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 advectiondiffusion 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.

Pollutant Dissipation at the Sediment‐Water Interface: A Robust Discrete Continuum Numerical Model and Recirculating Laboratory Experiments

“…The developed model relies on the numerical scheme suggested by Younes et al. (2014) to solve the coupled Navier‐Stokes and heat transfer equations (Younes et al., 2014). The same numerical scheme was adopted here to solve the merged form of Navier‐Stokes and Brinkman model (Equation ) coupled with the mass transport equations of the reactive system (Equation ).…”

“…Specific numerical methods were implemented for flow, advection, diffusion-dispersion, and reaction operators to obtain efficient simulations (in terms of computation time) while maintaining high accuracy and continuity of the state variables and mass fluxes between the water and sediment layers. The developed model relies on the numerical scheme suggested by Younes et al (2014) to solve the coupled Navier-Stokes and heat transfer equations (Younes et al, 2014). The same numerical scheme was adopted here to solve the merged form of Navier-Stokes and Brinkman model (Equation 1) coupled with the mass transport equations of the reactive system (Equation 3).…”

“…This method was selected because of its local mass conservation property. Further, MPFA was condensed in a single system with the DG method to avoid splitting errors when solving advection and diffusion‐dispersion terms separately (Younes et al., 2014). The operator splitting approach was used to include the reactive term.…”

Modeling two-phase flow in heterogeneous porous media: part 1, the case of low interfacial tension

“…The DG method is used to solve for the saturation of the non-wetting phase. The DG method is known to be mass conservative and is readily applied to different grid types (Younes et al 2014. The DG method is of particular interest in the case of multiphase flow as it helps to capture the discontinuity of the phases.…”