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Vortex interactions and wake transitions for flexible foil flapping in flowing soap film

Published online by Cambridge University Press:  19 March 2026

Sarthak K. Patel
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, 208016, India
Sachin Yashavant Shinde*
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, 208016, India
*
Corresponding author: Sachin Yashavant Shinde, sachin@iitk.ac.in

Abstract

Understanding the vortex interactions and wake transitions for flapping flexible foils is important because of their increased usage in bioinspired aquatic and aerial robotic propulsors. Although wake transitions have been studied for rigid foils, we experimentally investigate how flexibility alters the transitions and vortex interactions for flexible foils, which are closer to the natural flapping foils in fish, birds and insects. We conduct the experiments in a flowing soap film on a pitching airfoil with a flexible filament at its trailing edge (TE). We find that, apart from the Strouhal number (${\textit{St}}$), flexural rigidity (${\textit{EI}}$) is important to determine the transitions. We vary ${\textit{EI}}$ of the flexible filament by three orders of magnitude and also investigate an extreme case of ${\textit{EI}} \rightarrow \infty$. Flexibility triggers the shedding of multiple small ‘secondary vortices’ (SVs) along with big ‘primary vortices’ (PVs), unlike only PVs for the rigid foil. Continuous deformations of the flexible filament play crucial roles in determining the interaction of boundary layer vortices and trailing edge vortices and, ultimately, the generation and evolution of PVs and SVs. We identify five vortex interaction mechanisms (VIMs). Depending on how SVs interact with PVs, the wake assumes different patterns. We construct the ${\textit{St}}$${\textit{EI}}$ phase maps for wake transitions and newly identified VIMs. We devise a non-dimensional parameter $\varUpsilon$, referred to as ‘Yashavant number’. One order increase in $\varUpsilon$ reduces the number of VIMs by one. Instead of following the usual transition route, the flexible foil reveals counterintuitive transition trends that strongly depend on the filament ${\textit{EI}}$.

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Type
JFM Papers
Copyright
© The Author(s), 2026. Published by Cambridge University Press

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References

Ait Abderrahmane, H., Paidoussis, M.P., Fayed, M. & Ng, H.D. 2011 Flapping dynamics of a flexible filament. Phys. Rev. E 84, 066604.10.1103/PhysRevE.84.066604CrossRefGoogle ScholarPubMed
Alben, S., Witt, C., Baker, T.V., Anderson, E. & Lauder, G.V. 2012 Dynamics of freely swimming flexible foils. Phys. Fluids 24, 051901.10.1063/1.4709477CrossRefGoogle Scholar
Andersen, A., Bohr, T., Schnipper, T. & Walther, J.H. 2017 Wake structure and thrust generation of a flapping foil in two-dimensional flow. J. Fluid Mech. 812, R4.10.1017/jfm.2016.808CrossRefGoogle Scholar
Beizaie, M. & Gharib, M. 1997 Fundamentals of a liquid (soap) film tunnel. Exp. Fluids 23, 130140.10.1007/s003480050094CrossRefGoogle Scholar
Bohl, D.G. & Koochesfahani, M.M. 2009 MTV measurements of the vortical field in the wake of an airfoil oscillating at high reduced frequency. J. Fluid Mech. 620, 6388.10.1017/S0022112008004734CrossRefGoogle Scholar
Chomaz, J.M. & Cathalau, B. 1990 Soap films as two-dimensional classical fluids. Phys. Rev. A 41, 22432245.10.1103/PhysRevA.41.2243CrossRefGoogle ScholarPubMed
Couder, Y. & Basdevant, C. 1986 Experimental and numerical study of vortex couples in two-dimensional flows. J. Fluid Mech. 173, 225251.10.1017/S0022112086001155CrossRefGoogle Scholar
Couder, Y., Chomaz, J.M. & Rabaud, M. 1989 On the hydrodynamics of soap films. Physica D 37, 384405.10.1016/0167-2789(89)90144-9CrossRefGoogle Scholar
DaDeppo, D.A. & Schmidt, R. 1971 Analysis of nonlinear deflections of fibers. Textile Res. J. 41, 911915.10.1177/004051757104101106CrossRefGoogle Scholar
Daniel, T.L. & Combes, S.A. 2002 Flexible wings and fins: bending by inertial or fluid-dynamic forces? Integr. Compar. Biol. 42, 10441049.10.1093/icb/42.5.1044CrossRefGoogle ScholarPubMed
Drake, D.J., Epperson, C.G. & Burks, S.L. 2020 A thin film interference laboratory experiment for introductory physics. Phys. Teacher 58, 272275.10.1119/1.5145477CrossRefGoogle Scholar
Feng, S., Chen, C., Huang, Q. & Zhou, T. 2019 Experimental study on the flapping dynamics for a single flexible filament in a vertical soap film. J. Fluids Struct. 86, 236246.10.1016/j.jfluidstructs.2019.02.019CrossRefGoogle Scholar
Georgiev, D. & Vorobieff, P. 2002 The slowest soap-film tunnel in the southwest. Rev. Sci. Instrum. 73, 11771184.10.1063/1.1446040CrossRefGoogle Scholar
Gharib, M. & Derango, P. 1989 A liquid film (soap film) tunnel to study two-dimensional laminar and turbulent shear flows. Physica D 37, 406416.10.1016/0167-2789(89)90145-0CrossRefGoogle Scholar
Godoy-Diana, R., Aider, J.L. & Wesfreid, J.E. 2008 Transitions in the wake of a flapping foil. Phys. Rev. E 77, 016308.10.1103/PhysRevE.77.016308CrossRefGoogle ScholarPubMed
Godoy-Diana, R., Marais, C., Aider, J.L. & Wesfreid, J.E. 2009 A model for the symmetry breaking of the reverse Bénard–von Kármán vortex street produced by a flapping foil. J. Fluid Mech. 622, 2332.10.1017/S0022112008005727CrossRefGoogle Scholar
Heathcote, S. & Gursul, I. 2007 Flexible flapping airfoil propulsion at low Reynolds numbers. AIAA J. 45, 10661079.10.2514/1.25431CrossRefGoogle Scholar
Jia, L.B. & Yin, X.Z. 2008 Passive oscillations of two tandem flexible filaments in a flowing soap film. Phys. Rev. Lett. 100, 228104.10.1103/PhysRevLett.100.228104CrossRefGoogle Scholar
Jones, K.D., Dohring, C.M. & Platzer, M.F. 1998 Experimental and computational investigation of the Knoller–Betz effect. AIAA J. 36, 12401246.10.2514/2.505CrossRefGoogle Scholar
Kim, M.J. & Lee, J.H. 2019 Wake transitions of flexible foils in a viscous uniform flow. Phys. Fluids 31, 111906.10.1063/1.5120050CrossRefGoogle Scholar
Koochesfahani, M.M. 1989 Vortical patterns in the wake of an oscillating airfoil. AIAA J. 27, 12001205.10.2514/3.10246CrossRefGoogle Scholar
Kundu, P.K., Cohen, I.M. & Dowling, D.R. 2016 Fluid Mechanics. Academic Press.Google Scholar
Lagopoulos, N.S., Weymouth, G.D. & Ganapathisubramani, B. 2019 Universal scaling law for drag-to-thrust wake transition in flapping foils. J. Fluid Mech. 872, R1.10.1017/jfm.2019.361CrossRefGoogle Scholar
Lauder, G.V. 2015 Fish locomotion: recent advances and new directions. Annu. Rev. Mar. Sci. 7, 521545.10.1146/annurev-marine-010814-015614CrossRefGoogle ScholarPubMed
Mackowski, A.W. & Williamson, C.H.K. 2015 Direct measurement of thrust and efficiency of an airfoil undergoing pure pitching. J. Fluid Mech. 765, 524543.10.1017/jfm.2014.748CrossRefGoogle Scholar
Marais, C., Thiria, B., Wesfreid, J.E. & Godoy-Diana, R. 2012 Stabilizing effect of flexibility in the wake of a flapping foil. J. Fluid Mech. 710, 659669.10.1017/jfm.2012.390CrossRefGoogle Scholar
Mountcastle, A.M. & Daniel, T.L. 2009 Aerodynamic and functional consequences of wing compliance. Exp. Fluids 46, 873882.10.1007/s00348-008-0607-0CrossRefGoogle Scholar
Muijres, F.T. & Lentink, D. 2007 Wake visualization of a heaving and pitching foil in a soap film. Exp. Fluids 43, 665673.10.1007/s00348-007-0379-yCrossRefGoogle Scholar
Newburgh, R. & Goodale, D. 2009 Student difficulties in analyzing thin-film interference. Phys. Teacher 47, 227230.10.1119/1.3098209CrossRefGoogle Scholar
Quinn, D. & Lauder, G. 2021 Tunable stiffness in fish robotics: mechanisms and advantages. Bioinspir. Biomim. 17, 011002.10.1088/1748-3190/ac3ca5CrossRefGoogle ScholarPubMed
Rivera, M., Vorobieff, P. & Ecke, R.E. 1998 Turbulence in flowing soap films: velocity, vorticity, and thickness fields. Phys. Rev. Lett. 81, 14171420.10.1103/PhysRevLett.81.1417CrossRefGoogle Scholar
Roushan, P. & Wu, X.L. 2005 Structure-based interpretation of the Strouhal–Reynolds number relationship. Phys. Rev. Lett. 94, 054504.10.1103/PhysRevLett.94.054504CrossRefGoogle ScholarPubMed
Rutgers, M.A., Wu, X., Bhagavatula, R., Petersen, A.A. & Goldburg, W.I. 1996 Two-dimensional velocity profiles and laminar boundary layers in flowing soap films. Phys. Fluids 8, 28472854.10.1063/1.869105CrossRefGoogle Scholar
Rutgers, M.A., Wu, X. & Daniel, W.B. 2001 Conducting fluid dynamics experiments with vertically falling soap films. Rev. Sci. Instrum. 72, 30253037.10.1063/1.1379956CrossRefGoogle Scholar
Saffman, P.G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Schnipper, T. 2011 Exotic wakes of flapping fins. PhD thesis, Technical University of Denmark.Google Scholar
Schnipper, T., Andersen, A. & Bohr, T. 2009 Vortex wakes of a flapping foil. J. Fluid Mech. 633, 411423.10.1017/S0022112009007964CrossRefGoogle Scholar
Shah, C.L., Majumdar, D., Bose, C. & Sarkar, S. 2022 Chordwise flexible aft-tail suppresses jet-switching by reinstating wake periodicity in a flapping foil. J. Fluid Mech. 946, A12.10.1017/jfm.2022.591CrossRefGoogle Scholar
Shinde, S.Y. & Arakeri, J.H. 2013 Jet meandering by a foil pitching in quiescent fluid. Phys. Fluids 25, 041701.10.1063/1.4800321CrossRefGoogle Scholar
Shinde, S.Y. & Arakeri, J.H. 2014 Flexibility in flapping foil suppresses meandering of induced jet in absence of free stream. J. Fluid Mech. 757, 231250.10.1017/jfm.2014.480CrossRefGoogle Scholar
Shinde, S.Y. & Arakeri, J.H. 2018 Physics of unsteady thrust and flow generation by a flexible surface flapping in the absence of a free stream. Proc. R. Soc. A: Math. Phys. Engng Sci. 474, 20180519.10.1098/rspa.2018.0519CrossRefGoogle Scholar
Shyy, W., Berg, M. & Ljungqvist, D. 1999 Flapping and flexible wings for biological and micro air vehicles. Prog. Aerosp. Sci. 35, 455505.10.1016/S0376-0421(98)00016-5CrossRefGoogle Scholar
Thulasi, H.M., Menon, A., Jaiswara, P. & Shinde, S.Y. 2024 Effect of flexible flap length on flow generation by an airfoil pitching in quiescent fluid. J. Flow Vis. Image Process. 31, 113.10.1615/JFlowVisImageProc.2024049118CrossRefGoogle Scholar
Triantafyllou, M.S., Triantafyllou, G.S. & Gopalkrishnan, R. 1991 Wake mechanics for thrust generation in oscillating foils. Phys. Fluids A 3, 28352837.10.1063/1.858173CrossRefGoogle Scholar
Triantafyllou, G., Triantafyllou, M.S. & Grosenbaugh, M.A. 1993 Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluids Struct. 7, 205224.10.1006/jfls.1993.1012CrossRefGoogle Scholar
Tzezana, G.A. & Breuer, K.S. 2019 Thrust, drag and wake structure in flapping compliant membrane wings. J. Fluid Mech. 862, 871888.10.1017/jfm.2018.966CrossRefGoogle Scholar
Van Buren, T., Floryan, D. & Smits, A.J. 2019 Scaling and performance of simultaneously heaving and pitching foils. AIAA J. 57, 36663677.10.2514/1.J056635CrossRefGoogle Scholar
Vorobieff, P. & Ecke, R.E. 1999 Cylinder wakes in flowing soap films. Phys. Rev. E 60, 29532956.10.1103/PhysRevE.60.2953CrossRefGoogle ScholarPubMed
Wang, Z.J. 2005 Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183210.10.1146/annurev.fluid.36.050802.121940CrossRefGoogle Scholar
Wootton, R.J. 1999 Invertebrate paraxial locomotory appendages: design, deformation and control. J. Expl Biol. 202, 33333345.10.1242/jeb.202.23.3333CrossRefGoogle ScholarPubMed
Young, J., Walker, S.M., Richard, J.B., Taylor, G.K. & Thomas, A.L.R. 2009 Details of insect wing design and deformation enhance aerodynamic function and flight efficiency. Science 325, 15491552.10.1126/science.1175928CrossRefGoogle ScholarPubMed
Zhang, J., Childress, S., Libchaber, A. & Shelley, M. 2000 Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind. Nature 408, 835839.10.1038/35048530CrossRefGoogle Scholar
Zhu, X., He, G. & Zhang, X. 2014 How flexibility affects the wake symmetry properties of a self-propelled plunging foil. J. Fluid Mech. 751, 164183.10.1017/jfm.2014.310CrossRefGoogle Scholar