Convection in slender Rayleigh–Bénard cells is a combination of wall and tube components

A model for obtaining scaling laws for Rayleigh–Bénard convection (RBC) at high Rayleigh numbers in tall, slender cells (cells with low aspect ratio, = d/H 1 ) is presented. Traditional RBC ( 1 ) is characterised by large-eddy circulation scaling with the height of the cell, a near-isothermal core and almost all the thermal resistance provided at the horizontal walls. In slender RBC cells, on the other hand, away from the horizontal walls, tube-like convection with eddies scaling with the tube diameter and a linear temperature gradient driving the convective flow is present. The crux of our approach is to split the cell into two components: (i) ‘wall convection’ near the top and bottom horizontal walls and (ii) ‘tube convection (TC)’ in the central part away from the walls. By applying the scaling relations for both wall convection and TC, and treating the total thermal resistance as a sum of their contributions, unified scaling relations for Nusselt number, Reynolds number and mean vertical temperature gradient in slender RBC cells are developed. Our model is applicable for high enough Rayleigh numbers, such that convection both at the wall and in the tube are turbulent. Our model predictions compare well with the data from various studies in slender RBC cells where these conditions are satisfied. In particular, the effects of and Prandtl number are well captured. We propose a scaled aspect ratio using which we obtain ‘universal’ correlations for the heat flux and for the fractional temperature drop in the tube that include the effects of Rayleigh and Prandtl numbers. The profiles of suitably scaled horizontal and vertical velocity fluctuations, along with estimates for boundary layer thickness near the horizontal walls, and the radial distribution of the velocity fluctuations in the tube part are also presented.