Reynolds-number scaling of the turbulence-triggering threshold in pipes
In plain words
The faster the flow in a pipe, the smaller the disturbance needed to make it turbulent. How exactly this minimum disturbance shrinks with flow speed is not agreed.
Precise statement
For pipe flow at $\mathrm{Re} \gg \mathrm{Re}_c$, the minimum perturbation that triggers sustained turbulence shrinks with Re. Experiments with localized jet injection give an injected-flux threshold $\sim \mathrm{Re}^{-1}$ (Hof, Juel, Mullin 2003), while nonlinear optimization of the perturbation kinetic energy (minimal seed) gives $E_c \sim \mathrm{Re}^{-3}$. Determine whether $E_c \sim \mathrm{Re}^{-3}$ persists as $\mathrm{Re} \to \infty$ over all divergence-free perturbations, and reconcile it with the experimental exponent by relating the two threshold measures.
What would settle it
Converged minimal-seed calculations over at least one decade of Re, together with experiments using matched perturbation shapes and a common threshold measure.
Status in the literature
Unverified note
Minimal-seed computations give $E_c \sim \mathrm{Re}^{-3}$, i.e. amplitude $\sim \mathrm{Re}^{-3/2}$ (arXiv:2606.23269, 2026), steeper than the experimental exponent -1 for jet injection; the residual question is reconciling the two measures.