Direct forcing immersed boundary methods: Improvements to the ghost-cell method

Abstract

The ghost-cell immersed boundary methods (IBMs) are widely used to implement boundary conditions on non-body fitted grids. It has been previously shown that they have two major drawbacks. Firstly, these methods tend to have a maximum stencil size larger than 1, yielding non-band matrices. Secondly, for a maximum stencil size of 2 the ghost-cell linear IBM [1] have a first-order convergence rate for Neumann immersed boundary conditions. To address these two shortcomings and in the pursuit of increased accuracy and increased order of convergence the current article proposes the linear/quadratic square shifting methods for the ghost-cell IBM for Cartesian grids. The linear square shifting method guarantees a maximum stencil size of 1 and also improves the accuracy and convergence while the quadratic square shifting method improves the accuracy and convergence while maintaining the same stencil size of 2 as the original linear method [1]. The quadratic ghost-cell method [2,3] further improves the accuracy and convergence whilst maintaining a maximum stencil size of 3 while the currently proposed quadratic square shifting method makes it possible to increase the Lagrange polynomial interpolation order whilst maintaining a maximum stencil of 2 resulting in an improved order of convergence for Neumann immersed boundary conditions. The proposed methods are evaluated by considering their accuracy and convergence thanks to a comprehensive verification and validation process. Firstly, the canonical verification 2D and 3D Poisson test problems for various analytical solutions and immersed boundaries are considered and are followed by various verification and validation test cases for the Navier-Stokes governing equations, with and without heat transfer, in 2D and 3D.