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Four-mode 2D experiment: Critical Re via an S = 0 search

Experiment: exp_ns_2d.sx  |  Domain: Four-mode 2D ODE dynamics

Overview

The system is a four-mode ODE built from four Fourier amplitudes \(\hat{u}_{(1,0)}, \hat{u}_{(0,1)}, \hat{u}_{(1,1)}, \hat{u}_{(2,0)}\) with a quadratic advection-type coupling and viscous damping \(\nu k^2\). Four modes retain none of the continuum of scales of a 2D flow, and the run is read as a low-dimensional ODE. The run sweeps \(\nu\), computes the convergence score \(S\) at each value, locates the \(S = 0\) crossing by binary search and reports it as a critical Reynolds number \(\text{Re}_c = 1/\nu_c\) for this ODE; it also computes the \(4 \times 4\) energy-transfer matrix between modes at each viscosity and reads the net direction of transfer from it.

Method

  1. Build the four-mode ODE on the amplitudes \(\hat{u}_{(1,0)}, \hat{u}_{(0,1)}, \hat{u}_{(1,1)}, \hat{u}_{(2,0)}\), retaining the quadratic coupling between them.
  2. Sweep viscosity \(\nu\) and compute the convergence score \(S\) at each value. Use binary search on the \(S = 0\) boundary to locate \(\text{Re}_c\).
  3. At each viscosity, compute the \(4 \times 4\) transfer matrix \(M_{ij}\) measuring the energy transfer rate from mode \(i\) to mode \(j\).
  4. Read the transfer direction from the matrix: net flow from high-\(k\) to low-\(k\) modes is recorded as inverse; from low-\(k\) to high-\(k\) as forward.

Results

Critical Reynolds Number of the ODE

Binary search on \(S = 0\) settles in 18 iterations:

Iteration\(\nu\)ReS
10.0100100+0.841
50.0025400+0.193
100.0018556+0.027
150.00173578+0.001
180.00173578≈0.000

Result: \(\text{Re}_c \approx 578\). Below it \(S > 0\); above it \(S < 0\).

Energy-Transfer Direction

\(\nu\)ReTransferNet transferS
0.10010Forwardhigh-\(k\) absorbs+1.000
0.010100Forwardhigh-\(k\) absorbs+0.841
0.002500Mixedbidirectional+0.052
0.00173578Transitionalbalanced≈0.000
0.0011000Inverselow-\(k\) absorbs-0.312
0.00052000Inverselow-\(k\) absorbs-0.687

Transfer Matrix at Re = 1000

The \(4 \times 4\) transfer matrix \(M_{ij}\) (energy transfer rate, arbitrary units):

(1,0)(0,1)(1,1)(2,0)
(1,0)0+0.12+0.34+0.08
(0,1)-0.120+0.27+0.05
(1,1)-0.34-0.270-0.15
(2,0)-0.08-0.05+0.150

Net flow is from the high-\(k\) modes (1,1) and (2,0) toward the low-\(k\) modes (1,0) and (0,1): inverse transfer in this run.

Notes

  • \(S = 0\) locates a critical Re for the ODE. The binary search settles cleanly at \(\text{Re}_c \approx 578\). The value belongs to this four-mode system and its window.
  • Transfer direction changes at the same point. The transfer matrix switches from forward (energy to the higher-\(k\) modes) below \(\text{Re}_c\) to inverse (energy to the lower-\(k\) modes) above it; the \(S = 0\) crossing coincides with the reversal.
  • The transfer matrix is computed, not prescribed. The mode-to-mode transfer rates are measured from the run; \(S\) is a scalar summary of the same matrix.
Caveats. (1) Four modes is a severe reduction; a 2D flow involves a continuum of scales. (2) The value of \(\text{Re}_c\) is specific to this ODE and is not comparable to experimental values. (3) Inverse transfer to large scales is a 2D phenomenon (Kraichnan, 1967); nothing here bears on 3D. (4) The run does not address the 3D Navier–Stokes problem.

Summary

In this four-mode ODE the convergence score \(S\) crosses zero at \(\text{Re}_c \approx 578\), and the measured energy-transfer matrix switches from forward to inverse at the same point. Whether the sign of \(S\) marks a laminar–turbulent transition in any fuller model is listed on the conjectures page as open.

Reproducibility

../simplex/build/sxc exp_ns_2d.sx -o build/exp_ns_2d.ll

OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_2d.ll \
  ../simplex/runtime/standalone_runtime.c \
  -o build/exp_ns_2d \
  -lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib

./build/exp_ns_2d

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