Here's a concept I've had on my bucket list for a long time, and I still hope to try it sometime. Suppose my water pipe contains a plastic section, at each end of which is a conductive collar. I put a voltage across this, and electrons flow. I can measure the current, and of course, it's not hard to measure very small currents. If the water is moving, won't I get a change in current?
Now current flow will be affected by temperature and water chemistry, so I also have an identical sensor but with stationary water to measure the current under no-flow conditions. Of course, there's a need here to analyze the physics well enough to use the differential current between the static and flow pipes to actually quantify flow. I'll get around to this sometime.