Blade Element Momentum Theory
• Blade Element Theory has a number of
assumptions.
• The biggest (and worst) assumption is that the
inflow is uniform.
• In reality, the inflow is non-uniform.
• It may be shown that uniform inflow yields the
lowest induced power consumption.
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 1
Model
• Annulus
dy
y
Helicopters / Filipe Szolnoky Cunha
–
–
–
–
At a distance y
Width dy:
area dA=2πydy
Mass flow rate
Blade Element Momentum Theory
Slide 2
Thrust and Power
• The thrust for the annulus is :
• Or in the non-dimensional form:
• The induced power coefficient is:
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 3
Thrust and Power
• For the hover case λC=0.
– The thrust coefficient is then:
– The power coefficient is:
– Valid for any form of λ
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 4
Thrust Coefficient
• Let’s assume that λ(r)= λtiprn for n≥0.
• We can then relate λtip with CT
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 5
Power Coefficient
• Calculating the power coefficient
• Substituting in the λtip equation :
• Since we know that
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 6
Induced Inflow
• In the previous equation when n=0 then κ=1,
uniform inflow which has the lowest induced
power.
• For n>0 then κ>1. With increasing n the inflow is
more heavily biased towards the blade tip.
• Therefore what we want to achieved is, through
the blade properties (pitch angle and chord), the
first situation
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 7
Blade Element Momentum Theory
• From BET we have found
• Then we have
• Which can be expressed:
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 8
Blade Element Momentum Theory
• Or in another way:
• This quadratic equation has the solution:
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 9
Ideal Twist
• In hover (λC=0) it simplifies to
• Since the objective is uniform inflow then we
need θr=const.= θtip. The ideal twist is given by:
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 10
Thrust coefficient in Ideal Twist
• We can then calculate the thrust coefficient with
this ideal twist θr= θtip
• Recalling that
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 11
Blade Pitch angle
• In hover
• We can then obtain θtip
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 12
Final notes
• Recall that:
• Therefore the thrust varies linearly with r. We
have therefore a linear (triangular) variation of lift
over the blade.
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 13
Ideal Rotor vs. Optimum Rotor
• Ideal rotor has a non-linear twist: θ=θtip/r
• 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 θ
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.
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 14
Optimum Hovering Rotor
• Minimum Pi requires λ=const. (uniform inflow)
• Minimum P0 requires α= α(min Cd/Cl)= α1
• Then for minimum induce power θ= θtip/r and
each blade element must operate at α1
• With (BEMT) dCT=4λ2rdr then:
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 15
Optimum Hovering Rotor
• We have seen that the minimum induced power
requires a uniform inflow. Therefore the previous
equation is constant over the disk.
• Let’s assume that α1 is the same for all airfoils
along the blade and is independent of Re and M
• From the equation since α1 =const and we now
that λ=const then σr must be constant too.
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 16
Optimum Hovering Rotor
• The previous situation is achieved when
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 17
Optimum Hovering Rotor
• Let’s now study the blade twist for uniform inflow
and constant AOA:
• Expressing now in terms of the blade pitch
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 18
Optimum Hovering Rotor
• The total thrust can be calculated
• The local solidity of the optimum rotor can be
written:
• And the blade twist
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 19
Optimum Hovering Rotor
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 20
Optimum Hovering Rotor
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 21
Power estimates
• In a real rotor the non uniformity of λ means that
we must calculate numerically:
• We also saw that we could then calculate the
induce power factor κ :
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 22
Power estimates
• Let’s now calculate the rotor profile power.
Remember that (BET):
• We can use for Cd the expression:
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 23
Power estimates
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 24
Power estimates
• The profile power is then:
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 25
Power estimates
• For a ideally twisted blade λ=const and θr= θtip:
• Since
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 26
Power estimates
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 27
Figure of Merit
• Since we now have the expressions for all power
coefficients we can calculate the rotor FM:
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 28
Prandtl’s Tip-loss Function
• The idea of tip losses was already been introduced
with the factor B
• Prandtl provided a solution to the problem of loss
of lift near the blade tip.
• In this solution the curved helical vortex sheets of
the rotor wake were replaced by a series of 2D
sheets, meaning that the blade tip radius of
curvature is large
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 29
Prandtl’s Tip-loss Function
• Prandtl’s final result can be expressed in terms of
an induced velocity correction factor F:
• Where f:
• Remember that
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 30
Prandtl’s Tip-loss Function
Non dimensional radial position r
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 31
Prandtl’s Tip-loss Function
• Note that when Nb→∞ (actuator disk) then F→1
• The function can be incorporated in the BEMT:
• Since from BET
• We can write
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 32
Prandtl’s Tip-loss Function
• With the solution
• Since F is a function of λ the solution must be
found numerically
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 33
Compressibility Corrections
• So far we have considered that all aerodynamic
characteristics independent of Mach number.
• To introduce a correction to take into account the
influence of M, let’s used Glauert’s rule:
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 34
Compressibility Corrections
• The local blade M is:
• Which gives a lift-curve-slope correction of:
• We had obtained:
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 35
Compressibility Corrections
• Assuming the ideal twist (uniform inflow):
• Calculating the total thrust coefficient:
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 36
Compressibility Corrections
• In the previous expression:
• If M→0 then K→1
incompressible result
Helicopters / Filipe Szolnoky Cunha
and
we
Blade Element Momentum Theory
obtain
the
Slide 37
Weighted Solidity
• We have seen that for a optimum rotor c varies
with r.
• In these cases where c varies along the blade span
the rotor solidity is different from the local blade
solidity:
• The objective of weighted solidity is to help
compare performance of different rotors with
different blade planforms.
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 38
Trust Weighted Solidity
• The thrust coefficient is:
• Assuming a constant Cl:
• With the equivalent chord
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 39
Power Weighted Solidity
• Extending the study to the power coefficient
• Therefore
Helicopters / Filipe Szolnoky Cunha
Blade Element Momentum Theory
Slide 40