With wind, what will a perfectly circular flight path relative to the surrounding air look like from the ground?
It will be this :

http://www.aeroexperiments.org/images/grndtrck.gif

Pilot's of slow-flying aircraft like hang gliders and paragliders see tracks like these all the time on their GPS displays, as they circle in a thermal updraft in the presence of strong wind.

Depending on the strength of the wind, the "curliques" may become so stretched out that they don't form enclosed loops at all, just "points", i.e. a "scalloped" line.  This happens when the windspeed is greater than the airspeed. That means that the aircraft never makes any progress in the upwind direction at any point in the circle. 

Related: 
http://www.aeroexperiments.org/nocss.shtml
http://www.aeroexperiments.org/introcircles.shtml
http://www.aeroexperiments.org/circles.shtml