really random but after bleeding mine like 6 times, I was still getting the whine. Turns out my belt tensioner had gone bad at the exact same time, but it sounds just like a bad steering pump. The key difference was that it made the sounds with the changes in RPM, not just when turning. Anyway, what you describe doesn't sound like that but thought I would share anyway.

But I'd take the pump back and get another new one first.

Best of luck