First - is it a flute, as in a concert flute, made from metal or wood with a mechanical action and a blow hole and lip plate, or a flute that is really a recorder type instrument. The two share a name, and totally different science!
One type has a permanent labium to split the airflow, and is an open ended cylindrical bore instrument, the other type being a cylindrical tube with a closed end, excited by blowing across a hole.
Recording the 'recorder' type instrument works nicely with a condenser slightly above and looking down on the instrument, while the transverse flute has quite a few problems in the recording department. Where you stick the mic, let alone it's type changes depending on the direction. In stream placement gives that windy and breathy sound, while looking down gives a more consistent sound but often a bit thin.
The worst thing is the damn clacking some flutes make. The pads that go down to seal the holes can do it with quite a speed, and they can be noisy. When the keys are released, it's common to find the little soft pads to stop the over-travel, are less than soft and they make a noise too. To make it worse, many players do not let their fingers rest on the keys, ready to depress them, they keep them above, and seem to fire them down. The flute I have here records dreadfully because it's not had a service for a while. In the room, not too bad, but the mics, especially those with decent amounts of top end, pick it up very easily! Oh yes - and some players cannot stand still!
All this makes compression extremely tricky, and I often avoid it unless I want the breathiness exaggerated.