Accurate discretization of diffusion in the LS-STAG cut-cell method using diamond cell techniques

Abstract

In cut-cell methods for fluid flow in complex geometries such as the LS-STAG method (Nikfarjam et al. Comput Phys Commun 226:67-80, 2018), mesh non-orthogonality near the immersed boundary makes inaccurate 2-point formulas for computing the face-normal gradients in diffusive fluxes. A way to improve the accuracy is to compute the whole solution gradient at the cut-cell faces, thus decomposing the flux as an orthogonal contribution (using a standard 2-point formula) and non-orthogonal correction (using data at cell vertices). The solution at cut-cell vertices is then interpolated from cell-centered data and boundary conditions. This gradient reconstruction technique is commonly denominated “diamond cell method”.The diamond cell method is implemented in the LS-STAG code for cell-centered data (temperature equation in 2D geometries) and face-centered data (Navier-Stokes equations in 3D-extruded geometries) using various interpolation schemes for vertex reconstruction. The accuracy of the diamond cell discretization is firmly assessed on a series of 2D benchmark problems (Taylor-Couette solution, natural convection from a cylinder in an enclosure) by inspecting the formal order of accuracy and the heat flux distribution at the immersed boundary. Finally, the diamond cell technique is employed for enhancing the accuracy of the velocity gradients in 3D-extruded geometries which were previously computed with 2-point formulas. These computations illustrate the versatility of the diamond cell technique, which can be applied to other cut-cell methods.