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The motion of an axisymmetric body falling in a tube at moderate Reynolds numbers

  • Nicolas Brosse (a1) (a2) and Patricia Ern (a1) (a2)


This study concerns the rectilinear and periodic paths of an axisymmetric solid body (short-length cylinder and disk of diameter $d$ and thickness $h$ ) falling in a vertical tube of diameter $D$ . We investigated experimentally the influence of the confinement ratio ( $S= d/ D\lt 0. 8$ ) on the motion of the body, for different aspect ratios ( $\chi = d/ h= 3$ , $6$ and $10$ ), Reynolds numbers ( $80\lt Re\lt 320$ ) and a density ratio between the fluid and the body close to unity. For a given body, the Reynolds number based on its mean vertical velocity is observed to decrease when $S$ increases. The critical Reynolds number for the onset of the periodic motion decreases with $S$ in the case of thin bodies ( $\chi = 10$ ), whereas it appears unaffected by $S$ for thicker bodies ( $\chi = 3$ and $6$ ). The characteristics of the periodic motion are also strongly modified by the confinement ratio. A thick body ( $\chi = 3$ ) tends to go back to a rectilinear path when $S$ increases, while a thin body ( $\chi = 10$ ) displays oscillations of growing amplitude with $S$ until it touches the tube (at about $S= 0. 5$ ). For a given aspect ratio, however, the amplitudes of the oscillations follow a unique curve for all $S$ , which depends only on the relative distance of the Reynolds number to the threshold of path instability. In parallel, numerical simulations of the wake of a body held fixed in a uniform confined flow were carried out. The simulations allowed us to determine in this configuration the effect of the confinement ratio on the thresholds for wake instability (loss of axial symmetry at $R{e}_{c 1} $ and loss of stationarity at $R{e}_{c 2} $ ) and on the maximal velocity ${V}_{w} $ in the recirculating region of the stationary axisymmetric wake. The evolution with $\chi $ and $S$ of ${V}_{w} $ at $R{e}_{c 1} $ was used to define a Reynolds number $R{e}^{\ast} $ . Remarkably, for a freely moving body, $R{e}^{\ast} $ remains almost constant when $S$ varies, regardless of the nature of the path.


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Auguste, F. 2010 Instabilités de sillage générées derrière un corps solide cylindrique, fixe ou mobile dans un fluide visqueux. PhD thesis, Université Paul Sabatier, Toulouse, France:
Auguste, F., Fabre, D. & Magnaudet, J. 2010 Bifurcations in the wake of a thick circular disk. Theor. Comput. Fluid Dyn. 24, 305313.
Bhattacharyya, S. & Maiti, D. 2006 Vortex shedding suppression for laminar flow past a square cylinder near a plane wall: a two-dimensional analysis. Acta Mechanica 184, 1531.
Camarri, S. & Giannetti, F. 2007 On the inversion of the von Kármán street in the wake of a confined square cylinder. J. Fluid Mech. 574, 169178.
Camarri, S. & Giannetti, F. 2010 Effect of confinement on three-dimensional stability in the wake of a circular cylinder. J. Fluid Mech. 642, 477487.
Chen, J. H., Pritchard, W. G. & Tavener, S. J. 1995 Bifurcation for flow past a cylinder between parallel planes. J. Fluid Mech. 284, 2341.
Chhabra, R. P., Agarwal, S. & Chaudhary, K. 2003 A note on wall effect on the terminal falling velocity of a sphere in quiescent Newtonian media in cylindrical tubes. Powder Technol. 129 (1–3), 5358.
Cliffe, K. A., Spence, A. & Tavener, S. J. 2000 O(2)-symmetry breaking bifurcation: with application to the flow past a sphere in a pipe. Intl J. Numer. Meth. Fluids 32 (2), 175200.
Clift, R., Grace, J. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic Press.
Deloze, T. 2011 Couplage fluide–solide appliqué à l’étude de mouvement d’une sphère libre dans un tube vertical. PhD thesis, Université Louis Pasteur, Strasbourg, France.
Ern, P., Fernandes, P. C., Risso, F. & Magnaudet, J. 2007 Evolution of wake structure and wake-induced loads along the path of freely rising axisymmetric bodies. Phys. Fluids 19 (11), 113302.
Ern, P., Risso, F., Fabre, D. & Magnaudet, J. 2012 Wake-induced oscillatory paths of freely rising or falling bodies. Annu. Rev. Fluid Mech. 44, 97121.
Feng, J., Hu, H. H. & Joseph, D. D. 1994 Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation. J. Fluid Mech. 261, 95134.
Fernandes, P. C., Ern, P., Risso, F. & Magnaudet, J. 2005 On the zigzag dynamics of freely moving axisymmetric bodies. Phys. Fluids 17 (9), 098107.
Fernandes, P. C., Risso, F., Ern, P. & Magnaudet, J. 2007 Oscillatory motion and wake instability of freely rising axisymmetric bodies. J. Fluid Mech. 573, 479502.
Figueroa-Espinoza, B., Zenit, R. & Legendre, D. 2008 The effect of confinement on the motion of a single clean bubble. J. Fluid Mech. 616, 476480.
Kim, I., Elghobashi, S. & Sirignano, W. A. 1993 Three-dimensional flow over two spheres placed side by side. J. Fluid Mech. 246, 465488.
Legendre, D. & Magnaudet, J. 1998 The lift force on a spherical bubble in a viscous linear shear flow. J. Fluid Mech. 368, 81126.
Legendre, D., Magnaudet, J. & Mougin, G. 2003 Hydrodynamic interactions between two spherical bubbles rising side by side in a viscous liquid. J. Fluid Mech. 497, 133166.
Maheshwari, A., Chhabra, R. P. & Biswas, G. 2006 Effect of blockage on drag and heat transfer from a single sphere and an in-line array of three spheres. Powder Technol. 168 (2), 7483.
Sahin, M. & Owens, R. G. 2004 A numerical investigation of wall effects up to high blockage ratios on two-dimensional flow past a confined circular cylinder. Phys. Fluids 16 (5), 13051320.
Takemura, F. & Magnaudet, J. 2003 The transverse force on clean and contaminated bubbles rising near a vertical wall at moderate Reynolds number. J. Fluid Mech. 495, 235253.
Tavener, S. J. 1994 Stability of the O(2)-symmetric flow past a sphere in a pipe. Phys. Fluids 6 (12), 38843892.
Wham, R. M., Basaran, O. A. & Byers, C. H. 1996 Wall effects on flow past solid spheres at finite Reynolds number. Ind. Engng Chem. Res. 35 (3), 864874.
Yu, Z., Phan-Thien, N. & Tanner, R. I. 2004 Dynamic simulation of sphere motion in a vertical tube. J. Fluid Mech. 518, 6193.
Zovatto, L. & Pedrizzetti, G. 2001 Flow about a circular cylinder between parallel walls. J. Fluid Mech. 440, 125.
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The motion of an axisymmetric body falling in a tube at moderate Reynolds numbers

  • Nicolas Brosse (a1) (a2) and Patricia Ern (a1) (a2)


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