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.
"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...
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.
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).
To add on that, for all cases really means a lot more than bends in pipe or air off a wing or smoke which we have difficulty with today. You'll need to hit everything, plasma on the Sun, exotic matter, it leads to quasiparticles for the many-body problem.
My interest in this subject is I hope new developments lead to cardiac therapies. I send some buzzing acoustic signal with a pacemaker, I can treat blockages in the brain or tumors in hard to reach organs. A lot better to me than building faster missiles to blow more people up.
Physicists normally care little whether their mathematical tools are formally proven correct. See: the Feynman path integral. It makes sense and gives correct answers, but as far as I know, it has no rigorous mathematical basis.
Indeed, natural science deals in models, not 'the truth'.
In most cases 'correct' isn't an option, rather a degree of accuracy. If it's consistent with the measurements, or even better if it has predictive power, then it's useful.
I wouldn't say they don't deal in truth. Scientific theories clearly capture a lot of true elements of how the universe works. But proving that the mathematical methods are correct, starting from basic axioms, is something for mathematicians. Most physicists will use a mathematical tool if it's useful, even if it's not been formally proven correct.
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.)
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...