
First mathematical proof for a key law of turbulence in fluid mechanics - vo2maxer
https://cmns.umd.edu/news-events/features/4520
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semi-extrinsic
I can't for the life of me understand why they don't explain what Batchelor's
result _actually is_ in these press releases.

So here goes: in turbulence, scaling laws are a big thing. Specifically, it's
extremely useful to know how the power density in a turbulent flow is
distributed between small and large whirls.

For the distribution of energy, we have the well-known Kolmogorov scaling law,
that the power density scales with wavenumber to the power of -5/3\. This
holds approximately for all except the very largest whorls, down to the scale
where viscosity begins to dominate, where there are no smaller whorls. This is
known as the Kolmogorov scale. Due to a phenomenon called intermittency, we
have strong reason to believe it is not exactly -5/3.

These papers are concerned with the power spectrum of scalar mixing. Imagine
dropping ink into a turbulent water flow, how does it distribute between small
and large whorls?

Batchelor [1] arrived at the result in 1958 that the power spectrum scales
with wavenumber to the power of -1. This result has always been thought to be
much more robust than the Kolmogorov scaling, since different approaches have
given exactly the same answer.

The papers linked here give a proof that this relationship is exact. From a
quick skim, they are extremely technical in the mathematical sense, invoking
measure theory and Sobolev spaces and Itô calculus.

Certainly these results are very interesting. In some sense the big question
is whether their approaches can transfer to solving outstanding questions
about other phenomena in turbulence.

[1] [https://www.cambridge.org/core/journals/journal-of-fluid-
mec...](https://www.cambridge.org/core/journals/journal-of-fluid-
mechanics/article/smallscale-variation-of-convected-quantities-like-
temperature-in-turbulent-fluid-part-1-general-discussion-and-the-case-of-
small-conductivity/A8AEA175A906F98CDDB0F9ED146BB9FE)

~~~
eindiran
I think this is a good lay-person's introduction to basis of Kolmogorov's
length scale, specifically the Reynold's number:
[https://www.youtube.com/watch?v=wtIhVwPruwY](https://www.youtube.com/watch?v=wtIhVwPruwY)

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fluffything
The actual proof:
[https://arxiv.org/abs/1911.11014](https://arxiv.org/abs/1911.11014)

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peter_d_sherman
Excerpts:

"To understand fluid flow, scientists must first understand turbulence."

[...]

"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."

My Comments: Classical physicists look at matter as being composed of atoms
(neutrons, protons, electrons), String Theorists look at matter as being
composed of vibrating strings, Quantum Physicists look at matter as being
composed of quanta, that is, the possibility of a measurement in a given place
and time. I personally have considered looking at matter as the presence or
absence of force in a given space at a given scale -- but perhaps there's
another possible view here -- looking at matter as solidified fluids of
various different scales where as you go to smaller scales, the fluids move
faster / are less viscuous... If that view turns out to have any truth to it
(not saying that it does; it's all theoretical), then understanding fluid
mechanics, and more specifically, turbulence, might turn out to have an
additional application in helping to understand those small spaces /
interactions in matter...

Related: "An Introduction To Fluid Dynamics by G.K. Batchelor, University Of
Cambridge": [https://www.cambridge.org/core/books/an-introduction-to-
flui...](https://www.cambridge.org/core/books/an-introduction-to-fluid-
dynamics/18AA1576B9C579CE25621E80F9266993)

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zwieback
That's awesome and clicking through the links I learned that the proof for
Navier-Stokes has a million dollar prize on it. I didn't even know that it's
not proven, back in engineering school we basically took that as a given and
starting point for so much useful stuff. I guess that's the difference between
science and engineering.

~~~
tnecniv
Re: Navier-Stokes

The PDE that most analyses start with does not need to be proven. This has
been rigorously derived since forever. The thing that actually needs to be
proven is whether or not smooth solutions exist to the PDE for all cases of
parameters. In practice, people tend to stick to working with simple cases
that can be solved analytically, or use computer simulations to compute
approximate solutions (to very high precision).

~~~
fluffything
What does it even mean "to prove a PDE"? The million dollars is for proving
certain properties of a PDE, in this case, whether smooth solutions exist.

~~~
tnecniv
I was just using the language of the parent. By proven I meant derived from
Newton's laws, and that was my interpretation of the parent's comment.

~~~
knolan
The NS momentum equation is just Newton’s second law written for all the
forces acting on a fluid. We just don’t know if it always works.

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Myrmornis
These university PR department pieces are very often fluff/misleading; is it
feasibly to have HN policy/custom to link to more independent/objective
discussion wherever possible? (That's not to say that this particular HN link
should not exist or that the research described in this one is not important.)

