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There is no practical, accurate way to measure the flow rate of water in a river that doesn't involve multiple samplings across the width and depth of it, with the width and depth being variable conditions that also must be considered at different stages (i.e. flooding) The flow of a river is not laminar enough for accurate sampling in any one location.
a paddle/counter will not suffice on its own. River flow is typically measured with v-shaped wiers.
But if you wanted an electronic method, you might consider hollow tubes placed equidistantly across the river and fixed horizontal to the horizon and WITH the flow of the water so that the mouth of the tube does not restrict the flow. A slight connical shape to the tube would reduce eddies (think aerodynamics). An orriface of known dimensions would be located in the center of the tube, fixed with a flange, to cause a restriction and backpressure on the upstream side. Two ports located on opposing ends of the tube would be fixed with sealed (waterproof) pressure transducers. The differential pressure caused by the restriction would be a SQUARE ROOT FUNCTION of the flow rate, precluding of course that the orriface plate selected was of the appropriate dimensions.
With knowledge of the SURFACE AREA of the diameter of the tube, the pressure and thus flow of the water passing through the orriface, taken as an average across the river, and knowing the width and depth of that sampling area, a reasonable accurate flow calculation would be made.
due to non linear and turbulant flow its not easy to measure it, but if you can make a spill way if its a small river, then when you bring it in steady flow in and out of the spill arrangement, it can be aproximated to be liner and you need to get velocity profile and cross sectional area of the flow to determine the exact rate....