Technical Note Two- and three-dimensional numerical simulations of natural convection in a cylindrical envelope with an internal concentric cylinder with slots Kun Zhang a,b , Mo Yang b,⇑ , Yuwen Zhang c a Key Laboratory of Railway Vehicle Thermal Engineering, College of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou 730010, China b College of Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China c Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA article info Article history: Received 19 June 2013 Received in revised form 4 November 2013 Accepted 6 November 2013 Available online 4 December 2013 Keywords: Two dimensional simulation Three-dimensional simulation Non-linear characteristic abstract Two-dimensional and three-dimensional numerical simulations were carried out to simulate the natural convection in a cylindrical envelope with an internal concentric cylinder with slots. For the case of steady flow, the numerical solutions in 2D and 3D simulations are in good agreement with the experimental results. When the convection in experiment becomes unsteady convection at the larger slot degree, the oscillated solutions in 3D simulation differ essentially from steady solutions in 2D simulation. The critical Rayleigh numbers from steady to unsteady flow in the 3D simulations are lower than those in the 2D simulations. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction The enclosed isolated-phase busbar used for transmitting large electric current consists of two horizontal concentric metal cylin- ders. The heat generated in the busbar from the Joule heating is transferred to the outer envelope by radiation and natural convec- tion. The unsteady natural convections are well studied in horizon- tal concentric cylindrical annuli. Cheddadi et al. [1] studied the two-dimensional natural convection bifurcation in a horizontal annulus, and showed that flow pattern was not unique and de- pended on the initial conditions at high Rayleigh number. Liu et al. [2] investigated the critical Rayleigh number dictates the transition from steady to unsteady. Yoo [3] investigated the bifur- cation sequences to the chaos for the natural convection in hori- zontal concentric annuli using two-dimensional simulation. Adachi and Imai [4] investigated three-dimensional linear stability of natural convective flow between concentric horizontal cylin- ders. Ridouane et al. [5] numerically studied the multiple 3-D flow regimes including conduction, steady convection, and unsteady convection in a loop as the Rayleigh number was increased. Comparatively, little works have been reported on unsteady natural convection in more complex domain, such as in a cylindri- cal envelope with an internal concentric cylinder with slots. Kuleek [6] assumed that the heat transfer enhancement of a cylindrical envelope with in internal slotted cylinder was expected to be about 30–40% higher than that of the concentric cylindrical annuli. Wang et al. [7] experimentally investigated natural convection in a cylin- drical envelope with an internal slotted cylinder and found that the convective heat transfer coefficient with slots could be enhanced by as much as 50%. Zhang et al. [8–9] investigated nonlinear char- acteristics of natural convection in two-dimensional numerical simulation and experiment. The results indicated that the oscilla- tory flow undergoes several bifurcations and ultimately evolves to a chaotic flow. However, their results covered only the two-dimensional cases. It is not known whether the results in two-dimensional simulations are able to predict the non-linear behavior obtained in real three-dimensional domain. The objective of this paper is to study natural convection heat transfer in a cylindrical envelope with an internal concentric cylin- der with slots using two-dimensional and three-dimensional ap- proaches. Two- and three-dimensional numerical simulations of the natural convection at the same parameters will be performed in order to study the differences between them. The numerical solutions from both approaches will be compared in detail for stea- dy and unsteady flow and heat transfer. 2. Problem formulation A schematic diagram of the physical model under consideration is shown in Fig. 1. The inner and outer cylinders are kept at uni- form but different temperatures T i and T o , respectively, with T i > T o . 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.11.015 ⇑ Corresponding author. E-mail address: yangm66@gmail.com (M. Yang). International Journal of Heat and Mass Transfer 70 (2014) 434–438 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt