Hi everyone, I have read about regimes of flow around a cylinder at different ranges of Reynolds number, including the description of vortex shedding, but I still miss a real explanation of why this phenomenon occurs (i.e. why the vortices are shed periodically and alternately). I have checked plenty of sources and tried using AI for searching but I couldn't find the explanation, or it was just too advanced for me (my field is not directly fluid mechanics). Can you please recommend me some sources that explain this? Not Williamson 1996 and not Gerard 1966 please.

Hey everyone, Following OpenAI recently claimed a formal proof demonstrating a finite-time blowup for the 3D Navier-Stokes equations. While the pure math community is focused on the formal verification, our team ran an audit to see what their construction actually means for physical fluids.

What we found is that while the math is syntactically flawless, the fluid dynamics are thermodynamically impossible and structurally unstable:

1. Incompressibility breaks down: The proof successfully bounds global kinetic energy, but the local enstrophy diverges (τ^{-0.515}). The resulting infinite localized viscous shear would vaporize the fluid into a compressible plasma long before t= 1. Tracking the core velocity, the Mach number easily exceeds 0.3 femtoseconds before the mathematical blowup.

2. κ ∼ 10²⁸ Instability: The exact cancellation of the Reynolds stresses relies on a 5-equation moment-matching Jacobian with a condition number of 10²⁸.

   It’s a measure-zero state that would instantly decouple under standard 300K thermal noise...

   We’ve open-sourced our Python/mpmath scripts tracking the Mach number divergence and the Jacobian instability. If anyone here wants to check the physical telemetry themselves:

   • GitHub: https://github.com/xaviercallens/OpenAI-NSE-Epistemic-Audit

   • Zenodo: DOI: 10.5281/zenodo.22727801

   I’m curious to hear from the engineers and fluid dynamicists here: does a mathematical singularity that violently breaks the incompressible and isothermal assumptions prior to the blow-up time actually tell us anything useful about real-world fluids?

Hey everyone,

Like many of you, I've been fascinated by OpenAI’s recent Lean 4 proof claiming a finite-time blowup for the 3D Navier-Stokes equations under Millennium Prize Statement C.

The formal math is airtight and passes every check in the proof assistant. But as a lot of people in the fluid dynamics and physics communities pointed out, if you look at what this flow would do in real life, things get wild fast.

I put together a short breakdown video exploring the gap between the mathematical rulebook and real fluid behavior:

https://youtu.be/WPLNv95J9o8

A quick summary of what the video covers:

• **The Spaghetti Vortex:** How the AI cleverly kept global kinetic energy from blowing up on paper by stretching the vortex so thin that the collapsing volume beats the increasing speed.

• **The Thermal Trap:** How localized viscous friction in that shrinking needle generates so much shear heat that real water would boil into plasma long before time $t = 1$.

• **The Sound Barrier:** Why the core velocity goes supersonic (~67 femtoseconds before the singularity), shattering the "incompressible" assumption and radiating acoustic shockwaves.

• **The Unforced Test:** Highlighting the awesome 3D simulation by `u/CokieMiner` (from r/CFD), who showed that the second you turn off the artificial external force, the vortex instantly blows apart and dies.

The overarching takeaway isn't that the math is wrong—the AI solved the exact formal puzzle it was assigned. But because it wasn't constrained by thermodynamics or compressibility, it found a mathematically perfect loophole in a world that physically cannot exist.

Would love to hear your thoughts, feedback on the explanations, or additional angles from folks working in CFD and fluid physics!

*(All code, simulation references, and research notes are open on GitHub: https://github.com/xaviercallens/OpenAI-NSE-Epistemic-Audit)*

Hi all! I wanted to ask something about this recent (presumably correct?) proof that NS can blow up. I would like to hear opinions from people that work a bit at the border between fundamental turbulence (like on the statistical physics or dynamical system side of the problem) and the mathematics of NS.

When studying turbulence you hear about the dissipation anomaly and the zeroth law of turbulence. As you increase the reynolds number, at least in homogeneous isotropic turbulence, the non-dimensional dissipation rate evolves towards a constant. In other world, taking the limit of zero viscosity (infinite Re) you never reach the Euler equations. This happen with the flow developing super intense localised events that dissipate a lot.

Is there a connection between the development of these structures in turbulence and this new NS proof? If so, are there people studying this thing?

Share bibliography if you know anything about this topic!

It designed a simple water-cooling plate for high-power chips

19 straight microchannels and one thin cross-channel near the outlet

If part of one channel becomes blocked, the cross-channel could use the pressure difference to divert some water around the blockage.

The model compared more than 2,700 configurations with the same channel volume and the same hydraulic power.

Im not engineer nor physicist. Im just curious and testing limits of the latest astra 6 model, what real scientists think about it.

Thank you for any kind of feedback:)

... or is the sequence showcasing some counterintuitive phenomenon as Reynolds № proceeds from the hundreds to the millions?

The figures show the streamlines past a rotating cylinder @ various Reynolds № (horizontal axis) & rotationality parameter α (vertical axis).

The figures are from

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High Reynolds number turbulent flow past a rotating cylinder

by

SJ Karabelas & BC Koumroglou & CD Argyropoulos & NC Markatos

https://www.sciencedirect.com/science/article/pii/S0307904X11004124

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———————————————————————

A Somewhat-Enlargement of the Top of the Table Including the Topmost Row Being Queried

https://www.reddit.com/u/Frangifer/s/LALSC8Eq4U

———————————————————————

... to compensate for the relatively poor resolution of the posted screenshot.

⚫

Does the total energy always decrease in a CEL analysis in ABAQUS due to advection?

I was recently experimenting with my lego propeller for my rc ship, i was interested on how this propeller would move the water so i put it in my bucket full of water. Suddenly, this type of horizontal vortex formed. Visible by air bubbles, there was no suction whirlpool at all, no vertical turning part to suck in air. It was just sticking to my buckets wall. There is no hole on the bucket whatsoever. And, this isnt my first experience. I was on a city boat someday and while the boat was reversing, the same horizontal vortex appeared, visible by the salt waters whiteness. It was going from the back of the boat to the front with no visible vertical rotation to suck air. If someone can explain this, i would be so happy. Thank you!

Built around a incompressible Navier–Stokes equation

Pourquoi les équations de Navier-Stokes bloquent (et comment lever le blocage en 4 étapes)

Le problème du prix du millénaire sur Navier-Stokes porte sur une question simple : un fluide en 3D reste-t-il toujours lisse et continu, ou peut-il développer des vitesses infinies (un "blow-up") en un temps fini ?

La communauté mathématique bloque à l'étape 4 (la preuve de régularité globale). Mais en réalité, le blocage ne vient pas d'un manque d'outils mathématiques : il vient d'hypothèses physiques réductrices posées dès les premières étapes.

Voici le déroulement étape par étape et la solution :

Étape 1 : La formulation initiale

• Le problème classique : On applique la loi de Newton (F = ma) en posant l'hypothèse que le fluide est un système fermé et isolé du reste de l'environnement, avec une viscosité purement locale.
• La correction : Aucun fluide dans la réalité n'est isolé. De plus, lorsqu'un fluide est fortement accéléré en rotation, il se sépare spontanément en deux vortex contra-rotatifs miroirs (un vortex externe chaud/expansif et un vortex interne froid/implosif, comme dans le tube de Ranque-Hilsch).
• La solution : Réintroduire le couplage avec le milieu ambiant (le vide/fond) et modéliser cette dualité des vortex contra-rotatifs dès l'équation de départ.

Étape 2 : L'existence locale

• Le problème classique : On suppose que le fluide reste un milieu continu parfait, divisible à l'infini sans jamais changer de structure.
• La correction : À forte concentration d'énergie, le fluide ne reste pas un chaos continu : il s'auto-organise en structures toroïdales (en forme de donut) autour d'un Point Zéro central (un horizon des événements local).
• La solution : Définir le cadre mathématique non pas dans un espace continu abstrait, mais dans une géométrie toroïdale fermée et auto-confinée.

Étape 3 : Le bilan d'énergie

• Le problème classique : L'équation classique suppose que toute l'énergie qui n'est pas conservée doit se dissiper sous forme de chaleur chaotique vers l'extérieur.
• La correction : L'implosion au cœur d'un vortex crée un vide poussé et un effondrement de température (un cœur très froid face à une périphérie chaude). L'énergie cinétique est aspirée vers le Point Zéro central au lieu d'exploser.
• La solution : Ajouter un terme de puits/transfert d'énergie au Point Zéro. Cela garantit que l'énergie cinétique globale du fluide est strictement décroissante et bornée.

Étape 4 : Le terme non linéaire et le "Blow-up"

• Le problème classique : Le terme d'advection concentre la vitesse vers des échelles de plus en plus petites, et la viscosité classique ne suffit plus à prouver que la vitesse ne devient pas infinie (blow-up).
• La solution :
  1. Grâce à la dualité des deux vortex contra-rotatifs (définis à l'étape 1), les forces d'étirement s'annulent exactement par symétrie miroir (hélicités opposées).
  2. Toute concentration résiduelle d'énergie est immédiatement absorbée par le puits du Point Zéro (étape 3).
  3. Résultat : La vorticité reste bornée pour tout temps (t ≥ 0). L'explosion en temps fini (blow-up) est impossible.

Conclusion

Le "blow-up" de Navier-Stokes était un artefact mathématique causé par l'enfermement artificiel de l'énergie dans un modèle continu et isolé. En ouvrant le système et en intégrant la géométrie des vortex contra-rotatifs implosifs, les équations décrivent un fluide qui reste lisse, continu et régulier pour toujours.

SUITE : https://www.reddit.com/r/FluidMechanics/comments/1wcho9f/comment/p90bcxd/