A combination of Crouzeix-Raviart, Discontinuous Galerkin and MPFA methods for buoyancy-driven flows

Abstract

Purpose – The purpose of this paper is to develop an efficient non-iterative model combining advanced numerical methods for solving buoyancy-driven flow problems. Design/methodology/approach – The solution strategy is based on two independent numerical procedures. The Navier-Stokes equation is solved using the non-conforming Crouzeix-Raviart (CR) finite element method with an upstream approach for the non-linear convective term. The advection-diffusion heat equation is solved using a combination of Discontinuous Galerkin (DG) and Multi-Point Flux Approximation (MPFA) methods. To reduce the computational time due to the coupling, the authors use a non-iterative time stepping scheme where the time step length is controlled by the temporal truncation error. Findings – Advanced numerical methods have been successfully combined to solve buoyancy-driven flow problems on unstructured triangular meshes. The accuracy of the results has been verified using three test problems: first, a synthetic problem for which the authors developed a semi-analytical solution; second, natural convection of air in a square cavity with different Rayleigh numbers (103-108); and third, a transient natural convection problem of low Prandtl fluid with horizontal temperature gradient in a rectangular cavity. Originality/value – The proposed model is the first to combine advanced numerical methods (CR, DG, MPFA) for buoyancy-driven flow problems. It is also the first to use a non-iterative time stepping scheme based on local truncation error control for such coupled problems. The developed semi analytical solution based on Fourier series is also novel.