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arxiv: 2604.02440 · v1 · submitted 2026-04-02 · ⚛️ physics.flu-dyn

Parametric Reduced-Order modeling and Closed-Loop Control of Tandem-Cylinder Wakes

Tea Vojkovic , Dimitris Boskos , Abel-John Buchner This is my paper

Pith reviewed 2026-05-13 20:57 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords tandem cylindersvortex sheddingreduced-order modelmodel predictive controlclosed-loop flow controlwake suppressionNavier-Stokesfluid dynamics
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The pith

A parametric reduced-order model enables closed-loop suppression of vortex shedding in tandem-cylinder wakes at low Reynolds numbers.

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

The paper establishes a closed-loop control method for tandem cylinder flows in the co-shedding regime, where wake interactions produce strong unsteady loads. It derives a parametric reduced-order model from global weakly nonlinear analysis of the incompressible Navier-Stokes equations and extends the model to handle time-dependent forcing for real-time prediction. A model predictive controller is then synthesized and applied to the full-order system through velocity measurements and volumetric forcing, producing complete suppression of vortex shedding in both the gap and downstream wake for Reynolds numbers 50, 60, and 70 at eight-diameter spacing, together with a marked drop in unsteadiness at Re=80. The same framework succeeds with only one or two sensor points.

Core claim

A parametric reduced-order model obtained from global weakly nonlinear analysis of the incompressible Navier-Stokes equations, generalized to time-dependent forcing, accurately predicts controlled flow evolution and serves as the basis for a model predictive controller that, when applied to the full-order system via volumetric forcing and limited velocity measurements, fully suppresses vortex shedding in the gap region and downstream wake for Re=50, 60, and 70 at a cylinder spacing of eight diameters.

What carries the argument

The parametric reduced-order model from global weakly nonlinear analysis, extended to accommodate time-dependent volumetric forcing and used for real-time prediction inside a model predictive controller.

If this is right

  • Vortex shedding is eliminated in both the inter-cylinder gap and the wake of the second cylinder for Re=50, 60, and 70.
  • The same controller reduces flow unsteadiness substantially at Re=80.
  • Suppression is achieved with only one velocity measurement at Re=50 and two measurements at Re=60 and 70.
  • The reduced-order model supports real-time closed-loop actuation without requiring full-state sensing.

Where Pith is reading between the lines

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

  • The same modeling route could be tested on tandem arrangements at different spacings or with additional cylinders.
  • Sparse-sensor success points to possible use in experiments where only a few hot-wire or pressure probes are available.
  • If the model retains accuracy at modestly higher Reynolds numbers, the approach may offer a scalable route to wake control in engineering flows.

Load-bearing premise

The parametric reduced-order model remains faithful to the full-order Navier-Stokes dynamics once time-dependent forcing is introduced.

What would settle it

Direct simulation of the model predictive controller on the full-order Navier-Stokes equations at Re=70 shows persistent vortex shedding in the gap or wake despite the reduced-order predictions.

Figures

Figures reproduced from arXiv: 2604.02440 by Abel-John Buchner, Dimitris Boskos, Tea Vojkovic.

Figure 1
Figure 1. Figure 1: Schematic of the closed-loop architecture implemented here to control the oscillating flow around [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: 3 Reduced-Order Model In the following, we outline the methodology for constructing a forced reduced-order model (8) of the in￾compressible Navier–Stokes equations (2). We build on the weakly nonlinear analysis of Sipp and Lebedev [2007], Sipp [2012] to obtain a two-dimensional forced Stuart–Landau model. In contrast to the formula￾tion in Sipp [2012], where the reduced-order input is assumed to be constan… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the model-predictive controller. At each time step [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic defining the computational domain. The streamwise and transverse coordinates [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Velocity fields illustrating the steady base flow and the oscillatory flow on the limit cycle at [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Hydrodynamic force coefficients for fully developed limit-cycle flow over two cylinders in tan [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spatial structure of the modes appearing in the expansion (16) up to the order [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Prediction of the (scaled) mode amplitude [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Spatial distribution of the norm of the adjoint mode [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Time evolution of the global mode amplitude and its suppression. For [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Evolution of energy K = 0.5⟨u ′ ,u ′ ⟩ of the unsteady velocity u ′ relative to the base flow ub, as u ′ = u − ub. The flow evolves to the limit cycle before t = 500, at which time control is turned on. The solid red curve represents results obtained by applying control using the full measurements for the feedback, while the dashed blue curve is the result with state estimation via point measurements only… view at source ↗
Figure 11
Figure 11. Figure 11: Time evolution and suppression of lift coefficient [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Time evolution and suppression of drag coefficient [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Time evolution of the global mode amplitude and its suppression. For [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 6
Figure 6. Figure 6: This justifies the use of the linear mapping (23), which retains only terms proportional to [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 14
Figure 14. Figure 14: Time evolution and suppression of lift coefficient [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Time evolution and suppression of drag coefficient [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
read the original abstract

The flow around two circular cylinders arranged in a tandem exhibits complex wake interactions that lead to amplified unsteady loads, particularly in the co-shedding regime where a fully developed wake forms in the gap between the cylinders. Although various control strategies have been proposed to mitigate these effects, most prior studies have focused primarily on load alleviation. Complete suppression of vortex shedding, both in the gap region and in the wake of the second cylinder, has so far only been achieved using open-loop approaches. In this work, we propose a closed-loop control framework for suppressing vortex shedding in tandem cylinder flows in the co-shedding regime. Focusing on low Reynolds numbers and sufficiently large spacings, we derive a parametric reduced-order model using a global weakly nonlinear analysis of the incompressible Navier-Stokes equations. The model is generalized to account for time dependent forcing and facilitates the real time prediction of the flow evolution. Using this model, we design a model predictive controller and apply it to the full-order system via velocity measurements and volumetric forcing. The approach is demonstrated for a cylinder spacing of eight diameters. Vortex shedding is fully suppressed in both the gap region and the downstream wake for Reynolds numbers $Re=50$, $60$, and $70$, while a significant reduction in flow unsteadiness is achieved at $Re=80$. We further show that effective control is possible with limited sensing: suppression is achieved using a single measurement point for $Re=50$ and two-point measurements for $Re=60$ and $70$.

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 claims to derive a parametric reduced-order model from global weakly nonlinear analysis of the incompressible Navier-Stokes equations for tandem cylinder wakes in the co-shedding regime. The model is generalized to include time-dependent volumetric forcing and is used to design a model predictive controller that achieves closed-loop suppression of vortex shedding via limited velocity sensing. Full suppression is reported for Re=50, 60, and 70, with significant reduction in unsteadiness at Re=80, for a cylinder spacing of eight diameters.

Significance. If the central claims are substantiated with quantitative validation, the work would advance closed-loop flow control by showing that a physics-derived parametric ROM can enable complete vortex shedding suppression in a complex interacting wake, extending beyond prior open-loop methods. The limited-sensing aspect and direct application to the full-order system via MPC could inform practical strategies for load mitigation in bluff-body flows.

major comments (3)
  1. [Results] Results section: the claims of full suppression at Re=50, 60, 70 and reduction at Re=80 are stated without quantitative error metrics (e.g., RMS lift/drag coefficients, kinetic energy in the gap/wake) or direct comparison of controlled full-order DNS trajectories against the ROM predictions under the applied MPC forcing.
  2. [Model derivation] Model derivation and generalization sections: the global weakly nonlinear analysis procedure is not detailed sufficiently to allow reproduction; specifically, the incorporation of parametric Re dependence, the form of the cubic terms, and the truncation order are not shown explicitly, leaving the accuracy of the time-dependent forcing extension unverified.
  3. [Control application] Control application: no ROM-vs-DNS validation is reported for the actual closed-loop trajectories at the tested Re values, which directly bears on the assumption that the parametric ROM accurately predicts the flow evolution under the volumetric forcing when applied to the full Navier-Stokes system.
minor comments (2)
  1. [Abstract] Abstract and introduction: clarify the precise locations of the single- and two-point measurements used for sensing at each Re, and confirm they are the only inputs to the MPC.
  2. [Notation] Notation: ensure the forcing term and the reduced-order coefficients are defined consistently between the weakly nonlinear derivation and the MPC formulation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed review. The comments highlight important opportunities to strengthen the quantitative validation and reproducibility of the parametric reduced-order model and its closed-loop application. We address each major comment below and will incorporate the suggested revisions.

read point-by-point responses

  1. Referee: [Results] Results section: the claims of full suppression at Re=50, 60, 70 and reduction at Re=80 are stated without quantitative error metrics (e.g., RMS lift/drag coefficients, kinetic energy in the gap/wake) or direct comparison of controlled full-order DNS trajectories against the ROM predictions under the applied MPC forcing.

    Authors: We agree that additional quantitative metrics will improve substantiation of the suppression claims. In the revised manuscript we will add RMS lift and drag coefficients for the uncontrolled and controlled cases at each Re, together with kinetic energy integrals over the gap and wake regions. We will also include direct trajectory comparisons between the controlled DNS and the ROM predictions under the MPC forcing to quantify model fidelity during actuation. revision: yes

  2. Referee: [Model derivation] Model derivation and generalization sections: the global weakly nonlinear analysis procedure is not detailed sufficiently to allow reproduction; specifically, the incorporation of parametric Re dependence, the form of the cubic terms, and the truncation order are not shown explicitly, leaving the accuracy of the time-dependent forcing extension unverified.

    Authors: We accept that the derivation section requires greater explicitness for reproducibility. The revised manuscript will expand this section to present the full weakly nonlinear expansion procedure, the explicit parametric dependence of the base flow and linear operators on Re, the retained cubic terms in component form, the chosen truncation order, and the precise extension to time-dependent volumetric forcing, including the resulting forced ROM equations. revision: yes

  3. Referee: [Control application] Control application: no ROM-vs-DNS validation is reported for the actual closed-loop trajectories at the tested Re values, which directly bears on the assumption that the parametric ROM accurately predicts the flow evolution under the volumetric forcing when applied to the full Navier-Stokes system.

    Authors: We acknowledge the value of explicit closed-loop validation. While the current results already demonstrate successful suppression when the ROM-designed MPC is applied to the full-order DNS, the revised manuscript will add side-by-side comparisons of ROM-predicted modal amplitudes and selected flow quantities against the corresponding DNS time series for the controlled trajectories at Re = 50, 60, 70, and 80. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The parametric reduced-order model is obtained via global weakly nonlinear analysis applied directly to the incompressible Navier-Stokes equations and then extended for time-dependent volumetric forcing. The model-predictive controller is subsequently applied to the full-order system using velocity measurements. No load-bearing step in the abstract or described approach reduces a claimed prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness result, or renames an input as an output. The derivation therefore remains self-contained against the underlying PDEs and standard weakly nonlinear techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the central claim rests on the validity of the weakly nonlinear analysis and the accuracy of the reduced-order model when used for prediction under forcing.

axioms (1)
  • standard math The incompressible Navier-Stokes equations govern the flow
    Invoked as the starting point for the global weakly nonlinear analysis

pith-pipeline@v0.9.0 · 5578 in / 1253 out tokens · 35964 ms · 2026-05-13T20:57:01.072463+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Deep Reinforcement Learning Discovers a Novel Control Algorithm for Mitigating Flow-Induced Vibrations in Underactuated Tandem Cylinders

    physics.flu-dyn 2026-05 unverdicted novelty 7.0

    Deep reinforcement learning discovers high-frequency bang-bang and low-frequency lock-on rotary controls that suppress vibrations in fully and underactuated tandem cylinders by 70-95%.

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    B. S. Carmo, J. R. Meneghini, and S. J. Sherwin. Possible states in the flow around two circular cylinders in tandem with separations in the vicinity of the drag inversion spacing.Physics of Fluids, 22(5), 2010. C. C. Chicone.Ordinary differential equations with applications, volume 34. Springer, 2006. T.A. Davis. Algorithm 832: Umfpack v4. 3—an unsymmetr...

    work page 2010