Rayleigh-Benard convection in a vertical cylinder

Numerical solutions of the three-dimensional equations for Rayleigh-Benard convection in a vertical cylinder are presented. The governing non-linear coupled equations (energy and vorticity-vector potential) are approximated using finite differences. The energy and vorticity transport equations are solved using the Samarskii-Andreyev ADI scheme. A fast Fourier transform algorithm is used to solve the elliptic partial differential equations. Solutions are presented for aspect ratios (radius to height) 2 and 4, Prandtl number (Pr=7) and Rayleigh numbers 12 less than or equal to Ra less than or equal to 37500. A conductive (no motion) state exists when Ra less than or equal to Ra-c = 1860. For Ra > Ra-c, different flow patterns - concentric, radial, parallel and cross rolls are obtained.