide 12Momentum Theory in Hover

Momentum and energy equations

• The momentum rate of change is equal to the applied

force:

• The work done per unit time (power) done by the rotor is

equal to the energy rate of change

• Eliminating

Helicopters / Filipe Szolnoky CunhaSlide 13Momentum Theory in Hover

Conservation of Mass through the

rotor disk

• At control surface 1:

• At control surface ∞

• And:

Helicopters / Filipe Szolnoky CunhaSlide 14Momentum Theory in Hover

Conservation of Mass

• We can reach the conclusion that:

– The far wake induce velocity is twice the induce velocity at the disk

– The far wake area is half the rotor disk area

– In reality

Helicopters / Filipe Szolnoky CunhaSlide 15Momentum Theory in Hover

Bernoulli equation• Consider a particle that goes from Station 0

to station ∞

• We can apply Bernoulli equation between:

– Stations 0 and 1,

– Stations 2 and ∞.

• Recall assumptions that the flow is steady,

irrotational, inviscid.

0

1

2

∞

vh

w

Helicopters / Filipe Szolnoky CunhaSlide 16Momentum Theory in Hover

Bernoulli equation

• From the previous expressions we have:

Flow field

p∞

Pressure Velocity

Disc∆p

v

w

p∞

Helicopters / Filipe Szolnoky CunhaSlide 17Momentum Theory in Hover

Induced Velocity at the rotor disk

• We can now compute the induced velocity at the

rotor disk in terms of the thrust T

and

Helicopters / Filipe Szolnoky CunhaSlide 18Momentum Theory in Hover

Induced Velocity at the rotor disk

• And the following expression can be obtained:

Helicopters / Filipe Szolnoky CunhaSlide 19Momentum Theory in Hover

Ideal Power

• Power consumed=Energy rate flow out-Energy

rate flow in

• So:

Or in terms of the induced velocity:

Helicopters / Filipe Szolnoky CunhaSlide 20Momentum Theory in Hover

Disk Loading

• Disk loading is defined as the ratio of the thrust by the disk area:

• The expression of the induced velocity at the rotor can then be expressed in terms of the disk loading:

• Remember that in hover T=W

Helicopters / Filipe Szolnoky CunhaSlide 21Momentum Theory in Hover

Power Loading

• Power Loading is defined as:

• On the other hand the induced velocity at the rotor

can be obtained from:

• We can then write:

Helicopters / Filipe Szolnoky CunhaSlide 22Momentum Theory in Hover

Induced inflow ratio

• The induced velocity at the rotor can be expressed

in the following manner:

• λh is called the induced inflow ratio

• For rotating-wing aircraft it is the convention to

nondimensionalize all velocities by the blade tip

speed in hover

Helicopters / Filipe Szolnoky CunhaSlide 23Momentum Theory in Hover

Thrust coefficient

• Since the convention is to nondimensionalize the

velocities by the blade tip speed, we can define

the thrust coefficient:

• The inflow ratio can then be expressed

Helicopters / Filipe Szolnoky CunhaSlide 24Momentum Theory in Hover

Power coefficient

• The rotor power coefficient is defined as:

• Since the power is related to the rotor shaft torque by P=ΩQ and the rotor shaft torque is defined by:

• We can conclude that CP=CQ

Helicopters / Filipe Szolnoky CunhaSlide 25Momentum Theory in Hover

Thrust and power coefficient

• The two coefficient can be related using the

momentum theory.

• Therefore

Helicopters / Filipe Szolnoky CunhaSlide 26Momentum Theory in Hover

Figure merit

• All the previous expression were calculated for an

ideal rotor in an ideal fluid

• There is the necessity to calculate the rotor

efficiency

• In 1940 Prewitt of Kellett Aircraft introduce the

Figure of Merit

Helicopters / Filipe Szolnoky CunhaSlide 27Momentum Theory in Hover

Figure of Merit

• The ideal power is calculated using the

momentum theory so we can write

Helicopters / Filipe Szolnoky CunhaSlide 28Momentum Theory in Hover

Figure of merit

• 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).

Helicopters / Filipe Szolnoky CunhaSlide 29Momentum Theory in Hover

Non Ideal effects

• Until now we have considered ideal situation

• We did not take into account situations like:

– Non-uniform inflow

– Tip losses

– Wake swirl

– Non ideal wake contraction

– Finite number of blades

• We can then take into account these factors andcompute more accurately the necessary rotorpower

Helicopters / Filipe Szolnoky CunhaSlide 30Momentum Theory in Hover

Non Ideal effects

• First let’s correct the power coefficient using a

correction factor (induced power coefficient):

• Where κ is the induced power correction factor

• Typical value of κ is 1.15

Helicopters / Filipe Szolnoky CunhaSlide 31Momentum Theory in Hover

Non Ideal effects

• Secondly let’s take into account the blade drag:

– D is the drag per unit span

– Nb is the number of blades

– y is the blade element distance to the rotor hub

• The power necessary to overcame the blade drag

is:

Helicopters / Filipe Szolnoky CunhaSlide 32Momentum Theory in Hover

Non Ideal effects

• The drag force per unit span can be obtained using

the drag coefficient of the section profile

• It is assumed that:

– Cd0 is independent of Re and M

– The blade is not tapered or twisted

Helicopters / Filipe Szolnoky CunhaSlide 33Momentum Theory in Hover

Non Ideal effects

• The profile power is:

• With it’s associated power coefficient

Helicopters / Filipe Szolnoky CunhaSlide 34Momentum Theory in Hover

Non Ideal effects

• The rotor solidity is defined as:

• With typical values of 0.07 to 0.12

Helicopters / Filipe Szolnoky CunhaSlide 35Momentum Theory in Hover

Non Ideal effects

• The actual rotor power can then be expressed as:

• Using the modified form of the momentum theory

with the non ideal approximation for power the

rotor figure of merit can be written as:

Helicopters / Filipe Szolnoky CunhaSlide 36Momentum Theory in Hover

Induced Tip losses

• A portion of therotor near the tipdoes not producemuch lift due to theleakage of air fromthe bottom of thedisk to the top

• We can account forit by using a smallermodified radius BR

R

BR

Helicopters / Filipe Szolnoky CunhaSlide 37Momentum Theory in Hover

Induced Tip losses

• So the effective blade radius Re that produces liftis smaller than the blade radius R:

• Where B<1. The effective rotor disk area is:

• Which is smaller the the actual rotor disk are bya factor of B2.

Helicopters / Filipe Szolnoky CunhaSlide 38Momentum Theory in Hover

Induced Tip losses

• There are several propositions to calculate the

factor B:

– Prandtl theory

– Helicopters Rotor approximation

Since λi (inflow ratio) is small and in hover related to CT

Helicopters / Filipe Szolnoky CunhaSlide 39Momentum Theory in Hover

Induced Tip losses

• Empirical geometric calculations:

– Gessow & Meyers

c is the tip chord

– Sissingh

c0 is the root chord and τr is the blade tapper ratio

Helicopters / Filipe Szolnoky CunhaSlide 40Momentum Theory in Hover

Blade Loading Coefficient

• The blade loading coefficient is defined as:

– Where Ab is area of the all the blades

• The maximum realizable value is about 0.12 due

to the occurrence of blade stall

Helicopters / Filipe Szolnoky CunhaSlide 41Momentum Theory in Hover

Power Coefficient

• We have defined power loading as:

• Since

– T depends on (ΩR)2

– P depends on (ΩR)3

• To maximize PL →ΩR should be minimum

Helicopters / Filipe Szolnoky CunhaSlide 42Momentum Theory in Hover

Power Coefficient

• We have already reach to the relations:

• Using the modified momentum theory:

Helicopters / Filipe Szolnoky CunhaSlide 43Momentum Theory in Hover

Power Coefficient

• We can also write:

Helicopters / Filipe Szolnoky CunhaSlide 44Momentum Theory in Hover

Power Coefficient

• Or alternatively:

• That is