Colloquium Mathematics - Prof. O. Botella (LEMTA, Université de Lorraine, France)

Title:
The LS-STAG method with diamond cell techniques for flow computation in fully 3D complex geometries

Keywords : Navier-Stokes equations, Cut-Cell method, Gradient Discretisation, 3D.

Abstract:
The 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, following the principles of skew-symmetric discretization presented in [2]. Originally developed for 2D geometries, where only 3 types of generic cut-cells are present, its extension to 3D geometries is a challenge due to the large number of cut-cells (108) to consider. Recent extensions to 3D-extruded geometries [3, 4] have highlighted 2 issues that prevent the successful development of a totally 3D cut-cell method. First, the discretization of the viscous fluxes of the Navier- Stokes equations with a simple 2-point scheme is insufficiently accurate, due to mesh non-orthogonality near the immersed boundary. Secondly, the implementation of the fluxes at the immersed boundary is too complex to be successfully extended to the numerous types of 3D cut-cells. The first issue is overcomed by the use of “diamond schemes” [5, 6], which compute the whole velocity gradient at the cut-cell faces, thus decomposing the viscous flux as an orthogonal contribution (using a standard 2-point formula) and a non-orthogonal correction (using data at cell vertices). Moreover, the framework of the diamond schemes enables us to thoroughly revisit the discretization of the viscous fluxes at the immersed boundary, that no more necessitates a case by case treatment according to the positioning of the solid face of the cut-cells. This numerical algorithm allows a computationally efficient and accurate discretization in arbitrary 3D geometries. Validation will be presented for a series of 3D benchmark problems, such as the Stokes flow between concentric rotating spheres, for which an analytical solution exists [7], and the laminar and turbulent flow past steady and rotating spheres [8].

References
[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 con- servation properties. J. Comput. Phys., 229, 2010.
[2] R. W. C. P. Verstappen and A. E. P. Veldman. Symmetry-preserving discretization of turbulent flow. J. Comput. Phys., 187, 2003.
[3] H.J.L. van der Heiden. Modelling Viscous Effects in Offshore Flow Problems - A Numerical Study. PhD Thesis, University of Groningen, 2019.
[4] F. Nikfarjam, Y. Cheny and O. Botella. The LS-STAG immersed boundary/cut-cell method for non-Newtonian flows in 3D extruded geometries. Computer Physics Communications, 226:67–80, 2018.
[5] Y. Coudi`ere and G. Manzini. Benchmark 3D: The cell-centered finite volume method using least squares vertex reconstruction (“diamond scheme”). In Finite Volumes for Complex Applications VI Problems & Perspectives, Springer, 2011.
[6] B. Portelenelle, O. Botella and Y. Cheny. Accurate discretization of diffusion in the LS-STAG cut-cell method using diamond cell techniques. ECCM ECFD 2018.
[7] P.S. Marcus and L.S. Tuckerman. Simulation of flow between concentric rotating spheres. Part 1. Steady states. Journal of Fluid Mechanics, 185: 1987.
[8] D. Kim and H. Choi. Laminar flow past a sphere rotating in the streamwise direction. Journal of Fluid Mechanics, 461:365–386, 2002.

Olivier BOTELLA, LEMTA - Université de Lorraine- 2 avenue de la Fort de Haye, TSA 60604 - 54518 Vandœu- vre olivier.botella@univ-lorraine.fr