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An inviscid model for vortex shedding from a deforming body

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Abstract

An inviscid vortex sheet model is developed in order to study the unsteady separated flow past a two-dimensional deforming body which moves with a prescribed motion in an otherwise quiescent fluid. Following Jones (J Fluid Mech 496, 405–441, 2003) the flow is assumed to comprise of a bound vortex sheet attached to the body and two separate vortex sheets originating at the edges. The complex conjugate velocity potential is expressed explicitly in terms of the bound vortex sheet strength and the edge circulations through a boundary integral representation. It is shown that Kelvin’s circulation theorem, along with the conditions of continuity of the normal velocity across the body and the boundedness of the velocity field, yields a coupled system of equations for the unknown bound vortex sheet strength and the edge circulations. A general numerical treatment is developed for the singular principal value integrals arising in the solution procedure. The model is validated against the results of Jones (J Fluid Mech 496, 405–441, 2003) for computations involving a rigid flat plate and is subsequently applied to the flapping foil experiments of Heathcote et al. (AIAA J, 42, 2196–2204, 2004) in order to predict the thrust coefficient. The utility of the model in simulating aquatic locomotion is also demonstrated, with vortex shedding suppressed at the leading edge of the swimming body.

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References

  1. Anderson C.R., Chen Y.-C. and Gibson J.S. (2000). Control and identification of vortex wakes. J. Dyn. Syst. Meas. Contr. 122: 298–305

    Article  Google Scholar 

  2. Birkhoff, G.: Helmholtz and Taylor instability. In: Proceedings of the Symposium on Applied Mathematics, pp. 55–76, American Mathematical Society, Providence, RI (1962)

  3. Boyd J.P. (2000). Singular integral equations. Dover, Mineola, USA

    Google Scholar 

  4. Brady M., Leonard A. and Pullin D.I. (1998). Regularized vortex sheet evolution in three dimensions. J. Comput. Phys. 146: 520–545

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Caflisch R. and Orellana O. (1989). Singular solutions and ill-posedness of the evolution of vortex sheets. SIAM J. Math. Appl. 20: 417–430

    MathSciNet  Google Scholar 

  6. Carrier J., Greengard L. and Rokhlin V. (1988). A fast adaptive multipole algorithmfor particle simulations. SIAM J. Sci. Stat. Comput. 9: 669–686

    Article  MATH  MathSciNet  Google Scholar 

  7. Chorin A.J. and Bernard P.S. (1973). Discretisation of a vortex sheet with an example of roll-up. J. Comput. Phys. 13: 785–796

    Article  Google Scholar 

  8. Clenshaw C.W. and Curtis A.R. (1960). A method for numerical integration on an automatic computer. Num. Math. 2: 197–205

    Article  MATH  MathSciNet  Google Scholar 

  9. Clements R.R. (1973). An inviscid model of two-dimensional vortex shedding. J. Fluid Mech. 57: 321–336

    Article  MATH  ADS  Google Scholar 

  10. Clements R.R. and Maull D.J. (1975). The representation of sheets of vorticity by discrete vortices. Prog. Aerospace Sci. 16: 129–146

    Article  ADS  Google Scholar 

  11. Cortelezzi L. and Leonard A. (1993). Point vortex model of the unsteady separated flow past a semi-infinite plate with transverse motion. Fluid Dyn. Res. 11: 263–295

    Article  ADS  Google Scholar 

  12. Cortelezzi L., Chen Y.-C. and Chang H.-L. (1997). Nonlinear feedback control of the wake past a plate: From a low-order model to a higher-order model. Phys. Fluids 9: 2009–2021

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Didden N. (1979). On the formation of vortex rings: rolling up and production of circulation. Z. Angew. Math. Phys. 30: 101–69

    Article  Google Scholar 

  14. Draghicescu C.I. and Draghicescu M. (1995). A fast algorithm for vortex blob interactions. J. Comput. Phys. 116: 69–78

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Eldredge J.D. (2007). Numerical simulation of the fluid dynamics of 2D rigid body motion with the vortex particle method. J. Comput. Phys. 221: 626–648

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Fish F.E. and Lauder G.V. (2006). Passive and active flow control by swimming fishes and mammals. Annu. Rev. Fluid Mech. 38: 193–224

    Article  ADS  MathSciNet  Google Scholar 

  17. Graham J.M.R. (1983). The lift on an aerofoil in starting flow. J. Fluid Mech. 133: 413–425

    Article  MATH  ADS  Google Scholar 

  18. Heathcote S., Martin D. and Gursul I. (2004). Flexible flapping airfoil propulsion at zero freestream velocity. AIAA J. 42: 2196–2204

    Article  ADS  Google Scholar 

  19. Heathcote, S., Gursul, I.: Flexible flapping airfoil propulsion at low Reynolds numbers. AIAA 2005-1405

  20. Holm D.D., Nitsche M. and Putkaradze V. (2006). Euler-alpha and vortex blob regularization of vortex filament and vortex sheet motion. J. Fluid Mech. 555: 149–176

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Jones M.A. and Shelley M.J. (2005). Falling cards. J. Fluid Mech. 540: 393–425

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. Jones M.A. (2003). The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496: 405–441

    Article  MATH  ADS  MathSciNet  Google Scholar 

  23. Krasny R. (1986). Desingularization of periodic vortex sheet Roll-up. J. Comput. Phys. 65: 292–313

    Article  MATH  ADS  Google Scholar 

  24. Kress R. (1999). Linear integral equations. Springer, New York

    MATH  Google Scholar 

  25. Krasny R. (1986). A study of singularity formation in a vortex sheet by the point-vortex approximation. J. Fluid Mech. 167: 65–93

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Krasny R. (1991). Vortex sheet computations: Roll-up, wakes, separation. Lect. Appl. Math. 28: 385–401

    MathSciNet  Google Scholar 

  27. Krasny R. (1987). Computation of vortex sheet roll-up in the Trefftz plane. J. Fluid Mech. 184: 123–155

    Article  ADS  Google Scholar 

  28. Keulegan G.H. and Carpenter L.H. (2003). Forces on cylinders and plates in an oscillating fluid. J. Res. Natl. Bureau Stand. 60: 423–440

    Google Scholar 

  29. Koumoutsakos P. (2005). Multiscale flow simulations using particles. Annu. Rev. Fluid. Mech. 37: 457–487

    Article  ADS  MathSciNet  Google Scholar 

  30. Krasny R. and Nitsche M. (2002). The onset of chaos in vortex sheet flow. J. Fluid Mech. 454: 47–69

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. Lindsay K. and Krasny R. (2001). A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow. J. Comput. Phys. 172: 879–907

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  33. Lighthill M.J. (1971). Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. B. 179: 125–138

    ADS  Google Scholar 

  34. Moore D.W. (1979). The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. Lond. A. 365: 105–119

    Article  MATH  ADS  Google Scholar 

  35. Majda A.J. and Bertozzi A.L. (2002). Vorticity and incompressible flow. University Press, Cambridge

    MATH  Google Scholar 

  36. Mushkhelishvilli, A.: Singular integral equations. Moscow (1946)

  37. Moore D.W. (1975). The rolling up of a semi-infinite vortex sheet. Proc. R. Soc. Lond. A. 345: 417–430

    MATH  ADS  Google Scholar 

  38. Nitsche M. (2001). Singularity formation in a cylindrical and a spherical vortex sheet. J. Comput. Phys. 173: 208–230

    Article  MATH  ADS  Google Scholar 

  39. Nitsche M. and Krasny R. (1994). A numerical study of vortex ring formation at the edge of a circular tube. J. Fluid Mech. 276: 139–161

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. Pullin D.I. and Wang Z.J. (2004). Unsteady forces on an accelerating plate and application to hovering insect flight. J. Fluid. Mech. 509: 1–21

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Pedley T.J. and Hill S.J. (1999). Large-amplitude undulatory fish swimming: fluid dynamics coupled to internal mechanics. J. Exp. Biol. 202: 3431–3438

    Google Scholar 

  42. Pullin D.I. (1978). The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88: 401–430

    Article  MATH  ADS  MathSciNet  Google Scholar 

  43. Rott N. (1956). Diffraction of a weak shock with vortex generation. J. Fluid Mech. 1: 111–128

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. Sarpkaya T. (1975). An inviscid model of two-dimensional vortex shedding for transient and asymptotically steady separated flow over an inclined plate. J. Fluid Mech. 68: 109–128

    Article  MATH  ADS  Google Scholar 

  45. Shelley M.J. (1992). A study of singularity formation in vortex sheet motion by a spectrally accurate vortex method. J. Fluid Mech. 244: 493–526

    Article  MATH  ADS  MathSciNet  Google Scholar 

  46. Sarpkaya T. (1989). Computational methods with vortices- The 1988 Freeman scholar lecture. Trans. ASME J. Fluids Engng. 111: 5–52

    Article  ADS  Google Scholar 

  47. Saffman P.G. (1992). Vortex Dynamics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  48. Tytell E.D. and Lauder G.V. (2004). The hydrodynamics of eel swimming I Wake structure. J. Exp. Biol. 207: 1825–1841

    Article  Google Scholar 

  49. Triantafyllou M.S., Triantafyllou G.S. and Yue D.K.P. (2000). Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32: 33–53

    Article  ADS  MathSciNet  Google Scholar 

  50. Toomey, J., Eldredge, J.D.: Numerical and experimental investigation of the role of flexibility in flapping wing flight. AIAA 2006-3211

  51. Wu T.Y.-T. (1961). Swimming of a waving plate. J. Fluid Mech. 10: 321–344

    Article  MATH  ADS  MathSciNet  Google Scholar 

  52. Wu T.Y.-T. (1971). Hydrodynamics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plaet at variable forward speeds in an inviscid fluid. J. Fluid Mech. 46: 337–355

    Article  ADS  Google Scholar 

  53. Wolfgang M.J. and Anderson J.M. (1999). Near-body flow dynamics in swimming fish. J. Exp. Biol. 202: 2303–2327

    Google Scholar 

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Authors and Affiliations

  1. Mechanical and Aerospace Engineering Department, University of California, 420 Westwood Plaza, Los Angeles, CA, 90095-1597, USA

    Ratnesh K. Shukla & Jeff. D. Eldredge

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  1. Ratnesh K. Shukla
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  2. Jeff. D. Eldredge
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Correspondence to Ratnesh K. Shukla.

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Communicated by T. Colonius.

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Shukla, R.K., Eldredge, J.D. An inviscid model for vortex shedding from a deforming body. Theor. Comput. Fluid Dyn. 21, 343–368 (2007). https://doi.org/10.1007/s00162-007-0053-2

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