Pith. sign in

REVIEW 4 major objections 5 minor 59 references

Reinforcement-learning-based control of turbulent channel flows at high Reynolds numbers

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read A deep-reinforcement-learning policy beats classical opposition control for turbulent channel drag reduction at all tested Reynolds numbers, reaching 27.7% at Re_tau ≈ 1000.

desk verdict First DRL control at Re_tau=1000, with a self-consistent mechanism story; the headline DR numbers are real for the simulated box, but the short domain and missing seed statistics make them conditional. read the letter →

arxiv 2501.01573 v1 pith:CWQ3THTQ submitted 2025-01-02 physics.flu-dyn

Zisong Zhou , Mengqi Zhang , Xiaojue Zhu This is my paper · ORCID
classification physics.flu-dyn
keywords deepreinforcementlearningturbulentchannelflowdragreductionoppositioncontrolvirtualwallReynoldsstressbudgetamplitudemodulationdirectnumericalsimulation
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

The paper claims that a deep-reinforcement-learning (DRL) agent can find wall blowing-and-suction control laws for fully developed turbulent channel flow that remove 35.6% of the drag at Re_tau ≈ 180, 30.4% at Re_tau ≈ 550, and 27.7% at Re_tau ≈ 1000, outperforming the classical opposition control benchmark in each case. The learned policy senses streamwise velocity fluctuations at y+ = 15 and acts by blowing beneath high-speed near-wall streaks and sucking beneath low-speed streaks. The authors trace the drag reduction to a kinematic mechanism (elevating the virtual wall) and a dynamic mechanism (suppressing the redistribution term that feeds wall-normal velocity fluctuations, reducing Reynolds-stress production and skin friction). The declining benefit at higher Reynolds numbers is blamed on amplitude modulation of outer large-scale structures, which raises the residual Reynolds stress on the virtual wall.

What carries the argument

Two pieces carry the argument. The virtual wall theory of Hammond et al. (1998) provides the kinematic lens: blowing and suction create a height y_vw at which wall-normal velocity fluctuations are minimal, and drag reduction is governed by how high that wall sits and how much residual Reynolds stress -<u'v'> leaks through it. The budget-equation analysis provides the dynamic lens: the redistribution term Phi_22 = (2/rho)<p' ∂v'/∂y> in the transport equation for <v'v'>, which normally transfers turbulent kinetic energy into wall-normal fluctuations as part of the near-wall self-sustaining cycle, is suppressed by the learned control in the buffer layer, lowering <v'v'>, then Reynolds-stress production, then skin friction.

What would settle it

Run the C1000-3 policy in the same code with a streamwise domain of at least 8πh at Re_tau ≈ 1000; if the drag-reduction rate moves by more than a few percent from 27.7%, or if the residual Reynolds stress at the virtual wall no longer tracks large-scale high-speed regions, then the finite box size is materially shaping the conclusion.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that a TD3-trained DRL policy using near-wall streamwise velocity fluctuations at y+ = 15, with wall blowing and suction clipped to [-3u_tau, 3u_tau], outperforms opposition control at all three Reynolds numbers studied and remains effective at Re_tau ≈ 1000, a regime where DRL turbulence control had not previously been demonstrated. The policy's action is not v'-opposition: its blowing and suction correlate with near-wall u' (R ≈ 0.71) rather than with v' (R ≈ -0.10), blowing under high-speed streaks. The paper further argues that the drag reduction arises because the control redistributes turbulent kinetic energy away from wall-normal fluctuations—observed as a drop in the redistribution term Phi_22 in the wall-normal kinetic energy budget—which weakens Reynolds-stress production P_12 and reduces skin friction through the FIK identity. At high Reynolds number this mechanism is partially defeated by amplitude modulation of large-scale structures, which concentrates residual Reynolds stress at the virtual wall beneath large-scale high-speed regions.

Load-bearing premise

The reported drag-reduction numbers and the amplitude-modulation explanation assume that the simulation box used at Re_tau ≈ 1000 (streamwise length 2πh) is long enough and the grid fine enough to resolve the large-scale outer structures whose modulation of the near-wall field is blamed for the loss of effectiveness.

Editorial extensions

If this is right

  • At every Reynolds number tested, an expanded wall-action range raises the drag reduction, so the learned policy exploits wider actuation authority rather than a single optimal amplitude.
  • The power-saving ratio for the DRL policy is higher than opposition control at Re_tau ≈ 550 but lower at Re_tau ≈ 180, so actuator energy cost, not just drag reduction, decides which regime benefits practically.
  • Because the policy suppresses the redistribution term that feeds wall-normal fluctuations, it weakens the near-wall self-sustaining cycle, which should make the control compatible with other streak-suppression techniques.
  • At Re_tau ≈ 1000, the virtual-wall residual Reynolds stress is more than ten times its value at Re_tau ≈ 180, which quantitatively explains why drag reduction declines even while the learned policy remains effective.

Reading between the lines

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

  • Beyond the paper: if amplitude modulation is the cause of the high-Re loss, giving the DRL agent an input that senses or anticipates the outer large-scale high-speed regions should recover part of the lost drag reduction.
  • Beyond the paper: the strong u' correlation suggests the learned policy approximates a thresholded streak-targeting law; a simple rule such as blowing with clipped positive u' could be tested to see how much of the 27.7% comes from the policy's nonlinearity.
  • Beyond the paper: because the residual Reynolds stress at the virtual wall grows more than tenfold from Re_tau ≈ 180 to 1000, a longer-domain study at Re_tau ≈ 1000 is the natural next check on whether the 27.7% figure is box-size independent.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper trains TD3 deep-reinforcement-learning policies for wall blowing and suction in direct numerical simulations of turbulent channel flow at Re_tau ≈ 180, 550, and 1000, using the streamwise velocity fluctuation u' at y+ = 15 as the state and the integrated turbulent kinetic energy reduction as the reward. The authors report drag reductions of 35.6%, 30.4%, and 27.7% respectively, exceeding their own opposition-control baseline. They interpret the results through virtual-wall kinematics and budget-equation analysis, concluding that the DRL policy elevates the virtual wall and suppresses the redistribution term Phi_22, while at higher Reynolds numbers amplitude modulation of outer large-scale structures increases residual Reynolds stress on the virtual wall and reduces the achievable drag reduction. Appendix A documents tests with alternative states and rewards and a grid-refinement check at Re_tau ≈ 550.

Significance. If the results hold, this is the first DRL-based control of turbulent channel flow at Re_tau ≈ 1000, and it demonstrates that a policy trained on TKE reduction, not drag reduction, can outperform classical opposition control. The paper is careful to use a reward that is not the headline metric, which partially addresses circularity concerns, and it provides an opposition-control baseline from the same code as well as Appendix A checks of alternative inputs/rewards. Its main significance lies in the combination of high Reynolds number, a learned nonlinear policy, and a mechanistic explanation in terms of virtual-wall height and pressure redistribution. The headline numbers and the amplitude-modulation mechanism, however, rest on assumptions about domain size and statistical robustness that are not yet verified.

major comments (4)
  1. [Table 1; §3.3, Figs. 8–9; Appendix A] The Re_tau ≈ 1000 headline and the amplitude-modulation mechanism rest on a box with Lx = 2πh ≈ 6.3h and Lz = πh. Published channel and boundary-layer DNS show that log-region large-scale and very-large-scale motions have streamwise extents of order 10h or more, so a 6.3h periodic domain is likely to truncate or alias the structures the paper invokes in Figs. 8–9. Appendix A refines the grid only at Re_tau ≈ 550 and does not vary the domain size at any Reynolds number; no resolution check is reported at Re_tau ≈ 1000. The observed growth of residual Reynolds stress on the virtual wall and the decline of DR with Re may therefore be partly a numerical-box effect rather than a physical Reynolds-number effect. I recommend adding a domain-size test at Re_tau ≈ 1000 (e.g., Lx = 4πh or 6πh) with at least the same wall resolution, or explicitly softening the mechanism claim.
  2. [§3.1, Fig. 2, Table 4] Each configuration is represented by a single training run and the model is selected at episode 20 without reporting random seeds, initial conditions, or error bars on DR. Figure 2 shows noticeable episode-to-episode oscillation, and Table 5 indicates that alternative choices (C550-v15, C550-u20) give DR values within a few percent of the headline values. Without repeated seeds, the differences among the -1, -2, -3 action ranges and the comparison against opposition control cannot be distinguished from selection noise. Please provide at least three independent seeds per case with mean ± std for DR (or equivalent convergence diagnostics), and state the selection criterion objectively.
  3. [§3.3, Table 4, Eq. (3.2)] The virtual-wall residual Reynolds stress −⟨u′v′⟩vw is computed from the full controlled velocity field, which includes the direct kinematic effect of the imposed wall blowing and suction; at the wall v′ = v′w, and with |v′w| up to 3u0τ the actuation can contribute to ⟨u′v′⟩ at y+ = O(10). The paper then interprets the growth of −⟨u′v′⟩vw with Re as evidence for amplitude modulation of outer structures. This interpretation is established only if the actuation-induced contribution is removed or shown negligible; otherwise the 'residual' stress in Table 4 conflates the control itself with the physical mechanism being inferred. I suggest quantifying this contribution (e.g., by decomposing the field into actuation-induced and turbulent parts, or by evaluating the budget of ⟨u′v′⟩ across y+vw).
  4. [§3.3, Fig. 9] The joint p.d.f. in Fig. 9 is a correlational diagnostic: it shows that large values of |H(⟨u′v′⟩vw)| tend to occur beneath large-scale high-speed regions. This does not establish that amplitude modulation causes the residual stress; the same pattern could arise from superposition of the outer footprint at y+ = y+vw (the paper argues against this on scale grounds, but the statistical test is not shown) or from the actuation pattern itself. The causal phrasing 'significantly increases' in the abstract and §3.3 goes beyond the p.d.f. evidence. A conditional test — e.g., amplitude-modulation coefficient of the reconstructed small-scale envelope conditioned on large-scale sign, or a phase-averaged comparison — would make the mechanism claim load-bearing.
minor comments (5)
  1. [§2.1] There is a typo: 'AFiD ... was utilized to carried out the DNS' should read 'was utilized to carry out the DNS'.
  2. [Fig. 2 caption] The caption appears to be missing the line-style markers for the three cases; the glyphs after ':' are blank in the typeset version, so the reader cannot identify which line corresponds to suffix '-1', '-2', or '-3'.
  3. [Appendix A, first paragraph] The grid-refinement test at Re_tau ≈ 550 is described only as 'refined by a factor of 2'; please specify the actual grid sizes and the friction Reynolds number of the refined case so the test is reproducible.
  4. [§3.4, after Eq. (3.5)] The sentence 'the sum of the redistribution terms for the three velocity components ... is 0' should explicitly state Φ11 + Φ22 + Φ33 = 0 rather than 'is 0', to avoid the impression that each term individually vanishes.
  5. [§3.3, Fig. 9] The choice θ_L = 13° is stated to be robust in the range 11°–15°, but no sensitivity plot is shown; a brief statement of the range of results across θ_L would make the diagnostic more convincing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the drag-reduction numbers are independently measured outcomes of a policy trained on a TKE reward, with no fitted constants equating the two.

full rationale

The paper's central claim is that DRL-trained wall blowing and suction achieve measured drag reduction rates (35.6%, 30.4%, 27.7% at Re_tau ≈ 180, 550, 1000). The training reward is integrated TKE reduction (r = 1 − e/e0, §2.2, eq. 2.6), not drag reduction; the reported DR values are then computed from the controlled DNS friction velocity. No parameter is fit to the DR values, so the headline is not a fitted input called a prediction. Appendix A explicitly tests the surrogate-reward question: retraining with the drag-reduction rate as reward gives DR = 30.3% versus 30.4% for the TKE reward, showing the DR outcome is not forced by the choice of reward definition. The mechanistic chain (virtual wall height, residual Reynolds stress, redistribution term Φ22, production P12) is inferred post hoc from the controlled DNS and budget equations; these are physical diagnostics, not inputs to the training. The paper cites prior DRL work (Lee et al. 2023) for the TD3 algorithm and hyperparameters, but this is a methodological borrowing, not a self-citation of a load-bearing uniqueness or existence result; the authors' own prior code (AFiD, Zhu et al. 2018) is used only as a numerical solver. There is no self-definitional step, no ansatz smuggled in via citation that determines the conclusion, and no renaming of a known result as a new mechanism. The main caveat—that the Re_tau ≈ 1000 domain size (Lx = 2πh) and resolution are not validated at that Reynolds number, since the grid-refinement test in Appendix A is only at Re_tau ≈ 550—is a numerical-convergence and physical-domain concern, not circular reasoning. It affects the reliability of the high-Re mechanism attribution but does not make the derivation equivalent to its inputs. Therefore the circularity score is 0.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard turbulence modeling and DNS plus a set of hand-chosen DRL hyperparameters and analysis thresholds. No physically invented entities are introduced. The most consequential choices are the state plane, the action amplitude bound, and the 20-episode stopping rule, all of which can affect the reported drag-reduction numbers.

free parameters (5)
  • Detection plane y+ = 15 = 15 wall units
    State input location chosen following Sonoda et al. (2023); Appendix A shows moving it to y+ = 20 reduces DR from 30.4% to 24.9% at Re about 550, so the headline results depend on this hand-chosen location.
  • Action amplitude bound = 3 u_tau (pre-control)
    Headline DR values correspond to the widest tested action bound; DR increases monotonically with the bound, so the maximum rates are tied to this design choice rather than an optimized amplitude.
  • Episode stopping rule = 20 episodes
    Models selected at 20 episodes based on reward curves; the paper states longer training did not improve performance but does not show it, and no seed-averaged stopping criterion is reported.
  • Inclination angle theta_L = 13 degrees
    Used to compute downstream displacement in the amplitude-modulation joint p.d.f. analysis; paper states results are robust for 11 to 15 degrees.
  • State step dt+ = 48.4 to 59.3 wall units
    Time between control updates chosen to satisfy dt+ greater than 25 from optimal control; this temporal discretization affects the achievable DR and is inherited from prior design reasoning, not optimized.
assumptions (6)
  • standard math Incompressible Navier-Stokes equations with a body force maintaining constant bulk velocity
    Governing equations in section 2.1; standard physical model for the DNS.
  • domain assumption Periodic streamwise and spanwise directions and no-slip walls, with lower-wall transpiration control
    Channel setup in section 2.1; the domain size is part of this assumption and affects the large-scale structures captured.
  • domain assumption DNS grid resolutions in Table 1 are adequate
    Grid refinement is tested only for Re_tau about 550 in Appendix A, not for Re_tau about 1000.
  • domain assumption TD3 DRL training with n=5 and gamma=0.95 reaches a stable policy within 20 episodes
    Supported by reward curves in Figure 2, but no seed variation or retraining statistics are reported.
  • standard math Virtual wall theory of Hammond et al. (1998), FIK identity, and Reynolds-stress budget equations apply to the controlled flow
    Used in sections 3.3 and 3.4; these are established tools in the turbulence-control literature.
  • domain assumption Amplitude modulation of near-wall turbulence by outer large-scale structures is the correct causal framework, with outer signal at yO+ about 3.9 sqrt(Re_tau) and inclination angle 13 degrees
    Invoked in section 3.3 to attribute residual Reynolds stress on the virtual wall; depends on the domain capturing those outer structures.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Reinforcement-learning-based control of turbulent channel flows at high Reynolds numbers." pith.science (2026). https://pith.science/paper/CWQ3THTQ

@misc{pith2026250101573,
  author       = {Pith},
  title        = {Pith review of: Reinforcement-learning-based control of turbulent channel flows at high Reynolds numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWQ3THTQ}},
  note         = {Machine review of arXiv:2501.01573}
}
read the original abstract

Deep reinforcement learning (DRL) is employed to develop control strategies for drag reduction in direct numerical simulations (DNS) of turbulent channel flows at high Reynolds numbers. The DRL agent uses near-wall streamwise velocity fluctuations as input to modulate wall blowing and suction velocities. These DRL-based strategies achieve significant drag reduction, with maximum rates of 35.6% at Re_{\tau}=180, 30.4% at Re_{\tau}=550, and 27.7% at Re_{\tau}=1000, outperforming traditional opposition control methods. Expanded range of wall actions further enhances drag reduction, although effectiveness decreases at higher Reynolds numbers. The DRL models elevate the virtual wall through blowing and suction, aiding in drag reduction. However, at higher Reynolds numbers, the amplitude modulation of large-scale structures significantly increases the residual Reynolds stress on the virtual wall, diminishing the drag reduction. Analysis of budget equations provides a systematic understanding of the drag reduction dynamics behind. DRL models reduce skin friction by inhibiting the redistribution of wall-normal turbulent kinetic energy. This further suppresses the wall-normal velocity fluctuations, reducing the production of Reynolds stress, thereby decreasing skin friction. This study showcases the successful application of DRL in turbulence control at high Reynolds numbers and elucidates the nonlinear control mechanisms underlying the observed drag reduction.

Figures

Figures reproduced from arXiv: 2501.01573 by the authors.

Figure 1
Figure 1. The flow chart of reinforcement-learning-driven c [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The evolution of the normalized reward over episod [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Mean velocity profile under different control strat [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Wall-normal distributions of the velocity fluctua [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Wall-normal distributions of the averaged Reynol [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Joint probability density function of the wall blo [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Premultiplied spanwise energy spectra kzEuu of streamwise velocity fluctuations u ′ under different control strategies. For Re0 τ ≈ 180, (a) C180-0, (b) C180-opp, (c) C180-3. For Re0 τ ≈ 550, (d) C550-0, (e) C550-opp, (f ) C550-3. For Re0 τ ≈ 1000, (g) C1000-0, (h) C10…
Figure 8
Figure 8. Figure 8: Instantaneous distributions of u ′ on (x, z) plane at (a)(c)(e)(g) y + = y + vw and (b)(d)(f )(h) y + = 150. (a)(b) case C1000-opp, (c)(d) C1000-1, (e)(f) C1000-2, (g)(h) C1000-3. The black rectangles represent some sample areas on the virtual wall where velocity fluct…
Figure 9
Figure 9. Figure 9: Joint probability density function of the streamw [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Wall-normal distributions of the budget terms of [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Wall-normal distributions of the budget terms of [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 12
Figure 12. Figure 12: Instantaneous distributions of u ′ on (x, z) plane at y + = 20. (a) case C180-0, (b) C180-3, (c) C1000-0, (d) C1000-3. components, namely Φ11 = (2/ρ)hp ′∂u′/∂xi, Φ22 = (2/ρ)hp ′∂v′/∂yi, and Φ33 = (2/ρ)hp ′∂w′/∂zi, is 0. Above the viscous sublayer, the turbulent kineti…
Figure 13
Figure 13. Figure 13: Wall-normal distributions of the budget terms of [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Schematic diagram of the dynamic mechanism throu [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 51 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    del \'A lamo, J. C. & Jim \'e nez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels . Physics of Fluids 15 (6), L41--L44

    work page 2003

  4. [4]

    del \'A lamo, J. C. , Jim \'e nez, J. , Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels . Journal of Fluid Mechanics 500 , 135--144

    work page 2004

  5. [5]

    Balakumar, B. J. & Adrian, R. J. 2007 Large-and very-large-scale motions in channel and boundary-layer flows . Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 365 (1852), 665--681

    work page 2007

  6. [6]

    Journal of Fluid Mechanics 447 , 179--225

    Bewley, Thomas R , Moin, Parviz & Temam, Roger 2001 Dns-based predictive control of turbulence: an optimal benchmark for feedback algorithms . Journal of Fluid Mechanics 447 , 179--225

    work page 2001

  7. [7]

    Applied Mechanics Reviews 67 (5), 050801

    Brunton, Steven L & Noack, Bernd R 2015 Closed-loop turbulence control: Progress and challenges . Applied Mechanics Reviews 67 (5), 050801

    work page 2015

  8. [8]

    Physics of Fluids 14 (11), 4069--4080

    Chang, Yong , Collis, S Scott & Ramakrishnan, Srinivas 2002 Viscous effects in control of near-wall turbulence . Physics of Fluids 14 (11), 4069--4080

    work page 2002

Show all 59 references
  1. [9]

    Journal of Fluid Mechanics 262 , 75--110

    Choi, Haecheon , Moin, Parviz & Kim, John 1994 Active turbulence control for drag reduction in wall-bounded flows . Journal of Fluid Mechanics 262 , 75--110

    1994

  2. [10]

    Physics of Fluids 23 (2)

    Chung, Yongmann M & Talha, Tariq 2011 Effectiveness of active flow control for turbulent skin friction drag reduction . Physics of Fluids 23 (2)

    2011

  3. [11]

    & Xu, C.-X

    Deng, B.-Q. & Xu, C.-X. 2012 Influence of active control on STG -based generation of streamwise vortices in near-wall turbulence . Journal of Fluid Mechanics 710 , 234--259

    2012

  4. [12]

    Physics of fluids 14 (11), L73--L76

    Fukagata, Koji , Iwamoto, Kaoru & Kasagi, Nobuhide 2002 Contribution of reynolds stress distribution to the skin friction in wall-bounded flows . Physics of fluids 14 (11), L73--L76

    2002

  5. [13]

    International journal of heat and fluid flow 25 (3), 341--350

    Fukagata, Koji & Kasagi, Nobuhide 2004 Suboptimal control for drag reduction via suppression of near-wall reynolds shear stress . International journal of heat and fluid flow 25 (3), 341--350

    2004

  6. [14]

    , Hommema, S

    Guala, M. , Hommema, S. E. & Adrian, R. J. 2006 Large-scale and very-large-scale motions in turbulent pipe flow . Journal of Fluid Mechanics 554 , 521--542

    2006

  7. [15]

    The European Physical Journal E 46 (4), 27

    Guastoni, Luca , Rabault, Jean , Schlatter, Philipp , Azizpour, Hossein & Vinuesa, Ricardo 2023 Deep reinforcement learning for turbulent drag reduction in channel flows . The European Physical Journal E 46 (4), 27

    2023

  8. [16]

    Theoretical and Computational Fluid Dynamics 30 , 497--510

    Gu \'e niat, Florimond , Mathelin, Lionel & Hussaini, M Yousuff 2016 A statistical learning strategy for closed-loop control of fluid flows . Theoretical and Computational Fluid Dynamics 30 , 497--510

    2016

  9. [17]

    AIAA journal 27 (3), 308--314

    Gad-el Hak, Mohamed & Blackwelder, Ron F 1989 Selective suction for controlling bursting events in a boundary layer . AIAA journal 27 (3), 308--314

    1989

  10. [18]

    , Kim, J

    Hamilton, J. , Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures . Journal of Fluid Mechanics 287 , 317--348

    1995

  11. [19]

    Physics of Fluids 10 (9), 2421--2423

    Hammond, Edward P , Bewley, Thomas R & Moin, Parviz 1998 Observed mechanisms for turbulence attenuation and enhancement in opposition-controlled wall-bounded flows . Physics of Fluids 10 (9), 2421--2423

    1998

  12. [20]

    Physics of Fluids 32 (9)

    Han, Bing-Zheng & Huang, Wei-Xi 2020 Active control for drag reduction of turbulent channel flow based on convolutional neural networks . Physics of Fluids 32 (9)

    2020

  13. [21]

    Journal of Fluid Mechanics 683 , 57--93

    Hasegawa, Y & Kasagi, N 2011 Dissimilar control of momentum and heat transfer in a fully developed turbulent channel flow . Journal of Fluid Mechanics 683 , 57--93

    2011

  14. [22]

    & Jim \'e nez, J

    Hoyas, S. & Jim \'e nez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to Re _ =2003 . Physics of Fluids 18 , 011702

    2006

  15. [23]

    & Marusic, I

    Hutchins, N. & Marusic, I. 2007a Evidence of very long meandering features in the logarithmic region of turbulent boundary layers . Journal of Fluid Mechanics 579 , 1--28

  16. [24]

    & Marusic, I

    Hutchins, N. & Marusic, I. 2007b Large-scale influences in near-wall turbulence . Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 365 (1852), 647--664

  17. [25]

    Journal of Fluid Mechanics 723 , 264--288

    Hwang, Yongyun 2013 Near-wall turbulent fluctuations in the absence of wide outer motions . Journal of Fluid Mechanics 723 , 264--288

    2013

  18. [26]

    International journal of heat and fluid flow 23 (5), 678--689

    Iwamoto, Kaoru , Suzuki, Yuji & Kasagi, Nobuhide 2002 Reynolds number effect on wall turbulence: toward effective feedback control . International journal of heat and fluid flow 23 (5), 678--689

    2002

  19. [27]

    1998 The largest scales of turbulence

    Jim \'e nez, J. 1998 The largest scales of turbulence. In CTR Annual Research Briefs\/ , pp. 137--154 . Stanford University

    1998

  20. [28]

    & Moin, P

    Jim \'e nez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence . Journal of Fluid Mechanics 225 , 213--240

    1991

  21. [29]

    & Pinelli, A

    Jim \'e nez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence . Journal of Fluid Mechanics 389 , 335--359

    1999

  22. [30]

    Science 349 (6245), 255--260

    Jordan, Michael I & Mitchell, Tom M 2015 Machine learning: Trends, perspectives, and prospects . Science 349 (6245), 255--260

    2015

  23. [31]

    Journal of large-scale research facilities 7 (A138)

    J\" u lich Supercomputing Centre 2021 JUWELS Cluster and Booster: Exascale Pathfinder with Modular Supercomputing Architecture at Juelich Supercomputing Centre . Journal of large-scale research facilities 7 (A138)

    2021

  24. [32]

    Kim, K. C. & Adrian, R. J. 1999 Very large-scale motion in the outer layer . Physics of Fluids 11 (2), 417--422

    1999

  25. [33]

    Physics of Fluids A: Fluid Dynamics 5 (12), 3307--3309

    Kravchenko, Arthur G , Choi, Haecheon & Moin, Parviz 1993 On the relation of near-wall streamwise vortices to wall skin friction in turbulent boundary layers . Physics of Fluids A: Fluid Dynamics 5 (12), 3307--3309

    1993

  26. [34]

    Physics of Fluids 9 (6), 1740--1747

    Lee, Changhoon , Kim, John , Babcock, David & Goodman, Rodney 1997 Application of neural networks to turbulence control for drag reduction . Physics of Fluids 9 (6), 1740--1747

    1997

  27. [35]

    Journal of Fluid Mechanics 358 , 245--258

    Lee, Changhoon , Kim, John & Choi, Haecheon 1998 Suboptimal control of turbulent channel flow for drag reduction . Journal of Fluid Mechanics 358 , 245--258

    1998

  28. [36]

    Journal of Fluid Mechanics 860 , 886--938

    Lee, Myoungkyu & Moser, Robert D 2019 Spectral analysis of the budget equation in turbulent channel flows at high reynolds number . Journal of Fluid Mechanics 860 , 886--938

    2019

  29. [37]

    Physical Review Fluids 8 (2), 024604

    Lee, Taehyuk , Kim, Junhyuk & Lee, Changhoon 2023 Turbulence control for drag reduction through deep reinforcement learning . Physical Review Fluids 8 (2), 024604

    2023

  30. [38]

    Journal of Fluid Mechanics 932 , A44

    Li, Jichao & Zhang, Mengqi 2022 Reinforcement-learning-based control of confined cylinder wakes with stability analyses . Journal of Fluid Mechanics 932 , A44

    2022

  31. [39]

    arXiv preprint arXiv:1509.02971

    Lillicrap, Timothy P , Hunt, Jonathan J , Pritzel, Alexander , Heess, Nicolas , Erez, Tom , Tassa, Yuval , Silver, David & Wierstra, Daan 2015 Continuous control with deep reinforcement learning . arXiv preprint arXiv:1509.02971

    2015 arXiv

  32. [40]

    , Mathis, R

    Marusic, I. , Mathis, R. & Hutchins, N. 2010a High R eynolds number effects in wall turbulence . International Journal of Heat and Fluid Flow 31 (3), 418--428

  33. [41]

    , Mathis, R

    Marusic, I. , Mathis, R. & Hutchins, N. 2010b Predictive model for wall-bounded turbulent flow . Science 329 (5988), 193--196

  34. [42]

    , Hutchins, N

    Mathis, R. , Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers . Journal of Fluid Mechanics 628 , 311--337

    2009

  35. [43]

    , Hutchins, N

    Mathis, R. , Hutchins, N. & Marusic, I. 2011 A predictive inner--outer model for streamwise turbulence statistics in wall-bounded flows . Journal of Fluid Mechanics 681 , 537--566

    2011

  36. [44]

    Monty, J. P. , Hutchins, N. , Ng, H. C. H. , Marusic, I. & Chong, M. S. 2009 A comparison of turbulent pipe, channel and boundary layer flows . Journal of Fluid Mechanics 632 , 431--442

    2009

  37. [45]

    Physics of fluids 19 (10)

    Pami \`e s, Mathieu , Garnier, Eric , Merlen, Alain & Sagaut, Pierre 2007 Response of a spatially developing turbulent boundary layer to active control strategies in the framework of opposition control . Physics of fluids 19 (10)

    2007

  38. [46]

    Journal of Fluid Mechanics 913 , A25

    Paris, Romain , Beneddine, Samir & Dandois, Julien 2021 Robust flow control and optimal sensor placement using deep reinforcement learning . Journal of Fluid Mechanics 913 , A25

    2021

  39. [47]

    Journal of fluid mechanics 865 , 281--302

    Rabault, Jean , Kuchta, Miroslav , Jensen, Atle , R \'e glade, Ulysse & Cerardi, Nicolas 2019 Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control . Journal of fluid mechanics 865 , 281--302

    2019

  40. [48]

    Journal of Fluid Mechanics 960 , A30

    Sonoda, Takahiro , Liu, Zhuchen , Itoh, Toshitaka & Hasegawa, Yosuke 2023 Reinforcement learning of control strategies for reducing skin friction drag in a fully developed turbulent channel flow . Journal of Fluid Mechanics 960 , A30

    2023

  41. [49]

    arXiv preprint arXiv:2405.17210

    Su \'a rez, Pol , \'A lcantara- \'A vila, Francisco , Rabault, Jean , Mir \'o , Arnau , Font, Bernat , Lehmkuhl, Oriol & Vinuesa, Ricardo 2024 Flow control of three-dimensional cylinders transitioning to turbulence via multi-agent reinforcement learning . arXiv preprint arXiv:...

    2024 arXiv

  42. [50]

    Journal of Fluid Mechanics 693 , 150--200

    Touber, Emile & Leschziner, Michael A 2012 Near-wall streak modification by spanwise oscillatory wall motion and drag-reduction mechanisms . Journal of Fluid Mechanics 693 , 150--200

    2012

  43. [51]

    Computers & Fluids 116 , 10--16

    Van Der Poel, Erwin P , Ostilla-M \'o nico, Rodolfo , Donners, John & Verzicco, Roberto 2015 A pencil distributed finite difference code for strongly turbulent wall-bounded flows . Computers & Fluids 116 , 10--16

    2015

  44. [52]

    In Actuators\/ , , vol

    Varela, Pau , Su \'a rez, Pol , Alc \'a ntara- \'A vila, Francisco , Mir \'o , Arnau , Rabault, Jean , Font, Bernat , Garc \' a-Cuevas, Luis Miguel , Lehmkuhl, Oriol & Vinuesa, Ricardo 2022 Deep reinforcement learning for flow control exploits different physics for increasing ...

    2022

  45. [53]

    Journal of Computational Physics 123 (2), 402--414

    Verzicco, R & Orlandi, Paolo 1996 A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates . Journal of Computational Physics 123 (2), 402--414

    1996

  46. [54]

    Physics of Fluids 17 (1), 018101--018101

    Xu, Chun-Xiao & Huang, Wei-Xi 2005 Transient response of reynolds stress transport to spanwise wall oscillation in a turbulent channel flow . Physics of Fluids 17 (1), 018101--018101

    2005

  47. [55]

    Physical Review E 104 (1), 014210

    Zeng, Kevin & Graham, Michael D 2021 Symmetry reduction for deep reinforcement learning active control of chaotic spatiotemporal dynamics . Physical Review E 104 (1), 014210

    2021

  48. [56]

    Computer physics communications 229 , 199--210

    Zhu, Xiaojue , Phillips, Everett , Spandan, Vamsi , Donners, John , Ruetsch, Gregory , Romero, Joshua , Ostilla-M \'o nico, Rodolfo , Yang, Yantao , Lohse, Detlef , Verzicco, Roberto & others 2018 Afid-gpu: a versatile navier--stokes solver for wall-bounded turbulent flows on ...

    2018

  49. [57]

    @esa (Ref

    \@ifclassloaded aguplus natbib The aguplus class already includes natbib coding, so you should not add it explicitly Type <Return> for now, but then later remove the command natbib from the document \@ifclassloaded nlinproc natbib The nlinproc class already includes natbib cod...

  50. [58]

    @stdbsttrue NAT@ctr \@lbibitem[ NAT@ctr ] \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 [ @natanchorstart #2\@extra@b@citeb \@biblabel @num @natanchorend] @ifc...

  51. [59]

    @open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifundefined NAT@sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifundefined bib@heading @heading NAT@ctr thebibliography [1] @ \@biblabel NAT@ctr \@bib...

    1996

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.