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Three-dimensional numerical simulation of the wake flow of an afterbody at subsonic speeds


We numerically investigate the wake flow of an afterbody at low Reynolds number in the incompressible and compressible regimes. We found that, with increasing Reynolds number, the initially stable and axisymmetric base flow undergoes a first stationary bifurcation which breaks the axisymmetry and develops two parallel steady counter-rotating vortices. The critical Reynolds number (Re cs) for the loss of the flow axisymmetry reported here is in excellent agreement with previous axisymmetric BiGlobal linear stability (BiGLS) results. As the Reynolds number increases above a second threshold, Re co, we report a second instability defined as a three-dimensional peristaltic oscillation which modulates the vortices, similar to the sphere wake, sharing many points in common with long-wavelength symmetric Crow instability. Both the critical Reynolds number for the onset of oscillation, Re co, and the Strouhal number of the time-periodic limit cycle, Stsat, are substantially shifted with respect to previous axisymmetric BiGLS predictions neglecting the first bifurcation. For slightly larger Reynolds numbers, the wake oscillations are stronger and vortices are shed close to the afterbody base. In the compressible regime, no fundamental changes are observed in the bifurcation process. It is shown that the steady state planar-symmetric solution is almost equal to the incompressible case and that the break of planar symmetry in the vortex shedding regime is retarded due to compressibility effects. Finally, we report the developments of a low frequency which depends on the afterbody aspect ratio, as well as on the Reynolds and on the Mach number, prior to the loss of the planar symmetry of the wake.

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  1. 1

    Barkley D.: Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75(5), 750–756 (2006)

  2. 2

    Bohorquez P., Sanmiguel-Rojas E., Sevilla A., Jiménez-González J., Martínez-Bazán C.: Stability and dynamics of the laminar wake past a slender blunt-based axisymmetric body. J. Fluid Mech. 676, 110–144 (2011)

  3. 3

    Bouchet G., Mebarek M., Dušek J.: Hydrodynamic forces acting on a rigid fixed sphere in early transition regimes. Eur. J. Mech. B/Fluids 25, 321–336 (2006)

  4. 4

    Brès G.A., Colonius T.: Three-dimensional instabilities in compressible flow over open cavities. J. Fluid. Mech. 599, 309–339 (2008)

  5. 5

    Brion V., Sipp D., Jacquin L.: Optimal amplification of the Crow instability. Phys. Fluids 19, 111703 (2007)

  6. 6

    Colonius T.: Modeling artificial boundary conditions for compressible flow. Annu. Rev. Fluid Mech. 36, 315–345 (2004)

  7. 7

    Crow S.: Stability theory for a pair of trailing vortices. AIAA J. 8(12), 2173–2179 (1970)

  8. 8

    Ferziger J.H., Perić M.: Computational Methods for Fluid Dynamics. Springer, Berlin (2002)

  9. 9

    Ghidersa B., Dušek J.: Breaking of axisymmetry and onset of unsteadiness in the wake of a sphere. J. Fluid Mech. 423, 33–69 (2000)

  10. 10

    Greenshields C.J., Weller H.G., Gasparini L., Reese J.M.: Implementation of semi-discrete, non-staggered central schemes in a colocated, polyhedral, finite volume framework, for high-speed viscous flows. Int. J. Numer. Meth. Fluids 63(1), 1–21 (2010)

  11. 11

    Gumowski K., Miedzik J., Goujon-Durand S., Jenffer P., Wesfreid J.E.: Transition to a time-dependent state of fluid flow in the wake of a sphere. Phys. Rev. E 77(5), 055308 (2008)

  12. 12

    Johnson T.A., Patel V.C.: Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 19–70 (1999)

  13. 13

    Leontini J.S., Thompson M.C., Hourigan K.: A numerical study of global frequency selection in the time-mean wake of a circular cylinder. J. Fluid. Mech. 645, 435–446 (2010)

  14. 14

    Leweke T., Williamson C.: Experiments on long-wavelength instability and reconnection of a vortex pair. Phys. Fluids 23, 024101 (2011)

  15. 15

    Meliga, P.: A theoretical approach for the onset and control of unsteadiness in compressible afterbody flows. Ph.D. thesis, École Polytechnique (2008)

  16. 16

    Meliga P., Sipp D., Chomaz J.M.: Effect of compressibility on the global stability of axisymmetric wake flows. J. Fluid Mech. 660, 499–526 (2010)

  17. 17

    Meliga P., Sipp D., Chomaz J.M.: Open-loop control of compressible afterbody flows using adjoint methods. Phys. Fluids 22, 054109 (2010)

  18. 18

    Moore D., Saffman P.: The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425 (1975)

  19. 19

    Morzyński, M., Thiele, F.: Finite element method for global stability analysis of 3D flows. In: 4th AIAA Flow Control Conference, art. no. 2008-3865 (2008)

  20. 20

    Natarajan R., Acrivos A.: The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323–344 (1993)

  21. 21

    Ormières D., Provansal M.: Transition to turbulence in the wake of a sphere. Phys. Rev. Lett. 83(1), 80–83 (1999)

  22. 22

    Pier B.: Local and global instabilities in the wake of a sphere. J. Fluid Mech. 603, 39–61 (2008)

  23. 23

    Rodriguez I., Borrell R., Lehmkuhl O., Perez-Segarra C.D., Oliva A.: Direct numerical simulation of the flow over a sphere at Re = 3700. J. Fluid Mech. 679, 263–287 (2011)

  24. 24

    Roy C., Leweke T., Thompson M., Hourigan K.: Experiments on the elliptic instability in vortex pairs with axial core flow. J. Fluid Mech. 677, 383–416 (2011)

  25. 25

    Schouveiler L., Provansal M.: Self-sustained oscillations in the wake of a sphere. Phys. Fluids 14(11), 3846–3854 (2002)

  26. 26

    Sevilla A., Martínez-Bazán C.: Vortex shedding in high Reynolds number axisymmetric bluff-body wakes: Local linear instability and global bleed control. Phys. Fluids 16, 3460–3469 (2004)

  27. 27

    Sipp D., Lebedev A.: Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid. Mech. 593, 333–358 (2007)

  28. 28

    Stewart B.E., Thompson M.C., Leweke T., Hourigan K.: Numerical and experimental studies of the rolling sphere wake. J. Fluid Mech. 643, 137–162 (2010)

  29. 29

    Storti M.A., Nigro N.M., Paz R.R., Dalcín L.D.: Dynamic boundary conditions in computational fluid dynamics. Comput. Methods Appl. Mech. Eng. 197, 1219–1232 (2008)

  30. 30

    Tezuka A., Suzuki K.: Three-dimensional global linear stability analysis of flow around a spheroid. AIAA J. 44(8), 1697–1708 (2006)

  31. 31

    Theofilis V.: Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352 (2011)

  32. 32

    Theofilis V., Colonius T.: Special issue on global flow instability and control. Theor. Comput. Fluid Dyn. 25(1–4), 1–6 (2011)

  33. 33

    Thompson M., Leweke T., Provansal M.: Kinematics and dynamics of sphere wake transition. J. Fluids Struct. 15, 575–585 (2001)

  34. 34

    Tomboulides A.G., Orzsag S.A.: Numerical investigation of transitional and weak turbulent flow past a sphere. J. Fluid Mech. 416, 45–73 (2000)

  35. 35

    Tsai C.Y., Widnall S.E.: Stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73, 721–733 (1976)

  36. 36

    Viswanath P.R.: Flow management techniques for base and afterbody drag reduction. Prog. Aerospace Sci. 32, 79–129 (1996)

  37. 37

    Xin X., Liu Y., Ma D., Sun D.: Three-dimensional global linear stability analysis of compressible flow around a sphere. Chin. J. Comput. Phys. 28(1), 10–18 (2011)

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Correspondence to Patricio Bohorquez.

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Communicated by Colonius.

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Bohorquez, P., Parras, L. Three-dimensional numerical simulation of the wake flow of an afterbody at subsonic speeds. Theor. Comput. Fluid Dyn. 27, 201–218 (2013).

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  • Bluff body
  • Hydrodynamic stability
  • Laminar wake
  • Vortex instability
  • OpenFOAM