{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"2af659cd1e7663b34d46e87015a675ed8fdcbaeda176c8edaebc230ea88ce56b","created":"2026-10-03T07:18:04Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"36341dc25224758695d05542b73a2913eac8407b078eaa0e716e80c922f4ad4e","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"fluid.small-scale.decay-exponent","kind":"well-posed","literature_status":"contested","n":"1","parents":[],"plain":"When stirring stops, turbulence in a box dies away with its energy falling as a power of time. Measured powers differ by up to a factor of two between experiments, and theory gives competing values that depend on how the largest swirls were set up.","posed_since":"","precise":"Freely decaying homogeneous isotropic Navier-Stokes turbulence at high Re has kinetic energy $K(t) \\sim t^{-n}$. Kolmogorov's argument with a conserved Loitsyansky integral gives $n = 10/7$ for an initial spectrum $E(k) \\sim k^4$ at small $k$; Saffman's conserved integral gives $n = 6/5$ for $E(k) \\sim k^2$. Determine whether n is fixed by the low-wavenumber exponent of the initial spectrum alone as $\\mathrm{Re} \\to \\infty \\text{ and } t \\to \\infty$ in a domain much larger than the integral scale, whether the Loitsyansky integral is conserved, and the value of n for each class.","problem_ref":null,"references":"","settled_by":"DNS in domains much larger than the integral scale over long decay times at high Re, measuring n and the time dependence of the Loitsyansky and Saffman integrals for controlled initial spectra.","status_note":"2022 DNS observe both exponents approximately and over limited times for suitably arranged initial conditions, but the low-wavenumber spectrum drifts, so the invariants are not strictly conserved (Panickacheril John, Donzis, Sreenivasan, Phil. Trans. R. Soc. A 2022).","title":"Decay exponent of freely decaying homogeneous isotropic turbulence","topic_ref":"45474d2178078204b5f1dbc4ca64d4b74ada936d74d13eddb4d83cc4b1e3c985"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"3b41e448af57c560246feaa5e21974f2287945430658468c2b50b0763c9b16f6","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"6e12717579f2df3f0a9865bd7ca0af56767ea77fa4e2f06e69ccb6c8f00d53dc","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"MYAcqw0RBwZrgYFH-d9t73wv3no2Wdu0tJXtSiB5Mem-grYYImGZG3-OiHzdY0Ekf3E3Us8pr_T6oFSJwYhMBA"},"schema":"pubphys.envelope/1"},"record_hash":"3b41e448af57c560246feaa5e21974f2287945430658468c2b50b0763c9b16f6","leaf_index":1434}