When you run the tap in your washbasin, the stream of water hits the basin and spreads out in a very thin film of water moving at high speed. After a short distance, there is a line where the flow changes depth and travels more slowly. That wave front where the water changes from fast and shallow to slow and deep is an hydraulic jump. Although the jump in your washbasin is only a couple of millimetres high, these waves can reach three or four metres on a river.
They are fascinating and wonderful things: not only for surfers, who want to remain standing on their boards for a long, long time, but also for water engineers who need to control the flow and energy of their rivers and canals. One of the most spectacular natural examples is the bore on a river such as the Severn in the UK.
Canoeists are warned against them because of the extreme turbulence behind the wavefront, which does very unexpected things to people and boats. Canoeists drown in hydraulic jumps daily.
Next time you look at a river near a dam or a weir, look for the tell-tale signs of rapid, shallow flow. The first is the surface of the water is smooth, because the water is flowing faster than a surface wave can propagate. The second is that somewhere downstream is a standing wave where the flow changes from fast and smooth to deep and turbulent.
This phenomenon is governed by the Froude Number, which balances the inertial forces in a fluid against the gravitational forces. At any given Froude number, there are only two possible depths of water. One represents the depth ahead of the hydraulic jump, the other is the depth immediately behind the hydraulic jump. It is difficult to calculate the two permissible depths, except in a a smooth, regular, rectangular channel, but the keys are the Froude number, the cross-sectional area and the specific energy within the flowing fluid. Basically, the faster the fluid flows, the bigger the hydraulic jump.
If V2 > gD, then the flow is fast and shallow (rapid flow). This is called super-critical flow, and is very aggressive and abrasive to a river bed
If V2 < gD, then the flow is deep and probably turbulent. Sub-critical flow is much more benign
V is the speed of flow
g is the acceleration due to gravity
D is the depth of water
The case of a circular hydraulic jump, such as you might find in your washbasin, or when a jet of water hits a flat horizontal plate, is one of those problems which looks simple, but is very difficult to analyse properly
Looking at a large-scale application, such as a dam, the water flows a bit like the following diagram:
#### | \\p
#### | \\i
#### | \\d
#### | \\ sub-critical flow->
#### | \\flow_______JWWWWWWWWW Flow direction ---------->
#### | ===============================
our "hydraulic jump" is represented by the letter "J". Engineers will shape the base of the dam (between the 'f' of flow and the J) so that the falling water is guided into a horizontal trajectory, ensuring the jump forms at the desired place.
Note 1 Thanks to Gorgonzola for, well, for being a hero
Note 2 Thanks to Frankie for advice on canoing
Note 3 This piece written, formatted and edited in Dann's E2 offline scratchpad