Couple of notes:
- I believe there's a flowRateOutletVelocity patch you can use for outlets. I haven't taken a look at the code to see if it's equivalent to using inlet with a negative value, but might be worth a look
- what's the difference between door and door_outlet? If you're expecting inflow through either, then a pressureInletOutletVelocity BC will generally be a better idea than zeroGradient.
- What are your pressure boundary conditions? It's important they be appropriate for your velocity conditions.
- That seems like quite a low value to initialise omega with, what happens if you change it from 1e-5 to 1? For that matter, what did you set k as? FYI, omega should be root(k)/l, where l is your length scale (could just take something like half the width of your duct, it's not too important)