What if engineers could design a better jet with mathematical equations that drastically reduce the need for experimental testing? Or what if weather prediction models could predict details in the movement of heat from the ocean into a hurricane? These things are impossible now, but could be possible in the future with a more complete mathematical understanding of the laws of turbulence.
University of Maryland mathematicians Jacob Bedrossian, Samuel Punshon-Smith and Alex Blumenthal have developed the first rigorous mathematical proof explaining a fundamental law of turbulence. The proof of Batchelor's law will be presented at a meeting of the Society for Industrial and Applied Mathematics on December 12, 2019.
[...] Since its introduction in 1959, physicists have debated the validity and scope of Batchelor's law, which helps explain how chemical concentrations and temperature variations distribute themselves in a fluid. For example, stirring cream into coffee creates a large swirl with small swirls branching off of it and even smaller ones branching off of those. As the cream mixes, the swirls grow smaller and the level of detail changes at each scale. Batchelor's law predicts the detail of those swirls at different scales.
The law plays a role in such things as chemicals mixing in a solution, river water blending with saltwater as it flows into the ocean and warm Gulfstream water combining with cooler water as it flows north. Over the years, many important contributions have been made to help understand this law, including work at UMD by Distinguished University Professors Thomas Antonsen and Edward Ott. However, a complete mathematical proof of Batchelor's law has remained elusive.
"Before the work of Professor Bedrossian and his co-authors, Batchelor's law was a conjecture," said Vladimir Sverak, a professor of mathematics at the University of Minnesota who was not involved in the work. "The conjecture was supported by some data from experiments, and one could speculate as to why such a law should hold. A mathematical proof of the law can be considered as an ideal consistency check. It also gives us a better understanding of what is really going on in the fluid, and this may lead to further progress."
"We weren't sure if this could be done," said Bedrossian, who also has a joint appointment in UMD's Center for Scientific Computation and Mathematical Modeling. "The universal laws of turbulence were thought to be too complex to address mathematically. But we were able to crack the problem by combining expertise from multiple fields."
Journal References:
- Jacob Bedrossian, Alex Blumenthal, Samuel Punshon-Smith. Almost-sure exponential mixing of passive scalars by the stochastic Navier-Stokes equations. submitted to arXiv, 2019 [link]
- Jacob Bedrossian, Alex Blumenthal, Samuel Punshon-Smith. Almost-sure enhanced dissipation and uniform-in-diffusivity exponential mixing for advection-diffusion by stochastic Navier-Stokes. submitted to arXiv, 2019 [link]
(Score: 3, Informative) by c0lo on Friday December 13 2019, @02:48AM (3 children)
Maybe you are luckier.
Lagrangian chaos and scalar advection in stochastic fluid mechanics [arxiv.org]
The Batchelor spectrum of passive scalar turbulence in stochastic fluid mechanics [arxiv.org]
Tried to look into all 4 of them.
Got some absolutely fascinating blank stares on my face, caused by brain-meltdown-like symptoms.
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 0) by Anonymous Coward on Friday December 13 2019, @05:37AM
I managed to read the summary of your first link okay. I even understood most of the words up until I hit "white-in-time". After that things got iffy. Thanks for the links, by the way.
(Score: 5, Informative) by Anonymous Coward on Friday December 13 2019, @08:07AM (1 child)
not sure what your background is.
In brief, a practical example: say you're in a big dance hall, and someone wearing strong perfume walks in.
people are dancing to some rave/hard rock/running around and bumping into each other etc.
you are interested in the statistical properties of the perfume concentration.
the air in the dance hall is turbulent (drafts of air move around chaotically).
it is in fact forced turbulence, because the dancers keep moving around.
the force is basically random (dance is disorganized), i.e. "noise". "white noise" is a particular mathematical term that pins down some statistical properties of the force. In particular it makes it easier to prove various theorems if you use assume white noise rather than something else.
It's this sort of setting where Batchelor made a prediction of the statistics of perfume ("passive scalar" in turbulence-speak).
This prediction has been somewhat verified experimentally and numerically (but it's technically hard to do it in all relevant parameter ranges).
The UMD team has rigurously proved that the prediction is valid in a specific case of white noise forcing, and this is a very big deal.
(Score: 0) by Anonymous Coward on Friday December 13 2019, @12:46PM
Thank you for the plain English explanation. I had assumed they meant something in that general direction just due to the subject matter but the terminology was beyond me.
(Score: 0) by Anonymous Coward on Saturday December 14 2019, @07:48PM (1 child)
All this math and yet nobody can explain who farted in the elevator.
(Score: 0) by Anonymous Coward on Saturday December 14 2019, @11:47PM
Ah, but they CAN explain that now and even generate pretty false-color graphics...it just takes awhile to run the CFD code and by then you will have left the elevator.