# What are the relative efficiencies of the various stages of a counter rotating propeller?

After going through the CFD results of a drone project I am involved in, I noticed that the data shows significantly increased (+10%) thrust generation at the second stage of a counter-rotating, 2-stage, ducted fan. Both stages use identical airfoils and spin at the same rpm. They also have low solidity, meaning they look more like propellers than airliner fans, as the blades take up a small proportion of the disc area.

Generally, propellers are less efficient when the incoming airflow has a high velocity, so I would expect, in any N-stage fan using otherwise identical stages, for the N stage to produce less thrust than the N-1 stage, but apparently I am missing something.

In a previous answer here Peter Kämpf mentioned that the first stage of the Kuznetsov NK-12 turboprop pre-swirls the flow to improve thrust creation conditions at the second stage. How does this process work? Does it require a specific design of the stages to work in tandem?

The key thing to think about is that the first stage doesn't just accelerate air downwards, it also causes it to rotate ("swirl") in the same direction as the first stage propeller.

This happens because the first stage propeller, being an airfoil, generates both lift and drag. The lift corresponds to air being accelerated down the axis of the duct; the drag corresponds to air being accelerated circularly around it. The amount of swirl force is therefore related to the thrust and the effective L/D ratio of the propeller.

The airfoil of the second stage propeller is then encountering air that is moving towards it at the tangential velocity of the propeller, plus the tangential velocity of the air that has now been circularly accelerated ("pre-swirled") in the opposite direction by the first stage.

That increased velocity means that the angle of attack for the second stage airfoil will be increased compared to that encountered by the first stage, and hence greater lift is produced at the same RPM.

To balance the load on the two stages you would need to adjust the pitch or RPM of the second stage propeller.

Note that after the second stage, the air will be "unswirled" back to being closer to a non-rotating flow, and this is one of the reasons that a contra-rotating pair can be more efficient than a single propeller: overall, the system wastes less power in swirling the air rather than accelerating it axially.

• I've wondered about that; if you want to have two counter-rotating propellers like on a drone, that are fixed pitch, it wouldn't do to have the same pitch on both propellers. The second propeller should have a bit less pitch then, if it's going to operate at the same RPM as the first one and you want it to make the same thrust? Jul 9 '19 at 21:48
• Thanks for the answer. I understand what you mean to say regarding the induced velocity from the first stage on an intuitive level, but I feel it could be clearer. Also in your second and third paragraph, you use radial to describe the swirl component. Radial is usually used to denote something along the radial vector, I think you meant to use azimuthal or angular. Jul 10 '19 at 9:39
• @JohnK you could give the second propeller less pitch, or you could also just operate it at slightly lower RPM. On a typical electric multirotor, adjusting the RPM would be far easier, since then you could stick to all your propellers being standard types with the same pitch. Jul 14 '19 at 20:42
• @AEhere thanks for the feedback. I've replaced the misuse of 'radial' velocities with 'tangential'. Angular velocity would not be correct - that refers to a rotational velocity in e.g. RPM or radians/second, whereas what matters here is the linear velocity of any given point along the propeller blade, tangential to its circular path. Jul 14 '19 at 20:44

The aft propeller operates in the higher pressure environment that the front propeller generates. Thrust is a function of the stagnation pressure of the incoming flow - same principle as translational lift in helicopters.