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Mathematicians Prove Universal Law of Turbulence

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the University of Wisconsin
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Vladimir Sverak
George Batchelor
Jacob Bedrossian
Alex Blumenthal
Samuel Punshon-Smith
Jean-Luc Thiffeault
Qizheng Yan
David Saintillan



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Subsequent drops of black paint will be in different stages of the same transformation: stretching, elongating, getting incorporated into the graying body of the paint.Get Quanta Magazine delivered to your inboxThe way black paint mixes into white in this simulation demonstrates “passive scalar turbulence.” Batchelor’s law describes how such turbulent systems behave.Qizheng Yan and David Saintillan (UCSD)In the same way that the velocity varies from point to point in the churning sink, the concentration of black paint will vary from point to point within the mixing paint: more concentrated in some places (the thicker sinews) and less in others.This variation is an example of “passive scalar turbulence.” You can think of it as what happens when you mix one fluid, considered the “passive scalar,” into another — milk into coffee, say, or black paint into white.Passive scalar turbulence also characterizes many phenomena in the natural world, like the dramatic temperature variations between nearby points in the ocean. In that environment, the ocean currents “mix” temperatures the way stirring mixes black paint into white.Batchelor’s law is a prediction about the ratio of large-scale phenomena (thick tendrils of paint, or thick bands of ocean water at the same temperature) to phenomena at smaller scales (thinner tendrils) when a fluid is mixed into another. You’ll see thick tendrils of black paint (paint that’s been stirred for the least amount of time), along with thinner tendrils (paint that’s been stirred longer) and even thinner tendrils (paint that’s been stirred even longer).Batchelor’s law predicts that the number of thick tendrils, thinner tendrils and thinnest tendrils conforms to an exact ratio — similar to the way the nested figurines that comprise a Russian doll follow an exact ratio (in that case, one figurine per length scale).“In a given patch of fluid, I’ll see stripes of different scales because some droplets have barely begun to mix, while others have been mixing for a while,” Blumenthal said. The complicated nesting of phenomena at different length scales makes it impossible to exactly describe the emergence of Batchelor’s law in a single fluid flow.But the authors of the new work figured out how to get around this difficulty and prove the law anyway.Bedrossian, Blumenthal and Punshon-Smith adopted an approach that considers the average behavior of fluids across all turbulent systems. It allows mathematicians to adopt a high-level statistical view and to examine what happens in these types of systems in general, without getting bogged down in the specifics of every detail.“A little randomness allows you to smear out the difficulties,” Punshon-Smith said.And that’s what finally allowed the three mathematicians to prove Batchelor’s law.One way to prove a physical law is to think about the circumstances that would void the law. They proved that even if you tried to concoct a fluid perfectly engineered to defeat Batchelor’s law, the pattern would still emerge.“The main thing to understand is that the fluid can’t conspire against you,” Bedrossian said.For example, Batchelor’s law would fail if the mixing process produced permanent vortices, or whirlpools, in the paint. This further established that the turbulent fluid doesn’t form the kinds of local imperfections (vortices) that would prevent the elegant global picture described by Batchelor’s law from being true.In these first three papers, the authors did the hard mathematics required to prove that the paint mixes in a thorough, chaotic fashion.

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