On the role of discrete mass conservation for non-Boussinesq flow simulations in enclosures

We discuss the role of discrete mass conservation on the simulations of thermobuoyant flows in enclosures through a detailed numerical investigation in the Boussinesq and non-Boussinesq regimes. Employing a low Mach number formulation in conjunction with a pressure-based approach, we devise two unified algorithms that are capable of handling a wide range of temperature differences. These algorithms can be broadly categorized as discretely conservative and discretely consistent algorithms based on their implementation. While the discretely conservative algorithm computes the density by solving the mass conservation equation in each control volume (along with other conservation laws), the discretely consistent algorithm employs the algebraic equation of state to evaluate the local density. We show that the discretely consistent algorithm is prone to a discrete mass conservation error which can be detrimental to numerical stability in steady non-Boussinesq flows and can significantly influence the computed flowfield in case of unsteady non-Boussinesq convection. We also show that although the discretely conservative algorithm does not satisfy the discrete equation of state it introduces no significant errors in thermodynamic pressure for closed systems. Several numerical experiments over a range of temperature differences, Rayleigh and Prandtl numbers are carried out and the results whilst supporting the theoretical arguments also indicate that the discrete mass conservation errors are strongly linked respectively to numerical stability and solution accuracy in steady and unsteady non-Boussinesq flows.