Convective Rolls in a Compressible Fluid

Abstract

Rayleigh-Bénard convection occurs between two horizontally infinite plates when the lower plate is heated with respect to the upper one. The temperature gradient between the two plates causes convective rolls to form as the warmer fluid below becomes more buoyant than the cooler fluid above it. The standard (classical) analysis uses the Boussinesq approximation, which neglects the variations of fluid density except in relation to buoyancy forces. This approximation is not accurate for some real-world applications. This project is inspired by the atmosphere, so we will consider the onset of convection in a vertically stratified layer of fluid which we model using the anelastic equations. The standard analysis is presented in many textbooks and is used as a comparison to the analysis for the compressible fluid presented here. Using linear stability analysis, we compute the critical temperature differences required to induce convection. It is not possible to find analytical solutions, and therefore, numerical methods implemented in Python using LAPACK subroutines are used. Results of the critical Rayleigh number at the half-height of the fluid for a range of plate separation distances are computed. For all the cases that are considered, the solution for the compressible problem follows the standard solution for plate separation distances smaller than some viscosity-dependent value that increases with the viscosity of the fluid. For larger d, it is observed that the stratification inhibits the onset of convection. Our solution is far more idealized than any actual convection happening in the atmosphere. However, it does demonstrate, in this context, the limits of the Boussinesq approximation.