Same framework, applied to Fluid Dynamics
The framework's value lies in its universality across disparate domains. The brake operator \(\mathcal{B}\), dispersion \(\mathcal{S}\), consensus \(\mathcal{M}\), spectral primitive \(\mathcal{P}\), anti-shadow detector \(\mathfrak{A}\), and scope-reporter \(\mathscr{A}\) — together with Theorems 1–13 — are applied here exactly as on every other domain. Source code: github.com/senuamedia/uniformity. No per-domain calibration. No imported threshold. No bespoke fit.
Cross-domain catalogue — \(\beta\)-strip
Every brake-exponent reading across every domain. The same operator \(\mathcal{B}\) produced every point. This domain's points are highlighted; the rest of the catalogue is visible for cross-domain context.
Click any point for the full reading: instance, domain, \(\beta\) value, and a link to the source code.
What the framework provides for fluid dynamics
Navier–Stokes regularity has been studied for over a century with PDE machinery. What that body of work has not produced is a quantitative empirical signature that holds across structurally different formulations of cascade physics — DNS, Burgers, MHD, shell models, spectral closures — in the same dimensionless language. The σ-Uniformity Framework provides exactly that: a single brake exponent \(\beta\) that takes the same value (within tight cross-shadow dispersion) in all cascading-nonlinear formulations, and correctly fails (\(\beta \approx 1\)) in conservative-linear formulations where no brake mechanism exists. The framework's contribution converts NS regularity from a single-PDE conjecture into a universal cascade-physics statement with empirical backing across eight mathematical formulations and Reynolds numbers spanning eight orders of magnitude.
Headline results (catalogue instance 1)
~190 cells, 8 shadow classes, \(\mathrm{Re} = 10^1\) to \(10^{10}\)
- Brake constant \(1 - \beta \approx 0.45 \pm 0.05\) across 8 orders of magnitude in Re (Sabra shell, instance 1)
- NS-Galerkin DNS: \(\beta < 1\) in 28/30 cells; outliers traced to under-resolution and restored at \(N = 14\)
- Burgers brake saturates at 0.15 for \(\mathrm{Re} \ge 10^3\); shock-formation gives a Re-independent brake
- NLS-1D and Maxwell legitimately yield \(\beta \approx 1\). Conservative systems have no brake by construction; framework correctly identifies absence
- LES Smagorinsky yields \(\beta \approx 1.02\) at all Re. Linear-in-\(E\) dissipation cannot brake — diagnostic finding invisible to spectrum-level analyses
- K41 spectrum reproduced within 0.022 of literature; anomalous dissipation \(\bar\varepsilon\) recovered
Experiments
Scripts: domains/fluid-dynamics/experiments/ (16 scripts).
Shadow solvers: applications/ — ns_galerkin.py, burgers_1d.py, mhd.py, edqnm.py, klein_gordon.py, klein_gordon_nonlinear.py, nls_1d.py, maxwell_1d.py, les_dynamic.py, les_nonlinear_sgs.py, les_spectral.py.
Detail — eight shadow classes
| Shadow class | Cells | Re range | \(\beta < 1\) rate | Note |
|---|---|---|---|---|
| 3D NS-Galerkin DNS | 30 | \(10^2\)–\(10^4\) | 28/30 (93%) | 2 outliers at under-resolution |
| Burgers 1D | 21 | \(10^1\)–\(10^4\) | 21/21 (100%) | brake saturates at 0.15 |
| MHD 1D | 24 | \(10^2\)–\(10^4\) | 24/24 (100%) | — |
| Klein–Gordon nonlinear | 9 | 5 \(\nu\) values | 9/9 (100%) | — |
| NLS-1D integrable | 9 | 5 \(\nu\) values | — | \(\beta \approx 1.03\) by construction |
| Maxwell linear | 9 | 5 \(\nu\) values | — | \(\beta \approx 1.02\) by construction |
| Sabra shell | 24 | \(10^3\)–\(10^{10}\) | 24/24 (100%) | brake \(\approx 0.45 \pm 0.05\) |
| EDQNM closure | 21 | \(10^2\)–\(10^8\) | 21/21 (100%) | — |
| LES Smagorinsky | 12 | \(10^3\)–\(10^6\) | 4/12 (33%) | structural defect: linear-in-\(E\) dissipation |
Framework reading
The brake principle \(\beta < 1\) holds robustly across cascading-nonlinear shadows. Conservative-dynamics shadows (NLS-1D, Maxwell) correctly yield \(\beta \approx 1\) by physics. The engineering closure LES-Smagorinsky yields \(\beta \approx 1.02\) at all Re because its linear-in-\(E\) dissipation form structurally cannot brake — invisible to standard turbulence diagnostics, visible to the framework. The remaining work is the analytical proof that the same brake mechanism must hold for the actual NS PDE.