Momentum Theory
•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
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 1
Momentum Theory
• 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
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 2
Momentum Theory
• 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.
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 3
Momentum Theory
• Let’s simplify our first approach and develop a
simple method capable of predicting the rotor
thrust and power
Momentum Theory
• First developed by Rankine (1895) for marine
propellers and developed further and generalized
by several other authors
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 4
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 Cunha
Momentum Theory in Hover
Slide 5
Assumptions
• 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.
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 6
Assumptions
• 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.
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 7
Representation and notation
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 8
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 Cunha
Momentum Theory in Hover
Slide 9
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 Cunha
Momentum Theory in Hover
Slide 10
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 Cunha
Momentum Theory in Hover
Slide 11
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 Cunha
Momentum Theory in Hover
Slide 12
Conservation of Mass through the
rotor disk
• At control surface 1:
• At control surface ∞
• And:
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 13
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 Cunha
Momentum Theory in Hover
Slide 14
Bernoulli equation
• Consider a particle that goes from Station 0
to station ∞
• We can apply Bernoulli equation between:
0
1
vh
2
∞
– Stations 0 and 1,
– Stations 2 and ∞.
• Recall assumptions that the flow is steady,
irrotational, inviscid.
w
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 15
Bernoulli equation
• From the previous expressions we have:
p∞
Disc
∆p
Flow field
p∞
Pressure
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
v
w
Velocity
Slide 16
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 Cunha
Momentum Theory in Hover
Slide 17
Induced Velocity at the rotor disk
• And the following expression can be obtained:
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 18
Ideal Power
• Power consumed=Energy rate flow out-Energy
rate flow in
• So:
Or in terms of the induced velocity:
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 19
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 Cunha
Momentum Theory in Hover
Slide 20
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 Cunha
Momentum Theory in Hover
Slide 21
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 Cunha
Momentum Theory in Hover
Slide 22
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 Cunha
Momentum Theory in Hover
Slide 23
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 Cunha
Momentum Theory in Hover
Slide 24
Thrust and power coefficient
• The two coefficient can be related using the
momentum theory.
• Therefore
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 25
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 Cunha
Momentum Theory in Hover
Slide 26
Figure of Merit
• The ideal power is calculated
momentum theory so we can write
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
using
the
Slide 27
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 Cunha
Momentum Theory in Hover
Slide 28
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 and
compute more accurately the necessary rotor
power
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 29
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 Cunha
Momentum Theory in Hover
Slide 30
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 Cunha
Momentum Theory in Hover
Slide 31
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 Cunha
Momentum Theory in Hover
Slide 32
Non Ideal effects
• The profile power is:
• With it’s associated power coefficient
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 33
Non Ideal effects
• The rotor solidity is defined as:
• With typical values of 0.07 to 0.12
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 34
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 Cunha
Momentum Theory in Hover
Slide 35
Induced Tip losses
R
BR
Helicopters / Filipe Szolnoky Cunha
• A portion of the
rotor near the tip
does not produce
much lift due to the
leakage of air from
the bottom of the
disk to the top
• We can account for
it by using a smaller
modified radius BR
Momentum Theory in Hover
Slide 36
Induced Tip losses
• So the effective blade radius Re that produces lift
is smaller than the blade radius R:
• Where B<1. The effective rotor disk area is:
• Which is smaller the the actual rotor disk are by
a factor of B2.
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 37
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 Cunha
Momentum Theory in Hover
Slide 38
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 Cunha
Momentum Theory in Hover
Slide 39
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 Cunha
Momentum Theory in Hover
Slide 40
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 Cunha
Momentum Theory in Hover
Slide 41
Power Coefficient
• We have already reach to the relations:
• Using the modified momentum theory:
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 42
Power Coefficient
• We can also write:
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 43
Power Coefficient
• Or alternatively:
• That is
Helicopters / Filipe Szolnoky Cunha
Momentum Theory in Hover
Slide 44