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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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)
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)
Auguste, F., Magnaudet, J., Fabre, D.: Falling styles of disks. J. Fluid Mech. 719, 388–405 (2013)
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)
Johnson, T.A., Patel, V.C.: Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 19–70 (1999)
Mordant, N., Pinton, J.F.: Velocity measurement of a settling sphere. Eur. Phys. J. B 18, 343–352 (2000)
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)
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)
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)
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)
Fabre, D., Tchoufag, J., Magnaudet, J.: The steady oblique path of buoyancy-driven disks and spheres. J. Fluid Mech. 707, 24–36 (2012)
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)
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)
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)
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)
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)
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)
Hecht, F.: New development in FreeFem\(++\). J. Numer. Math. 20, 251–265 (2012)
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)
Fabre, D., Auguste, F., Magnaudet, J.: Bifurcations and symmetry breakings in the wake of axisymmetric bodies. Phys. Fluids 20(5), 051702 (2008)
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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