Direct numerical simulation of a differentially heated cavity of aspect ratio 4 with Rayleigh numbers up to 10 11 – Part I: Numerical methods and time-averaged flow F.X. Trias, A. Gorobets, M. Soria, A. Oliva * Centre Tecnològic de Transferència de Calor (CTTC), Technical University of Catalonia (UPC), ETSEIAT, c/Colom 11, 08222 Terrassa, Spain article info Article history: Received 4 February 2009 Accepted 25 August 2009 Available online 5 November 2009 Keywords: Direct numerical simulation Differentially heated cavity Natural convection Turbulence abstract A set of direct numerical simulations of a differentially heated cavity of aspect ratio 4 with adiabatic horizontal walls is presented. The five configurations selected here (Rayleigh numbers based on the cav- ity height Ra ¼ 6:4  10 8 ; 2  10 9 ; 10 10 ; 3  10 10 and 10 11 ; Pr ¼ 0:71) cover a relatively wide range of Ra from weak to fully developed turbulence. A short overview of the numerical methods and the method- ology used to verify the code and the simulations is presented. The time-averaged flow results are pre- sented and discussed in this first part. Significant changes are observed for the two highest Ra for which the transition point at the boundary layers clearly moves upstream. Such displacement increases the top and bottom regions of disorganisation shrinking the area in the cavity core where the flow is stratified. Consequently, thermal stratification values are significantly greater than unity (1.25 and 1.41, respectively). Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Natural convection in differentially heated cavities (DHC) has been the subject of numerous studies over the past decades. This configuration models many engineering applications such as venti- lation of rooms, cooling of electronics devices or air flow in build- ings. Simultaneously, this configuration has served as a prototype for the development of numerical algorithms. A summary of previ- ous direct numerical simulations (DNS) of air-filled ðPr ¼ 0:71Þ DHC relevant in the context of this paper is presented in the fol- lowing paragraphs. The coordinate system used here is: x 1 for the periodic direction and x 2 (horizontal) and x 3 (vertical) for the two wall-normal directions. Ra is the Rayleigh number based on the cavity height. A 3 ¼ L 3 =L 2 and A 1 ¼ L 1 =L 2 are the height and depth aspect ratios, respectively (see Fig. 1). Unless otherwise sta- ted, they used the Boussinesq approximation. The early numerical studies concentrated on configurations characterised by small Ra in the steady laminar regime. After the pioneering work of Vahl Davis and Jones [1], where the original benchmark formulation was establish for a set of square 2D cavi- ties with 10 3 6 Ra 6 10 6 , Hortmann et al. [2] used a multigrid method to solve the problem with finer meshes up to 640  640. Later, solutions for the full range of 2D steady-state solutions ðRa 6 10 8 Þ have been obtained using different methods by Le Quéré [3], Ravi et al. [4] and Wan et al. [5]. The 3D cubic cavity ðA 1 ¼ A 3 ¼ 1Þ, with adiabatic horizontal walls and solid vertical walls in the third direction is also a well-known configuration, but it has received comparatively little attention (see Fusegi et al. [6]; Tric et al. [7]; Wakashima and Saitoh [8]). For large height as- pect ratio cavities, within a certain range of Ra, a steady-state mul- ticellular flow is obtained (see Lartigue et al. [9]; Le Quéré [10]; Schweiger et al. [11]). Beyond a critical Ra, the 2D DHC flows become time-dependent (periodic, chaotic and eventually fully turbulent). Due to the pres- ence of high temperature areas at the bottom of the cavity, the con- figuration with perfectly conducting horizontal walls (linear distribution of temperature at the top and bottom walls) is more unstable than the configuration with adiabatic walls. The transition to unsteadiness of this configuration was studied by Winters [12], obtaining a critical number of Ra ¼ 2:109  10 6 , later confirmed by Henkes [13]. For the square cavity with adiabatic horizontal walls, Le Quéré and Behnia [14] identified the critical number as Ra ¼ 1:82  0:01  10 8 and studied the time-dependent chaotic flows up to Ra ¼ 10 10 . In the case of cavities also with adiabatic hor- izontal walls and height aspect ratio A 3 ¼ 4, Le Quéré [15] deter- mined that there is a Hopf bifurcation at Ra ¼ 1:03  10 8 and that a chaotic behaviour is first observed at Ra ¼ 2:3  10 8 . 2D chaotic flows have been studied by Farhangnia et al. [16], who carried out a direct simulation for Ra ¼ 6:4  10 10 and by Xin and Le Quéré [17], who studied the situations with Ra ¼ 6:4  10 8 ; 2  10 9 and 10 10 . A cavity with A 3 ¼ 8 and Ra ¼ 1:7408  10 8 (unsteady) has 0017-9310/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2009.10.026 * Corresponding author. Tel.: +34 93 739 81 92; fax: +34 93 739 81 01. E-mail address: cttc@cttc.upc.edu (A. Oliva). International Journal of Heat and Mass Transfer 53 (2010) 665–673 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt