zbMATH — the first resource for mathematics

Path oscillations and enhanced drag of light rising spheres. (English) Zbl 1419.76357
Summary: The dynamics of light spheres rising freely under buoyancy in a large expanse of viscous fluid at rest at infinity is investigated numerically. For this purpose, the computational approach developed by G. Mougin and the second author [Int. J. Multiphase Flow 28, No. 11, 1837–1851 (2002; Zbl 1137.76687)] is improved to account for the instantaneous viscous loads induced by the translational and rotational sphere accelerations, which play a crucial role in the dynamics of very light spheres. A comprehensive map of the rise regimes encountered up to Reynolds numbers (based on the sphere diameter and mean rise velocity) of the order of \(10^{3}\) is set up by varying independently the body-to-fluid density ratio and the relative magnitude of inertial and viscous effects in approximately 250 distinct combinations. These computations confirm or reveal the presence of several distinct periodic regions on the route to chaos, most of which only exist within a finite range of the sphere relative density and Reynolds number. The wake structure is analysed in these various regimes, evidencing the existence of markedly different shedding modes according to the style of path. The variation of the drag force with the flow parameters is also examined, revealing that only one of the styles of path specific to very light spheres yields a non-standard drag behaviour, with drag coefficients significantly larger than those measured on a fixed sphere under equivalent conditions. The outcomes of this investigation are compared with available experimental and numerical results, evidencing points of consensus and disagreement.

76F65 Direct numerical and large eddy simulation of turbulence
76F06 Transition to turbulence
Full Text: DOI
[1] Auguste, F.2010 Instabilités de sillage et trajectoires d’un corps solide cylindrique immergé dans un fluide visqueux. PhD thesis, Université Paul Sabatier, Toulouse, France (available online at http://thesesups.ups-tlse.fr/1186/).
[2] Auguste, F.; Magnaudet, J.; Fabre, D., Falling styles of disks, J. Fluid Mech., 719, 388-405, (2013) · Zbl 1284.76130
[3] Bergé, P.; Pomeau, Y.; Vidal, C., Order Within Chaos, (1984), Wiley
[4] Calmet, I.; Magnaudet, J., Large-eddy simulation of high-Schmidt number mass transfer in a turbulent channel flow, Phys. Fluids, 9, 438-455, (1997)
[5] Cano-Lozano, J. C.; Martínez-Bazán, C.; Magnaudet, J.; Tchoufag, J., Paths and wakes of deformable nearly spheroidal rising bubbles close to the transition to path instability, Phys. Rev. Fluids, 1, (2016)
[6] Chrust, M.; Bouchet, G.; Dušek, J., Numerical simulation of the dynamics of freely falling discs, Phys. Fluids, 25, (2013)
[7] Chrust, M.; Bouchet, G.; Dušek, J., Effect of solid body degrees of freedom on the path instabilities of freely falling or rising flat cylinders, J. Fluids Struct., 47, 55-70, (2014)
[8] Clift, R.; Grace, J. R.; Weber, M. E., Bubbles, Drops and Particles, (1978), Academic Press
[9] Eckert, M., The Dawn of Fluid Dynamics, (2006), Wiley
[10] Fabre, D.; Tchoufag, J.; Magnaudet, J., The steady oblique path of buoyancy-driven disks and spheres, J. Fluid Mech., 707, 24-36, (2012) · Zbl 1275.76064
[11] Feuillebois, F.; Lasek, A., Rotational historic term in nonstationary Stokes flow, Q. J. Mech. Appl. Maths, 31, 435-443, (1978) · Zbl 0391.76029
[12] Gatignol, R., The Faxen formulas for a rigid particle in an unsteady non-uniform Stokes-flow, J. Méc. Théor. Appl., 2, 143-160, (1983) · Zbl 0544.76032
[13] Ghidersa, B.; Dušek, J., Breaking of axisymetry and onset of unsteadiness in the wake of a sphere, J. Fluid Mech., 423, 33-69, (2000) · Zbl 0977.76028
[14] Horowitz, M.; Williamson, C. H. K., Critical mass ratio and a new periodic four-ring vortex wake mode for freely rising and falling spheres, Phys. Fluids, 20, (2008) · Zbl 1182.76328
[15] 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) · Zbl 1189.76152
[16] Jenny, M.; Dušek, J., Efficient numerical method for the direct numerical simulation of the flow past a single light moving spherical body in transitional regimes, J. Comput. Phys., 194, 215-232, (2004) · Zbl 1136.76365
[17] Jenny, M.; Dušek, J.; Bouchet, G., Instabilities and transition of a sphere falling or ascending in a Newtonian fluid, J. Fluid Mech., 508, 201-239, (2004) · Zbl 1065.76068
[18] Jeong, J.; Hussain, F., On the identification of a vortex, J. Fluid Mech., 285, 69-94, (1995) · Zbl 0847.76007
[19] Johnson, T. A.; Patel, V. C., Flow past a sphere up to Reynolds number of 300, J. Fluid Mech., 378, 10-70, (1999)
[20] Karamanev, D. G., The study of the rise of buoyant spheres in gas reveals the universal behaviour of free rising rigid spheres in fluid in general, Intl J. Multiphase Flow, 27, 1479-1486, (2001) · Zbl 1388.76408
[21] Karamanev, D. G.; Chavarie, C.; Mayer, R. C., Dynamics of the free rise of a light solid sphere in liquid, AIChE J., 42, 1789-1792, (1996)
[22] Karamanev, D. G.; Nikolov, L. N., Free rising spheres do not obey Newton’s law for free settling, AIChE J., 33, 1843-1846, (1992)
[23] Magnaudet, J.; Mougin, G., Wake instability of a fixed spheroidal bubble, J. Fluid Mech., 572, 311-337, (2007) · Zbl 1188.76203
[24] Magnaudet, J.; Rivero, M.; Fabre, J., Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow, J. Fluid Mech., 284, 97-135, (1995) · Zbl 0848.76063
[25] Manneville, P.; Pomeau, Y., Intermittency and the Lorentz model, Phys. Lett. A, 75, 1-2, (1979) · Zbl 0985.37503
[26] Mittal, R., Planar symmetry in the unsteady wake of a sphere, AIAA J., 37, 388-390, (1999)
[27] Mougin, G.; Magnaudet, J., The generalized Kirchhoff equations and their application to the interaction between a rigid body and an arbitrary time-dependent viscous flow, Intl J. Multiphase Flow, 28, 1837-1851, (2002) · Zbl 1137.76687
[28] Mougin, G.; Magnaudet, J., Path instability of a rising bubble, Phys. Rev. Lett., 88, (2002) · Zbl 1137.76687
[29] Natarajan, R.; Acrivos, A., The instability of the steady flow past spheres and disks, J. Fluid Mech., 254, 323-344, (1993) · Zbl 0780.76027
[30] Orr, T. S.; Domaradzki, J. A.; Spedding, G. R.; Constantinescu, G. S., Numerical simulations of the near wake of a sphere moving in a steady, horizontal motion through a linearly stratified fluid at Re = 1000, Phys. Fluids, 27, (2015)
[31] Ostmann, S.; Chaves, H.; Brüker, C., Path instabilities of light particles rising in a liquid with background rotation, J. Fluids Struct., 70, 403-416, (2017)
[32] Pomeau, Y.; Manneville, P., Intermittent transition to turbulence in dissipative dynamical systems, Commun. Math. Phys., 74, 189-197, (1980)
[33] Poon, E. K. W.; Ooi, A. S. H.; Giacobello, M.; Iaccarino, G.; Chung, D., Flow past a transversely rotating sphere at Reynolds numbers above the laminar regime, J. Fluid Mech., 759, 751-781, (2014)
[34] Rivero, M.; Magnaudet, J.; Fabre, J., New results on the forces exerted on a spherical body by an accelerated flow, C. R. Acad. Sci. Paris II-B, 312, 1499-1506, (1991) · Zbl 0724.76089
[35] Tomboulides, A. G.; Orszag, S. A., Numerical investigation of transitional and weak turbulent flow past a sphere, J. Fluid Mech., 416, 45-73, (2000) · Zbl 1156.76419
[36] Turton, R.; Levenspiel, O., A short note on the drag correlation for spheres, Powder Technol., 47, 83-86, (1986)
[37] Uhlmann, M.; Dušek, J., The motion of a single heavy sphere in ambient fluid: a benchmark for interface-resolved particulate flow simulations with significant relative velocities, Intl J. Multiphase Flow, 59, 221-243, (2014)
[38] Veldhuis, C. H. J.; Biesheuvel, A., An experimental study of the regimes of motion of spheres falling or ascending freely in Newtonian fluid, Intl J. Multiphase Flow, 33, 1074-1087, (2007)
[39] Veldhuis, C. H. J.; Biesheuvel, A.; Lohse, D., Freely rising light solid spheres, Intl J. Multiphase Flow, 35, 312-322, (2009)
[40] Zhang, W.; Stone, H. A., Oscillatory motions of circular disks and nearly spherical particles in viscous flows, J. Fluid Mech., 367, 329-358, (1998) · Zbl 0912.76015
[41] Zhou, W.; Dušek, J., Chaotic states and order in the chaos of the paths of freely falling and ascending spheres, Intl J. Multiphase Flow, 75, 205-223, (2015)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.