A Systematic Procedure for Achieving Higher-Order Spatial Accuracy in Ghost Fluid and Other Embedded Boundary Methods for Fluid-Structure Interaction Problems

Abstract

A systematic procedure is presented for enhancing the spatial accuracy of ghost fluid and other types of embedded boundary methods for CFD in general, and fluid-structure interaction problems in particular. Such methods are gaining popularity because they simplify a number of computational issues. These range from gridding the fluid domain, to designing and implementing Eulerian-based algorithms for challenging fluid-structure applications characterized by large structural motions and deformations or topological changes. However, because they typically operate on non body-fitted grids, ghost fluid and other embedded boundary methods also complicate other issues such as the treatment of wall boundary conditions in general, and fluid-structure transmission conditions in particular. These methods also tend to be at best first-order space-accurate at the embedded interfaces. In some cases, they are also provably inconsistent at these locations. A theory is presented in this paper for addressing this issue. It is developed for a model problem where the fluid is represented by the advection equation and the dynamic motion of the structure is prescribed. For the sake of clarity, but without any loss of generality, this theory is presented in one and two dimensions. However, its extension to three dimensions is straigthforward. This theory leads to a departure from the current practice of populating ghost fluid values independently from the chosen spatial discretization scheme. It shows that by taking into account the pattern and properties of a preferred higher-order discretization scheme, ghost values can be attributed as to preserve the formal order of spatial accuracy of this scheme. The procedure is illustrated in this paper by its application to various finite difference methods. Its impact is also demonstrated by two-dimensional numerical experiments that confirm its theoretically proven ability to preserve higher-order spatial accuracy, including in the vicinity of embedded interfaces.