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Helicopters / Filipe Szolnoky CunhaSlide 1Momentum Theory in Forward Flight
•In helicopter forward flight the rotor moves
through the air with an edgewise velocity
component that is parallel to the rotor plane
•Since the helicopter rotor has to produced
both the lifting force and the propulsion force.
•The rotor must be tilted forward at an AOA
relatively to the oncoming flow.
Momentum Theory in
Forward Flight
Helicopters / Filipe Szolnoky CunhaSlide 2Momentum Theory in Forward Flight
The Dynamic Pressure varies
Radially and Azimuthally
Vtip=ΩR+V∞
Vtip=ΩR
Vtip=ΩR-V∞
Vtip=ΩR
Helicopters / Filipe Szolnoky CunhaSlide 3Momentum Theory in Forward Flight
Force Balance in Hover
Thrust
Weight
Drag
Drag
Rotor Disk
•In hover, T= W
•The drag forces on the individual blades cancel
each other out, when summed up.
Helicopters / Filipe Szolnoky CunhaSlide 4Momentum Theory in Forward Flight
Glauert’s flow model
• To start this effort, we will need a very simpleinflow model.
• A model proposed by Glauert (1926) is used.
• This model is phenomenological, notmathematically well founded.
• It gives reasonable estimates of inflow velocity atthe rotor disk, and is a good starting point.
• It also gives the correct results for an ellipticallyloaded wing.
Helicopters / Filipe Szolnoky CunhaSlide 5Momentum Theory in Forward Flight
Force Balance in Forward Flight
Flight Direction
Thrust, T
Vehicle Drag, D
Weight, W
Helicopters / Filipe Szolnoky CunhaSlide 6Momentum Theory in Forward Flight
Force Balance in Forward Flight
Flight Direction
α
Rotor DiskT
W
D
c.g.
Helicopters / Filipe Szolnoky CunhaSlide 7Momentum Theory in Forward Flight
Glauert’s flow model
Freestream, V∞ Freestream, V
∞
Induced velocity, vi
Freestream, V∞
Induced velocity, w
Total velocity, V
Total velocity, V
Helicopters / Filipe Szolnoky CunhaSlide 8Momentum Theory in Forward Flight
Total Velocity at the Rotor Disk
Helicopters / Filipe Szolnoky CunhaSlide 9Momentum Theory in Forward Flight
Conservation laws
• Conservation of momentum in the direction
normal to the disk:
• Conservation of energy in the same direction
Helicopters / Filipe Szolnoky CunhaSlide 10Momentum Theory in Forward Flight
Conservation laws
• From the two previous equations we can write:
• And reach the conclusion that w=2vi , the same
result was in the previous cases.
• Knowing that the mass flow at the disk is ρAU:
• In high speed forward flight V∞>>vi so:
Helicopters / Filipe Szolnoky CunhaSlide 11Momentum Theory in Forward Flight
Induce velocity
• We know for the hover case that:
• Then from the previous equation:
Helicopters / Filipe Szolnoky CunhaSlide 12Momentum Theory in Forward Flight
Non dimensional forms
• The non-dimensional form using the tip speed ΩR:
• So that the non-dimensional induced velocity
equation can be written as:
Helicopters / Filipe Szolnoky CunhaSlide 13Momentum Theory in Forward Flight
Non dimensional forms
• Since we already know that:
• We can write:
• Which requires a numerical solution
Helicopters / Filipe Szolnoky CunhaSlide 14Momentum Theory in Forward Flight
Approximate Form at
High Speed Forward Flight
• If the advance ratio µ is higher than 0.2 and α is
small, µ far exceeds the inflow ratio λ:
Helicopters / Filipe Szolnoky CunhaSlide 15Momentum Theory in Forward Flight
Variation of Non-Dimensional Inflow
with Advance Ratio
• Notice that inflow velocity rapidly decreases withadvance ratio
Helicopters / Filipe Szolnoky CunhaSlide 16Momentum Theory in Forward Flight
Variation of Non-Dimensional
Inflow with Advance Ratio
Helicopters / Filipe Szolnoky CunhaSlide 17Momentum Theory in Forward Flight
Power Consumption in
Forward Flight
• The ideal power from Glauert’s theory is
• For the actual power we have to take into account the blade profile power
• From the equilibrium of forces Tsinα=D so:
• Where Tvi is the induce power and DV∞ is the Parasitic power
Helicopters / Filipe Szolnoky CunhaSlide 18Momentum Theory in Forward Flight
Power Consumption in
Forward Flight• The induce power decrease with the advance ratio
µ
• The Parasite power can be calculated:
• The Parasite power increases with the cube of the
forward velocity (or advance ratio µ)
Helicopters / Filipe Szolnoky CunhaSlide 19Momentum Theory in Forward Flight
Power in Forward Flight
Induced Power, Tvi
Profile power
Available Power
Helicopters / Filipe Szolnoky CunhaSlide 20Momentum Theory in Forward Flight
Power Coefficient
Induced power
Parasite Power
Profile Power
1.15
CD is the vehicle parasite drag coefficient and S the reference
area. Because there is no agreement on a common reference
area it is customary to supply the product CDS=f equivalent
flat plate area