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The flow past a freely rotating sphere

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Abstract

We consider the flow past a sphere held at a fixed position in a uniform incoming flow but free to rotate around a transverse axis. A steady pitchfork bifurcation is reported to take place at a threshold \(Re^\mathrm{OS}=206\) leading to a state with zero torque but nonzero lift. Numerical simulations allow to characterize this state up to \(Re\approx 270\) and confirm that it substantially differs from the steady-state solution which exists in the wake of a fixed, non-rotating sphere beyond the threshold \(Re^\mathrm{SS}=212\). A weakly nonlinear analysis is carried out and is shown to successfully reproduce the results and to give substantial improvement over a previous analysis (Fabre et al. in J Fluid Mech 707:24–36, 2012). The connection between the present problem and that of a sphere in free fall following an oblique, steady (OS) path is also discussed.

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References

  1. Ern, P., Risso, F., Fabre, D., Magnaudet, J.: Wake-induced oscillatory paths of bodies freely rising or falling in fluids. Annu. Rev. Fluid Mech. 44, 97–121 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Assemat, P., Fabre, D., Magnaudet, J.: The onset of unsteadiness of two-dimensional bodies falling or rising freely in a viscous fluid: a linear study. J. Fluid Mech. 690, 173–202 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Auguste, F., Magnaudet, J., Fabre, D.: Falling styles of disks. J. Fluid Mech. 719, 388–405 (2013)

    Article  MATH  Google Scholar 

  4. Citro, V., Tchoufag, J., Fabre, D., Giannetti, F., Luchini, P.: Linear stability and weakly nonlinear analysis of the flow past rotating spheres. J. Fluid Mech (under review)

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

    Article  Google Scholar 

  6. Mordant, N., Pinton, J.F.: Velocity measurement of a settling sphere. Eur. Phys. J. B 18, 343–352 (2000)

    Article  Google Scholar 

  7. Horowitz, M., Williamson, C.H.K.: The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres. J. Fluid Mech. 651, 251–294 (2010)

    Article  MATH  Google Scholar 

  8. Obligado, M., Machicoane, N., Chouippe, A., Volk, R., Uhlmann, M., Bourgoin, M.: Path instability on a sphere towed at constant speed. J. Fluids Struct. 58, 99–108 (2015)

    Article  Google Scholar 

  9. Jenny, M., Dusek, J., Bouchet, G.: Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid. J. Fluid Mech. 508, 201–239 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Uhlmann, M., Dusek, J.: The motion of a single heavy sphere in ambient fluid: a benchmark for interface-resolved particulate flow simulations with significant relative velocities. Int. J. Multiph. Flow 59, 221–243 (2014)

    Article  Google Scholar 

  11. Fabre, D., Tchoufag, J., Magnaudet, J.: The steady oblique path of buoyancy-driven disks and spheres. J. Fluid Mech. 707, 24–36 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Tchoufag, J., Fabre, D., Magnaudet, J.: Weakly nonlinear model with exact coefficients for the fluttering and spiraling motion of buoyancy-driven bodies. Phys. Rev. Lett. 115, 114501 (2015)

    Article  Google Scholar 

  13. Citro, V., Giannetti, F., Luchini, P., Auteri, F.: Global stability and sensitivity analysis of boundary-layer flows past a hemispherical roughness element. Phys. Fluids 27, 084110 (2015)

    Article  Google Scholar 

  14. Tammisola, O., Giannetti, F., Citro, V., Juniper, M.: Second-order perturbation of global modes and implications for spanwise wavy actuation. J. Fluid Mech. 755, 314–335 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lashgari, I., Tammisola, O., Citro, V., Juniper, M., Brandt, L.: The planar X-junction flow: stability analysis and control. J. Fluid Mech. 753, 1–28 (2014)

    Article  Google Scholar 

  16. 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)

    Article  MATH  Google Scholar 

  17. Meliga, P., Chomaz, J.M., Sipp, D.: Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion. J. Fluid Mech. 633, 159–189 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hecht, F.: New development in FreeFem\(++\). J. Numer. Math. 20, 251–265 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Tchoufag, J., Fabre, D., Magnaudet, J.: Global linear stability analysis of the wake and path of buoyancy-driven disks and thin cylinders. J. Fluid Mech. 740, 278–311 (2014)

    Article  MathSciNet  Google Scholar 

  20. Fabre, D., Auguste, F., Magnaudet, J.: Bifurcations and symmetry breakings in the wake of axisymmetric bodies. Phys. Fluids 20(5), 051702 (2008)

    Article  MATH  Google Scholar 

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Correspondence to Vincenzo Citro.

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Communicated by Dr. Vassilios Theofilis.

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Fabre, D., Tchoufag, J., Citro, V. et al. The flow past a freely rotating sphere. Theor. Comput. Fluid Dyn. 31, 475–482 (2017). https://doi.org/10.1007/s00162-016-0405-x

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  • DOI: https://doi.org/10.1007/s00162-016-0405-x

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