Skip to main content
SpringerLink
Log in

A numerical study of linear and nonlinear kinematic models in fish swimming with the DSD/SST method

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

Flow over two fish (modeled by two flexible plates) in tandem arrangement is investigated by solving the incompressible Navier–Stokes equations numerically with the DSD/SST method to understand the differences between the geometrically linear and nonlinear models. In the simulation, the motions of the plates are reconstructed from a vertically flowing soap film tunnel experiment with linear and nonlinear kinematic models. Based on the simulations, the drag, lift, power consumption, vorticity and pressure fields are discussed in detail. It is found that the linear and nonlinear models are able to reasonably predict the forces and power consumption of a single plate in flow. Moreover, if multiple plates are considered, these two models yield totally different results, which implies that the nonlinear model should be used. The results presented in this work provide a guideline for future studies in fish swimming.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Triantafyllou MS, Triantafyllou GS, Yue DKP (2000) Hydrodynamics of fish swimming. Annu Rev Fluid Mech 32:33–53

    Article  MathSciNet  Google Scholar 

  2. Fish FE, Lauder GV (2006) Passive and active flow control by swimming fishes and mammals. Annu Rev Fluid Mech 38:193–224

    Article  MathSciNet  Google Scholar 

  3. Shyy W, Aono H, Chimakurthi SK, Trizila P, Kang CK, Cesnik CES, Liu H (2010) Recent progress in flapping wing aerodynamics and aeroelasticity. Prog Aerosp Sci 46:284–327

    Article  Google Scholar 

  4. Wu TY (2011) Fish swimming and bird/insect flight. Annu Rev Fluid Mech 43:25–58

    Article  Google Scholar 

  5. Deng HB, Xu YQ, Chen DD, Dai H, Wu J, Tian FB (2013) On numerical modeling of animal swimming and flight. Comput Mech 52:1221–1242

    Article  MATH  MathSciNet  Google Scholar 

  6. Taylor G (1951) Analysis of the swimming of microscopic organisms. Proc R Soc Lond A 209:447–461

    Article  MATH  Google Scholar 

  7. Wu TY (1961) Swimming of a waving plate. J Fluid Mech 10:321–344

    Article  MATH  MathSciNet  Google Scholar 

  8. Cheng JY, Zhuang LX, Tong BG (1991) Analysis of swimming three-dimensional waving plates. J Fluid Mech 232:341–355

    Article  MATH  MathSciNet  Google Scholar 

  9. Jones RT (1946) Properties of low-aspect-ratio pointed wings at speeds below and above the speed of sound. NACA rep. 835

  10. Lighthill MJ (1960) Note on the swimming of slender fish. J Fluid Mech 9:305–317

    Article  MathSciNet  Google Scholar 

  11. Lighthill MJ (1970) Aquatic animal propulsion of high hydromechanical efficiency. J Fluid Mech 44:265–301

    Article  MATH  Google Scholar 

  12. Mittal S, Tezduyar T (1992) A finite element study of incompressible flows past oscillating cylinders and airfoils. Int J Numer Methods Fluids 15:1073–1118

    Article  Google Scholar 

  13. Mittal S, Tezduyar TE (1995) Parallel finite element simulation of 3D incompressible flows-fluid–structure interactions. Int J Numer Methods Fluids 21:933–953

    Article  MATH  Google Scholar 

  14. Tezduyar TE, Behr M, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces-the deforming-spatial-domain/space–time procedure: I. The concept and the preliminary numerical tests. Comput Methods Appl Mech Eng 94:339–351

    Article  MATH  MathSciNet  Google Scholar 

  15. Tezduyar TE, Behr M, Mittal S, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces-the deforming-spatial-domain/space–time procedure: II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders. Comput Methods Appl Mech Eng 94:353–371

    Article  MATH  MathSciNet  Google Scholar 

  16. Tezduyar TE (1992) Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech 28:1–44

    Article  MATH  MathSciNet  Google Scholar 

  17. Liu H, Wassersug R, Kawachi K (1996) A computational fluid dynamics study of tadpole swimming. J Exp Biol 199:1245–1260

    Google Scholar 

  18. Liu H, Wassersug R, Kawachi K (1997) The three-dimensional hydrodynamics of tadpole locomotion. J Exp Biol 200:2807–2819

    Google Scholar 

  19. Shen L, Zhang X, Yue DKP, Triantafyllou MS (2003) Turbulent flow over a flexible wall undergoing a streamwise travelling wave motion. J Fluid Mech 484:197–221

    Article  MATH  Google Scholar 

  20. Dong GJ, Lu XY (2005) Numerical analysis on the propulsive performance and vortex shedding of fish-like travelling wavy plate. Int J Numer Methods Fluids 48:1351–1373

    Article  MATH  Google Scholar 

  21. Wu JZ, Pan ZL, Lu XY (2005) Unsteady fluid-dynamic force solely in terms of control-surface integral. Phys Fluids 17:098102

    Article  Google Scholar 

  22. Tian FB, Lu XY, Luo H (2012) Propulsive performance of a body with a traveling wave surface. Phys Rev E 86:016304

    Article  Google Scholar 

  23. Dong GJ, Lu XY (2007) Characteristics of flow over traveling-wavy foils in a side-by-side arrangement. Phys Fluids 19:057107

    Article  Google Scholar 

  24. Deng J, Shao XM, Yu ZS (2007) Hydrodynamic studies on two traveling wavy foils in tandem arrangement. Phys Fluids 19:113104

    Article  Google Scholar 

  25. Wang S, Zhang X, He G (2012) Numerical simulation of a three-dimensional fish-like body swimming with finlets. Commun Comput Phys 11:1323–1333

    MathSciNet  Google Scholar 

  26. 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 408:835–839

    Article  Google Scholar 

  27. Zhu L, Peskin CS (2003) Interaction of two flapping filaments in a flowing soap film. Phys Fluids 15:1954–1960

    Article  MathSciNet  Google Scholar 

  28. Tezduyar TE, Sathe S, Keedy R, Stein K (2006) Space–time finite element techniques for computation of fluid–structure interactions. Comput Methods Appl Mech Eng 195:2002–2027

    Article  MATH  MathSciNet  Google Scholar 

  29. Jia LB, Li F, Yin XZ, Yin XY (2007) Coupling modes between two flapping filaments. J Fluid Mech 581:199–220

    Article  MATH  MathSciNet  Google Scholar 

  30. Jia LB, Yin XZ (2008) Passive oscillations of two tandem flexible filaments in a flowing soap film. Phys Rev Lett 100:228104

    Article  Google Scholar 

  31. Zhu L (2009) Interaction of two tandem deformable bodies in a viscous incompressible flow. J Fluid Mech 635:455–475

    Article  MATH  MathSciNet  Google Scholar 

  32. Alben S (2009) Wake-mediated synchronization and drafting in coupled flags. J Fluid Mech 641:489–496

    Article  MATH  MathSciNet  Google Scholar 

  33. Kim S, Huang WX, Sung HJ (2010) Constructive and destructive interaction modes between two tandem flexible flags in viscous flow. J Fluid Mech 661:511–521

    Article  MATH  Google Scholar 

  34. Tian FB, Luo H, Zhu L, Lu XY (2010) Interaction between a flexible filament and a downstream rigid body. Phys Rev E 82:026301

  35. Tian FB, Luo H, Zhu L, Liao JC, Lu XY (2011) An immersed boundary-lattice Boltzmann method for elastic boundaries with mass. J Comput Phys 230:7266–7283

    Article  MATH  MathSciNet  Google Scholar 

  36. Tian FB, Luo H, Zhu L, Lu XY (2011) Coupling modes of three filaments in side-by-side arrangement. Phys Fluids 23:111903

    Article  Google Scholar 

  37. Tian FB (2013) Role of mass on the stability of flag/flags in uniform flow. Appl Phys Lett 103:034101

    Article  Google Scholar 

  38. Carling J, Williams TL, Bowtell G (1998) Self-propelled anguilliform swimming: simultaneous solution of the two-dimensional Navier–Stokes equations and Newton’s laws of motion. J Exp Biol 201:3143–3166

    Google Scholar 

  39. Kern S, Koumoutsakos P (2006) Simulations of optimized anguilliform swimming. J Exp Biol 209:4841–4857

    Article  Google Scholar 

  40. Ristroph L, Zhang J (2008) Anomalous hydrodynamic drafting of interacting flapping flags. Phys Rev Lett 101:194502

    Article  Google Scholar 

  41. Liao Q, Dong GJ, Lu XY (2004) Vortex formation and force characteristics of a foil in the wake of a circular cylinder. J Fluids Struct 19:491–510

    Article  Google Scholar 

  42. Tian FB, Lu XY, Luo H (2012) Onset of instability of a flag in uniform flow. Theor Appl Mech Lett 2:022005

    Article  Google Scholar 

  43. Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43:555–575

    Article  MATH  MathSciNet  Google Scholar 

  44. Tezduyar TE, Sathe S (2007) Modeling of fluid–structure interactions with the space–time finite elements: solution techniques. Int J Numer Methods Fluids 54:855–900

    Article  MATH  MathSciNet  Google Scholar 

  45. Takizawa K, Tezduyar TE (2011) Multiscale space–time fluid–structure interaction techniques. Comput Mech 48:247–267

    Article  MATH  MathSciNet  Google Scholar 

  46. Takizawa K, Tezduyar TE (2012) Space–time fluid–structure interaction methods. Math Models Methods Appl Sci 22(supp02):1230001

    Article  MathSciNet  Google Scholar 

  47. Tezduyar TE, Sathe S, Stein K (2006) Solution techniques for the fully-discretized equations in computation of fluid–structure interactions with the space–time formulations. Comput Methods Appl Mech Eng 195:5743–5753

    Article  MATH  MathSciNet  Google Scholar 

  48. Tezduyar TE (2006) Interface-tracking and interface-capturing techniques for finite element computation of moving boundaries and interfaces. Comput Methods Appl Mech Eng 195:2983–3000

  49. Tezduyar TE (2007) Finite elements in fluids: stabilized formulations and moving boundaries and interfaces. Comput Fluids 36:191–206

    Article  MATH  MathSciNet  Google Scholar 

  50. Wang SY, Tian FB, Jia LB, Lu XY, Yin XZ (2010) The secondary vortex street in the wake of two tandem circular cylinders at low Reynolds number. Phys Rev E 81:036305

    Article  Google Scholar 

  51. Bazilevs Y, Takizawa K, Tezduyar TE (2013) Challenges and directions in computational fluid–structure interaction. Math Models Methods Appl Sci 23:215–221

    Article  MATH  MathSciNet  Google Scholar 

  52. Takizawa K, Montes D, McIntyre S, Tezduyar TE (2013) Space–time VMS methods for modeling of incompressible flows at high Reynolds numbers. Math Models Methods Appl Sci 23:223–248

    Article  MATH  MathSciNet  Google Scholar 

  53. Hirt CW, Amsden AA, Cook JL (1974) An arbitrary Lagrangian–Eulerian computing method for all flow speeds. J Comput Phys 14:227–253

    Article  MATH  Google Scholar 

  54. Tian FB, Luo H, Song J, Lu XY (2013) Force production and asymmetric deformation of a flexible flapping wing in forward flight. J Fluids Struct 36:149–161

    Article  Google Scholar 

  55. Xu YQ, Tian FB, Deng YL (2013) An efficient red blood cell model in the frame of IB-LBM and its application. Int J Biomath 6:1250061

    Article  MathSciNet  Google Scholar 

  56. Tian FB, Chang S, Luo H, Rousseau B (2013) Computational modeling of flow-induced vocal fold vibration. In: Annual ORNL biomedical science and engineering conference (BSEC 2013), Oak Ridge

  57. Tian FB, Chang S, Luo H, Rousseau B (2013) A 3D numerical simulation of wave propagation on the vocal fold surface. In: Proceedings of the 10th international conference on advances in quantitative laryngology, Voice and Speech Research, Cincinnati, p 94921483

  58. Tian FB, Dai H, Luo H, Doyle JF, Rousseau B (2013) Computational fluid–structure interaction for biological and biomedical flows. In: Proceedings of the ASME 2013 fluids engineering division summer meeting, Incline Village, Nevada, p 16408

  59. Tian FB, Dai H, Luo H, Doyle JF, Rousseau B (2014) Fluid–structure interaction involving large deformations: 3D simulations and applications to biological systems. J Comput Phys 258:451–469

    Article  MathSciNet  Google Scholar 

  60. Johnson AA, Tezduyar TE (1999) Advanced mesh generation and update methods for 3D flow simulations. Comput Mech 23:130–143

    Article  MATH  Google Scholar 

  61. Takizawa K, Henicke B, Puntel A, Kostov N, Tezduyar TE (2012) Space–time techniques for computational aerodynamics modeling of flapping wings of an actual locust. Comput Mech 50:743–760

    Article  MATH  Google Scholar 

  62. Takizawa K, Kostov N, Puntel A, Henicke B, Tezduyar TE (2012) Space–time computational analysis of bio-inspired flapping-wing aerodynamics of a micro aerial vehicle. Comput Mech 50:761–778

    Article  MATH  Google Scholar 

  63. Takizawa K, Henicke B, Puntel A, Kostov N, Tezduyar TE (2013) Computer modeling techniques for flapping-wing aerodynamics of a locust. Comput Fluids 85:125–134

    Article  MATH  MathSciNet  Google Scholar 

  64. Takizawa K, Henicke B, Puntel A, Spielman T, Tezduyar TE (2012) Space–time computational techniques for the aerodynamics of flapping wings. J Appl Mech 79:010903

    Article  Google Scholar 

  65. Tian FB (2014) FSI modeling with the DSD/SST method for the fluid and finite difference method for the structure. Comput Mech 54:581–589

    Article  MATH  MathSciNet  Google Scholar 

  66. Takizawa K, Tezduyar TE, Kostov N (2014) Sequentially-coupled space–time FSI analysis of bio-inspired flapping-wing aerodynamics of an MAV. Comput Mech 54:213–233

    Article  MATH  MathSciNet  Google Scholar 

  67. Kalro V, Aliabadi S, Garrard W, Tezduyar T, Mittal S, Stein K (1997) Parallel finite element simulation of large ram-air parachutes. Int J Numer Methods Fluids 24:1353–1369

    Article  MATH  Google Scholar 

  68. Tezduyar TE, Sathe S, Pausewang J, Schwaab M, Christopher J, Crabtree J (2008) Interface projection techniques for fluid-structure interaction modeling with moving-mesh methods. Comput Mech 43:39–49

    Article  MATH  Google Scholar 

  69. Tezduyar TE, Sathe S, Schwaab M, Pausewang J, Christopher J, Crabtree J (2008) Fluid–structure interaction modeling of ringsail parachutes. Comput Mech 43:133–142

    Article  MATH  Google Scholar 

  70. Takizawa K, Spielman T, Tezduyar TE (2011) Space–time FSI modeling and dynamical analysis of spacecraft parachutes and parachute clusters. Comput Mech 48:345–364

    Article  MATH  Google Scholar 

  71. Takizawa K, Fritze M, Montes D, Spielman T, Tezduyar TE (2012) Fluid–structure interaction modeling of ringsail parachutes with disreefing and modified geometric porosity. Comput Mech 50:835–854

    Article  MATH  Google Scholar 

  72. Takizawa K, Tezduyar TE (2012) Computational methods for parachute fluid–structure interactions. Arch Comput Methods Eng 19:125–169

    Article  MathSciNet  Google Scholar 

  73. Takizawa K, Spielman T, Moorman C, Tezduyar TE (2012) Fluid–structure interaction modeling of spacecraft parachutes for simulation-based design. J Appl Mech 79:010907

    Article  Google Scholar 

  74. Takizawa K, Montes D, Fritze M, McIntyre S, Boben J, Tezduyar TE (2013) Methods for FSI modeling of spacecraft parachute dynamics and cover separation. Math Models Methods Appl Sci 23:307–338

    Article  MATH  MathSciNet  Google Scholar 

  75. Takizawa K, Tezduyar TE, Boswell C, Kolesar R, Montel K (2014) FSI modeling of the reefed stages and disreefing of the Orion spacecraft parachutes. Comput Mech 54:1203–1220

    Article  MATH  MathSciNet  Google Scholar 

  76. Takizawa K, Tezduyar TE, Kolesar R, Boswell C, Kanai T, Montel K (2014) Multiscale methods for gore curvature calculations from FSI modeling of spacecraft parachutes. Comput Mech 54:1461–1476

    Article  MATH  MathSciNet  Google Scholar 

  77. Takizawa K, Tezduyar TE, Boswell C, Tsutsui Y, Montel K (2014) Special methods for aerodynamic-moment calculations from parachute FSI modeling. Comput Mech. doi:10.1007/s00466-014-1074-5

  78. Tezduyar TE, Takizawa K, Brummer T, Chen PR (2011) Space–time fluid–structure interaction modeling of patient-specific cerebral aneurysms. Int J Numer Methods Biomed Eng 27:1665–1710

    Article  MATH  MathSciNet  Google Scholar 

  79. Takizawa K, Bazilevs Y, Tezduyar TE (2012) Space–time and ALE-VMS techniques for patient-specific cardiovascular fluid–structure interaction modeling. Arch Comput Methods Eng 19:171–225

    Article  MathSciNet  Google Scholar 

  80. Takizawa K, Schjodt K, Puntel A, Kostov N, Tezduyar TE (2012) Patient-specific computer modeling of blood flow in cerebral arteries with aneurysm and stent. Comput Mech 50:675–686

    Article  MATH  MathSciNet  Google Scholar 

  81. Takizawa K, Schjodt K, Puntel A, Kostov N, Tezduyar TE (2013) Patient-specific computational analysis of the influence of a stent on the unsteady flow in cerebral aneurysms. Comput Mech 51:1061–1073

    Article  MATH  MathSciNet  Google Scholar 

  82. Takizawa K, Tezduyar TE, Buscher A, Asada S (2014) Space–time fluid mechanics computation of heart valve models. Comput Mech 54:973–986

    Article  MATH  MathSciNet  Google Scholar 

  83. Takizawa K, Henicke B, Tezduyar TE, Hsu M-C, Bazilevs Y (2011) Stabilized space–time computation of wind-turbine rotor aerodynamics. Comput Mech 48:333–344

    Article  MATH  Google Scholar 

  84. Takizawa K, Henicke B, Montes D, Tezduyar TE, Hsu M-C, Bazilevs Y (2011) Numerical-performance studies for the stabilized space–time computation of wind-turbine rotor aerodynamics. Comput Mech 48:647–657

    Article  MATH  Google Scholar 

  85. Bazilevs Y, Hsu M-C, Takizawa K, Tezduyar TE (2012) ALE-VMS and ST-VMS methods for computer modeling of wind-turbine rotor aerodynamics and fluid–structure interaction. Math Models Methods Appl Sci 22(supp02):1230002

    Article  Google Scholar 

  86. Takizawa K, Tezduyar TE, McIntyre S, Kostov N, Kolesar R, Habluetzel C (2014) Space–time VMS computation of wind-turbine rotor and tower aerodynamics. Comput Mech 53:1–15

    Article  MATH  Google Scholar 

  87. Tian FB, Bharti RP, Xu YQ (2014) Deforming-spatial-domain/stabilized space–time (DSD/SST) method in computation of non-Newtonian fluid flow and heat transfer with moving boundaries. Comput Mech 54:581–589

    Article  MATH  MathSciNet  Google Scholar 

  88. Tezduyar T, Aliabadi S, Behr M, Johnson A, Mittal S (1993) Parallel finite element computation of 3D flows. Computer 26:27–36

    Article  Google Scholar 

  89. Johnson AA, Tezduyar TE (1994) Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interface. Comput Methods Appl Mech Eng 119:73–94

    Article  MATH  Google Scholar 

  90. Triantafyllou GS, Triantafyllou MS, Grosenbaugh MA (1993) Optimal thrust development in oscillating foils with application to fish propulsion. J Fluids Struct 7:205–224

    Article  Google Scholar 

Download references

Acknowledgments

We thank Professors X.-Z. Yin and L.-B. Jia at the University of Science and Technology of China for their assistance in reconstructing the plate motions. This research was partially supported by the UNSW Canberra’s Early Career Researcher Grants Scheme 2015, the Australian Research Council’s Discovery Project Funding Scheme (No. DP130103850), and the National Natural Science Foundation of China (No. 11202175). Simulations were partially undertaken with computational resources on the National Computational Infrastructure National Facility through the National Computational Merit Allocation Scheme supported by the Australian Government.

Author information

Authors and Affiliations

  1. School of Engineering and Information Technology, University of New South Wales, Canberra, ACT, 2600, Australia

    Fang-Bao Tian

Authors
  1. Fang-Bao Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fang-Bao Tian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tian, FB. A numerical study of linear and nonlinear kinematic models in fish swimming with the DSD/SST method. Comput Mech 55, 469–477 (2015). https://doi.org/10.1007/s00466-014-1116-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-014-1116-z

Keywords

Search

Navigation