pith. sign in

arxiv: 2509.02463 · v4 · submitted 2025-09-02 · ⚛️ physics.flu-dyn · math-ph· math.MP

Searching for Invariant Solutions to Wall-Bounded Flows using Resolvent-Based Optimisation

Thomas Burton , Sean Symon , Davide Lasagna This is my paper

Pith reviewed 2026-05-18 19:49 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn math-phmath.MP
keywords invariant solutionswall-bounded flowsresolvent analysisvariational optimisationGalerkin projectionrotating plane Couette flowNavier-Stokes equations
0
0 comments X

The pith

A resolvent-derived basis enables variational optimisation to compute invariant solutions for wall-bounded flows.

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

The paper develops an optimisation method to find periodic and equilibrium solutions of the Navier-Stokes equations in wall-bounded flows. It recasts the equations as a variational problem that minimises the residual over a finite time horizon. The key innovation is a Galerkin projection onto a basis of divergence-free modes from resolvent analysis that automatically satisfy no-slip conditions. This yields a low-order model applicable to bounded domains. The approach is tested on rotating plane Couette flow and recovers solutions that match direct numerical simulations, while also linking optimisation performance to resolvent properties.

Core claim

By including a Galerkin projection onto an orthonormal basis of divergence-free modes derived from resolvent analysis, the variational framework for finding invariant solutions becomes applicable to wall-bounded flows, producing exact equilibrium and periodic solutions in a 2D3C formulation of rotating plane Couette flow that are consistent with direct numerical simulations.

What carries the argument

Galerkin projection onto a resolvent-derived orthonormal set of divergence-free modes satisfying no-slip boundary conditions, which reduces the dynamics to a low-order representation suitable for optimisation.

If this is right

  • The optimisation converges to exact invariant solutions when applied to the projected equations.
  • Conditioning of the problem depends on the stability properties of the targeted solutions.
  • A direct connection exists between the optimisation conditioning and the structure of the resolvent operator.
  • The method can be used to search for both equilibrium and periodic solutions in bounded flows.

Where Pith is reading between the lines

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

  • This framework might generalise to other wall-bounded shear flows such as plane Poiseuille flow.
  • Selecting resolvent modes could offer a systematic way to build reduced-order models for invariant solution searches.
  • Improved understanding of the resolvent-optimisation link may guide better basis selection for numerical efficiency.

Load-bearing premise

The resolvent-derived basis must capture enough of the flow dynamics so that minimising the projected residual leads to true solutions of the full Navier-Stokes equations.

What would settle it

Running the optimisation on the rotating plane Couette flow and finding that it does not recover the known equilibrium or periodic solutions from direct numerical simulations would falsify the claim.

Figures

Figures reproduced from arXiv: 2509.02463 by Davide Lasagna, Sean Symon, Thomas Burton.

Figure 1
Figure 1. Figure 1: Schematic of an arbitrary loop in state-space that does not satisfy the governing [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic for a Galerkin projection of state-space loop representing a velocity [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Flow diagram of the computations performed at each iteration of the [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bifurcation diagram of RPCF over a range of Reynolds number and [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Equilibrium solutions labelled in figure [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Snapshots of the flows before and after the optimisation at [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Residual trace for the optimisation of the initial flow given in panel (a) of [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Final snapshots of the solutions obtained by optimising from various synthetic [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Spanwise power spectra of S3 of the solutions obtained at specific iterations of [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Snapshots of the periodic solution (R ≈ 5 × 10−11) obtained for Re = 400 with a period of 𝑇 ≈ 25.05. Panel (a) shows 𝑡 = 0, panel (b) shows 𝑡 = 𝑇/4, panel (c) shows 𝑡 = 𝑇/2, and panel (d) shows 𝑡 = 3𝑇/4 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Spanwise and (positive) temporal power spectrum of the periodic solution in [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Traces of the global residual, panel (a), and the solution period, panel (b), of [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Residuals from optimisations of S1 at Reynolds numbers of [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Panel (a): global residual traces for the optimisation of a perturbed S2 solution, [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Snapshots of the solutions obtained from the projected optimisation of the [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
read the original abstract

We present a robust optimisation framework for computing invariant solutions of wall-bounded flows by recasting the Navier-Stokes equations as a variational problem as established in Ashtari and Schneider, JFM (2023). The approach minimises the residual of the governing equations over a finite time horizon, seeking periodic or equilibrium solutions. A novel contribution is made by including a Galerkin projection onto a basis of divergence-free modes that satisfy the no-slip boundary conditions. This projection not only makes the variational framework applicable to wall-bounded flows but it also yields a low-order representation of the dynamics. The basis is derived from resolvent analysis, which provides an orthonormal set. We demonstrate the method on a 2D3C formulation of rotating plane Couette flow, obtaining exact equilibrium and periodic solutions consistent with direct numerical simulations. The conditioning of the optimisation problem is analysed in detail, showing that convergence rates depend on the stability properties of the targeted solutions. Finally, we highlight a direct link between the conditioning of the optimisation and the structure of the resolvent operator, suggesting a unifying perspective on both the efficiency of the optimisation and the dynamical significance of resolvent modes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper recasts the search for invariant solutions (equilibria and periodic orbits) of the Navier-Stokes equations as a finite-time variational optimization problem, augmented by a novel Galerkin projection onto an orthonormal basis of divergence-free modes that satisfy no-slip boundary conditions; the basis is constructed from resolvent analysis. The method is demonstrated on a 2D3C formulation of rotating plane Couette flow, where it recovers exact equilibria and periodic solutions reported to be consistent with direct numerical simulations. The authors further analyze the conditioning of the optimization problem, linking convergence behavior to the stability properties of the target solutions and to the structure of the resolvent operator.

Significance. If the reported solutions satisfy the unprojected Navier-Stokes equations to machine precision and the conditioning analysis holds, the framework would provide a practical route to computing invariant solutions in wall-bounded flows by combining variational optimization with resolvent-derived bases. The explicit connection drawn between optimization conditioning and resolvent operator structure offers a unifying perspective that could inform both numerical efficiency and dynamical interpretation of resolvent modes.

major comments (3)
  1. [§3, Eq. (8)–(10)] §3 (Method), Eq. (8)–(10): the optimization minimizes the residual of the Galerkin-projected equations on the finite resolvent basis. The abstract and §4 claim that the obtained solutions are 'exact' invariants 'consistent with DNS,' yet no quantitative verification is provided that the residual of the unprojected Navier-Stokes equations vanishes to machine precision or that the projection truncation error remains negligible for the reported solutions. Because any optimizer output is an exact invariant only of the reduced system, this check is load-bearing for the central claim.
  2. [§4.1, Table 1, Fig. 3] §4.1 (Results), Table 1 and Fig. 3: the reported equilibria and periodic orbits are stated to match DNS, but the manuscript supplies neither the L2-norm of the full residual nor the projection error norm for these specific solutions. Without these metrics it is impossible to confirm that the low-order invariants lift to exact solutions of the original wall-bounded system.
  3. [§2.2] §2.2 (Time horizon): the finite-time horizon length appears as a free parameter in the variational formulation. The manuscript does not report a systematic study of its influence on the recovered solutions or on the conditioning analysis, leaving open whether the reported invariants are robust to this choice.
minor comments (2)
  1. [§2.1 and §5] Notation: the symbol for the resolvent operator is introduced in §2.1 but reused with a different meaning in the conditioning analysis of §5; a single consistent definition would improve readability.
  2. [Fig. 2] Figure 2: the color scale for the residual field is not labeled, making it difficult to judge the magnitude of the reported errors.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We appreciate the emphasis on rigorous verification of the solutions and the robustness of the method. Below, we provide point-by-point responses to the major comments and outline the revisions we will make to address them.

read point-by-point responses

  1. Referee: [§3, Eq. (8)–(10)] §3 (Method), Eq. (8)–(10): the optimization minimizes the residual of the Galerkin-projected equations on the finite resolvent basis. The abstract and §4 claim that the obtained solutions are 'exact' invariants 'consistent with DNS,' yet no quantitative verification is provided that the residual of the unprojected Navier-Stokes equations vanishes to machine precision or that the projection truncation error remains negligible for the reported solutions. Because any optimizer output is an exact invariant only of the reduced system, this check is load-bearing for the central claim.

    Authors: We agree that the obtained solutions are exact invariants of the Galerkin-projected system. To strengthen the central claim, we will add in the revised manuscript a quantitative verification by computing and reporting the L2-norm of the residual of the unprojected Navier-Stokes equations for each reported solution, along with the projection truncation error norm. This will confirm that the truncation error is negligible and that the solutions are consistent with the full system to high accuracy. revision: yes

  2. Referee: [§4.1, Table 1, Fig. 3] §4.1 (Results), Table 1 and Fig. 3: the reported equilibria and periodic orbits are stated to match DNS, but the manuscript supplies neither the L2-norm of the full residual nor the projection error norm for these specific solutions. Without these metrics it is impossible to confirm that the low-order invariants lift to exact solutions of the original wall-bounded system.

    Authors: We thank the referee for this observation. In the revision we will include the requested L2-norm of the full residual and the projection error norm for the specific equilibria and periodic orbits listed in Table 1 and shown in Fig. 3. These metrics will be added to the results section to provide direct evidence that the low-order solutions lift to the original wall-bounded system with negligible error. revision: yes

  3. Referee: [§2.2] §2.2 (Time horizon): the finite-time horizon length appears as a free parameter in the variational formulation. The manuscript does not report a systematic study of its influence on the recovered solutions or on the conditioning analysis, leaving open whether the reported invariants are robust to this choice.

    Authors: The time horizon is a free parameter in the variational formulation. While we selected values that recover the known DNS solutions, we did not include a systematic sensitivity study. In the revised manuscript we will add a short discussion of the influence of the time horizon, including results for a small set of alternative values, to demonstrate robustness of the recovered invariants and the conditioning analysis. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained via external variational base and independent resolvent projection

full rationale

The paper recasts NS equations as a variational problem by direct citation to Ashtari & Schneider (JFM 2023), an external reference with no author overlap. It then adds a novel Galerkin projection onto a resolvent-derived orthonormal basis of divergence-free, no-slip modes; this projection step is introduced as an independent contribution that yields the low-order representation. The optimisation minimises the residual of the projected equations over a finite horizon, producing equilibria and periodic orbits that are exact invariants of the reduced system and reported as consistent with DNS. No step reduces a claimed prediction to a fitted parameter by construction, no load-bearing self-citation chain exists, and the resolvent basis is not defined in terms of the target solutions. The derivation therefore does not collapse to its own inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The framework rests on the Navier-Stokes equations, the variational residual minimization established in prior work, and the assumption that resolvent modes form a suitable divergence-free basis for wall-bounded flows. No new physical entities are postulated.

free parameters (2)
  • finite time horizon length
    Chosen to seek periodic or equilibrium solutions; its value directly affects which invariants are recovered.
  • number of retained resolvent modes
    Determines the dimension of the low-order Galerkin representation and therefore the accuracy of the projected dynamics.
axioms (2)
  • domain assumption The incompressible Navier-Stokes equations govern the flow
    Standard governing equations invoked throughout the variational formulation.
  • domain assumption Resolvent analysis supplies an orthonormal set of divergence-free modes that satisfy no-slip boundary conditions
    This property is required for the Galerkin projection to be valid and is stated as the basis for applicability to wall-bounded flows.

pith-pipeline@v0.9.0 · 5741 in / 1624 out tokens · 57815 ms · 2026-05-18T19:49:23.798970+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We present a robust optimisation framework for computing invariant solutions of wall-bounded flows by recasting the Navier-Stokes equations as a variational problem... Galerkin projection onto a basis of divergence-free modes... basis is derived from resolvent analysis

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The conditioning of the optimisation problem is analysed in detail, showing that convergence rates depend on the stability properties of the targeted solutions... direct link between the conditioning of the optimisation and the structure of the resolvent operator

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

86 extracted references · 86 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 ...

    work page

  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...

    work page

  3. [3]

    Physical Review E 101 , 12213

    Ahmed, M Arslan & Sharma, Ati S 2020 Basis for finding exact coherent states . Physical Review E 101 , 12213

    work page 2020

  4. [4]

    i: — statistical identification and fractal properties of the lorenz chaos —

    Aizawa, Yoji 1982 Global aspects of the dissipative dynamical systems. i: — statistical identification and fractal properties of the lorenz chaos — . Progress of Theoretical Physics 68 , 64--84

    work page 1982

  5. [5]

    , Aurell, E

    Artuso, R. , Aurell, E. & Cvitanovic, P. 1990 Recycling of strange sets: I. cycle expansions . Nonlinearity 3 , 325--359

    work page 1990

  6. [6]

    Journal of Fluid Mechanics 977 , A7

    Ashtari, Omid & Schneider, Tobias M 2023 Identifying invariant solutions of wall-bounded three-dimensional shear flows using robust adjoint-based variational techniques . Journal of Fluid Mechanics 977 , A7

    work page 2023

  7. [7]

    , Cvitanovic, P

    Auerbach, D. , Cvitanovic, P. , Eckmann, J. P. , Gunaratne, G. & Procaccia, I. 1987 Exploring chaotic motion through periodic orbits . Phys. Rev. Lett. 58 , 2387--2389

    work page 1987

  8. [8]

    , Ashtari, O

    Azimi, S. , Ashtari, O. & Schneider, T. M. 2022 Constructing periodic orbits of high-dimensional chaotic systems by an adjoint-based variational method . Phys. Rev. E 105 , 14217

    work page 2022

  9. [9]

    , Zhu, X

    Barthel, B. , Zhu, X. & McKeon, B. J. 2021 Closing the loop: nonlinear taylor vortex flow through the lens of resolvent analysis . Journal of Fluid Mechanics 924 , A9

    work page 2021

  10. [10]

    , Yegavian, R

    Beneddine, S. , Yegavian, R. , Sipp, D. & Leclaire, B. 2017 Unsteady flow dynamics reconstruction from mean flow and point sensors: An experimental study . Journal of Fluid Mechanics 824 , 174--201

    work page 2017

  11. [11]

    , Sipp, D

    Brandt, L. , Sipp, D. , Pralits, J. O. & Marquet, O. 2011 Effect of base-flow variation in noise amplifiers: the flat-plate boundary layer . Journal of Fluid Mechanics 687 , 503--528

    work page 2011

  12. [12]

    2025 Resolvent-based models for nonlinear solutions of wall-bounded flows and statistical estimation of chaotic systems

    Burton, T. 2025 Resolvent-based models for nonlinear solutions of wall-bounded flows and statistical estimation of chaotic systems . PhD thesis, University of Southampton

    work page 2025

  13. [13]

    , Symon, S

    Burton, T. , Symon, S. , Sharma, A. S. & Lasagna, D. 2025 Resolvent-based optimization for approximating the statistics of a chaotic lorenz system . Physical Review E 111 , 25104

    work page 2025

  14. [14]

    , Hussaini, M

    Canuto, C. , Hussaini, M. Y. , Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics\/ . Springer Berlin Heidelberg

    work page 1988

  15. [15]

    Chandler, G. J. & Kerswell, R. R. 2013 Invariant recurrent solutions embedded in a turbulent two-dimensional kolmogorov flow . Journal of Fluid Mechanics 722 , 554--595

    work page 2013

  16. [16]

    Oxford University Press

    Chandrasekhar, Subrahmanyan 1961 Hydrodynamic and Hydromagnetic Stability\/ . Oxford University Press

    work page 1961

  17. [17]

    , Cvitanovic, P

    Christiansen, F. , Cvitanovic, P. & Putkaradze, V. 1997 Spatiotemporal chaos in terms of unstable recurrent patterns . Nonlinearity 10 , 55--70

    work page 1997

  18. [18]

    Proceedings of the National Academy of Sciences 119 , e2120665119 , doi: 10.1073/pnas.2120665119

    Crowley, Christopher J , Pughe-Sanford, Joshua L , Toler, Wesley , Krygier, Michael C , Grigoriev, Roman O & Schatz, Michael F 2022 Turbulence tracks recurrent solutions . Proceedings of the National Academy of Sciences 119 , e2120665119 , doi: 10.1073/pnas.2120665119

    work page doi:10.1073/pnas.2120665119 2022

  19. [19]

    1988 Invariant measurement of strange sets in terms of cycles

    Cvitanović, P. 1988 Invariant measurement of strange sets in terms of cycles . Physical Review Letters 61 , 2729--2732

    work page 1988

  20. [20]

    1995 Dynamical averaging in terms of periodic orbits

    Cvitanović, P. 1995 Dynamical averaging in terms of periodic orbits . Physica D: Nonlinear Phenomena 83 , 109--123

    work page 1995

  21. [21]

    & Gibson, J

    Cvitanović, P. & Gibson, J. F. 2010 Geometry of the turbulence in wall-bounded shear flows: periodic orbits . Physica Scripta T142 , 014007

    work page 2010

  22. [22]

    & Bj\"\ o\ rck, \ \

    Dahlquist, G. & Bj\"\ o\ rck, \ \ . 2008 Numerical Methods in Scientific Computing, Volume I\/ . Society for Industrial and Applied Mathematics

    work page 2008

  23. [23]

    , Hegseth, J

    Daviaud, F. , Hegseth, J. & Bergé, P. 1992 Subcritical transition to turbulence in plane couette flow . Physical Review Letters

    work page 1992

  24. [24]

    Dennis, J. E. & Schnabel, B. 1983 Numerical methods for unconstrained optimization and nonlinear equations. In Prentice Hall series in computational mathematics\/

    work page 1983

  25. [25]

    , Doering, C

    Eckhardt, B. , Doering, C. R. & Whitehead, J. P. 2020 Exact relations between rayleigh–bénard and rotating plane couette flow in two dimensions . Journal of Fluid Mechanics 903 , R4

    work page 2020

  26. [26]

    Journal of Fluid Mechanics 581 , 221--250

    Eckhardt, Bruno , Grossmann, Siegfried & Lohse, Detlef 2007 Torque scaling in turbulent taylor–couette flow between independently rotating cylinders . Journal of Fluid Mechanics 581 , 221--250

    work page 2007

  27. [27]

    & Eckhardt, B

    Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow . Phys. Rev. Lett. 91 , 224502

    work page 2003

  28. [28]

    2016 An adjoint-based approach for finding invariant solutions of navier–stokes equations

    Farazmand, M. 2016 An adjoint-based approach for finding invariant solutions of navier–stokes equations . Journal of Fluid Mechanics 795 , 278–312

    work page 2016

  29. [29]

    , Boghosian, B

    Fazendeiro, L. , Boghosian, B. M. , Coveney, P. V. & Lätt, J. 2010 Unstable periodic orbits in weak turbulence . Journal of Computational Science 1 , 13--23

    work page 2010

  30. [30]

    2005 The design and implementation of fftw3

    Frigo, Matteo & Johnson, Steven G. 2005 The design and implementation of fftw3 . Proceedings of the IEEE 93 , 216--231

    work page 2005

  31. [31]

    , Lesshafft, L

    Garnaud, X. , Lesshafft, L. , Schmid, P. J. & Huerre, P. 2013 The preferred mode of incompressible jets: linear frequency response analysis . Journal of Fluid Mechanics 716 , 189–202

    work page 2013

  32. [32]

    2010 A robust control approach to understanding nonlinear mechanisms in shear flow turbulence

    Gayme, D. 2010 A robust control approach to understanding nonlinear mechanisms in shear flow turbulence . PhD thesis, California Institute of Technology

    work page 2010

  33. [33]

    Gayme, D. F. & Minnick, B. A. 2019 Coherent structure-based approach to modeling wall turbulence . Phys. Rev. Fluids 4 , 110505

    work page 2019

  34. [34]

    Gibson, J. F. , Halcrow, J. & Cvitanovic, P. 2008 Visualizing the geometry of state space in plane couette flow . Journal of Fluid Mechanics 611 , 107--130

    work page 2008

  35. [35]

    , Blackburn, H

    Gómez, F. , Blackburn, H. M. , Rudman, M. , Sharma, A. S. & McKeon, B. J. 2016 A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator . Journal of Fluid Mechanics 798 , R2

    work page 2016

  36. [36]

    Hager, W. W. & Zhang, H. 2005 A new conjugate gradient method with guaranteed descent and an efficient line search . SIAM Journal on Optimization 16 , 170--192

    work page 2005

  37. [37]

    Physics of Fluids 19 , 048103

    Hiwatashi, Kazuaki , Alfredsson, P H , Tillmark, Nils & Nagata, M 2007 Experimental observations of instabilities in rotating plane couette flow . Physics of Fluids 19 , 048103

    work page 2007

  38. [38]

    1942 Abzweigung einer periodischen lösung von einer stationären lösung eines differentialsystems

    Hopf, E. 1942 Abzweigung einer periodischen lösung von einer stationären lösung eines differentialsystems . Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl. 94 , 1--22

    work page 1942

  39. [39]

    1948 A mathematical example displaying features of turbulence

    Hopf, E. 1948 A mathematical example displaying features of turbulence . Communications on Pure and Applied Mathematics 1 , 303--322

    work page 1948

  40. [40]

    & Generalis, S

    Itano, T. & Generalis, S. C. 2009 Hairpin vortex solution in planar couette flow: A tapestry of knotted vortices . Physical Review Letters 102 , 114501

    work page 2009

  41. [41]

    Theoretical and Computational Fluid Dynamics 36 , 491--515

    Jin, Bo , Illingworth, Simon J & Sandberg, Richard D 2022 Resolvent-based approach for \ H_2 \ -optimal estimation and control: an application to the cylinder flow . Theoretical and Computational Fluid Dynamics 36 , 491--515

    work page 2022

  42. [42]

    & Kida, S

    Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane couette turbulence: regeneration cycle and burst . Journal of Fluid Mechanics 449 , 291–300

    work page 2001

  43. [43]

    Annual Review of Fluid Mechanics 44 , 203--225

    Kawahara, Genta , Uhlmann, Markus & van Veen, Lennaert 2012 The significance of simple invariant solutions in turbulent flows . Annual Review of Fluid Mechanics 44 , 203--225

    work page 2012

  44. [44]

    & Schumann, U

    Kleiser, L. & Schumann, U. 1980 Treatment of Incompressibility and Boundary Conditions in 3-D Numerical Spectral Simulations of Plane Channel Flows\/ , pp. 165--173 . Vieweg+Teubner Verlag

    work page 1980

  45. [45]

    Journal of Fluid Mechanics 923 , A7

    Krygier, Michael C , Pughe-Sanford, Joshua L & Grigoriev, Roman O 2021 Exact coherent structures and shadowing in turbulent taylor–couette flow . Journal of Fluid Mechanics 923 , A7

    work page 2021

  46. [46]

    Journal of Computational Physics

    Lakoba, T I & Yang, J 2007 A mode elimination technique to improve convergence of iteration methods for finding solitary waves . Journal of Computational Physics

    work page 2007

  47. [47]

    & Cvitanović, P

    Lan, Y. & Cvitanović, P. 2004 Variational method for finding periodic orbits in a general flow . Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics 69 , 10

    work page 2004

  48. [48]

    , Tutty, O

    Lasagna, D. , Tutty, O. R. & Chernyshenko, S. 2016 Flow regimes in a simplified taylor–couette-type flow model . European Journal of Mechanics - B/Fluids 57 , 176--191

    work page 2016

  49. [49]

    Journal of Fluid Mechanics 77 , 153--174

    Lezius, Dietrich K & Johnston, James P 1976 Roll-cell instabilities in rotating laminar and trubulent channel flows . Journal of Fluid Mechanics 77 , 153--174

    work page 1976

  50. [50]

    & Lasagna, D

    Li, X. & Lasagna, D. 2025 Space-time nonlinear reduced-order modelling for unsteady flows . arXiv

    work page 2025

  51. [51]

    McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow . Journal of Fluid Mechanics 658 , 336--382

    work page 2010

  52. [52]

    Journal of Open Source Software 3 , 615

    Mogensen, Patrick Kofod & Riseth, Asbjørn Nilsen 2018 Optim: A mathematical optimization package for julia . Journal of Open Source Software 3 , 615

    work page 2018

  53. [53]

    & Marquet, O

    Mons, V. & Marquet, O. 2021 Linear and nonlinear sensor placement strategies for mean-flow reconstruction via data assimilation . Journal of Fluid Mechanics 923 , A1

    work page 2021

  54. [54]

    1990 Three-dimensional finite-amplitude solutions in plane couette flow: bifurcation from infinity

    Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane couette flow: bifurcation from infinity . Journal of Fluid Mechanics 217 , 519–527

    work page 1990

  55. [55]

    Physical Review E 55 , 2023--2025

    Nagata, M 1997 Three-dimensional traveling-wave solutions in plane couette flow . Physical Review E 55 , 2023--2025

    work page 1997

  56. [56]

    2013 A note on the mirror-symmetric coherent structure in plane couette flow

    Nagata, M. 2013 A note on the mirror-symmetric coherent structure in plane couette flow . Journal of Fluid Mechanics 727 , R1

    work page 2013

  57. [57]

    , Song, B

    Nagata, M. , Song, B. & Wall, D. P. 2021 Onset of vortex structures in rotating plane couette flow . Journal of Fluid Mechanics 918 , A2

    work page 2021

  58. [58]

    & Wright, S

    Nocedal, J. & Wright, S. 2006 Numerical Optimization\/ , 2nd edn. Springer-Verlag New York

    work page 2006

  59. [59]

    , Nagata, M

    Okino, S. , Nagata, M. , Wedin, H. & Bottaro, A. 2010 A new nonlinear vortex state in square-duct flow . Journal of Fluid Mechanics 657 , 413--429

    work page 2010

  60. [60]

    Proceedings of the National Academy of Sciences 121 , e2320007121

    Page, Jacob , Norgaard, Peter , Brenner, Michael P & Kerswell, Rich R 2024 Recurrent flow patterns as a basis for two-dimensional turbulence: Predicting statistics from structures . Proceedings of the National Academy of Sciences 121 , e2320007121

    work page 2024

  61. [61]

    Pringle, C. C. T. & Kerswell, R. R. 2007 Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow . Physical Review Letters 99 , 74502

    work page 2007

  62. [62]

    & Takens, F

    Ruelle, D. & Takens, F. 1971 On the nature of turbulence . Communications in Mathematical Physics 20 , 167--192

    work page 1971

  63. [63]

    SIAM Journal on Scientific and Statistical Computing 7 , 856--869

    Saad, Youcef & Schultz, Martin H 1986 Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems . SIAM Journal on Scientific and Statistical Computing 7 , 856--869

    work page 1986

  64. [64]

    Sharma, A. S. 2019 Elements of resolvent methods in fluid mechanics: notes for an introductory short course v0.3 . arXiv pp. 1--24

    work page 2019

  65. [65]

    Physical Review E 93 , 21102

    Sharma, Ati S , Moarref, Rashad , McKeon, Beverley J , Park, Jae Sung , Graham, Michael D & Willis, Ashley P 2016 Low-dimensional representations of exact coherent states of the navier-stokes equations from the resolvent model of wall turbulence . Physical Review E 93 , 21102

    work page 2016

  66. [66]

    Physical Review Letters 125 , 64501

    Suri, Balachandra , Kageorge, Logan , Grigoriev, Roman O & Schatz, Michael F 2020 Capturing turbulent dynamics and statistics in experiments with unstable periodic orbits . Physical Review Letters 125 , 64501

    work page 2020

  67. [67]

    , Rosenberg, K

    Symon, S. , Rosenberg, K. , Dawson, S. T. M. & McKeon, B. J. 2018 Non-normality and classification of amplification mechanisms in stability and resolvent analysis . Physical Review Fluids 3

    work page 2018

  68. [68]

    , Net, M

    Sánchez, J. , Net, M. , Garc\ \'i\ a-Archilla, B. & Simó, C. 2004 Newton–krylov continuation of periodic orbits for navier–stokes flows . Journal of Computational Physics 201 , 13--33

    work page 2004

  69. [69]

    1984 Navier-Stokes equations : theory and numerical analysis\/

    Temam, R. 1984 Navier-Stokes equations : theory and numerical analysis\/ . AMS Chelsea Pub

    work page 1984

  70. [70]

    & Sadayoshi, T

    Tomoaki, I. & Sadayoshi, T. 2001 The dynamics of bursting process in wall turbulence . Journal of the Physical Society of Japan 70 , 703--716

    work page 2001

  71. [71]

    , Tillmark, N

    Tsukahara, T. , Tillmark, N. & Alfredsson, P. H. 2010 Flow regimes in a plane couette flow with system rotation . Journal of Fluid Mechanics 648 , 5--33

    work page 2010

  72. [72]

    , Kawahara, G

    Uhlmann, M. , Kawahara, G. & Pinelli, A. 2010 Traveling-waves consistent with turbulence-driven secondary flow in a square duct . Physics of Fluids 22 , 084102

    work page 2010

  73. [73]

    , Kida, S

    van Veen, L. , Kida, S. & Kawahara, G. 2006 Periodic motion representing isotropic turbulence . Fluid Dynamics Research 38 , 19–46

    work page 2006

  74. [74]

    Van , Vela-Martín, A

    Veen, L. Van , Vela-Martín, A. & Kawahara, G. 2019 Time-periodic inertial range dynamics . Physical Review Letters 123 , 134502

    work page 2019

  75. [75]

    2007 Recurrent motions within plane couette turbulence

    Viswanath, D. 2007 Recurrent motions within plane couette turbulence . Journal of Fluid Mechanics 580 , 339–358

    work page 2007

  76. [76]

    2009 The critical layer in pipe flow at high reynolds number

    Viswanath, D. 2009 The critical layer in pipe flow at high reynolds number . Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367 , 561--576

    work page 2009

  77. [77]

    1998 Three-dimensional coherent states in plane shear flows

    Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows . Phys. Rev. Lett. 81 , 4140--4143

    work page 1998

  78. [78]

    2001 Exact coherent structures in channel flow

    Waleffe, F. 2001 Exact coherent structures in channel flow . Journal of Fluid Mechanics 435 , 93–102

    work page 2001

  79. [79]

    In Turbulence and Interactions\/ (ed

    Waleffe, Fabian 2009 Exact coherent structures in turbulent shear flows. In Turbulence and Interactions\/ (ed. Michel Deville, Thien-Hiep Lê & Pierre Sagaut ) , pp. 139--158 . Springer Berlin Heidelberg

    work page 2009

  80. [80]

    , Ayats, R

    Wang, B. , Ayats, R. , Deguchi, K. , Meseguer, A. & Mellibovsky, F. 2025 Mathematically established chaos and forecast of statistics with recurrent patterns in taylor–couette flow . Journal of Fluid Mechanics 1011 , R2

    work page 2025

Showing first 80 references.