When reading anything about ancient Greece that mentions "the flute", there are very high odds that it should say "the aulos". Aulos is so frequently mistranslated as flute that you almost have to assume flute means aulos in any English text. The aulos is a double reed instrument played vertically, sometimes in pairs and sometimes not. The flute has no reeds and is played horizontally/transverse and almost no one is crazy enough to try to play two at once.
|Pan Playing Double Aulos Among the White Violets|
Now just to confuse things, there was a transverse flute in use in ancient Greece. It was considered a country instrument, not very sophisticated and linked to shepherds. There are almost no mythological stories that feature this instrument and the only reference to a God playing one (that shouldn't actually read aulos that is) that I have run across is Pan and I'm not sure about that one. It is possible that the original Greek text said panpipes or syrinx instead of flute, another common mistranslation. Although since Pan was a God of shepherds, it is not impossible that in this case, they actually meant the transverse flute.
|Pan Playing Transverse Flute Among the Wild Columbine|