Whenever the 15th century polymath Leonardo da Vinci wasn’t painting masterpieces or coming up with new ways to launch humans into the sky, he could often be found outdoors, quietly contemplating the eddies of water flowing downstream.
What amazed the Renaissance master has since puzzled countless scientists. Half a millennium later, we’re still scratching our heads over a thing called a hydraulic jump. Now physicists from the University of Cambridge might finally have it solved.
Hydraulic jumps are such a familiar sight, we could be forgiven for thinking we fully understand how they form.
Turn on a tap, and watch the water flow across the bottom of the sink. As the growing puddle slows at the edges, the water almost seems to pile up into a ‘step’ that remains in place until at last the sink begins to fill.
This standing shockwave can also be found at the bottom of weirs, waterfalls, tidal bores… practically anywhere there’s a meeting of currents flowing at sufficiently different rates.
Their beauty has no doubt captured philosophical minds for far longer than 500 years, but it’s in Leonardo’s notes on the nature of water that we find the first detailed considerations on how liquids behave under different kinds of flow.
To da Vinci, it was purely the nature of water to behave in such a manner. He didn’t have much more of an explanation.
In following centuries, the 18th century Italian physicist Giovanni Battista Guglielmini and 19th century Italian mathematician George Bidone added mathematical detail to the watery step. Still, they didn’t really attempt to argue why it rippled this way.
Finally, in 1914, a physicist with the rather long-winded name of John William ‘Just Call Me Lord Rayleigh’ Strutt, the third Baron Rayleigh, ventured a suggestion in a paper on bores and liquid shock waves.
His theoretical explanation took into account things like viscosity, kinetic energy, and potential energy.
Surface tension, on the other hand, “doubtless plays a considerable part, but this may be minimised by increasing the stream, and correspondingly the depth of the water over the plate, so far as may be convenient”.
Other researchers since Lord Rayleigh have also dismissed surface tension as trivial, favouring models describing the link between the radius of the faster flowing liquid and height of the hydraulic jump as a combination of viscosity, inertia, and gravity.
As water flows along a surface, friction overcomes its inertia and slows the fluid down. If the change in speed is abrupt enough, a shockwave develops, where the liquid piles up over a short distance into a jump.
The size of the step has been assumed to be determined by the tug of potential energy balanced by the push of the mass of water at its foot.
There’s been contention over the years whether gravity really does play an important role in determining the height of the jump, and so the cause of this strange watery cliff that drew da Vinci’s interest all those years ago remains up for debate.
In a new study, chemical engineering researcher Rajesh Bhagat thinks previous scientists may have been a little too quick in ruling out the influence of surface tension.
“We show that, at the jump, surface tension and viscous forces balance the momentum in the liquid film and gravity plays no significant role,” Bhagat and his team write in their report.
Being able to ignore the effect of gravity and concentrate on surface tension allows for other ways to manipulate the hydraulic jump, such as by adding surfactants.
“Understanding this process has big implications and could reduce industrial water use dramatically,” says Bhagat.
“People can use this theory to find new ways to clean everything from cars to factory equipment.”
Whether Lord Rayleigh would be impressed is hard to say. But we’d like to think da Vinci would be happy knowing more about the nature of water and its mesmerising flow.
This research was published in the Journal of Fluid Mechanics.