On the Volume Conservation of the Immersed Boundary Method

The immersed boundary (IB) method is an approach to problems of fluid-structure interaction in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the fluid. It is well known that some versions of the IB method can suffer from poor volume conservation. Methods have been introduced to improve the volume-conservation properties of the IB method, but they either have been fairly specialized, or have used complex, nonstandard Eulerian finite-difference discretizations. In this paper, we use quasi-static and dynamic benchmark problems to investigate the effect of the choice of Eulerian discretization on the volume-conservation properties of a formally second-order accurate IB method. We consider both collocated and staggered-grid discretization methods. For the tests considered herein, the staggered-grid IB scheme generally yields at least a modest improvement in volume conservation when compared to cell-centered methods, and in many cases considered in this work, the spurious volume changes exhibited by the staggered-grid IB method are more than an order of magnitude smaller than those of the collocated schemes. We also compare the performance of cell-centered schemes that use either exact or approximate projection methods. We find that the volume-conservation properties of approximate projection IB methods depend strongly on the formulation of the projection method. When used with the IB method, we find that pressure-free approximate projection methods can yield extremely poor volume conservation, whereas pressure-increment approximate projection methods yield volume conservation that is nearly identical to that of a cell-centered exact projection method.