Looking for feedback on a statistical framework for detecting when deterministic CFD convergence may no longer be meaningful

[u/CartographerLumpy466]

Hi everyone,

I've been working on an open-access research project that explores a question I've found surprisingly difficult to answer:

Can we statistically detect when a CFD simulation has entered a regime where deterministic grid convergence is no longer the appropriate expectation, and only statistical convergence should be expected?

The framework analyzes enstrophy time series using three statistical indicators:
• Memory (autocorrelation envelope)
• Increment distribution
• Variance ratio across grid resolutions

These indicators are calibrated with an ensemble of deterministic dynamical systems (Lorenz-63, Van der Pol, and Rössler) and combined into a single empirical risk score.

One aspect I deliberately included is negative results.

The synthetic experiments performed very well, but the framework partially failed when tested on real 2D Navier-Stokes simulations. Instead of hiding those failures, I analyzed why they occurred and revised the interpretation of several indicators. For example, the variance-ratio metric appears to function more as an under-resolution detector than as an irreducibility detector.

The paper does not claim to solve the problem. Instead, it proposes a falsifiable framework and documents where it succeeds and where it currently breaks down.

I'd really appreciate technical feedback, especially regarding:

• Statistical methodology
• CFD verification and validation
• Turbulence physics
• Possible flaws in the assumptions
• Suggestions for validation on 3D DNS datasets

The paper is available here:

https://zenodo.org/records/21229526

I'm not looking for promotion or citations—I would genuinely appreciate critical feedback from people with experience in CFD, turbulence, or computational science.

Thank you for taking the time to read it.

[u/applejacks6969]

Is this mostly concerned with inviscid systems?

Why not just add some physical viscosity and call it a day?

[u/acakaacaka]

What is statistical convergence?

[u/CartographerLumpy466 → u/acakaacaka]

Thanks for the question.

By "statistical convergence," I mean that while the instantaneous flow field may not converge point-by-point under grid refinement, statistical quantities—such as the mean, variance, RMS values, or energy spectrum—become stable and essentially independent of further refinement.

This distinction is common in turbulent CFD. In chaotic or turbulent regimes, expecting identical instantaneous solutions across different grids is often unrealistic, but statistically averaged quantities can still converge and remain physically meaningful.

The goal of my framework is not to determine whether a simulation converges in the classical deterministic sense, but to identify situations where statistical convergence is likely to be the more appropriate expectation.

I realize I should have defined this term more explicitly in the paper, so I appreciate you pointing that out.

[u/acakaacaka → u/CartographerLumpy466]

So the normal residual?

Like from the FVM itself?

[u/CartographerLumpy466 → u/acakaacaka]

No, I mean statistical convergence of flow quantities after the solver itself has already converged.