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Intermittent fluctuations determine the nature of chaos in turbulence

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Chaos in fully developed turbulence scales as Re^0.59, set by intermittent strain fluctuations.

desk verdict Reports a clean empirical case for Re^0.59 Lyapunov scaling with a plausible intermittency mechanism, but the causal step rests on an untested delta-correlation ansatz; worth refereeing. read the letter →

arxiv 2505.09538 v1 pith:2BUTUW5A submitted 2025-05-14 physics.flu-dyn cond-mat.stat-mechnlin.CD

Aikya Banerjee , Ritwik Mukherjee , Sugan Durai Murugan , Subhro Bhattacharjee , Samriddhi Sankar Ray This is my paper · ORCID
classification physics.flu-dyncond-mat.stat-mechnlin.CD MSC 76F2076F0537D45
keywords LyapunovexponentturbulenceintermittencyReynolds-numberscalingstrain-ratetensordecorrelatorshellmodelchaos
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the rate of chaos in fully developed turbulence—the largest Lyapunov exponent $\lambda$—grows with Reynolds number as $\lambda \propto \mathrm{Re}^{0.59 \pm 0.04}$, not as the long-standing mean-field $\sqrt{\mathrm{Re}}$ scaling. The exponent is set not by the average strain rate but by the intermittent fluctuations of the velocity gradient, specifically the standard deviation of the most compressive strain-rate eigenvalue. The same scaling emerges from direct numerical simulations of the Navier–Stokes equation, from a local cascade model, and from a short-time theory of the decorrelator, covering almost seven decades in Reynolds number. If correct, it settles the long disagreement between multifractal estimates near $\mathrm{Re}^{0.459}$ and simulations reporting values around $\mathrm{Re}^{0.53}$.

What carries the argument

The load-bearing object is the integrated decorrelator $\Phi(t) = \frac{1}{V}\int d\mathbf{x}\, |\delta\mathbf{u}(\mathbf{x},t)|^2/2$ for two nearly identical velocity fields, whose evolution reads $d\Phi/dt = \beta_S + \beta_\eta$, with $\beta_S = -\langle\delta\mathbf{u}\cdot S \cdot \delta\mathbf{u}\rangle$ from the strain tensor and $\beta_\eta = \nu\langle\delta\mathbf{u}\cdot\nabla^2 \delta\mathbf{u}\rangle$ from dissipation. The strain term is expanded in the eigenbasis of $S$, giving $\Gamma(\mathbf{x},t) \approx \gamma_3 \cos^2\theta_3$, and after writing $\Phi(t)$ via an integrating factor $\Theta$ the short-time exponential growth is obtained from a cumulant expansion $\Phi(t) \lesssim \Phi(0)\exp(-\langle\Delta\Theta\rangle + \tfrac{1}{2}\langle\Delta\Theta^2\rangle)$. Assuming approximate delta-correlation in time gives $\langle\Delta\Theta^2\rangle \propto \gamma_3^{\mathrm{std}}$, which carries the $\mathrm{Re}^{0.59}$ scaling and dominates over $\langle\Delta\Theta\rangle \sim \sqrt{\mathrm{Re}}$ in the high-Reynolds limit.

What would settle it

Measure the standard deviation of the compressive strain eigenvalue $\gamma_3^{\mathrm{std}}$ and the Lyapunov exponent $\lambda$ over a wide Reynolds-number range in the same flow (or in a shell model) and test whether $\lambda/\gamma_3^{\mathrm{std}}$ is constant and whether both scale with the same exponent; a measurably Reynolds-dependent ratio, or a change in the $\gamma_3$ exponent when the flow's intermittency is suppressed by filtering, would falsify the identification.

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Extended reading notes

Core claim

The central discovery is an identity: $\lambda = \lambda_{\mathrm{DNS}} = \gamma_3^{\mathrm{std}} \sim \mathrm{Re}^{0.59}$, where $\gamma_3^{\mathrm{std}}$ is the standard deviation of the most compressive eigenvalue of the strain-rate tensor, while the mean compressive eigenvalue obeys $-\langle\gamma_3\rangle \sim \sqrt{\mathrm{Re}}$. The measured Lyapunov exponent coincides with the fluctuation amplitude of the strain field, not with its mean, so the departure from mean-field chaos is a direct signature of intermittency. The paper derives this by decomposing the strain term in the decorrelator equation into the eigendirections of the strain tensor, Taylor-expanding the short-time integrated decorrelator to second order, and mapping the accumulated strain to the variance of the compressive eigenvalue.

Load-bearing premise

The derivation assumes that over short times the compressive strain fluctuations are effectively delta-correlated in time, so that the variance of the accumulated strain, $\langle\Delta\Theta^2\rangle$, is directly proportional to the standard deviation $\gamma_3^{\mathrm{std}}$; if that temporal decorrelation fails, the claimed $\mathrm{Re}^{0.59}$ derivation collapses even though the measured scaling may stand.

Editorial extensions

If this is right

  • The Lyapunov exponent can be read off from single-time statistics of the strain-rate tensor: measure the standard deviation of the compressive eigenvalue and you know $\lambda$.
  • The multifractal prediction $\lambda \sim \mathrm{Re}^{0.459}$ is superseded: intermittency enters not only through a spread of Hölder exponents but through the variance of the strain eigenvalue itself.
  • The same decorrelator machinery transfers from Hamiltonian systems to driven-dissipative fluids, giving a general route to chaos exponents in steady non-equilibrium flows.
  • Because the scaling survives in a local shell model, the large-Reynolds result does not depend on the non-local pressure interaction; locality suffices for the intermittent origin of $\alpha$.
  • At lower Reynolds numbers the mean-field $\sqrt{\mathrm{Re}}$ term should dominate, predicting a crossover in $\lambda(\mathrm{Re})$ that could be tested directly.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the identity $\lambda = \gamma_3^{\mathrm{std}}$ holds generally, then any mechanism that changes the intermittency of the compressive strain—rotation, stratification, magnetic fields—should change the chaos exponent in a predictable way, making $\lambda$ a diagnostic of intermittency rather than a separate quantity.
  • The delta-correlation Ansatz implies a specific testable prediction: the two-time correlation of the compressive strain eigenvalue should decay on a time scale independent of Reynolds number (in Kolmogorov units); if that correlation time grows with Reynolds number, the derivation overestimates $\langle\Delta\Theta^2\rangle$ and the exponent would change.
  • The crossover to mean-field $\sqrt{\mathrm{Re}}$ at low Reynolds number could be mapped by measuring $\lambda$ and $\gamma_3^{\mathrm{std}}$ for $\mathrm{Re}\approx 10\text{–}50$; the paper's data start at $\mathrm{Re}=50$, so this crossover is untested.
  • One could test the mechanism without solving dynamics: in any steady turbulent flow with measured strain statistics, compute $\gamma_3^{\mathrm{std}}$ and predict $\lambda$; a discrepancy would point to nonlocal contributions beyond the shell model.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the Reynolds-number dependence of the largest Lyapunov exponent in forced three-dimensional turbulence, claiming λ ∝ Re^α with α = 0.59 ± 0.04. Using twin simulations of the Navier-Stokes equation and of the GOY shell model, the authors construct a spatially integrated velocity-difference decorrelator Φ, observe a short-time exponential-growth phase and a later saturation at 2E0, and extract Lyapunov exponents from the exponential phase. They report that their measured exponents are consistent with the standard deviation of the compressional eigenvalue of the strain-rate tensor, γ_3^std, while the mean eigenvalue −⟨γ3⟩ follows the mean-field Ruelle scaling √Re. The central mechanistic claim is that intermittent strain fluctuations, quantified by γ_3^std, set the Lyapunov exponent and explain the departure from √Re. Appendix B develops this connection through a cumulant expansion of the decorrelator, and the shell-model simulations are used to extend the claimed scaling to higher Reynolds numbers.

Significance. If established, the central claim would provide a microscopically motivated explanation for the long-standing discrepancy between Ruelle's mean-field result λ ∝ √Re and numerical/experimental exponents α ≲ 0.6, and it would directly connect turbulence intermittency with chaos. The paper has several genuine strengths: the short-time exponential growth and long-time saturation of the decorrelator are convincingly demonstrated in Figs. 1–3; the long-time balance between the viscous contribution βη → −2⟨ε⟩ and the forcing contribution is clean and internally consistent; and the shell-model computation provides an independent numerical model that reproduces the same empirical exponent. The authors are also transparent that the key theoretical step relies on an Ansatz, which is to their credit. However, the load-bearing theoretical link between intermittency and α is currently a consistency relation rather than a derivation, and the causality implied by the title is not yet supported by the evidence presented.

major comments (3)
  1. [Appendix B, Eq. (B-10)] The mechanistic conclusion rests on two uncontrolled approximations. The step from Eq. (B-9) to Eq. (B-10) truncates the cumulant expansion at second order without an estimate of higher cumulants, and the very next step invokes 'a further Ansatz of an approximate delta-correlation in time' to conclude ⟨∆Θ²⟩ ∝ γ_3^std. Neither approximation is tested anywhere in the manuscript. Since γ_3^std is measured from the same DNS runs that produce λ, the agreement in Fig. 4 is a consistency relation within the same dataset, not an independent confirmation that intermittency causes the Re^0.59 scaling. To make the causal claim load-bearing, the authors should either estimate the correlation time of Γ and check whether ⟨∆Θ²⟩ computed directly from the DNS strain history agrees with the delta-correlation Ansatz, or provide evidence that the fourth cumulant is subdominant. Without such a test, the title's assertion that intermittent fluctuations 'determine' the nature of chaos is not established.
  2. [Fig. 4 and Appendix B] The exponent α = 0.59 is not derived from first principles; it is the measured slope of γ_3^std versus Re, and λ is then shown to share that slope. The paper therefore explains the departure from √Re only in the sense of identifying a fluctuating strain statistic with the same empirical exponent; it does not explain why γ_3^std itself scales as Re^0.59. The sentences in the main text stating that 'the dominant scaling in the statistics of γ3 is due to γ_3^std ∼ Re^α' and that the departure 'must stem from intermittent fluctuations' should be reframed as a conjecture supported by data, or the paper should provide a separate argument fixing α. This is not a fatal flaw if the paper is read as reporting an empirical law, but the current text overstates the theoretical content.
  3. [Fig. 4 and Sec. GOY model, Appendix A.2] The claim that the scaling holds for nearly seven decades relies on rescaling λ_GOY by an arbitrary constant factor and on combining two disjoint Reynolds-number ranges, roughly 50–1400 from DNS and 10^7–10^8 from the shell model. The shell model is a local, phenomenological cascade model, and the connection between its Lyapunov exponent and the strain-eigenvalue mechanism derived in Appendix B is not made. The shell-model data therefore provide a useful consistency check of the exponent, but they cannot validate the specific Navier-Stokes intermittency mechanism, and the arbitrary rescaling means that the seven-decade statement is not a parameter-free prediction. This limitation should be stated explicitly when the universal-scaling claim is made.
minor comments (5)
  1. [Fig. 4 caption] The word 'Lypanov' should be 'Lyapunov'.
  2. [Equation (C-1)] The first bracket in Eq. (C-1) contains the term δu_{n+2}u^A_{n+1} twice; this appears to be a typographical duplication from the derivation of the shell-model perturbation equation.
  3. [Main text, paragraph containing Fig. 4] The equality 'λ = λ_DNS = λ_GOY = γ_3^std' is stated without qualification, but the GOY exponents are rescaled by a constant and the DNS exponents are extracted from different procedures; the statement should read 'consistent up to error bars and model-dependent prefactors.'
  4. [Inset of Fig. 1] The notation 'n^2_i direction cosines' is unclear; it should be 'the squared direction cosines n_i^2' or similar.
  5. [Conclusions] The final paragraph mentions power-law tails in the local exponent distributions and states that their origin will not be explored; since these tails are used as qualitative evidence of intermittency, a sentence clarifying that they are not part of the quantitative mechanism would help the reader distinguish evidence from illustration.

Circularity Check

1 steps flagged · score 6.0 of 10

The Lyapunov exponent scaling alpha = 0.59 is imported from the measured gamma_3^std scaling in the same DNS runs, making the intermittency mechanism a consistency check rather than a parameter-free prediction.

  1. fitted input called prediction [Appendix B, paragraph after Eq. (B-10); main text discussion of Fig. 4]
    "The determination of ⟨∆Θ2⟩ requires a further Ansatz of an approximate delta-correlation in time, at least over time-scales of the order associated with strain-rate, to yield ⟨∆Θ2⟩∝ γstd_3 ∼ Re0.59 ... Hence λ =λDNS∼λS∼ Reα with α = 0.59± 0.04. ... In Fig. 4 we show a plot of γstd_3 from our DNSs and find a near perfect agreement with the Lyapunov exponent confirming ... λ = λDNS = γstd_3 ∼ Re0.59."

    The derivation stops at the point where the claimed result is inserted: γstd_3 ∼ Re0.59 is the measured scaling of the strain eigenvalue fluctuation from the same DNS fields that produce λ_DNS (Fig. 4). The cumulant truncation in Eq. (B-10) and the unverified delta-correlation ansatz convert this measured scaling into the reported Lyapunov scaling. No independent computation of α is offered; the theoretical curve is the measured strain-statistic scaling relabeled as λ. The agreement in Fig. 4 therefore tests internal consistency between two quantities from the same dataset, not a parameter-free prediction of the intermittency mechanism.

full rationale

Most of the paper is not circular by self-citation: the decorrelator evolution is re-derived from the Navier-Stokes equation in Appendix B, and the GOY shell model provides an independent though local consistency check. The circular content is concentrated in the step from Eq. (B-10) to α = 0.59. Eq. (B-10) itself is a cumulant expansion that predicts no Reynolds-number exponent; the exponent enters only through the assertion that ⟨ΔΘ²⟩ ∝ γstd_3 ∼ Re0.59, and that scaling is read off the same DNS data from which λ_DNS is extracted. Thus the paper does not derive α from first principles; it imports the measured γstd_3 scaling and then reports the observed λ scaling as confirmation. The equality λ ≈ γstd_3 is a nontrivial empirical relation, and the claim that fluctuations rather than the mean set the Lyapunov scale has independent content, so this is partial circularity rather than a vacuous paper. However, the central claim that intermittency determines α = 0.59 is a fitted input called a prediction, and the delta-correlation and second-cumulant truncation assumptions on which it rests are never independently tested.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper contributes a new measurement and a mechanistic relation, but the central exponent is fitted from data, and the theoretical link relies on assumptions about the strain-field fluctuations (cumulant truncation, delta correlation) that are not independently justified.

free parameters (3)
  • Lyapunov exponent scaling exponent alpha = 0.59 +/- 0.04
    Obtained by linear fits of log lambda versus log Re for DNS and shell model data (Fig. 4). This is the central fitted value of the paper.
  • lambda_GOY rescaling factor = Not specified (constant factor)
    Main text states lambda_GOY is 'rescaled by a constant factor to overlap with the DNS data since the Reynolds number ranges are different'. This factor affects the offset but not the slope, and it is a free parameter in the composite plot.
  • Scaling exponent of gamma_3_std = 0.59
    The scaling gamma_3_std ~ Re^0.59 is measured from DNS and inserted into the theoretical expression for lambda. It is not derived from first principles.
assumptions (5)
  • ad hoc to paper The second-order cumulant truncation Phi(t) approximately Phi(0) exp[-<Delta_Theta> + (1/2)<Delta_Theta^2>] adequately captures the exponential growth rate.
    Appendix B Eq. (B-10); no small parameter justifies dropping higher cumulants, and the error is uncontrolled.
  • ad hoc to paper Delta_Theta(t) is delta-correlated in time at strain-rate timescales, allowing <Delta_Theta^2> proportional to gamma_3_std.
    Appendix B: 'The determination of <Delta_Theta^2> requires a further Ansatz of an approximate delta-correlation in time, at least over time-scales of the order associated with strain-rate.'
  • domain assumption The GOY shell model reproduces the same Lyapunov-exponent scaling as the Navier-Stokes equation despite its local interactions.
    Used in Fig. 4 to extend the scaling to Reynolds numbers near 10^8; the shell model is a simplified cascade model and is not derived from the Navier-Stokes equation.
  • domain assumption The flow is incompressible, homogeneous, isotropic, and in a statistically steady state maintained by constant energy injection.
    Standard setup described in Appendix A and the main text; required for spatial averaging and for the decorrelator formalism.
  • domain assumption The localized perturbation delta_u_0 is small enough that the linearized evolution governs the exponential-growth phase.
    epsilon_0 = 10^-5 for DNS and 10^-3 for shell model; required for the measured lambda to be the true largest Lyapunov exponent.

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Cite this review

Pith. "Pith review of Intermittent fluctuations determine the nature of chaos in turbulence." pith.science (2026). https://pith.science/paper/2BUTUW5A

@misc{pith2026250509538,
  author       = {Pith},
  title        = {Pith review of: Intermittent fluctuations determine the nature of chaos in turbulence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2BUTUW5A}},
  note         = {Machine review of arXiv:2505.09538}
}
abstract

We adapt recent ideas for many-body chaos in nonlinear, Hamiltonian fluids [Murugan \textit{et al.}, Phys. Rev. Lett. 127, 124501 (2021)] to revisit the question of the Reynolds number Re dependence of the Lyapunov exponent $\lambda\propto{\rm Re}^\alpha$ in fully developed turbulence. The use of such decorrelators allow us to investigate the interplay of the competing effects of viscous dissipation and nonlinearity. We obtain a precise value of $\alpha = 0.59 \pm 0.04$ and show that departure from the Kolmogorov mean field result $\lambda \propto \sqrt{{\rm Re}}$ is a consequence of the intermittent fluctuations in the velocity-gradient tensor. The robustness of our results are further confirmed in a local, dynamical systems model for turbulence.

Figures

Figures reproduced from arXiv: 2505.09538 by the authors.

Figure 1
Figure 1. FIG. 1. Representative plots of the normalised spatially av [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Semilog plots of Φ vs [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Semilog plot of the shell model decorrelator Φ [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. A loglog plot of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Plots of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Uncertainty Growth in Stably Stratified Turbulence

    physics.flu-dyn 2025-12 conditional novelty 6.0 of 10

    In forced stratified turbulence, increasing stratification monotonically lowers the Lyapunov exponent and slows vertical uncertainty spread, while the decay–growth–saturation sequence persists.

Reference graph

Works this paper leans on

35 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [1]

    Pandit, P

    R. Pandit, P. Perlekar, and S. S. Ray, Pramana 73, 157 (2009)

    work page 2009

  2. [2]

    Frisch, Turbulence: The Legacy of A

    U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, 1995)

    1995

  3. [3]

    Ruelle, Physics Letters A 72, 81 (1979)

    D. Ruelle, Physics Letters A 72, 81 (1979)

    work page 1979

  4. [4]

    Crisanti, M

    A. Crisanti, M. H. Jensen, A. Vulpiani, and G. Paladin, Phys. Rev. Lett. 70, 166 (1993)

    1993

  5. [5]

    A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299–303 (1941)

    work page 1941

  6. [6]

    Benzi, G

    R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani, Journal of Physics A: Mathematical and General 17, 3521 (1984)

    work page 1984

  7. [7]

    Frisch and G

    U. Frisch and G. Parisi, Proceedings of the International School of Physics Enrico Fermi, Course LXXXVIII, Varenna, 1985 (1985)

    work page 1985

  8. [8]

    have recently reported α = 0.53. This lack of consen- sus between the theoretical estimate and measurements — as well as the lack of agreement between different sim- ulations [4, 8–13] — suggests that the origins of chaos in turbulence is far from settled. But is there a way to address the microscopic origin of α from the Navier-Stokes equation itself and...

    arXiv 2025

Show all 35 references
  1. [9]

    Berera and R

    A. Berera and R. D. J. G. Ho, Phys. Rev. Lett. 120, 024101 (2018)

    2018

  2. [10]

    Yamada and K

    M. Yamada and K. Ohkitani, Journal of the Physical Society of Japan 56, 4210 (1987)

    1987

  3. [11]

    Mukherjee, J

    S. Mukherjee, J. Schalkwijk, and H. J. J. Jonker, Journal of the Atmospheric Sciences 73, 2715 (2016)

    2016

  4. [12]

    Boffetta and S

    G. Boffetta and S. Musacchio, Phys. Rev. Lett. 119, 054102 (2017)

    2017

  5. [13]

    Mohan, N

    P. Mohan, N. Fitzsimmons, and R. D. Moser, Phys. Rev. Fluids 2, 114606 (2017)

    2017

  6. [14]

    J. Ge, J. Rolland, and J. C. Vassilicos, Journal of Fluid Mechanics 977, A17 (2023)

    2023

  7. [15]

    A. Das, S. Chakrabarty, A. Dhar, A. Kundu, D. A. Huse, R. Moessner, S. S. Ray, and S. Bhattacharjee, Phys. Rev. Lett. 121, 024101 (2018)

    2018

  8. [16]

    Bilitewski, S

    T. Bilitewski, S. Bhattacharjee, and R. Moessner, Phys. Rev. B 103, 174302 (2021)

    2021

  9. [17]

    Bilitewski, S

    T. Bilitewski, S. Bhattacharjee, and R. Moessner, Phys. Rev. Lett. 121, 250602 (2018)

    2018

  10. [18]

    S. D. Murugan, D. Kumar, S. Bhattacharjee, and S. S. Ray, Phys. Rev. Lett. 127, 124501 (2021)

    2021

  11. [19]

    A. G. Lamorgese, D. A. Caughey, and S. B. Pope, Physics of Fluids 17, 015106 (2004)

    2004

  12. [20]

    Mukherjee, R

    S. Mukherjee, R. K. Singh, M. James, and S. S. Ray, Nature Physics 19, 891 (2023)

    2023

  13. [21]

    Supplemental video (2025), spatio-temporal evolution of an initially localised perturbation

    2025

  14. [22]

    Biferale, Annual Review of Fluid Mechanics 35, 441 (2003)

    L. Biferale, Annual Review of Fluid Mechanics 35, 441 (2003)

    2003

  15. [23]

    P. D. Ditlevsen, Turbulence and Shell Models (Cambridge University Press, Cambridge, UK, 2010)

    2010

  16. [24]

    E. B. Gledzer, Soviet Physics Doklady 18, 216 (1973)

    1973

  17. [25]

    Ohkitani and M

    K. Ohkitani and M. Yamada, Progress of Theoretical Physics 81, 329 (1989)

    1989

  18. [26]

    S. H. Shenker and D. Stanford, Journal of High Energy Physics 2014 (2014)

    2014

  19. [27]

    J. Bec, L. Biferale, G. Boffetta, M. Cencini, S. Musacchio, and F. Toschi, Physics of Fluids 18, 091702 (2006)

    2006

  20. [28]

    S. S. Ray, Phys. Rev. Fluids 3, 072601 (2018)

    2018

  21. [29]

    S. S. Ray, D. Mitra, and R. Pandit, New Journal of Physics 10, 033003 (2008). 6 Appendix A: Direct Numerical Simulations

    2008

  22. [30]

    (A-1) on a periodic box of size L = 2π withN3 collocation points

    The Navier-Stokes Equation We perform direct numerical simulations (DNSs), by using a pseudospectral method, of the three-dimensional (3D), incompressible, unit density Navier Stokes equation, with the pressure field P ∂u ∂t + u·∇ u =ν∇2u−∇P + f. (A-1) on a periodic box of siz...

  23. [31]

    with A0 = k0 = 1. The system is driven to a non-equilibrium steady state (NESS) through a large-scale forcing with a constant energy injection at large scales corresponding to wavenumber 1 ≤kforce≤ 2. This statistically steady velocity fieldu is taken as the initial field for ...

  24. [32]

    The Gledzer-Ohkitani-Y amada (GOY) Model The Gledzer-Ohkitani-Yamada or the GOY shell model d dtun = ιkn[un+2un+1− 1 4un+1un−1− 1 8un−1un−2]∗ − νk2 nun +f (A-2) is one of several cascade models which mimic Navier-Stokes turbulence and are designed to achieve very high Reynolds...

  25. [33]

    Evolution equation for ϕ(r,t ) and Φ In this section we provide the derivation of the equation of motion of the decorrelator for the Navier-Stokes equation and in particular estimate the long and short time limits of βS and βη. 7 We begin by recalling that the pressure term in...

  26. [34]

    Hence, in what follows, we drop the contribution from βη, to simplify notation

    Short and long-time asymptotics of βS At early times, ⟨δuiδfi⟩ = 0 (since the forcing terms are nearly identical), the pre saturation exponential growth of the decorrelator with the Lyapunov exponent λ = λDNS∼ λS is controlled by the short time asymptotics of βS. Hence, in wha...

  27. [35]

    (B-11) We find at short times, in the exponential-growth phase, this terms scales as βη = O(1/τη), where τη = q ν ⟨ε⟩ corresponds to the smallest Kolmogorov time scales

    Short and long-time asymptotics of βη The viscous term can be expanded as βη = ν⟨δui∇2δui⟩ =−ν⟨∂jδui∂jδui⟩ = −ν⟨(∂iuB j ∂iuB j +∂iuA j∂iuA j − 2∂iuB j ∂iuA j )⟩ = −4ν Tr(S2) + 2ν⟨∂iuA j∂iuB j⟩ = 2⟨ε⟩ + 2ν⟨∂iuA j∂iuB j⟩. (B-11) We find at short times, in the exponential-growth ...

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