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REVIEW 4 major objections 5 minor 45 references

A parameter-free closure reproduces the decay laws and mixing constants of decaying isotropic turbulence.

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2026-08-01 05:46 UTC pith:V4UYHFD2

load-bearing objection A self-consistent closure framework, but the validation claim outruns the evidence: the core identity (7) is never tested against independent data. the 4 major comments →

arxiv 2607.22105 v1 pith:V4UYHFD2 submitted 2026-07-24 physics.flu-dyn

Numerical Validation of Lyapunov-Liouville Theory and Non-Diffusive Closures in Decaying Isotropic Fluid and Scalar Turbulence

Nicola de Divitiis This is my paper
classification physics.flu-dyn MSC 76F0576F55 PACS 47.27.Gs47.27.E-
keywords freely decaying turbulenceLyapunov–Liouville theorynon-diffusive closurevon Kármán–Howarth equationCorrsin equationBatchelor constantObukhov–Corrsin constantscalar intermittency
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims that the Lyapunov–Liouville theory, applied to the relative motion of fluid elements, supplies closed forms for the triple correlations in the von Kármán–Howarth and Corrsin equations without any fitted constants. Integrating these closed equations from three classical initial states yields the known decay exponents (Saffman–Birkhoff m ≃ n ≃ −1.25; Loitsiansky m ≃ −1.51, n ≃ −0.89; Gaussian m ≃ −2.7), the 4/5-law maximum, the Batchelor constant C_B ≃ 3.5, the Obukhov–Corrsin constant C_OC ≃ 1.8, and a Prandtl-number-dependent transition from Gaussian to intermittent temperature-increment PDFs. If correct, one internally consistent theory with no tuning parameters accounts for the main observed inventory of freely decaying isotropic turbulence.

Core claim

On the author's own terms, the central discovery is that the non-diffusive closures K = u^3 sqrt((1−f)/2) ∂f/∂r and G = u θ^2 sqrt((1−f)/2) ∂fθ/∂r, derived from the link ⟨Λ_L²(r)⟩_L r² = 2u²(1−f) between the finite-scale Lagrangian Lyapunov exponent and the longitudinal velocity correlation, close the von Kármán–Howarth and Corrsin equations. Numerical integration of the closed system from Saffman–Birkhoff, Loitsiansky, and Gaussian initial conditions reproduces the empirically known decay exponents for each case, the Kolmogorov and Yaglom functions peaking near 4/5 and 2/3, the spectral constants C_B ≃ 3.5 and C_OC ≃ 1.8, and the growth of small-scale intermittency in temperature increments

What carries the argument

The load-bearing link is Eq. (7): the mean-square finite-scale Lagrangian Lyapunov exponent times the squared separation equals twice the velocity variance times (1 − f(r)). Combined with the spectral-gap hierarchy (1) separating Lagrangian from Eulerian fluctuation rates, this yields algebraic closures (6) for the triple correlations in terms of the double correlations only. These closures fix the small-scale skewness to −3/7, the Rλ = 10 validity threshold, and the parametric family of increment PDFs via the characteristic function χ(Rλ).

Load-bearing premise

The entire validation rests on Eq. (7), the asserted proportionality between the squared finite-scale Lagrangian Lyapunov exponent and the correlation defect 2u²(1−f); the paper neither derives it in this work nor tests it directly against DNS or experiments, so if that link is wrong the closure (6) and all resulting exponents and constants are a self-consistent simulation of a false premise.

What would settle it

A direct numerical simulation of forced or freely decaying HIT that computes both sides of Eq. (7) across the inertial range, or that compares the simulated triple correlation k(r) with the closure value u³√((1−f)/2) ∂f/∂r. A systematic disagreement at moderate Reynolds numbers would invalidate the premise; agreement would strongly support the closed system.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the closure stands, the decay exponents of freely decaying homogeneous isotropic turbulence are not universal but follow directly from the initial correlation profile, matching Saffman–Birkhoff and Loitsiansky classes without ad hoc constants.
  • The 4/5-law and 2/3-law maxima in decaying flows are obtained at the moment the Taylor microscale stabilizes (t ≈ 2 in Lyapunov units), not as extended plateaus, consistent with nonstationary experiments.
  • The framework sets a sharp validity boundary at Rλ = 10, below which the non-diffusive closure ceases to be applicable and energy decays far faster than the power-law regime.
  • At P r ≥ 1000, the thermal microscale drops below the Kolmogorov scale and the compensated spectrum yields C_B ≃ 3.5, in agreement with Batchelor scaling; the Obukhov–Corrsin constant stabilizes at C_OC ≃ 1.8.
  • The increment PDFs built from the theory predict that intermittency of temperature increments is controlled by the Péclet number, transitioning from near-Gaussian at low P r to scale-dependent non-Gaussian tails at high P r.

Where Pith is reading between the lines

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

  • Editorial inference: because the closure (6) is written entirely in terms of the double correlation f, the same form should extend to any passive scalar governed by a diffusion equation; the paper's temperature results are the first test of that generality.
  • Editorial inference: a direct falsifiable check the paper does not perform is to measure the finite-scale Lagrangian Lyapunov exponent and the longitudinal correlation simultaneously in DNS, or to compare the predicted triple correlation k(r) from Eq. (6) with the DNS-computed k(r). If Eq. (7) fails there, the decay exponents and spectral constants would be self-consistent artifacts of the closure
  • Editorial inference: the PDF family (10)–(13) predicts higher-order moments (e.g., flatness at r = 0 for velocity) that go beyond the skewness values quoted; these could be compared with existing high-Reynolds DNS, and a mismatch would localize where the non-observability assumption fails.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. This paper integrates the von Kármán–Howarth and Corrsin equations closed by the author's non-diffusive closures, Eq. (6), for three initial correlation profiles (Saffman–Birkhoff, Loitsiansky, Gaussian) and for Prandtl numbers from 10^-3 to 1000. It reports a two-stage evolution—an initial nonlinear development regime of about two Lyapunov times followed by a diffusive decay regime—and extracts decay exponents, integral scales, dissipation coefficients, spectral constants, Kolmogorov/Yaglom functions, and synthesized velocity/temperature increment PDFs. The central claim is that the results numerically validate the Lyapunov–Liouville framework developed in refs. [18,19] and reproduce the main phenomenology of decaying HIT without fitted constants. The internal algebra is coherent and the distinction between the development and diffusive regimes is clearly derived, but the validation evidence amounts to comparisons of emergent outputs with literature values, while the fundamental closure identity (7) is assumed rather than directly tested.

Significance. If the framework were validated, this would be a significant result: a single closure scheme producing decay laws, approximate Kolmogorov/Obukhov–Corrsin constants, Batchelor scaling, and Pr-dependent intermittency from a unified Lyapunov–Liouville argument. The analytical derivation in Section 3 of the two-regime evolution of λ_T and λ_θ, including the inviscid τ≈2 developing time and the √(νt), √(κt) growth, is transparent and internally consistent. The paper is also honest in reporting the absence of genuine spectral plateaus and the t<33 validity limit for the Gaussian initial condition. However, the significance is conditional because the paper's only direct validation target is the closure (6), which is itself a restatement of the untested relation (7), and several headline agreements (notably the −3/7 skewness and the PDF amplitude Φ(0)) are fixed by the construction. Thus the paper is best read as a self-consistent application of a proposed closure, not as independent numerical validation of the underlying theory.

major comments (4)
  1. [§2, Eq. (7) and closure (6)] The validation claim is not yet supported. Equation (7), ⟨Λ_L^2(r)⟩_L r^2 = 2u^2(1−f(r)), is the physical input that yields closure (6), but it is never checked against DNS or experimental data. All reported agreements—decay exponents, spectral constants, PDF shapes—come from integrating Eqs. (3) with closure (6), so they test the closure assumption, not the underlying Lyapunov–Liouville identity. Moreover, some headline agreements are fixed by construction: substituting (6) into (8) gives H_u^(3)(0)=−3/7 algebraically, and Eq. (16) determines Φ(0) by imposing the same value at Rλ*=10. A direct, matched-initial-condition comparison of Eq. (7), or of the implied K(r)/G(r), against DNS or laboratory turbulence is necessary before the phrase 'numerical validation' is warranted. Without such a test, a failure of Eq. (7) would invalidate the entire exercise.
  2. [§2, Eq. (16) and §4, Figs. 11, 27] The PDF construction (10)–(13) is presented as predicting a transition from quasi-Gaussian statistics at low Pr to intermittency at high Pr, but it is constrained to reproduce the closure's own skewness. The amplitude Φ(0)=0.1409 is not an independent parameter: it is chosen so that H_u^(3)(0)=−3/7 at Rλ*=10. The same Rλ*=10 is then used in Section 4 as the operational validity threshold (e.g., after Fig. 20). This circularity should be explicitly acknowledged and either broken by an independent determination of Φ(0) or presented as a consistency check rather than as a predictive validation of the PDF statistics.
  3. [§4, Figs. 8–9, 17–18, 23–24, 28–30] The reported spectral constants are extracted from compensated spectra that the text itself describes as having no extended plateau, but only 'relatively flat maxima' or 'oblique inflection points.' Without a defined fitting window, a precise fitting procedure, or an uncertainty estimate, statements such as 'the Batchelor constant converges to C_B = 3.5', 'C_OC matches 1.8', and 'C_K = 1.72–1.75' are overstated. This is especially clear for C_B, which is quoted as ≈5 at Pr=10 and ≈3.5 at Pr=1000. Please provide the fitting formulas, wavenumber ranges, and sensitivity, or explicitly soften these claims to reflect the absence of a true plateau.
  4. [§4, Gaussian initial condition (Eq. 25) and Fig. 21] The abstract and conclusions quote m≃−2.7 as one of the main validated decay exponents, but the text states that n never reaches a stationary value within t∈(2,33) and that the simulation is terminated at t≈33 because Rλ drops below 10. It is therefore unclear whether m=−2.7 is an asymptotic exponent or a transient value truncated by the validity limit. The fitting interval, the method used to extract the exponent, and the associated uncertainty should be reported; otherwise the claim should be limited to a statement about the accessible window.
minor comments (5)
  1. [§2, Eq. (10)] Φ(r) is used in Eqs. (10) but only Φ(0) is defined. Please specify the r-dependence of Φ and its role in the PDF synthesis.
  2. [§5] The word 'ultra-rapidamente' is Italian; it should be 'ultra-rapidly' in English.
  3. [§4, Figs. 4 and 27] The text reports Rλ≈165 at t≈2 in one place and Rλ≈164 at t≈2.2 in another; please reconcile the values or clarify the time instants.
  4. [§4, Fig. 30] The text refers to 'Figs. 30a and 30b' but the figure has left/right panels without (a)/(b) labels. Please standardize the panel citations.
  5. [Abstract / Conclusions] The statement that the model works 'without the need for empirical tuning' is misleading: Φ(0) and Rλ* are input parameters fixed by the theory, and the spectral constants are obtained by fitting non-flat compensated spectra. Please qualify this claim.

Circularity Check

3 steps flagged

Partial circularity: skewness and PDF intermittency are built into the assumed closure, while decay/spectral outputs remain emergent.

specific steps
  1. self definitional [Section 2, Eqs. (6)-(9) and Eq. (16)]
    "Based on the Liouville–Lyapunov analysis of Ref. [18], these closures are given by: K(r) = u3√((1 − f)/2) ∂f/∂r, G(r) = uθ2√((1 − f)/2) ∂fθ/∂r. ... In the small-scale limit, these expressions satisfy: H(3)u(0) = lim r→0 H(3)u(r) = −3/7, H(3)θ(0) = lim r→0 H(3)θ(r) = −1/5, which stands in excellent agreement with established literature [26,27,28,29,30,31]."

    With closure (6), the triple correlation k is fixed through Eq. (5), and Eq. (8) defines H(3)u = 6k/(2(1−f))^{3/2}. Expanding f near r=0 gives H(3)u(0)=−3/7 identically. The same constant is then imposed through Eq. (16) to determine Φ(0), so the synthesized PDFs are forced to carry exactly this skewness. Presenting the result as 'excellent agreement with established literature' is therefore a consistency check of the assumed closure, not an independent numerical prediction.

  2. fitted input called prediction [Section 4, PDF computation, after Eqs. (10)-(11)]
    "This code requires as inputs the velocity increment skewness H(3)u(r), Rλ, and P r obtained from the preceding simulations."

    The PDFs are not independent outputs: they are constructed to match the simulation's own third-order skewness H(3)u(r). The advertised Pr-dependent transition (quasi-Gaussian at low Pr, growing kurtosis at high Pr) is a built-in property of Eqs. (10)-(13), since Ψθ ∝ √Pe, rather than an emergent validation against separate DNS or experimental data. Thus the claim that the PDFs 'validate the predictive robustness of the theory' is a self-consistency test of the parametric ansatz, not a prediction independent of its inputs.

  3. self citation load bearing [Section 2, Eq. (7); Section 4, all simulations]
    "These expressions, determined through the link between finite scale lagrangian lyapunov exponent and velocity correlation [18, 19] ⟨Λ2L(r)⟩Lr2 = 2u2(1 − f(r)), represent non-diffusive closures..."

    The central premise of the whole numerical exercise, Eq. (7) together with closure (6), is imported from same-author Refs. [18,19] and is never directly tested in this paper against DNS or experiments. Every simulation integrates this assumed closure, and the reported agreements (decay exponents, spectral constants, PDF shapes) are consequences of that premise. The validation chain therefore rests on an untested self-citation: if Eq. (7) is wrong, all headline outputs would be artifacts of the same self-consistent system. This is load-bearing, although the decay exponents and spectral constants are not literally encoded as inputs.

full rationale

The manuscript's central claim is a 'numerical validation' of the Lyapunov–Liouville closure. Most headline outputs—decay exponents, spectral constants, characteristic scales—are not literally fixed by the closure's algebraic structure and could in principle disagree with literature, so they retain independent content. However, three load-bearing steps are substantially circular. First, the skewness −3/7 at r→0 is an exact algebraic consequence of closure (6) via Eq. (8); Eq. (16) then uses that same value to fix Φ(0), feeding the PDF construction. Second, the PDFs are computed from Eq. (10) using the simulation's own H(3)u(r), Rλ, and Pr as inputs, and the Pr-dependent kurtosis growth is already built into Ψθ∝√Pe, so the claimed 'validation' of intermittency transitions is a self-consistency check of the parametric ansatz, not an independent prediction. Third, closure (6) and Eq. (7) are imported from the author's own Refs. [18,19] and are never checked directly against DNS or experiments in this paper; every simulation integrates that premise, so the entire validation chain rests on an untested self-citation. Because of these elements, the paper is partially circular: the skewness and PDF transitions reduce by construction to the assumed closure and its own simulation data, while the decay-law and spectral-constant agreements remain emergent outputs. Score 6 reflects this partial circularity, not a fully definitional equivalence.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 3 invented entities

The central claim rests entirely on closure forms and structural ansätze imported from the author's own prior work ([18,19]), with no independent measurement of any free constant: Φ(0) and Rλ* are fixed by internal consistency conditions of the same theory. The only external anchors are qualitative comparisons of emergent outputs (exponents, spectral constants) to literature values, with no matched-initial-condition DNS, no error bars, and no code. The ledger therefore shows a theory that is internally parameter-free but externally uncalibrated.

free parameters (2)
  • Φ(0) (PDF amplitude at r = 0) = 0.1409
    Determined by Eq. (16) from H_u^(3)(0) = −3/7 and Rλ* = 10; it inherits both from the author's closure theory [18,19] rather than being measured independently. It sets the strength of the intermittency term Ψ_u(r) = Φ(r)√Rλ in the PDF model (10).
  • Rλ* (transitional Taylor-scale Reynolds number) = 10
    From [19]: the threshold at which the closure is deemed invalid, used to stop the Gaussian run at t = 33 and to calibrate Φ(0). It is an output of the author's own bifurcation analysis of the same equation, so its role as an independent validity check is limited.
axioms (5)
  • domain assumption Finite-scale Lyapunov-correlation link: ⟨Λ_L²(r)⟩_L r² = 2u²(1−f) (Eq. 7)
    Imported from [18,19]; it is the physical input from which both closures (6) are built. If it fails, all downstream 'validations' are ungrounded.
  • domain assumption Spectral-gap hierarchy (Eq. 1): SL >> sup{ΛL} >> ⟨ΛL⟩ ≳ sup{ΛE} >> ⟨ΛE⟩ >> SE
    Stated as the regime of validity of the theory; asserted to hold for t > 2 in the Saffman and Loitsiansky runs and until t = 33 in the Gaussian run. Not independently tested.
  • ad hoc to paper Trivariate-Gaussian ansatz for ζ, ζ+, ζ− in the increment model (Eqs. 10–12)
    The PDF shapes of velocity and temperature increments follow entirely from this structural ansatz from [19]; the paper uses it to synthesize PDFs and then presents the resulting intermittency as validation.
  • domain assumption Closure forms (6): K = u³√((1−f)/2) ∂f/∂r, G = uθ²√((1−f)/2) ∂fθ/∂r
    Taken as given from [18]; every simulation studies the dynamics under these closures, and several benchmark numbers (−3/7 skewness, 4/5-law peak) are algebraic consequences of this choice.
  • standard math HIT benchmark truths: Kolmogorov 4/5 law, k^−5/3 and k^−1 scalings, literature constants
    Used as reference values; the paper compares model outputs to them. The 4/5 law is exact in stationary HIT; the comparison here is to a flat maximum in decaying flow.
invented entities (3)
  • Non-observable bifurcation modes / quasi-PDFs no independent evidence
    purpose: Conceptual mechanism from [19] used to derive the increment statistics (10)–(13) and claimed to enforce Kolmogorov scaling and Pr-dependent intermittency.
    No direct observable handle; their support is the internal consistency of the author's own framework, with the PDF outputs calibrated by the model's own skewness.
  • Eulerian and Lagrangian bifurcation rates SL, SE (Eq. 2) no independent evidence
    purpose: Internal diagnostics used to check the hierarchy (1) and to declare the validity window of each simulation.
    Computed from model quantities within the same framework; not externally measurable.
  • Liouville spectral gap no independent evidence
    purpose: Justifies the non-diffusive closures; invoked as the physical origin of (6).
    From [18]; no independent evidence is presented in this paper.

pith-pipeline@v1.3.0-alltime-deepseek · 22005 in / 25265 out tokens · 234540 ms · 2026-08-01T05:46:05.999174+00:00 · methodology

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This work presents a comprehensive numerical validation of the Lyapunov-Liouville theoretical framework and its non-diffusive turbulence closures under freely decaying homogeneous isotropic turbulence (HIT). The closed system of von Karman-Howarth and Corrsin equations is integrated via an autonomous, high-accuracy architecture across three initial states (Saffman-Birkhoff, Loitsiansky, and Gaussian correlation profiles) and Prandtl numbers from Pr = 10^-3 to 1000. The analysis scrutinizes the transient phase, the self-preserving diffusive regime, and the internal structure of turbulence via velocity and temperature increment probability density functions (PDFs). Our findings reveal that the closure accurately captures distinct decay paths. The Saffman-Birkhoff case yields asymptotic exponents m = -1.25 and n = -1.25. The Loitsiansky condition accelerates mechanical decay (m = -1.51) due to higher dissipation but exhibits higher thermal persistence (n = -0.89). Conversely, the Gaussian profile induces ultra-rapid decay (m = -2.7), reaching its operational limit at t = 33 as R_lambda drops below 10. Furthermore, the model replicates the non-equilibrium evolution of characteristic scales. At Pr = 1000, the thermal microscale drops below the Kolmogorov scale, confirming Batchelor's scaling where the Batchelor constant converges to C_B = 3.5 and the Obukhov-Corrsin constant matches C_OC = 1.8. Finally, the synthesized PDFs capture a sharp transition from quasi-Gaussian statistics at low Pr to enhanced, scale-dependent small-scale intermittency at high Pr, validating the predictive robustness of the theory for multi-scale scalar mixing.

Figures

Figures reproduced from arXiv: 2607.22105 by Nicola de Divitiis.

Figure 1
Figure 1. Figure 1: Characteristic function χ as a function of the Taylor-scale Reynolds number, Rλ. which physically dictates that χ must remain strictly positive. Thus, the limit χ = 0 is assumed to occur at Rλ = R∗ λ = 10, representing the threshold for homogeneous isotropic turbulence. This allows for the identification of Φ(0) via Eq. (15): Φ(0) = 1 p R∗ λ vuut H (3) u0 2/3 4 − 2H (3) u0 2/3 = 0.1409... (16) The resultin… view at source ↗
Figure 2
Figure 2. Figure 2: Taylor and Corrsin microscales, along with the root-mean- [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the Lagrangian Lyapunov exponents and bifu [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of physical quantities as a function of time start [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of physical quantities as a function of time start [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spatial variation laws at different time instances for the Sa [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Spatial variation laws at different time instances for the Sa [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Turbulent kinetic energy spectra (a) and compensated K [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) Temperature spectra at different time instances. (b [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Local scaling exponents of (a) velocity correlation and ( [PITH_FULL_IMAGE:figures/full_fig_p035_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of the PDF of the longitudinal velocity increment [PITH_FULL_IMAGE:figures/full_fig_p036_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Evolution of the Lagrangian Lyapunov exponents and bifu [PITH_FULL_IMAGE:figures/full_fig_p037_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Evolution of physical quantities as a function of time star [PITH_FULL_IMAGE:figures/full_fig_p038_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of physical quantities as a function of time star [PITH_FULL_IMAGE:figures/full_fig_p039_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Spatial variation laws at different time instances for the L [PITH_FULL_IMAGE:figures/full_fig_p040_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Spatial variation laws at different time instances for the L [PITH_FULL_IMAGE:figures/full_fig_p041_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Turbulent kinetic energy spectra (a) and compensated [PITH_FULL_IMAGE:figures/full_fig_p042_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: (a) Temperature spectra at different time instances. ( [PITH_FULL_IMAGE:figures/full_fig_p043_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Local scaling exponents of (a) velocity correlation and ( [PITH_FULL_IMAGE:figures/full_fig_p044_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Evolution of the Lagrangian Lyapunov exponents and bifu [PITH_FULL_IMAGE:figures/full_fig_p045_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Evolution of physical quantities as a function of time star [PITH_FULL_IMAGE:figures/full_fig_p046_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Evolution of physical quantities as a function of time star [PITH_FULL_IMAGE:figures/full_fig_p047_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Turbulent kinetic energy spectra (a) and compensated [PITH_FULL_IMAGE:figures/full_fig_p048_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: (a) Temperature spectra at different time instances. ( [PITH_FULL_IMAGE:figures/full_fig_p049_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Evolution of physical quantities as a function of time star [PITH_FULL_IMAGE:figures/full_fig_p050_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Characteristic scales vs time at different Prandtl numbe [PITH_FULL_IMAGE:figures/full_fig_p051_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: PDF of the temperature increment ∆ϑ at different Prandtl numbers: P r = 10−3 , 10−2 , 10−1 , 1, 10, 102 , and 103 . 0 and r = 10λT at the end of the development regime (i.e., at t ≃ 2.2), which corresponds to a Taylor-scale Reynolds number of approximately 164. Alongside these curves, the PDF of the velocity increment is also reported (indicated by the red line). Specifically, the figure includes the PDFs… view at source ↗
Figure 28
Figure 28. Figure 28: Temperature spectrum evolution (left) and compensat [PITH_FULL_IMAGE:figures/full_fig_p053_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Temperature spectrum evolution (left) and compensat [PITH_FULL_IMAGE:figures/full_fig_p054_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Obukhov–Corrsin spectral function evolution for [PITH_FULL_IMAGE:figures/full_fig_p055_30.png] view at source ↗

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Reference graph

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