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arxiv: 2605.15438 · v1 · pith:4KTGLVXQnew · submitted 2026-05-14 · 🧮 math.OC · physics.flu-dyn

Control of the Fluidic Pinball using the Quadratic-Quadratic Regulator

Ali Bouland , Jeff Borggaard This is my paper

Pith reviewed 2026-05-19 14:34 UTC · model grok-4.3

classification 🧮 math.OC physics.flu-dyn
keywords fluidic pinballquadratic-quadratic regulatorinterpolatory model order reductionwake stabilizationnonlinear flow controlNavier-Stokes equationsReynolds number
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The pith

A quadratic-quadratic regulator stabilizes the fluidic pinball wake at Re=50 where linear control fails.

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

The paper tries to establish that combining interpolatory model order reduction with a quadratic-quadratic regulator produces a feedback law that stabilizes the complex wake behind three cylinders. A sympathetic reader would care because the approach directly addresses the quadratic nonlinearities present in the Navier-Stokes equations, allowing control where standard linear feedback cannot. At Re_D=30 the QQR design meets its performance target 40.1 percent faster than linear control; at Re_D=50 it fully suppresses vortex shedding and eliminates lift oscillations while the linear controller does not.

Core claim

The IMOR-QQR framework provides an effective model-based control strategy that can manage nonlinear hydrodynamic instabilities in complex wake flows such as the fluidic pinball. At Re_D=50 the QQR controller successfully stabilizes the wake whereas the linear controller fails to overcome the nonlinearity of the flow. The QQR control suppresses vortex shedding, resulting in the elimination of lift oscillations and a reduction in the drag coefficient.

What carries the argument

The quadratic-quadratic regulator (QQR) designed on a reduced-order model obtained via interpolatory model order reduction (IMOR) from a finite-element discretization of the actuated Navier-Stokes equations.

If this is right

  • At Re_D=50 the controller eliminates lift oscillations by suppressing vortex shedding.
  • The drag coefficient decreases under successful QQR stabilization.
  • At Re_D=30 the QQR law reaches the desired performance 40.1 percent faster than linear feedback.
  • The framework handles the mutual interactions among the three cylinder wakes.

Where Pith is reading between the lines

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

  • The same quadratic accounting may improve control of other bluff-body wakes that exhibit similar quadratic nonlinearities.
  • If the reduced model is small enough, the resulting feedback could support real-time implementation on embedded hardware.
  • Testing the method at Reynolds numbers above 50 would indicate how far the quadratic approximation remains useful before higher-order terms dominate.

Load-bearing premise

The reduced-order model from interpolatory model order reduction must accurately represent the input-output dynamics of the full fluidic pinball system so the controller transfers to the original high-dimensional flow.

What would settle it

Apply the QQR feedback law computed from the reduced-order model directly to the full-order finite-element simulation at Re_D=50 and check whether vortex shedding and lift oscillations are suppressed or persist.

Figures

Figures reproduced from arXiv: 2605.15438 by Ali Bouland, Jeff Borggaard.

Figure 1
Figure 1. Figure 1: The velocity magnitude of the steady-state solution [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the mesh used for the fluidic pinball. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Vorticity Plot for 𝑅𝑒𝐷 = 50. 2.1.2 Boundary Conditions The boundary conditions closely follow what was used in [15]. They consist of Dirichlet boundary conditions for the three cylinders (𝒗 = 0), free stream boundary conditions for the inlets and walls (𝒗 = 𝒆𝑥). For the outlet, stress-free boundary conditions are assumed. The multi-input multi-output (MIMO) controller can directly adjust the Dirichlet boun… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Bode plot of the IRKA ROM model with [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Vorticity fields for 𝑅𝑒𝐷 = 30 with QQR controller 𝒖 (2) at three different times; (a) t = 10, (b) t = 30, (c) t = 80. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: 𝑅𝑒𝐷 = 30 case (a) The top figure shows a comparison between the 𝐿2 error in velocity for the linear and QQR controllers, while the bottom one shows a comparison of the integrated running cost for both controllers. (b) Shows the control efforts for the three cylinders and both controllers, represented by their tangential velocity. In figures 7a and 7b, the lift and drag coefficients are plotted, respectivel… view at source ↗
Figure 7
Figure 7. Figure 7: 𝑅𝑒𝐷 = 30 case (a) The top figure presents the lift coefficients on each individual cylinder while the bottom figure provides the evolution of 𝐶𝐿 on the system of cylinders. (b) Figures corresponding to the drag coefficient 𝐶𝐷 evolution. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Vorticity fields for 𝑅𝑒𝐷 = 50 with the QQR controller 𝒖 (2) at three different times; (a) t = 10, (b) t = 30, (c) t = 80. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: 𝑅𝑒𝐷 = 50 case (a) The top figure shows a comparison between the 𝐿2 error in velocity for the linear and QQR controllers, while the bottom one shows a comparison between the ideal cost function for both controllers. (b) Shows the control efforts for the three cylinders and both controllers, represented by their tangential velocity In figures 10a and 10b, the lift and drag coefficients are plotted, respectiv… view at source ↗
Figure 10
Figure 10. Figure 10: 𝑅𝑒𝐷 = 50 case (a) The top figure presents the lift coefficients on each individual cylinder while the bottom figure provides the evolution of 𝐶𝐿 on the system of cylinders. (b) Figures corresponding to the drag coefficient 𝐶𝐷 evolution. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
read the original abstract

The fluidic pinball presents a significant benchmark for nonlinear flow control, managing the complex interactions of three cylinder wakes. This study addresses the stabilization of the fluidic pinball to its unstable steady-state solution using a model-based nonlinear feedback strategy. We propose a framework that combines interpolatory model order reduction (IMOR) with the quadratic-quadratic regulator (QQR), a feedback control methodology that is specifically suited to the quadratic nonlinearity of the Navier-Stokes equations. A finite element model (FEM) of the problem coupled with IMOR is used to produce a reduced-order model (ROM) that accurately represents the input-output dynamics of the actuated wake. The performance of the QQR control is evaluated against the traditional linear feedback control for two different Reynolds numbers, $Re_D = 30$ and $Re_D = 50$. At $Re_D = 30$, the QQR controller is able to stabilize the wake and reaches the desired performance criteria 40.1\% faster than using a linear feedback controller. More significantly, at $Re_D = 50$, the QQR controller successfully stabilizes the wake, whereas the linear controller fails to overcome the nonlinearity of the flow. The QQR control effectively suppresses vortex shedding, resulting in the elimination of lift oscillations and a reduction in the drag coefficient. These results demonstrate that the IMOR-QQR framework provides an effective model-based control strategy that can manage nonlinear hydrodynamic instabilities in such complex wake 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

1 major / 2 minor

Summary. The paper proposes combining interpolatory model order reduction (IMOR) with the quadratic-quadratic regulator (QQR) to stabilize the fluidic pinball wake to its unstable steady state. Using a finite-element discretization of the actuated Navier-Stokes equations, an IMOR-reduced quadratic model is constructed and used to design the QQR feedback law. Performance is compared to linear feedback at Re_D=30 (where QQR reaches criteria 40.1% faster) and Re_D=50 (where QQR stabilizes the wake and suppresses vortex shedding while linear control fails).

Significance. If the ROM-to-full-order transfer is rigorously validated, the work would demonstrate a practical model-based nonlinear control strategy for a standard benchmark where linear methods are insufficient, highlighting the value of retaining quadratic terms in reduced-order flow control. The IMOR-QQR combination itself is a methodological contribution worth noting if the closed-loop accuracy is confirmed.

major comments (1)
  1. [Abstract and §4] Abstract and §4 (results): The central claim that the QQR controller designed on the IMOR quadratic ROM successfully stabilizes the wake at Re_D=50 while linear feedback fails rests on the unverified assumption that the reduced quadratic system faithfully reproduces the input-output map of the full-order actuated Navier-Stokes equations under closed-loop operation. No closed-loop lift/drag time series, residual norms, or direct ROM-vs-FEM trajectory comparisons under the same feedback law are reported, which is load-bearing for the headline result.
minor comments (2)
  1. [§2] Notation for the quadratic-quadratic cost and the IMOR projection operators should be introduced with explicit equations early in §2 to avoid ambiguity when the QQR Riccati solution is presented.
  2. [Figures in §4] Figure captions for the closed-loop wake visualizations should include the specific gain matrices or control effort levels used so that the suppression of vortex shedding can be directly linked to the reported metrics.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comments on validation. We address the major point below and will strengthen the manuscript with additional closed-loop comparisons.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (results): The central claim that the QQR controller designed on the IMOR quadratic ROM successfully stabilizes the wake at Re_D=50 while linear feedback fails rests on the unverified assumption that the reduced quadratic system faithfully reproduces the input-output map of the full-order actuated Navier-Stokes equations under closed-loop operation. No closed-loop lift/drag time series, residual norms, or direct ROM-vs-FEM trajectory comparisons under the same feedback law are reported, which is load-bearing for the headline result.

    Authors: We agree that explicit closed-loop validation between the IMOR quadratic ROM and the full-order FEM is essential to support the Re_D=50 result. The current manuscript validates the ROM primarily in open-loop and reports QQR performance on the ROM itself. In the revised manuscript we will add full-order closed-loop simulations in which the QQR feedback law (designed on the ROM) is applied directly to the actuated Navier-Stokes FEM model. We will include lift and drag time series for both the full-order system and the ROM prediction under identical feedback, together with quantitative error metrics and residual norms. These additions will be placed in Section 4 and referenced in the abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on numerical transfer from ROM to full-order system

full rationale

The paper's central result is a numerical demonstration that a QQR controller designed on an IMOR-reduced quadratic model stabilizes the fluidic pinball wake at Re_D=50 while linear feedback does not. This is evaluated by applying the designed gain to the original FEM discretization and observing lift/drag suppression. No equations or steps in the abstract reduce the stabilization claim to a fitted parameter or self-citation by construction; the IMOR projection and QQR Riccati solution are standard operations whose output is then tested externally on the unreduced system. The derivation chain is therefore self-contained against the reported simulations rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard fluid-dynamics modeling assumptions and the premise that the chosen ROM preserves the essential input-output behavior needed for control synthesis.

axioms (2)
  • domain assumption The incompressible Navier-Stokes equations accurately describe the fluid flow around the cylinders.
    Invoked implicitly as the governing physics for the FEM model.
  • domain assumption The quadratic-quadratic regulator is an appropriate feedback law for systems whose nonlinearity is quadratic.
    Central methodological choice justified by the structure of the Navier-Stokes equations.

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Reference graph

Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    Numerical simulation of the flow behind a rotary oscillating circular cylinder.PhysicsofFluids, 10(869), 1998

    Seung-Jin Baek and Hyung Jin Sung. Numerical simulation of the flow behind a rotary oscillating circular cylinder.PhysicsofFluids, 10(869), 1998

    work page 1998

  2. [2]

    Closed-loop control of an open cavity flow using reduced- order models.Journal of FluidMechanics, 641:1–50, 2009

    A Barbagallo, D Sipp, and Peter J Schmid. Closed-loop control of an open cavity flow using reduced- order models.Journal of FluidMechanics, 641:1–50, 2009

    work page 2009

  3. [3]

    Theproperorthogonaldecompositionintheanalysis of turbulent flows.Annual ReviewofFluidMechanics, 25(1):539–575, 1993

    GalBerkooz,PhilipHolmes,andJohnLLumley. Theproperorthogonaldecompositionintheanalysis of turbulent flows.Annual ReviewofFluidMechanics, 25(1):539–575, 1993

    work page 1993

  4. [4]

    Borggaard and Serkan Gugercin

    Jeff. Borggaard and Serkan Gugercin. Model reduction for daes with an application to flow control. In Rudibert King, editor,Active Flow and Combustion Control 2014, pages 381–396, Cham, 2015. Springer International Publishing

    work page 2014

  5. [5]

    A goal-oriented reduced-order modeling approach for nonlinear systems.ComputersandMathematics withApplications, 71(11):2155–2169, 2016

    Jeff Borggaard, Zhu Wang, and Lizette Zietsman. A goal-oriented reduced-order modeling approach for nonlinear systems.ComputersandMathematics withApplications, 71(11):2155–2169, 2016

    work page 2016

  6. [6]

    The quadratic-quadratic regulator problem: Approximating feedback controls for quadratic-in-state nonlinear systems

    Jeff Borggaard and Lizette Zietsman. The quadratic-quadratic regulator problem: Approximating feedback controls for quadratic-in-state nonlinear systems. In2020 American Control Conference (ACC), pages 818–823, 2020

    work page 2020

  7. [7]

    First instability and structural sensitivity of the flow past two side-by-side cylinders.Journal of FluidMechanics, 749:22, 2014

    M Carini, F Giannetti, and F Auteri. First instability and structural sensitivity of the flow past two side-by-side cylinders.Journal of FluidMechanics, 749:22, 2014

    work page 2014

  8. [8]

    On the origin of the flip-flop instability of two side-by-side cylinder wakes.Journal of FluidMechanics, 742:552–576, 2014

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

    work page 2014

  9. [9]

    Vortex suppression and drag reduction in the wake of counter-rotating cylinders.Journal ofFluidMechanics, 679:343–382, 2011

    Andre S Chan, Peter A Dewey, Antony Jameson, Chunlei Liang, and Alexander J Smits. Vortex suppression and drag reduction in the wake of counter-rotating cylinders.Journal ofFluidMechanics, 679:343–382, 2011

    work page 2011

  10. [10]

    An experimental study on a suction flow control method to reduce theunsteadinessofthewindloadsactingonacircularcylinder

    Wen-Li Chen, Hui Li, and Hui Hu. An experimental study on a suction flow control method to reduce theunsteadinessofthewindloadsactingonacircularcylinder. ExperimentsinFluids,55:1–20,2014

    work page 2014

  11. [11]

    Numericalsimulationofpassive-suction-jetcontrol of flow over two side-by-side circular cylinders.OceanEngineering, 257:111624, 2022

    Wen-LiChen,Xiang-WeiMin,andYan-JiaoGuo. Numericalsimulationofpassive-suction-jetcontrol of flow over two side-by-side circular cylinders.OceanEngineering, 257:111624, 2022

    work page 2022

  12. [12]

    Artificial intelligence control applied to drag reduction of the fluidic pinball.PAMM, 19(1):e201900268, 2019

    Guy Y Cornejo Maceda, Bernd R Noack, François Lusseyran, Nan Deng, Luc Pastur, and Marek Morzynski. Artificial intelligence control applied to drag reduction of the fluidic pinball.PAMM, 19(1):e201900268, 2019

    work page 2019

  13. [13]

    Low-order model for successive bifurcations of the fluidic pinball.Journal of FluidMechanics, 884:A37, 2020

    Nan Deng, Bernd R Noack, Marek Morzyński, and Luc R Pastur. Low-order model for successive bifurcations of the fluidic pinball.Journal of FluidMechanics, 884:A37, 2020

    work page 2020

  14. [14]

    Noack, Marek Morzyński, and Luc R

    Nan Deng, Bernd R. Noack, Marek Morzyński, and Luc R. Pastur. Galerkin force model for transient and post-transient dynamics of the fluidic pinball.Journal of FluidMechanics, 918(A4), 2021

    work page 2021

  15. [15]

    Routetochaosinthefluidicpinball

    NanDeng,LucRPastur,MarekMorzyński,andBerndRNoack. Routetochaosinthefluidicpinball. In FluidsEngineeringDivisionSummerMeeting,volume51555,pageV001T01A005.AmericanSociety of Mechanical Engineers, 2018. 18

    work page 2018

  16. [16]

    Control techniques in flow past a cylinder-a review

    Lalith Kumar Doreti and L Dineshkumar. Control techniques in flow past a cylinder-a review. InIOP Conference Series: materials science and engineering, volume 377, page 012144. IOP Publishing, 2018

    work page 2018

  17. [17]

    Vortex shedding resonance from a rotationally oscillating cylinder

    N Fujisawa, K Ikemoto, and K Nagaya. Vortex shedding resonance from a rotationally oscillating cylinder. Journal ofFluids andStructures, 12:1041–1053, 1998

    work page 1998

  18. [18]

    Gmsh: A 3-d finite element mesh generator with built-inpre-andpost-processingfacilities

    Christophe Geuzaine and Jean-François Remacle. Gmsh: A 3-d finite element mesh generator with built-inpre-andpost-processingfacilities. InternationalJournalforNumericalMethodsinEngineering, 79(11):1309–1331, 2009

    work page 2009

  19. [19]

    A multistep technique with implicit difference schemes for calculating two-or three- dimensional cavity flows.Journal of ComputationalPhysics, 30(1):76–95, 1979

    Katuhiko Goda. A multistep technique with implicit difference schemes for calculating two-or three- dimensional cavity flows.Journal of ComputationalPhysics, 30(1):76–95, 1979

    work page 1979

  20. [20]

    Model Reduction of Descriptor Systems by Interpolatory Projection Methods.SIAMJournal on ScientificComputing, 35(5):B1010–B1033, 2013

    Serkan Gugercin, T Stykel, and S A Wyatt. Model Reduction of Descriptor Systems by Interpolatory Projection Methods.SIAMJournal on ScientificComputing, 35(5):B1010–B1033, 2013

    work page 2013

  21. [21]

    How to control hydrodynamic force on fluidic pinball via deep reinforcement learning.PhysicsofFluids, 35(4):045137, April 2023

    Feng Haodong, Yue Wang, Xiang Hui, Jin Zhiyang, and Fan Dixia. How to control hydrodynamic force on fluidic pinball via deep reinforcement learning.PhysicsofFluids, 35(4):045137, April 2023

    work page 2023

  22. [22]

    Suppression of vortex shedding for flow around a circular cylinder using optimal control.International Journal for Numerical Methods in Fluids, 38(1):43–69, 2002

    Chris Homescu, IM Navon, and Zhijin Li. Suppression of vortex shedding for flow around a circular cylinder using optimal control.International Journal for Numerical Methods in Fluids, 38(1):43–69, 2002

    work page 2002

  23. [23]

    Kazufumi Ito and S. S. Ravindran. A reduced basis method for control problems governed by PDEs. In W. Desch, F. Kappel, and K. Kunisch, editors,Control and Estimation of Distributed Parameter Systems, volume 16, pages 153–168. Birkhäuser, 1998

    work page 1998

  24. [24]

    Flow control around a circular cylinder using pulsed dielectric barrier discharge surface plasma.Physicsof Fluids, 21(8), 2009

    Timothy N Jukes and Kwing-So Choi. Flow control around a circular cylinder using pulsed dielectric barrier discharge surface plasma.Physicsof Fluids, 21(8), 2009

    work page 2009

  25. [25]

    Scalable computation of energy functionsfornonlinearbalancedtruncation

    Boris Kramer, Serkan Gugercin, Jeff Borggaard, and Linus Balicki. Scalable computation of energy functionsfornonlinearbalancedtruncation. ComputerMethodsinAppliedMechanicsandEngineering, 427:117011, 2024

    work page 2024

  26. [26]

    Flow past two rotating cylinders.Physicsof Fluids, 23(1), 2011

    S Kumar, B Gonzalez, and O Probst. Flow past two rotating cylinders.Physicsof Fluids, 23(1), 2011

    work page 2011

  27. [27]

    Springer Nature, 2017

    Hans Petter Langtangen and Anders Logg.SolvingPDEs in Python: the FEniCS TutorialI. Springer Nature, 2017

    work page 2017

  28. [28]

    Flow structure of wake behind a rotationally oscillating circular cylinder

    Sang-Joon Lee and Jung-Yeop Lee. Flow structure of wake behind a rotationally oscillating circular cylinder. Journal ofFluids andStructures, 22(8):1097–1112, 2006

    work page 2006

  29. [29]

    Yiqing Li, Wenshi Cui, Qing Jia, Qiliang Li, Zhigang Yang, Marek Morzyński, and Bernd R. Noack. Explorative gradient method for active drag reduction of the fluidic pinball and slanted ahmed body. Journal ofFluid Mechanics, 932(A7), 2022

    work page 2022

  30. [30]

    Dolfin: Automated finite element computing.ACMTransactionson MathematicalSoftware(TOMS), 37(2):1–28, 2010

    Anders Logg and Garth N Wells. Dolfin: Automated finite element computing.ACMTransactionson MathematicalSoftware(TOMS), 37(2):1–28, 2010. 19

    work page 2010

  31. [31]

    The structure of inhomogeneous turbulent flows

    John L Lumley. The structure of inhomogeneous turbulent flows. InAtmospheric Turbulence and RadioWavePropagation, pages 166–178, 1967

    work page 1967

  32. [32]

    Stabilization of the fluidic pinball with gradient-enriched machine learning control.Journal of Fluid Mechanics, 917(A42), 2021

    Guy Y Cornejo Maceda, Yiqing Li, François Lusseyran, Marek Morzyński, and Bernd R Noack. Stabilization of the fluidic pinball with gradient-enriched machine learning control.Journal of Fluid Mechanics, 917(A42), 2021

    work page 2021

  33. [33]

    Self-tuning model predictive control for wake flows.Journal ofFluidMechanics, 983(A26), 2024

    Luigi Marra, Andrea Meilán-Vila, and Stefano Discetti. Self-tuning model predictive control for wake flows.Journal ofFluidMechanics, 983(A26), 2024

    work page 2024

  34. [34]

    B. Moore. Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEETransactionsonAutomaticControl, 26(1):17–32, 1981

    work page 1981

  35. [35]

    Mullis and R

    C. Mullis and R. Roberts. Synthesis of minimum roundoff noise fixed point digital filters.IEEE Transactionson CircuitsandSystems, 23(9):551–562, 1976

    work page 1976

  36. [36]

    Solution of Hamilton Jacobi Bellman Equations

    Carmeliza Luna Navasca and Arthur J Krener. Solution of Hamilton Jacobi Bellman Equations. In IEEEConferenceonDecisionandControl, volume 39, pages 570–574, 2000

    work page 2000

  37. [37]

    Viscous flow around a rotationally oscil- lating circular cylinder.ISASreport, 40(12):311–338, 1975

    Atsushi Okajima, Hiroyuki Takata, and Tsuyoshi Asanuma. Viscous flow around a rotationally oscil- lating circular cylinder.ISASreport, 40(12):311–338, 1975

    work page 1975

  38. [38]

    Control of oscillatory forces on a circular cylinder by rotation

    Yuh-Roung Ou. Control of oscillatory forces on a circular cylinder by rotation. Technical report, ICASE, 1991. number 91–67

    work page 1991

  39. [39]

    Reduced-order modeling of the fluidicpinball

    Luc R Pastur, Nan Deng, Marek Morzyński, and Bernd R Noack. Reduced-order modeling of the fluidicpinball. In11thChaoticModelingandSimulationInternationalConference11,pages205–213. Springer, 2019

    work page 2019

  40. [40]

    Artificial neural networkstrainedthroughdeepreinforcementlearningdiscovercontrolstrategiesforactiveflowcontrol

    Jean Rabault, Miroslav Kuchta, Atle Jensen, Ulysse Réglade, and Nicolas Cerardi. Artificial neural networkstrainedthroughdeepreinforcementlearningdiscovercontrolstrategiesforactiveflowcontrol. Journal ofFluid Mechanics, 865:281–302, April 2019

    work page 2019

  41. [41]

    Clarence W. Rowley. Model reduction for fluids using balanced proper orthogonal decomposition. International Journal ofBifurcationandChaos, 15(3):997–1013, 2005

    work page 2005

  42. [42]

    Rowley and David R

    Clarence W. Rowley and David R. Williams. Dynamics and control of high-Reynolds-number flow over open cavities.Annual Reviewof FluidMechanics, 38(1):251–276, January 2006

    work page 2006

  43. [43]

    Numericalsimulation of flow over two side-by-side circular cylinders.Journal of Hydrodynamics, Ser

    HesamSarvghad-Moghaddam,NavidNooredin,andBehzadGhadiri-Dehkordi. Numericalsimulation of flow over two side-by-side circular cylinders.Journal of Hydrodynamics, Ser. B, 23(6):792–805, 2011

    work page 2011

  44. [44]

    Balancingfornonlinearsystems

    JacquelienMAScherpen. Balancingfornonlinearsystems. Systems&ControlLetters,21(2):143–153, 1993

    work page 1993

  45. [45]

    Computation investigation of drag reduction on a rotationally oscillating cylinder.ESAIM:Proceedings, 1:307–323, 1996

    D Shiels, A Leonard, and A Stagg. Computation investigation of drag reduction on a rotationally oscillating cylinder.ESAIM:Proceedings, 1:307–323, 1996

    work page 1996

  46. [46]

    Smith, Jeff Moehlis, and Philip Holmes

    Troy R. Smith, Jeff Moehlis, and Philip Holmes. Low-dimensional modelling of turbulence using the proper orthogonal decomposition: A tutorial.NonlinearDynamics, 41(1–3):275–307, August 2005. 20

    work page 2005

  47. [47]

    PhD Thesis, Virginia Polytechnic Institute and State University, 2009

    Miroslav Stoyanov.ReducedOrder Methods forLargeScale Riccati Equations. PhD Thesis, Virginia Polytechnic Institute and State University, 2009

    work page 2009

  48. [48]

    S. Taneda. Visual observations of the flow past a circular cylinder performing a rotatory oscillation. Journal ofthePhysicalSocietyof Japan, 45(3):1038–1043, 1978

    work page 1978

  49. [49]

    Control of wakes and vortex-induced vibrations of a single circular cylinder using synthetic jets.Journal of FluidsandStructures, 60:160–179, 2016

    Chenglei Wang, Hui Tang, Fei Duan, and CM Simon. Control of wakes and vortex-induced vibrations of a single circular cylinder using synthetic jets.Journal of FluidsandStructures, 60:160–179, 2016

    work page 2016

  50. [50]

    Vortex Formation in the Wake of an Oscillating Cylinder.Journal of FluidsandStructures, 2:355–381, 1988

    C H K Williamson and A Roshko. Vortex Formation in the Wake of an Oscillating Cylinder.Journal of FluidsandStructures, 2:355–381, 1988

    work page 1988

  51. [51]

    Stremler

    Wenchao Yang, Emad Masroor, and Mark A. Stremler. A new definition of vortex formation length. Fluid-Structure-SoundInteractionsAnd Control, pages 109–114, 2024. 21

    work page 2024