I was just giving myself a refresher course on the 15th AF and noted something I wanted to pass along to you--also in context to what you originally posted.
The patch on the LEFT shoulder of a uniform is the outfit that a soldier/airman serves in PRESENTLY--the outfit that is paying him.
When a soldier/airman changes commands or is transfered or is put on the Enlisted Reserve or for some reason goes to another unit outside the responsibility of the one he HAD been serving with, that patch comes OFF the LEFT SHOULDER and the NEW outfit insignia goes there.
IF the previous unit can be classified as a COMBAT UNIT, then the soldier/airman has the right to put his "former combat unit" patch on his RIGHT SHOULDER.This is done today and has its roots in WW2.
So if the uniform you described has the 2nd Air Force on the LEFT and the 15th on the RIGHT then this was the uniform he wore AFTER he got back to the states. His previous COMBAT UNIT insignia is on the right shoulder and he is PRESENTLY being paid by the 2nd USAAF in some training capacity stateside.Current unit Left Shoulder.
ALSO--My ready reference shows that there were six (6) B-17 equipt Bomb Groups attached to the 15th Air Force.One of them happens to be the 2nd Bomb Group---DO NOT CONFUSE THE 2ND BOMB GROUP WITH THE SECOND AIR FORCE---They are two completely different things!