Enthusiasm is updated after every game, and at the daily update each day.
So 13 enthusiasm might drop down to 11 enthusiasm after 1 day. But 7 enthusiasm might take 3 days to drop to 6. And then another week to drop to 5.
Enthusiasm will be updated right after a game depending on the effort put in to the game.
Enthusiasm will be updated at the daily update to implement the daily increase or decrease depending on what your enthusiasm is.
If you are in the cup, I would guess that 5 games of TIE in a row will get you to about 12 enthusiasm.
If you are not in the cup I do not think you can get to enthusiasm 12 on game day, but you could get to enthusiasm 10. This would take about 6 TIE games in a row I think.