Assessment of the Oberbeck–Boussinesq approximation for buoyancy-driven turbulence in air

Abstract

The full mathematical representation of natural convection is very complex, as it involves, besides continuity and the equations for the transport of momentum and energy, one state equation for density and three laws for the dependency of the thermophysical parameters on pressure and temperature. In addition it requires the representation of pressure work and viscous dissipation in the energy equation. Most numerical simulations and theoretical studies of natural convection use a simplified model based on the Oberbeck–Boussinesq approximation. With respect to the general formulation, the simplified problem is characterized by a divergence-free velocity field, uses constant thermophysical parameters and neglects viscous dissipation and pressure work. Although the Oberbeck–Boussinesq equations have become a physical case in themselves, in certain flow conditions non-Oberbeck–Boussinesq phenomena are non-negligible thus significantly affecting the flow solution. The aim of the present work is to quantitatively identify the flow conditions that give rise to non-negligible non-Oberbeck–Boussinesq phenomena. We demonstrate that the use of direct numerical simulation data combined with the theoretical framework provided by Gray and Giorgini (1976) represents a sound practice to address this issue. The test-case selected is the Rayleigh–Bénard problem at Ra$=0.7\times 10^6$ with air as working fluid. Direct numerical simulations carried out using the compressible, variable property formulation and the Oberbeck–Boussinesq approximation highlight that a 5% tolerance on variations of the thermophysical properties of air around the reference state $({\tilde{\Theta }}_0,{\tilde{P}}_0)$ = (30 °C, 1 atm) only marginally affects the statistical values of both global and local quantities. However, this tolerance represents a very stringent condition that for a tank of height $\tilde{H}=2$ m filled with air at a reference state $({\tilde{\Theta }}_0,{\tilde{P}}_0)$ = (30 °C, 1 atm) leads to a rather low maximum Rayleigh number of the order of $10^{10}$ that can be investigated without considering the influence of non-Oberbeck–Boussinesq effects. Hence, some doubts about the use of the Oberbeck–Boussinesq approximation for the study of high Rayleigh numbers are envisaged at least for air as working fluid.