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-<(< theta>)over bar>(2)-epsilon-epsilon (theta) turbulence closure was performe d to detect the existence, origin, creation and behavior of convective roll s and associated wall Nusselt (Nu) number variation in thermal convection i n 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 3 2:1. The time evolution of the convective rolls and the formation of the co rner vortices were analyzed using numerical how visualization, and the corr elation between roll structures and heat transfer established. A major cons equence-of the imposed two dimensionality appeared in the persistence of re gular roll structures at higher Ra numbers that approach a steady state for all configurations considered. This finding contradicts the full three-dim ensional direct numerical simulations (DNS), large eddy simulations (LES), and three-dimensional transient Reynolds-averaged Navier-Stokes (TRANS) com putations, which all show continuously changing unsteady patterns. However, the final-stage roll structures, long-term averaged mean temperature and t urbulence moments, and the Nusselt number (both local and integral), are al l reproduced in good agreement with the ensemble-averaged 3D DNS, TRANS, an d several recent experimental results. These findings justified the 2D appr oach as an acceptable method for ensemble average analysis of fully 3D flow s 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 correla tion 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 Ra-0.309 proposed recently by Niemela et al. [Nature (London) 404 , 837 (2000)] for 10(6)less than or equal to Ra less than or equal to 10(17 ).