Knots and Fluid Flow - Finding Patterns within Turbulence

Could Knots Unravel Mysteries of Fluid Flow?

Irvine Lab

A trefoil vortex knot in water generated by a hydrofoil.

By: Natalie Wolchover

December 9, 2013

Spaghetti-thin shoelaces, sturdy hawsers, silk cravats — all are routinely tied in knots. So too, physicists believe, are water, air and the liquid iron churning in Earth’s outer core. Knots twist and turn in the particle pathways of turbulent fluids, as stable in some cases as a sailor’s handiwork. For decades, scientists have suspected the rules governing these knots could offer clues for untangling turbulence — one of the last great unknowns of classical physics — but any order exhibited by the knots was lost in the surrounding chaos.

Now, with deft new tools at their fingertips, physicists are beginning to master the art of tying knots in fluids and other flowable entities, such as electromagnetic fields, enabling controlled study of their behavior. “Now that we have these knots, we can measure the shape of them in 3-D; we can look at the flow field around them,” said William Irvine, a physicist at the University of Chicago. “We can really figure out what the rules of the game are.”

Knots and linked loops exist in turbulent fluids like Earth’s outer core because they arise when a rotation coincides with a flow. (As the fluid rotates, the particle pathways, or “streamlines,” get dragged around and entangled in an effect similar to tying a shoelace.) Investigating knotted fluids both on paper and in the lab could provide a much richer picture of how these tangles, once formed, affect the future evolution of the fluids. The researchers say this new means of probing fluid flow could eventually advance the scientific understanding of the plasma rising off the surface of the sun, thermonuclear fusion, Earth’s interior and atmosphere, and other systems embroiled in turbulence.

“This is all a realization of this dream of understanding fluids in terms of the knots and links of the streamlines,” said Randy Kamien, a professor of physics and astronomy at the University of Pennsylvania.

Illustrations of knots and links, including a trefoil knot, top left, in an 1869 paper by Lord Kelvin on his knotted vortex theory of atoms. Related Video: Knot Possible?

The dream began in the 1860s with an ingenious knot theory of nature. Lord Kelvin proposed that atoms were knotted vortexes swirling in the ether, an invisible, fluidlike medium believed at the time to fill space. One element would be the simplest knot, called a trefoil, another a figure-eight knot, and so on — and none could transform into another. Though incorrect, Kelvin’s idea spawned the branch of mathematics known as knot theory and ultimately led to the realization that knots do more than passively form in fluids; they can have a pivotal, though as yet poorly understood, influence on turbulent fluid dynamics. In seminal work published in 1969, Keith Moffatt, then a young Cambridge University lecturer, proved that the measure of the total knottedness and linkage in ideal fluids — ones, like liquid helium, that lack viscosity — stays constant over time. In viscous fluids, this measure, called “helicity,” fluctuates, and knots can transform or unravel. But scientists still don’t know when and why helicity dissipates.

“There is a vast literature about what happens to knottedness in fluids, but it has been really hard to do experiments for a long time,” Irvine said. “It wasn’t until recently that we got these great tools for making and measuring things in 3-D, which is essential for knots.”

Earlier this year, Irvine’s team used water displacing objects called hydrofoils, created through 3-D printing, to fashion a trefoil knot out of a water vortex — the first vortex knot ever created in the lab. Using lasers, Kamien’s group constructed a knotlike structure in liquid crystals, the self-aligning fluids found in LCD television screens. And a third group — led by Mark Dennis, a theoretical physicist at the University of Bristol in the United Kingdom — tied knots in filaments of darkness swirling inside laser beams.

Alongside the experimental advances, researchers have also formulated new mathematical descriptions of knotted fluids and fields that can be analyzed on paper rather than in the lab.
Electromagnetic fields — entities that fill space and oscillate at different frequencies, some of which our eyes perceive as light — are mathematical solutions to a set of laws known as Maxwell’s equations. As reported in October in Physical Review Letters, Irvine and his colleagues Hridesh Kedia, Iwo Bialynicki-Birula and Daniel Peralta-Salas discovered a large class of solutions in which the contours of the electromagnetic fields, called “field lines,” twist and turn in knots.

A static, knotted electromagnetic field was derived in the 1990s, but “the new work is much more general,” said Moffatt, now a professor emeritus of mathematical physics at Cambridge. “They provide a technique for finding a really huge variety of knots.”

Irvine and coauthors will show in forthcoming work that there are corresponding knotted solutions to Euler’s equations, which govern ideal fluids. Because they have zero viscosity, these fluids flow perfectly smoothly, much like the light fields studied by the researchers. “It illustrates that we can be talking about very different physical systems with the same sorts of solutions,” Dennis noted. This equivalence means that if physicists discover the principles behind knots in Earth’s core, the same rules should apply to the tangled vortexes near an airplane wing.

A Knotty Picture

To visualize a knotted electromagnetic field, imagine three-dimensional space partitioned into doughnut-shaped tori of a continuous range of sizes, nested like Russian dolls. Collections of field lines (shown above in orange and blue) form the surface of each torus. In the trefoil knot field, each line wraps twice around the perimeter of the torus and three times through the center. As a result, any two lines on the same torus are knotted, and any two lines from different tori are knotted too. “It gives you a way of filling space with knots,” Irvine said. Then, as the field propagates through space and time, “these knots travel with the light.” (Illustration: Irvine Lab)

The knotted light fields that Irvine and his colleagues derived on paper may be realizable experimentally, he said, within a tightly focused and polarized laser beam. By shining the knotted beam onto another material, such as plasma, it should also be possible to “transfer the knottedness onto that thing,” he said, enabling controlled study of knots in a range of settings.

At present, almost nothing is experimentally proven about how knots in fluids and fields evolve over time despite decades of speculation and extensive computer simulations.

“Suppose William [Irvine] made two trefoil knots in a fluid and shot them at each other,” Kamien said. “What do they do? How do they interact? That’s completely beyond the scope of what we understand.” The answers to these seemingly simple questions, he added, are central to “how fluids work.”

For starters, when do knots unravel and when do they not? Moffatt proved that helicity stays constant in zero-viscosity fluids — a law of nature analogous to the conservation of energy in frictionless systems. But just as friction saps energy from a car, particle collisions suck helicity out of viscous fluids like water and plasma. “We know helicity is not exactly conserved, but how is it not exactly conserved?” Kamien asked. “Nobody really knows.”

Gary A. Glatzmaier, Los Alamos National Laboratory, U.S. Department of Energy
A supercomputer model of flow patterns in Earth’s liquid iron outer core. The fluid’s complex topology is believed to play a central role in the “dynamo effect,” which generates the planet’s magnetic field.

The most pressing question is what happens when knotted or linked vortices in a viscous fluid cross and separate — a common process called reconnection. Some researchers hypothesize that link or knot helicity is converted into “twist helicity,” or faster swirling of the vortices, keeping the total helicity constant. However, preliminary work by Moffatt and Yoshifumi Kimura, a professor of fluid dynamics at Nagoya University in Japan, suggests that helicity dissipates during reconnection. “It’s an open question,” Moffatt said.
Reconnection is central to many turbulent processes, such as feedback between large and small eddies in Earth’s atmosphere, the heating of the solar corona and the generation of Earth’s magnetic field. In thermonuclear fusion — a solar process in which atoms fuse together, releasing massive amounts of energy — a turbulent plasma constantly undergoes reconnection as it relaxes to its minimum energy state. Understanding whether helicity remains constant during this process will help researchers correctly model and replicate fusion in the laboratory. “That’s why it’s an important issue to try to understand,” Moffatt said. “The long-term hope for mankind is to produce energy from fusion.”

Quantities that are “conserved,” or stay constant in time, “give you powerful ways to look at complicated problems,” Irvine explained. “Understanding a new conserved quantity, helicity, could have a huge impact on how we understand flows. It’s one of those holy grails.”

Once the rules of knottedness are established, some scientists say it might be possible to harness them through clever system design to control turbulence. The findings might suggest, for example, a better shape for airplane wings. “Could you braid the turbulence, and would that make it possible for planes to fly closer together?” Kamien asked. “Turbulence appears to be random. But is there some way to keep it from being random?”