In the file you provided the curved “surface” is actually a polysurface comprised of two trimmed surfaces. The two trimmed surfaces appear to be the result of trimming a single, closed surface. It looks like FlowAlongSrf was mapping to the untrimmed, not the trimmed surface. I replaced the two trimmed surfaces with a single untrimmed surface. Proceedure to do that was:
DupEdge for the trimming curves.
Explode the polysurface.
Delete one of the surfaces.
Untrim the other surface. (A closed surface results.)
Use SrfSeam to adjust the location of the seam in the closed surface to coincide with one of the trimming curves.
Trim the surface using the trimming curves.
ShrinkTrimmedSrfToEdge the trimmed surface.
Then use FlowAlongSrf