Three-dimensional natural convection in an enclosure with a sphere at different vertical locations H.S. Yoon a , D.H. Yu b , M.Y. Ha b, * , Y.G. Park b a Advanced Ship Engineering Research Center, Pusan National University, San 30, Jangjeon Dong, Geumjeong Gu, Busan 609-735, Republic of Korea b School of Mechanical Engineering, Pusan National University, San 30, Jangjeon Dong, Geumjeong Gu, Busan 609-735, Republic of Korea article info Article history: Received 11 March 2009 Received in revised form 23 January 2010 Accepted 24 January 2010 Available online 27 March 2010 Keywords: Natural convection Sphere Cubic enclosure Nusselt number abstract Numerical calculations are carried out for the three-dimensional natural convection induced by a temperature difference between a cold outer cubic enclosure and a hot inner sphere. The immersed- boundary method (IBM) to model a sphere based on the ﬁnite volume method is used to study a three-dimensional natural convection for different Rayleigh numbers varying in the range of 10 3 –10 6 . This study investigates the effect of the inner sphere location on the heat transfer and ﬂuid ﬂow. The ﬂow and thermal ﬁelds eventually reach the steady state for all Rayleigh numbers regardless of the sphere location. For Rayleigh numbers of 10 5 and 10 6 , the variation of local Nusselt number of the sphere along the circumferential direction is large, showing the strong three dimensionality of the natural convection in the enclosure unlike to the cases of lower Rayleigh numbers of 10 3 and 10 4 . For the highest Rayleigh number, the local peaks of the Nusselt number on the top wall of the enclosure shows the sinusoidal dis- tribution along the circumferential direction. The ﬂow and thermal ﬁelds, and the local and surface-aver- aged Nusselt numbers on the sphere and the enclosure are highlighted in detail. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction The natural convection in an enclosure is relevant to many industrial and environmental applications such as heat exchangers, nuclear and chemical reactors, cooling of electronic equipments, and stratiﬁed atmospheric boundary layers. In engineering applica- tions, the geometries that arise, however, are more complicated than a simple enclosure ﬁlled with convective ﬂuid. A geometric conﬁguration of interest is with the presence of bodies embedded within the enclosure. This study investigates the natural convec- tion in the annuli between an inner sphere and an outer cubical enclosure. Many investigations have dealt with the inﬂuence of the pres- ence of a body with various thermal conditions on two-dimen- sional natural convection within a square enclosure with either horizontally [1–5] or vertically [6–11] imposed temperature differ- ence or heat ﬂux. Fusegi et al. [12] studied numerically the three-dimensional steady-state natural convection, for the Rayleigh number range of 10 3 6 Ra 6 10 6 , in a cubical enclosure, which is heated differen- tially at two vertical side walls. It was found that the convective activities are intensiﬁed when the Rayleigh number is increased. Pallarès et al. [13] performed numerically to characterize the natural convection, in the range of 3500 6 Ra 6 10,000, in a cubical cavity where buoyancy has been induced by imposing a moderate temperature difference between the heated bottom and the cooled top plates with perfectly adiabatic vertical walls. They showed that there are four different structures and the existence of four struc- tures makes the surface-averaged Nusselt number dependent on the pattern adopted by the ﬂow. Ozoe et al. [14] obtained two convective structures in the super- critical region 4000 6 Ra 6 8000 for the cubical cavity with adia- batic lateral walls. Both patterns may be described as single circulating rolls, one with axis parallel to two opposite vertical walls and the other with axis parallel to the diagonal of the hori- zontal walls. Hernández and Frederick [15] studied numerically the effect of the enclosure’s aspect ratio for the supercritical Rayleigh number (Ra =8  10 3 ). They found a new convective structure which exhibits a characteristic toroidal like form, with ﬂow descending near the four vertical edges and ascending at the central vertical axis of the cube and the overall Nusselt number changed continu- ously when the aspect ratio is increased. However, there is little information about three-dimensional natural convection when a heated sphere exists within the cooled cubical enclosure and especially, the location of inner heated sphere is changed along the vertical centerline of the vertical cubic enclosure. The purpose of this study is to examine how the position of inner heated sphere relative to the outer cubic enclosure affects the natural convection phenomena for different Rayleigh numbers when a hot inner sphere is located at different positions along the 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.03.013 * Corresponding author. Tel.: +82 51 510 2440; fax: +82 51 512 9835. E-mail address: myha@pusan.ac.kr (M.Y. Ha). International Journal of Heat and Mass Transfer 53 (2010) 3143–3155 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt