SaltaraHydraulic Jump / Froude NumberSticker

Saltara is a long-jumper who lives in a kitchen sink.

Turn on the tap. The water spreads from the stream in a thin, fast sheet, then at a sharp ring it suddenly thickens into a slow, deep layer. That ring is a circular hydraulic jump, and Saltara’s leap is the moment fast and shallow becomes slow and deep.

The cleanest version is in a straight channel, where flow speed is compared with the speed of shallow-water waves, the square root of g times depth. Their ratio is the Froude number, the first line on this page. Above one, the flow is too fast for any ripple to travel upstream and warn the water ahead, the shallow-water analogue of supersonic. Below one, it is slow and deep. Where the two meet, a planar jump forms, and conserving momentum across it fixes how high she lands, the second line: run in at a Froude number of 3 and the depth rises about 3.77 times. The energy lost goes into the churning roller. Jean-Baptiste Bélanger derived this momentum form in 1841; his 1828 essay had given the backwater equation, as Hubert Chanson showed in 2009.

The sink is harder, and the question is open. The classical view is that gravity and viscosity set the radius of the ring (Watson 1964; Bohr, Dimon and Putkaradze 1993), with surface tension as a refinement (Bush and Aristoff 2003). In 2018 Rajesh Bhagat and colleagues at Cambridge argued that at sink scale surface tension and viscosity dominate and gravity barely matters, after firing jets down, up and sideways onto flat plates and finding the same jump radius each time. Duchesne, Andersen and Bohr disputed this in 2019, as did Bohr and Scheichl in 2021. It is not settled.

Fast and shallow. Then slow and deep.