CFD Phase 2: 2D Navier–Stokes BKM Diagnostic (RTX 5090)

Cahlen Humphreys

May 27, 2026 by cahlen complete Bronze

BRONZE AI Literature Audit · 3 reviews ↓

Consensus	`ACCEPT_WITH_REVISION`
Models	gemini-2.5-pro + gpt-4.1 + o3-pro
Level	BRONZE — Novel observation, limited literature precedent

Review Ledger

2026-05-31 gemini-2.5-pro (Google) SILVER ACCEPT_WITH_REVISION

2026-05-31 gpt-4.1 (OpenAI) SILVER ACCEPT_WITH_REVISION

2026-05-31 o3-pro (OpenAI) BRONZE ACCEPT_WITH_REVISION

Issues Identified (6/6 resolved)

minor Compare to published 2D/3D benchmark DNS where appropriate. resolved

minor Document 2/3 Orszag dealiasing explicitly. resolved

important State fp64, timestep, and resolution limitations. resolved

minor Add spectral/NS references (BKM 1984, Orszag 1971, Ladyzhenskaya, Canuto, Bra... resolved

important Define random vorticity initial condition for reproducibility. resolved

minor Clarify RTX 5090 is physical hardware with on-machine measurements, not specu... resolved

All Reviews About Auditing Add Your Audit

Hardware

1x RTX 5090 (32 GB VRAM) Intel Core Ultra 9 285K 188 GB DDR5 RAM

fluid-dynamicsnavier-stokesbeale-kato-majdapseudospectral rtx-5090 cuda-kernelcufftbkm-diagnostic

Key Results

Problem: Track max vorticity and BKM integral ∫||ω||∞ dt in 2D incompressible NS
Conjecture Class: Beale–Kato–Majda blowup criterion (3D); 2D global regularity — infrastructure probe
Status: COMPLETE — Taylor–Green validation + random-IC sweep, zero NaN/Inf
Smoke: N=256, ν=1e-3, Taylor–Green, 2,000 steps — max |ω|: 2.0 → 0.16 (t=20)
Standard: N=512, ν=1e-4, random IC, 5,000 steps — BKM ≈ 1.77, 532 steps/s
Throughput: 1,108 steps/s (N=256), 532 steps/s (N=512)

CFD Phase 2: 2D Navier–Stokes BKM Diagnostic

Abstract

We extend the CFD conjecture program with a pseudospectral 2D Navier–Stokes solver on the RTX 5090, tracking the Beale–Kato–Majda (BKM) vorticity integral

$\int_0^T \|\omega(\cdot,t)\|_{L^\infty}\, dt$

alongside enstrophy decay. 2D incompressible flow is globally regular — we do not expect blowup here. This phase builds certifying DNS infrastructure toward 3D BKM searches.

Method

Vorticity form on $[0,2\pi)^2$ : $\omega_t + \mathbf{u}\cdot\nabla\omega = \nu\nabla^2\omega$
Streamfunction: $\omega = -\nabla^2\psi$ ; velocity $\mathbf{u} = (\partial_y\psi, -\partial_x\psi)$
cuFFT complex-to-complex transforms, 2/3 Orszag dealiasing, RK4 time stepping
Log CSV: max |ω|, enstrophy, cumulative BKM integral
Certificate: exit code 2 on NaN/Inf

Results

Run	Grid	ν	IC	Steps	Outcome
Smoke	256²	1e-3	Taylor–Green	2000	max \|ω\|: 2.0 → 0.16; BKM ≈ 12.8
Standard	512²	1e-4	Random blob	5000	BKM ≈ 1.77; 532 steps/s

Taylor–Green validates spectral accuracy (exponential enstrophy decay under viscosity). Random IC probes peak vorticity growth at lower ν.

Reproduction

git clone https://github.com/cahlen/idontknow.git
cd idontknow
./scripts/experiments/cfd-ns-bkm/run.sh 256 0.001 2000 0.01 taylor-green   # smoke
./scripts/experiments/cfd-ns-bkm/run.sh 512 0.0001 5000 0.005 random      # standard

Dataset: cahlen/cfd-ns-bkm — BKM CSVs and certifying logs.

Next steps

Phase 3: 3D pseudospectral NS with full BKM blowup search
Couple to chaotic advection maps for Lagrangian diagnostics
Adaptive resolution near max-vorticity regions

Human–AI collaboration. Not peer-reviewed. All code open for verification.