Natural convection in a differentially heated horizontal cylinder: Effects of Prandtl number on flow structure and instability

Abstract

Natural convection in a differentially heated horizontal cylinder is investigated numerically and analytically. Particular attention is paid to the structure of steady convection, the nature of the transients and the onset of unsteadiness for a range of Prandtl numbers extending from 0.7 to infinity. The numerical algorithm integrates the 2-D Navier-Stokes equations in velocity-pressure formulation with a Chebyshev-Fourier spatial approximation. A gradual shift from the conduction to the boundary layer regime is observed for increasing Rayleigh number and the steady flow structure becomes rapidly independent of Pr. Whereas classical scalings are obtained for the azimuthal velocity and the thermal boundary layer thickness, the dynamic boundary layer thickness is found to be independent of the Prandtl number. A simplified semi-analytical model derived from projecting the governing equations on the lowest Fourier modes is proposed, which explains this property. Its solutions are in good quantitative agreement with the full nonlinear solutions in particular for large Prandtl numbers. For large enough Rayleigh values, the transients are found to be dominated by internal waves and the approach to steady state is achieved in an oscillatory manner by decay of internal wave motion. In the steady boundary layer regime, the average Nusselt number classically scales like Ra1/4 and a correlation valid over the range of Prandtl numbers considered is 0.28Ra1/4. The onset of unsteadiness is investigated either by direct numerical integration or by linear stability analysis which combines Newton’s iterations to determine the unstable steady states and Arnoldi’s method to compute the eigenvalues of largest real part of the linearized evolution operator about a steady state. It is thus found that the steady state solution undergoes a Hopf bifurcation and that depending on the Prandtl number the most unstable eigenvector may break or keep the symmetry of the base flow. The critical Rayleigh number is found to achieve an asymptotic value for large enough Prandtl number. The location of the hottest point is also shown to have a very large effect on the critical value. Finally, time integration of the unsteady nonlinear equations indicates that the Hopf bifurcation seems of supercritical type for values of the Prandtl number up to 9 and possibly subcritical for larger Pr values.