NATURAL-CONVECTION IN A DIFFERENTIALLY HEATED HORIZONTAL CYLINDER - EFFECTS OF PRANDTL NUMBER ON FLOW STRUCTURE AND INSTABILITY

Natural convection in a differentially heated horizontal cylinder is i nvestigated numerically and analytically. Particular attention is paid to the structure of steady convection, the nature of the transients a nd the onset of unsteadiness for a range of Prandtl numbers extending from 0.7 to infinity. The numerical algorithm integrates the 2-D Navie r-Stokes equations in velocity-pressure formulation with a Chebyshev-F ourier spatial approximation. A gradual shift from the conduction to t he boundary layer regime is observed for increasing Rayleigh number an d the steady flow structure becomes rapidly independent of Pr. Whereas classical scalings are obtained for the azimuthal velocity and the th ermal boundary layer thickness, the dynamic boundary layer thickness i s found to be independent of the Prandtl number. A simplified semi-ana lytical model derived from projecting the governing equations on the l owest Fourier modes is proposed, which explains this property. Its sol utions are in good quantitative agreement with the full nonlinear solu tions in particular for large Prandtl numbers. For large enough Raylei gh 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 Ra-1/4 and a corre lation valid over the range of Prandtl numbers considered is 0.28Ra(1/ 4) The onset of unsteadiness is investigated either by direct numerica l integration or by linear stability analysis which combines Newton's iterations to determine the unstable steady states and Amoldi's method to compute the eigenvalues of largest real part of the linearized evo lution 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 sy mmetry of the base flow. The critical Rayleigh number is found to achi eve 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 eq uations indicates that the Hopf bifurcation seems of supercritical typ e for values of the Prandtl number up to 9 and possibly subcritical fo r larger Pr values. (C) 1997 American Institute of Physics.