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Transition to turbulence in the wake of a fixed sphere in mixed convection

Published online by Cambridge University Press:  14 April 2009

MIROSLAV KOTOUČ
Affiliation:
Institut de Mécanique des Fluides et des Solides, Université de Strasbourg, France
GILLES BOUCHET
Affiliation:
Institut de Mécanique des Fluides et des Solides, Université de Strasbourg, France
JAN DUŠEK*
Affiliation:
Institut de Mécanique des Fluides et des Solides, Université de Strasbourg, France
*
Email address for correspondence: dusek@imfs.u-strasbg.fr

Abstract

The thermal effect on axisymmetry breaking and transition to turbulence in the wake of a fixed heated sphere is investigated in the mixed convection configurations commonly known as ‘assisting’ and ‘opposing’ flows in which the buoyancy tends, respectively, to accelerate and decelerate the flow. The study is carried out in the RiRe parameter plane (Ri being the mixed convection parameter – the Richardson number) for two values of Prandtl number – 0.72 (≈air and many gases) and 7 (≈water). We show that convection affects considerably the transition (as compared to that observed in the wake of an unheated sphere) even at moderate Richardson numbers. The latter are taken to be positive in assisting flow and negative in opposing one. In this notation, it can be said that convection shifts the primary-instability threshold to higher Reynolds numbers with increasing Richardson number. In assisting flow, the primary bifurcation is always regular, but at Ri ≥ 0.6 it appears in azimuthal subspaces associated with higher azimuthal wavenumbers m > 1. The transition scenario is characterized by a large variety of regimes explainable by nonlinear interactions between different azimuthal subspaces. On the side of higher (positive) Richardson numbers the axisymmetric flow is found stable up to Re = 1400 at Pr = 0.72 and Ri = 0.7. In opposing flow, the m = 1 subspace is always the most unstable, but the regular bifurcation gives way to a Hopf one at Ri < −0.1. Close to the junction of both bifurcations a similar variety of regimes precedes the transition to chaos as in assisting flow. On the side of negative Richardson numbers the primary (Hopf) bifurcation threshold is found as low as Re = 100 at Ri = −0.25 and at both investigated Prandtl numbers. After a primary periodic regime characterized by vortex shedding with a symmetry plane, the transition proceeds via a series of increasingly irregular helical regimes.

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Papers
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Ayyaswamy, P. S. 1999 Combustion Dynamics of Moving Droplets, vol. 1. Gulf Publishing.Google Scholar
Bar-Ziv, E., Zhao, B., Mograbi, E., Katoshevski, D. & Ziskind, G. 2002 Experimental validation of the Stokes law at nonisothermal conditions. Phys. Fluids 14, 20152018.CrossRefGoogle Scholar
Bhattacharyya, S. & Singh, A. 2008 Mixed convection from an isolated spherical particle. Intl J. Heat Mass Transfer 51, 10341048.CrossRefGoogle Scholar
Bouchet, G., Mebarek, M. & Dušek, J. 2006 Hydrodynamic forces acting on a rigid fixed sphere in early transitional regimes. Euro. J. Mech. 25, 321336.CrossRefGoogle Scholar
Chen, T. S. & Mucoglu, A. 1977 Analysis of mixed forced and free convection about a sphere. Intl J. Heat Mass Transfer 20, 867875.CrossRefGoogle Scholar
Chiang, C. H. & Sirignano, W. A. 1993 Interacting, convecting, vaporizing fuel droplets with variable properties. Intl J. Heat Mass Transfer 36, 875886.CrossRefGoogle Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic.Google Scholar
Danaila, I., Dušek, J. & Anselmet, F. 1998 Nonlinear dynamics at a Hopf bifurcation with axisymmetry breaking in a jet. Phys. Rev. E 57, 36953698.CrossRefGoogle Scholar
Dudek, D. R., Fletcher, T. H., Longwell, J. P. & Sarofim, A. F. 1988 Natural convection induced drag forces on spheres at low Grashof numbers: comparison of theory with experiment. IntL J. Heat Mass Transfer 31, 863873.CrossRefGoogle Scholar
Gan, H., Chang, J., Feng, J. J. & Hu, H. H. 2003 Direct numerical simulation of the sedimentation of solid particles with thermal convection. J. Fluid Mech. 481, 385411.CrossRefGoogle Scholar
Geoola, F. & Cornish, A. R. H. 1982 Numerical simulation of free convective heat transfer from a sphere. IntL J. Heat Mass Transfer 25, 16771687.CrossRefGoogle Scholar
Ghidersa, B. & Dušek, J. 2000 Breaking of axisymetry and onset of unsteadiness in the wake of a sphere. J. Fluid Mech. 423, 3369.CrossRefGoogle Scholar
Hieber, C. A. & Gebhart, B. 1969 Mixed convection from a sphere at small Reynolds and Grashof numbers. J. Fluid Mech. 38, 137159.CrossRefGoogle Scholar
Jenny, M. & Dušek, J. 2004 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, 215232.CrossRefGoogle Scholar
Jenny, M., Dušek, J. & Bouchet, G. 2004 Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid. J. Fluid Mech. 508, 201239.CrossRefGoogle Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.CrossRefGoogle Scholar
Jia, H. & Gogos, G. 1996 Laminar natural convection heat transfer from isothermal spheres. Intl J. Heat Mass Transfer 39, 16031615.CrossRefGoogle Scholar
Johnson, T. A. & Patel, V. C. 1999 Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 1970.CrossRefGoogle Scholar
Katoshevski, D., Zhao, B., Ziskind, G. & Bar-Ziv, E. 2001 Experimental study of the drag force acting on a heated particle. J. Aerosol Sci. 32, 7386.CrossRefGoogle Scholar
Klyachko, L. S. 1963 Heat transfer between a gas and a spherical surface with the combined action of free and forced convection. J. Heat Transfer 85, 355357.CrossRefGoogle Scholar
Kotouč, M. 2008 Transition à la turbulence du sillage d'une sphre fiwe ou libre en convection mixte. PhD thesis, Université Louis Pasteur, Strasbourg I.Google Scholar
Kotouč, M., Bouchet, G. & Dušek, J. 2008 Loss of axisymmetry in flow past a heated sphere - assisting flow. Intl J. Heat Mass Transfer 51, 26862700.CrossRefGoogle Scholar
Lecordier, J. C., Hamma, L. & Paranthoen, P. 1991 The control of vortex shedding behind heated circular cylinders at low Reynolds numbers. Exp. Fluids 10, 224229.CrossRefGoogle Scholar
McLeod, P., Riley, D. S. & Sparks, R. S. J. 1996 Melting of a sphere in hot fluid. J. Fluid Mech. 327, 393409.CrossRefGoogle Scholar
Michalke, A. 1984 Survey on jet instability theory. Prog. Aerospace Sci. 21, 159199.CrossRefGoogle Scholar
Mittal, R. 1999 A Fourier-Chebyshev spectral collocation method for simulating flow past spheres and spheroids. Intl J. Numer. Meth. Fluids 30, 921937.3.0.CO;2-3>CrossRefGoogle Scholar
Mograbi, E. & Bar-Ziv, E. 2005 a Dynamics of a spherical particle in mixed convection flow field. J. Aerosol Sci. 36, 387409.CrossRefGoogle Scholar
Mograbi, E. & Bar-Ziv, E. 2005 b On the mixed convection hydrodynamic force on a sphere. J. Aerosol Sci. 36, 11771181.CrossRefGoogle Scholar
Mograbi, E., Ziskind, G., Katoshevski, D. & Bar-Ziv, E. 2002 Experimental study of the forces associated with mixed convection from a heated sphere at small Reynolds and Grashof numbers. Part II. Assisting and opposing flows. Intl J. Heat Mass Transfer 45, 24232430.CrossRefGoogle Scholar
Mucoglu, A. & Chen, T. S. 1978 Mixed convection about a sphere with uniform surface heat flux. J. Heat Transfer 100, 542544.CrossRefGoogle Scholar
Najjar, F. M. & Balachandar, S. 1996 Low-frequency unsteadiness in the wake of a normal flat plate. J. Fluid Mech. 370, 101147.CrossRefGoogle Scholar
Natarajan, R. & Acrivos, A. 1993 The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323344.CrossRefGoogle Scholar
Nguyen, H. D., Paik, S. & Chung, J. N. 1993 Unsteady mixed convection heat transfer from a solid sphere: the conjugate problem. Intl J. Heat Mass Transfer 36, 44434453.CrossRefGoogle Scholar
Ormières, D. & Provansal, M. 1999 Transition to turbulence in the wake of a sphere. Phys. Rev. Lett. 83, 8083.CrossRefGoogle Scholar
Patera, A. 1984 A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54 (3), 468488.CrossRefGoogle Scholar
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos. Addison-Wesley.Google Scholar
Tang, L. & Johnson, A. T. 1990 Flow visualization of mixed convection about a sphere. Intl Comm. Heat Mass Transfer 17, 6777.CrossRefGoogle Scholar
Thompson, M., Hourigan, K. & Sheridan, J. 1996 Three-dimensional instabilities in the wake of a circular cylinder. Exp. Therm. Fluid Sci. 12, 190196.CrossRefGoogle Scholar
Wang, A.-B., Trávníček, Z. & Chia, K.-C. 2000 On the relationship of effective Reynolds number and Strouhal number for the laminar vortex shedding of a heated circular cylinder. Phys. Fluids 12, 14011410.CrossRefGoogle Scholar
Wong, K. L., Lee, S. C. & Chen, C. K. 1986 Finite element solution of laminar combined convection from a sphere. J. Heat Transfer 108, 860865.CrossRefGoogle Scholar
Wu, M.-H. & Wang, A.-B. 2007 On the transitional wake behind a heated circular cylinder. Phys. Fluids 19, 084102–19.CrossRefGoogle Scholar
Wu, S. J., Miau, J. J., Hu, C. C. & Chou, J. H. 2005 On low-frequency modulations and three-dimensionality in vortex shedding behind an normal plate. J. Fluid Mech. 526, 117146.CrossRefGoogle Scholar
Yuge, T. 1960 Experiments on heat transfer from spheres including combined natural and forced convection. J. Heat Transfer 82, 214220.CrossRefGoogle Scholar