How to Surf on a River: Hydraulic Jumps and Integral Control-Volume Analysis

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Water rushing along a river rapids or exiting a dam spillway always has an uneven surface structure -- the water bounces up and down as it moves downstream. Sometimes the water elevates because it is going over a rock, but also we routinely find water jumping up and creating a wave, when there is no obstacle to flow over. This creates the distinctive look of river rapids and the distinctive pattern of a dam spillway.

The subtlety of what is happening is often hard to see -- the mechanism is not very closely related to waves crashing on a beach, and the water that flows over a rock might not look that different from the water that is forming a wave pattern despite flowing over a smooth riverbed. Certainly if you kayak through rapids, you can tell the difference.

Anyway, if you reproduce the flow in a clear tank, you can see the wave pattern much more clearly, and it helps to show that something pretty cool is happening. We can also use an integral control volume analysis to describe many parts of this flow, and unlike the rapids in a river, we can make obstacles appear and disappear, and the transients associated with the changes actually clue you in to the wave nature of the process much better than the steady-state solution does.

Although the steady-state solution doesn't link too clearly to the wave mechanics, the steady-state solution does allow for an integral control-volume analysis that predicts the speed change and the height change in the jump, and it requires a trick that in turn requires a surface integral proof, and everything gets better when we prove that the surface integral of the unit normal around a closed 2D contour is equal to zero.

This video series is called "Professor Kirby's Fluid Mechanics Kitchen" because I got dinged for trademark infringement when I called it "National Committee for Fluid Mechanics Films: In Color!". That's a fluids professor joke. This video was recorded in the Taylor Design Studio at Cornell. If you are wondering why there is a little pink helicopter flying around, it is a Kirby helicopter (like the Nintendo thing) that we had to put in there as an easter egg for the students. There are actually dozens of easter eggs for the student scavenger hunt. Does the internet want to do my "find the crazy stuff in my videos" scavenger hunt? I suspect no -- the internet does not want that. But seriously, just try to find the two-dollar bill. It is hard to spot.

The music in the series is the music I listened to when I was doing my fluid mechanics homework back when I was in school. In this video, the intro and outro are both excerpts from the famous Babu beat juggle of "Blind Alley" that he used to win the ITF Beat Juggle World Championships when I was in grad school. A landmark moment in beat juggle-ry. Kidz these days. None of them can beat juggle.