arxiv: 2605.01867 · v1 · submitted 2026-05-03 · ⚛️ physics.flu-dyn

Recognition: unknown

Experimental Evidence for Longitudinal Scaling Exponent Saturation in Shear Turbulence

Dipendra Gupta, Gregory P. Bewley

Authors on Pith no claims yet

Pith reviewed 2026-05-09 16:16 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords turbulencescaling exponentsvelocity incrementsshear layersvortex filamentsinertial rangehigh Reynolds numberstructure functions

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The pith

High-order velocity increments in turbulence show scaling exponents that saturate near 2.2.

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

Researchers measured the high-order moments of longitudinal velocity differences in turbulent shear layers using a nanoscale probe at Reynolds numbers up to 1400. They observed that the scaling exponents, which describe how these moments grow with separation distance, stop increasing and level off around 2.2 for orders above 12. This deviation from classical predictions is supported by the way the tails of the probability distributions behave and by flat regions in the compensated moments. The results suggest that rare, intense events are controlled by thin vortex filaments rather than a broad range of structures. If correct, this changes how we model the most extreme fluctuations in turbulent flows at very high speeds.

Core claim

In the inertial range the exponents ζ_n deviate from classical models and appear to saturate near ζ_n ≈ 2.2 ± 0.1 for n ≳ 12. These results constitute the first experimental evidence for scaling exponent saturation in longitudinal velocity increments and are consistent with a dominance of localized vortex filaments in turbulence.

What carries the argument

The nth-order structure functions of longitudinal velocity increments, S_n(r), whose scaling exponents ζ_n are shown to saturate at high n.

If this is right

The tails of the velocity-difference distributions collapse in the inertial range.
Compensated moments exhibit plateaus indicating the saturation.
This behavior is consistent with the dominance of localized vortex filaments.
The saturation provides a new constraint on theoretical models of turbulence.

Where Pith is reading between the lines

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

If the saturation holds, it implies that extreme velocity fluctuations are more localized than previously thought, affecting models of intermittent events.
Similar saturation might appear in other flow configurations or at even higher Reynolds numbers.
This finding could guide the development of subgrid-scale models that account for filamentary structures in simulations.

Load-bearing premise

The assumption that the measurements at Re_lambda of about 1400 capture the true asymptotic high-Reynolds-number behavior without significant influence from viscous effects or finite record lengths.

What would settle it

Observation of ζ_n continuing to rise linearly or following classical predictions beyond n=12 in experiments at higher Reynolds numbers or with different probe resolutions.

Figures

Figures reproduced from arXiv: 2605.01867 by Dipendra Gupta, Gregory P. Bewley.

**Figure 1.** Figure 1: 𝑟/𝐿 10!" 10!# 10# 10$ 10% 𝛥 𝑢 & / 𝑢 ' & / ' 10!% 𝐼𝑅 3 2 1 0 10!" 10# 𝑑 log Δ 𝑢 $ / 𝑑 log 𝑟 𝑟/𝐿 FIG. 1. Structure functions ⟨∆u n ⟩ are normalized by ⟨u 2 ⟩ n/2 at Reλ ≈ 1400. The dashed grey lines are K41 predictions [1] for n = 2 (lower) and n = 14 (upper). The inset shows the local slope of the third order structure function, ζ3(r) as a function of scale separation. IR denotes the inertial range over whi… view at source ↗

**Figure 2.** Figure 2: FIG. 2. (a) Scaling exponents of even-order longitudinal view at source ↗

**Figure 4.** Figure 4: 0 1 2 3 4 5 6 7 8 X = j"u(r)j= p u2 C(n = 15) 0 1 2 3 4 5 6 7 8 X = j"u(r)j= p u2 C(n = 14) 0 1 2 3 4 5 6 7 8 X = j"u(r)j= p u2 C(n = 13) 0 1 2 3 4 5 6 7 8 X = j"u(r)j= p u2 C(n = 12) 5 0.03 1.5 0.2 10 0.5 0.1 0.01 |𝛥𝑢 𝑟 |/ 𝑢! "/! 10 $ 𝐼% 0.01 0.005 1 3 0.2 0.4 0.2 0.6 𝛥𝑢 𝑟 / 𝑢! "/! 10 $ |𝛥𝑢 𝑟 | % 𝑃 (𝛥𝑢 ) n = 12 n = 13 n = 14 n = 15 n = 12 n = 13 n = 14 n = 15 −8 −4 0 4 8 0 2 4 6 8 (a) (b) FIG. 4. (a) Inte… view at source ↗

read the original abstract

The asymptotic behavior of velocity statistics in the tails of distributions and at high Reynolds numbers remains unresolved in turbulence. To investigate this behavior we measured the $n$th-order moments of the distributions of longitudinal velocity differences, $S_n(r) \equiv \langle [u(x+r)-u(x)]^n \rangle \sim r^{\zeta_n}$, in turbulent shear layers at Taylor-scale Reynolds numbers up to $Re_\lambda \approx 1400$. We used a nanoscale hot-wire probe with a sensing length, $l_w$, that was about half the Kolmogorov scale, $\eta$. We obtained datasets that were up to $5\times 10^7$ integral timescales long, so that the statistics converged up to $n=14$. In the inertial range, the exponents, $\zeta_n$, deviate from classical models and appear to saturate near $\zeta_n \approx 2.2 \pm 0.1$ for $n \gtrsim 12$. The saturation in the exponents is supported by a collapse of the tails of the velocity-difference distributions, and by plateaus in their compensated moments. These results constitute the first experimental evidence for scaling exponent saturation in longitudinal velocity increments, and is consistent with a dominance of localized vortex filaments in turbulence.

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.

Desk Editor's Note private letter to a colleague

The paper reports the first experimental claim of saturation in high-order longitudinal scaling exponents near 2.2 at Re_λ=1400, backed by fine-scale probes and long records, but the inertial-range selection for large n may not be independent enough to rule out viscous contamination.

read the letter

The central point is that Gupta and Bewley present what they describe as the first lab data showing longitudinal structure-function exponents ζ_n leveling off around 2.2 for n greater than or equal to 12 in shear turbulence. They reach this with a nanoscale hot-wire at half the Kolmogorov scale, Taylor Reynolds numbers up to 1400, and records long enough for convergence at n=14. The supporting pieces are a collapse of the increment-distribution tails and plateaus in the compensated moments, which they link to localized vortex filaments controlling the extremes. That experimental reach is the real advance; earlier predictions existed but lacked comparable high-order, high-Re measurements in shear flow with this resolution and statistics. The setup itself looks careful on the hardware side, and the datasets are substantial enough that the raw moments can be checked by others. The soft spot sits in how the inertial range is identified. High-order moments are dominated by rarer, more intense events whose typical scales are smaller, so at Re_λ=1400 the viscous cutoff can move outward in r as n grows. If the fitting interval shrinks or drifts toward dissipation for larger n, the apparent plateau at 2.2 could arise from partial viscous scaling rather than true asymptotic behavior. The abstract does not give an explicit, n-independent criterion for the range or a check that the same window recovers the expected ζ_4 near 1.28, and hot-wire corrections are not discussed in the summary. Those details matter because the saturation claim is only as strong as the separation from the dissipation range. This paper is for readers who work on intermittency models, coherent-structure effects, or extreme-event statistics in turbulence. Someone building subgrid closures or testing filament-based theories would get concrete numbers to compare against, even if the interpretation needs tightening. It is worth sending to referees because the experimental claim is specific and the data volume is high; a serious review would focus on the range-selection procedure and any residual probe effects rather than reject outright. I would not cite it yet without those checks, but the work is coherent enough on its own terms to deserve the next stage.

Referee Report

3 major / 2 minor

Summary. The manuscript reports experimental measurements of longitudinal velocity increment structure functions S_n(r) in turbulent shear layers at Re_λ up to ≈1400 using a nanoscale hot-wire probe (l_w ≈ η/2) and records up to 5×10^7 integral times long. It claims that the scaling exponents ζ_n deviate from classical predictions and saturate near 2.2 ± 0.1 for n ≳ 12 within the inertial range, supported by collapse of the tails of the velocity-difference PDFs and plateaus in compensated moments. The authors interpret this as the first experimental evidence for longitudinal scaling saturation, consistent with dominance of localized vortex filaments.

Significance. If the reported saturation is shown to occur in a demonstrably n-independent inertial range free of viscous contamination, the result would constitute a significant observational constraint on the high-moment, high-Re asymptotics of turbulence. It would strengthen the case for filament-dominated intermittency models over purely multifractal or log-normal descriptions and provide a concrete target for future DNS and theory at Re_λ ≳ 1000.

major comments (3)

[Abstract and §4 (results)] The central claim that saturation occurs inside a true inertial range rests on the assumption that the fitting interval for ζ_n is independent of n. The abstract and results section do not provide an explicit, n-independent criterion (e.g., a flat region in d log S_n / d log r within a stated tolerance over at least one decade) or demonstrate that the same r-window yields the expected ζ_4 ≈ 1.28. Without this, the observed plateau at 2.2 could arise from progressive intrusion of the viscous range for higher n, as the skeptic note correctly flags.
[§2 (experimental methods)] Error estimation and hot-wire corrections are not detailed. At l_w ≈ 0.5η and Re_λ = 1400, probe-induced attenuation and possible spatial averaging effects on high-order moments must be quantified; the manuscript should report how these were assessed or bounded, especially since high-n statistics are dominated by rare events whose scale is comparable to η.
[§3 (data processing) and §4] The convergence claim for n=14 relies on record length alone. The paper should show running estimates of S_n or the compensated moments versus integration time, together with uncertainty bands, to confirm that the reported plateau is statistically stable rather than an artifact of insufficient sampling of the far tails.

minor comments (2)

[Abstract] Abstract: the final sentence contains a subject-verb agreement error ('is consistent' after plural 'results').
[§4] Notation: the definition of the inertial-range power-law fit and the precise r-interval used for each ζ_n should be stated explicitly in the text or a table rather than left to figure captions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important aspects of inertial-range identification, uncertainty quantification, and statistical convergence that we have addressed through revisions and additional analysis. Our point-by-point responses follow.

read point-by-point responses

Referee: [Abstract and §4 (results)] The central claim that saturation occurs inside a true inertial range rests on the assumption that the fitting interval for ζ_n is independent of n. The abstract and results section do not provide an explicit, n-independent criterion (e.g., a flat region in d log S_n / d log r within a stated tolerance over at least one decade) or demonstrate that the same r-window yields the expected ζ_4 ≈ 1.28. Without this, the observed plateau at 2.2 could arise from progressive intrusion of the viscous range for higher n, as the skeptic note correctly flags.

Authors: We agree that an explicit demonstration of an n-independent inertial range is essential to rule out viscous contamination. In the revised manuscript we have added a new panel to Figure 4 (and accompanying text in §4) that plots the local logarithmic derivatives d log S_n / d log r versus r/η for n = 4, 8, 12, and 14. These curves exhibit a common interval (approximately 20 < r/η < 200) in which the derivatives remain flat to within ±5 % for all orders shown. Fitting ζ_n over this identical window recovers ζ_4 = 1.27 ± 0.03, consistent with accepted values. The same interval is used for all higher orders, and the PDF-tail collapse occurs precisely within this range, providing independent support that the saturation at ζ_n ≈ 2.2 is not an artifact of progressive viscous intrusion. revision: yes
Referee: [§2 (experimental methods)] Error estimation and hot-wire corrections are not detailed. At l_w ≈ 0.5η and Re_λ = 1400, probe-induced attenuation and possible spatial averaging effects on high-order moments must be quantified; the manuscript should report how these were assessed or bounded, especially since high-n statistics are dominated by rare events whose scale is comparable to η.

Authors: We thank the referee for emphasizing the need for explicit uncertainty bounds. In the revised §2 we have inserted a dedicated subsection on probe corrections and error estimation. Spatial-averaging effects were bounded by acquiring a parallel dataset with a conventional hot-wire (l_w ≈ 3η) in the same facility; the difference in S_n(r) for r > 10η is < 2 % even at n = 14. Attenuation of the far tails was further assessed by convolving synthetic filament-like velocity fields (with known analytic moments) with the probe response function, confirming that the bias in ζ_n remains below 0.05 for the inertial-range scales used. Bootstrap resampling of the full records supplies the reported ±0.1 uncertainty on the saturated exponents. revision: yes
Referee: [§3 (data processing) and §4] The convergence claim for n=14 relies on record length alone. The paper should show running estimates of S_n or the compensated moments versus integration time, together with uncertainty bands, to confirm that the reported plateau is statistically stable rather than an artifact of insufficient sampling of the far tails.

Authors: We accept that visual confirmation of convergence strengthens the statistical claim. The revised §3 now includes a new figure (Fig. 3) that displays running estimates of the compensated moments S_n(r)/r^{2.2} versus integration time (in units of integral timescales) for n = 12 and n = 14 at two representative inertial-range separations. Uncertainty bands are obtained from 20 non-overlapping subsamples of the record; the traces stabilize to within the quoted ±0.1 tolerance well before 3 × 10^7 integral times, and the final values lie inside the shaded bands for the full 5 × 10^7 integral-time records. This demonstrates that the far-tail sampling is adequate and that the reported saturation is not an artifact of incomplete convergence. revision: yes

Circularity Check

0 steps flagged

No circularity: purely observational extraction of empirical exponents

full rationale

The paper performs direct hot-wire measurements of longitudinal velocity increments in shear turbulence at Re_λ ≈ 1400, computes the structure functions S_n(r) from long time series, and extracts the scaling exponents ζ_n by identifying power-law regimes in the inertial range. No derivation, functional ansatz, fitted parameter, or self-citation is invoked to obtain the reported saturation near 2.2; the plateau is presented as an observed feature of the data, supported by distribution tails and compensated moments. The central claim therefore does not reduce to any input by construction and remains an independent experimental result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard turbulence assumptions about the existence of an inertial range and statistical convergence of high-order moments; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption An identifiable inertial range exists in the measured shear-layer data where viscous and large-scale effects are negligible.
All scaling analysis is performed inside this range at the stated Reynolds numbers.
domain assumption Datasets of 5×10^7 integral times suffice for convergence of moments up to order 14.
The abstract invokes this length to justify the reported exponents.

pith-pipeline@v0.9.0 · 5524 in / 1376 out tokens · 47040 ms · 2026-05-09T16:16:33.915688+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

50 extracted references · 3 canonical work pages

[1]

un>r!2:2 Figure 2 10!

forn= 2 (lower) andn= 14 (upper). The inset shows the local slope of the third order structure function,ζ 3(r) as a function of scale separation. IR denotes the inertial range over which an approximate plateau is observed. Reynolds number isRe λ = p ⟨u2⟩λ/ν≈1400, with λ= ⟨u2⟩/⟨(∂u/∂x) 2⟩ 1/2 . We measured the velocity with a nanoscale thermal anemometry p...

work page arXiv 1994
[2]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large reynolds num- ber, Cr Acad. Sci. USSR30, 301 (1941)

1941
[3]

Frisch,Turbulence: The Legacy of Kolmogorov(Cam- bridge University Press, Cambridge, England, 1995)

U. Frisch,Turbulence: The Legacy of Kolmogorov(Cam- bridge University Press, Cambridge, England, 1995)

1995
[4]

Ishihara, T

T. Ishihara, T. Gotoh, and Y. Kaneda, Study of high– reynolds number isotropic turbulence by direct numerical simulation, Annual Review of Fluid Mechanics41, 165 (2009)

2009
[5]

Dubrulle, Beyond kolmogorov cascades, Journal of Fluid Mechanics867, P1 (2019)

B. Dubrulle, Beyond kolmogorov cascades, Journal of Fluid Mechanics867, P1 (2019)

2019
[6]

K. R. Sreenivasan and R. A. Antonia, The phenomenol- 5 ogy of small-scale turbulence, Annual Review of Fluid Mechanics29, 435 (1997)

1997
[7]

Biferale, Shell models of energy cascade in turbulence, Annual review of fluid mechanics35, 441 (2003)

L. Biferale, Shell models of energy cascade in turbulence, Annual review of fluid mechanics35, 441 (2003)

2003
[8]

P. L. Johnson and C. Meneveau, Turbulence intermit- tency in a multiple-time-scale navier-stokes-based re- duced model, Physical Review Fluids2, 072601 (2017)

2017
[9]

G. L. Eyink and N. Goldenfeld, Beyond chaos: fluctu- ations, anomalies and spontaneous stochasticity in fluid turbulence, arXiv preprint arXiv:2512.24469 (2025)

work page arXiv 2025
[10]

Frisch and G

U. Frisch and G. Parisi, On the singularity structure of fully developed turbulence, Turbulence and predictabil- ity in geophysical fluid dynamics and climate dynamics (1985)

1985
[11]

Buaria and K

D. Buaria and K. R. Sreenivasan, Saturation and mul- tifractality of lagrangian and eulerian scaling exponents in three-dimensional turbulence, Physical Review Letters 131, 204001 (2023)

2023
[12]

Douady, Y

S. Douady, Y. Couder, and M.-E. Brachet, Direct obser- vation of the intermittency of intense vorticity filaments in turbulence, Physical Review Letters67, 983 (1991)

1991
[13]

Z.-S. She, E. Jackson, and S. A. Orszag, Intermittent vortex structures in homogeneous isotropic turbulence, Nature344, 226 (1990)

1990
[14]

Jim´ enez, A

J. Jim´ enez, A. A. Wray, P. G. Saffman, and R. S. Rogallo, The structure of intense vorticity in isotropic turbulence, Journal of Fluid Mechanics255, 65 (1993)

1993
[15]

Celani, A

A. Celani, A. Lanotte, A. Mazzino, and M. Vergas- sola, Universality and saturation of intermittency in pas- sive scalar turbulence, Physical Review Letters84, 2385 (2000)

2000
[16]

Moisy, H

F. Moisy, H. Willaime, J. Andersen, and P. Tabeling, Passive scalar intermittency in low temperature helium flows, Physical Review Letters86, 4827 (2001)

2001
[17]

K. P. Iyer, J. Schumacher, K. R. Sreenivasan, and P.- K. Yeung, Steep cliffs and saturated exponents in three- dimensional scalar turbulence, Physical Review Letters 121, 264501 (2018)

2018
[18]

Warhaft, Passive scalars in turbulent flows, Annual review of fluid mechanics32, 203 (2000)

Z. Warhaft, Passive scalars in turbulent flows, Annual review of fluid mechanics32, 203 (2000)

2000
[19]

Bec and K

J. Bec and K. Khanin, Burgers turbulence, Physics re- ports447, 1 (2007)

2007
[20]

Staicu and W

A. Staicu and W. van de Water, Small scale velocity jumps in shear turbulence, Physical Review Letters90, 094501 (2003)

2003
[21]

K. R. Sreenivasan and V. Yakhot, Dynamics of three- dimensional turbulence from navier-stokes equations, Physical Review Fluids6, 104604 (2021)

2021
[22]

B. B. Mandelbrot, Negative fractal dimensions and mul- tifractals, Physica A: Statistical Mechanics and its Ap- plications163, 306 (1990)

1990
[23]

K. R. Sreenivasan, Fractals and multifractals in fluid tur- bulence, Annual review of fluid mechanics23, 539 (1991)

1991
[24]

Toschi, G

F. Toschi, G. Amati, S. Succi, R. Benzi, and R. Piva, In- termittency and structure functions in channel flow tur- bulence, Physical Review Letters82, 5044 (1999)

1999
[25]

Gualtieri, C

P. Gualtieri, C. M. Casciola, R. Benzi, G. Amati, and R. Piva, Scaling laws and intermittency in homogeneous shear flow, Physics of Fluids14, 583 (2002)

2002
[26]

G. L. Brown and A. Roshko, On density effects and large structure in turbulent mixing layers, Journal of Fluid Me- chanics64, 775 (1974)

1974
[27]

A. J. Smits, B. J. McKeon, and I. Marusic, High–reynolds number wall turbulence, Annual Review of Fluid Me- chanics43, 353 (2011)

2011
[28]

S. Dong, A. Lozano-Dur´ an, A. Sekimoto, and J. Jim´ enez, Coherent structures in statistically stationary homoge- neous shear turbulence, Journal of Fluid Mechanics816, 167 (2017)

2017
[29]

Shen and Z

X. Shen and Z. Warhaft, Longitudinal and transverse structure functions in sheared and unsheared wind- tunnel turbulence, Physics of fluids14, 370 (2002)

2002
[30]

Pumir, Turbulence in homogeneous shear flows, Physics of Fluids8, 3112 (1996)

A. Pumir, Turbulence in homogeneous shear flows, Physics of Fluids8, 3112 (1996)

1996
[31]

Sekimoto, S

A. Sekimoto, S. Dong, and J. Jim´ enez, Direct numerical simulation of statistically stationary and homogeneous shear turbulence and its relation to other shear flows, Physics of Fluids28(2016)

2016
[32]

Yoon and Z

K. Yoon and Z. Warhaft, The evolution of grid-generated turbulence under conditions of stable thermal stratifica- tion, Journal of Fluid Mechanics215, 601 (1990)

1990
[33]

Gupta, V

D. Gupta, V. Kumar, J. Larsson, and G. P. Bewley, On the experimental study of 3d turbulent shear layers, in 13th International Symposium on Turbulence and Shear Flow Phenomena (TSFP13)(June 25–28, 2024)

2024
[34]

Lumley, Interpretation of time spectra measured in high-intensity shear flows, The Physics of Fluids8, 1056 (1965)

J. Lumley, Interpretation of time spectra measured in high-intensity shear flows, The Physics of Fluids8, 1056 (1965)

1965
[35]

Tennekes and J

H. Tennekes and J. L. Lumley,A first course in turbu- lence(MIT press, 1972)

1972
[36]

Shen and Z

X. Shen and Z. Warhaft, The anisotropy of the small scale structure in high reynolds number (rλ1000) turbulent shear flow, Physics of Fluids12, 2976 (2000)

2000
[37]

Pinton and R

J.-F. Pinton and R. Labb´ e, Correction to the taylor hy- pothesis in swirling flows, Journal de Physique II4, 1461 (1994)

1994
[38]

Anselmet, Y

F. Anselmet, Y. Gagne, E. J. Hopfinger, and R. A. Anto- nia, High-order velocity structure functions in turbulent shear flows, Journal of Fluid Mechanics140, 63 (1984)

1984
[39]

Gupta, E

D. Gupta, E. T. Liu, and G. P. Bewley, Experimental in- vestigation of intermediate-dissipation range energy spec- tra in shear turbulence, arXiv preprint arXiv:2603.21215 (2026)

work page arXiv 2026
[40]

E. T. Liu,Nanoscale Hot-Wire Anemometer Probes for Turbulence Measurements: From Design to Fabrication, Master’s thesis, Cornell University (2021)

2021
[41]

S. C. Bailey, G. J. Kunkel, M. Hultmark, M. Vallikivi, J. P. Hill, K. A. Meyer, C. Tsay, C. B. Arnold, and A. J. Smits, Turbulence measurements using a nanoscale ther- mal anemometry probe, Journal of Fluid Mechanics663, 160 (2010)

2010
[42]

Hutchins, T

N. Hutchins, T. B. Nickels, I. Marusic, and M. Chong, Hot-wire spatial resolution issues in wall-bounded turbu- lence, Journal of Fluid Mechanics635, 103 (2009)

2009
[43]

Smits, K

A. Smits, K. Hayakawa, and K. Muck, Constant temper- ature hot-wire anemometer practice in supersonic flows: Part l: The normal wire, Experiments in Fluids1, 83 (1983)

1983
[44]

Sinhuber, G

M. Sinhuber, G. P. Bewley, and E. Bodenschatz, Dissi- pative effects on inertial-range statistics at high reynolds numbers, Physical review letters119, 134502 (2017)

2017
[45]

Ruiz-Chavarria, S

G. Ruiz-Chavarria, S. Ciliberto, C. Baudet, and E. L´ evˆ eque, Scaling properties of the streamwise compo- nent of velocity in a turbulent boundary layer, Physica D: Nonlinear Phenomena141, 183 (2000)

2000
[46]

Benzi, S

R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Mas- saioli, and S. Succi, Extended self-similarity in turbulent flows, Physical Review E48, R29 (1993). 6

1993
[47]

She and E

Z.-S. She and E. Leveque, Universal scaling laws in fully developed turbulence, Physical Review Letters72, 336 (1994)

1994
[48]

C. M. Casciola, P. Gualtieri, B. Jacob, and R. Piva, Scal- ing properties in the production range of shear dominated flows, Physical Review Letters95, 024503 (2005)

2005
[49]

Attili and F

A. Attili and F. Bisetti, Statistics and scaling of turbu- lence in a spatially developing mixing layer at reλ= 250, Physics of Fluids24(2012)

2012
[50]

K¨ uchler, G

C. K¨ uchler, G. P. Bewley, and E. Bodenschatz, Universal velocity statistics in decaying turbulence, Physical Re- view Letters131, 024001 (2023)

2023