Fun with fluids: Hydraulic jump

We usually think of shock waves as exotic - something that happens when a huge explosion goes off, or when a supersonic plane flies by.  A shock in a gas is a relatively abrupt boundary between relatively cold gas moving faster than the speed of sound in the gas (that is, with a Mach number \(M \equiv v/c_{s} > 1\), where \(c_{s}\) is the sound speed and \(v\) is the speed of the gas) and warmer gas moving slower than sound (\(M < 1\)).  A shock that moves on its own relative to a stationary environment is a shock wave, while one that remains fixed in place relative to its surroundings is a "standing shock".   The details of the gas motion within the shock itself are very complicated, but the constraints of mass and momentum conservation make it possible to understand a lot about the relationship between upstream and downstream gas conditions even without knowing the nitty gritty.

  

It turns out that you have very likely seen a fluid analog of a standing shock in your sink!   Run the tap so that a stream of water hits the flat bottom of a typical kitchen sink.  You will see a disk-shaped region with a radius of a few cm (depending on flow rate) where the water is fast-moving but thin, surrounded by a turbulent ring, outside of which the water layer is thicker but slower-moving.   This is called a hydraulic jump.   The fast-moving water has a speed \(v\) greater than the speed of gravity-driven ripples in a thin fluid layer, \(\sqrt{g h}\), where \(g\) is the gravitational acceleration and \(h\) is the fluid depth.  The Froude number \(Fr \equiv v/\sqrt{gh}\) is greater than one on the fast-moving side of the jump, and less than one on the slow moving side.  Like the gas shock case, the boundary is a mess, but mass and momentum conservation can let you calculate the flow speed and fluid depth downstream if you know the flow speed and fluid depth upstream.   

The receding floodwaters in my neighborhood Tuesday provided me with a great example of a hydraulic jump, shown in the brief video clip above.  I still think it's cool that you can see an analog of a sonic boom in your sink, or in a nearby street if you're unlucky.