Natural convection of Boussinesq and non-Boussinesq airflows simulated in a tall annular cavity

Abstract

Natural convection of air in a tall annular cavity is explored by numerical solutions of the fully compressible Navier­Stokes equations in transient, axisymmetric (two-dimensional) form. A temperature difference is imposed on the two vertical walls, each isothermal, with the two horizontal boundaries adiabatic; the outer cylindrical surface is heated and the inner cylindrical surface cooled. Thermophysical properties of air, including density, viscosity, thermal conductivity, and specific heat are all variable with temperature. Laminar flow inside a tall annular cavity is prone to an instability which takes the form of a multicellular cat’s eye pattern drifting downward in the cavity. In this study, non-Boussinesq effects on the cat’s eye instability are explored by simulating flows in an annular cavity of aspect ratio A = 40 and diameter ratio η = 0.8, for Rayleigh numbers less than 14,000. A range of dimensionless wall temperature differences are investigated, 0 < ε ≤ 0.2, where ε is the temperature difference between the two vertical walls, Th – Tc , divided by the sum of the absolute temperatures, Th + Tc . The dimensionless wall temperature difference is shown to have pronounced effects on: the onset of the cat’s eye instability, subsequent flow bifurcations, wave speeds, oscillation time-periods as well as local heat transfer rates. The critical Rayleigh number, Rac , signifies a transition from steady unicellular flow to transient multicellular flow; Rac is found to increase with ε. At higher values of dimensionless wall temperature difference ε, mono-periodic oscillatory flows are predicted to occur over a smaller range of Rayleigh numbers, prior to the onset of quasiperiodic flow.