Simple Performance Prediction Methods Momentum Theory © L. Sankar Helicopter Aerodynamics

© L. Sankar Helicopter Aerodynamics Background Developed for marine propellers by Rankine (1865), Froude (1885). Used in propellers by Betz (1920) This theory can give a first order estimate of HAWT performance, and the maximum power that can be extracted from a given wind turbine at a given wind speed. This theory may also be used with minor changes for helicopter rotors, propellers, etc. © L. Sankar Helicopter Aerodynamics

© L. Sankar Helicopter Aerodynamics 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 far away from the rotor. This theory makes a number of simplifying assumptions. © L. Sankar Helicopter Aerodynamics

Assumptions (Continued) 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. © L. Sankar Helicopter Aerodynamics

© L. Sankar Helicopter Aerodynamics Control Volume V Station1 Disk area is A Station 2 V- v2 Station 3 V-v3 Stream tube area is A4 Velocity is V-v4 Station 4 Total area S © L. Sankar Helicopter Aerodynamics

© L. Sankar Helicopter Aerodynamics Conservation of Mass © L. Sankar Helicopter Aerodynamics

Conservation of Mass through the Rotor Disk V-v2 V-v3 Thus v2=v3=v There is no velocity jump across the rotor disk The quantity v is called velocity deficit at the rotor disk © L. Sankar Helicopter Aerodynamics

Global Conservation of Momentum Mass flow rate through the rotor disk times velocity loss between stations 1 and 4 © L. Sankar Helicopter Aerodynamics

Conservation of Momentum at the Rotor Disk V-v Due to conservation of mass across the Rotor disk, there is no velocity jump. Momentum inflow rate = Momentum outflow rate Thus, drag D = A(p2-p3) p2 p3 V-v © L. Sankar Helicopter Aerodynamics

Conservation of Energy Consider a particle that traverses from station 1 to station 4 We can apply Bernoulli equation between Stations 1 and 2, and between stations 3 and 4. Not between 2 and 3, since energy is being removed by body forces. Recall assumptions that the flow is steady, irrotational, inviscid. 1 V-v 2 3 4 V-v4 © L. Sankar Helicopter Aerodynamics

© L. Sankar Helicopter Aerodynamics From an earlier slide, drag equals mass flow rate through the rotor disk times velocity deficit between stations 1 and 4 Thus, v = v4/2 © L. Sankar Helicopter Aerodynamics

© L. Sankar Helicopter Aerodynamics Induced Velocities V The velocity deficit in the Far wake is twice the deficit Velocity at the rotor disk. To accommodate this excess Velocity, the stream tube has to expand. V-v V-2v © L. Sankar Helicopter Aerodynamics

Power Produced by the Rotor © L. Sankar Helicopter Aerodynamics

© L. Sankar Helicopter Aerodynamics Summary According to momentum theory, the velocity deficit in the far wake is twice the velocity deficit at the rotor disk. Momentum theory gives an expression for velocity deficit at the rotor disk. It also gives an expression for maximum power produced by a rotor of specified dimensions. Actual power produced will be lower, because momentum theory neglected many sources of losses- viscous effects, tip losses, swirl, non-uniform flows, etc. © L. Sankar Helicopter Aerodynamics