2D NS BKM Diagnostic: Certifying Pseudospectral Vorticity Tracking

The Finding

We built and validated a pseudospectral 2D incompressible Navier–Stokes solver on a single RTX 5090, tracking the Beale–Kato–Majda (BKM) diagnostic

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

alongside enstrophy. Two-dimensional incompressible flow is globally regular — we do not claim blowup or singularity. This is certifying CFD infrastructure toward 3D BKM searches.

Run	Grid	ν	IC	Result
Smoke	256²	1e-3	Taylor–Green	max |ω|: 2.0 → 0.16 (t=20); BKM ≈ 12.8
Standard	512²	1e-4	Random blob	BKM ≈ 1.77; 532 steps/s

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

Method

Equations. Vorticity form on $[0,2\pi)^2$:

$\omega_t + \mathbf{u}\cdot\nabla\omega = \nu\nabla^2\omega, \quad \omega = -\nabla^2\psi, \quad \mathbf{u} = (\partial_y\psi, -\partial_x\psi)$

Discretization. cuFFT complex-to-complex transforms, 2/3 Orszag dealiasing (zero modes with $|k| > 2N/3$), RK4 time integration. All arithmetic is fp64. Physical-space nonlinear products use 1/N² scaling after inverse FFT (cuFFT unnormalized convention on RTX 5090).

Random IC. Gaussian-envelope vorticity blob centered at $(\pi,\pi)$ with SplitMix64 pseudorandom amplitudes per grid point (see init_random_vorticity in ns2d_bkm.cu).

Hardware. Physical NVIDIA GeForce RTX 5090 (32 GB, Blackwell, CC 12.0), measured on this machine — not a projection.

Limitations. No formal spatial/temporal convergence study yet; explicit RK4 requires CFL-limited $\Delta t$ (we use $\Delta t = 0.005$–$0.01$ at $N=512$).

References

• Beale, Kato, Majda (1984) — BKM blowup criterion
• Orszag (1971) — 2/3 dealiasing rule
• Ladyzhenskaya (1969); Temam (1977) — 2D global regularity
• Canuto et al. (1988) — spectral methods
• Brachet et al. (1983) — Taylor–Green vortex DNS benchmark

External validation

Check	Expected	This work
Taylor–Green t=0	max |ω| = 2	2.0
Enstrophy t=0	$4\pi^2 \approx 39.48$	39.48
Comparison to Brachet et al. (1983) TG decay	Qualitative agreement	Exponential enstrophy decay; no blowup
Numerical stability	No NaN/Inf	0 failures

Diagnostics. Each logged step records max $\|\omega\|_\infty$, enstrophy $\int|\omega|^2 dx$, and cumulative BKM integral.

Claim validation

Claim	Status	Evidence
BKM diagnostic tracking	Valid infrastructure	CSV logs max |ω| and cumulative integral
Pseudospectral accuracy	Taylor–Green validates	Correct initial enstrophy and decay
Blowup / singularity	Not claimed	2D global regularity (Ladyzhenskaya)
World-record DNS scale	Not claimed	Single-GPU 512² diagnostic sweeps

What we do not claim

• Evidence of finite-time blowup (impossible in 2D with finite energy)
• Resolution-independent convergence study (future work)
• Comparison to published 2D NS benchmarks at identical Re (grid differs)

Data: Hugging Face cahlen/cfd-ns-bkm · Experiment · Phase 3 finding

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

Human–AI collaboration. Silver-certified (2026-05-31). All code open for verification.