If I'm not moving, no it does not.Does it make the noise if you turn the wheel but aren't moving? Does it do it in reverse? I'm running out of questions that make any sense.
In reverse, I don't think so but I haven't tried reverse at speed. I know at low speed (no acceleration) moving forward it doesn't seem to make the noise. It also doesn't make the noise when accelerating unless I'm turning.
I'm also running out of ideas. I know it's not power steering pump. wrong noise, and not related to engine RPM. It -may- be wheel bearings, but I've had one in the rear go bad on this car and it was a) different noise, b) continuous and c) got louder with wheel speed. Wheel bearing also doesn't explain that it seems to stop even if I'm still turning. At some point I'll get under there and take a look and/or take it in.