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An open channel is simply a natural or manmade course through which water flows with a free surface. 

water flowing through an open channel

A flume is a specially shaped, fixed structure used to measure the flow of water.  Flumes can be used in line with or inserted into open channels to measure the flow of water through the channels.

galvanized steel parshall flume measuring flwo through a natural open channel

 

Image:  MDPI