A Weird Water Behavior That Intrigued Da Vinci Finally Has an Explanation

I'll go out on a limb here and guess that it's the surface friction causing turbulence...

I'll go out on a limb here and say that Leo had too much time on his hands.

I've always wondered why the water formed this pattern.

I'm not sure I completely understand it, but this is what I got from the article.

The water stream has some momentum and after it hits the surface and spreads out, this momentum becomes weaker per unit area. This change in momentum represents a force per unit area.

Water also has a surface tension, force per unit area, due to the attraction between water molecules.

The third force is due to viscosity, a force dependent on the speed of the water.

The viscosity and surface tension oppose the change in momentum. When the sum of the viscosity force and the surface tension force exceeds the change in momentum, the water motion abruptly changes, becoming turbulent and forming the "step".

They came to this conclusion by noting how the distance to the step varied due to added substances that change the water's surface tension (detergent) and viscosity (propanol).

I think it might be interesting to vibrate a round sink at different frequencies to see if you could calm the water surface....

http://archive.nmartmuseum.org/site/art/virtual-gallery/art-on-the-edge-20101/the-artists/eric-tillinghast/two-tanks1.html

What a clean sink! Where are all the dishes????

I may be missing something, but the hydraulic jump is well-known. It's seen eg downstream of a flow measurement flume.

It occurs when flow changes from super- to sub-critical.

You're right, it's the same phenomenon seen in rivers and spillways.

https://en.wikipedia.org/wiki/Hydraulic_jump

I think on the smaller scale, surface tension takes the place of gravity in the same way that ripples governed by surface tension differ from the water waves controlled by gravity.

Yes, I was thinking of cases with a substantial depth of water.

The general case of hydrodynamics is a hard problem, impossible to solve rigorously I believe. I read somewhere about physicists and mathematicians in the 18th and 19th centuries (Euler among others) writing down partial differential equations for fluid flow, but finding they were impossible to solve. They tried putting various things like surface tension or viscosity as zero, to try to get a toe-hold. They got a solution, but it was so far from the properties of real fluids that witty mathematicians called it the "Theory of the flow of dry water".

I don't really see what is mysterious about this.

If you break it down to the smallest part, how does a single drop of water behave? It forms a standing wave that radiates out from the point of contact to the point where counterbalancing forces start to push back.

Given the steady stream of water instead of one drop, you have the same standing wave effect as long as there is water of a sufficient depth for waves to push back in the opposite direction.

In a sink, it's obvious where the return waves are coming from as the original motion is reflected back from the sides. A turbulent pattern is not surprising. The point where the outward motion meets a reverse force is where the standing wave will occur. You can alter the radius of the standing wave by reducing or increasing the flow of the falling water.

OTOH an equation to describe this is, ok, fair enough, a mystery to me.

The Navier-Stokes equations describe how a liquid behaves, but are difficult to solve without making some simplifying assumptions. (Definitely beyond my pay grade.)

https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations

Anything that has applications on the subject of weather modeling is well worth it in my books!! (And now that I think of it, I can see the same patterns in eg tornadoes... whoosh! )

Yeah, that's the sot of thing I was thinking of.

https://www.researchgate.net/figure/Hairpin-vortices-roll-up-and-are-woven-in-around-the-primary-vortex-whose-tips-creep_fig7_230906552

I'd have thought it was glaringly obvious, like how water runs back off the sand on a shallow beach and undercuts the next advancing ripple forming a step in the water level.
Don't need a "scientist" to "work it out"
Del

Spoken like a man who knows his catspaws.

https://en.wikipedia.org/wiki/Capillary_wave

"Big whorls have bigger whorls that feed upon velocity.

Small whorls have smaller whorls and so-on, 'til viscosity."

_{- with apologies to Anonymous Poster #0.}