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arxiv: 2604.10840 · v1 · submitted 2026-04-12 · ⚛️ physics.flu-dyn

Wake-Induced Drag and Phase-Reconstructed Dynamics of a Flexible Plate in Normal Flow

Maryam Boukor , Pedro Tall\'on Marr\'on , Richard Phat The Nguyen , J\'er\^ome V\'etel , \'Eric Laurendeau , Fr\'ed\'erick P. Gosselin This is my paper

Pith reviewed 2026-05-10 15:03 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords flexible platewake dynamicsvortex sheddingfluid-structure interactiondrag forceparticle image velocimetryoscillation symmetryS-2S mode
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The pith

Structural oscillation symmetry dictates wake topology and drag on a flexible plate.

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

The study focuses on a flexible plate in normal airflow to understand how its vibrations influence the surrounding flow. It finds that the symmetry of these oscillations controls the form of vortex shedding behind the plate. Symmetric vibrations lead to two parallel rows of same-sign vortices, while antisymmetric vibrations create alternating vortex pairs. This difference in wake structure directly affects the average drag force experienced by the plate. Such insights help model the behavior of flexible natural elements like tree branches or aquatic plants under wind or water flow.

Core claim

Structural oscillation symmetry directly dictates the wake topology. The symmetric vibration regime is characterised by two parallel 2S-type vortex shedding patterns on either side of the plate—herein termed the S-2S mode—whereas the antisymmetric regime exhibits a classic 2P-type shedding pattern. Furthermore, an impulse-based force analysis links these wake circulations to drag, revealing an additional mean drag penalty in the antisymmetric regime.

What carries the argument

The S-2S mode of two parallel 2S-type vortex shedding patterns, which ties the symmetry of plate vibrations to specific wake structures and enables drag quantification via impulse-based analysis.

If this is right

  • Wake topology follows directly from whether vibrations are symmetric or antisymmetric.
  • Mean drag increases in the antisymmetric regime due to the 2P shedding pattern.
  • Impulse analysis of wake circulations provides a route to compute the additional drag penalty.
  • Coherent wake structures can be extracted even from non-time-resolved data in fluid-structure problems.

Where Pith is reading between the lines

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

  • The symmetry-to-wake link may apply to other flexible bodies such as flags or membranes in flow.
  • Controlling vibration symmetry could offer a way to reduce drag on reconfigurable structures.

Load-bearing premise

Non-time-resolved particle image velocimetry data combined with proper orthogonal decomposition, robust principal component analysis, and singular value decomposition can accurately reconstruct the periodic coherent vortex structures and their phases without significant artifacts.

What would settle it

Time-resolved particle image velocimetry measurements that either confirm or refute the reconstructed S-2S vortex patterns in the symmetric regime and 2P patterns in the antisymmetric regime.

Figures

Figures reproduced from arXiv: 2604.10840 by \'Eric Laurendeau, Fr\'ed\'erick P. Gosselin, J\'er\^ome V\'etel, Maryam Boukor, Pedro Tall\'on Marr\'on, Richard Phat The Nguyen.

Figure 1
Figure 1. Figure 1: Schematic of the experimental PIV setup. The specimen is positioned in the wind tunnel test section and illuminated by a laser sheet defining the measurement plane. Particle images recorded by the camera are processed using cross-correlation to obtain velocity fields. Flow visualization was performed using a PIV system (nanopiv, LaVision) equipped with an Nd:YAG pulsed laser (Model NANO L 200-15, Litron La… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the modal decomposition methodology. The procedure involves RPCA for separating coherent dynamics from noise, SVD for extracting dominant modes, and POD with snapshot sorting to organize snapshots in phase. The reconstruction step yields mode shapes, temporal coefficients, and phase-plane portraits that characterize the flow dynamics. 2.2. Flow reconstruction The methodology adopted to postpro… view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of plate deformation and mean wake organization across flow regimes. (a) Mean 𝑦-position of the tip with respect to the symmetry plane 𝑦𝑡 𝑖 𝑝. The lower and upper limits of the tip motion are defined by the 10th and 90th percentiles. (b-g) Deformation of the flexible plate green plastic 8-4 (𝐿 = 8 cm, 𝑤 =4 cm) and the corresponding mean streamwise velocity field in its wake, for different regimes… view at source ↗
Figure 4
Figure 4. Figure 4: Deformed shape of a reconfigured plate and its recirculation zone in the static regime with: a) dimensions normalized by the rigid plate length 𝐿, b) dimensions stretched by 𝜉 = 𝐶 1/3 𝑦 . The recirculation zone is delimited by the time-averaged 𝑢 = 0 contour line. reconfigured plate length. When applying the stretching factor in [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The first four POD modes of the streamwise velocity 𝑢, denoted as 𝝍𝒖, of the green plastic 8-4. a) The first line represents the static reconfiguration regime at 𝑈 =8.2 m s−1 . b) The second line represents the symmetrical vibration at 𝑈 = 11.2 m s−1 . c) The third line represents the transition from the symmetrical vibration to the antisymmetrical vibration at 𝑈 = 12.7 m s−1 . d) The fourth line represent… view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of modal dynamics and energy distribution across flow regimes. Top: Phase portraits constructed from the temporal coefficients (modal amplitudes) of the first four POD modes of the streamwise velocity field. Each subplot shows the projection in the (𝛷𝐼 ,𝛷𝐼 𝐼) or (𝛷𝐼 𝐼 𝐼,𝛷𝐼𝑉 ) plane for four flow regimes: (a) static reconfiguration (𝑈 = 8.2 m s−1 ), (b) symmetric vibration (𝑈 = 11.2 m s−1 ), (c) t… view at source ↗
Figure 7
Figure 7. Figure 7: Spanwise vorticity fields 𝜔𝑧 at five successive time instants (𝑡0, 𝑡0 + 𝑇/4, 𝑡0 + 𝑇/2, 𝑡0 + 3𝑇/4, 𝑡0 + 𝑇) for three flow regimes: a) static reconfiguration; b) symmetric vibration; and c) anti-symmetric vibration. Here, 𝑡0 marks the initial instant chosen for visualization, and 𝑇 denotes the oscillation period. the oscillating plate and the incoming flow leads to the periodic shedding of counter￾rotating v… view at source ↗
Figure 8
Figure 8. Figure 8: Schematics of the vortex shedding patterns found for: a) symmetrical vibration regime; and b) anti￾symmetrical vibration regime. P: vortex pair, S: Single vortex. observed in oscillating cylinders in the cross-stream direction, where two single vortices of opposite sign are shed alternately from each side of the body during each oscillation cycle (Williamson & Roshko 1988). In the present configuration, th… view at source ↗
Figure 9
Figure 9. Figure 9: Mean reconfiguration number 𝑅 versus Cauchy number 𝐶𝑦𝐶𝐷, with and without impulse-based correction. ◦: force balance measurements; □: values corrected by subtracting the wake mean drag estimated from the impulse theory. The black curve ( ) represents the model of Gosselin et al. (2010). I: symmetric vibration, II: antisymmetric vibration. 4. Conclusion In this study, we investigated the wake dynamics of fl… view at source ↗
Figure 10
Figure 10. Figure 10: Wake breathing during the symmetric shedding regime.Left: Streamwise evolution of the normalized wake width 𝑊wake/𝐿 for snapshots around the half-cycle of oscillation. The wake width is defined as the vertical extent of the velocity-deficit region identified by the contour 𝑢/𝑈 = 0.8 (black contour). Right: Instantaneous streamwise velocity field at the phase 𝑗 = 𝑇/2, where the wake contraction is most pro… view at source ↗
Figure 11
Figure 11. Figure 11: Correlation analysis of POD temporal coefficients using the Randomized Dependence Coefficient (upper triangle) and phase portraits (lower triangle). Symmetric modes are shown in blue, antisymmetric modes in orange, and the corresponding harmonics are indicated by dashed outlines. These results correspond to the symmetrical regime at 𝑈 = 11.2𝑚/𝑠 0 A0-23 [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Correlation analysis of POD temporal coefficients using the Randomized Dependence Coefficient (upper triangle) and phase portraits (lower triangle). Symmetric modes are shown in blue, antisymmetric modes in orange, and the corresponding harmonics are indicated by dashed outlines. These results correspond to the transition regime at 𝑈 = 12.7𝑚/𝑠 0 A0-24 [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Correlation analysis of POD temporal coefficients using the Randomized Dependence Coefficient (upper triangle) and phase portraits (lower triangle). Symmetric modes are shown in blue, antisymmetric modes in orange, and the corresponding harmonics are indicated by dashed outlines. These results correspond to the antisymmetrical regime at 𝑈 = 14.2𝑚/𝑠 0 A0-25 [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
read the original abstract

Flexible structures in an incoming perpendicular flow typically undergo elastic reconfiguration that reduces drag; however, at higher velocities, they are prone to dynamical instabilities that entail complex wake dynamics and fluctuating loads. In this study, we investigate the wake of a thin, flexible plate clamped at its midpoint and oriented normal to an airflow, modelling reconfigurable natural systems such as trees and sea-grasses. By combining Proper Orthogonal Decomposition, Robust Principal Component Analysis, and Singular Value Decomposition with non-time-resolved Particle Image Velocimetry, we reconstruct the periodic coherent flow structures across both static and vibrating regimes. We demonstrate that structural oscillation symmetry directly dictates the wake topology. The symmetric vibration regime is characterised by two parallel 2S-type vortex shedding patterns on either side of the plate-herein termed the S-2S mode-whereas the antisymmetric regime exhibits a classic 2P-type shedding pattern. Furthermore, an impulse-based force analysis links these wake circulations to drag, revealing an additional mean drag penalty in the antisymmetric regime. Our approach offers a practical framework to extract and interpret coherent wake structures from limited temporal data, enhancing our understanding of fluid-structure interactions and informing aerodynamic load predictions.

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 / 1 minor

Summary. The manuscript presents an experimental study of the wake behind a flexible plate in normal airflow, using non-time-resolved PIV combined with POD, RPCA, and SVD to reconstruct periodic coherent structures. The central claim is that the symmetry of the plate's structural oscillations determines the wake topology: symmetric vibrations result in an S-2S mode consisting of two parallel 2S-type vortex shedding patterns, while antisymmetric vibrations produce a 2P-type shedding pattern. An impulse-based analysis of the wake circulations is used to link these topologies to drag, showing an additional mean drag penalty associated with the antisymmetric regime.

Significance. If the reconstruction method accurately captures the phase and topology without artifacts, this work provides a useful framework for analyzing fluid-structure interactions in flexible systems from limited temporal data. It highlights how oscillation symmetry influences wake dynamics and drag, which could have implications for understanding and predicting loads on natural and engineered reconfigurable structures. The approach strengthens the connection between structural dynamics and wake topology in experimental settings where full time resolution is challenging.

major comments (2)
  1. The distinction between S-2S and 2P modes, which underpins the claim that structural oscillation symmetry dictates wake topology, depends on the fidelity of the POD-RPCA-SVD reconstruction from non-time-resolved PIV. The paper does not include validation tests (e.g., against synthetic data or time-resolved benchmarks) or quantitative assessment of phase errors, making it unclear whether the reported topologies are physical or reconstruction artifacts.
  2. The impulse-based force analysis linking wake circulations to an additional mean drag penalty in the antisymmetric regime lacks reported quantitative values, error estimates, or direct comparison to measured forces. Without these, the magnitude of the drag difference and its attribution to the 2P vs S-2S topologies cannot be rigorously evaluated.
minor comments (1)
  1. The abstract mentions 'an additional mean drag penalty' but provides no numerical value or relative magnitude, which would help contextualize the finding.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment below and have revised the manuscript accordingly to improve the validation and quantitative rigor of our analyses.

read point-by-point responses

  1. Referee: The distinction between S-2S and 2P modes, which underpins the claim that structural oscillation symmetry dictates wake topology, depends on the fidelity of the POD-RPCA-SVD reconstruction from non-time-resolved PIV. The paper does not include validation tests (e.g., against synthetic data or time-resolved benchmarks) or quantitative assessment of phase errors, making it unclear whether the reported topologies are physical or reconstruction artifacts.

    Authors: We agree that explicit validation of the reconstruction is important for confirming that the reported wake topologies are physical. The POD-RPCA-SVD method follows established procedures for phase reconstruction from non-time-resolved data, and the topologies align with expected symmetry-based patterns. However, to address this concern directly, the revised manuscript includes new validation tests on synthetic vortex fields that mimic the observed shedding modes. These tests provide quantitative phase error estimates and reconstruction fidelity metrics, confirming that the S-2S and 2P patterns are reliably recovered without introducing artifacts. revision: yes

  2. Referee: The impulse-based force analysis linking wake circulations to an additional mean drag penalty in the antisymmetric regime lacks reported quantitative values, error estimates, or direct comparison to measured forces. Without these, the magnitude of the drag difference and its attribution to the 2P vs S-2S topologies cannot be rigorously evaluated.

    Authors: We acknowledge that the original presentation of the impulse analysis would benefit from more explicit quantification. The revised manuscript now reports the specific mean drag increments derived from the wake circulations for each regime, together with error estimates obtained from cycle-to-cycle variability in the reconstructed fields. Direct force measurements were not acquired in the present experiments, so a side-by-side comparison with load-cell data is not possible; instead, the inferred drag differences are placed in context with existing literature on similar bluff-body wakes to support the attribution to the distinct vortex topologies. revision: partial

Circularity Check

0 steps flagged

No significant circularity in experimental wake reconstruction

full rationale

The paper reports an observational experimental study that processes non-time-resolved PIV snapshots via POD, RPCA and SVD to classify observed wake topologies (S-2S for symmetric plate motion, 2P for antisymmetric) and links them to drag via impulse analysis. No derivation chain exists in which a claimed prediction or first-principles result reduces by the paper's own equations to quantities fitted from the same dataset. The topologies are presented as direct outputs of the decomposition applied to measured data, not as self-definitional or fitted-input predictions. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work are invoked to force the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no free parameters, invented entities, or non-standard axioms are stated. Relies on established incompressible flow and vortex dynamics assumptions common to PIV-based fluid mechanics studies.

axioms (1)
  • standard math Incompressible flow and vortex dynamics govern the wake structures observed via PIV
    Implicit foundation for all vortex shedding analysis and impulse-based force calculations in the abstract.

pith-pipeline@v0.9.0 · 5545 in / 1285 out tokens · 50521 ms · 2026-05-10T15:03:32.458392+00:00 · methodology

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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    Alben, Silas, Shelley, Michael & Zhang, Jun2002 Drag reduction through self-similar bending of a flexible body.Nature420(6915), 479–481. Baskaran, Mrudhula, Hutin, Louis & Mulleners, Karen2023 Reconfiguring it out: How flexible structures interact with fluid flows.Physical Review Fluids8(11), 110509. Birch, James M & Dickinson, Michael H2003 The influence...

    work page arXiv 2013

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    Graham, William Richard, Ford, CW Pitt & Babinsky, Holger2017 An impulse-based approach to estimating forces in unsteady flow.Journal of Fluid Mechanics815, 60–76

    Gosselin, Fr ´ed´erick, De Langre, Emmanuel & Machado-Almeida, Bruno A2010 Drag reduction of flexible plates by reconfiguration.Journal of Fluid Mechanics650, 319–341. Graham, William Richard, Ford, CW Pitt & Babinsky, Holger2017 An impulse-based approach to estimating forces in unsteady flow.Journal of Fluid Mechanics815, 60–76. Heathcote, Sam, Wang, Z &...

    work page 2003

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    Roshko, Anatol1955 On the wake and drag of bluff bodies.Journal of the aeronautical sciences22(2), 124–132. Scherl, Isabel, Strom, Benjamin, Shang, Jessica K, Williams, Owen, Polagye, Brian L & Brunton, Steven L2020 Robust principal component analysis for modal decomposition of corrupt fluid flows.Physical Review Fluids5(5), 054401. Schnipper, Teis, Ander...

    work page 2017