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arxiv: 2508.05892 · v1 · submitted 2025-08-07 · ⚛️ physics.flu-dyn · nlin.CD

Lyapunov stability analysis of the chaotic flow past two square cylinders

Sidhartha Sahu , George Papadakis This is my paper

Pith reviewed 2026-05-19 00:05 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn nlin.CD
keywords Lyapunov stabilitychaotic flowsquare cylinderscovariant Lyapunov vectorsglobal linear stability analysisvortex sheddingjet flappingwake instability
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The pith

Lyapunov analysis shows two unstable directions in the chaotic flow past two square cylinders, one of which global linear stability analysis on the mean flow incorrectly marks neutral.

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

The paper applies Lyapunov stability analysis to the irregular flow past two side-by-side square cylinders at Reynolds number 200 and gap ratio 1. Linearizing the Navier-Stokes equations around the time-dependent base flow obtained from simulation yields two positive Lyapunov exponents. The associated covariant Lyapunov vectors localize the leading instability to the near wake with frequencies tied to vortex shedding and jet flapping, while the second vector captures a subharmonic instability further downstream. Global linear stability analysis performed on the time-averaged flow recovers a neutral eigenmode whose structure and frequency closely match the leading covariant vector, yet the Lyapunov calculation demonstrates positive growth in that same direction.

Core claim

The flow past two side-by-side square cylinders is chaotic and possesses two positive Lyapunov exponents. Covariant Lyapunov vectors computed from the linearized Navier-Stokes equations around the irregular base flow show that the leading vector has its footprint in the near wake and exhibits the two dominant frequencies present in the drag coefficient spectrum, corresponding to vortex shedding and jet flapping. The second unstable vector captures the subharmonic of the shedding frequency. Global linear stability analysis of the time-averaged flow identifies a neutral eigenmode that resembles the leading SPOD mode of the first covariant vector in both structure and frequency, yet the full-tf

What carries the argument

Covariant Lyapunov vectors obtained by linearizing the Navier-Stokes equations around the irregular time-dependent base flow from direct simulation.

If this is right

  • The near-wake region contains the primary source of the leading instability.
  • Frequencies extracted from the leading covariant vector match the dominant peaks observed in the drag coefficient spectrum.
  • The second unstable direction corresponds to subharmonic instability of the shedding frequency.
  • Different covariant vectors occupy spatially distinct regions, with the leading one concentrated upstream of the others.

Where Pith is reading between the lines

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

  • The same linearization approach could be used on other bluff-body wakes to locate hidden unstable directions that mean-flow analysis overlooks.
  • Flow control targeting the near-wake footprint of the leading covariant vector might suppress the dominant chaotic fluctuations.
  • The discrepancy between neutral global modes and positive Lyapunov exponents indicates that averaging the base flow systematically underestimates growth rates in chaotic regimes.

Load-bearing premise

Linearizing the Navier-Stokes equations around the irregular time-dependent base flow from the simulation accurately captures the growth rates of disturbances without higher-order nonlinear effects dominating the tangent-space dynamics.

What would settle it

A recomputation of the largest Lyapunov exponent on the same flow using an independent numerical scheme or substantially refined grid that returns a non-positive value would falsify the reported instability.

Figures

Figures reproduced from arXiv: 2508.05892 by George Papadakis, Sidhartha Sahu.

Figure 1
Figure 1. Figure 1: (a) Flow configuration and boundary conditions, (b) Global and zoomed-in views of mesh 2. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mesh independence study (a) u¯(y) across the gap between the cylinders at x = 0, (b) Spectrum of C ′ L for the bottom cylinder. 3 Flow characteristics 1 2 1 2 1 2 1 2 1+2 1+2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Contour plots of instantaneous vorticity ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Frequency spectrum of the fluctuating force coefficients for (a) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spectra of the transverse velocity v(x, y) at x = 2 (left column), 10 (middle column) and 20 (right column). Top row: along the gap centerline (y = 0); bottom row: along the centerline behind the top prism (y = 1). The shaded regions mark the frequency bands defined in figure 4. To explain the physical mechanisms behind St1 and St2, we compute the spectra of the transverse velocity v(x, y) at three locatio… view at source ↗
Figure 6
Figure 6. Figure 6: Streamwise evolution of the dominant St of the v velocity spectrum along (a) y = 0 and (b) y = 1. The shaded regions mark the frequency bands defined in figure 4. 0 2 4 6 8 10 12 x -4 -2 0 2 4 y -0.27 0.00 0.50 1.00 1.46 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Contours of the time-averaged streamwise velocity [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Energy spectrum of the first 6 SPOD modes of the flow. [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Contour plots of streamwise (left column) and cross-stream (right column) velocities of the real part of [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Convergence of LEs and forward-backward evolution to calculate CLVs for [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Lyapunov spectrum of the flow. [20] proposed a stable algorithm to extract CLVs, leveraging the geometric understanding of the tangent space of [42]. The algorithm is based on a forward-backward evolution of the tangent space. A brief overview is presented in Algorithm 2 below; the detailed methodology of [21] is presented in Appendix A. The 4 phases of the algorithm are shown in figure 10. The algorithm … view at source ↗
Figure 12
Figure 12. Figure 12: Contour plots of the magnitude of the leading CLV [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Contour plot of the second unstable CLV: (a) Instantaneous [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Contour plot of the stable CLV: (a) Instantaneous [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Variation of the angle between V3 and V4 over time (left), contour plots of instantaneous |V3| and |V4| at = 3000 (right). (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Probability density functions (PDFs) of angles between CLVs spanning (a) the unstable and neutral [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Energy spectrum of the first six SPOD modes of [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Contour plots of streamwise velocity (left column) and cross-stream velocity (right column) of the real part [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Contour plots of streamwise velocity (left column) and cross-stream velocity (right column) of the imaginary [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: figure 20. While the energy of [PITH_FULL_IMAGE:figures/full_fig_p021_20.png] view at source ↗
Figure 20
Figure 20. Figure 20: Energy spectrum of the first six SPOD modes of [PITH_FULL_IMAGE:figures/full_fig_p022_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Contour plots of streamwise velocity (left column) and cross-stream velocity (right column) of the real part [PITH_FULL_IMAGE:figures/full_fig_p022_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Eigenvalues from GLSA of time-averaged flow (left) Comparison of LEs and the real parts of the eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p023_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Contour plots of streamwise velocity (left column) and cross-stream velocity (right column) of the real [PITH_FULL_IMAGE:figures/full_fig_p024_23.png] view at source ↗
read the original abstract

We investigate the stability of the flow past two side-by-side square cylinders (at Reynolds number 200 and gap ratio 1) using tools from dynamical systems theory. The flow is highly irregular due to the complex interaction between the flapping jet emanating from the gap and the vortices shed in the wake. We first perform Spectral Proper Orthogonal Decomposition (SPOD) to understand the flow characteristics. We then conduct Lyapunov stability analysis by linearizing the Navier-Stokes equations around the irregular base flow and find that it has two positive Lyapunov exponents. The Covariant Lyapunov Vectors (CLVs) are also computed. Contours of the time-averaged CLVs reveal that the footprint of the leading CLV is in the near-wake, whereas the other CLVs peak further downstream, indicating distinct regions of instability. SPOD of the two unstable CLVs is then employed to extract the dominant coherent structures and oscillation frequencies in the tangent space. For the leading CLV, the two dominant frequencies match closely with the prevalent frequencies in the drag coefficient spectrum, and correspond to instabilities due to vortex shedding and jet-flapping. The second unstable CLV captures the subharmonic instability of the shedding frequency. Global linear stability analysis (GLSA) of the time-averaged flow identifies a neutral eigenmode that resembles the leading SPOD mode of the first CLV, with a very similar structure and frequency. However, while GLSA predicts neutrality, Lyapunov analysis reveals that this direction is unstable, exposing the inherent limitations of the GLSA when applied to chaotic flows.

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

2 major / 2 minor

Summary. The manuscript applies dynamical-systems tools to the chaotic wake of two side-by-side square cylinders at Re = 200 and gap ratio 1. After SPOD characterization of the base flow, the Navier-Stokes equations are linearized about the instantaneous simulated velocity field; two positive Lyapunov exponents are reported together with their covariant Lyapunov vectors (CLVs). SPOD of the unstable CLVs yields dominant frequencies that match features in the drag spectrum. Global linear stability analysis performed on the time-averaged flow recovers a neutral eigenmode whose spatial structure and frequency closely resemble the leading CLV, leading the authors to conclude that GLSA misses an unstable direction present in the full tangent dynamics.

Significance. If the reported positive Lyapunov exponents and the sign difference with the GLSA neutral mode prove robust, the work supplies concrete evidence that time-averaged linearization can fail to capture instability mechanisms that are visible only when the tangent dynamics are integrated along the chaotic trajectory. The frequency correspondence between CLV SPOD modes and drag spectra offers a direct link between tangent-space structures and observable forces, which is a useful methodological contribution for chaotic bluff-body flows.

major comments (2)
  1. [§4] §4 (Lyapunov exponent computation): No convergence diagnostics are presented for the leading Lyapunov exponents with respect to integration length, reorthogonalization interval, or spatial resolution of the tangent-linear operator. In a high-dimensional chaotic discretization these parameters directly control whether the sign of the largest exponent is physical or numerical; without them the central claim that the leading direction is unstable (while GLSA finds neutrality) rests on unverified numerical fidelity.
  2. [§5] §5 (GLSA comparison): The time-averaged base flow used for GLSA is not specified with respect to averaging interval or convergence of the mean; likewise, the growth rate of the reported neutral eigenmode is not quantified to machine precision or shown to remain neutral under modest changes in averaging window. These details are required to make the contrast with the positive Lyapunov exponent rigorous rather than qualitative.
minor comments (2)
  1. [Figures 6-7] Figure captions for the time-averaged CLV contours should explicitly indicate the locations of the two cylinders and the flow direction to aid readers who are not already familiar with the geometry.
  2. [Methods] The linearization step in the methods section would benefit from an explicit statement that the base flow is the instantaneous DNS velocity field at each time step rather than a filtered or interpolated field.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments correctly identify areas where additional numerical validation will strengthen the manuscript, and we address each point below with plans for revision.

read point-by-point responses

  1. Referee: [§4] §4 (Lyapunov exponent computation): No convergence diagnostics are presented for the leading Lyapunov exponents with respect to integration length, reorthogonalization interval, or spatial resolution of the tangent-linear operator. In a high-dimensional chaotic discretization these parameters directly control whether the sign of the largest exponent is physical or numerical; without them the central claim that the leading direction is unstable (while GLSA finds neutrality) rests on unverified numerical fidelity.

    Authors: We agree that explicit convergence diagnostics are necessary to rigorously establish that the positive leading Lyapunov exponent is physical rather than numerical. Although the computations were performed using integration lengths, reorthogonalization intervals, and resolutions selected according to standard practices for high-dimensional fluid systems to ensure convergence, these tests were not documented in the manuscript. In the revised version we will add a new subsection (or appendix) presenting convergence plots for the leading exponents versus integration length, reorthogonalization interval, and spatial resolution of the tangent operator. These diagnostics will confirm that the sign remains positive and stable within the tested ranges. revision: yes

  2. Referee: [§5] §5 (GLSA comparison): The time-averaged base flow used for GLSA is not specified with respect to averaging interval or convergence of the mean; likewise, the growth rate of the reported neutral eigenmode is not quantified to machine precision or shown to remain neutral under modest changes in averaging window. These details are required to make the contrast with the positive Lyapunov exponent rigorous rather than qualitative.

    Authors: We acknowledge that more precise documentation of the averaging procedure and a quantitative statement of the growth rate are needed to make the GLSA–Lyapunov comparison fully rigorous. In the revised manuscript we will specify the exact time interval used to compute the time-averaged base flow, report the growth rate of the neutral eigenmode to machine precision (confirming it lies within numerical tolerance of zero), and include a brief sensitivity test demonstrating that the mode remains neutral under modest changes to the averaging window. These additions will allow a clearer quantitative contrast with the positive Lyapunov exponent. revision: yes

Circularity Check

0 steps flagged

No circularity: standard linearization and established Lyapunov algorithms applied to externally simulated base flow

full rationale

The derivation chain consists of (1) DNS simulation of the chaotic flow, (2) linearization of the Navier-Stokes operator around the instantaneous irregular base flow, (3) integration of the tangent-linear system with reorthogonalization to extract Lyapunov exponents and CLVs, and (4) direct comparison with GLSA performed on the time-averaged field. None of these steps reduce by construction to fitted parameters, self-defined quantities, or load-bearing self-citations. The positive Lyapunov exponents are outputs of the tangent-space integration, not inputs; the contrast with the neutral GLSA eigenmode follows from the distinct base flows (instantaneous vs. averaged) rather than any tautological renaming or ansatz smuggling. The paper remains self-contained against external numerical and dynamical-systems benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim depends on the validity of linearizing the Navier-Stokes equations around a numerically generated irregular base flow and on the numerical accuracy of the Lyapunov exponent and CLV algorithms; no new physical entities are postulated.

axioms (2)
  • standard math The incompressible Navier-Stokes equations govern the fluid motion at the stated Reynolds number.
    Invoked when the paper states it linearizes the Navier-Stokes equations.
  • domain assumption The irregular base flow obtained from direct numerical simulation is sufficiently long and representative for tangent-space linearization.
    Stated in the abstract when describing the linearization step around the irregular base flow.

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    Relation between the paper passage and the cited Recognition theorem.

    We then conduct Lyapunov stability analysis by linearizing the Navier-Stokes equations around the irregular base flow and find that it has two positive Lyapunov exponents. ... Global linear stability analysis (GLSA) of the time-averaged flow identifies a neutral eigenmode ... while GLSA predicts neutrality, Lyapunov analysis reveals that this direction is unstable

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Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    Zhou, and X

    Md Mahbub Alam, Y. Zhou, and X. Wang. The wake of two side-by-side square cylinders. Journal of Fluid Mechanics, 669:432 – 471, February 2011

    work page 2011

  2. [2]

    Schmid, and Dan S

    Shervin Bagheri, Philipp Schlatter, Peter J. Schmid, and Dan S. Henningson. Global stability of a jet in crossflow.Journal of Fluid Mechanics, 624:33–44, April 2009

    work page 2009

  3. [3]

    D. Barkley. Linear analysis of the cylinder wake mean flow.Europhysics Letters, 75(5):750, July 2006

    work page 2006

  4. [4]

    Henderson

    Dwight Barkley and Ronald D. Henderson. Three-dimensional floquet stability analysis of the wake of a circular cylinder.Journal of Fluid Mechanics, 322:215–241, 1996

    work page 1996

  5. [5]

    Gabriela M

    Dwight Barkley, M. Gabriela M. Gomes, and Ronald D. Henderson. Three-dimensional instability in flow over a backward-facing step.J. Fluid Mech., 473:167–190, December 2002

    work page 2002

  6. [6]

    Conditions for validity of mean flow stability analysis.Journal of Fluid Mechanics, 798:485–504, 2016

    Samir Beneddine, Denis Sipp, Anthony Arnault, Julien Dandois, and Lutz Lesshafft. Conditions for validity of mean flow stability analysis.Journal of Fluid Mechanics, 798:485–504, 2016. 28

    work page 2016

  7. [7]

    Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them

    Giancarlo Benettin, Luigi Galgani, Antonio Giorgilli, and Jean-Marie Strelcyn. Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory.Meccanica, 15(1):9–20, March 1980

    work page 1980

  8. [8]

    Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them

    Giancarlo Benettin, Luigi Galgani, Antonio Giorgilli, and Jean-Marie Strelcyn. Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application.Meccanica, 15(1):21–30, March 1980

    work page 1980

  9. [9]

    Thomas R. Bewley. Linear control and estimation of nonlinear chaotic convection: Harnessing the butterfly effect.Physics of Fluids, 11(5):1169–1186, May 1999

    work page 1999

  10. [10]

    Bhattacharya and James W

    S. Bhattacharya and James W. Gregory. Investigation of the cylinder wake under spanwise periodic forcing with a segmented plasma actuator.Physics of Fluids, 27(1):014102, January 2015

    work page 2015

  11. [11]

    Birkhoff

    George D. Birkhoff. Proof of the ergodic theorem.Proceedings of the National Academy of Sciences, 17(12):656–660, 1931

    work page 1931

  12. [12]

    Toward a chaotic adjoint for LES.Proceedings of the 2016 Center for Turbulence Research Summer Program, pages 385–394, January 2017

    P Blonigan, Pablo Fernandez, S Murman, Qiqi Wang, Georgios Rigas, and L Magri. Toward a chaotic adjoint for LES.Proceedings of the 2016 Center for Turbulence Research Summer Program, pages 385–394, January 2017

    work page 2016

  13. [13]

    Carini, F

    M. Carini, F. Giannetti, and F. Auteri. On the origin of the flip–flop instability of two side-by-side cylinder wakes.Journal of Fluid Mechanics, 742:552–576, 2014

    work page 2014

  14. [14]

    Chiarini, M

    A. Chiarini, M. Quadrio, and F. Auteri. Linear stability of the steady flow past rectangular cylinders. Journal of Fluid Mechanics, 929:A36, 2021

    work page 2021

  15. [15]

    The stability of time-periodic flows.Annual Review of Fluid Mechanics, 8:57–74,

    S H Davis. The stability of time-periodic flows.Annual Review of Fluid Mechanics, 8:57–74,

    work page

  16. [16]

    J. P. Eckmann and D. Ruelle. Ergodic theory of chaos and strange attractors.Rev. Mod. Phys., 57:617–656, July 1985

    work page 1985

  17. [17]

    Edwards, L.S

    W.S. Edwards, L.S. Tuckerman, R.A. Friesner, and D.C. Sorensen. Krylov Methods for the Incompressible Navier-Stokes Equations.Journal of Computational Physics, 110(1):82–102, January 1994

    work page 1994

  18. [18]

    Fernandez and Q

    P. Fernandez and Q. Wang. Lyapunov spectrum of the separated flow around the NACA 0012 airfoil and its dependence on numerical discretization.Journal of Computational Physics, 350: 453–469, 2017

    work page 2017

  19. [19]

    Structural sensitivity of the first instability of the cylinder wake

    Flavio Giannetti and Paolo Luchini. Structural sensitivity of the first instability of the cylinder wake. Journal of Fluid Mechanics, 581:167–197, 2007

    work page 2007

  20. [20]

    Ginelli, P

    F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi, and A. Politi. Characterizing Dynamics with Covariant Lyapunov Vectors.Physical Review Letters, 99(13):130601, September 2007

    work page 2007

  21. [21]

    Covariant Lyapunov vectors

    Francesco Ginelli, Hugues Chaté, Roberto Livi, and Antonio Politi. Covariant Lyapunov vectors. Journal of Physics A: Mathematical and Theoretical, 46(25):254005, June 2013. 29

    work page 2013

  22. [22]

    Gonçalves, Guilherme F

    Rodolfo T. Gonçalves, Guilherme F. Rosetti, André L.C. Fujarra, and Allan C. Oliveira. Experimental study on vortex-induced motions of a semi-submersible platform with four square columns, part i: Effects of current incidence angle and hull appendages.Ocean Engineering, 54:150–169, 2012

    work page 2012

  23. [23]

    Three-dimensional wake transition of a square cylinder.Journal of Fluid Mechanics, 842:102–127, 2018

    Hongyi Jiang, Liang Cheng, and Hongwei An. Three-dimensional wake transition of a square cylinder.Journal of Fluid Mechanics, 842:102–127, 2018

    work page 2018

  24. [24]

    Kantarakias and George Papadakis

    Kyriakos D. Kantarakias and George Papadakis. Sensitivity analysis of chaotic systems using a frequency-domain shadowing approach.Journal of Computational Physics, 474:111757, 2023

    work page 2023

  25. [25]

    Kantarakias and George Papadakis

    Kyriakos D. Kantarakias and George Papadakis. Uncertainty quantification of time-average quantities of chaotic systems using sensitivity-enhanced polynomial chaos expansion.Phys. Rev. E, 109:044208, April 2024

    work page 2024

  26. [26]

    H. J. Kim and P. A. Durbin. Investigation of the flow between a pair of circular cylinders in the flopping regime.Journal of Fluid Mechanics, 196:431–448, 1988

    work page 1988

  27. [27]

    Kuptsov and Ulrich Parlitz

    Pavel V. Kuptsov and Ulrich Parlitz. Theory and Computation of Covariant Lyapunov Vectors. Journal of Nonlinear Science, 22(5):727–762, October 2012

    work page 2012

  28. [28]

    R. B. Lehoucq, D. C. Sorensen, and C. Yang.ARPACK Users’ Guide. Society for Industrial and Applied Mathematics, 1998

    work page 1998

  29. [29]

    Wake of two side-by-side square cylinders at low Reynolds numbers.Physics of Fluids, 29(3):033604, March 2017

    Shengwei Ma, Chang-Wei Kang, Teck-Bin Arthur Lim, Chih-Hua Wu, and Owen Tutty. Wake of two side-by-side square cylinders at low Reynolds numbers.Physics of Fluids, 29(3):033604, March 2017

    work page 2017

  30. [30]

    E. Meiburg. On the role of subharmonic perturbations in the far wake.Journal of Fluid Mechanics, 177:83–107, 1987

    work page 1987

  31. [31]

    Flow-induced vibrations in arrays of cylinders.Annual Review of Fluid Mechanics, 25:99–114, 1993

    P M Moretti. Flow-induced vibrations in arrays of cylinders.Annual Review of Fluid Mechanics, 25:99–114, 1993. ISSN 1545-4479. doi: https://doi.org/10.1146/annurev.fl.25.010193.000531

    work page doi:10.1146/annurev.fl.25.010193.000531 1993

  32. [32]

    Hyperbolicity, shadowing directions and sensitivity analysis of a turbulent three- dimensional flow

    Angxiu Ni. Hyperbolicity, shadowing directions and sensitivity analysis of a turbulent three- dimensional flow. Journal of Fluid Mechanics, 863:644–669, March 2019

    work page 2019

  33. [33]

    Sensitivity analysis on chaotic dynamical systems by non-intrusive least squares shadowing (nilss).J

    Angxiu Ni and Qiqi Wang. Sensitivity analysis on chaotic dynamical systems by non-intrusive least squares shadowing (nilss).J. Comput. Phys., 347:56–77, 2017

    work page 2017

  34. [34]

    Characteristics of the leading Lyapunov vector in a turbulent channel flow

    Nikolay Nikitin. Characteristics of the leading Lyapunov vector in a turbulent channel flow. Journal of Fluid Mechanics, 849:942–967, 2018

    work page 2018

  35. [35]

    V. I. Oseledets. A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems. Trudy Moskov. Mat. Obšč., 19:179–210, 1968

    work page 1968

  36. [36]

    Flow instabilities in the wake of a rounded square cylinder

    Doohyun Park and Kyung-Soo Yang. Flow instabilities in the wake of a rounded square cylinder. Journal of Fluid Mechanics, 793:915–932, 2016

    work page 2016

  37. [37]

    On the frequency selection of finite-amplitude vortex shedding in the cylinder wake

    Benoît Pier. On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. Journal of Fluid Mechanics, 458:407–417, 2002

    work page 2002

  38. [38]

    Pilyugin.Shadowing in Dynamical Systems

    Sergei Yu. Pilyugin.Shadowing in Dynamical Systems. Springer Berlin, Heidelberg, 1999. 30

    work page 1999

  39. [39]

    Stephen B. Pope. Turbulent Flows. Cambridge University Press, 2000

    work page 2000

  40. [40]

    Flow effect around two square cylinders arranged side by side using Lattice Boltzmann method.International Journal of Modern Physics C, 19 (11):1683–1694, 2008

    Yong Rao, Yushan Ni, and Chaofeng Liu. Flow effect around two square cylinders arranged side by side using Lattice Boltzmann method.International Journal of Modern Physics C, 19 (11):1683–1694, 2008

    work page 2008

  41. [41]

    W. C. Reynolds and A. K. M. F. Hussain. The mechanics of an organized wave in turbulent shear flow. part 3. Theoretical models and comparisons with experiments.Journal of Fluid Mechanics, 54(2):263–288, 1972

    work page 1972

  42. [42]

    Ergodic theory of differentiable dynamical systems.Publications Mathématiques de l’IHÉS, 50:27–58, 1979

    David Ruelle. Ergodic theory of differentiable dynamical systems.Publications Mathématiques de l’IHÉS, 50:27–58, 1979

    work page 1979

  43. [43]

    Strange attractors.The Mathematical Intelligencer, 2(3):126–137, September 1980

    David Ruelle. Strange attractors.The Mathematical Intelligencer, 2(3):126–137, September 1980

    work page 1980

  44. [44]

    Peter J. Schmid. Nonmodal Stability Theory. Annu. Rev. Fluid Mech., 39(1):129–162, January 2007

    work page 2007

  45. [45]

    Vortex pairing in jets as a global Floquet instability: modal and transient dynamics.Journal of Fluid Mechanics, 862:951–989, 2019

    Léopold Shaabani-Ardali, Denis Sipp, and Lutz Lesshafft. Vortex pairing in jets as a global Floquet instability: modal and transient dynamics.Journal of Fluid Mechanics, 862:951–989, 2019

    work page 2019

  46. [46]

    A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems.Progress of Theoretical Physics, 61(6):1605–1616, June 1979

    Ippei Shimada and Tomomasa Nagashima. A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems.Progress of Theoretical Physics, 61(6):1605–1616, June 1979

    work page 1979

  47. [47]

    Sipp and A

    D. Sipp and A. Lebedev. Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows.J. Fluid Mech, 593:333–358, 2007

    work page 2007

  48. [48]

    Global linear instability.Annual Review of Fluid Mechanics, 43:319–352,

    Vassilios Theofilis. Global linear instability.Annual Review of Fluid Mechanics, 43:319–352,

    work page

  49. [49]

    Schmidt, and Tim Colonius

    Aaron Towne, Oliver T. Schmidt, and Tim Colonius. Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis.Journal of Fluid Mechanics, 847:821–867, July 2018

    work page 2018

  50. [50]

    Periodic orbits, Lyapunov vectors, and singular vectors in the Lorenz system.Journal of the Atmospheric Sciences, 55(3):390 – 398, 1998

    Anna Trevisan and Francesco Pancotti. Periodic orbits, Lyapunov vectors, and singular vectors in the Lorenz system.Journal of the Atmospheric Sciences, 55(3):390 – 398, 1998

    work page 1998

  51. [51]

    Kasun Wijesooriya, Damith Mohotti, Chi-King Lee, and Priyan Mendis. A technical review of computational fluid dynamics (cfd) applications on wind design of tall buildings and structures: Past, present and future.Journal of Building Engineering, 74:106828, 2023

    work page 2023

  52. [52]

    Xu and M

    M. Xu and M. R. Paul. Covariant Lyapunov vectors of chaotic Rayleigh-Bénard convection. Phys. Rev. E, 93:062208, June 2016

    work page 2016

  53. [53]

    Flow past a square cylinder with an angle of incidence.Physics of Fluids, 22(4):043603, April 2010

    Dong-Hyeog Yoon, Kyung-Soo Yang, and Choon-Bum Choi. Flow past a square cylinder with an angle of incidence.Physics of Fluids, 22(4):043603, April 2010

    work page 2010

  54. [54]

    Extreme events and non- Kolmogorov−5/3 spectra in turbulent flows behind two side-by-side square cylinders.Journal of Fluid Mechanics, 874:677–698, 2019

    Yi Zhou, Koji Nagata, Yasuhiko Sakai, and Tomoaki Watanabe. Extreme events and non- Kolmogorov−5/3 spectra in turbulent flows behind two side-by-side square cylinders.Journal of Fluid Mechanics, 874:677–698, 2019. 31

    work page 2019

  55. [55]

    Energy transfer in turbulent flows behind two side-by-side square cylinders.Journal of Fluid Mechanics, 903:A4, 2020

    Yi Zhou, Koji Nagata, Yasuhiko Sakai, Tomoaki Watanabe, Yasumasa Ito, and Toshiyuki Hayase. Energy transfer in turbulent flows behind two side-by-side square cylinders.Journal of Fluid Mechanics, 903:A4, 2020. 32

    work page 2020