A stability analysis and some numerical computations for thermal convection with a variable buoyancy factor

Linear and nonlinear analyses of thermal convection with a variable "buoyancy factor", which is defined as the product of thermal expansion coefficient and gravitational acceleration, are investigated for a fluid layer between two infinite horizontal plates. An isothermal boundary condition is applied for both boundaries, and the buoyancy factor throughout the fluid layer is chosen to be a function of depth. For various profiles of variable buoyancy factor, the associated eigenvalue problem for the linear regime is solved numerically using a spectral method. It is found that for the case of buoyancy factor gain, where the vertical rate of change of the buoyancy factor is positive, the results are reversed. A formula for the critical Rayleigh number as a function of the statistical features of the buoyancy factor is developed. For the nonlinear regime, computations based on a spectral Fourier-Chebyshev collocation method are carried out for six parabolic profiles of buoyancy factor. Flow patterns are found to be dominated by two-dimensional rolls for the Rayleigh numbers considered. The computed Nusselt numbers indicate that buoyancy factor deficit (gain) yields lower (higher) heat flux when compared with the corresponding constant buoyancy factor case. When the buoyancy factor deficit is sufficiently large to produce sign changes in the profile, our numerical simulations show that multiple layering in the vertical direction can be produced.published or submitted for publicationis peer reviewe