A decoupled monolithic projection method for natural convection problems

Over the decades, the natural convection phenomenon has received considerable attention because of a variety of real-world applications. It is important to investigate time-dependent dynamics in natural convection, relying on a strong coupling between incompressible flows and heat transfers. An implicit treatment for the coupling requires for an iterative procedure while an explicit treatment induces a timestep restriction due to numerical instability. To overcome this issue, inspired by our previous work [1], which developed an efficient monolithic numerical procedure based on a projection method (DMPM1) for solving natural convection problems, we propose a new decoupled monolithic projection method (DMPM2). In the present monolithic method, the divergence-free velocity field is obtained from the pressure gradient after solving only one Poisson equation per time step. Accordingly, the temperature is updated with velocity changes. We further verify second-order temporal accuracy of the DMPM2 for velocity, pressure, and temperature based on global error estimates in terms of a discrete l-norm. Considering the energy evolution, we prove that the DMPM2 is stable when the time step is less than or equal to a constant. Finally, we demonstrate that the proposed method is more stable and computationally efficient than DMPM1 and other semi-implicit methods for two-dimensional Rayleigh–Bénard convection problems.