Looked up the dictionary, and it seems there's no entry for the phrasal verb "bite down", so it's not a common usage; however, there are a few entries for "bite down a towel" on Google. I also used a colloquation dictionary and it doesn't appear there either.
He bit down a towel in anger.
The question is what "bite down" means exactly? I thought it meant "bite hard into", but I am not sure.