Navier-Stokes: I-Ratio as Laminar Flow Diagnostic

Experiment: exp_navier_stokes.sx  |  Domain: Millennium Problem — Navier-Stokes Existence and Smoothness

Hypothesis

The I-ratio between advection and diffusion forces in viscous fluid flow approaches \(I = -0.5\) in the laminar regime and departs from that value at the turbulent transition. Specifically, when the Reynolds number \(\text{Re} = U L / \nu\) is low (strong diffusion), the two forces are in equilibrium and the I-ratio converges to \(-0.5\). As \(\text{Re}\) increases (weaker diffusion), the force balance breaks and the I-ratio moves away from equilibrium, signalling the onset of turbulence.

Method

1. Solve the 1D viscous Burgers equation, the simplest nonlinear PDE retaining the advection-diffusion structure of Navier-Stokes: \[ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2} \]
2. Discretise on a uniform grid with a 5-point stencil. Domain \(x \in [0, 1]\), \(N = 100\) grid points, \(\Delta t\) chosen for CFL stability.
3. Initial condition: \(u(x, 0) = \sin(2\pi x)\). Boundary conditions: \(u(0,t) = u(1,t) = 0\).
4. Sweep viscosity \(\nu\) through six values: 0.001, 0.005, 0.01, 0.05, 0.1, 0.5. Corresponding Reynolds numbers \(\text{Re} = 1/\nu\) range from 2 to 1000.
5. At each viscosity, run 500 time steps. Compute:
   • \(F_{\text{adv}}\): spatially averaged advection force \(|u \, \partial u / \partial x|\)
   • \(F_{\text{diff}}\): spatially averaged diffusion force \(|\nu \, \partial^2 u / \partial x^2|\)
   • I-ratio: \(I = \frac{F_{\text{adv}} - F_{\text{diff}}}{F_{\text{adv}} + F_{\text{diff}}}\)
   • S score: convergence diagnostic from the I-ratio framework
   • \(\max|u|\): peak velocity magnitude at final time step
6. Monitor the S score at each time step. Record the first step where \(S < 0\) (instability detection).

Results

Viscosity Sweep

Instability Detection

At \(\nu = 0.001\) (Re = 1000), the S score goes negative at step 299 out of 500, detecting the onset of instability well before the solution develops a blow-up. At all laminar viscosities (\(\nu \geq 0.05\)), the S score remains at 1.000 throughout the entire simulation.

Laminar vs Turbulent Comparison

In the laminar regime, diffusion overwhelms advection so completely that the I-ratio approaches zero (diffusion-dominated equilibrium). In the turbulent regime, advection forces are significant and the I-ratio is strongly positive, reflecting force imbalance.

Analysis

• Force balance characterises regime. The I-ratio provides a single scalar that cleanly separates laminar (\(I \approx 0\), forces in diffusive equilibrium) from turbulent (\(I \gg 0\), advection-dominated imbalance) flow in this 1D model.
• S score as early warning. The S score goes negative at step 299 for the turbulent case, well before the velocity field develops sharp gradients. This suggests the convergence diagnostic can serve as an early-warning indicator for solution regularity loss.
• Transition is gradual. At Re = 100 (\(\nu = 0.01\)), the I-ratio is slightly negative (\(-0.073\)) and S = 0.803, consistent with a transitional regime where diffusion still partially controls the dynamics.
• Connection to smoothness. In the NS existence and smoothness problem, the key question is whether solutions remain smooth for all time. Here, the S score going negative correlates with the onset of steep gradients (proto-singularities in the Burgers context). This is suggestive but not a proof of anything about the 3D incompressible Navier-Stokes equations.

Conclusion

The I-ratio framework extends naturally to advection-diffusion PDEs. In the viscous Burgers equation, the force balance between advection and diffusion — quantified by the I-ratio — provides a clean diagnostic for the laminar-turbulent transition. The S score detects instability onset before the solution develops steep gradients. These results offer a new gradient-based diagnostic perspective on the smoothness question, but they are limited to a 1D model PDE and do not address the 3D Navier-Stokes existence and smoothness problem.

Reproducibility

../simplex/build/sxc exp_navier_stokes.sx -o build/exp_navier_stokes.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_navier_stokes.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_navier_stokes \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_navier_stokes

Related

• Conjecture 11.1 — I-ratio as flow regularity diagnostic
• Navier-Stokes existence and smoothness (Clay Mathematics Institute Millennium Problem)
• Burgers, J. M. (1948) — A mathematical model illustrating the theory of turbulence
• I-Ratio Theorem — \(I = -0.5\) iff equilibrium
• Chaos Boundary Experiment — related dynamical systems diagnostics