Convective rolls and heat transfer in finite-length rayleigh-Benard convection: A two-dimensional numerical study

A two-dimensional (2D) numerical study using a single-point algebraic k-straight theta;(2)-varepsilon-varepsilon(straight theta) turbulence closure was performed to detect the existence, origin, creation and behavior of convective rolls and associated wall Nusselt (Nu) number variation in thermal convection in 2D horizontal slender enclosures heated from below. The study covered the Rayleigh (Ra) numbers from 10(5) to 10(12) and aspect ratios from 4:1 to 32:1. The time evolution of the convective rolls and the formation of the corner vortices were analyzed using numerical flow visualization, and the correlation between roll structures and heat transfer established. A major consequence of the imposed two dimensionality appeared in the persistence of regular roll structures at higher Ra numbers that approach a steady state for all configurations considered. This finding contradicts the full three-dimensional direct numerical simulations (DNS), large eddy simulations (LES), and three-dimensional transient Reynolds-averaged Navier-Stokes (TRANS) computations, which all show continuously changing unsteady patterns. However, the final-stage roll structures, long-term averaged mean temperature and turbulence moments, and the Nusselt number (both local and integral), are all reproduced in good agreement with the ensemble-averaged 3D DNS, TRANS, and several recent experimental results. These findings justified the 2D approach as an acceptable method for ensemble average analysis of fully 3D flows with at least one homogeneous direction. Based on our 2D computations and adopting the low and high Ra number asymptotic power laws of Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)], new prefactors in the Nu-Ra correlation for Pr=O(1) were proposed that fit better several sets of data over a wide range of Ra numbers and aspect ratios: Nu=0.1Ra(1/4)+0.05Ra(1/3). Even better agreement of our computations was achieved with the new correlation Nu=0.124 Ra0.309 proposed recently by Niemela et al. [Nature (London) 404, 837 (2000)] for 10(6)