Find the diameter of your turret. This should be in the floor plans or perhaps even the plat diagram of your home. However, if it is not, a crude method would be running a tape measure from window to window, as long as the windows are directly opposite one another.
Order radius guttering for the diameter you need. These are custom-fabricated from metal, which is a process generally beyond the do-it-yourself homeowner.
Mark the high and low points of the guttering run. Your guttering should drop about 1/2 inch for every foot of distance. Radius guttering may or may not make a complete circle, depending on your preferences and the manufacturer, so find out the distance your gutter will travel around the circle and mark accordingly.
Attach brackets on the fascia board at intervals around the turret so that your gutter will drop consistently and meet the starting and ending points. Connect the guttering to the brackets.
Run the downspout from the hole in one end of the guttering. This end should be the lower end.