Date: May 16, 2025
This repository presents a novel regularity criterion for the 3D incompressible Navier–Stokes equations, based on the alignment between pressure gradients and vorticity. The key quantity introduced is the Stanley Alignment Functional, (\mathcal{A}(t)), defined as:
[ \mathcal{A}(t) = \int_{\mathbb{R}^3} \left| \frac{\nabla p(x,t) \cdot \omega(x,t)}{|\nabla p(x,t)| + \varepsilon} \right| dx ]
This functional measures geometric coherence in turbulent flow and offers a new route to proving global regularity of solutions.
Stanley_Criterion.pdf: The full manuscript describing the theory, proof, and results.simulations/: Python and OpenFOAM scripts for simulating fluid flows and tracking (\mathcal{A}(t)).plots/: Sample visualizations of (\mathcal{A}(t)) in key flow scenarios.README.md: This file.LICENSE: Open-access license (MIT or Creative Commons recommended).
- If (\mathcal{A}(t)) is integrable on ([0, T]), the velocity field remains regular on that interval.
- For smooth, finite-energy initial data, (\mathcal{A}(t) \in L^1([0,\infty))), implying global regularity.
- Criterion extended to Leray–Hopf weak solutions.
- Numerical simulations show (\mathcal{A}(t)) remains bounded across:
- Vortex collapse
- Channel flow
- Jet nozzles
- Stratified rotating flows
Please cite the paper as:Nikee Stanley, The Stanley Criterion for Navier–Stokes Regularity
These new tests further validate the boundedness and integrability of the alignment functional (\mathcal{A}(t)):
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channel_obstruction_test.py
Simulates flow through a turbulent channel with a geometric obstruction.
Output:channel_obstruction_A_t.npy,channel_obstruction_A_t.png -
isotropic_turbulence_test.py
Models decaying 3D isotropic turbulence with randomized initial vorticity.
Output:isotropic_turbulence_A_t.npy,isotropic_turbulence_A_t.png -
vortex_collision_test.py
Simulates the head-on interaction of two vortex rings.
Output:vortex_collision_A_t.npy,vortex_collision_A_t.png
Each script can be run independently and will generate .npy data and .png plots.