Linear and weakly nonlinear analysis of Rayleigh–Bénard convection of perfect gas with non-Oberbeck–Boussinesq effects

Abstract

The influences of non-Oberbeck–Boussinesq (NOB) effects on flow instabilities and bifurcation characteristics of Rayleigh–Bénard convection are examined. The working fluid is air with reference Prandtl number $Pr=0.71$ and contained in two-dimensional rigid cavities of finite aspect ratios. The fluid flow is governed by the low-Mach-number equations, accounting for the NOB effects due to large temperature difference involving flow compressibility and variations of fluid viscosity and thermal conductivity with temperature. The intensity of NOB effects is measured by the dimensionless temperature differential $\unicode[STIX]{x1D716}$. Linear stability analysis of the thermal conduction state is performed. An $\unicode[STIX]{x1D716}^{2}$ scaling of the leading-order corrections of critical Rayleigh number $Ra_{cr}$ and disturbance growth rate $\unicode[STIX]{x1D70E}$ due to NOB effects is identified, which is a consequence of an intrinsic symmetry of the system. The influences of weak NOB effects on flow instabilities are further studied by perturbation expansion of linear stability equations with regard to $\unicode[STIX]{x1D716}$, and then the influence of aspect ratio $A$ is investigated in detail. NOB effects are found to enhance (weaken) flow stability in large (narrow) cavities. Detailed contributions of compressibility, viscosity and buoyancy actions on disturbance kinetic energy growth are identified quantitatively by energy analysis. Besides, a weakly nonlinear theory is developed based on centre-manifold reduction to investigate the NOB influences on bifurcation characteristics near convection onset, and amplitude equations are constructed for both codimension-one and -two cases. Rich bifurcation regimes are observed based on amplitude equations and also confirmed by direct numerical simulation. Weakly nonlinear analysis is useful for organizing and understanding these simulation results.