Hi. I went to a few stores today looking for a case for my folk flute (music, sporting goods, army-navy). What I ended up with is a $5 heavy canvas machete sheath. It’s 24 inches long and almost exactally the right size for the folk flute when assembled. The sheath even has a belt loop so you can practice flute Iaido. 
This isn’t a final solution since I probably need a more case like container to put the flute in at home, but it’s better than the towel I had it wrapped up in.
-Brett