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arxiv: 2604.24342 · v1 · submitted 2026-04-27 · ⚛️ physics.flu-dyn

Numerical Investigation of Elastically-Mounted tandem Cylinders using an ALE Runge-Kutta Discontinuous Galerkin method

Alexios Papadimitriou , Spyridon Zafeiris , George Papadakis This is my paper

Pith reviewed 2026-05-08 01:35 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords vortex-induced vibrationsdiscontinuous Galerkinarbitrary Lagrangian-Euleriantandem cylindersfluid-structure interactionhigh-order methodswake dynamicshp-refinement
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The pith

High-order ALE discontinuous Galerkin method efficiently captures multi-body wake dynamics in tandem cylinder vortex-induced vibrations by favoring polynomial over mesh refinement.

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

The paper develops a high-order Arbitrary-Lagrangian-Eulerian Runge-Kutta discontinuous Galerkin framework for fluid-structure interaction problems involving elastically mounted cylinders. The method incorporates discrete geometric conservation law enforcement and radial basis function mesh deformation to accommodate large structural motions while preserving free-stream accuracy. It is tested on two-cylinder cross-flow oscillations at Reynolds number 200 and a three-cylinder arrangement with two degrees of freedom at Reynolds number 150, reproducing benchmark oscillation patterns and uncovering irregular trajectories driven by wake interference. A central demonstration shows that elevating the polynomial order preserves vortical structures over long distances more effectively and at lower cost than mesh refinement, due to the method's low numerical diffusion on relatively coarse grids. This matters because wake-driven structural responses in multi-body configurations require accurate long-range transport of flow features without excessive computational expense.

Core claim

The framework accurately models the coupled dynamics of tandem cylinders undergoing vortex-induced vibrations, with the two-cylinder case reproducing established Lissajous curves, Poincaré maps, power spectra, and vortex shedding modes at Re=200. In the three-cylinder case at Re=150 the trajectories display highly irregular behavior governed by a periodic attract-and-release mechanism acting on the trailing cylinder's streamwise motion. An hp-refinement study establishes that increasing polynomial order outperforms mesh refinement for these wake-dominated interactions because the low-diffusion property of the high-order discretization maintains coherent vortical structures across coarse mesh

What carries the argument

Arbitrary-Lagrangian-Eulerian Runge-Kutta discontinuous Galerkin solver with discrete geometric conservation law enforcement and radial basis function mesh deformation

If this is right

  • Two-cylinder cross-flow oscillations at Re=200 match literature benchmarks via Lissajous curves, Poincaré maps, and power spectra.
  • Three-cylinder trajectories exhibit irregular motion governed by a periodic attract-and-release process in the wake interference.
  • Polynomial-order elevation preserves distant vortical structures more efficiently than mesh refinement on coarse grids.
  • Vortex shedding modes are classified consistently with established benchmarks for both configurations.

Where Pith is reading between the lines

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

  • The demonstrated low-diffusion behavior suggests the method could scale to simulations involving more than three cylinders or longer wake convection distances.
  • Engineers modeling offshore riser arrays or heat-exchanger tube bundles may obtain reliable vibration predictions at reduced cost by adopting polynomial refinement over mesh refinement.
  • The framework's handling of two-degree-of-freedom motion opens the door to testing whether similar irregular attract-and-release patterns appear in other multi-body fluid-structure systems at comparable Reynolds numbers.
  • Because the method preserves wake coherence on coarse meshes, it may enable parametric studies over wider ranges of structural stiffness and damping than low-order approaches allow.

Load-bearing premise

Discrete geometric conservation law enforcement together with radial basis function mesh deformation maintains free-stream preservation and accuracy for large displacements and complex wake interactions without introducing unquantified numerical artifacts.

What would settle it

A systematic discrepancy between the computed trajectories, power spectra, or vortex shedding modes in the three-cylinder configuration and independent high-fidelity reference data obtained at the same Reynolds numbers.

Figures

Figures reproduced from arXiv: 2604.24342 by Alexios Papadimitriou, George Papadakis, Spyridon Zafeiris.

Figure 1
Figure 1. Figure 1: Schematic of an arbitrary deformation of an element view at source ↗
Figure 2
Figure 2. Figure 2: Free Stream Preservation problem and corresponding computational grid. On the left, the computational view at source ↗
Figure 3
Figure 3. Figure 3: Results of the Free Stream Preservation simulations. view at source ↗
Figure 4
Figure 4. Figure 4: The two cylinders in tandem arrangement with 1 DoF in the cross-flow direction. view at source ↗
Figure 5
Figure 5. Figure 5: The initial computational mesh (left) and the deformed mesh for large grid deformations for different values view at source ↗
Figure 6
Figure 6. Figure 6: Normalized Power Spectra on the time series of the reduced displacements view at source ↗
Figure 7
Figure 7. Figure 7: Vorticity Contours for U ∗ = 7 at ∼ 146s for polynomial order of p = 1, 3, 5. The top image is p = 1 the middle image is p = 3 and the bottom image is p = 5. irregular, quasi-periodic phase orbits. At higher reduced velocities (U ∗ = 7, 8), strong periodic behavior is reestab￾lished. The ALE-DG framework captures these highly non-linear phase dynamics in strict accordance with the liter￾ature. To gain insi… view at source ↗
Figure 8
Figure 8. Figure 8: Plots of the Lissajous curves for the two cylinder case. The view at source ↗
Figure 9
Figure 9. Figure 9: The three Modes of vortex shedding observed in the specific range of reduced velocities. Mode 1 is demon view at source ↗
Figure 10
Figure 10. Figure 10: Time Series and Power Spectra results for the two cylinder case. The orange line ( view at source ↗
Figure 11
Figure 11. Figure 11: Frequencies corresponding to the peak of the power spectra for the two-cylinder case, with the view at source ↗
Figure 12
Figure 12. Figure 12: The three cylinders in tandem arrangement with 2 DoF in both in-flow and cross-flow directions. view at source ↗
Figure 13
Figure 13. Figure 13: Computational grid for the three cylinder in tandem case. view at source ↗
Figure 14
Figure 14. Figure 14: Cylinder trajectories results for the three cylinder case, for view at source ↗
Figure 15
Figure 15. Figure 15: Cylinder Trajectories for the three cylinder case with different reduced velocities view at source ↗
Figure 16
Figure 16. Figure 16: Phase Portraits and Poincare Maps for the three cylinder case, with a reduced velocity of ´ U ∗ = 10. Whenever they are available the reference points from Yu et al. [46] are included plotted using the ⃝ marker view at source ↗
Figure 17
Figure 17. Figure 17: Poincare Phase Portraits and Maps for the three cylinder case, with a reduced velocity of ´ U ∗ = 7. Whenever they are available the reference points from Yu et al. [46] are included plotted using the ⃝ marker. A highly irregular case is the one of U ∗ = 7, as seen in view at source ↗
Figure 18
Figure 18. Figure 18: Vortex shedding for the U ∗ = 5, 7, 10 cases, with the top representing a reduced velocity of 5, the middle a reduced velocity of 7 and the bottom a reduced velocity of 10. Another interesting feature observed in the U ∗ = 9 case is the periodic ”attract-and-release” cycle characterizing the stream-wise oscillations. In view at source ↗
Figure 19
Figure 19. Figure 19: Time series for reduced displacement x ∗ for U ∗ = 9. The markers refer to the time snapshots for the vortex shedding structures from view at source ↗
Figure 20
Figure 20. Figure 20: Vortex shedding patterns for U ∗ = 9 for different time snapshots. By contrast, at t ≈ 462 s, the response reaches a peak (see view at source ↗
Figure 21
Figure 21. Figure 21: Maximum amplitudes computed as A. = (maxt(.) − mint(.))/2 (filled blue circles). They are compared with the maximum amplitudes from Yu et al. [46] (hollow black squares) view at source ↗
Figure 22
Figure 22. Figure 22: Dense first-order mesh for h-refinement for the three tandem cylinder case. For this comparison, we consider the U ∗ = 10 case. The resulting trajectories for the coarse grid p = 3 and the dense p = 1 cases are illustrated in view at source ↗
Figure 23
Figure 23. Figure 23: Comparison of cylinder trajectories for U ∗ = 10 between first-order and third-order simulations, alongside the reference data from the literature. This illustrates how the higher-order discretization better preserves the physical characteristics of the wake, particularly in regions of strong vortex interaction and pairing. In contrast, the near-body flow field remains qualitatively similar in both simula… view at source ↗
Figure 24
Figure 24. Figure 24: Vortex wake comparison between the two orders at view at source ↗
read the original abstract

This work presents a high-order Arbitrary-Lagrangian-Eulerian (ALE) Discontinuous Galerkin framework for simulating multi-body Vortex-Induced Vibrations. The ALE formulation extends a Runge-Kutta Interior-Penalty nodal DG solver with minimal additional computational overhead, incorporating discrete enforcement of the Geometric Conservation Law (GCL) to ensure free-stream preservation and Radial Basis Function (RBF) mesh deformation to handle large structural displacements. The framework is applied to elastically-mounted tandem cylinder configurations: a two-cylinder arrangement with cross-flow oscillations at Re=200, and a three-cylinder arrangement with two degrees of freedom at Re=150. In the three-cylinder case, the trajectories exhibit highly irregular behavior driven by complex wake interference, including a periodic attract-and-release mechanism governing the trailing cylinder's stream-wise response. Results are verified against established benchmarks through Lissajous curves, Poincar\'{e} phase maps, power spectra, and vortex shedding mode classification. An hp-refinement comparison demonstrates that increasing the polynomial order is more effective and computationally efficient than mesh refinement for capturing multi-body wake dynamics, as the low numerical diffusion of the high-order method preserves vortical structures over long distances on relatively coarse meshes. These findings highlight the importance of high-order methods for CFD-FSI applications where wake interactions drive the structural response.

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 / 0 minor

Summary. The manuscript presents a high-order Arbitrary-Lagrangian-Eulerian (ALE) Runge-Kutta Discontinuous Galerkin framework for multi-body vortex-induced vibration simulations. It extends a nodal DG solver with discrete Geometric Conservation Law enforcement and Radial Basis Function mesh deformation to handle large structural displacements. The method is applied to a two-cylinder tandem configuration with cross-flow oscillations at Re=200 and a three-cylinder configuration with two degrees of freedom at Re=150, where irregular trajectories arise from wake interference including an attract-and-release mechanism. Results are verified against benchmarks using Lissajous curves, Poincaré maps, power spectra, and vortex shedding mode classification. An hp-refinement study concludes that increasing polynomial order is more effective and computationally efficient than mesh refinement for preserving multi-body wake dynamics due to the low numerical diffusion of the high-order scheme.

Significance. If the ALE components prove robust, the work would provide a useful high-order tool for CFD-FSI problems involving complex wake-body interactions in multi-cylinder systems. The hp-refinement comparison offers practical guidance on method selection for preserving vortical structures over long distances, and the benchmark verifications via multiple diagnostic tools strengthen the results.

major comments (1)
  1. [ALE formulation and verification] The assertion that discrete GCL enforcement ensures free-stream preservation with minimal overhead (stated in the ALE formulation description) lacks a targeted verification test using the exact RBF deformation sequence extracted from the three-cylinder 2DOF trajectories at Re=150. This is load-bearing for the central hp-refinement claim, as any degradation in free-stream preservation or introduction of artifacts under large irregular displacements could affect the observed wake preservation and the superiority of p-refinement over h-refinement on coarse meshes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comment. We address the point below and agree that the suggested verification will strengthen the presentation of the ALE framework.

read point-by-point responses

  1. Referee: The assertion that discrete GCL enforcement ensures free-stream preservation with minimal overhead (stated in the ALE formulation description) lacks a targeted verification test using the exact RBF deformation sequence extracted from the three-cylinder 2DOF trajectories at Re=150. This is load-bearing for the central hp-refinement claim, as any degradation in free-stream preservation or introduction of artifacts under large irregular displacements could affect the observed wake preservation and the superiority of p-refinement over h-refinement on coarse meshes.

    Authors: We agree that a targeted test using the precise RBF deformation sequence extracted from the three-cylinder 2DOF trajectories at Re=150 would provide direct evidence of free-stream preservation under the irregular, large-amplitude motions encountered in the simulations. While the manuscript already contains general free-stream preservation tests for the ALE-RKDG scheme with RBF deformation and discrete GCL enforcement, these do not replicate the exact sequence of mesh velocities from the 2DOF case. In the revised manuscript we will add a dedicated verification subsection (or appendix) that extracts the RBF deformation sequence from the Re=150 trajectories, applies it to a uniform free-stream flow, and demonstrates that the discrete GCL maintains preservation to machine accuracy with negligible additional cost. This addition will directly support the hp-refinement results by confirming that the observed wake preservation on coarse meshes is not an artifact of mesh-motion errors. revision: yes

Circularity Check

0 steps flagged

No circularity: framework extension and hp-comparison rest on external benchmarks and direct simulation results.

full rationale

The paper extends an existing Runge-Kutta IP-DG solver with ALE, discrete GCL enforcement, and RBF deformation, then applies the method to two FSI test cases. All central claims (free-stream preservation, wake preservation on coarse high-order meshes, superiority of p- over h-refinement) are supported by comparisons to published benchmarks (Lissajous curves, Poincaré maps, power spectra) and by side-by-side hp-refinement runs on the same problem. No equation or result is shown to equal its own input by construction, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness theorem is imported from self-citation. The derivation chain is therefore self-contained against external data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard DG stability and approximation theory plus the assumption that RBF interpolation preserves sufficient mesh quality for the reported displacements.

axioms (2)
  • standard math Standard interior-penalty DG stability and approximation properties hold under ALE mapping
    Invoked when extending the RK-IP nodal DG solver to moving domains
  • domain assumption RBF mesh deformation maintains positive Jacobian and sufficient quality for the reported cylinder motions
    Required for large structural displacements without re-meshing

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