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According to Energy Minimization for Wireless Communication With Rotary-Wing UAV, it is better to fly in circles than to hover at a single place (if hovering is required). Flight requires less power. As the math model is simple, I can not find what is the "perfect" radius of the flight. Can rotary-wing UAV fly in lets say 2m radius with static speed $s$, or is there some law of physics (for example centrifugal force) that enforces the UAV to fly in wider circle? This is purely theoretical question, math is welcome.

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  • $\begingroup$ I would argue that increasing the centripetal force required to navigate would increase the effective load and take away from the efficiency gain of flying at Vme. Physics wise, you could mathematically determine the additional navigational thrust required to maintain a given radius knowing Vme, the radius of the circle, and the associated mass of the aircraft $\endgroup$ Mar 30, 2022 at 15:01

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As Brydon says, you could work out the bank angle required to generate the centripetal force for various size circles. The tighter the circle, the more lift you lose. The 'optimum' radius would be an infinitely large circle (i.e. a straight line) so you'd need a reason to do tighter circles, and a way to balance that against the power required to fly.

However you'll also have to know the optimum flight speed, which will depend on the disk loading. If you're just hovering in still air, a huge, lightly loaded disk is most efficient (see the man-powered helicopter) but this has a very low top-speed. Most helicopters are optimised for forward flight, which is why they are more efficient when flying forward.

Then there's wind. You can't optimise for a static hover because you'll need to fight the wind. You might then find it's most efficient to circle when the wind is light, and match the windspeed when it approaches your design cruise speed. And you'll have to balance whether you want efficiency in still air, or the ability to keep flying in stronger winds. You'd need good data on expected wind speeds, and not a simple equation.

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  • $\begingroup$ I have a simulation with wind info based on RANS and I have measured the most efficient forward flight speed. I have also tested multiple radiuses of the flight and selected the most efficient. However, I was not able to estimate if there is lower bound for radius in real flight, simulations were fine with any number, I did not take centrifugal force into account tho. $\endgroup$
    – Ruli
    Mar 31, 2022 at 13:12

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