2D Four-Mode Navier-Stokes

Experiment: exp_ns_2d.sx  |  Validates: Conjecture 11.1

Hypothesis

A 2D four-mode Galerkin truncation of the Navier-Stokes equations admits a critical Reynolds number \(\text{Re}_c\) that can be located precisely via binary search on the convergence score \(S = 0\) surface. Furthermore, the interaction matrix between modes should reveal cascade direction: inverse energy cascade (large scales absorb energy) at low viscosity, forward cascade (energy dissipates to small scales) at high viscosity.

Method

1. Truncate the 2D incompressible Navier-Stokes equations to four Fourier modes \(\hat{u}_{(1,0)}, \hat{u}_{(0,1)}, \hat{u}_{(1,1)}, \hat{u}_{(2,0)}\), retaining the nonlinear advection coupling.
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 \(4 \times 4\) interaction matrix \(M_{ij}\) measuring energy transfer rate from mode \(i\) to mode \(j\).
4. Classify cascade direction from the interaction matrix: if net energy flows from high-\(k\) to low-\(k\) modes, the cascade is inverse; if from low-\(k\) to high-\(k\), forward.

Results

Critical Reynolds Number

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

Result: \(\text{Re}_c \approx 578\). Below this threshold \(S > 0\) (smooth dynamics); above it \(S < 0\) (instability onset).

Energy Cascade Direction

Interaction Matrix at Re = 1000

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

Net flow is from high-\(k\) modes (1,1) and (2,0) toward low-\(k\) modes (1,0) and (0,1), confirming inverse cascade.

Analysis

• \(S = 0\) locates critical Re. The binary search converges cleanly, giving \(\text{Re}_c \approx 578\). This is consistent with known critical Reynolds numbers for low-mode 2D NS truncations in the literature.
• Cascade direction flips at criticality. The interaction matrix transitions from forward (energy to small scales) below \(\text{Re}_c\) to inverse (energy to large scales) above it. The \(S = 0\) surface coincides with the cascade reversal point.
• Interaction matrix discovers structure. Without prescribing cascade theory, the mode-mode energy transfer matrix self-organises to reveal the cascade direction. The S score provides a scalar summary of this matrix-level structure.

Conclusion

The convergence score \(S\) provides a scalar criterion for locating the critical Reynolds number in a 2D four-mode NS truncation. At \(\text{Re}_c \approx 578\), the energy cascade reverses direction from forward to inverse, and \(S\) crosses zero. The interaction matrix between modes autonomously discovers the cascade structure. These results validate Conjecture 11.1 within the truncated 2D setting: the S score diagnoses flow regularity and its sign change marks the laminar-turbulent transition.

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

Related

• Conjecture 11.1 — S score as flow regularity diagnostic
• 1D Navier-Stokes Experiment — viscous Burgers baseline
• Blow-Up Countdown — S as early warning before singularity
• S-Smoothness Contingency — 25-case statistical validation
• I-Ratio Theorem — \(I = -0.5\) iff equilibrium
• Kraichnan, R. H. (1967) — Inertial ranges in two-dimensional turbulence