6-Momentum Theory in Forward Flight

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