An Etymological Dictionary of the German Language/Annotated/Flöte
Flöte, feminine, from the equivalent Middle High German floite, vloite, feminine, ‘flute’; corresponds to Dutch fluit, from Old French flaüte, Modern French flûte (whence also English flute, Dutch fluit); compare Italian flauto, ‘flute.’ In the idiom flötengehen, ‘to come to nothing,’ a Low German fleuten, ‘to flow’ (Old Low German fliotan), appears; it meant originally (in the 18th century) ‘to go through, run away.’