Burgers-equation experiment: I-ratio across a viscosity sweep

Experiment: exp_navier_stokes.sx  |  Domain: Burgers-equation dynamics

Question

The I-ratio between the advection and diffusion terms of viscous Burgers is near \(-0.5\) when the two terms balance and moves away from that value when they do not. When the Reynolds number \(\text{Re} = U L / \nu\) is low (strong diffusion) the two terms are in balance; as \(\text{Re}\) increases the balance breaks and the I-ratio moves away, with \(S\) falling. The run measures where along the sweep this happens.

Method

1. Solve the 1D viscous Burgers equation, the simplest nonlinear PDE with an advection term and a diffusion term: \[ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2} \]
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 term \(|u \, \partial u / \partial x|\)
   • \(F_{\text{diff}}\): spatially averaged diffusion term \(|\nu \, \partial^2 u / \partial x^2|\)
   • I-ratio: \(I = \frac{F_{\text{adv}} - F_{\text{diff}}}{F_{\text{adv}} + F_{\text{diff}}}\)
   • S score: the convergence score computed from the same two terms
   • \(\max|u|\): peak velocity magnitude at the final time step
6. Monitor the S score at each time step. Record the first step where \(S < 0\).

Results

Viscosity Sweep

Where \(S\) Goes Negative

At \(\nu = 0.001\) (Re = 1000), the S score goes negative at step 299 of 500, before the solution develops its steepest gradients. At all diffusion-dominated viscosities (\(\nu \geq 0.05\)), the S score stays at 1.000 throughout the run.

Diffusion-dominated vs Advection-dominated

At high viscosity diffusion overwhelms advection so completely that the I-ratio sits at zero. At low viscosity the advection term is significant and the I-ratio is strongly positive.

Notes

• The I-ratio separates the two regimes. A single scalar separates the diffusion-dominated runs (\(I \approx 0\)) from the advection-dominated runs (\(I \gg 0\)) in this 1D model.
• \(S\) as an early signal. At Re = 1000 the S score goes negative at step 299, before the velocity field develops its sharpest gradients.
• The 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 partly controls the dynamics.
• What the steep gradients are. In Burgers, \(S\) going negative coincides with the onset of shock steepening. This is an observation about a 1D scalar equation and says nothing about the 3D incompressible Navier–Stokes equations.

Summary

On 1D viscous Burgers across a viscosity sweep from \(\nu = 0.5\) to \(\nu = 0.001\), the I-ratio between the advection and diffusion terms moves from 0 to 0.322, and the convergence score \(S\) from 1.000 to \(-0.102\), with \(S\) going negative at step 299 of 500 at the lowest viscosity. The results describe this 1D model PDE only.

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

• Conjectures — the I-ratio as a flow diagnostic is listed there as open
• Burgers, J. M. (1948) — A mathematical model illustrating the theory of turbulence
• I-Ratio Identity — \(\|\sum g_i\|^2 = D(1 + 2I)\)
• Exploratory experiment: logistic-map sweep — the S score on a discrete map