cfd-ns3d-bkm

# 3D Navier–Stokes BKM Blowup Search **Three-dimensional vorticity-form incompressible Navier–Stokes** with **vortex stretching** \\((\boldsymbol{\omega}\cdot\nabla)\mathbf{u}\\), pseudospectral **cuFFT C2C** on a periodic cube, and BKM integral tracking. Custom **CUDA** kernel on **NVIDIA RTX 5090**. $$ \partial_t \omega + (\mathbf{u}\cdot\nabla)\omega = (\boldsymbol{\omega}\cdot\nabla)\mathbf{u} + \nu \nabla^2 \omega $$ $$ \hat{\mathbf{u}} = \frac{i(\mathbf{k}\times\hat{\boldsymbol{\omega}})}{\lVert \mathbf{k} \rVert^2} $$ > Phase 3 of the [bigcompute.science](https://bigcompute.science) CFD program. Extends the [2D BKM diagnostic](https://huggingface.co/datasets/cahlen/cfd-ns-bkm). **No blowup observed** at tested resolution/Re — certifying infrastructure at moderate grid sizes. ## Quick Start ```python 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 | Column | Type | Description | |--------|------|-------------| | `step` | int | Time-step index | | `time` | float | Physical time \\(t\\) | | `max_vorticity` | float | \\(\lVert \omega \rVert_{L^\infty}\\) | | `enstrophy` | float | \\(\tfrac{1}{2}\int \lVert \omega \rVert^2 \, dV\\) | | `bkm_cumulative` | float | BKM integral \\(\int_0^t \lVert \omega \rVert_{L^\infty}\, ds\\) | Certifying logs in `logs/`. Metadata in `metadata.json`. ### Configurations | Config | Grid | \\(\nu\\) | IC | Steps | \\(\Delta t\\) | Final max \\(\lVert \omega \rVert_{L^\infty}\\) | Final BKM | |--------|------|---|-----|-------|-----|-----------------|-----------| | `smoke_taylor_green` | \\(64^3\\) | 0.01 | Taylor–Green | 200 | 0.002 | 4.17 at \\(t=0.4\\) | 1.63 | | `standard_random` | \\(128^3\\) | \\(10^{-3}\\) | Random blob | 1000 | 0.002 | 0.614 at \\(t=2.0\\) | 1.24 | | `blowup_search` | \\(256^3\\) | \\(10^{-4}\\) | Random blob | 500 | 0.001 | 0.878 at \\(t=0.5\\) | 0.44 | | `blowup_search_long` | \\(256^3\\) | \\(10^{-4}\\) | Random blob | 2000 | 0.001 | 0.887 at \\(t=2.0\\) | 1.76 | | `blowup_search_5000` | \\(256^3\\) | \\(10^{-4}\\) | Random blob | 5000 | 0.001 | 0.903 at \\(t=5.0\\) | 4.45 | | `taylor_green_256` | \\(256^3\\) | \\(10^{-3}\\) | Taylor–Green | 1000 | 0.001 | 4.44 at \\(t=1.0\\) | 4.23 | | `kerr_nu1e-04` | \\(256^3\\) | \\(10^{-4}\\) | Kerr tubes | 2000 | 0.001 | 5.0 at \\(t=2.0\\) | 9.99 | | `kerr_nu1e-03` | \\(256^3\\) | \\(10^{-3}\\) | Kerr tubes | 1000 | 0.001 | 5.0 at \\(t=1.0\\) | 4.99 | All runs: **zero NaN/Inf**. ### Method (summary) - 3D vorticity form with explicit vortex-stretching term in physical space - Velocity from Fourier relation \\(\hat{\mathbf{u}} = i(\mathbf{k}\times\hat{\boldsymbol{\omega}})/\lVert \mathbf{k} \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\\): BKM **≈ 1.24** by \\(t=2\\); vorticity remains bounded at tested Re - **256³ blowup search** at \\(\nu=10^{-4}\\): BKM **≈ 0.44** by \\(t=0.5\\); extended to **≈ 1.76** by \\(t=2.0\\); **≈ 4.45** by \\(t=5.0\\) (5000 steps); **2.3 steps/s**; no blowup signal - **256³ Taylor–Green** at \\(\nu=10^{-3}\\): BKM **≈ 4.23** by \\(t=1.0\\); max \\(\lVert \omega \rVert_{L^\infty} \approx 4.44\\) - **Phase 5 Kerr IC** at \\(256^3\\), \\(\nu=10^{-4}\\): BKM **≈ 9.99** by \\(t=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 ```bash 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](https://github.com/cahlen/idontknow/blob/main/scripts/experiments/cfd-ns3d-bkm/ns3d_bkm.cu) ## Related - Phase 2 (2D): [cahlen/cfd-ns-bkm](https://huggingface.co/datasets/cahlen/cfd-ns-bkm) - CFD program: [cfd-chaotic-advection](https://bigcompute.science/experiments/cfd-chaotic-advection/) - Experiment page: [cfd-ns3d-bkm](https://bigcompute.science/experiments/cfd-ns3d-bkm/) - Finding (3D): [cfd-ns3d-bkm-infrastructure](https://bigcompute.science/findings/cfd-ns3d-bkm-infrastructure/) - Finding (2D): [cfd-ns-bkm-diagnostic](https://bigcompute.science/findings/cfd-ns-bkm-diagnostic/) - Code: [idontknow/scripts/experiments/cfd-ns3d-bkm](https://github.com/cahlen/idontknow/tree/main/scripts/experiments/cfd-ns3d-bkm) ## Citation ```bibtex @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.