Pith. sign in

REVIEW 4 major objections 6 minor 41 references

Experimental investigation of the turbulent cascade development by injection of single large-scale Fourier modes

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Injecting a single Fourier mode into a round jet produces a cascade of exact harmonic peaks, and a one-dimensional Navier-Stokes projection reproduces their growth and absorption.

desk verdict Solid experimental observation of harmonic cascade from a single injected mode; the claimed simulation support is too thin. read the letter →

arxiv 1908.05613 v3 pith:3PQYBBWH submitted 2019-08-15 physics.flu-dyn

Margherita Dotti , Rasmus Korslund Schlander , Preben Buchhave , Clara Marika Velte This is my paper · ORCID
classification physics.flu-dyn
keywords turbulentcascadetriadinteractionssingleFouriermodeinjectionroundjetvelocitypowerspectrumharmonicgenerationNavier-Stokesnonlinearterminitialconditions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

To see how turbulent energy moves between scales, the paper injects a single-frequency velocity oscillation into a round jet and watches what happens downstream. It finds that exact multiples of the injected frequency appear one after another, as if each harmonic is created by the repeated action of the nonlinear term in the Navier-Stokes equation, and that these peaks eventually dissolve into the background turbulence. The same harmonic development and cascade timing are reproduced by a one-dimensional projection of the Navier-Stokes equation applied to the measured initial velocity trace. If this picture is right, the shape of a turbulent spectrum depends on its initial conditions, and the time it takes for injected modes to be absorbed is a finite, measurable quantity that turbulence models should respect.

What carries the argument

The central object is a single injected Fourier mode, realized experimentally as a narrow-band oscillation at one frequency in a round jet. The mechanism that carries the argument is the quadratic nonlinear term $(\mathbf{u}\cdot\nabla)\mathbf{u}$ of the Navier-Stokes equation: substituting a wave at frequency $f$ yields components at $2f$ and at sum and difference frequencies, so repeated actions generate a cascade of exact harmonics. The computational machinery is a one-dimensional projection of Navier-Stokes onto the instantaneous velocity direction, in which the measured initial velocity time trace is passed in small incremental steps through a fluid control volume with downstream distance converted to evolution time; the finite interaction region is what suppresses odd harmonics such as the third.

What would settle it

Run the same jet experiment with two injected frequencies $f_1$ and $f_2$ that are not integer multiples of each other. The quadratic mechanism predicts new narrow peaks at $f_1+f_2$ and $|f_1-f_2|$ and at their combinations, appearing with the same sequential delays seen in the single-mode case; if those cross-frequency peaks are absent and only a linear superposition of the two separate harmonic cascades is seen, the repeated-triad cascade claim is refuted.

Watch

Extended reading notes

Core claim

The paper claims that the turbulent cascade can be observed directly by injecting a narrow-band oscillation, effectively a single Fourier mode, into a round jet and measuring the velocity power spectrum at increasing downstream distances. The measured spectra show successive generation of higher harmonics at exact integer multiples of the injected frequency, with energy passing back and forth between harmonics while drifting toward higher frequencies, and finally the absorption of all injected structure into the fully developed background turbulence. The paper further claims that this entire development, including the delays between cascade events, is reproduced by a one-dimensional solution of the Navier-Stokes equation in which the measured initial time trace is repeatedly passed through small fluid control volumes along the flow direction. The match supports the view that the repeated action of the nonlinear term, frequency doubling plus sum and difference interactions, is the operative mechanism, and that initial-condition memory persists far downstream rather than being erased by universal small-scale statistics.

Load-bearing premise

The comparison rests on treating downstream distance as the time a fluid parcel has been evolving, using the local mean flow speed, and on believing that a one-dimensional version of the Navier-Stokes equation with pressure left out still captures how energy moves between scales in the real three-dimensional jet.

Editorial extensions

If this is right

  • Higher harmonics appear successively downstream at exact integer multiples of the injected frequency and remain sharp even when submerged in high-intensity turbulence.
  • The injected structure becomes fully absorbed at roughly the same downstream position regardless of Reynolds number, so the absorption time scales inversely with the jet velocity.
  • Initial-condition memory persists far downstream, arguing against a purely universal, equilibrium description of the spectrum.
  • The close match between measured and simulated spectra implies that the omitted pressure-gradient term is not required to capture this harmonic cascade.
  • The observed cascade delays align with delays reported by others, giving an independent experimental handle on the time scales of triad interactions.

Reading between the lines

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

  • A two-frequency injection experiment, using two incommensurate frequencies, should produce all sum and difference peaks with interaction-order delays; measuring those delays would provide a direct test of whether the cascade is driven by quadratic triads rather than by some other transfer process.
  • If the one-dimensional projection indeed captures spectral transfer, the same experimental setup could be used to map triad-interaction time scales as a function of frequency separation, data that could constrain turbulence closures without resolving all triads.
  • The Reynolds-number-independent absorption distance with velocity-dependent absorption time suggests a convective rather than diffusive absorption mechanism; comparing absorption distances across jet diameters and nozzle geometries would test how universal that distance is.
  • For active flow control, the observation implies that a periodic actuator's injected frequency may persist as sharp harmonics far downstream, so control effects could outlast the actuator region and should be included in control-oriented models.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper reports experiments in which a narrow-band oscillation, nominally a single Fourier mode, is injected into a round jet flow via an oscillating airfoil or a vortex-shedding rod, and the downstream development of the streamwise velocity power spectrum is measured. The authors observe the successive appearance of exact harmonic multiples of the injected frequency, a downstream drift of energy toward higher harmonics, and eventual absorption of the peaks into the background turbulent spectrum. In the vortex-shedding configuration, the measured spectra are compared with a one-dimensional projection of the Navier-Stokes equation developed in companion papers, using the measured initial time trace as the simulation input. The paper reports close agreement between measured and simulated spectral development, derives Reynolds-number-dependent absorption times (Table 1), and argues that these results demonstrate initial-condition memory and dynamic cascade delays relevant to turbulence modeling.

Significance. If the claims are substantiated, the experiments provide a clean and visually compelling demonstration of triad-interaction dynamics in a laboratory jet, including the persistence of exact harmonic frequencies and the finite time scale of the cascade. The observations are an independent experimental benchmark that could be valuable for testing spectral-transfer models and for questioning strict universality of the Kolmogorov picture. However, the paper's central quantitative claim—that the spectral development matches a one-dimensional solution of the Navier-Stokes equation—is not currently established because the model omits the pressure gradient and the comparison is only visual. The experimental harmonic-cascade observations themselves are likely sound and are worthy of publication once the validation claims are properly qualified and quantified.

major comments (4)
  1. [§2 and §3.3] There is an internal inconsistency about the pressure gradient. Section 2 states that the main problem of the one-dimensional model is "the inability of computing the pressure gradient" and describes the method as a projection of forces onto the instantaneous velocity direction, which omits pressure. Yet Section 3.3 states that the simulation "uses only the terms in the Navier-Stokes equation without further approximations." Since the pressure gradient is one of the terms of the Navier-Stokes equation, this is a direct contradiction. The claim that the measured spectra closely match a solution of the Navier-Stokes equation is load-bearing for the abstract and conclusions, and it cannot be supported while the pressure term is omitted. The authors should either include a pressure model or explicitly recharacterize the simulation as a reduced model, and they should test the sensitivity of the harmonic cascade and absorption times to the pressure omission.
  2. [§3.3, Fig. 11] The agreement between measurement and simulation is established only visually, at two downstream positions and a single Reynolds number. No error bars, goodness-of-fit statistic, or quantitative spectral distance metric is provided, and the simulation uses the measured initial time trace as its input, so shared spectral features may simply reflect the common initial condition rather than dynamical fidelity. To support the "closely matching" claim, the authors should provide a quantitative comparison across the full downstream evolution, include uncertainty estimates, and ideally test the simulation with different initial conditions, injection amplitudes, or Reynolds numbers to show that the agreement reflects the model dynamics rather than the input.
  3. [§3.2, Table 1] The absorption times in Table 1 depend on converting downstream distance to convection time, yet the paper itself notes in Section 1 that Taylor's hypothesis cannot be assumed to be valid in high-intensity flows. The conversion is described as "integration of the downstream decaying velocity over the downstream distance," but the specific velocity used and the justification for this conversion are not given. Moreover, the statement that peaks are "completely absorbed" around 44D is not backed by an objective absorption criterion or threshold. Without a defined criterion and a validated time conversion, the Reynolds-number-dependence of the absorption time and the comparison to the cascade delays of Josserand et al. [7] are not quantitatively established.
  4. [Figs. 6, 7, 11] The spectral measurements are presented without uncertainty quantification. Given the finite record lengths (400 records of 2 s for LDA, 120 s for HWA, 100 records of 0.2 s for vortex shedding), ensemble or statistical error estimates are feasible and necessary to support claims such as the "center of gravity" of the peaks moving downstream, the relative dominance of harmonics, and the eventual absorption into the background spectrum. The absence of error bars is particularly problematic for the visual comparison in Fig. 11, where the claimed close agreement could be sensitive to statistical fluctuations in the spectral estimates.
minor comments (6)
  1. [§3.1] The phrase "observed to be large enough to scatter enough light for a to produce a satisfactory signal quality" contains a typographical error ("for a to produce").
  2. [References] Reference [5] is an informal URL collected on a specific date rather than a citable publication; a proper source for the DNS visualization should be provided.
  3. [Fig. 7] The caption states that spectra are offset by 10–15 dB between curves, but the precise offset for each curve is not given, making quantitative comparison between downstream positions difficult.
  4. [§3.2] The dimensions of the larger airfoil are given as 50 and 210 mm, but it is unclear which is the chord and which is the span; please specify both dimensions explicitly.
  5. [§3.3] The acquisition parameters are inconsistent: the text mentions a sampling rate of 20 kHz and a sampling interval of 50 µs, which are consistent, but also a "range of 50kHz" for the Hamming window; please clarify the relationship between the window range and the sampling rate.
  6. [§3.2] The statement that the relative absorption distance is "independently of Reynolds number, consistently completely absorbed around approximately the same downstream spatial position, i.e., ∼44D" would benefit from a definition of the absorption threshold used to determine that position.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the downstream spectra are an external experimental benchmark, and the companion 1D simulation is used as an initial-value prediction rather than as a fit to the downstream data.

full rationale

The paper's central measurements are independent of the theoretical claim: the injected narrow-band mode, the downstream spectra, and the observed harmonic peaks are experimentally obtained quantities. The prediction that repeated actions of the quadratic nonlinear term generate exact sum and difference frequencies follows directly from substituting a travelling cosine mode into (u·∇)u, and the experiment is then compared against that consequence rather than being used to define it. The companion 1D simulation is cited from the same authors' prior work, but the current paper gives it a falsifiable role: the measured time trace nearest to the rod is used as the initial condition, and the computed downstream spectrum is compared with a later measured spectrum (Section 3.3, Fig. 11). That is an initial-value comparison, not a fitted-parameter reproduction; the simulation could disagree with the measurement, so the agreement is not guaranteed by construction. The acknowledged omission of the pressure gradient in Section 2 is a modeling approximation and a genuine correctness risk, but it does not make the claimed agreement circular, because the simulation output is not defined in terms of the downstream experimental spectra. Likewise, the conversion of downstream distance to evolution time via the local convection velocity is an interpretive assumption, and the paper itself notes the general limitations of Taylor's hypothesis, yet this assumption is not a hidden fit of the harmonic cascade or absorption times. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no known result is merely relabeled. The visual, single-case nature of the Fig. 11 comparison weakens the strength of the validation, but that is an evidence-quality concern, not circularity. Overall, the derivation chain is not equivalent to its inputs, so the circularity score is 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the 1D projection model from the authors' companion papers, on a Taylor-type convection mapping, and on several unquantified choices (filter cutoff, absorption threshold, interaction region size). The experimental measurements themselves are the independent part; the simulation agreement adds little independent support because the model is by the same authors and is not shipped.

free parameters (4)
  • Spectral window interaction length = not specified
    The suppression of odd harmonics, especially the third harmonic, is explained in the companion paper [20] as a finite interaction region effect; the size of this region is not given and it directly shapes the predicted spectrum.
  • 1D simulation low-pass filter cutoff = not specified
    Section 3.3 states the program uses a rectangular low-pass filter to remove DC and high-frequency noise; the cutoff frequency is not reported, and it affects the generated spectra.
  • Absorption threshold = 44D downstream (approximate)
    The criterion for 'completely absorbed' is not defined; the assertion that absorption occurs at approximately 44 jet diameters for all Reynolds numbers lacks an objective rule.
  • Jet mean velocity decay model = not specified
    Absorption times in Table 1 are computed by integrating downstream velocity over distance; the jet decay law used for the conversion is not stated, introducing a model dependence.
assumptions (6)
  • standard math Incompressible Navier-Stokes equations govern the jet flow.
    All analysis and simulation are based on the NSE; no compressibility or thermal effects are considered.
  • domain assumption The 1D projection of NSE onto the instantaneous velocity direction preserves the spectral energy transfer relevant to cascade development.
    Used throughout Section 3.3; this projection abandons the full 3D velocity field.
  • ad hoc to paper The pressure gradient can be neglected in the 1D model without materially changing the spectral evolution.
    Section 2 admits the pressure term cannot be computed in the model, yet the model is claimed to match experiments.
  • domain assumption Downstream position maps to evolution time via local mean convection velocity (Taylor-type hypothesis).
    Section 2 states downstream position is interpreted as time for a fluid element; the paper notes Taylor's hypothesis is generally invalid in high-intensity flows.
  • domain assumption The injected narrow-band temporal oscillation represents a single Fourier mode.
    Section 3 treats the oscillation as a plane wave, acknowledging this is an approximation for a jet flow.
  • ad hoc to paper The finite interaction region spectral window from companion paper [20] correctly describes the measured spectrum shape.
    Used to explain the weakness of the third harmonic; not independently validated here.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Experimental investigation of the turbulent cascade development by injection of single large-scale Fourier modes." pith.science (2026). https://pith.science/paper/3PQYBBWH

@misc{pith2026190805613,
  author       = {Pith},
  title        = {Pith review of: Experimental investigation of the turbulent cascade development by injection of single large-scale Fourier modes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3PQYBBWH}},
  note         = {Machine review of arXiv:1908.05613}
}
read the original abstract

The current work presents an experimental investigation of the dynamic interactions between flow scales caused by repeated actions of the nonlinear term of the Navier-Stokes equation. Injecting a narrow band oscillation, representing a single Fourier mode, into a round jet flow allows the measurement of the downstream generation and development of higher harmonic spectral components and to measure when these components are eventually absorbed into fully developed turbulence. Furthermore, the dynamic evolution of the measured power spectra observed corresponds well to the measured cascaded delays reported by others. Closely matching spectral development and cascade delays have also been derived directly from a one-dimensional solution of the Navier-Stokes equation described in a companion paper. The results in the current work provide vital information about how initial conditions influence development of the shape of the spectrum and about the extent of the time scales in the triad interaction process, which should be of significance to turbulence modelers.

Figures

Figures reproduced from arXiv: 1908.05613 by the authors.

Figure 1
Figure 1. And even though the net effect of the energy [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 34 canonical work pages

  1. [7]

    Tur- bulence: Does Energy Cascade Exist?

    Josserand C., Le Berre M., Lehner T., Pomeau, Y., “Tur- bulence: Does Energy Cascade Exist?”. Journal of Statisti- cal Physics, 167, 596–625. https://doi.org/10.1007/s10955- 016-1642-5 (2017)

    doi:10.1007/s10955- 2017

  2. [1]

    The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers

    Kolmogorov AN., “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers”, Proc. R. Soc. Lond. A, (1991), 434, 9–13. https://doi.org/10.1098/rspa.1991.0075 (1991)

    arXiv 1991

  3. [2]

    Hydrodynamic turbulence: a 19th century problem with a challenge for the 21st century

    L’vov V., Procaccia I., “Hydrodynamic turbulence: a 19th century problem with a challenge for the 21st century”. In: Boratav O., Eden A., Erzan A. (eds) Turbulence Modeling and Vortex Dynamics. Lecture Notes in Physics, vol 491. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0105026 (1997)

    work page doi:10.1007/bfb0105026 1997

  4. [3]

    Weather Prediction by Numerical Process

    Richardson LF., Lynch P., “Weather Prediction by Numerical Process”. 2nd edn. Cambridge: Cambridge University Press (Cambridge Mathematical Library). https://doi.org/10.1017/CBO9780511618291 (2007)

    work page doi:10.1017/cbo9780511618291 2007

  5. [4]

    The structure of isotropic turbulence at very high Reynolds numbers

    Kraichnan RH., “The structure of isotropic turbulence at very high Reynolds numbers”, Journal of Fluid Mechan- ics. Cambridge University Press, 5(4), pp. 497–543, doi: 10.1017/S0022112059000362 (1959)

    doi:10.1017/s0022112059000362 1959

  6. [5]

    From https : //www.mashayek.com/research collected on 15.03.2020

    work page 2020

  7. [6]

    A 50-Year Retrospective and the Fu- ture

    George WK., “A 50-Year Retrospective and the Fu- ture”, in: Whither Turbulence and Big Data in the 21st Century?, Springer International Publishing, https : //doi.org/10.1007/978− 3− 319− 41217− 7 2 (2017)

    doi:10.1007/978 2017

  8. [8]

    Interacting scales and energy transfer in isotropic turbulence

    Zhou Y., “Interacting scales and energy transfer in isotropic turbulence”, Physics of Fluids A: Fluid Dynamics, 5:10, 2511–2524, https://doi.org/10.1063/1.858764 (1993)

    work page doi:10.1063/1.858764 1993

Show all 41 references
  1. [9]

    Spectrum and energy transfer in steady Burgers turbulence

    Girimaji SS., Zhou Y., “Spectrum and energy transfer in steady Burgers turbulence”, Physics Letters A, 202:4, 279–287, https://doi.org/10.1016/0375-9601(95)00317-V (1995)

    1995 doi

  2. [10]

    Dynamics of direct large-small scale couplings in coherently forced turbulence: concurrent physical- and Fourier-space views

    Yeung PK., Brasseur JG., Wang Q., “Dynamics of direct large-small scale couplings in coherently forced turbulence: concurrent physical- and Fourier-space views”, Journal of Fluid Mechanics. Cambridge University Press, 283, pp. 43–

  3. [11]

    Spectral evolution of the Navier-Stokes equations for low order couplings of Fourier modes

    Brasseur JG., Corrsin S., “Spectral evolution of the Navier-Stokes equations for low order couplings of Fourier modes”, In: Comte-Bellot G., Mathieu J. (eds) Advances in Turbulence. Springer, Berlin, Heidelberg, pp. 152-162, (1987)

    1987

  4. [12]

    Local energy transfer and nonlocal interactions in homogeneous, isotropic tur- bulence

    Domaradzki A., Rogallo RS., “Local energy transfer and nonlocal interactions in homogeneous, isotropic tur- bulence”, Physics of Fluids A: Fluid Dynamics 2, 413, https://doi.org/10.1063/1.857736 (1990)

    1990 doi

  5. [13]

    Triad interactions in a forced turbulence

    Ohkitani K., Kida S., “Triad interactions in a forced turbulence”, Physics of Fluids A: Fluid Dynamics 4, 794, https://doi.org/10.1063/1.858296 (1992)

    1992 doi

  6. [14]

    The Sim- ilarity of Non-local Triad Interactions in Energy Transfer of Isotropic Turbulence

    Bo-Bin W., Gui-Xiang C., Xu C., Zhang ZS., “The Sim- ilarity of Non-local Triad Interactions in Energy Transfer of Isotropic Turbulence”, Chinese Physics Letters, 29:10, 104701, https://doi.org/10.1088/0256-307X/29/10/104701 (2012)

    2012 doi

  7. [15]

    The response of isotropic turbulence to isotropic and anisotropic forcing at large scales

    Yeung K., Brasseur JG., “The response of isotropic turbulence to isotropic and anisotropic forcing at large scales”, Physics of Fluids A: Fluid Dynamics 3, 884. https://doi.org/10.1063/1.857966 (1991)

    1991 doi

  8. [17]

    Rapid distortion theory and the ‘problems’ of turbulence

    Hunt JCR., Carruthers DJ., “Rapid distortion theory and the ‘problems’ of turbulence”, Journal of Fluid Me- chanics. Cambridge University Press, 212, pp. 497–532. doi: 10.1017/S0022112090002075 (1990)

    1990 doi

  9. [18]

    Turbulence without Richardson-Kolmogorov cascade

    Mazellier N., Vassilicos JC., “Turbulence without Richardson-Kolmogorov cascade”, Physics of Fluids 22, 075101, https://doi.org/10.1063/1.3453708 (2010)

    2010 doi

  10. [19]

    Kolmogorov’s hypotheses and global energy spectrum of turbulence

    Liao ZJ., Su WD., “Kolmogorov’s hypotheses and global energy spectrum of turbulence”, Physics of Fluids 27, 041701, https://doi.org/10.1063/1.4916964 (2015)

    2015 doi

  11. [20]

    Dynamic triad interactions and evolving turbulence spectra

    Buchhave P., Velte CM., “Dynamic triad interactions and evolving turbulence spectra”, submitted for review, arXiv:1906.04756 [physics.flu-dyn] (2019) 12 Margherita Dotti et al

    2019

  12. [21]

    A 1D Navier-Stokes Machine and its application to turbulence studies

    Buchhave P., Velte CM., “A 1D Navier-Stokes Machine and its application to turbulence studies”, submitted for review, arXiv:2002.10184 [physics.flu-dyn] (2020)

    2020 arXiv

  13. [22]

    Experimental observa- tion of resonances in modulated turbulence

    Cadot O., Titon JH., Bonn D., “Experimental observa- tion of resonances in modulated turbulence”, Journal of Fluid Mechanics, 485, 161-170, (2003)

    2003

  14. [23]

    Periodically kicked turbulence

    Lohse D., “Periodically kicked turbulence”, Physical Re- view E, 62(4), 4946, (2000)

    2000

  15. [24]

    Staircase scaling of short- time energy transfer in turbulence

    Fang L., Wu T., Bos WJT., “Staircase scaling of short- time energy transfer in turbulence”, Journal of Turbulence, Volume 21, Issue 4, pp. 234-242, (2020)

    2020

  16. [25]

    43:247-272 (2010) https://doi.org/10.1146/annurev-fluid- 122109-160634

    Cattafesta III L.N., Sheplak M., “Actuators for Active Flow Control, Annual Review of Fluid Mechanics, Vol. 43:247-272 (2010) https://doi.org/10.1146/annurev-fluid- 122109-160634

    2010 doi

  17. [26]

    Flow Control: Passive, Active, and Re- active Flow Management

    Gad-el-Hak M., “Flow Control: Passive, Active, and Re- active Flow Management”. Cambridge: Cambridge Univer- sity Press. (2000) doi: 10.1017/CBO9780511529535

    2000 doi

  18. [27]

    (ed.), “Boundary Layer and Flow Control Its Principles and Application, eBook ISBN: 9781483226668, Pergamon (1961)

    Lachmann GV. (ed.), “Boundary Layer and Flow Control Its Principles and Application, eBook ISBN: 9781483226668, Pergamon (1961)

    1961

  19. [28]

    Evolution of a spectrally local disturbance in grid-generated, nearly isotropic turbulence

    Kellogg RM., Corrsin S., “Evolution of a spectrally local disturbance in grid-generated, nearly isotropic turbulence”, Journal of Fluid Mechanics. Cambridge University Press, 96(4), pp. 641669. doi: 10.1017/S0022112080002297 (1980)

    1980 doi

  20. [29]

    Measurement of turbulent spatial structure and kinetic energy spectrum by exact temporal-to-spatial mapping

    Buchhave P., Velte CM., “Measurement of turbulent spatial structure and kinetic energy spectrum by exact temporal-to-spatial mapping”. Physics of Fluids 29, 085109, https://doi.org/10.1063/1.4999102 (2017)

    2017 doi

  21. [30]

    Stochastic Tools in Turbulence

    Lumley JL., “Stochastic Tools in Turbulence”, Academic Press, ISBN: 9780323162258 (1970)

    1970

  22. [31]

    Turbulence spec- tra from a tidal channel

    Grant HL., Stewart RW., Moilliet A., “Turbulence spec- tra from a tidal channel”, Journal of Fluid Mechanics, 12(2), 241-268, (1962)

    1962

  23. [32]

    Spectra of Turbulence at High Reynolds Number

    Gibson MM., “Spectra of Turbulence at High Reynolds Number”, Nature 195, 1281–1283, https://doi.org/10.1038/1951281a0 (1962)

    1962 doi

  24. [33]

    Turbulence measure- ments in axisymmetric jets of air and helium. Part 1. Air jet

    Panchapakesan NR., Lumley JL., “Turbulence measure- ments in axisymmetric jets of air and helium. Part 1. Air jet”, Journal of Fluid Mechanics, Cambridge University Press, 246, pp. 197–223, doi: 10.1017/S0022112093000096 (1993)

    1993 doi

  25. [34]

    Velocity measure- ments in a high-Reynolds-number, momentum-conserving, axisymmetric, turbulent jet

    Hussein HJ., Capp SP., George WK., “Velocity measure- ments in a high-Reynolds-number, momentum-conserving, axisymmetric, turbulent jet”, Journal of Fluid Mechan- ics. Cambridge University Press, 258, pp. 31–75. doi: 10.1017/S002211209400323X (1994)

    1994 doi

  26. [35]

    Turbulence: An Introduction to Its Mecha- nism and Theory

    Hinze JO., “Turbulence: An Introduction to Its Mecha- nism and Theory”, 2nd ed., Mcgraw-Hill College, (1975)

    1975

  27. [36]

    Two-point Similarity in the Round Jet

    Ewing D., Frohnapfel B., George WK., Pedersen JM., Westerweel J., “Two-point Similarity in the Round Jet”, Journal of Fluid Mechanics, Cambridge University Press, 577, pp. 309330. doi: 10.1017/S0022112006004538 (2007)

    2007 doi

  28. [37]

    Experimental Evaluation of Kolmogorov’s −5/3 and 2 /3 Power Laws in the Developing Turbulent Round Jet

    Yaacob MR., Schlander RK., Buchhave P., Velte CM., “Experimental Evaluation of Kolmogorov’s −5/3 and 2 /3 Power Laws in the Developing Turbulent Round Jet”. Jour- nal of Advanced Research in Fluid Mechanics and Thermal Sciences, 45(1), 14-21, (2018)

    2018

  29. [38]

    Validation of improved laser Doppler anemometer (LDA) based on the fully developed turbulent round jet

    Yaacob MR., Schlander RK., Buchhave P., Velte CM., “Validation of improved laser Doppler anemometer (LDA) based on the fully developed turbulent round jet”. In M. Md Ghazaly, A. Zaki Shukor, G. Chin Kim, R. Ranom, Z. Rasin, M. Azri, A. Anas Yusof, S. Ahmad Radzi, N. Nordin, M....

    2018

  30. [39]

    Mapping of the turbulent round jet develop- ing region using a constant temperature anemome- ter (CTA)

    Yaacob MR., Schlander RK., Buchhave P., Velte CM., “Mapping of the turbulent round jet develop- ing region using a constant temperature anemome- ter (CTA)”. Malaysian Journal of Fundamental and Applied Sciences, (Special Issue on Natural Sci- ences and Mathematics (ESCon 2018)...

    2018 doi

  31. [40]

    Contraction design for small low- speed wind tunnels

    Bell JH., Mehtaz RD., “Contraction design for small low- speed wind tunnels”, Technical Report NASA-CR-177488, NAS 1.26:177488 (1988)

    1988

  32. [41]

    Experimental Investigation of the Turbulent Axisymmetric Jet

    Capp SP., “Experimental Investigation of the Turbulent Axisymmetric Jet”. PhD-dissertation, University at Buf- falo, May (1983)

    1983

  33. [95]

    doi: 10.1017/S0022112095002230 (1995)

    1995 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.