A family of second order time stepping methods for the Darcy–Brinkman equations

Deferred defect‐correction finite element method for the Darcy‐Brinkman model

“…In contrast, concerning  > 1.0, the flow is dominated by solutal buoyancy effect, and a counterclockwise solutal recirculation exists in the center of the rectangular cavity along with two clockwise thermal recirculations occurring near the right top and left bottom corners of the cavity. These graphics are in agreement with the results in the investigations of [11,19,20].…”

“…In Figure 3, we present profiles for the isotherm line, isoconcentration line and velocity streamline. From this figure, we find that it has an excellent agreement between the presented method and the previous work in [20]. Hence, the presented method is efficient for solving the Darcy-Brinkman equations at some high Rayleigh numbers and can capture this classical model well.…”

“…Besides, when 𝐿𝑒 = 1.0, it means that the temperature and concentration equal diffusivity and behave the same way. In fact, from the figure, we can find that the isotherm line and isoconcentration line become some straight lines, which are pointed out in [19,20].…”

“…The computational domain we tested is a rectangular enclosure with an aspect ratio of 𝐴 = 𝐻∕𝐿, where 𝐿 is cavity length. In this numerical experiment, we compute the current model by setting 𝐴 = 2, which is a tall rectangular cavity and is analyzed in [1,11,15,19,20]. We impose no-slip boundary conditions for the velocity on the whole boundary.…”

“…Moreover, a projection-based stabilized finite element error analysis has been presented by Çıbık and Kaya [3]. Further, based on the idea of curvature stabilization, Çıbık et al [20] have proposed and analyzed a family of second order time stepping methods. Besides, Eroglu et al [1] have extended the post-processing implementation of a projection based variational multiscale method with proper orthogonal decomposition to flows governed by double diffusive convection.…”