Finite element analysis of a projection-based stabilization method for the Darcy–Brinkman equations in double-diffusive convection

The modified characteristics finite element method for time-dependent Darcy-Brinkman problem

“…Suppose Γ D = ∂Ω\Γ B , where Γ B is a regular open subset of ∂Ω . Consider the following time-dependent Darcy–Brinkman equations in a dimensionless form (Çıbık and Kaya, 2013; Goyeau et al , 1996): where u , p , T and C are the velocity, pressure, temperature and concentration fields, respectively; f u , f T and f C are three source terms. Besides, ν represents the kinematic viscosity; γ T and γ C the thermal diffusivity and mass diffusivity, respectively; Da the Darcy number; g the gravitational acceleration vector; and β T and β C the thermal and solutal expansion coefficients, respectively.…”

“…For our study, we introduce some important dimensionless parameters used in (Çıbık and Kaya, 2013): the thermal Grashof number GrT=gβT△TH3ν2, the solutal Grashof number GrC=gβC△CH3ν2, the buoyancy ratio N=βC△CβT△T, the Prandtl number Pr=νγT, the Schmidt number Sc=νγC, the Lewis number Le=ScPr and the Darcy number Da=KH2, where H , K , △ T and △ C denote the cavity height, permeability, characteristics temperature and concentration differences along the enclosure, respectively. We will use another dimensionless parameter, namely, the thermal Rayleigh number Ra = Gr T PrDa for comparison issues.…”

“…The Darcy–Brinkman problem (1), which does not only incorporate the velocity vector field as well as the pressure field but also contains the temperature field and concentration field, plays an important role in many current scientific, engineering and industrial applications (Çıbık and Kaya, 2013; Huang et al , 2013; Huang et al , 2014; Yang and Jiang, 2018). This problem constitutes an important system of equations in metallurgy, electrochemistry, contaminant transport in ground water and so on.…”

“…Yang and Jiang (2018) have studied a full explicitly uncoupled variational multiscale stabilization finite element method for solving the considered equations in double-diffusive convection. Recently, to overcome spurious oscillations along sharp layers, which are obtained by some numerical methods, Çıbık and Kaya (2013) have presented a projection-based stabilization method of the double-diffusive convection in the Darcy–Brinkman flow.…”

A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior

“…Standard numerical methods for solving singularly perturbed problems are unstable and do not give accurate results due to the perturbation parameter  . If suitable numerical methods such as finite difference method and finite element method [1][2][3][4][5][6][7][8] for solving these problems are developed, then the stable and accurate results are obtained. Therefore, we prefer to apply finite difference method for this problem in this paper.…”