On laminar natural convection inside multi-layered spherical shells Yosef Gulberg, Yuri Feldman ⇑ Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel article info Article history: Received 11 January 2015 Received in revised form 2 July 2015 Accepted 7 July 2015 Available online 27 August 2015 Keywords: Laminar natural convection Multi-layered spherical shells Nu–Ra functional relation Immersed boundary method abstract Laminar natural convection flow inside multi-layered spherical shells with internal hot and external cold boundaries was investigated. Direct numerical simulations (DNS), which were performed by utilizing the immersed boundary method, addressed the fully 3D natural convection flow inside spherical shells with concentric, eccentric, equi-spaced and non-equi-spaced zero-thickness internal baffles. The insulation efficiency of the spherical shell was studied for up to four equi-spaced concentric internal layers. A unified functional dependency correlating modified Nu  and Ra  numbers was derived for spherical shells with up to four equi-spaced concentric internal layers. The effects of both vertical and horizontal eccentricity of the internal layers and of the width variation of concentric layers on the overall insulating performance of the spherical shell were investigated and quantified in terms of the Nu–Ra functionality. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Buoyancy-driven flow developing inside spherical annuli has been the subject of considerable research, both theoretical and experimental for the past fifty years. Typically, the buoyancy-driven flow between two isothermal concentric spheres (where each sphere is held at a different temperature) has been investigated as a function of the diameter ratio, / ¼ D i =D o , and the Rayleigh, Ra, and Prandtl, Pr, numbers. The pioneering experi- mental studies of Bishop et al. [1,2], which focused on visualization of the flow, indicated three distinct types of flow pattern – ‘‘crescent eddy’’, ‘‘kidney-shaped’’ and ‘‘falling vortices’’ – that depend on the diameter ratio, /, of the shells. Their experimental results were confirmed by the study of Mack and Hardee [3], who derived a low-Rayleigh-numbers analytical solution for the natural convection of air between two concentric spheres. More recently, the natural convection flow of working fluids other than air (namely, water and silicone oils) was experimentally addressed by Scanlan et al. [4] and visualized by Yin et al.[5]. The later group described naturally induced flow patterns and categorized the type of the flow for each fluid in terms of the inverse of the relative gap width and the Rayleigh number. Subsequent numerical studies on steady and transient natural convection flow inside spherical shells extended the state of the art to an even wider range of Pr (0:71 6 Pr 6 100) [6,7] and Ra (10 2 6 Ra 6 5  10 5 ) [7] numbers and to the analysis of vertically eccentric configurations [8]. The theoretical analysis of unsteady natural convection inside a differentially heated spherical annulus is a challenging problem, since different flow regimes can dominate locally in its different regions, taking the form of Rayleigh-Bènard convection at the top of the shell, of a differentially heated cavity at the near-equatorial region, and of a thermally stable flow regime at the bottom of the shell. Moreover, instabilities and transition scenarios are sensitive to the value of the Pr number and to the ratio of the internal to external diameter / [9,10]. For shells with an internal hot boundary and an external cold boundary, the flow patterns vary with the ratio /: Powe et al. [11] described a ‘‘mod- ified kidney shaped eddy’’ for wide shells (/ 6 0:5), an ‘‘interior e xpansion–contraction’’ for 0:5 6 /  0:65, a ‘‘three dimensional spiral’’ flow for 0:65 6 /  0:85, and a ‘‘falling vortices’’ pattern for narrow shells (0:85 6 /). Futterer et al. [12] reported that the flow inside shells of large and moderate widths (0:41 6 /  0:71) with a cold internal boundary and a hot external boundary exhibited an unsteady ‘‘dripping blob’’ phenomenon for Pr ¼1. Natural convection inside a spherical annulus comprises an essential heat transfer mechanism in various engineering design problems, such as in solar energy collectors, storage tanks, thermal energy storage (TES) systems and nuclear reactors. Another poten- tial application of spherical annuli is related to the design of the Titan Montgolfiere hot air balloon, which was recently chosen by NASA as the air-robot vehicle of choice for the exploration of Titan’s atmosphere. Given Titan’s low gravity (one-seventh that of Earth) and its cryogenic atmospheric temperatures (72–94 K), heat transfer by radiation can safely be neglected, and natural convection can be regarded as the only heat transfer mechanism for the stationary suspended balloon. Such a balloon, designed to http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.07.032 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. International Journal of Heat and Mass Transfer 91 (2015) 908–921 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt