Seems to me tnat an end-flute, like the chalil Pete Seeger used to play, would be a lot easier to make with a carrot. Just bore the veg out as a straight pipe, and blow across the end as you would a soda bottle. It is trickier, really. You hold the pipe transversely, as with a flute, but not below your lower lip. Tuck it into one corner of your mouth, so that the airstream you are blowing strikes the inside edge of the opposite side of the pipe (the side away from you). As Barbara says, the edge needs to be sharp. Start with a tough carrot ? If you can get a tone with a straight pipe, then try some finger holes.

You can try the technique on a recorder, if you have one. A standard pennywhistle is too skinny (for me, I'm sure someone can do it). A fatter, low whistle would work. A low D is probably too fat. Once you yank the mouthpiece off your recorder or whistle, of course, it won't be in tune. You've lost some length. But, hey! You've created a new scale!

Knew a guy who played tunes on his tent poles this way at festivals.