A Spectral method determination of the first critical Rayleigh number in a cylindrical container

Abstract

The onset of laminar axisymmetric Rayleigh–Bénard convection is investigated analytically for fluid in a cylindrical container. All surfaces are considered to be solid and no-slip for the flow, whereas for the thermal boundary conditions both a perfectly conducting and an insulated side wall are considered. The governing Boussinesq equations are perturbed and an approximate solenoidal flow field and a temperature field are determined, using the assumption of separation of variables. Subsequently, a Chebysev–Galerkin spectral method is employed to reduce the equations to a system of first-order nonlinear ordinary differential equations. The approximate representation of the flow and temperature fields make it possible to perform the calculations analytically. The first critical Rayleigh number ( Ra cr ) is then calculated using local stability analysis. The resulting value of Ra cr compares favorably with previous numerical and experimental studies. The analytical solution presented here allows for deeper insights into the physics of this extensively studied problem to be identified.