Most of the time I have recorded flute I have used a LDC even though in that area of mic placement there is not a lot of room. I have used an AKG 451 as well. In both these cases I was always happy with the bottom end of the flute in the mix process. A SDC may make it sound a tiny bit thin.
Depends on what flutes they play too. A lot of the time the player worked with played the standard flute size but also played right up to large wooden flutes with low end. So the LDC came into its own. A ribbon mike would sound good I would imagine.
A LDC does work well because as we know it is good to match the mic to the sound you are recording. If a flute had a higher more present sound that was bright, I would tend to choose a microphone that is warmer and smoother sounding. That way the combination of mic and sound will yield a much better recorded sound needing less processing later.
Some flute players may move around a bit so you might have to be prepared for that and it is one of those things that can go off axis quite quickly. If they are sight reading then they tend to stay put much more. Some flute players have microphones attached to the instrument and it is always good to capture that signal at the same time too.
Watch out for the high notes as well. They can change their sound radically and become super bright. Also level can go up large amounts too. If you are worried, set up a compressor set for limiting with a very high threshold as a guard. But you should be able to set the gain in such a way you can capture their full playing dynamic range without clipping.