Adequate boundary conditions for compressible pipe flow
 

Dear community,

I am struggling with my SIMPLE implementation. First of all, I implemented the method on a staggered grid for 2D axisymmetric coordinates. However, my problem also occurs when going to a 1D problem.
I would like to simulate the flow in a pipe (nothing fancy, I know) with a defined set of input parameters like inflow velocity and mass flow into the system. I followed the implementation rules from Versteeg's book and I'm pretty sure that at least the inflow cell and the interior cells and their respective momentum equations are implemented correctly. However, concerning the outflow boundary and the corresponding pressure (or better density) correction, I accumulated some problems - which is why I seek your advice.
First of all, what I actually want to know is the development of velocity and density throughout the pipe/nozzle. I cannot, however, state that the density at the outflow boundary is fixed. In my problem the temperature is constant throughout the domain but the density is assumed to change since I simulate a gaseous flow. I tried to implement the zero velocity gradient boundary condition at the outlet but there seems to be a big misunderstanding or wrong implementation because the resulting velocity field iteratively decays to zero (except the inflow velocity which is fixed). Maybe I was mistaken when substituting the pressure in the momentum and continuity equation by the density (using the ideal gas law p=nkT). Because now, even if I know that the outlet pressure will be 0 Pa (the outlet is in a vacuum), I cannot say that the density at the outlet will be zero because the temperature in the vacuum also has an effect. I hope you understand my problem...
In other words: the flow I want to simulate is known at the inlet, unknown at the outlet and expected to have variable density but constant temperature. Can anyone comment on appropriate boundary conditions for this setup? I would deeply appreciate any help or hints. I will gladly provide more information if needed.

Thanks in advance,
Chris