When I'm teaching Leaving Certificate river studies we do an experiment to measure the river's velocity. It's very simple: just time how long a tennis ball takes to float 10m (and repeat a few times to improve the accuracy of the result). Here's what happened when I used a #GoPro to get a 'tennis ball's eye view' of the river bed! It's the River Liffey up in its middle course on Kippure Estate.
Here the river is shallow and it's safe to get in and do the experiment, but you can do the same thing on any stretch of river by measuring out a set distance (10m works well) on the bank and throwing in a floating object to time. Just make sure the object is biodegradable so that you don't pollute the river! A stick or a handfull of dry leaves or grass would work well. As with all experiments the more times you can replicate it the more accurate your results will be, and remember river's don't flow at the same speed all the way across their width. Try taking measurements in the centre of the river and nearer the banks and see how they compare.
Since river's usually flow faster in the centre where the channel is deepest (less friction against the river bed), this has a big effect on the course a rowing boat would ideally take too! Have a look at the video of our colours rowing race and watch how the cox steers close to the bank whenever she can to keep us 'out of the flow' as we race upstream against it.