3D Navier–Stokes BKM Blowup Search

Three-dimensional vorticity-form incompressible Navier–Stokes with vortex stretching $(\mathit{\omega }\cdot \mathrm{\nabla })\mathbf{u}(\backslash boldsymbol\{\backslash omega\}\backslash cdot\backslash nabla)\backslash mathbf\{u\}$, pseudospectral cuFFT C2C on a periodic cube, and BKM integral tracking. Custom CUDA kernel on NVIDIA RTX 5090.

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

$$\widehat{\mathbf{u}}=\frac{i(\mathbf{k}\times \widehat{\mathit{\omega }})}{\parallel \mathbf{k}{\parallel }^2}\backslash hat\{\backslash mathbf\{u\}\}\; =\; \backslash frac\{i(\backslash mathbf\{k\}\backslash times\backslash hat\{\backslash boldsymbol\{\backslash omega\}\})\}\{\backslash lVert\; \backslash mathbf\{k\}\; \backslash rVert^2\}$$

Phase 3 of the bigcompute.science CFD program. Extends the 2D BKM diagnostic. No blowup observed at tested resolution/Re — certifying infrastructure at moderate grid sizes.

Quick Start

from datasets import load_dataset

ds = load_dataset("cahlen/cfd-ns3d-bkm", "standard_random", split="train")
row = ds[-1]
print(f"t={row['time']:.3f}, max|ω|={row['max_vorticity']:.4f}, BKM={row['bkm_cumulative']:.4f}")

What's In This Dataset

Certifying logs in logs/. Metadata in metadata.json.

Configurations

All runs: zero NaN/Inf.

Method (summary)

• 3D vorticity form with explicit vortex-stretching term in physical space
• Velocity from Fourier relation $\widehat{\mathbf{u}}=i(\mathbf{k}\times \widehat{\mathit{\omega }})\mathrm{/}\parallel \mathbf{k}{\parallel }^2\backslash hat\{\backslash mathbf\{u\}\}\; =\; i(\backslash mathbf\{k\}\backslash times\backslash hat\{\backslash boldsymbol\{\backslash omega\}\})/\backslash lVert\; \backslash mathbf\{k\}\; \backslash rVert^2$; 2/3 dealiasing; RK4; fp64
• BKM integral accumulated each step for blowup-criterion monitoring

Key Results

• First certifying 3D pseudospectral BKM runs on RTX 5090
• Random IC at ${128}^{3}128^3$: BKM ≈ 1.24 by $t=2t=2$; vorticity remains bounded at tested Re
• 256³ blowup search at $\nu ={10}^{-4}\backslash nu=10^\{-4\}$: BKM ≈ 0.44 by $t=0.5t=0.5$; extended to ≈ 1.76 by $t=2.0t=2.0$; ≈ 4.45 by $t=5.0t=5.0$ (5000 steps); 2.3 steps/s; no blowup signal
• 256³ Taylor–Green at $\nu ={10}^{-3}\backslash nu=10^\{-3\}$: BKM ≈ 4.23 by $t=1.0t=1.0$; max $\parallel \omega {\parallel }_{L^{\mathrm{\infty }}}\approx 4.44\backslash lVert\; \backslash omega\; \backslash rVert\_\{L^\backslash infty\}\; \backslash approx\; 4.44$
• Phase 5 Kerr IC at ${256}^{3}256^3$, $\nu ={10}^{-4}\backslash nu=10^\{-4\}$: BKM ≈ 9.99 by $t=2t=2$ vs random ≈ 1.76 (5.7×); structured vortex tubes drive stronger BKM growth
• No finite-time blowup signal at this resolution — consistent with viscous DNS at moderate Re

Reproduction

git clone https://github.com/cahlen/idontknow.git
cd idontknow
./scripts/experiments/cfd-ns3d-bkm/run.sh 64 0.01 200 0.002 taylor-green
./scripts/experiments/cfd-ns3d-bkm/run.sh 128 0.001 1000 0.002 random
python3 scripts/experiments/cfd-ns3d-bkm/upload_hf.py

CUDA kernel: ns3d_bkm.cu

Related

• Phase 2 (2D): cahlen/cfd-ns-bkm
• CFD program: cfd-chaotic-advection
• Experiment page: cfd-ns3d-bkm
• Finding (3D): cfd-ns3d-bkm-infrastructure
• Finding (2D): cfd-ns-bkm-diagnostic
• Code: idontknow/scripts/experiments/cfd-ns3d-bkm

Citation

@misc{humphreys2026cfdns3dbkm,
  author = {Humphreys, Cahlen},
  title = {3D Navier–Stokes BKM Blowup Search (GPU Pseudospectral DNS)},
  year = {2026},
  publisher = {Hugging Face},
  howpublished = {\\url{https://huggingface.co/datasets/cahlen/cfd-ns3d-bkm}}
}

Human–AI collaborative research. All code and data open for verification.