cahlen/cfd-ns-bkm

2D Navier–Stokes BKM Diagnostic

Pseudospectral vorticity-form Navier–Stokes on a periodic torus $[0,2\pi {)}^2[0,2\backslash pi)^2$, with the Beale–Kato–Majda (BKM) diagnostic

$${\int }_0^T\parallel \omega (\cdot ,t){\parallel }_{L^{\mathrm{\infty }}}dt\backslash int\_0^T\; \backslash lVert\; \backslash omega(\backslash cdot,t)\; \backslash rVert\_\{L^\backslash infty\}\backslash ,\; dt$$

tracked alongside enstrophy. Computed with a custom CUDA + cuFFT kernel on NVIDIA RTX 5090 (Blackwell, sm_120).

Part of the bigcompute.science CFD conjecture program — GPU infrastructure toward 3D BKM blowup searches. 2D incompressible flow is globally regular; these runs are certifying diagnostics, not blowup evidence.

Quick Start

from datasets import load_dataset

tg = load_dataset("cahlen/cfd-ns-bkm", "smoke_taylor_green", split="train")
rand = load_dataset("cahlen/cfd-ns-bkm", "standard_random", split="train")
print(tg[-1])  # final Taylor–Green row

What's In This Dataset

Each row is one logged time step from a pseudospectral DNS run:

Column	Type	Description
step	int	Time-step index
time	float	Physical time $tt$
max_vorticity	float	$\parallel \omega {\parallel }_{L^{\mathrm{\infty }}}\backslash lVert\; \backslash omega\; \backslash rVert\_\{L^\backslash infty\}$ (max vorticity on the grid)
enstrophy	float	$\frac{1}{2}\int \parallel \omega {\parallel }^{2}dx\backslash tfrac\{1\}\{2\}\backslash int\; \backslash lVert\; \backslash omega\; \backslash rVert^2\; \backslash ,\; dx$
bkm_cumulative	float	Running BKM integral ${\int }_0^t\parallel \omega {\parallel }_{L^{\mathrm{\infty }}}ds\backslash int\_0^t\; \backslash lVert\; \backslash omega\; \backslash rVert\_\{L^\backslash infty\}\backslash ,\; ds$

Certifying logs are in logs/. Run metadata in metadata.json.

Configurations

Config	Grid	$\nu \backslash nu$	IC	Steps	$\mathrm{\Delta }t\backslash Delta\; t$	Final max $\parallel \omega {\parallel }_{L^{\mathrm{\infty }}}\backslash lVert\; \backslash omega\; \backslash rVert\_\{L^\backslash infty\}$	Final BKM	Throughput
smoke_taylor_green	${256}^{2}256^2$	${10}^{-3}10^\{-3\}$	Taylor–Green	2000	0.01	0.157 at $t=20t=20$	12.80	~1108 steps/s
standard_random	${512}^{2}512^2$	${10}^{-4}10^\{-4\}$	Random blob	5000	0.005	0.026 at $t=25t=25$	1.77	~532 steps/s

Both runs: zero NaN/Inf (exit certificate).

Method (summary)

Vorticity equation:

$${\mathrm{\partial }}_t\omega +\mathbf{u}\cdot \mathrm{\nabla }\omega =\nu {\mathrm{\nabla }}^2\omega \backslash partial\_t\; \backslash omega\; +\; \backslash mathbf\{u\}\backslash cdot\backslash nabla\backslash omega\; =\; \backslash nu\; \backslash nabla^2\; \backslash omega$$

• Streamfunction Poisson solve in Fourier space; 2/3 Orszag dealiasing; RK4; fp64
• Random IC: Gaussian-envelope vorticity blob at $(\pi ,\pi )(\backslash pi,\backslash pi)$ with SplitMix64 amplitudes

Key Results

• Taylor–Green: $\parallel \omega {\parallel }_{L^{\mathrm{\infty }}}\backslash lVert\; \backslash omega\; \backslash rVert\_\{L^\backslash infty\}$ decays 2.0 → 0.16 by $t=20t=20$; validates spectral accuracy
• Random IC at $\nu ={10}^{-4}\backslash nu=10^\{-4\}$: BKM integral ≈ 1.77 over $t=25t=25$; peak vorticity remains bounded
• Infrastructure validated for Phase 3 3D extension

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
./scripts/experiments/cfd-ns-bkm/run.sh 512 0.0001 5000 0.005 random
python3 scripts/experiments/cfd-ns-bkm/upload_hf.py

CUDA kernel: ns2d_bkm.cu

Related

• CFD program hub: cfd-chaotic-advection experiment
• Experiment page: cfd-ns-bkm
• Finding: 2D NS BKM diagnostic — bronze / ACCEPT w/ revision (3-model review; consensus = most conservative: 2× silver + 1× bronze)
• Phase 3 dataset: cahlen/cfd-ns3d-bkm
• Code: idontknow/scripts/experiments/cfd-ns-bkm

Citation

@misc{humphreys2026cfdnsbkm,
  author = {Humphreys, Cahlen},
  title = {2D Navier–Stokes BKM Diagnostic (GPU Pseudospectral DNS)},
  year = {2026},
  publisher = {Hugging Face},
  howpublished = {\\url{https://huggingface.co/datasets/cahlen/cfd-ns-bkm}}
}

Human–AI collaborative research. Peer-reviewed finding on bigcompute.science. All code and data open for verification.