Comparison of two and three-dimensional Rayleigh-Bénard convection of power-law fluids in cylindrical and annular enclosures

Abstract

In the present study, three-dimensional steady-state laminar Rayleigh-Bénard convection (RBC) of power-law fluids in cylindrical and annular enclosures with a square cross-section has been investigated numerically. The simulations have been carried out for a range of power-law exponents for different non-dimensional parameters such as nominal Rayleigh number (Ra) (from 103 to 105), internal radius to enclosure height ratio (ri/L) (from 0 to 2), for a single representative value of Prandtl number (i.e. Pr=102) and an aspect ratio of unity together with Dirichlet boundary conditions. It has been found that convection augments with decreasing n values due to the shear-thinning character of viscosity. As a reflection of this, the mean Nusselt number (Nu¯) increases with decreasing n values and the flow patterns show multiple torus like structures. These findings have been compared with previous findings where the same analysis has been conducted for two-dimensional axisymmetric simulations of cylindrical enclosures. It has been revealed that the results of the two-dimensional analysis are not representative for a three-dimensional configuration in terms of flow pattern and the Nusselt number at Ra > 104 for both the Newtonian (i.e. n=1.0) and shear-thinning (i.e. n < 1) fluids for high values of Rayleigh number. Additionally, the values of critical Rayleigh numbers Racrit (RaNu¯=1) (where Nu¯ values deviate from unity for n ≤ 1 (n > 1)) have been found to be almost insensitive to the change in ri/L, but they decrease with decreasing value of n. Finally, a new correlation for Nu¯, Racrit and RaNu¯=1 has been proposed for the range of parameters considered in this study, based on three-dimensional analysis, which can be very useful for practical applications.