{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"1bde80a8b90ca2dbaf47dbe47af06d39be1b9086c73d032bae90a446ececbfaa","created":"2026-10-03T07:18:04Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"800d03cd10d58cd5ad0a410456731f202612621b819294c04dd0bf7e0044e63d","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"fluid.shear-transition.threshold-scaling","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"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.","posed_since":"2003","precise":"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.","problem_ref":null,"references":"","settled_by":"Converged minimal-seed calculations over at least one decade of Re, together with experiments using matched perturbation shapes and a common threshold measure.","status_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.","title":"Reynolds-number scaling of the turbulence-triggering threshold in 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