Randomness rules in turbulent flows

Jun 01, 2011
Gregory Eyink is a professor of applied mathematics and statistics at the Johns Hopkins University. Credit: The Johns Hopkins University

It seems perfectly natural to expect that two motorists who depart from the same location and follow the same directions will end up at the same destination. But according to a Johns Hopkins University mathematical physicist, this is not true when the "directions" are provided by a turbulent fluid flow, such as you find in a churning river or stream. Verifying earlier theoretical predictions, Gregory Eyink's computer experiments reveal that, in principle, two identical small beads dropped into the same turbulent flow at precisely the same starting location will end up at different – and entirely random – destinations.

"This result is as astonishing and unexpected as if I told you that I fired a gun aimed at precisely the same point on a target but the bullet went in a completely different direction each and every time. It's surprising because, even though the beads are exactly the same and the flow of water is exactly the same, the result is different," said Eyink, professor of applied mathematics and statistics at The Whiting School of Engineering. "It is crucial here that the flow is turbulent — as in whitewater rapids or a roiling volcanic plume — and not smooth, regular flow as in a quiet-running stream."

An article about the phenomenon appears in a recent issue of Physical Review E and is available online.

To conduct his study, Eyink used a virtual "stream" that is part of an online public database of created with Whiting School colleagues Charles Meneveau and Randal Burns, as well as with physicist Alexander Szalay of the Krieger School of Arts and Sciences. Into this "stream" Eyink tossed virtual "" at precisely the same point and let them drift within the . The researcher then randomly "kicked" each of the particles as they moved along, with different "kicks" at different points along the way. The particles, as one would expect when subjected to different "kicks," followed different paths.

"But here's the surprising thing," Eyink explained. "As the kicks got weaker and weaker, the particles still followed random – and different – paths. In the end, the computer experiment seemed to show that the particles would follow different paths even if the kicks vanished completely."

This phenomenon is called "spontaneous stochasticity," which basically means that objects placed in a turbulent flow – even objects that are identical and which are dropped into the same spot – will end up in different places.

"Thus, we know that 'God plays dice' not only with subatomic particles, but also with everyday particles like soot or dust carried by a turbulent fluid," Eyink said.

Eyink's study also revealed that the magnetic lines of force that are carried along in a moving magnetized fluid (like a stream of molten metal) move in a completely random way when the fluid flow is turbulent. This contradicts the fundamental principle of "magnetic flux-freezing" formulated by Nobel Prize-winning astrophysicist Hannes Alfvéen in 1942, which states that magnetic lines of force are carried along in a moving fluid like strands of thread cast into a flow.

"This principle of Alfveen's is fundamental to our understanding of how fluid motions in the Earth's core and in the sun generate those bodies' magnetic fields, and my study may provide a solution to the longstanding puzzle of why flux freezing seems to fail in violent solar flares and in other turbulent plasma flows," Eyink said.

Explore further: First in-situ images of void collapse in explosives

More information: Eyink's home page: www.ams.jhu.edu/~eyink/

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barakn
3 / 5 (2) Jun 01, 2011
"As the kicks got weaker and weaker, the particles still followed random and different paths. In the end, the computer experiment seemed to show that the particles would follow different paths even if the kicks vanished completely."

As the kicks got smaller and smaller the change in velocity they imparted got closer and closer to the limit of precision of the variable type used to store the velocity. This finite data precision and rounding error added noise to the particle trajectories, noise which could have approached the same magnitude as the random kicks themselves. I hope they accounted for this, but suspect they didn't.
CSharpner
not rated yet Jun 01, 2011
I must have misread something. Is it not completely obvious to everyone that particles in a turbulent flow should follow different paths? Is this not, in essence, the plain old butterfly effect? I mean, if the flow is "turbulent", then by definition, not all water paths are fixed at every position in the stream. The turbulence itself is the random change of the flow, which means anything flowing in it is subject to random movement.

Was there something else I missed?
marraco
not rated yet Jun 01, 2011
Wait. He was experimenting with a simulation?. Wat is "virtual" particle?

A simulation only reflects the numerical solution of an equation. Not real fluid.

And since flow is chaotic, the numerical solution necessarily is different that the exact solution.

We not even know if there are exact solutions for Navier Stokes on turbulent flows.
hush1
not rated yet Jun 01, 2011
Great comments.
"Thus, we know that 'God plays dice' not only with subatomic particles, but also with everyday particles like soot or dust carried by a turbulent fluid," Eyink said.
lol
Taking sides, again, are we?

One coin for unification! Go team, go. lol