I hope you can help me in finding solution to the following problem. I'm trying to find a centerline of a pipe (or cavity) with a complex shaped profile, which can be find in the control valves, for instance. If a pipe/cavity has a constant profile throughout, the solution is trivial. But if its boundaries consist of number of arcs and lines, as can be found in the valves, the solution is much more difficult. Well, for me at least. So far I could not find it and did not find and references to the similar problem in the books. I need this centerline to be able to find the correct cross-sections of the valve cavity later on. The picture included.The equations that are used for flow in a pipe are not really valid over the typical cross section in a valve, so finding the transverse areas or centerline won't really do you a lot of good.What you have to do is solve the flow equations for a large number of small volumes of fluid, and integrate them over the volume of the cavity. The number of equations that you need to solve at once is very large. Fortunately, that's just the sort of thing that computers can do well.

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