D. Fabre

GIANNETTI, FLAVIO

J. Tchoufag

LUCHINI, Paolo

English

Archivio della Ricerca - Università di Salerno

The steady oblique path of buoyancy-driven rotating spheresFlavio Giannetti∗, David Fabre †, Jöel Tchoufag†, and Paolo Luchini∗The dynamics of freely falling or rising bodies under the effect of boyancy is anactive field of research1. A large variety of paths has been reported in experiments andnumerical simulations, including fluttering, tumbling, spiral and chaotic motions. Forlight spheres, in particular2, a number of regimes characterized by weak deviationswith respect to the vertical have been noticed, the simplest one being a steady oblique(SO) trajectory, with the sphere rotating around an axis perpendicular to the path.In this work, we investigate this situation using two different numerical approaches.The first method3 is a weakly nonlinear asymptotic expansion in terms of the rotationrate ω. The approach makes use of a modal expansion in the azimuthal direction toreduce the three-dimensional problem into a set of two-dimensional equations, whichare solved at each order by the finite-element FreeFem++ software, in the line of ref.4.An expression for the lift and torque coefficients as functions of Reynolds number Reand rotation rate ω is derived (see fig (a) for an example). By imposing the torqueto vanish, the approach allows to describe the steady oblique (SO) path, and predictsthat the latter emerges from a supercritical bifurcation at a critical Reynolds numberReSO = 206.075, regardless of the body-to-fluid density ratio (fig (b)).The second numerical approach is a fully three-dimensional numerical simulationobtained with a combined immersed-boundary multigrid code. Steady solutions forboth the flow around the sphere with fixed ω and the SO path with zero torque areobtained using Newton iteration. Results compare well with the weakly nonlinearprediction in the vicinity of the critical Reynolds number and allow to investigate theSO path in the fully nonlinear range. The approach also allows to investigate thestability of the zero-torque SO solutions.-0.0003-0.0002-0.0001 0 0.0001 0.0002 0.0003 180  190  200  210  220  230Num. (3d)Asymp.ReTorque(a) Torque Coefficient for imposed rotation rate(ω = 10−3): asymptotic and 3d numerical data 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 200  205  210  215  220  225  230Zero TorqueReω(b) Bifurcation diagram ω − Re for zero-torque so-lutions (3d numerical simulations)∗DIIN, Universitá degli studi di Salerno, via Ponte Don Melillo 1, 84084 Fisciano (SA), Italy†Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, 2 Allée duProfesseur Camille Soula, 31400, Toulouse, France1Ern, P. et al. , Annu. Rev. Fluid Mech. 44, 97 (2012)2Jenny M. et al., J. Fluid Mech. 508, 201 (2004); Veldhuis C.H.J. & Biesheuvel, Int. J. Multi-phase Flow 33, 1074 (2007); Horowitz M. & Williamson C.H.K., J. Fluid Mech. 651, 251 (2010)3Fabre, Tchoufag and Magnaudet, J. Fluid Mech. submitted.4Meliga, P. et al., J. Fluid Mech. 633, 159 (2009)

The steady oblique path of buoyancy-driven rotating spheres

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The steady oblique path of buoyancy-driven rotating spheres

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