Well, you would quibble over a name? What do you expect the armed forces of the United States to be wielding? Muskets? The U.S. Air Force is an outgrowth of the Army Air Corps. We'll just roll back its name and make you happy.
Clause 12: To raise and support Armies, but no Appropriation of Money to that Use shall be for a longer Term than two Years;
Clause 18: To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof.
You have a stronger case against the Department of Education.
Does the Air Force comply with the restriction on multi-year appropriations? I would think that many of the fancier air-weapons projects would fall afoul of that.
Not to say I'd be opposed to an amendment authorizing such an exception if need be.