Bootstrap DNS

Pseudospectral Direct Numerical Simulation for testing predictions from the Bootstrap Universe Programme

A lightweight Python DNS solver for incompressible Navier-Stokes, designed to extract strain-rate diagnostics that test specific predictions from Papers 163–166 of the Bootstrap Programme.

What This Tests

The Bootstrap Programme derives turbulence properties from the geometry of the Poincaré homology sphere S³/2I and the Steinbach polynomial t³ − 7t/3 + 7/27 = 0. This solver provides computational evidence for or against those predictions.

Paper 165 — The Heptagonal Constraint

Prediction: The strain-rate tensor in developed turbulence is constrained by the Steinbach polynomial. The joint PDF of the Q-R invariants should cluster along the Steinbach discriminant curve Δ = 49A⁶.

Test: Extract the velocity gradient tensor ∂uᵢ/∂xⱼ at every grid point, compute strain-rate eigenvalues and Q-R invariants, compare with the Steinbach constraint.

Paper 166 — The Cascade Exponent

Prediction: Kolmogorov's −5/3 exponent derives from the Newton-Girard power sums of the Steinbach polynomial: the prime 5 emerges at exactly p₅, giving −(quintic emergence order)/(eigenvalue count) = −5/3.

Test: Verify k⁻⁵/³ energy spectrum develops in the inertial range (sanity check — well established, but confirms the solver is correct).

Specific Diagnostics

Test	What We Measure	What Paper 165/166 Predicts
Q-R joint PDF	Shape of (Q,R) scatter	Clustering along Steinbach discriminant
Eigenvalue ratios	λ₂/λ₁ and λ₃/λ₁	Steinbach: λ₂/λ₁ ≈ 0.076, λ₃/λ₁ ≈ −1.076
Energy spectrum	E(k) vs k	k⁻⁵/³ inertial range
Vieillefosse tail	Q-R teardrop shape	Standard benchmark (validates solver)

Quick Start

# Clone
git clone https://github.com/DrCliffKeeble/bootstrap-dns.git
cd bootstrap-dns

# Install dependencies
pip install -r requirements.txt

# Run with defaults (32³, Re~125 — runs in ~1 min)
python scripts/dns_solver.py

# Run production (128³, Re~1600 — needs decent hardware, ~hours)
python scripts/dns_solver.py --N 128 --Re 1600 --T 20.0 --dt 0.001

# Generate diagnostic plots from saved data
python scripts/plot_diagnostics.py results/

Configuration

All parameters are set via command-line arguments:

Argument	Default	Description
--N	32	Grid points per dimension (32, 64, 128, 256)
--Re	125	Reynolds number
--T	5.0	Final time
--dt	0.004	Time step
--outdir	results/	Output directory
--print-every	50	Diagnostic print interval (steps)

Recommended Configurations

Config	Grid	Re	Time	Hardware	Purpose
Quick test	32³	125	5.0	Any laptop	Verify solver works
Development	64³	400	10.0	Laptop	Strain statistics, partial spectrum
Production	128³	1600	20.0	Desktop/workstation	Full inertial range, Steinbach test
Research	256³	6000	30.0	HPC cluster	Publication-quality statistics

Output

The solver produces:

results/
├── dns_state.npz          # Final velocity field (Fourier coefficients)
├── diagnostics.json       # Time series: energy, enstrophy, max|u|
├── spectrum.npz           # Energy spectra at saved times
├── strain_data.npz        # Q, R, Q_S, R_S, eigenvalues (at peak dissipation)
├── fig1_energy.png        # Energy and enstrophy evolution
├── fig2_spectrum.png      # Energy spectrum with k⁻⁵/³ reference
├── fig3_QR.png            # Q-R joint PDF with Steinbach line
└── fig4_eigenvalues.png   # Strain eigenvalue ratio distributions

Method

• Spatial discretisation: Fourier pseudospectral with 2/3 dealiasing
• Temporal integration: Classical RK4
• Nonlinear term: Rotational (Lamb) form: −(ω × u) with Leray projection
• Initial condition: Taylor-Green vortex: u = sin(x)cos(y)cos(z), v = −cos(x)sin(y)cos(z), w = 0
• Boundary conditions: Triply periodic on [0, 2π]³

Requirements

• Python ≥ 3.9
• NumPy
• Matplotlib
• SciPy (optional, for advanced post-processing)

Context

This code is part of the Bootstrap Universe Programme, a mathematical physics research programme deriving physical constants and structures from the geometry of S³/2I.

Papers tested by this code:

• Paper 165: "The Heptagonal Constraint" — Steinbach polynomial constrains strain tensor
• Paper 166: "The Cascade Exponent" — Kolmogorov −5/3 from Newton-Girard power sums

Author: Dr. Clifford Keeble, PhD
ORCID: 0009-0003-6828-2155
Location: Woodbridge, Suffolk, UK

Licence

MIT — see LICENCE

Status

🟡 Active development. The solver is validated against standard Taylor-Green benchmarks. The Steinbach diagnostic is preliminary — higher Reynolds numbers (Re ≥ 1600) are needed for meaningful strain statistics. Contributions and independent verification welcome.

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"Bootstrap is telling us it's about integers."

🐕☕⬡