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arxiv: 2606.27921 · v1 · pith:WKUS2NMCnew · submitted 2026-06-26 · ⚛️ physics.flu-dyn

Effect of an aligned current on the stability of oscillatory incompressible flow past a circular cylinder

Geng Chen , Lian Gan , Philip H. Gaskell This is my paper

Pith reviewed 2026-06-29 03:03 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords cylinder wakeoscillatory flowsteady currentFloquet stabilityperiod-doubling bifurcationKeulegan-Carpenter numbervelocity ratiowake re-stabilization
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The pith

A steady current with m > 0.5 introduces period-doubling bifurcations in oscillatory flow past a cylinder and produces a re-stabilization region at high m.

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

The paper maps the linear stability of combined steady and oscillatory flow past a circular cylinder using two-dimensional Floquet analysis across ranges of Keulegan-Carpenter number, steady-to-oscillatory velocity ratio m, and oscillatory Reynolds number. It shows that m above 0.5 activates a subharmonic period-doubling mode that is absent when the flow is purely oscillatory. At Re_m = 100 the neutral curve in (KC, m) space becomes strongly non-monotonic, with a distinct interval beyond m ≈ 0.9 in which the wake regains Z2 symmetry even though the peak Reynolds number reaches 190. Direct numerical simulations confirm that the linear predictions match the saturated nonlinear states when only one mode is unstable.

Core claim

Within the two-dimensional Floquet framework a steady current component with m > 0.5 produces a period-doubling subharmonic bifurcation that does not occur for m = 0. For Re_m = 100 the neutral stability boundary in (KC, m) space is strongly non-monotonic, separating regions of intrinsic stability from single-mode instability and containing a re-stabilization band beyond m ≈ 0.9 where the flow recovers a Z2-symmetric state at peak Reynolds number ≈ 190 despite both forcing components being individually supercritical; a separate regime permits coexistence of two distinct unstable modes.

What carries the argument

Two-dimensional Floquet stability analysis that tracks the loci of leading multipliers in (KC, m) parameter space to identify synchronous, quasi-periodic, and subharmonic bifurcation modes.

If this is right

  • The neutral stability curve in (KC, m) space becomes strongly non-monotonic once m exceeds 0.5.
  • A sub-region of mode re-stabilization appears for m greater than approximately 0.9, recovering Z2 symmetry at peak Reynolds number near 190.
  • A distinct parameter regime permits simultaneous instability of two modes of different type.
  • Direct numerical simulations confirm that the linear Floquet predictions correctly forecast the saturated nonlinear state when only one mode is unstable.

Where Pith is reading between the lines

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

  • The reported re-stabilization may be sensitive to spanwise perturbations that commonly appear in cylinder wakes at these Reynolds numbers.
  • The non-monotonic stability boundary could be used to select operating conditions that suppress vortex-induced forces in combined wave-current environments.
  • Repeating the Floquet scan at lower Re_m would test whether the period-doubling threshold remains near m = 0.5 when the flow is closer to the onset of instability.

Load-bearing premise

The entire stability diagram is obtained under the assumption that the flow remains strictly two-dimensional.

What would settle it

A three-dimensional simulation or experiment that shows the re-stabilization band disappearing or shifting to a different m value at the same peak Reynolds number would falsify the reported two-dimensional bifurcation loci.

Figures

Figures reproduced from arXiv: 2606.27921 by Geng Chen, Lian Gan, Philip H. Gaskell.

Figure 1
Figure 1. Figure 1: Purely oscillatory flow. (a) Stability map in (𝛽, 𝐾𝐶) space, indicating the onset of synchronous (𝑆, solid line) and quasi-periodic (𝑄𝑃, dashed line) instability. The blue dot-dashed line signifies Re𝑚 = 100. (b) Possible Floquet multiplier 𝜇 loci in the complex plane with increasing Re𝑚. Arrows depict the two characteristic pathways identified in Elston et al. (2006): S, where a complex conjugate pair exi… view at source ↗
Figure 2
Figure 2. Figure 2: (𝑎) Schematic of the 2D circular computational solution domain, of total radius 𝑅𝑑 = 70𝐷 (not to scale), with the time-dependent free-stream velocity 𝑢∞ (𝑡) imposed at the outer boundary. (𝑏) Domain discretisation showing the structured quadrilateral mesh employed, together with an insert showing an exploded￾view in the vicinity of the cylinder surface and revealing also the high-order internal nodal distr… view at source ↗
Figure 3
Figure 3. Figure 3: Leading Floquet multipliers (𝜇0, 𝜇1) in the complex plane for the three representative unstable Floquet mode types evaluated when 𝑅𝑒𝑚 = 100: (𝑎) synchronous; (𝑏) quasi-periodic; (𝑐) subharmonic, or period-doubling. At a Re𝑚 higher than the critical value, more than one multipliers may lie outside the unit circle in the upper half-plane, and with 4 possible combinations of the three mode types listed above:… view at source ↗
Figure 4
Figure 4. Figure 4: Floquet stability map in the (𝐾𝐶, 𝑚) space at Re𝑚 = 100. The right vertical axis is Re𝑝 = Re𝑚 (1+𝑚). Marker colours indicate the stability state of the flow: stable (blue), synchronous (𝑆, red), quasi-periodic (𝑄𝑃, green), and subharmonic (𝑆𝐻, purple). Split markers denote two Floquet multipliers located outside the unit circle, with left and right halves representing the leading and sub-leading multiplier… view at source ↗
Figure 5
Figure 5. Figure 5: The three possible types of modal coexistence at 𝐾𝐶 = 11, which demonstrate the rich dependence on 𝑚 of the instability mode selection. (𝑎) 𝑚 = 0.1: 𝑆 + 𝑆; (𝑏) 𝑚 = 0.2: 𝑆 + 𝑄𝑃; (𝑐) 𝑚 = 0.6: 𝑆𝐻 + 𝑆𝐻 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Vorticity field of stable cases at Re𝑚 = 100 and 𝜑 = 1.60𝜋. Low-𝑚 cases (4, 0.2), (5, 0.3), (6, 0.4) and (7, 0.4) show strong cycle-to-cycle vortex interaction with the cylinder; the higher-𝑚 case (7, 0.7) show the shed structures carried downstream and away from the cylinder, with small cycle-to-cycle interaction. The corresponding free-stream velocity 𝑈 ( ) and drag coefficient 𝐶𝑑 ( ) are inset. A striki… view at source ↗
Figure 7
Figure 7. Figure 7: Loci of the leading Floquet multiplier 𝜇0 for the given (𝐾𝐶, 𝑚) condition as Re𝑚 increases from subcritical values to Re𝑚 = 100, with arrows indicating the direction of increasing Re𝑚. Values of Re𝑚 are annotated alongside the loci, although not all values examined are labelled for the purposes of clarity. The dashed curve marks the boundary of the unit circle. Marker colouring follows the instability clas… view at source ↗
Figure 8
Figure 8. Figure 8: Phase-resolved evolution of the leading Floquet mode over one fundamental oscillation period 𝑇 for the case (𝐾𝐶, 𝑚) = (9, 0.3) evaluated when Re𝑚 = 100. Contours show the real part of the vorticity mode, R𝑒[𝜔ˆ 𝑧 ]. The solid and dashed lined boxes indicate the vortices generated from two consecutive cycles, respectively. envelope of 𝐶𝑙 grows exponentially before saturating into strictly 𝑇-periodic cycles. … view at source ↗
Figure 9
Figure 9. Figure 9: 𝐶𝑙 of the case (𝐾𝐶, 𝑚) = (9, 0.3) evaluated when Re𝑚 = 100 by DNS. (𝑎) Time history of 𝐶𝑙 , fitted with an exponential function for its envelope under transient growth. (𝑏) Frequency spectrum of 𝐶𝑙 computed after its oscillation saturates, showing dominant peaks at 𝑓0 and its harmonics [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Fully developed 𝜔𝑧 distribution for the case (𝐾𝐶, 𝑚) = (9, 0.3) evaluated when Re𝑚 = 100 by DNS, at 𝜑 = 0, 𝜋 and 2𝜋. Colour-map key as in figure 6. The corresponding lift coefficient 𝐶𝑙 ( ) and drag coefficient 𝐶𝑑 ( ) are inset. This net deflection of the vortex structure within a cycle is associated with a negative time-averaged lift 𝐶𝑙 , as confirmed by the inset 𝐶𝑙 panels. The direction of wake deflect… view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of R𝑒[𝜔ˆ 𝑧 ] (top row) and I𝑚[𝜔ˆ 𝑧 ] (bottom row) of the leading Floquet mode for (𝐾𝐶, 𝑚) = (7, 0.5) when Re𝑚 = 100, shown over six forcing periods. The two rows share the same spatial structure but are shifted in phase for 2𝑇. Colour-map key as in figure 8. 0 10 20 30 40 t/T −6 0 6 Cl ×10−1 (a) 0 1 2 3 4 f / f0 0.00 0.25 C A l 0.264 0.737 1.266 (b) [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: 𝐶𝑙 of the case (𝐾𝐶, 𝑚) = (7, 0.5) evaluated when Re𝑚 = 100 by DNS. (a) 𝐶𝑙 development in time￾domain. (b) Frequency spectrum of 𝐶𝑙 computed after its oscillation saturates, showing peaks at 𝑛 ± 𝛥 𝑓 / 𝑓0, where the incommensurate secondary frequency 𝑓 DNS 𝑠 = 𝛥 𝑓 ≈ 0.264 𝑓0. DNS simulation at Re𝑚 = 100 confirms flow quasi-periodicity, supported by the time domain 𝐶𝑙(𝑡) for 𝑡 > 30𝑇, figure 12 (a), and the s… view at source ↗
Figure 13
Figure 13. Figure 13: Phase-evolution of R𝑒[𝜔ˆ 𝑧 ] for case (𝐾𝐶, 𝑚) = (10, 0.7) evaluated at Re𝑚 = 100 over one oscillation cycle. The identical structure, differing only by a sign inversion and intensity between 𝑡 and 𝑡 + 𝑇, is a characteristic feature of period-doubling. Colour-map key as in figure 8. at 𝑚 = 0, can be unlocked when the steady component 𝑈𝑐 becomes sufficiently large, i.e. for 𝑚 ≳ 0.6. The two representative c… view at source ↗
Figure 14
Figure 14. Figure 14: 𝐶𝑙 of the case (𝐾𝐶, 𝑚) = (10, 0.7) evaluated when Re𝑚 = 100 by DNS. (a) 𝐶𝑙 development in time-domain, showing alternating speaks repeating in 2𝑇. (b) Frequency spectrum of 𝐶𝑙 computed after its oscillation saturates, with the dominant peak at the period-doubled frequency 𝑓 / 𝑓0 = 0.5 [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Fully developed 𝜔𝑧 contours for (𝐾𝐶, 𝑚) = (10, 0.7) evaluated at Re𝑚 = 100 by DNS, showing perfect period-2 phase-locking over 2𝑇 intervals. Colour-map key as is figure 6. The corresponding lift coefficient 𝐶𝑙 ( ) and drag coefficient 𝐶𝑑 ( ) are inset. to exp(𝜎𝑇) ≈ 1.92, in agreement with the prediction from the linear analysis. That is, 𝜇0 = −1.92 at Re𝑚 = 100. The nonlinearly saturated wake shown in fig… view at source ↗
Figure 16
Figure 16. Figure 16: Loci of the leading Floquet multiplier 𝜇0 for representative re-stabilisation cases. (𝑎): pathway 𝑄𝑃 → re-stabilisation, (𝐾𝐶, 𝑚) = (11, 0.9); (𝑏): 𝑆 → 𝑄𝑃 → re-stabilisation, (12, 0.9); (𝑐): 𝑆𝐻 → 𝑄𝑃 → re-stabilisation, (12, 0.9). Arrows indicating the direction of increasing Re𝑚 to 100. Marker colouring follows the instability classifications in figure 4 and the dashed curve marks the boundary of the unit … view at source ↗
Figure 17
Figure 17. Figure 17: Cycle variation of fully developed 𝜔𝑧 contour for (𝐾𝐶, 𝑚) = (12, 0.9), Re𝑚 = 100. The flow field remains symmetric with respect to the wake centreline. Colour-map key as in figure 6. the previous cycle transversely, with only a negligible displacement in the −𝑥 direction. Inter-cycle vortex interaction is therefore greatly reduced, which facilitates the formation of distinct vortex pairs and well-organise… view at source ↗
Figure 18
Figure 18. Figure 18: Loci of the leading Floquet multipliers in the complex plane for three cases as Re𝑚 increases at given (𝐾𝐶, 𝑚) condition. Labels give Re𝑚 values; arrows show the direction of increasing Re𝑚. (𝑎) (12, 0.3): mode coexisting 𝑆 + 𝑄𝑃 (the complex-conjugate locus is not shown). (𝑏) (11, 0.1): mode coexisting 𝑆 + 𝑆. (𝑐) (10, 0.1): a special transitional case 𝑄𝑃 at Re𝑚 = 100. The range 75 ≤ Re𝑚 ≤ 94 is shifted sl… view at source ↗
Figure 19
Figure 19. Figure 19: DNS time history of the lift coefficient 𝐶𝑙 and its frequency spectrum at Re𝑚 = 100. (a,b) (𝐾𝐶, 𝑚) = (12, 0.3), with quasi-periodic peaks at 𝑓 / 𝑓0 = 𝑛 ± 0.243; (c,d) (𝐾𝐶, 𝑚) = (11, 0.1), with peaks at 𝑓 / 𝑓0 = 𝑛 ± 0.042, 𝑛 ∈ N. modes of different multiplier magnitudes, also be associated with the lower instantaneous Reynolds number attained within each cycle according to equation (2.1) permitting a 𝑄𝑃 mo… view at source ↗
Figure 20
Figure 20. Figure 20: Pathway patterns of the leading Floquet multiplier as Re𝑚 increases from a subcritical value to Re𝑚 = 100 for a given (𝐾𝐶, 𝑚) condition. (𝑎): No transition, (𝑏): Unstable transition, and (𝑐) Re-stabilising transition. Coloured arrows indicate the classifications of the Floquet mode: synchronous (𝑆, red), quasi-periodic (𝑄𝑃, green), subharmonic (𝑆𝐻, purple), and stable (blue). QP characteristics (figure no… view at source ↗
Figure 21
Figure 21. Figure 21: Loci of the leading Floquet multipliers in the complex plane for three cases as Re𝑚 increases at given (𝐾𝐶, 𝑚) condition. Labels give Re𝑚 values; arrows show the direction of increasing Re𝑚. (𝑎):(12, 0.8), (𝑏): (8, 0.7), (𝑐): (12, 0.4). In (b) two ranges of Re𝑚 are shifted away from the negative real axis for clarity. Marker colouring follows the instability classifications in figure 4 and the dashed curv… view at source ↗
Figure 22
Figure 22. Figure 22: Estimated dependence of critical Reynolds number 𝑅𝑒𝑐 𝑚 on (𝐾𝐶, 𝑚). The field is interpolated onto a fine mesh using the modified Akima piecewise cubic Hermite scheme (Akima 1970). Filled contours and thin black lines show 𝑅𝑒𝑐 𝑚 value. The cases in the stable region on the left of the heavy dark-red neutral stability curve do not bifurcate for 𝑅𝑒𝑚 ≤ 100. Dotted white lines crudely delineate the bifurcation… view at source ↗
read the original abstract

The stability of incompressible flow past a circular cylinder under collinear steady and oscillatory forcing is investigated within a two-dimensional Floquet framework. The flow is parameterised by the Keulegan-Carpenter number $KC \in [4,12]$, the steady-to-oscillatory velocity ratio $m \in [0,1]$, and the oscillatory Reynolds number $Re_m \in [20,100]$. The loci of the leading Floquet multipliers, and hence case-specific bifurcation modes, are examined by progressively reducing $Re_m$ to subcritical values for prescribed $m$. A steady current with $m > 0.5$ gives rise to a period-doubling subharmonic bifurcation that does not occur in purely oscillatory flow, where only synchronous and quasi-periodic modes arise. For $Re_m = 100$, three key features are discernible. First, the neutral stability curve in $(KC,m)$ space is strongly non-monotonic in $m$, separating intrinsically stable regions from those with single unstable modes; a sub-region of striking mode re-stabilisation appears beyond $m \approx 0.9$, where the flow recovers a $Z_2$-symmetric state at peak Reynolds number $\approx 190$, despite the steady and oscillatory components each being individually unstable. Second, a distinct regime supports the coexistence of two unstable modes of different types. Third, complementary direct numerical simulations show that, for a single unstable mode, the linear analysis successfully predicts the saturated nonlinear state even when $Re_m = 100$ substantially exceeds the critical Reynolds number, whereas under mode coexistence the quasi-periodic attractor tends to dominate the developed dynamics.

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 paper examines the two-dimensional stability of incompressible flow past a circular cylinder under collinear steady and oscillatory forcing, parameterized by KC ∈ [4,12], m ∈ [0,1], and Re_m ∈ [20,100]. Using Floquet analysis, it identifies a period-doubling subharmonic bifurcation for m > 0.5 (absent in purely oscillatory flow), a strongly non-monotonic neutral curve in (KC,m) space at Re_m=100 with a re-stabilization region beyond m≈0.9 that recovers Z2 symmetry at peak Re≈190 despite individual component instability, a regime of coexisting unstable modes, and consistency between linear predictions and 2D DNS for single-mode cases.

Significance. If the 2D results hold, the non-monotonic neutral curve and re-stabilization phenomenon would provide new insight into the interaction between steady and oscillatory wake instabilities, including mode coexistence. The use of Floquet multipliers to track bifurcation loci and complementary DNS for nonlinear saturation are strengths, but the strictly 2D scope limits applicability given known 3D onsets near Re=190.

major comments (2)
  1. [Abstract] Abstract and § on numerical methods: the re-stabilization claim (recovery of Z2-symmetric state at peak Re≈190 for m≳0.9) occurs precisely where the steady-cylinder mode-A instability onsets (Re_crit≈190). The analysis is confined to a strictly two-dimensional Floquet framework with 2D DNS; no discussion or test addresses whether 3D perturbations would destroy the reported re-stabilization or alter the neutral curve in this window.
  2. [Abstract] Abstract: the statement that 'the linear analysis successfully predicts the saturated nonlinear state' for single-mode cases at Re_m=100 rests on DNS whose discretization, domain size, grid-convergence, and error estimates are not supplied, leaving the quantitative support for the bifurcation loci and re-stabilization only partially verifiable.
minor comments (1)
  1. The parameter ranges and the precise definition of the base flow (steady plus oscillatory components) should be stated explicitly in the introduction or methods section for reproducibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their detailed and constructive review. We address each major comment below and will incorporate revisions to improve clarity and completeness while respecting the two-dimensional scope of the study.

read point-by-point responses

  1. Referee: [Abstract] Abstract and § on numerical methods: the re-stabilization claim (recovery of Z2-symmetric state at peak Re≈190 for m≳0.9) occurs precisely where the steady-cylinder mode-A instability onsets (Re_crit≈190). The analysis is confined to a strictly two-dimensional Floquet framework with 2D DNS; no discussion or test addresses whether 3D perturbations would destroy the reported re-stabilization or alter the neutral curve in this window.

    Authors: We agree that the re-stabilization occurs near the known onset of three-dimensional mode-A instability for a steady cylinder and that our work employs a strictly two-dimensional Floquet analysis together with 2D DNS. The manuscript does not examine three-dimensional perturbations. In revision we will add an explicit discussion of this limitation in the conclusions, stating that the reported neutral curve and re-stabilization are two-dimensional results and that three-dimensional effects could modify the stability boundaries near Re≈190. The non-monotonic neutral curve and period-doubling bifurcation remain valid findings within the two-dimensional framework. revision: yes

  2. Referee: [Abstract] Abstract: the statement that 'the linear analysis successfully predicts the saturated nonlinear state' for single-mode cases at Re_m=100 rests on DNS whose discretization, domain size, grid-convergence, and error estimates are not supplied, leaving the quantitative support for the bifurcation loci and re-stabilization only partially verifiable.

    Authors: The referee correctly notes that the manuscript omits detailed DNS parameters. We will revise the numerical-methods section (and, if space permits, the abstract) to include domain size, grid resolution, convergence tests, and error estimates for the direct numerical simulations, thereby strengthening the quantitative support for the comparison between linear predictions and nonlinear saturation. revision: yes

standing simulated objections not resolved
  • Whether three-dimensional perturbations would destroy the reported re-stabilization or alter the neutral curve cannot be answered without performing three-dimensional stability analysis, which lies outside the present two-dimensional study.

Circularity Check

0 steps flagged

No circularity: numerical Floquet analysis is self-contained

full rationale

The paper performs stability analysis via direct numerical solution of the 2D incompressible Navier-Stokes equations and computation of Floquet multipliers for the time-periodic base flow. No analytic derivations, fitted parameters renamed as predictions, or self-citation chains are present that reduce any claimed bifurcation locus or re-stabilization region to its own inputs by construction. The reported neutral curves, period-doubling modes for m>0.5, and non-monotonic behavior in (KC,m) space are outputs of the numerical eigenvalue problem, not tautological redefinitions. Complementary DNS are used only for nonlinear saturation checks and do not alter the linear stability results. This is the standard, non-circular workflow for such hydrodynamic stability studies.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard incompressible Navier-Stokes equations solved under a two-dimensional assumption; no free parameters are fitted to data and no new entities are postulated.

axioms (2)
  • standard math Incompressible Navier-Stokes equations govern the flow
    Standard governing equations invoked for the fluid motion.
  • domain assumption Flow remains strictly two-dimensional
    Analysis performed inside a two-dimensional Floquet framework as stated in the abstract.

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