Non-Oberbeck-Boussinesq effects due to large temperature differences in a differentially heated square cavity filled with air

Abstract

We numerically investigate non-Oberbeck-Boussinesq (NOB) effects due to large temperature differences in a two-dimensional differentially heated square cavity using low-Mach-number equations. The working fluid is air and the Prandtl number Pr is 0.71 for the reference state. The considered Rayleigh numbers Ra range from 105 to 109. Various temperature differences ΔT̂ between the hot and cold plates are considered and the maximum value is up to 360K. The critical Rayleigh number for the onset of unsteadiness decrease with increasing temperature differences. The NOB effects on the temperature and velocity fields are investigated. It is found that both the thermal and velocity boundary layers become thicker near the hot plate while they get thinner near the cold plates under NOB conditions. The central temperature is increased compared to Oberbeck-Boussinesq (OB) cases considering NOB effects. The normalized center temperature θc roughly increases linearly with increasing temperature differential ∊. The horizontal velocity near the top plate is normally enhanced by NOB effects while it is normally decreased near the bottom plate under NOB conditions. In spite of these marked qualitative differences in the NOB flow relative to OB flow, the overall integral quantities like the Nusselt number Nu and Reynolds number Re are insensitive to NOB effects and retain their Ra-scaling exponents. The Nu has a minor decrease under NOB conditions. The maximum decrease is only about 3% for temperature difference as large as 360K. Within OB approximation, the Nusselt number Nu is found to scale as ∼Ra0.27, and the inverse thermal boundary layer thickness λθ-1 also scale as ∼Ra0.27. These scaling exponents do not change under NOB conditions. The Reynolds number based on the root mean square (r.m.s) velocity Rerms increases slightly for NOB cases. We also find Rerms∼Ra0.37,Rew∼Ra0.50,Rev∼Ra0.45 for OB cases, where Rew/Rev are Reynolds number based on maximum magnitude of vertical/horizontal velocity over the whole cell. The NOB effects almost have no influence on these scaling exponents.