From steady to unsteady horizontal gradient-driven convection at high Prandtl number K.E. Uguz a,⇑ , G. Labrosse b , R. Narayanan a,1 , F. Pigeonneau c a Department of Chemical Engineering, University of Florida, FL 32611 Gainesville, USA b Département de Physique, Université Paris-Sud 11, 91400 Orsay Cedex, France c Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 39 Quai Lucien Lefranc, BP 135 93303 Aubervilliers Cedex, France article info Article history: Received 20 June 2013 Received in revised form 28 November 2013 Accepted 2 December 2013 Available online xxxx Keywords: Natural convection Spectral Chebyshev Unsteady High Prandtl abstract A 2-D rectangular cavity subject to a horizontal temperature gradient imposed on its upper boundary induces a naturally convective flow, leading to two surprising results. First, at large Rayleigh numbers (Ra) two boundary layers appear driving the flow in such a way that heat is removed from the cold part of the fluid and released into the hot region of the fluid, generating a dynamical ‘‘heat pump’’. Second, a narrow region of pulsating oscillatory flow appears in the vicinity of the boundary layer which is at the vertical wall adjoining the cold edge of the upper boundary. These pulsations occur when Ra exceeds a critical value, Ra c , and are associated with large downward and upward vertical flows that appear simul- taneously near this vertical wall. The rest of the cavity is principally a ‘‘dead’’ cold region with weak flows. This region is the heat source of the heat pump. It occupies the main part of the cavity which progres- sively gets colder while the horizontal thermal boundary layer progressively gets hotter with increasing Ra. We explain the physics of these phenomena via numerical calculations that employ a spectral Cheby- shev method. It is shown, for the first time, that the curve of Ra c versus the aspect ratio is non-monotonic. It is also shown that the fundamental frequency of the oscillations, x, increases monotonically with the aspect ratio of the cavity that we have considered. Physical reasons are advanced to explain our observations. Ó 2013 Published by Elsevier Ltd. 1. Introduction This study aims to explain the reason for, and the nature of oscillations that occur in forced horizontal convection [1] in a 2-D cavity, when uneven heating is only applied along the upper horizontal boundary. These oscillations have never been observed earlier in the present configuration. Gramberg et al. [2, Chiu-Webster et al. [3] and more recently Pigeonneau and Flesselles [4] studied horizontal convection mainly in shallow cavities heated from above where the Prandtl number was taken to be large (greater than one hundred). Hughes and Griffiths [5] reviewed horizontal convection for different geophysical situations, principally applica- ble where temperature or heat fluxes are applied either on the bot- tom or on the top boundaries. They concluded that horizontal convection exhibits asymmetric convective flow where a vertical plume is observed. Simultaneously, a boundary layer is established adjacent to the horizontal boundary at which the uneven temper- ature profile is applied. In the configuration of interest wherein the upper boundary is differentially heated the flow is principally advective and the resulting flow dynamics generates narrow re- gions with substantially vertical flow and oscillations. Our work is motivated by industrial processes such as glass processing, how- ever we are not deterred from using a two dimensional configura- tion as much of the physics associated with advective heat transport can be extracted without excessive numerical expense. To this end we use a numerical method with a high degree of accu- racy for the spatial discretization, guaranteeing that the time behav- ior is correctly described. The method, a Chebyshev spectral technique, has been used in the past [6] to successfully decouple the pressure field from the velocity field yielding an equation for the pressure field akin to a diffusion equation which can be solved without imposing boundary conditions on the pressure field. The advantage of doing this is that the method of Successive Diagonaliza- tion [6,7] can then be employed so that multi-dimensional problems can be solved in terms of successive one-dimensional problems. 2. Problem description The problem is depicted in Fig. 1. It shows a two-dimensional rectangular cavity of height H along the vertical direction i z , and 0017-9310/$ - see front matter Ó 2013 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.12.002 ⇑ Corresponding author. Tel.: +1 3522221108. E-mail addresses: erdem.uguz@ufl.edu (K.E. Uguz), ranga@ufl.edu (R. Naraya- nan). 1 Principal corresponding author. International Journal of Heat and Mass Transfer 71 (2014) 469–474 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt