Inertial-range power-law scaling: admissibility constraints on anomalous scaling exponents from moment inequalities and the four-fifths law

In homogeneous isotropic turbulence, the inertial-range ansatz $\overline{(\delta u)^n}\sim r^{\zeta_n}$ for the longitudinal velocity structure functions, with $\zeta_3=1$ fixed by Kolmogorov's four-fifths law, underpins both the Kolmogorov 1941 (K41) exponents $\zeta_n=n/3$ and all intermittency-corrected (Kolmogorov 1962, K62-type) exponent sets. A 2021 Letter argued that Hölder's inequality forces $\zeta_{2p}=p\zeta_2$ and, with the four-fifths law, $\zeta_2=2/3$, so that anomalous exponents are inadmissible and most probably finite-Reynolds-number artifacts; that derivation was disputed by a published Comment and reframed by its authors as a 'plausibility' argument in 2023. The open problem is the admissibility question this dispute leaves unresolved: determine the exact set of constraints that moment inequalities together with $\zeta_0=0$ and $\zeta_3=1$ impose on $\{\zeta_n\}$, in particular whether corrections $\alpha_n\neq0$ to $n/3$ are excluded, and adjudicate the disputed claims $\zeta_{2p}=p\zeta_2$, $\zeta_2\le2/3$ and $\zeta_2=2/3$. The question remains open: no published Reply or independent adjudication of the dispute has appeared, and high-Reynolds-number experiments and simulations through 2026 continue to report anomalous exponents whose admissibility is exactly what is contested.

Background

Homogeneous isotropic turbulence (HIT) is the idealized state of a turbulent flow whose statistics are invariant under translations, rotations and reflections of the coordinate system; it is approximated by wind-tunnel grid turbulence and by direct numerical simulation (DNS) in periodic boxes. The central objects of its small-scale theory are the longitudinal velocity structure functions: for a separation $r$ along the mean-flow direction, $\delta u=u(x+r)-u(x)$ is the longitudinal increment and $S_n(r)=\overline{(\delta u)^n}$ is its $n$-th moment, the overbar denoting an ensemble (or long-time) average. Kinetic energy is transferred from the large scales, of size the integral length scale $L$, down to the dissipative scales of size the Kolmogorov length scale $\eta=(\nu^3/\bar\epsilon)^{1/4}$ ($\nu$ is the kinematic viscosity and $\bar\epsilon$ the mean dissipation rate per unit mass); the inertial range is the intermediate range $\eta\ll r\ll L$ in which both viscosity and the details of forcing are supposed to be irrelevant.

Kolmogorov's 1941 theory (K41) predicts, in the limit of infinite Reynolds number, the self-similar laws $S_n(r)=C_n(\bar\epsilon r)^{n/3}$, i.e. $\zeta_n=n/3$. One of these is exact: the four-fifths law $S_3(r)=-\frac{4}{5}\bar\epsilon r$, derived from the Kármán--Howarth equation (the exact budget equation for the two-point velocity correlation) under local homogeneity and local isotropy, so that $\zeta_3=1$ is the one exponent fixed rigorously by the Navier--Stokes equations. Kolmogorov's 1962 refinement (K62) and subsequent intermittency models replace $n/3$ by undetermined exponents $\zeta_n$ to account for the spatio-temporal intermittency of the dissipation rate; the resulting anomalous exponents (measured values near $\zeta_2\approx0.70$, $\zeta_4\approx1.28$ instead of $2/3$ and $4/3$) are the empirical basis of the multifractal picture of turbulence. In practice exponents are often extracted by extended self-similarity (ESS), in which $S_n$ is plotted against $\overline{|\delta u|^3}$ rather than against $r$; this presumes the absolute third moment scales linearly, $\overline{|\delta u|^3}\sim r$, an assumption for which no exact derivation exists (the four-fifths law governs the signed moment).

Two facts make the status of the exponent set a genuine problem rather than a mere fitting question. First, moment inequalities constrain any exponent set coming from a power-law ansatz: the textbook consequence of Hölder's inequality (see the standard monograph of Frisch) is that $\zeta_n$ must be a concave function of $n$, which together with $\zeta_0=0$ and $\zeta_3=1$ gives $\zeta_2\ge2/3$; whether moment inequalities plus the four-fifths law exclude anomalous exponents altogether is disputed. Second, both K41 and K62 are asymptotic statements for infinite Reynolds number, whereas all measurements are made at finite Reynolds number; the finite-Reynolds-number (FRN) effect, studied since Qian, contaminates scaling estimates. A 2021 Letter by Djenidi, Antonia and Tang sharpened both points: it claimed that Hölder's inequality forces $\zeta_{2p}=p\zeta_2$, that the Cauchy--Schwarz inequality with the four-fifths law gives $\zeta_2\le2/3$, and that the absolute-moment form with $\zeta_3=1$ gives $\zeta_2=2/3$, concluding that intermittency-model exponents 'most probably' reflect finite-Reynolds-number contamination and calling for new theory consistent with these constraints. A subsequent Comment by Kopyev, Il'yn, Sirota and Zybin argued that the proof contains a technical error, and the authors' 2023 follow-up restyled the relations as 'plausibility constraints' while maintaining that $\zeta_n=n/3+\alpha_n$ is implausible unless $\alpha_n=0$. Meanwhile the FRN literature (the 'dual scaling' framework of Tang, Antonia and Djenidi, and EDQNM closure calculations, where EDQNM denotes the eddy-damped quasi-normal Markovian spectral closure model) indicates that an inertial range in which both the power laws and the four-fifths law hold is established, if at all, only at Taylor-scale Reynolds numbers $\mathrm{Re}_\lambda$ (the Reynolds number based on the Taylor microscale $\lambda=(\overline{u^2}/\overline{(\partial u/\partial x)^2})^{1/2}$, the length scale formed from the velocity variance and its mean-square streamwise gradient) of order $10^4$--$10^5$, beyond current experiments and DNS, while high-Reynolds-number DNS and shear-flow experiments continue to report anomalous exponents coexisting with a satisfied four-fifths law. The question of what the inertial-range framework actually admits is therefore open on both its mathematical and its empirical face.

Problem Statement

Determine the exact set of constraints that Hölder- and Cauchy--Schwarz-type inequalities for the moments of $\delta u$, combined with $\zeta_0=0$ and the four-fifths-law value $\zeta_3=1$, impose on the exponent sequence $\{\zeta_n\}$ of the inertial-range power-law ansatz $\overline{(\delta u)^n}=A_n(r/L)^{\zeta_n}$ for $\eta\ll r\ll L$ in homogeneous isotropic turbulence. In particular, decide whether intermittency-corrected (Kolmogorov-1962-type) exponents $\zeta_n=n/3+\alpha_n$ with $\alpha_n\neq0$ are admissible under these constraints, and adjudicate the disputed claims of Djenidi, Antonia and Tang (Phys. Fluids 33, 031703 (2021)) that Hölder's inequality applied to the power-law form gives $\zeta_{2p}=p\zeta_2$, that Cauchy--Schwarz applied to $|\overline{(\delta u)^3}|$ with $\zeta_3=1$ gives $\zeta_2\le2/3$, and that Hölder's inequality applied to the absolute-moment power-law form while assuming $\zeta_3=1$ gives $\zeta_2=2/3$, against the objection of Kopyev, Il'yn, Sirota and Zybin (Phys. Fluids 33, 119101 (2021)) that the proof contains a technical error rendering that conclusion ungrounded. All of these parts are required: the exact constraint set is the primary deliverable, and the admissibility question, the three disputed claims and the Comment's objection must also be adjudicated explicitly, each as a distinct verdict.

References

1. L. Djenidi, R. A. Antonia, and S. L. Tang, 'Mathematical constraints on the scaling exponents in the inertial range of fluid turbulence', Physics of Fluids 33(3), 031703 (2021). DOI: 10.1063/5.0039643, https://doi.org/10.1063/5.0039643
2. A. V. Kopyev, A. S. Il'yn, V. A. Sirota, and K. P. Zybin, 'Comment on "Mathematical constraints on the scaling exponents in the inertial range of fluid turbulence" [Phys. Fluids 33(3), 031703 (2021)]', Physics of Fluids 33(11), 119101 (2021). DOI: 10.1063/5.0070027, https://doi.org/10.1063/5.0070027
3. L. Djenidi, R. A. Antonia, and S. L. Tang, 'Scaling of turbulent velocity structure functions: plausibility constraints', Journal of Fluid Mechanics 965, A14 (2023). DOI: 10.1017/jfm.2023.416, https://doi.org/10.1017/jfm.2023.416
4. L. Djenidi, R. A. Antonia, and S. L. Tang, 'Scale invariance in finite Reynolds number homogeneous isotropic turbulence', Journal of Fluid Mechanics 864, 244-272 (2019). DOI: 10.1017/jfm.2019.28, https://doi.org/10.1017/jfm.2019.28
5. R. A. Antonia, S. L. Tang, L. Djenidi, and Y. Zhou, 'Finite Reynolds number effect and the 4/5 law', Physical Review Fluids 4, 084602 (2019). DOI: 10.1103/PhysRevFluids.4.084602, https://doi.org/10.1103/PhysRevFluids.4.084602
6. M. Meldi, L. Djenidi, and R. A. Antonia, 'Sensitivity analysis of the second and third-order velocity structure functions to the Reynolds number in decaying and forced isotropic turbulence using the EDQNM model', European Journal of Mechanics B/Fluids 88, 229-242 (2021). DOI: 10.1016/j.euromechflu.2021.04.003, https://doi.org/10.1016/j.euromechflu.2021.04.003
7. S. L. Tang, R. A. Antonia, and L. Djenidi, 'Dual scaling and the n-thirds law in grid turbulence', Journal of Fluid Mechanics 975, A32 (2023). DOI: 10.1017/jfm.2023.888, https://doi.org/10.1017/jfm.2023.888
8. S. Tang, L. Danaila, and R. A. Antonia, 'Finite Reynolds Number Effect on Small-Scale Statistics in Decaying Grid Turbulence', Atmosphere 15(5), 540 (2024). DOI: 10.3390/atmos15050540, https://doi.org/10.3390/atmos15050540
9. K. P. Iyer, K. R. Sreenivasan, and P. K. Yeung, 'Scaling exponents saturate in three-dimensional isotropic turbulence', Physical Review Fluids 5, 054605 (2020). DOI: 10.1103/PhysRevFluids.5.054605, https://doi.org/10.1103/PhysRevFluids.5.054605, arXiv:2002.11900, https://arxiv.org/abs/2002.11900
10. C. Küchler, G. P. Bewley, and E. Bodenschatz, 'Universal Velocity Statistics in Decaying Turbulence', Physical Review Letters 131, 024001 (2023). DOI: 10.1103/PhysRevLett.131.024001, https://doi.org/10.1103/PhysRevLett.131.024001
11. D. Gupta and G. P. Bewley, 'Experimental Evidence for Longitudinal Scaling Exponent Saturation in Shear Turbulence', arXiv:2605.01867 (2026). https://arxiv.org/abs/2605.01867
12. W. D. McComb, S. R. Yoffe, M. F. Linkmann, and A. Berera, 'Spectral analysis of structure functions and their scaling exponents in forced isotropic turbulence', Physical Review E 90, 053010 (2014). DOI: 10.1103/PhysRevE.90.053010, https://doi.org/10.1103/PhysRevE.90.053010
13. S. R. Yoffe and W. McComb, 'Does intermittency affect the inertial transfer rate in stationary isotropic turbulence?', arXiv:2107.09112 (2021). https://arxiv.org/abs/2107.09112
14. Y. Xie, K. Wu, X.-L. Xiong, K. Nagata, T. Watanabe, and Y. Zhou, 'Finite-Reynolds-number effects on the structure functions in turbulent flow with external intermittency', Journal of Fluid Mechanics 1037, A46 (2026). DOI: 10.1017/jfm.2026.11642, https://doi.org/10.1017/jfm.2026.11642
15. J. Z. Shi, 'Qian Jian (1939-2018) and his contribution to small-scale turbulence studies', Physics of Fluids 33, 041301 (2021). DOI: 10.1063/5.0043566, https://doi.org/10.1063/5.0043566
16. R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli, and S. Succi, 'Extended self-similarity in turbulent flows', Physical Review E 48, R29-R32 (1993). DOI: 10.1103/PhysRevE.48.R29, https://doi.org/10.1103/PhysRevE.48.R29
17. U. Frisch, 'Turbulence: The Legacy of A. N. Kolmogorov', Cambridge University Press (1995). DOI: 10.1017/CBO9781139170666, https://doi.org/10.1017/CBO9781139170666
18. A. N. Kolmogorov, 'The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers', Proceedings of the Royal Society of London A 434, 9-13 (1991 translation of the 1941 Dokl. Akad. Nauk SSSR article). DOI: 10.1098/rspa.1991.0075, https://doi.org/10.1098/rspa.1991.0075
19. A. N. Kolmogorov, 'Dissipation of energy in the locally isotropic turbulence', Proceedings of the Royal Society of London A 434, 15-17 (1991 translation of the 1941 Dokl. Akad. Nauk SSSR article). DOI: 10.1098/rspa.1991.0076, https://doi.org/10.1098/rspa.1991.0076
20. A. N. Kolmogorov, 'A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number', Journal of Fluid Mechanics 13, 82-85 (1962). DOI: 10.1017/S0022112062000518, https://doi.org/10.1017/S0022112062000518