A complete heatline analysis on mixed convection within a square cavity: Effects of thermal boundary conditions via thermal aspect ratio D. Ramakrishna a , Tanmay Basak b , S. Roy a , I. Pop c, * a Department of Mathematics, Indian Institute of Technology Madras, Chennai - 600036, India b Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai - 600036, India c Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania article info Article history: Received 23 May 2011 Received in revised form 12 January 2012 Accepted 13 January 2012 Available online 6 March 2012 Keywords: Streamlines Heatlines Mixed convection Square cavity Thermal aspect ratio Penalty finite element method abstract In this article, numerical investigation is carried out for mixed convection heat transfer within square cavities for various thermal boundary conditions on bottom and side walls based on thermal aspect ratio (A). A penalty finite element analysis with bi-quadratic elements has been used to investigate the results in terms of isotherms, streamlines, heatlines and average Nusselt numbers for a wide range of param- eters (1  Re100, 0.015  Pr10, 10 3  Gr10 5 ). A detailed analysis of flow pattern shows that natural convection or forced convection depends on both parameters: Ri (Ri ¼ Gr/Re 2 ) and Pe (Pe ¼ Re$Pr). Results indicate that, at low Pr (Pr ¼ 0.015) with low Gr (Gr ¼ 10 3 ), isotherms are decoupled with flow profile and conduction dominant heat transfer is observed irrespective of Re, due to low Peclet number. At Gr ¼ 10 3 , lid-driven force dominates and the non-symmetric flow distribution occurs irrespective of Re (1,10 and 100), Pr (0.015, 0.7 and 10) and thermal aspect ratio (0.1, 0.5 and 0.9). At Gr ¼ 10 5 with Re ¼ 1, natural convection dominates the flow irrespective of Pr and A. Considerably smaller dominance of lid- driven force is observed over buoyancy force at Gr ¼ 10 5 with Re ¼ 10 irrespective of Pr for A ¼ 0.1 and 0.5, whereas strong effect of lid-driven force is found at Gr ¼ 10 5 with Re ¼ 100 irrespective of Pr and A. Multiple circulations are found in streamlines and heatlines especially for A ¼ 0.5 and 0.9 at high Rey- nolds number (Re ¼ 100) with Pr ¼ 10 and Gr ¼ 10 5 . It is found that, streamlines and heatlines circulation cells follow qualitatively similar pattern for higher Pr (Pr  0.7) at Gr ¼ 10 5 irrespective of Re. Thermal gradient is found to be high at the center of the bottom wall for A ¼ 0.1 due to highly dense heatlines at that zone whereas that is low for A ¼ 0.9 irrespective of Re, Pr and Gr. It is also found that, as thermal aspect ratio increases, the average Nusselt number decreases for Pr ¼ 0.015 and Pr ¼ 0.7 irrespective of Re. Finally, it is concluded that overall heat transfer rates are higher for A ¼ 0.1 as compared to other thermal aspect ratios (A ¼ 0.5, A ¼ 0.9) irrespective of Pr (0.015  Pr10), Re (1  Re100) and Gr (10 3  Gr10 5 ). Ó 2012 Elsevier Masson SAS. All rights reserved. 1. Introduction The study of mixed convection heat transfer within closed cavities has been regarded as one of the most important research topics due to its wide spectrum of scientific and engineering applications such as chemical vapour deposition processes [1], electronic cooling devices [2e4], polymer processing [5], green- house covering [6], energy extraction [7], manufacturing processes [8], material processing [9], wedge flow processes [10], stretching sheet extraction processes [11] etc. In mixed convection flows, the forced convection and free convection effects are of comparable magnitudes. The non-dimensional parameters which influence the heat transfer during mixed convection are Grashof number (Gr), Reynolds number (Re) and Prandtl number (Pr) where Gr and Re represent the strength of the natural convection and forced convection, respectively. Another dimensionless parameter, Richardson number Ri ¼ Gr/Re n , also involves in the analysis of mixed convection which represents the importance of natural convection relative to the forced convection. The exponent n, depends on the geometry, thermal boundary condition and the fluid. The natural convection is negligible as Ri/0 and forced convection is negligible as Ri/N. Extensive investigations were carried out numerically to describe mixed convection heat transfer in lid-driven cavities. Al- Amiri et al. [12] analyzed the mixed convection heat transfer in * Corresponding author. Tel: þ40 264 594315; fax: þ40 264 591906. E-mail addresses: tanmay@iitm.ac.in (T. Basak), sjroy@iitm.ac.in (S. Roy), pop.ioan@yahoo.co.uk (I. Pop). Contents lists available at SciVerse ScienceDirect International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts 1290-0729/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2012.01.010 International Journal of Thermal Sciences 57 (2012) 98e111