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

Abstract

International audienceThe LS-STAG method [1] is a Cartesian method for incompressible flow computations in irregular ge-ometries which aims at discretizing accurately the flow equations in the cut-cells, i.e. cells of complex polyhedral shape formed by the intersection of the Cartesian mesh with the immersed boundary. The LS-STAG method has recently been extend to non-Newtonian and heat transfer computations in 2D and 3D extruded geometries [5]. An issue encountered in 2D heat transfer is related to the discretization of heat diffusion in the cut-cells. In effect, due to the non-orthogonality of the cut-cells, the use of 2-point formulas for computing cell-face fluxes proves to be inaccurate. A way to improve the accuracy is to compute the whole temperature 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 temperature at cut-cell vertices are then interpolated from cell-centered data and boundary conditions. This gradient reconstruction technique is commonly denom-inated "secondary gradients" [4] in the CFD community and "diamond cell method" [2] in the applied mathematics community. The diamond cell method is first implemented in the LS-STAG code for 2D heat transfer problems using various interpolation schemes (inverse distance weighting, least-squares, Delaunay triangulation). The accuracy of the discretization is firmly assessed on a series of benchmark problems (Taylor-Couette flow, natural convection from a cylinder in an enclosure [3]) by inspecting the formal order of accuracy and the heat flux distribution at the immersed boundary. The impact of the cut-cells quality (smoothness, orthogonal quality) on the solution accuracy will be firmly analysed. In a companion presentation by Brice Portelenelle, the diamond cell discretization will be employed to formulate a systematic discretization of the viscous fluxes of the Navier-Stokes equations in 3D cut-cells of arbitrary shape. Références [1] Y. Cheny and O. Botella. The LS-STAG method : A new immersed boundary / level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties. [2] Y. Coudière, J.-P. Vila, and P. Villedieu. Convergence rate of a finite volume scheme for a twodimensional convection-diffusion problem.ESAIM: Mathematical Modelling and Numerical Analysis,33:493–516, 1999. [3] I Demirdžić,Ž Lilek, and M Perić. Fluid flow and heat transfer test problems for non-orthogonalgrids: Bench-mark solutions.International Journal for Numerical Methods in Fluids, 15(3):329–354,1992. [4] S.R. Mathur and J.Y. Murthy. A pressure-based method for unstructured meshes.Numerical HeatTransfer, 31:195–215, 1997. [5] F. Nikfarjam, Y. Cheny, and O. Botella. The LS-STAG immersed boundary/cut-cell method fornon-Newtonian flows in 3D extruded geometries.Computer Physics Communications, 226:67–80,2018