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