Pressure jump and momentum source
I am trying to understand the relation between the pressure jump that occurs, for example, across a fan and the source of momentum. Consider a very long, constant area, cylindrical pipe open to the atmosphere on both sides. Suppose that a very thin fan is placed in the middle of the pipe. In Fluent I can model this system by setting an isothermal, 2D, axisymmetrical problem, using water, with a boundary face of type "fan" extending from the axis to the wall of the pipe, and specifying a constant pressure jump across this boundary. This will produce a Hagen-Poiseuille-like flow, except that the pressure will drop in the axial direction with a constant gradient from the inlet of the pipe to some negative value at the fan location, then the fan will rise the pressure to some positive value on the downstream side of the fan, and finally the pressure will show the same negative gradient up to the outlet where the pressure will be atmospheric again. The pressure in the radial direction will be pretty uniform. The velocity profile in the radial direction will be parabolic (provided the pressure jump is not too large).
Now, how can I model this very problem, but using a momentum source term in the axial direction, instead of the fan boundary condition?
In orther to set up a "volumetric" source, I created three regions, a very thin region at the center of the pipe (say 0.01% of the total lenght of the pipe) and the upstream and downstream regions. I used a non-uniform grid in the axial direction to get a good cell aspect ratio near the thin region. In the radial direction I also specified a non-uniform grid so as to have the same volume for all the cells in the thin region. In this thin region, I specified a constant source term. The solution converged with second order interpolation in both pressure and momentum to less than 1E-06 in pressure and velocities, but the results were unexpected; with flow getting away from the thin region in any direction!!!!! It should be mentioned that I have tried many values for the source term.
So, the question really is, given a pressure jump across a thin region, how should I set the equivalent momentum source term?
The source term = power of the fan / zone_volume
= Pressure * Volumetric_flow_rate / zone_volume
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