Direct forcing immersed boundary method: Improvements to the Ghost Node Method
Résumé
It has been previously shown that the ghost-cell immersed boundary methods (IBMs) with a maximum stencil size larger than 1 do not yield band matrices and as a result cannot use the more efficient geometric multi-grid algorithms and instead must rely on the more generic and less efficient algebraic multi-grid algorithms. To address these shortcomings and in the pursuit of smaller total run times, smaller memory requirements, and increased accuracy the current article proposes the linear square shifting and the quadratic ghost node methods for the ghost-cell IBM for Cartesian grids. The linear square shifting method guarantees a maximum stencil size of 1 for all immersed boundaries and the increases in accuracy and convergence of the proposed method are comprehensively verified with the canonical verification Poisson test problem. A comprehensive analysis of the effect of the quadratic ghost node method together with the shifting approach for various immersed boundary conditions is also performed with the Poisson test problem. The improved computational efficiency of these methods, and their various combinations, is also verified through the canonical validation test cases of laminar pipe flow and laminar flow past a sphere for various Reynolds numbers, wherein speed-ups of approximately three are achieved.
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