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arxiv: 2606.02900 · v1 · pith:7RWBYSWRnew · submitted 2026-06-01 · ⚛️ physics.flu-dyn

Energy Transfer Mechanisms in Wake-Modulated Transonic Flutter

Vedasri Godavarthi , Jacob Turner , Jung-Hee Seo , Rajat Mittal This is my paper

Pith reviewed 2026-06-28 12:23 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords transonic flutterforce partitioningenergy transferwake modulationaeroelasticitydirect numerical simulationNACA0012 airfoilgap flow
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0 comments X

The pith

The gap flow between a wing and an underwing cylinder dominates energy transfer driving transonic flutter.

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

High-fidelity simulations of a sinusoidally pitching NACA0012 airfoil with an underwing cylinder at transonic Mach numbers show that the cylinder expands flutter boundaries relative to an airfoil alone. The study extends the force partitioning method to compressible flows and applies it regionally to the flow domain. This partitioning identifies the gap flow between the wing and cylinder as the primary source of net energy transfer from the fluid to the wing. Blockage from the cylinder accelerates flow over the wing, increasing the flutter tendency, but this effect appears only when the cylinder sits upstream of the airfoil pivot point.

Core claim

By extending the force partitioning method to compressible flows and applying it to distinct regions in direct numerical simulations, the gap flow between the wing and the cylinder emerges as the dominant contributor to energy transfer from the flow to the wing.

What carries the argument

The force partitioning method extended to compressible flows, which isolates power transfer contributions from specific flow regions such as the gap between wing and cylinder.

If this is right

  • Cylinder placement upstream of the airfoil pivot point enhances flutter, while downstream placement does not.
  • Blockage-induced flow acceleration on the wing surface increases the tendency for flutter onset.
  • Flutter boundaries expand significantly when the cylinder is present compared with an isolated airfoil.
  • Regional application of the partitioning method can locate energy sources in other unsteady high-speed flows.

Where Pith is reading between the lines

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

  • Designs that alter gap geometry or introduce local flow control could shift flutter boundaries without changing the overall configuration.
  • The same regional partitioning could be tested on three-dimensional or higher-Reynolds-number cases to check whether gap dominance persists.
  • Energy maps from the method offer a diagnostic for predicting flutter in multi-body systems such as aircraft with external stores.

Load-bearing premise

The extended force-partitioning method correctly isolates the dominant energy-transfer region without significant contamination from numerical dissipation or domain-boundary effects.

What would settle it

A controlled simulation in which the gap flow is blocked or removed while all other parameters remain fixed, followed by measurement of whether net energy input to the wing and flutter amplitude decrease.

Figures

Figures reproduced from arXiv: 2606.02900 by Jacob Turner, Jung-Hee Seo, Rajat Mittal, Vedasri Godavarthi.

Figure 1
Figure 1. Figure 1: Various flow mechanisms observed in a sinusoidally pitching cylinder-airfoil system in transonic Mach [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Computational domain and mesh setup for the current study. (b) Setup of cylinder placed underneath the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Instantaneous vorticity-fields of a sinusoidally pitching airfoil-only (a)-(l) and cylinder-airfoil systems [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Energy Maps for (a) a pitching airfoil-only system (reproduced from [Turner et al., 2024a]) and (b) pitching [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Time history of (a)-(b) the pitching motion [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Influence fields corresponding to power (𝜙) (a)-(c) for a pitching cylinder-airfoil system with 𝑓 ★ = 0.1 and 𝐴𝜃 = 10◦ at three different time instances of a pitching cycle. Note that 𝜙 = 0 at 𝑡 = 0 as the body velocity is zero at the maximum pitch-up phase. By projecting the compressible momentum equation (equation 1) on ∇𝜙 and integrating over the fluid volume V we obtain, ∫ V ∇𝑝 · ∇𝜙𝑑𝑉 = ∫ V  ∇ · 𝝉 − 𝜕… view at source ↗
Figure 7
Figure 7. Figure 7: Application of compressible power partitioning method to the pitching cylinder-airfoil system at [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Application of regional power partitioning method for flow over a pitching NACA0012 airfoil at [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Quadrant-wise contribution of energy transfer [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Instantaneous vorticity-fields of a sinusoidally pitching airfoil-only (a)-(f) and cylinder-airfoil (g)-(l) systems [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Dominant effects of underwing cylinder on airfoil flutter. Power coefficient contribution from (a)-(c) [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Dominant effect of underwing cylinder on airfoil flutter for [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Sub-regional partitioning in 𝑄3 to isolate energy extraction contribution from distinct flow structures and phenomenologies. (a) Instantaneous sub-region division of 𝑄3 for a pitching cylinder-airfoil system at (𝑀∞, 𝐴𝜃 ) = (0.8, 5 ◦ ). (b) Phase-averaged contribution from the shock-train, cylinder and airfoil shear layers to W𝑄3 . (c) Cumulative integral of the asymmetric component W𝑄3 for the subregions.… view at source ↗
Figure 14
Figure 14. Figure 14: Effect of underwing cylinder placement on energy extraction from fluid to the airfoil. (a)-(d) In [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Grid and statistical convergence. (a)-(b) Grid dependence test for pitching cylinder-airfoil system at [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Effect of underwing cylinder on shock-induced separation in [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
read the original abstract

Transonic flutter is a detrimental aeroelastic instability that can generate large-amplitude structural oscillations, leading to severe vibration, fatigue damage, reduced operational limits, and potentially catastrophic structural failure. Incoming wake disturbances can further amplify this instability, making it critical to identify the underlying aerodynamic mechanisms responsible for predicting and controlling flutter onset. The underlying flow physics is complex with nonlinear interactions between the wake and the wing, shock motion, shock-induced flow separation, vortex shedding and the wing motion. In this study, we perform high-fidelity direct numerical simulations of a sinusoidally pitching NACA0012 airfoil with an underwing cylinder at various transonic Mach numbers and a Reynolds number of 10,000. Through energy maps, we identify that the addition of the cylinder significantly expands flutter boundaries compared to an airfoil-only system. We extend the force partitioning method to partition the power transferred between the flow and the airfoil for compressible flows. Application of this approach to distinct regions of the flow domain indicates that the gap flow between the wing and the cylinder is the dominant contributor to the energy transfer from flow to the wing. The blockage effects due to the cylinder cause flow acceleration on the wing which further enhances the tendency for flutter. We investigate cylinder placement relative to the airfoil to reveal that flutter is enhanced only when the cylinder is placed upstream of the pivot point on the airfoil. The current study highlights how such partitioning methods can parse force and energy transfer mechanisms in complex, unsteady high-speed 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

3 major / 2 minor

Summary. The manuscript reports 2D DNS of a sinusoidally pitching NACA0012 airfoil with an underwing cylinder at Re=10,000 and transonic Mach numbers. It extends the force-partitioning method to compressible flows, generates energy maps showing expanded flutter boundaries relative to the airfoil-only case, and concludes that the gap flow between wing and cylinder is the dominant contributor to energy transfer from flow to wing. Cylinder placement upstream of the pivot is found to enhance flutter via blockage-induced acceleration.

Significance. If the compressible extension of the force-partitioning method can be shown to conserve total power and correctly attribute regional contributions without contamination from shocks or numerical dissipation, the work would supply a useful diagnostic for dissecting wake-induced energy transfer in unsteady transonic aeroelasticity, complementing existing surface-integral approaches.

major comments (3)
  1. [Abstract/Methods] Abstract and Methods: The central claim that gap-flow partitioning identifies the dominant energy-transfer region rests on an unshown compressible extension of the force-partitioning method; no derivation of the volume integrals, no check that partitioned power sums to the surface integral on the same mesh, and no benchmark against known compressible test cases are supplied, directly undermining in the regional ranking.
  2. [Results] Results (energy maps): No error bars, mesh-sensitivity data, or domain-boundary sensitivity tests are reported for the partitioned energy contributions; at transonic Mach and Re=10,000 with shock-induced separation, any numerical dissipation at discontinuities could leak into the gap-flow attribution and reverse the reported dominance.
  3. [Cylinder placement study] § on cylinder placement: The conclusion that flutter is enhanced only for upstream placement is based on the same unvalidated partitioning; without a direct comparison of total aerodynamic power (surface integral) versus partitioned sum for each placement, the causal link between gap-flow dominance and expanded flutter boundaries remains unverified.
minor comments (2)
  1. [Introduction] The 2D assumption and Re=10,000 are stated but not justified against expected 3D effects in real transonic flutter; a brief discussion of this limitation would strengthen the manuscript.
  2. [Figures] Figure captions for energy maps should explicitly state the integration domains used for each regional partition.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript to provide the requested derivations, validations, and sensitivity analyses.

read point-by-point responses

  1. Referee: [Abstract/Methods] Abstract and Methods: The central claim that gap-flow partitioning identifies the dominant energy-transfer region rests on an unshown compressible extension of the force-partitioning method; no derivation of the volume integrals, no check that partitioned power sums to the surface integral on the same mesh, and no benchmark against known compressible test cases are supplied, directly undermining in the regional ranking.

    Authors: We agree that the compressible extension requires explicit presentation. In the revised manuscript we will add a dedicated Methods subsection deriving the volume integrals, accounting for variable density and compressibility effects. We will also show that the partitioned power sums exactly to the surface-integral power on the same mesh and include benchmark comparisons against standard compressible unsteady test cases (e.g., oscillating airfoils with known analytic or high-fidelity reference data). revision: yes

  2. Referee: [Results] Results (energy maps): No error bars, mesh-sensitivity data, or domain-boundary sensitivity tests are reported for the partitioned energy contributions; at transonic Mach and Re=10,000 with shock-induced separation, any numerical dissipation at discontinuities could leak into the gap-flow attribution and reverse the reported dominance.

    Authors: We acknowledge the need for quantitative uncertainty assessment. The revision will include mesh-convergence studies for the partitioned energies, error bars derived from grid-refinement differences, and domain-boundary sensitivity tests. These additions will confirm that the reported gap-flow dominance is robust to numerical dissipation at shocks and separation regions. revision: yes

  3. Referee: [Cylinder placement study] § on cylinder placement: The conclusion that flutter is enhanced only for upstream placement is based on the same unvalidated partitioning; without a direct comparison of total aerodynamic power (surface integral) versus partitioned sum for each placement, the causal link between gap-flow dominance and expanded flutter boundaries remains unverified.

    Authors: We will add, for every cylinder placement examined, a direct side-by-side comparison of the total aerodynamic power obtained from the surface integral against the sum of all partitioned regional contributions. This verification will be presented in the revised cylinder-placement section to substantiate the link between gap-flow dominance and the observed flutter-boundary expansion. revision: yes

Circularity Check

0 steps flagged

No circularity: central claim is a direct computational result from regional application of the extended method

full rationale

The paper conducts DNS of the pitching airfoil-cylinder system at transonic conditions and extends the force-partitioning method to compressible flows. The dominant-contributor claim for the gap-flow region is obtained by partitioning power on sub-domains of the simulated flow field. No equation in the provided material reduces this attribution to a fitted parameter, a self-citation chain, or a definitional identity; the result is an output of the numerical experiment rather than an input renamed as a prediction. Minor self-citation risk, if present in the full text for the original force-partitioning method, is not load-bearing for the regional finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on the accuracy of the compressible force-partitioning extension and the assumption that low-Re 2D DNS captures the dominant energy pathways without missing 3D or high-Re effects; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Navier-Stokes equations govern the compressible flow around the airfoil-cylinder system
    Invoked implicitly as the basis for the DNS described in the abstract.
  • domain assumption The extended force-partitioning method correctly decomposes power transfer in the presence of shocks and compressibility
    Central to the regional energy analysis; location: abstract paragraph on method extension.

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discussion (0)

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Reference graph

Works this paper leans on

99 extracted references

  1. [1]

    and Mai, H

    Schewe, G. and Mai, H. and Dietz, G. , title =. J. Fluid. Struct. , volume =. 2003 , doi =

    2003

  2. [2]

    Kholodar, D. B. and Dowell, E. H. and Thomas, J. P. and Hall, K. C. , title =. J. Aircr. , volume =. 2004 , doi =

    2004

  3. [3]

    and Schewe, G

    Dietz, G. and Schewe, G. and Mai, H. , title =. J. Fluid. Struct. , volume =. 2006 , doi =

    2006

  4. [4]

    Bendiksen, O. O. , title =. Prog. Aerosp. Sci. , volume =. 2011 , doi =

    2011

  5. [5]

    Bendiksen, O. O. , title =. AIAA Paper , year =

  6. [6]

    Lee, B. H. K. , title =. Prog. Aerosp. Sci. , volume =. 2001 , doi =

    2001

  7. [7]

    Karnick, P. T. and Venkatraman, K. , title =. J. Fluid Mech. , volume =. 2017 , doi =

    2017

  8. [8]

    and Knight, D

    Ghosh Choudhuri, P. and Knight, D. D. , title =. J. Fluid Mech. , volume =. 1996 , doi =

    1996

  9. [9]

    Corke, T. C. and Thomas, F. O. , title =. Annu. Rev. Fluid Mech. , volume =. 2015 , doi =

    2015

  10. [10]

    and Mittal, R

    Menon, K. and Mittal, R. , title =. J. Fluid Mech. , volume =. 2019 , doi =

    2019

  11. [11]

    Beran, P. S. and Khot, N. S. and Eastep, F. E. and Snyder, R. D. and Zweber, J. V. , title =. J. Aircr. , volume =. 2004 , doi =

    2004

  12. [12]

    Parker, G. H. and Maple, R. C. and Beran, P. S. , title =. J. Aircr. , volume =. 2007 , doi =

    2007

  13. [13]

    and Raveh, D

    Iovnovich, M. and Raveh, D. E. and Michaels, D. and Adar, M. , title =. J. Aircr. , volume =. 2017 , doi =

    2017

  14. [14]

    Denegri, C. M. Jr. and Dubben, J. A. and Maxwell, D. L. , title =. J. Aircr. , volume =. 2005 , doi =

    2005

  15. [15]

    Lambourne, N. C. , title =

  16. [16]

    and Isogai, K

    Yamasaki, M. and Isogai, K. and Uchida, T. and Yukimura, I. , title =. AIAA J. , volume =. 2004 , doi =

    2004

  17. [17]

    and Koch, S

    Braune, M. and Koch, S. , title =. Exp. Fluids , volume =. 2020 , doi =

    2020

  18. [18]

    Barnes, C. J. and Visbal, M. R. , title =. AIAA J. , volume =. 2023 , doi =

    2023

  19. [19]

    Thomas, J. P. and Dowell, E. H. and Hall, K. C. , title =. J. Aircr. , volume =. 2004 , doi =

    2004

  20. [20]

    Turner, J. M. and Seo, J.-H. and Mittal, R. , title =. AIAA Paper , year =

  21. [21]

    Turner, J. M. and Seo, J.-H. and Mittal, R. , title =. AIAA J. , volume =. 2024 , doi =

    2024

  22. [22]

    and Napolitano, M

    Quartapelle, L. and Napolitano, M. , title =. AIAA J. , volume =. 1983 , doi =

    1983

  23. [23]

    , title =

    Chang, C.-C. , title =. Proc. R. Soc. Lond. A , volume =. 1992 , doi =

    1992

  24. [24]

    and Lei, S.-Y

    Chang, C.-C. and Lei, S.-Y. , title =. Proc. R. Soc. Lond. A , volume =. 1996 , doi =

    1996

  25. [25]

    and Su, J.-Y

    Chang, C.-C. and Su, J.-Y. and Lei, S.-Y. , title =. Theor. Comput. Fluid Dyn. , volume =. 1998 , doi =

    1998

  26. [26]

    and Lu, X.-Y

    Wu, J. and Lu, X.-Y. and Zhuang, L.-X. , title =. J. Fluid Mech. , volume =. 2007 , doi =

    2007

  27. [27]

    and Tognaccini, R

    Marongiu, C. and Tognaccini, R. , title =. AIAA J. , volume =. 2010 , doi =

    2010

  28. [28]

    and Tognaccini, R

    Marongiu, C. and Tognaccini, R. and Ueno, M. , title =. AIAA J. , volume =. 2013 , doi =

    2013

  29. [29]

    and Zhang, X

    Wang, S. and Zhang, X. and He, G. and Liu, T. , title =. AIAA J. , volume =. 2015 , doi =

    2015

  30. [30]

    and Shi, Y

    Liu, L. and Shi, Y. and Zhu, J. and Su, W. and Zou, S. and Wu, J. , title =. J. Fluid Mech. , volume =. 2014 , doi =

    2014

  31. [31]

    and Wu, J

    Liu, L. and Wu, J. and Shi, Y. and Zhu, J. , title =. Fluid Dyn. Res. , volume =. 2014 , doi =

    2014

  32. [32]

    and Tognaccini, R

    Mele, B. and Tognaccini, R. , title =. Phys. Fluids , volume =. 2014 , doi =

    2014

  33. [33]

    and Ostieri, M

    Mele, B. and Ostieri, M. and Tognaccini, R. , title =. AIAA J. , volume =. 2016 , doi =

    2016

  34. [34]

    Spalart, P. R. , title =. J. Fluid Mech. , volume =. 2008 , doi =

    2008

  35. [35]

    , title =

    Kroo, I. , title =. Annu. Rev. Fluid Mech.s , volume =. 2001 , doi =

    2001

  36. [36]

    and Grosjean, N

    Boudet, J. and Grosjean, N. and Jacob, M. C. , title =. Int. J. Aeroacoust. , volume =. 2005 , doi =

    2005

  37. [37]

    and Thiele, F

    Greschner, B. and Thiele, F. and Casalino, D. and Jacob, M. C. , title =. Comput. Fluids , volume =. 2008 , doi =

    2008

  38. [38]

    Rolandi, L. V. and Fontane, J. and Jardin, T. and Gressier, J. and Joly, L. , title =. J. Fluid Mech. , volume =. 2023 , doi =

    2023

  39. [39]

    and Stober, J

    Kleinert, J. and Stober, J. and Lutz, T. , title =. CEAS Aeronaut. J. , volume =. 2023 , doi =

    2023

  40. [40]

    and Ehrle, M

    Kleinert, J. and Ehrle, M. and Waldmann, A. and Lutz, T. , title =. CEAS Aeronaut. J. , volume =. 2024 , doi =

    2024

  41. [41]

    Schauerte, C. J. and Schreyer, A.-M. , title =. CEAS Aeronaut. J. , volume =. 2024 , doi =

    2024

  42. [42]

    and Xiong, N

    Tao, Y. and Xiong, N. and Wang, X. and Lin, J. and Liu, Z. and Ma, S. and Wu, J. , title =. Chinese Journal of Aeronautics , volume =. 2021 , doi =

    2021

  43. [43]

    and Molton, P

    Jacquin, L. and Molton, P. and Deck, S. and Maury, B. and Soulevant, D. , title =. AIAA J. , volume =. 2009 , doi =

    2009

  44. [44]

    and Dandois, J

    Molton, P. and Dandois, J. and Lepage, A. and Brunet, V. and Bur, R. , title =. AIAA J. , volume =. 2013 , doi =

    2013

  45. [45]

    Bunton, R. W. and Denegri, C. M. , title =. J. Aircr. , volume =

  46. [46]

    Saffman, P. G. , title =

  47. [47]

    and Meier, G

    Nakagawa, T. and Meier, G. E. A. and Timm, R. and Lent, H.-M. , title =. Proc. R. Soc. Lond. A , volume =. 1987 , doi =

    1987

  48. [48]

    Jacob, M. C. and Boudet, J. and Casalino, D. and Michard, M. , title =. Theor. Comput. Fluid Dyn. , volume =. 2005 , doi =

    2005

  49. [49]

    and Miyazato, Y

    Matsuo, K. and Miyazato, Y. and Kim, H. D. , title =. Prog. Aerosp. Sci. , volume =. 1999 , doi =

    1999

  50. [50]

    Denegri, C. M. Jr. and Dubben, J. A. and Kernazhitskiy, S. L. , title =. J. Aircr. , volume =

  51. [51]

    and Chen, P

    Lee, D.-H. and Chen, P. C. , title =. 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference , year =

  52. [52]

    and Mittal, R

    Menon, K. and Mittal, R. , title =. J. Fluid Mech. , volume =. 2020 , doi =

    2020

  53. [53]

    and Kroo, I

    Ning, A. and Kroo, I. and Aftosmis, M. J. and Nemec, M. and Kless, J. E. , title =. J. Aircr. , volume =. 2014 , doi =

    2014

  54. [54]

    and Rudnik, R

    Sebastian, S. and Rudnik, R. , title =. CEAS Aeronaut. J. , volume =. 2024 , doi =

    2024

  55. [55]

    Impact of 2D Engine Nacelle Flow on Buffet , journal =

    L. Impact of 2D Engine Nacelle Flow on Buffet , journal =. 2024 , doi =

    2024

  56. [56]

    Wake-Airfoil Interaction as Broadband Noise Source: A Large-Eddy Simulation Study , journal =

    Boudet, J. Wake-Airfoil Interaction as Broadband Noise Source: A Large-Eddy Simulation Study , journal =. 2005 , doi =

    2005

  57. [57]

    Padmanabhan, M. A. and Dowell, E. H. and Thomas, J. P. and Pasiliao, C. L. , title =. J. Aircr. , volume =. 2016 , doi =

    2016

  58. [58]

    Padmanabhan, M. A. and Pasiliao, C. L. and Dowell, E. H. , title =. AIAA J. , volume =. 2014 , doi =

    2014

  59. [59]

    Dowell, E. H. and Thomas, J. P. and Hall, K. C. and Denegri, C. M. , title =. J. Aircr. , volume =. 2009 , doi =

    2009

  60. [60]

    Thomas, J. P. and Dowell, E. H. and Hall, K. C. and Denegri, C. M. , title =. AIAA Struct. Struct. Dyn. Mater. Conf , volume =

  61. [61]

    Parker, G. H. and Maple, R. C. and Beran, P. S. , title =. AIAA Struct. Struct. Dyn. Mater. Conf , year =

  62. [62]

    Chen, P. C. and Zhang, Z. and Zhou, Z. and Wang, X. Q. and Mignolet, M. P. , title =. AIAA Aerospace Sciences Meeting , year =

  63. [63]

    Denegri, C. M. and Sharma, V. K. and Northington, J. S. , title =. J. Aircr. , volume =. 2016 , month = jan, doi =

    2016

  64. [64]

    Denegri, C. M. and Dubben, J. A. and Maxwell, D. L. , title =. J. Aircr. , volume =. 2005 , month = mar, doi =

    2005

  65. [65]

    Beran, P. S. and Khot, N. S. and Eastep, F. E. and Snyder, R. D. and Zweber, J. V. , title =. J. Aircr. , volume =. 2004 , month = nov, doi =

    2004

  66. [66]

    Thomas, J. P. and Dowell, E. H. and Hall, K. C. , title =. AIAA J. , volume =. 2002 , month = apr, doi =

    2002

  67. [67]

    , title =

    Iovnovich, M. , title =. International Forum on Aeroelasticity and Structural Dynamics , address =

  68. [68]

    and Yuan, W

    Sandhu, R. and Yuan, W. and Poirel, D. , title =. 31st Congress of the International Council of the Aeronautical Sciences , year =

  69. [69]

    and Rimple, S

    Yuan, W. and Rimple, S. and Poirel, D. , title =. J. Aerosp. Eng. , volume =

  70. [70]

    Denegri, C. M. and Dubben, J. A. and Kernazhitskiy, S. L. , title =. J. Aircr. , volume =

  71. [71]

    Lechniak, J. A. and Bhamidipati, K. K. and Pasiliao, C. L. , title =. AIAA Struct. Struct. Dyn. Mater. Conf , address =. 2012 , paper =

    2012

  72. [72]

    and Maple, R

    Parker, G. and Maple, R. and Beran, P. , title =. AIAA Struct. Struct. Dyn. Mater. Conf , address =. 2005 , doi =

    2005

  73. [73]

    Transonic flow past oscillating airfoils , author=. Annu. Rev. Fluid Mech.s , volume=

  74. [74]

    PloS one , volume=

    Centripetal acceleration reaction: an effective and robust mechanism for flapping flight in insects , author=. PloS one , volume=. 2015 , publisher=

    2015

  75. [75]

    On the initiation and sustenance of flow-induced vibration of cylinders: insights from force partitioning , author=. J. Fluid Mech. , volume=. 2021 , publisher=

    2021

  76. [76]

    Force and moment in incompressible flows , author=. AIAA J. , volume=

  77. [77]

    A high-order sharp-interface immersed boundary solver for high-speed flows , author=. J. Comput. Phys. , volume=. 2024 , publisher=

    2024

  78. [78]

    2006 , publisher=

    Viscous fluid flow , author=. 2006 , publisher=

    2006

  79. [79]

    Compact finite difference schemes with spectral-like resolution , author=. J. Comput. Phys. , volume=. 1992 , publisher=

    1992

  80. [80]

    Gaitonde, D. V. and Shang, J. S. and Young, J. L. , title =. Int. J. Num. Methods Eng. , volume =

Showing first 80 references.