Freely decaying homogeneous isotropic turbulence: quantitative prediction of decay laws, regime transitions, and the evolution of the largest eddies from initial conditions

Heisenberg's 1948 theory of statistical and isotropic turbulence derived a self-similar decay solution for grid-generated turbulence — an energy decay proportional to $1/t$, with the smallest-wavenumber spectrum anchored by Batchelor's $k^{4}$ law — while leaving its quantitative core open: the turbulent-viscosity constant $\kappa \approx 0.8$ had to be taken from experiment, the approximate spectral solutions had uncontrolled accuracy, the transition from the $1/t$ law to the faster (eventually $t^{-5/2}$) decay was described only qualitatively, and the role of the largest eddies was heuristic. The underlying question survives seventy-five years later: given the initial energy spectrum of freely decaying homogeneous isotropic turbulence, predict quantitatively the decay exponent and its dependence on initial conditions, the growth of the integral scale, the dynamically conserved large-scale invariants, and the timing of regime transitions. Later work resolved pieces — the final-period decay law, the classification into Saffman ($k^{2}$) and Batchelor ($k^{4}$) spectral classes, and the approximate conservation of the corresponding invariants in large-box simulations — but the exponent's initial-condition dependence, the non-equilibrium evolution of the dissipation coefficient, and the largest-eddy dynamics remain actively contested in high-Reynolds-number experiments, record-length direct numerical simulations, and new exact-solution claims as of 2024-2026.

Background

Homogeneous isotropic turbulence is the idealized state of turbulence whose statistical properties are independent of position and of the orientation of the coordinate axes; it is approximated by the flow downstream of a uniform grid in a wind tunnel ('grid turbulence'). Its structure is described by the energy spectrum $E(k)$ (Heisenberg's $F(k)$), the kinetic energy per unit mass contained between wavenumber $k$ and $k+dk$, where $k$ is inversely proportional to eddy size. When stirring stops, the turbulence decays: energy cascades from large eddies to small ones, where viscosity dissipates it, and the total energy $u^{2}$ (the mean-square velocity fluctuation) falls as a power law $u^{2} \sim t^{-n}$. By 1948 the inertial-range spectral shape (the range of eddies large compared with the dissipative ones but small compared with the energy-containing ones), $E(k) \sim k^{-5/3}$, had been established by similarity arguments. Heisenberg (1948) — an English version of 'Zur statistischen Theorie der Turbulenz' — supplied a transfer closure: the back-reaction of the small eddies on the large ones was represented by a 'turbulent viscosity', an eddy viscosity $\eta_{k}$ built from the spectrum itself, with a single numerical constant $\kappa \approx 0.8$ taken from experiment. Applying this closure to the grid-decay measurements of Batchelor & Townsend, he derived a similarity (self-preserving) solution $F(k,t) = t^{-1/2} f(k\sqrt{t})$: during decay the integral length scale $\ell$ (a measure of the size of the energy-containing, largest eddies, defined from the two-point velocity correlation) grows like $\sqrt{t}$ and $u^{2}$ falls like $1/t$. He acknowledged that the spectrum cannot remain linear at the smallest wavenumbers: Batchelor's independent analysis forces $E(k) \propto k^{4}$ as $k \to 0$, tied to the conservation of the Loitsyansky integral (a weighted integral of the two-point velocity correlation), which at low Reynolds number (the ratio of inertial to viscous forces) produces the faster 'final period' decay $u^{2} \sim t^{-5/2}$. The paper left the quantitative core open: the cross-over from the $1/t$ law to the faster decay was described only qualitatively, the largest eddies were treated heuristically, $\kappa$ was not derived, and the approximate spectral solutions had uncontrolled accuracy. The later literature transformed the small-wavenumber question into a classification problem. Batchelor (1948, 1949), Proudman & Reid (1954) and Batchelor & Proudman (1956) tied the $k^{4}$ spectrum to the Loitsyansky integral and showed that pressure-induced long-range velocity correlations complicate the picture, while Saffman (1967) showed that initial conditions carrying a nonzero total turbulent linear impulse (the volume integral of the velocity field, which is time-independent in an unbounded flow) yield instead $E(k \to 0) \propto k^{2}$, protected by a different invariant — the Saffman integral, the integral of the two-point velocity correlation over all spatial separations, proportional to the mean square of that impulse — with a different decay law $u^{2} \sim t^{-6/5}$ versus the classical $t^{-10/7}$ of the $k^{4}$ class. The modern picture is therefore that the low-wavenumber spectral shape — and hence the decay exponent — is set by the initial conditions and is not universal: large-box direct numerical simulation (DNS, i.e. numerical solution of the full Navier-Stokes equations without turbulence modeling) by Ishida, Davidson & Kaneda (2006) finds the Loitsyansky integral approximately conserved and $u^{2} \sim t^{-10/7}$ for the $k^{4}$ class, and a dedicated experiment by Gorce & Falcon (2024) realizes the Saffman class with self-similar $k^{2}$ decay. At the same time, the dissipation coefficient $C_{\epsilon}$ in the law $\epsilon = C_{\epsilon} u^{3}/\ell$ (with $\epsilon$ the energy dissipation rate per unit mass), long treated as a universal constant of decaying turbulence, was shown by the Vassilicos school (Mazellier & Vassilicos 2008; Valente & Vassilicos 2012; Nedić, Vassilicos & Ganapathisubramani 2013; Dairay, Obligado & Vassilicos 2015) to depend on the initial large-scale conditions and to evolve during decay ('non-equilibrium' dissipation scaling), which feeds directly back into the decay exponents; self-preservation analyses by George (1992) and Djenidi & Antonia (2015) likewise find the exponent tied to the large-scale spectral shape. What no derivation yet supplies is the full quantitative map already requested in 1948: given $E(k,0)$, predict $E(k,t)$, the exponent $n$, the growth of the largest eddies, and the time at which each decay regime gives way to the next.

Problem Statement

Given the initial energy spectrum $E(k,0)$ of freely decaying homogeneous isotropic turbulence at high Reynolds number — in particular the shape of its small-wavenumber (largest-eddy) part and the associated large-scale structure — determine, from the equations of motion rather than from constants fitted to the decay data, and with every closure or invariance assumption made explicit: (1) the subsequent evolution $E(k,t)$ and the asymptotic energy-decay law $u^{2} \sim t^{-n}$, including the exponent $n$ for each admissible initial-spectral class and which features of the decay, if any, are universal; (2) which small-wavenumber spectral classes are dynamically admissible and which large-scale invariants (the Saffman integral, the Loitsyansky integral, or others) are conserved, drift, or are broken during decay; (3) the growth law of the integral length scale, i.e. how the largest eddies evolve and how they interact with the energy cascade; and (4) the time, as a function of the initial conditions, at which each decay regime gives way to the next — in the original 1948 formulation, the cross-over from the initial $1/t$ similarity decay to a faster law, and eventually to the final period with $u^{2} \sim t^{-5/2}$ for the $k^{4}$ class or $t^{-3/2}$ for the $k^{2}$ class. These four parts are facets of a single derivation rather than four independent problems: a complete solution of (1) that determines $E(k,t)$ also determines the integral-scale growth (3), the regime-transition times (4), and the admissible spectral classes together with their invariants (2). The problem was originally posed within Heisenberg's eddy-viscosity closure, where the same questions appeared as the demand to derive the constant $\kappa \approx 0.8$ theoretically, to justify the accuracy of the paper's approximate spectral solutions, and to predict the transition time; that closure is now superseded. The extension to inhomogeneous or anisotropic turbulence, raised in the same 1948 paper, is a field-scale program and not a completion condition of this problem.

References

1. W. Heisenberg (1948), 'On the theory of statistical and isotropic turbulence', Proceedings of the Royal Society of London A 195, 402-406. DOI: 10.1098/rspa.1948.0127, https://doi.org/10.1098/rspa.1948.0127
2. W. Heisenberg (1948), 'Zur statistischen Theorie der Turbulenz', Zeitschrift für Physik 124, 628-657. DOI: 10.1007/BF01668899, https://doi.org/10.1007/BF01668899
3. G. K. Batchelor and A. A. Townsend (1948), 'Decay of isotropic turbulence in the initial period', Proceedings of the Royal Society of London A 193, 539-558. DOI: 10.1098/rspa.1948.0061, https://doi.org/10.1098/rspa.1948.0061
4. G. K. Batchelor and A. A. Townsend (1948), 'Decay of turbulence in the final period', Proceedings of the Royal Society of London A 194, 527-543. DOI: 10.1098/rspa.1948.0095, https://doi.org/10.1098/rspa.1948.0095
5. G. K. Batchelor (1948), 'Energy decay and self-preserving correlation functions in isotropic turbulence', Quarterly of Applied Mathematics 6, 97-116. DOI: 10.1090/qam/28162, https://doi.org/10.1090/qam/28162
6. G. K. Batchelor (1949), 'The role of big eddies in homogeneous turbulence', Proceedings of the Royal Society of London A 195, 513-532. DOI: 10.1098/rspa.1949.0007, https://doi.org/10.1098/rspa.1949.0007
7. S. Chandrasekhar (1949), 'On Heisenberg's elementary theory of turbulence', Proceedings of the Royal Society of London A 200, 20-33. DOI: 10.1098/rspa.1949.0156, https://doi.org/10.1098/rspa.1949.0156
8. I. Proudman and G. K. Batchelor (1951), 'A comparison of Heisenberg's spectrum of turbulence with experiment', Mathematical Proceedings of the Cambridge Philosophical Society 47, 158-176. DOI: 10.1017/S0305004100026463, https://doi.org/10.1017/S0305004100026463
9. I. Proudman and W. H. Reid (1954), 'On the decay of a normally distributed and homogenous turbulent velocity field', Philosophical Transactions of the Royal Society of London A 247, 163-189. DOI: 10.1098/rsta.1954.0016, https://doi.org/10.1098/rsta.1954.0016
10. G. K. Batchelor and I. Proudman (1956), 'The large-scale structure of homogenous turbulence', Philosophical Transactions of the Royal Society of London A 248, 369-405. DOI: 10.1098/rsta.1956.0002, https://doi.org/10.1098/rsta.1956.0002
11. P. G. Saffman (1967), 'The large-scale structure of homogeneous turbulence', Journal of Fluid Mechanics 27, 581-593. DOI: 10.1017/S0022112067000552, https://doi.org/10.1017/S0022112067000552
12. W. K. George (1992), 'The decay of homogeneous isotropic turbulence', Physics of Fluids A 4, 1492-1509. DOI: 10.1063/1.858423, https://doi.org/10.1063/1.858423
13. T. Ishida, P. A. Davidson and Y. Kaneda (2006), 'On the decay of isotropic turbulence', Journal of Fluid Mechanics 564, 455-475. DOI: 10.1017/S0022112006001625, https://doi.org/10.1017/S0022112006001625
14. T. Tatsumi (2012), 'Statistical theories versus statistical mechanics of homogeneous isotropic turbulence', Journal of Turbulence 13, N31. DOI: 10.1080/14685248.2012.702910, https://doi.org/10.1080/14685248.2012.702910
15. N. Mazellier and J. C. Vassilicos (2008), 'The turbulence dissipation constant is not universal because of its universal dependence on large-scale flow topology', Physics of Fluids 20, 015101. DOI: 10.1063/1.2832778, https://doi.org/10.1063/1.2832778
16. P. C. Valente and J. C. Vassilicos (2012), 'Universal Dissipation Scaling for Nonequilibrium Turbulence', Physical Review Letters 108, 214503. DOI: 10.1103/PhysRevLett.108.214503, https://doi.org/10.1103/PhysRevLett.108.214503
17. J. Nedić, J. C. Vassilicos and B. Ganapathisubramani (2013), 'Axisymmetric Turbulent Wakes with New Nonequilibrium Similarity Scalings', Physical Review Letters 111, 144503. DOI: 10.1103/PhysRevLett.111.144503, https://doi.org/10.1103/PhysRevLett.111.144503
18. L. Djenidi and R. A. Antonia (2015), 'A general self-preservation analysis for decaying homogeneous isotropic turbulence', Journal of Fluid Mechanics 773, 345-365. DOI: 10.1017/jfm.2015.250, https://doi.org/10.1017/jfm.2015.250
19. T. Dairay, M. Obligado and J. C. Vassilicos (2015), 'Non-equilibrium scaling laws in axisymmetric turbulent wakes', Journal of Fluid Mechanics 781, 166-195. DOI: 10.1017/jfm.2015.493, https://doi.org/10.1017/jfm.2015.493
20. M. Sinhuber, E. Bodenschatz and G. P. Bewley (2015), 'Decay of Turbulence at High Reynolds Numbers', Physical Review Letters 114, 034501. DOI: 10.1103/PhysRevLett.114.034501, https://doi.org/10.1103/PhysRevLett.114.034501
21. M. Meldi and P. Sagaut (2017), 'Turbulence in a box: quantification of large-scale resolution effects in isotropic turbulence free decay', Journal of Fluid Mechanics 818, 697-715. DOI: 10.1017/jfm.2017.158, https://doi.org/10.1017/jfm.2017.158
22. J.-B. Gorce and E. Falcon (2024), 'Freely Decaying Saffman Turbulence Experimentally Generated by Magnetic Stirrers', Physical Review Letters 132, 264001. DOI: 10.1103/PhysRevLett.132.264001, https://doi.org/10.1103/PhysRevLett.132.264001
23. N. Zambrano and K. Duraisamy (2026), 'Two-point turbulence closures in physical space', Journal of Fluid Mechanics 1034, A12. DOI: 10.1017/jfm.2026.11485, https://doi.org/10.1017/jfm.2026.11485 (preprint arXiv:2511.00669, https://arxiv.org/abs/2511.00669)
24. A. Rodhiya and K. R. Sreenivasan (2026), 'The asymptotic state of decaying turbulence', Philosophical Transactions of the Royal Society A 384, 20250021. DOI: 10.1098/rsta.2025.0021, https://doi.org/10.1098/rsta.2025.0021
25. A. Migdal (2026), 'Geometric solution of turbulence as diffusion in loop space', Philosophical Transactions of the Royal Society A 384, 20250032. DOI: 10.1098/rsta.2025.0032, https://doi.org/10.1098/rsta.2025.0032