I am expecting at least 15 mins a game this season. As I keep saying, I think people will be surprsied a bit because the areas he has as strengths early on this team can really use. Back up SF to guard SFs (this was the most underrated things Turner did, which Brown will really have to give us minutes to do as we don't have other players to do it), a transition scorer, a guy who can finish at the rim (alley oops on the Celtics again maybe woooo), and I am expecting him to be a pretty good rebounder even this year at SF.
As his shooting grows he can earn more. I would give it two seasons as a guess. Hard to say though because so much could change with this team. Remember two seasons ago we were leaning on Jeff Green. Plus, as Pho said above, who knows maybe he ends up developing his shooting and ball handling and he can play SG some. Maybe he can rebound and defend bigger guys and play small ball PF too. It'll be interesting to watch.