REVIEW 3 major objections 4 minor 64 references
Turbulent puffs in transitional pulsatile pipe flow at moderate pulsation amplitudes
T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read Turbulent puffs survive pulsating pipe flow by actively using the linear instabilities of the instantaneous laminar profile, not just mean shear.
desk verdict A solid DNS map plus a suggestive but not airtight causal argument for inflectional-instability-driven puff survival, and a useful but partly fitted model extension. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central device is the master–slave simulation, in which a slave DNS runs in parallel with a full simulation while its instantaneous mean axial profile is overwritten by an artificial profile constructed to have the same kinetic energy as the master's mean profile but a shear that decreases monotonically from the wall to the centerline, so that all inflection points are erased. Puff decay in the slave marks the causal role of inflectional instability. The second device is the Extended Barkley Model (EBM), a two-variable advection–diffusion–reaction system whose turbulent production term contains a phase-lagged bulk velocity $\bar{U}(t+\varphi)$ and an instability term $\gamma\lambda(t)$, where $\lambda(t)$ is the instantaneous maximum growth rate of the laminar profile evaluated under a quasi-steady assumption and $\varphi(\mathit{Wo})$ is the Womersley phase lag. The EBM also rescales the noise intensity with Reynolds number to capture intermittent splitting states.
What would settle it
Run a master–slave pair at $\mathit{Re}=2100$, $\mathit{Wo}=11$, $A=0.5$ with a control slave profile that keeps the inflection points of the Sexl-Womersley profile but shifts their phase or depth while preserving kinetic energy; if puffs in that control slave decay as quickly as in the inflection-free slave, the decay is caused by a different feature of the profile manipulation rather than by the absence of inflection points.
Extended reading notes
Core claim
At Reynolds numbers $2100 \le \mathit{Re} \le 3000$, Womersley numbers $5 \lesssim \mathit{Wo} \lesssim 19$ and amplitudes $0.5 \le A \le 1$, turbulent puffs in pulsatile pipe flow are not merely modulated by the pulsation: they use the inflection-point instabilities of the time-dependent laminar Sexl-Womersley profile as a survival mechanism. In the master–slave simulations, when the instantaneous mean profile is replaced by an energy-matched, shear-minimized profile without inflection points, puffs that otherwise survive for long times decay within a few pulsation periods, and this happens exactly in the parameter range where helical perturbations have large transient growth on the laminar profile. The extended Barkley model, which adds a phase lag derived from the Womersley solution and a production term proportional to the instantaneous laminar growth rate $\lambda(t)$, reproduces the measured upstream front speeds and the survival/decay thresholds throughout the parameter space. The two production mechanisms are separable: the phase-lagged mean shear acts like a time-varying Reynolds number, and the inflectional instability adds turbulence production only during the phases when the laminar profile is linearly unstable.
Load-bearing premise
The causal conclusion rests on the assumption that the slave profile differs from the master profile only by lacking inflection points, so that the puff decay seen in slave simulations is caused by that loss and not by other side-effects of the energy-matching or shear-minimization procedure, and that the quasi-steady growth rate $\lambda(t)$ faithfully represents the instability that puffs exploit.
Editorial extensions
If this is right
- The survival/decay boundary for puffs becomes a predictable function $\mathit{Re}_c(\mathit{Wo}, A)$, and the Extended Barkley Model reproduces the DNS boundary reasonably well, with better agreement at $A=0.5$ than at $A=1$.
- At high Womersley numbers puffs behave as in steady pipe flow; at low Womersley numbers the dynamics are quasi-steady; at intermediate $5 \lesssim \mathit{Wo} \lesssim 19$ the inflectional instability is the decisive survival mechanism.
- The upstream front speed of puffs decreases as $\mathit{Re}$ and $A$ increase and approaches the steady-pipe value as $\mathit{Wo}$ increases, a trend captured by both DNS and the EBM.
- Removing the inflectional term (setting $\gamma=0$ in the EBM, or erasing inflection points in the slave DNS) makes puffs decay at parameters where they otherwise survive, identifying the instability as a continuous production source rather than a trigger-only effect.
- Because the EBM reduces to the original Barkley model at $A=0$, the extended model is a direct generalization that inherits the steady-pipe front-speed fits while adding the pulsatile mechanisms.
Reading between the lines
- By implication, the same master–slave diagnostic could be applied to other time-periodic shear flows with inflectional instabilities, such as pulsatile channels, oscillatory Stokes layers, or cardiac-type waveforms, to test whether localized turbulence survives by riding instantaneous instabilities there too.
- The phase lag $\varphi(\mathit{Wo})$ means that control or relaminarization strategies based on the instantaneous Reynolds number alone will be mistimed; interventions should be phased relative to the delayed turbulence response.
- Because the EBM overestimates puff lifetime at $\mathit{Re}\lesssim 2050$, $A=1$ when the $\gamma\lambda(t)$ term dominates, the quasi-steady assumption for $\lambda(t)$ likely degrades at low Reynolds numbers, and a non-quasi-steady correction could sharpen the model's decay threshold.
- A testable prediction of the two-source picture is that tailoring the pulsation waveform to suppress inflection points during the low-$\mathit{Re}$ phase, rather than merely reducing amplitude, should shift the survival threshold to higher mean Reynolds numbers.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper investigates localized turbulent puffs in pulsatile pipe flow at Re = 2100-3000, Wo = 5-19, and A = 0.5-1 by means of DNS. It classifies four behaviors (rapid decay, localized structures, stochastic decay, and highly intermittent states), performs master-slave DNS to test whether puffs rely on the inflectional instabilities of the Sexl-Womersley profile, and extends the Barkley model (EBM) with a phase-lagged shear term and an instability term proportional to the instantaneous growth rate of the laminar profile. The authors conclude that puffs survive through two mechanisms: mean-shear production as in statistically steady pipe flow, and the instabilities of the instantaneous pulsatile mean profile. The EBM reproduces front speeds and decay thresholds qualitatively in the explored parameter range.
Significance. The paper provides a substantial DNS dataset (71 simulations), a causal master-slave methodology inspired by recent work, and a low-order model extension that captures several observed features of pulsatile pipe flow. The central claim, if established, advances the mechanistic understanding of puff survival in pulsatile pipe flow by identifying two coexisting turbulence production mechanisms. The open-source code and detailed appendices are assets, and the parametric comparison between DNS and EBM is valuable. However, the causal attribution currently rests on a control that does not fully separate the removal of inflection points from a reduction in mean shear, and the EBM agreement is partly a result of calibrated parameters. The paper is therefore a solid step forward, but the central claim needs additional support before publication.
major comments (3)
- [§IV.B and Appendix A (eqs. A5, A8, A21)] The master-slave causal analysis does not cleanly isolate the removal of inflection points from a reduction in mean shear. The slave profile US is obtained by minimizing the mean shear S in eq. (A5) subject to matching the master's mean kinetic energy EM in eq. (A8), so the blunted profile produced by eq. (A21) differs from the master profile in the entire shear distribution, not only in the absence of inflection points. The transient-growth control in §IV.A is performed on the laminar slave profile US0 (eq. A6, parabolic), not on the blunted turbulent slave profile used in the master-slave DNS. As a result, the rapid decay of puffs in slave simulations could be caused by reduced mean-shear production rather than by the loss of inflectional instability. To support the central claim, the authors should demonstrate that the blunted slave profile preserves the transient growth of streamwise-constant perturbations relative to the master's instantaneous mean profile, or construct an alternative control that removes inflection points while keeping the mean shear unchanged.
- [§III (behavior classification) and Table I] The classification of the four behaviors rests on single DNS runs, and for the StD category the authors explicitly state that life-time statistics were not computed (§III, item 3). Several simulations in Table I are integrated for fewer than two periods (e.g., cases 19 and 33 have N_T = 1.7 and 1.5, respectively), and some 'survive' points in figure 9 correspond to runs of only 4.4-8.7 periods (cases 17, 36 and 37). Since the comparison with the EBM in §V.C and the decay thresholds in figure 9 rely on these classifications, the quantitative robustness of the parametric thresholds is not established. At least a few representative parameters should be repeated with several independent initial conditions to provide lifetime statistics and error bars.
- [§V and Appendix B (eqs. B9, B11, B17)] The agreement between the EBM and DNS is partly a fitting result. The parameters σ(Re) (eq. B11), ε, and γ (eq. B17) are calibrated to match the DNS data, and the paper explicitly states that γ must be correctly fitted or the model fails (Appendix B7). The abstract's claim of 'good agreement … in a broad parametric space' is therefore stronger than the evidence: at A = 1 the EBM decay threshold is near Wo ≈ 10, whereas in DNS it appears closer to Wo ≈ 8 (§V.C). The authors should clearly distinguish calibrated from ab initio parameters and present the EBM as a phenomenological model that is consistent with, rather than an independent confirmation of, the proposed mechanism.
minor comments (4)
- [Abstract and Introduction] The amplitude bounds are written inconsistently (0.5 < A < 1 in the abstract vs 0.5 ≤ A ≤ 1 in the text and conclusions); please unify the notation.
- [References] References [33] and [34] appear in the bibliography but are not cited in the text; either cite them where relevant or remove them from the reference list.
- [Figure 5 caption] The color scale is capped at G ≤ 10^3, which may hide the magnitude of the strongest helical growth; consider rescaling or explicitly noting the cap in the caption.
- [Throughout] The notation 'R e' with a space appears throughout the manuscript; please ensure the typesetting is uniform.
Circularity Check
The EBM's agreement with DNS is partly a fit of σ, γ and ϵ, so its confirmation of the two production mechanisms is partly circular; the master-slave DNS provides independent but confounded support.
-
fitted input called prediction
[Appendix B, §B4b–B4d, eqs. (B11), (B17); §§V.A–V.C]
"In order to better fit the model results to the DNS results we set ϵ = 0.1 and change σ depending on Re: σ = 6/5 · (Re − 1933)/1000 ... We find a good compromise with: γ = min (1, 0.28 log (T)). ... in order to find a better match with the DNS results, it should be slightly decreased ... ϵ is set to half its BM value in the EBM."
The EBM's two production terms, rŪ(t+ϕ) and γλ(t), are inserted into eq. (B9) to represent the hypothesized mechanisms, and their strengths (σ, γ, ϵ) are explicitly calibrated against the DNS. The paper then uses the EBM's agreement with those same DNS and the γ=0/ϕ=0 sensitivity tests to conclude that both mechanisms are required, e.g. 'According to our DNS, causal analysis and the EBM results, turbulence ... makes use of mainly two mechanisms to survive.' Because the parameters were fitted to the DNS being reproduced, the EBM comparison is a fitting exercise rather than an independent prediction, so the model-based part of the conclusion is partly an input of the model. The master-slave causal analysis is independent, which limits the circularity to the EBM branch.
full rationale
The central mechanistic claim—that puffs actively use the inflectional instabilities of the instantaneous Sexl-Womersley profile—rests primarily on the master-slave DNS, which is a controlled numerical experiment and not a fit-based prediction. The transient-growth analysis, the λ(t) computation, and citations to Morón et al. [17], Xu et al. [21], and Feldmann et al. [24] are published, externally checkable results; these self-citations are not load-bearing circularity. The genuinely circular ingredient is the Extended Barkley Model: σ(Re), γ, and ϵ are explicitly chosen 'to better fit the model results to the DNS results' (eqs. B11 and B17, §B4d), and the same DNS are then used to assess the model and to argue that both production mechanisms are necessary. That makes the EBM's agreement a fitting outcome, so the model-derived confirmation of the two-source picture reduces by construction. A separate non-circular limitation is that the slave profile is obtained by minimizing the mean shear S (eq. A5) while matching the master's energy EM (eq. A8); hence removing inflection points and reducing mean-shear production are the same operation, leaving the master-slave causal attribution confounded. This lowers confidence in the causal conclusion but is not a definitional tautology. Overall, the central claim retains independent content, so the circularity score is 4 rather than 6 or higher.
Assumptions & free parameters
free parameters (4)
- EBM noise intensity σ =
σ = (6/5)(Re − 1933)/1000, clamped to [0.2, 0.85]
- EBM relaxation coefficient ϵ =
0.1
- EBM inflectional-instability gain γ =
min(1, 0.28 log T)
- Phase lag approximation constants =
ϕ(Wo) ≈ 32.34° + 35.17° arctan(0.75(Wo − 2))
assumptions (5)
- domain assumption The Barkley model is a valid low-order representation of transitional pipe flow turbulence, and its variables q and u capture the essential physics.
- domain assumption The instantaneous linear stability (eigenvalues) of the Sexl-Womersley profile, computed under a quasi-steady assumption, is continuous in time and can be represented by λ(t) ≥ 0.
- domain assumption Slave profiles with monotonic shear and no inflection points, matched in kinetic energy to the master, isolate the effect of inflectional instability while preserving mean-shear production.
- domain assumption The optimal perturbation localized in a 5D section with |u'0| ≈ 3e-2 triggers a single puff representative of natural transition at these parameters.
- domain assumption DNS grid resolutions as given in tables I-II are sufficient for the observed dynamics.
Cite this review
Pith. "Pith review of Turbulent puffs in transitional pulsatile pipe flow at moderate pulsation amplitudes." pith.science (2026). https://pith.science/paper/WHSACC4P
@misc{pith2026250601494,
author = {Pith},
title = {Pith review of: Turbulent puffs in transitional pulsatile pipe flow at moderate pulsation amplitudes},
year = {2026},
howpublished = {\url{https://pith.science/paper/WHSACC4P}},
note = {Machine review of arXiv:2506.01494}
}
read the original abstract
We show that, in the transitional regime of pulsatile pipe flow, at moderate-to-high amplitudes 0.5 < A < 1, the first long-lived turbulent structures are localized and take the form of the puffs and slugs observed in statistically steady pipe flow. We perform direct numerical simulations at many pulsation frequencies, amplitudes and Re, and observe different dynamics of puffs and slugs. At certain flow parameters we find, using a causal analysis, that puffs actively make use of linear instabilities in the laminar Sexl-Womersley profile to survive the pulsation. Using all these lessons learned, we extend a low order model by Barkley et al., Nature (2015), to reproduce these dynamics. We find a good agreement between the extended model and our numerical results in a broad parametric space of pulsation amplitudes 0.5 < A < 1, frequencies Wo > 5 and 2100 < Re < 3000. With the help of our numerical results, causal analysis and model, we determine that turbulence production has two sources at these flow parameters: the mean shear as in statistically steady pipe flow, and the instabilities of the instantaneous pulsatile mean profile.
Figures
Figures from the paper (8 more)
Reference graph
Works this paper leans on
-
[1]
We classify these decay events as deterministic
First elongation, then rapid decay (RaD) : the initial helical perturbation, used to initialize the flow, first grows in length and magnitude, and then decays in less than one pulsation period (figure 1a). We classify these decay events as deterministic. They are different from decay events that happen (stochastically) after more than one pulsation period...
-
[2]
See an example of this behavior in figure 1b
Localized turbulent structures (Loc) : the initial helical perturbation localizes in a puff, that is then modulated in length and magnitude by the pulsation and survives for long times without splitting or decaying. See an example of this behavior in figure 1b
-
[3]
Localized structures, then stochastic decay (StD) : the initial helical perturbation localizes in structures that are modulated, in length and magnitude, by the pulsation. These structures, however, tend to suddenly decay after typically a short number of pulsation periods (figure 1c). Although we do not explicitly compute life-time statistics of these ca...
-
[4]
Highly intermittent state (Int) : the initial helical perturbation localizes in structures modulated by the pulsation. These structures, however, randomly split until the DNS reaches a highly intermittent state where turbulence aggregates in localized structures modulated by the pulsation and separated by laminar patches (figure 1d). In the following sect...
work page 1936
-
[5]
O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Philosophical Transactions of the Royal society of London , 935 (1883)
-
[6]
M. Avila, D. Barkley, and B. Hof, Transition to turbulence in pipe flow, Annual Review of Fluid Mechanics55, null (2023), https://doi.org/10.1146/annurev-fluid-120720-025957
-
[7]
A. Meseguer and L. N. Trefethen, Linearized pipe flow to reynolds number107, Journal of Computational Physics186, 178 (2003)
work page 2003
-
[8]
B. Eckhardt, T. M. Schneider, B. Hof, and J. Westerweel, Turbulence transition in pipe flow, Annu. Rev. Fluid Mech.39, 447 (2007)
work page 2007
Show all 64 references
-
[9]
Schmid and D
P. Schmid and D. Henningson,Stability and Transition in Shear Flows, Applied Mathematical Sciences (Springer New York, 2000)
2000
-
[10]
Barkley, B
D. Barkley, B. Song, V. Mukund, G. Lemoult, M. Avila, and B. Hof, The rise of fully turbulent flow, Nature526, 550 (2015)
2015
-
[11]
Barkley, Theoretical perspective on the route to turbulence in a pipe, Journal of Fluid Mechanics803, P1 (2016)
D. Barkley, Theoretical perspective on the route to turbulence in a pipe, Journal of Fluid Mechanics803, P1 (2016)
2016
-
[12]
Kühnen, B
J. Kühnen, B. Song, D. Scarselli, N. B. Budanur, M. Riedl, A. P. Willis, M. Avila, and B. Hof, Destabilizing turbulence in pipe flow, Nature Physics14, 386 (2018)
2018
-
[13]
B. Song, D. Barkley, B. Hof, and M. Avila, Speed and structure of turbulent fronts in pipe flow, Journal of Fluid Mechanics 813, 1045–1059 (2017)
2017
-
[14]
K. Chen, D. Xu, and B. Song, Propagation speed of turbulent fronts in pipe flow at high reynolds numbers, Journal of Fluid Mechanics935, A11 (2022)
2022
-
[15]
Frishman and T
A. Frishman and T. Grafke, Mechanism for turbulence proliferation in subcritical flows, Proceed- ings of the Royal Society A: Mathematical, Physical and Engineering Sciences 478, 20220218 (2022), https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2022.0218
2022
-
[16]
Romanò, A
F. Romanò, A. Charles, F. Dottori, and S. Amir Bahrani, Transition to turbulence in a heated non-newtonian pipe flow, Physics of Fluids33, 091702 (2021), https://doi.org/10.1063/5.0064456
2021 doi
-
[17]
Annulareffekt
T. Sexl, Über den von E. G. Richardson entdeckten "Annulareffekt", Zeitschrift für Physik61, 349 (1930)
1930
-
[18]
J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, The Journal of physiology127, 553 (1955)
1955
-
[19]
D. Xu, S. Warnecke, B. Song, X. Ma, and B. Hof, Transition to turbulence in pulsating pipe flow, Journal of Fluid Mechanics 831, 418 (2017)
2017
-
[20]
Xu and M
D. Xu and M. Avila, The effect of pulsation frequency on transition in pulsatile pipe flow, Journal of Fluid Mechanics857, 937 (2018)
2018
-
[21]
Morón, D
D. Morón, D. Feldmann, and M. Avila, Effect of waveform on turbulence transition in pulsatile pipe flow, Journal of Fluid Mechanics 948, A20 (2022)
2022
-
[22]
S. J. Cowley, High frequency rayleigh instability of stokes layers, inStability of time dependent and spatially varying flows (Springer, 1987) pp. 261–275
1987
-
[23]
J. R. A. Nebauer, On the stability and transition of time periodic pipe flow (2019)
2019
-
[24]
D. Xu, A. Varshney, X. Ma, B. Song, M. Riedl, M. Avila, and B. Hof, Nonlinear hydrodynamic instability and turbulence in pulsatile flow, Proceedings of the National Academy of Sciences117, 11233 (2020)
2020
-
[25]
D. Xu, B. Song, and M. Avila, Non-modal transient growth of disturbances in pulsatile and oscillatory pipe flows, Journal of Fluid Mechanics907 (2021)
2021
-
[26]
Pier and P
B. Pier and P. J. Schmid, Optimal energy growth in pulsatile channel and pipe flows, Journal of Fluid Mechanics926, A11 (2021)
2021
-
[27]
J. Kern, M. Beneitez, A. Hanifi, and D. Henningson, Transient linear stability of pulsating poiseuille flow using optimally time-dependent modes, Journal of Fluid Mechanics927, A6 (2021)
2021
-
[28]
Feldmann, D
D. Feldmann, D. Morón, and M. Avila, Spatiotemporal intermittency in pulsatile pipe flow, Entropy23, 10.3390/e23010046 (2021)
2021 doi
-
[29]
J. M. López, D. Feldmann, M. Rampp, A. Vela-Martín, L. Shi, and M. Avila, nscouette – a high-performance code for direct numerical simulations of turbulent taylor–couette flow, SoftwareX11, 100395 (2020)
2020
-
[30]
A. P. Willis, The openpipeflow navier–stokes solver, SoftwareX6, 124 (2017)
2017
-
[31]
C. Weng, S. Boij, and A. Hanifi, Numerical and theoretical investigation of pulsatile turbulent channel flows, Journal of Fluid Mechanics792, 98–133 (2016). 14
2016
-
[32]
Tuerke and J
F. Tuerke and J. Jiménez, Simulations of turbulent channels with prescribed velocity profiles, Journal of Fluid Mechanics 723, 587–603 (2013)
2013
-
[33]
Vela-Martín, The synchronisation of intense vorticity in isotropic turbulence, Journal of Fluid Mechanics913, R8 (2021)
A. Vela-Martín, The synchronisation of intense vorticity in isotropic turbulence, Journal of Fluid Mechanics913, R8 (2021)
2021
-
[34]
Pier and P
B. Pier and P. J. Schmid, Linear and nonlinear dynamics of pulsatile channel flow, Journal of Fluid Mechanics815, 435–480 (2017)
2017
-
[35]
M. C. Brindise and P. P. Vlachos, Pulsatile pipe flow transition: Flow waveform effects, Physics of Fluids30, 015111 (2018), https://doi.org/10.1063/1.5021472
2018 doi
-
[36]
Scarselli, J
D. Scarselli, J. M. Lopez, A. Varshney, and B. Hof, Turbulence suppression by cardiac-cycle-inspired driving of pipe flow, Nature 621, 70 (2023)
2023
-
[37]
Avila, A
M. Avila, A. P. Willis, and B. Hof, On the transient nature of localized pipe flow turbulence, Journal of Fluid Mechanics 646, 127–136 (2010)
2010
-
[38]
Avila, D
K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, The onset of turbulence in pipe flow, Science333, 192 (2011)
2011
-
[39]
Barkley, H
D. Barkley, H. M. Blackburn, and S. J. Sherwin, Direct optimal growth analysis for timesteppers, International journal for numerical methods in fluids57, 1435 (2008)
2008
-
[40]
Barkley, Modeling the transition to turbulence in shear flows, inJournal of Physics: Conference Series, Vol
D. Barkley, Modeling the transition to turbulence in shear flows, inJournal of Physics: Conference Series, Vol. 318 (IOP Publishing, 2011) p. 032001
2011
-
[41]
Barkley, Simplifying the complexity of pipe flow, Physical Review E84, 016309 (2011)
D. Barkley, Simplifying the complexity of pipe flow, Physical Review E84, 016309 (2011)
2011
-
[42]
Shih, T.-L
H.-Y. Shih, T.-L. Hsieh, and N. Goldenfeld, Ecological collapse and the emergence of travelling waves at the onset of shear turbulence, Nature Physics12, 245 (2016)
2016
-
[43]
B. Hof, A. de Lozar, M. Avila, X. Tu, and T. M. Schneider, Eliminating turbulence in spatially intermittent flows, Science 327, 1491 (2010), https://www.science.org/doi/pdf/10.1126/science.1186091. 15 Case R e W o A Nr Nθ Nx R eτ ∆r+ min ∆r+ max ∆Rθ+ ∆x+ N T Behavior 1 2100 8 ...
2010 doi
-
[44]
Condition 1: boundary condition The slave profile must comply with the no-slip boundary condition and thus vanish at the wall: US (r = R, t) = 0. (A1) 16 Case R e W o ANr Nθ Nx R eτ ∆r+ min ∆r+ max ∆Rθ+ ∆x+ N TBehavior 1 2100 9 0.50 80 64 1152 99.35 0.032 1.81 3.25 5.75 5.8 Lo...
-
[45]
Condition 2: time dependency and maximum energy The slave profile must be time dependent. Its bulk velocity¯US (t): ¯US (t) = 2 R2 Z R 0 USrdr, (A2) is set equal to ¯US (t) = r 3EL (t) 2 , (A3) being: EL (t) = 1 πR2 Z 2π 0 Z R 0 1 2 U 2 SW rdrdθ, (A4) the kinetic energy of the...
-
[46]
Given conditions (A1) and (A3), by minimizingS, we obtain profiles whose shear monotonically decreases from the wall to the center-line of the pipe, without inflection points
Condition 3: monotonic shear The average shear: S = 2 R2 Z R 0 1 2 ∂US ∂r 2 rdr, (A5) of the profile must be minimum. Given conditions (A1) and (A3), by minimizingS, we obtain profiles whose shear monotonically decreases from the wall to the center-line of the pipe, without in...
-
[47]
complies with these three initial requirements
Laminar slave mean profile The parabolic profile: US0 (r, t) = 2 ¯US (t) 1 − r R 2 , (A6) whose energy is exactlyEL (t), i.e. complies with these three initial requirements. By dropping the time dependence of ¯US and EL in the notation and settingR = 1, one finds: Z 1 0 U 2 S0...
-
[48]
(A8) If EM ≡ EL, the flow in the master simulation is laminar,US ≡ US0, and the energy of the slave mean profile will be maximum
Condition 4: energy of the slave mean profile The slave mean profile must have the same energy as the master mean profile: 2 R2 Z R 0 1 2 U 2 Srdr = EM (t) = 2 R2 Z R 0 1 2 ⟨ux⟩2 θ,x rdr. (A8) If EM ≡ EL, the flow in the master simulation is laminar,US ≡ US0, and the energy of...
-
[49]
We have dropped the time dependence of¯US, EM, US and therefore L, λL and µL in the notation for clarity
Method of small variations We can express mathematically conditions (A3)–(A8) in a functional: S = 2 Z R 0 L (r, US, U′ S) dr, (A9) to be minimised, beingU ′ S = ∂US ∂r , L = 1 2 U ′2 Sr + λL USr − ¯US 2R + µL 1 2 U 2 Sr − EM 2R (A10) the Lagrangian, andλL and µL two Lagrange ...
-
[50]
This flow field is the resultant field of a previous DNS at similar flow parameters, and has only one localized turbulent structure
Computational set-up of master-slave DNS We initialize each pair of master-slave DNS with the same flow field. This flow field is the resultant field of a previous DNS at similar flow parameters, and has only one localized turbulent structure. In order to integrate simultaneou...
-
[51]
The master simulation is integrated one time step
-
[52]
Using the instantaneous mean profileUM = ⟨ux⟩θ,x of the master, it computes the instantaneous energyEM, see eq. (A8). In our pseudo-spectral code,UM = ⟨ux⟩θ,x, corresponds to the (0, 0) Fourier mode of the axial velocity
-
[53]
Using the corresponding laminar pulsatile pipe flow kinetic energyEL it computes the desired¯US, see eq. (A3)
-
[54]
(A21), that complies with the desired¯US and EM
It uses a Newton-Raphson method to compute theUS profile, eq. (A21), that complies with the desired¯US and EM
-
[55]
(In the code, it overwrites the (0, 0) Fourier mode of the axial velocity of the slave simulation.)
It overwrites the mean profile of the slave simulation and imposesUS instead. (In the code, it overwrites the (0, 0) Fourier mode of the axial velocity of the slave simulation.)
-
[56]
19 R0 R1 ζ D q σ δ ϵ BM 1920 2250 0.79 0.13 ≤ 0.5 0.1 0.2 EBM 1920 2250 0.79 0.13 0.2 ≤ σ ≤ 0.85 0.1 0.1 TABLE III
It integrates one time step the slave simulation, ignoring the evolution of its mean profile. 19 R0 R1 ζ D q σ δ ϵ BM 1920 2250 0.79 0.13 ≤ 0.5 0.1 0.2 EBM 1920 2250 0.79 0.13 0.2 ≤ σ ≤ 0.85 0.1 0.1 TABLE III. BM parameters as described in Barkleyet al.[6] and the value of par...
1920
-
[57]
The former corre- sponds to the turbulence intensity at each axial location and time
The original BM The original BM considers two one-dimensional time-dependent variablesq (x, t) and u (x, t). The former corre- sponds to the turbulence intensity at each axial location and time. According to Barkleyet al. [6] q represents some form of the cross-section integra...
1920
-
[58]
Equations of the EBM In order to adapt the BM to pulsatile pipe flow we introduce several changes. The equations of the EBM read: ∂q ∂t = − u − ζ ¯U (t) ∂q ∂x + fEBM (q, u) + Dq ∂2q ∂x2 + σ (Re) τ (t, x) q, (B7) ∂u ∂t = −u ∂u ∂x + gEBM (q, u), (B8) with fEBM (q, u) = q h r ¯U ...
1933
-
[59]
While the bulk velocity is set by the pulsation, the evolution ofUc (t) can be obtained from the NSE
Extensions to the u equations In pulsatile pipe flow, the laminar center-line velocityUc (t) and bulk velocity¯U (t) are functions of time. While the bulk velocity is set by the pulsation, the evolution ofUc (t) can be obtained from the NSE. First we assume laminar flow, uuu (...
-
[60]
[6] assumed that the turbulence intensity is advected at the center-line velocity u, corrected with the parameterζ
Extensions to the q equations In the original BM, Barkleyet al. [6] assumed that the turbulence intensity is advected at the center-line velocity u, corrected with the parameterζ. In the case of the EBM we multiply the parameterζ in equation (B7) by the bulk velocity ¯U to acc...
-
[62]
Computational set-up of the EBM Equations (B7) and (B8) are integrated following Barkleyet al. [6]. The second order derivatives are discretized with second order central finite differences, and the first order derivatives with a first order upwind scheme. The system is integr...
-
[63]
The EBM at A = 0.0 Note that if one setsA = 0 in the EBM, except for the new definitions ofσ and ϵ, one recovers the original BM. At A = 0 the parameters derived from the laminar flow are constant in time:Uc = 2, ¯U = 1, PG + Fv0 = 0 and λ = 0 and the EBM equations (B7)–(B10) ...
-
[64]
Some of its limitations are inherited from the original BM
Limitations of the EBM When correctly fitted the EBM captures the dynamics of pulsatile pipe flow in a broad parametric regime, but it has some limitations that need to be mentioned. Some of its limitations are inherited from the original BM. One is the intermittent behavior o...
2000
-
[2500]
However, in full DNS, at theseRe the flow usually reaches a highly heterogeneous state, where localized turbulent patches coexist with laminar flow patches, see fig
In this regime, according to the BM, puffs elongate into slugs filling the whole pipe with turbulence, as seen in figure 11b, c and d. However, in full DNS, at theseRe the flow usually reaches a highly heterogeneous state, where localized turbulent patches coexist with laminar...
Reviewed August 7, 2026 · model on record in the stance chip above.
Discussion (0). Sign in to comment.