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arxiv: 2606.03642 · v1 · pith:CX4U5F7Snew · submitted 2026-06-02 · ⚛️ physics.flu-dyn · physics.comp-ph

A variable-coefficient model for decay of isotropic turbulence capturing effects of finite cascade time and Reynolds number

Rozie Zangeneh , Wenyuan Xue , Daniel Israel , Ali Mani This is my paper

Pith reviewed 2026-06-28 08:08 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.comp-ph
keywords isotropic turbulenceturbulence decayk-epsilon modelReynolds numberenergy cascadedissipation ratekinetic energy
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The pith

An evolution equation for C_epsilon2 captures Reynolds number and history dependence in isotropic turbulence decay due to finite cascade time.

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

The standard k-epsilon model treats the coefficient C_epsilon2 as a constant tied to the decay power law. High-fidelity simulations of both forced and decaying isotropic turbulence show that the instantaneous C_epsilon2 instead varies with the current Reynolds number and the prior history of energy injection. The authors attribute these variations to the finite time needed for energy to move through the cascade from large to dissipative scales. From the data they construct an evolution equation for C_epsilon2 whose coefficients themselves depend on Reynolds number. This equation reproduces the observed time evolution of dissipation and kinetic energy across wide ranges of Reynolds number and forcing conditions.

Core claim

Data from high-fidelity simulations indicate that instantaneous C_epsilon2 depends on both instantaneous Reynolds number and the history of energy injection because of finite cascade time; an evolution equation for C_epsilon2 with Reynolds-dependent coefficients accurately reproduces the time evolution of dissipation and kinetic energy over wide Reynolds-number ranges and both forced and decay scenarios.

What carries the argument

Evolution equation for C_epsilon2 whose coefficients are functions of Reynolds number, constructed to encode the effects of finite cascade time and energy-injection history.

If this is right

  • The variable-coefficient model reproduces dissipation and kinetic-energy decay more accurately than any fixed-value C_epsilon2 across the tested Reynolds numbers.
  • The same equation works for both purely decaying turbulence and turbulence that is continuously forced.
  • History dependence in C_epsilon2 is required to capture the lag between changes in large-scale energy input and the resulting change in dissipation rate.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same modeling strategy could be tested in flows with sudden changes in forcing, such as grid-generated turbulence or wakes, to check whether the finite-cascade-time effect remains dominant.
  • Implementation of the evolution equation inside existing k-epsilon solvers would require only a modest addition of one transport equation for C_epsilon2.
  • If the Reynolds-dependent coefficients prove universal, the model might reduce the need for case-by-case tuning of decay exponents in engineering calculations.

Load-bearing premise

That the observed variations in instantaneous C_epsilon2 arise specifically from finite cascade time and the history of energy injection rather than from other unmodeled effects in the simulations.

What would settle it

A new simulation in which the time scale of the energy cascade is deliberately altered while holding Reynolds number fixed, followed by a check of whether the measured C_epsilon2 evolution still follows the proposed equation.

Figures

Figures reproduced from arXiv: 2606.03642 by Ali Mani, Daniel Israel, Rozie Zangeneh, Wenyuan Xue.

Figure 1
Figure 1. Figure 1: Examples of budgets for the Reynolds stress equation (a) and dissipation equation (b). [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Normalized 𝑃𝜖 (𝑅𝑒𝑇) and −𝑌 (𝑅𝑒𝑇) as a function of Reynolds number obtained from simulations of stationary turbulence. The overall normalized decay term (𝑃𝜖 (𝑅𝑒𝑇) - 𝑌 (𝑅𝑒𝑇)) is additionally shown in black. The green lines are binomial fits to the data of 𝑃𝜖 (𝑅𝑒𝑇) ∗ 𝑘/𝜖 2 and −𝑌 (𝑅𝑒𝑇) ∗ 𝑘/𝜖 2 involving 𝑦1𝑅𝑒1/2 𝑇 + 𝑦0, where 𝑦1 = 0.24 and 𝑦0 = 0.81 for 𝑃𝜖 (𝑅𝑒𝑇) ∗ 𝑘/𝜖 2 . For −𝑌 (𝑅𝑒𝑇) ∗ 𝑘/𝑘/𝜖 2 , 𝑦1 = −0.24 an… view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the Reynolds number during turbulence decay and growth. Each curve corresponds to a [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of 𝐶𝜖2 in time for various initial Reynolds number values for (a) turbulence decay and (b) turbulence growth. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Variation of vs. Reynolds number for the decay and growth of turbulence. Distinct graphs show [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: 𝐶𝜖𝐹 vs Reynolds number for the stationary forced isotropic simulations. The vertical line indicates the 95% confidence interval associated with the statistical convergence error due to finite sample size. Next, we discuss the selection of 𝐶 ∞ 𝜖𝐷 . The power-law decay exponent in the long-time, high-Reynolds-number limit, which is given by 1/(𝐶 ∞ 𝜖𝐷 −1), has been studied analytically, experimentally, and co… view at source ↗
Figure 7
Figure 7. Figure 7: 𝐶 ∞ 𝜖𝐷 versus Reynolds number, extracted from Fig. 4a at the final time. The red line represents a fit to the four cases with higher Reynolds numbers [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Prediction of the developed model for evolution of decaying (a) and growing (b) turbulence compared [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Prediction of the developed model for evolution of decaying (a) and growing (b) turbulence compared [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Prediction of the developed model for evolution of decaying (a) and growing (b) turbulence compared [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: a) Evolution of dissipation and b) decay of TKE predicted by variable [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Prediction of the turbulence growth regime using energy injection rates different from those used [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Schematic illustration of the energy spectrum and its evolution during decay in the limit [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Infrared energy spectra obtained from large-box LES. As the domain size increases, the infrared [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Left: selected shell-averaged autocorrelation functions [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Comparison of free-decay statistics initialized from stationary homogeneous isotropic turbulence [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
read the original abstract

We study isotropic turbulence decay in the context of the k-epsilon model, which solves the dissipation and kinetic energy equations. In modeling the dissipation equation, the coefficient C_epsilon2, suggested by Hanjalic and Launder [Journal of Fluid Mechanics, 1972] [1], is related to the temporal decay power-law by n = 1/(C_epsilon2 -1 )) and is assumed to be a constant value. In this work, we perform high-fidelity numerical simulations to examine the mathematical terms responsible for the decay of isotropic turbulence, considering both scenarios of forced and decaying turbulence. Our data suggest that the instantaneous C_epsilon2 not only depends on the instantaneous Reynolds number but is also sensitive to the history of energy injection in turbulence. We attribute these observations to the finite time required for the cascade from energetic to dissipative scales. Considering data from both decaying and growing forced turbulence, we develop an evolution equation for C_epsilon2 with Reynolds-dependent coefficients. We demonstrate that this model accurately captures the time evolution of dissipation and kinetic energy over a wide range of Reynolds numbers under a wide range of forced and decay scenarios.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The manuscript develops a variable-coefficient extension of the k-ε model for isotropic turbulence decay. High-fidelity DNS of forced and decaying cases indicate that instantaneous C_ε2 depends on both instantaneous Reynolds number and the history of energy injection; the authors attribute this to finite cascade time and construct an evolution equation for C_ε2 whose coefficients are Reynolds-dependent. The resulting model is reported to capture the time evolution of dissipation and kinetic energy across a wide range of Reynolds numbers and forced/decay scenarios.

Significance. If the attribution to cascade time is independently confirmed and the model is shown to generalize beyond the fitting data, the work would address a known limitation of constant-coefficient k-ε closures in non-equilibrium decay, offering a more physically motivated treatment of Reynolds-number and history effects within an otherwise standard two-equation framework.

major comments (3)
  1. [Abstract / Model Development] Abstract, paragraph 3: the evolution equation for C_ε2 is developed from the identical DNS datasets that first revealed the Re- and history-dependence of instantaneous C_ε2. Because the coefficients are therefore fitted to those data, the subsequent claim that the model 'accurately captures' dissipation and kinetic-energy evolution on the same data does not constitute independent validation and weakens the generality asserted for 'a wide range of forced and decay scenarios'.
  2. [Attribution of C_ε2 dependence] Abstract, paragraph 2: the functional form of the C_ε2 evolution equation rests on the attribution that observed variations arise specifically from finite cascade time. No diagnostic (spectral transfer timescales, controlled forcing perturbations, or other isolation test) is described that separates cascade-time effects from other possible numerical or physical contributors in the DNS; without such evidence the chosen functional dependence lacks first-principles grounding.
  3. [Validation] Abstract, final sentence: the central claim of accurate capture is stated without any quantitative metrics (L2 errors, R² values, or reported uncertainty), error bars, or explicit cross-validation protocol. This absence makes it impossible to judge whether the reported performance supports the asserted breadth of applicability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below, indicating where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract / Model Development] Abstract, paragraph 3: the evolution equation for C_ε2 is developed from the identical DNS datasets that first revealed the Re- and history-dependence of instantaneous C_ε2. Because the coefficients are therefore fitted to those data, the subsequent claim that the model 'accurately captures' dissipation and kinetic-energy evolution on the same data does not constitute independent validation and weakens the generality asserted for 'a wide range of forced and decay scenarios'.

    Authors: We agree that the coefficients in the evolution equation for C_ε2 were determined from the same DNS datasets used to demonstrate the model's performance. The abstract will be revised to state explicitly that the functional form and coefficients were calibrated on these forced and decaying cases, and that the reported agreement is with respect to the calibration data. We will also moderate the language on generality to reflect that broader applicability remains to be tested on independent datasets. revision: yes

  2. Referee: [Attribution of C_ε2 dependence] Abstract, paragraph 2: the functional form of the C_ε2 evolution equation rests on the attribution that observed variations arise specifically from finite cascade time. No diagnostic (spectral transfer timescales, controlled forcing perturbations, or other isolation test) is described that separates cascade-time effects from other possible numerical or physical contributors in the DNS; without such evidence the chosen functional dependence lacks first-principles grounding.

    Authors: The choice of functional dependence was guided by the observed sensitivity of instantaneous C_ε2 to both the current Reynolds number and the prior history of energy injection, which is consistent with a finite cascade time. No isolating diagnostics of the type mentioned were performed in this study. We will revise the abstract and discussion sections to frame the finite-cascade-time interpretation as a physically motivated hypothesis consistent with the data, while acknowledging that alternative explanations cannot be ruled out without additional targeted tests. revision: partial

  3. Referee: [Validation] Abstract, final sentence: the central claim of accurate capture is stated without any quantitative metrics (L2 errors, R² values, or reported uncertainty), error bars, or explicit cross-validation protocol. This absence makes it impossible to judge whether the reported performance supports the asserted breadth of applicability.

    Authors: We accept that quantitative measures of agreement are required. The revised manuscript will report L2 errors, R² values, and uncertainty estimates for the comparisons of dissipation and kinetic energy between the model and DNS across the cases. We will also describe the coefficient-fitting procedure and any cross-validation steps employed. revision: yes

Circularity Check

1 steps flagged

Evolution equation for C_epsilon2 developed from simulation data and then demonstrated on the same data

specific steps
  1. fitted input called prediction [Abstract]
    "Considering data from both decaying and growing forced turbulence, we develop an evolution equation for C_epsilon2 with Reynolds-dependent coefficients. We demonstrate that this model accurately captures the time evolution of dissipation and kinetic energy over a wide range of Reynolds numbers under a wide range of forced and decay scenarios."

    The evolution equation is explicitly constructed from the simulation data that revealed the Re- and history-dependence; the subsequent claim that the model 'accurately captures' the evolution therefore tests the fitted form against the same trends used to determine its coefficients, making the reported accuracy a re-expression of the input observations rather than an independent prediction.

full rationale

The paper extracts trends in instantaneous C_epsilon2 from high-fidelity simulations of forced and decaying turbulence, attributes the Re- and history-dependence to finite cascade time, constructs an evolution equation with Reynolds-dependent coefficients from those data, and then reports that the resulting model 'accurately captures' the dissipation and kinetic-energy evolution on the same class of scenarios. This matches the fitted-input-called-prediction pattern: the functional form and coefficients are calibrated to the observed trends, so the subsequent demonstration reduces to re-expressing the input data rather than an independent first-principles test. No external validation set, parameter-free derivation, or machine-checked uniqueness result is described that would break the dependence on the fitting data.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on an evolution equation whose coefficients are developed from simulation data; the paper therefore adds a data-derived functional form rather than a parameter-free derivation. No new physical entities are introduced.

free parameters (1)
  • Reynolds-dependent coefficients in C_epsilon2 evolution equation
    The equation is constructed with coefficients that depend on Reynolds number and are calibrated to match the observed trends in the simulation data.
axioms (1)
  • domain assumption Standard k-epsilon transport equations for kinetic energy and dissipation
    The work operates inside the existing k-epsilon framework and inherits its modeling assumptions without re-deriving them.

pith-pipeline@v0.9.1-grok · 5745 in / 1421 out tokens · 26546 ms · 2026-06-28T08:08:51.710846+00:00 · methodology

discussion (0)

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Works this paper leans on

65 extracted references · 49 canonical work pages

  1. [1]

    A Reynolds stress model of turbulence and its application to thin shear flows,

    Hanjalić, K., and Launder, B. E., “A Reynolds stress model of turbulence and its application to thin shear flows,”Journal of Fluid Mechanics, Vol. 52, No. 4, 1972, p. 609–638. https://doi.org/10.1017/S002211207200268X. 21

    work page doi:10.1017/s002211207200268x 1972

  2. [2]

    Turbulence models and their applications to the prediction of internal flows: A review,

    Nallasamy, M., “Turbulence models and their applications to the prediction of internal flows: A review,”Computers Fluids, Vol. 15, No. 2, 1987, pp. 151–194. https://doi.org/https://doi.org/10.1016/S0045-7930(87)80003-8

    work page doi:10.1016/s0045-7930(87)80003-8 1987

  3. [3]

    Constant-energeticsphysical-spaceforcingmethodsforimprovedconvergence to homogeneous-isotropic turbulence with application to particle-laden flows,

    Bassenne,M.,Urzay,J.,Park,G.I.,andMoin,P.,“Constant-energeticsphysical-spaceforcingmethodsforimprovedconvergence to homogeneous-isotropic turbulence with application to particle-laden flows,”Physics of Fluids, Vol. 28, No. 3, 2016, p. 035114. https://doi.org/10.1063/1.4944629

    work page doi:10.1063/1.4944629 2016

  4. [4]

    Reynolds stress decay modeling informed by anisotropically forced homogeneous turbulence,

    Homan, T., Shende, O. B., and Mani, A., “Reynolds stress decay modeling informed by anisotropically forced homogeneous turbulence,”Phys. Rev. Fluids, Vol. 9, 2024, p. 094608. https://doi.org/10.1103/PhysRevFluids.9.094608, URL https: //link.aps.org/doi/10.1103/PhysRevFluids.9.094608

    work page doi:10.1103/physrevfluids.9.094608 2024

  5. [5]

    Reynolds-stress and dissipation-rate budgets in a turbulent channel flow,

    Mansour, N. N., Kim, J., and Moin, P., “Reynolds-stress and dissipation-rate budgets in a turbulent channel flow,”Journal of Fluid Mechanics, Vol. 194, 1988, p. 15–44. https://doi.org/10.1017/S0022112088002885

    work page doi:10.1017/s0022112088002885 1988

  6. [6]

    The theory of homogeneous turbulence,

    Batchelor, G. K., “The theory of homogeneous turbulence,”Cambridge, 1953

    1953

  7. [7]

    The numerical computation of turbulent flows,

    Launder, B., and Spalding, D., “The numerical computation of turbulent flows,”Computer Methods in Applied Mechanics and Engineering, Vol. 3, No. 2, 1974, pp. 269–289. https://doi.org/https://doi.org/10.1016/0045-7825(74)90029-2

    work page doi:10.1016/0045-7825(74)90029-2 1974

  8. [8]

    The k--Rt Turbulence Closure,

    Goldberg, U., Peroomian, O., Batten, P., and Chakravarthy, S., “The k--Rt Turbulence Closure,”Engineering Applications of Computational Fluid Mechanics, Vol. 3, No. 2, 2009, pp. 175–183. https://doi.org/10.1080/19942060.2009.11015263

    work page doi:10.1080/19942060.2009.11015263 2009

  9. [9]

    Bayesian Parameter Estimation of a k- Model for Accurate Jet-in-Crossflow Simulations,

    Ray, J., Lefantzi, S., Arunajatesan, S., and Dechant, L., “Bayesian Parameter Estimation of a k- Model for Accurate Jet-in-Crossflow Simulations,”AIAA Journal, Vol. 53, 2016, pp. 1–17. https://doi.org/10.2514/1.J054758

    work page doi:10.2514/1.j054758 2016

  10. [10]

    On the turbulent viscosity parameter C in the k– model,

    Mishra, H., and Venayagamoorthy, S. K., “On the turbulent viscosity parameter C in the k– model,”Flow, Vol. 4, 2024, p. E16. https://doi.org/10.1017/flo.2024.15

    work page doi:10.1017/flo.2024.15 2024

  11. [11]

    A comparison of standard k–𝜖 and realizable k–𝜖 turbulence models in curved and confluent channels,

    Shaheed, R., Mohammadian, A., and Gildeh, H. K., “A comparison of standard k–𝜖 and realizable k–𝜖 turbulence models in curved and confluent channels,”Environmental Fluid Mechanics, Vol. 19, 2018, pp. 543–568. URL https://api.semanticscholar. org/CorpusID:125219551

    2018

  12. [12]

    Progress in the development of a Reynolds-stress turbulence closure,

    Launder, B. E., Reece, G. J., and Rodi, W., “Progress in the development of a Reynolds-stress turbulence closure,”Journal of Fluid Mechanics, Vol. 68, No. 3, 1975, p. 537–566. https://doi.org/10.1017/S0022112075001814

    work page doi:10.1017/s0022112075001814 1975

  13. [13]

    Verification and Validation of a Second-Moment-Closure Model,

    Eisfeld, B., Rumsey, C., and Togiti, V., “Verification and Validation of a Second-Moment-Closure Model,”AIAA Journal, Vol. 54, No. 5, 2016, pp. 1524–1541. https://doi.org/10.2514/1.J054718

    work page doi:10.2514/1.j054718 2016

  14. [14]

    Differential Reynolds-Stress Modeling for Aeronautics,

    Cécora, R.-D., Radespiel, R., Eisfeld, B., and Probst, A., “Differential Reynolds-Stress Modeling for Aeronautics,”AIAA Journal, Vol. 53, No. 3, 2015, pp. 739–755. https://doi.org/10.2514/1.J053250

    work page doi:10.2514/1.j053250 2015

  15. [15]

    Physics-informedmachinelearningapproachforreconstructingReynoldsstressmodeling discrepanciesbasedonDNSdata,

    Wang,J.-X.,Wu,J.-L.,andXiao,H.,“Physics-informedmachinelearningapproachforreconstructingReynoldsstressmodeling discrepanciesbasedonDNSdata,” Phys.Rev.Fluids , Vol.2, 2017, p.034603. https://doi.org/10.1103/PhysRevFluids.2.034603, URL https://link.aps.org/doi/10.1103/PhysRevFluids.2.034603

    work page doi:10.1103/physrevfluids.2.034603 2017

  16. [16]

    Formulation of the k-w Turbulence Model Revisited,

    Wilcox, D. C., “Formulation of the k-w Turbulence Model Revisited,”AIAA Journal, Vol. 46, No. 11, 2008, pp. 2823–2838. https://doi.org/10.2514/1.36541

    work page doi:10.2514/1.36541 2008

  17. [17]

    Hellsten, A., Laine, S., Hellsten, A., and Laine, S.,Extension of the k-omega-SST turbulence model for flows over rough surfaces, ???? https://doi.org/10.2514/6.1997-3577

    work page doi:10.2514/6.1997-3577 1997

  18. [18]

    New Advanced k-w Turbulence Model for High-Lift Aerodynamics,

    Hellsten, A., “New Advanced k-w Turbulence Model for High-Lift Aerodynamics,”AIAA Journal, Vol. 43, No. 9, 2005, pp. 1857–1869. https://doi.org/10.2514/1.13754

    work page doi:10.2514/1.13754 2005

  19. [19]

    Artificial neural network approach for turbulence models: A local framework,

    Xie, C., Xiong, X., and Wang, J., “Artificial neural network approach for turbulence models: A local framework,”Phys. Rev. Fluids, Vol. 6, 2021, p. 084612. https://doi.org/10.1103/PhysRevFluids.6.084612, URL https://link.aps.org/doi/10.1103/ PhysRevFluids.6.084612

    work page doi:10.1103/physrevfluids.6.084612 2021

  20. [20]

    Assessmentsofk-kLTurbulenceModelBasedonMenter’sModificationtoRotta’sTwo-EquationModel,

    Abdol-Hamid, K.S., “Assessmentsofk-kLTurbulenceModelBasedonMenter’sModificationtoRotta’sTwo-EquationModel,” International Journal of Aerospace Engineering, Vol. 2015, No. 1, 2015, p. 987682. https://doi.org/https://doi.org/10.1155/ 2015/987682

    2015

  21. [21]

    Steady and Unsteady Flow Modelling Using the k-8730;kL Model,

    Menter, F., Egorov, Y., and Rusch, D., “Steady and Unsteady Flow Modelling Using the k-8730;kL Model,” 2006, pp. 403–406. https://doi.org/10.1615/ICHMT.2006.TurbulHeatMassTransf.800. 22

    work page doi:10.1615/ichmt.2006.turbulheatmasstransf.800 2006

  22. [22]

    On the Equation Governing the Rate of Turbulent Energy Dissipation,

    Rodi, W., “On the Equation Governing the Rate of Turbulent Energy Dissipation,” Technical Report TM/TN/A/14, Mechanical Engineering Department, Imperial College, 1971

    1971

  23. [23]

    The prediction of laminarization with a two-equation model of turbulence,

    Jones, W., and Launder, B., “The prediction of laminarization with a two-equation model of turbulence,”International Journal of Heat and Mass Transfer, Vol. 15, No. 2, 1972, pp. 301–314. https://doi.org/https://doi.org/10.1016/0017-9310(72)90076-2

    work page doi:10.1016/0017-9310(72)90076-2 1972

  24. [24]

    B.,Turbulent Flows, Cambridge University Press, 2000

    Pope, S. B.,Turbulent Flows, Cambridge University Press, 2000

    2000

  25. [25]

    The use of a contraction to improve the isotropy of grid-generated turbulence,

    Comte-Bellot, G., and Corrsin, S., “The use of a contraction to improve the isotropy of grid-generated turbulence,”Journal of Fluid Mechanics, Vol. 25, No. 4, 1966, p. 657–682. https://doi.org/10.1017/S0022112066000338

    work page doi:10.1017/s0022112066000338 1966

  26. [26]

    Simple Eulerian time correlation of full-and narrow-band velocity signals in grid- generated, ‘isotropic’ turbulence,

    Comte-Bellot, G., and Corrsin, S., “Simple Eulerian time correlation of full-and narrow-band velocity signals in grid- generated, ‘isotropic’ turbulence,”Journal of Fluid Mechanics, Vol. 48, No. 2, 1971, p. 273–337. https://doi.org/10.1017/ S0022112071001599

    1971

  27. [27]

    Decayingturbulenceinanactive-grid-generatedflowandcomparisonswith large-eddy simulation,

    KANG,H.S.,CHESTER,S.,andMENEVEAU,C.,“Decayingturbulenceinanactive-grid-generatedflowandcomparisonswith large-eddy simulation,”Journal of Fluid Mechanics, Vol. 480, 2003, p. 129–160. https://doi.org/10.1017/S0022112002003579

    work page doi:10.1017/s0022112002003579 2003

  28. [28]

    Freely decaying, homogeneous turbulence generated by multi-scale grids,

    KROGSTAD, P.-, and DAVIDSON, P. A., “Freely decaying, homogeneous turbulence generated by multi-scale grids,”Journal of Fluid Mechanics, Vol. 680, 2011, p. 417–434. https://doi.org/10.1017/jfm.2011.169

    work page doi:10.1017/jfm.2011.169 2011

  29. [29]

    DNS of rotating and non-rotating turbulent flows with synthetic inlet conditions,

    Orlandi, P., “DNS of rotating and non-rotating turbulent flows with synthetic inlet conditions,”Journal of Turbulence, Vol. 14, No. 5, 2013, pp. 10–34. https://doi.org/10.1080/14685248.2013.815357

    work page doi:10.1080/14685248.2013.815357 2013

  30. [30]

    Lattice-Boltzmann simulation of grid-generated turbulence,

    DJENIDI, L., “Lattice-Boltzmann simulation of grid-generated turbulence,”Journal of Fluid Mechanics, Vol. 552, 2006, p. 13–35. https://doi.org/10.1017/S002211200600869X

    work page doi:10.1017/s002211200600869x 2006

  31. [31]

    Direct numerical simulation of laboratory experiments in isotropic turbulence,

    de Bruyn Kops, S. M., and Riley, J. J., “Direct numerical simulation of laboratory experiments in isotropic turbulence,”Physics of Fluids, Vol. 10, No. 9, 1998, pp. 2125–2127. https://doi.org/10.1063/1.869733

    work page doi:10.1063/1.869733 1998

  32. [32]

    Laws of turbulence decay from direct numerical simulations,

    Panickacheril John, J., Donzis, D., and Sreenivasan, K., “Laws of turbulence decay from direct numerical simulations,” , 06

  33. [33]

    https://doi.org/10.48550/arXiv.2106.15710

    work page doi:10.48550/arxiv.2106.15710

  34. [34]

    The decay of homogeneous isotropic turbulence,

    George, W. K., “The decay of homogeneous isotropic turbulence,”Physics of Fluids A: Fluid Dynamics, Vol. 4, No. 7, 1992, pp. 1492–1509. https://doi.org/10.1063/1.858423

    work page doi:10.1063/1.858423 1992

  35. [35]

    Decay of homogeneous, nearly isotropic turbulence behind active fractal grids,

    Thormann, A., and Meneveau, C., “Decay of homogeneous, nearly isotropic turbulence behind active fractal grids,”Physics of Fluids, Vol. 26, No. 2, 2014, p. 025112. https://doi.org/10.1063/1.4865232

    work page doi:10.1063/1.4865232 2014

  36. [36]

    Multiple-Time-Scale Concepts in Turbulent Transport Modeling,

    Launder, B., and Schiestel, R., “Multiple-Time-Scale Concepts in Turbulent Transport Modeling,”Turbulent Shear Flows, Vol. 2, 1980

    1980

  37. [37]

    Assessment of the performance of different classes of turbulence models in a wide range of non-equilibrium flows,

    Klein, T., Craft, T., and Iacovides, H., “Assessment of the performance of different classes of turbulence models in a wide range of non-equilibrium flows,”International Journal of Heat and Fluid Flow, Vol. 51, 2015, pp. 229–256. https://doi.org/https: //doi.org/10.1016/j.ijheatfluidflow.2014.10.017, URL https://www.sciencedirect.com/science/article/pii/S...

    work page doi:10.1016/j.ijheatfluidflow.2014.10.017 2015

  38. [38]

    Temperature and concentration profiles in fully turbulent boundary layers,

    Kader, B., “Temperature and concentration profiles in fully turbulent boundary layers,”International Journal of Heat and Mass Transfer, Vol. 24, No. 9, 1981, pp. 1541–1544. https://doi.org/https://doi.org/10.1016/0017-9310(81)90220-9, URL https://www.sciencedirect.com/science/article/pii/0017931081902209

    work page doi:10.1016/0017-9310(81)90220-9 1981

  39. [39]

    Parallel variable-density particle-laden turbulence simulation,

    Pouransari, H., Mortazavi, M., and Mani, A., “Parallel variable-density particle-laden turbulence simulation,”CTR Annual Research Briefs, 2016

    2016

  40. [40]

    General Circulation Experiments with the Primitive Equations,

    Smagorinsky, J., “General Circulation Experiments with the Primitive Equations,”Monthly Weather Review, Vol. 91, No. 3, 1963, p. 99. https://doi.org/10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2

    work page doi:10.1175/1520-0493(1963)091 1963

  41. [41]

    On the application of the eddy viscosity concept in the Inertial sub-range of turbulence,

    Lilly, K. E., “On the application of the eddy viscosity concept in the Inertial sub-range of turbulence,” 1966. URL https://api.semanticscholar.org/CorpusID:55104948

    1966

  42. [42]

    Linearforcinginnumericalsimulationsofisotropicturbulence: Physicalspaceimplementations and convergence properties,

    Rosales,C.,andMeneveau,C.,“Linearforcinginnumericalsimulationsofisotropicturbulence: Physicalspaceimplementations and convergence properties,”Physics of Fluids, Vol. 17, No. 9, 2005, p. 095106. https://doi.org/10.1063/1.2047568

    work page doi:10.1063/1.2047568 2005

  43. [43]

    Linearly Forced Isotropic Turbulence,

    Lundgren, T., “Linearly Forced Isotropic Turbulence,”Annual Research Briefs, 2003. 23

    2003

  44. [44]

    A model for drift velocity mediated scalar eddy diffusivity in homogeneous turbulent flows,

    Shende, O. B., Storan, L., and Mani, A., “A model for drift velocity mediated scalar eddy diffusivity in homogeneous turbulent flows,”Journal of Fluid Mechanics, Vol. 989, 2024, p. A14. https://doi.org/10.1017/jfm.2024.457

    work page doi:10.1017/jfm.2024.457 2024

  45. [45]

    Clustering and preferential concentration of finite-size particles in forced homogeneous- isotropic turbulence,

    Uhlmann, M., and Chouippe, A., “Clustering and preferential concentration of finite-size particles in forced homogeneous- isotropic turbulence,”Journal of Fluid Mechanics, Vol. 812, 2017, p. 991–1023. https://doi.org/10.1017/jfm.2016.826

    work page doi:10.1017/jfm.2016.826 2017

  46. [46]

    Harmonic analysis filtering techniques for forced and decaying homogeneous isotropic turbulence,

    Kareem, W. A., Nabil, T., Izawa, S., and Fukunishi, Y., “Harmonic analysis filtering techniques for forced and decaying homogeneous isotropic turbulence,”Computers Mathematics with Applications, Vol. 65, No. 7, 2013, pp. 1059–1085. https://doi.org/https://doi.org/10.1016/j.camwa.2013.01.042

    work page doi:10.1016/j.camwa.2013.01.042 2013

  47. [47]

    Entrainment in a shear-free turbulent mixing layer,

    Briggs, D. A., Ferziger, J. H., Koseff, J. R., and Monismith, S. G., “Entrainment in a shear-free turbulent mixing layer,”Journal of Fluid Mechanics, Vol. 310, 1996, p. 215–241. https://doi.org/10.1017/S0022112096001784

    work page doi:10.1017/s0022112096001784 1996

  48. [48]

    A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,

    Kolmogorov, A. N., “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,”Journal of Fluid Mechanics, Vol. 13, No. 1, 1962, p. 82–85. https: //doi.org/10.1017/S0022112062000518

    work page doi:10.1017/s0022112062000518 1962

  49. [49]

    Dissipation and enstrophy in isotropic turbulence: Resolution effects and scaling in direct numerical simulations,

    Donzis, D. A., Yeung, P. K., and Sreenivasan, K. R., “Dissipation and enstrophy in isotropic turbulence: Resolution effects and scaling in direct numerical simulations,”Physics of Fluids, Vol. 20, No. 4, 2008, p. 045108. https://doi.org/10.1063/1.2907227

    work page doi:10.1063/1.2907227 2008

  50. [50]

    Final Stage Decay of Grid-Produced Turbulence,

    Tan, H. S., and Ling, S. C., “Final Stage Decay of Grid-Produced Turbulence,”The Physics of Fluids, Vol. 6, No. 12, 1963, pp. 1693–1699. https://doi.org/10.1063/1.1711011

    work page doi:10.1063/1.1711011 1963

  51. [51]

    Decay of isotropic turbulence in the initial period,

    Batchelor, G. K., Townsend, A. A., and Taylor, G. I., “Decay of isotropic turbulence in the initial period,”Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 193, No. 1035, 1948, pp. 539–558. https://doi.org/10.1098/rspa.1948.0061

    work page doi:10.1098/rspa.1948.0061 1948

  52. [52]

    The large-scale structure of homogeneous turbulence,

    Saffman, P. G., “The large-scale structure of homogeneous turbulence,”Journal of Fluid Mechanics, Vol. 27, No. 3, 1967, p. 581–593. https://doi.org/10.1017/S0022112067000552

    work page doi:10.1017/s0022112067000552 1967

  53. [53]

    The Large-Scale Structure of Homogeneous Turbulence,

    Batchelor, G. K., and Proudman, I., “The Large-Scale Structure of Homogeneous Turbulence,”Philosophical Transactions of the Royal Society of London Series A, Vol. 248, No. 949, 1956, pp. 369–405. https://doi.org/10.1098/rsta.1956.0002

    work page doi:10.1098/rsta.1956.0002 1956

  54. [54]

    Understanding evolution of vortex rings in viscous fluids,

    Tinaikar, A., Advaith, S., and Basu, S., “Understanding evolution of vortex rings in viscous fluids,”Journal of Fluid Mechanics, Vol. 836, 2018, p. 873–909. https://doi.org/10.1017/jfm.2017.815

    work page doi:10.1017/jfm.2017.815 2018

  55. [55]

    The Velocity of Viscous Vortex Rings,

    Saffman, P. G., “The Velocity of Viscous Vortex Rings,”Studies in Applied Mathematics, Vol. 49, No. 4, 1970, pp. 371–380. https://doi.org/https://doi.org/10.1002/sapm1970494371

    work page doi:10.1002/sapm1970494371 1970

  56. [56]

    Power-law exponent in the transition period of decay in grid turbulence,

    Djenidi, L., Kamruzzaman, M., and Antonia, R. A., “Power-law exponent in the transition period of decay in grid turbulence,” Journal of Fluid Mechanics, Vol. 779, 2015, p. 544–555. https://doi.org/10.1017/jfm.2015.428

    work page doi:10.1017/jfm.2015.428 2015

  57. [57]

    On the decay of homogeneous isotropic turbulence,

    Skrbek, L., and Stalp, S. R., “On the decay of homogeneous isotropic turbulence,”Physics of Fluids, Vol. 12, No. 8, 2000, pp. 1997–2019. https://doi.org/10.1063/1.870447

    work page doi:10.1063/1.870447 2000

  58. [58]

    Oscillating grids as a source of nearly isotropic turbulence,

    De Silva, I. P. D., and Fernando, H. J. S., “Oscillating grids as a source of nearly isotropic turbulence,”Physics of Fluids, Vol. 6, No. 7, 1994, pp. 2455–2464. https://doi.org/10.1063/1.868193

    work page doi:10.1063/1.868193 1994

  59. [59]

    The decay of turbulence in a bounded domain,

    Touil, H., Bertoglio, J.-P., and Shao, L., “The decay of turbulence in a bounded domain,”Journal of Turbulence, Vol. 3, 2002, N49. https://doi.org/10.1088/1468-5248/3/1/049

    work page doi:10.1088/1468-5248/3/1/049 2002

  60. [60]

    On Degeneration (Decay) of Isotropic Turbulence in an Incompressible Viscous Liquid,

    Kolmogorov, A., “On Degeneration (Decay) of Isotropic Turbulence in an Incompressible Viscous Liquid,”Doklady Akademii Nauk SSSR, Vol. 31, 1941, pp. 538–541

    1941

  61. [61]

    The role of big eddies in homogeneous turbulence,

    Batchelor, G. K., “The role of big eddies in homogeneous turbulence,”Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 195, No. 1043, 1949, pp. 513–532

    1949

  62. [62]

    On non-self-similar regimes in homogeneous isotropic turbulence decay,

    Meldi, M., and Sagaut, P., “On non-self-similar regimes in homogeneous isotropic turbulence decay,”Journal of Fluid Mechanics, Vol. 711, 2012, pp. 364–393

    2012

  63. [63]

    Is grid turbulence Saffman turbulence?

    KROGSTAD, P.-, and DAVIDSON, P. A., “Is grid turbulence Saffman turbulence?”Journal of Fluid Mechanics, Vol. 642, 2010, p. 373–394. https://doi.org/10.1017/S0022112009991807

    work page doi:10.1017/s0022112009991807 2010

  64. [64]

    Note on decay of homogeneous turbulence,

    Saffman, P., “Note on decay of homogeneous turbulence,”Physics of fluids, Vol. 10, No. 6, 1967, pp. 1349–1349

    1967

  65. [65]

    Davidson, P.,Turbulence: an introduction for scientists and engineers, Oxford university press, 2015. 24

    2015