A moving-least-squares reconstruction of the hydrodynamic loads on deformable bodies: Application to immersed boundary methods

Immersed boundary methods (IBMs) provide a convenient and efficient approach for fluid-structure interaction (FSI) problems, as they allow for flexible handling of moving boundaries without conforming mesh requirements. However, computing the hydrodynamic loads exerted by the fluid on the immersed body is a well-known issue, as fluid grid points are not directly available on the wet surface. In the literature, several procedures have been proposed, but they typically yield non-smooth stress distributions or have low accuracy. On the other hand, higher-order methods can only be applied to rigid bodies in the limit of boundary layer flow. In this paper, we propose a novel procedure capable of accurately evaluating the stresses acting on solid surfaces, which is based on a truncated Taylor series of the whole fluid stress tensor, without relying on any assumption on the underlying flow or body dynamics. The terms of the Taylor series are evaluated directly from the flow solution using a versatile moving-least-squares (MLS) interpolation with a small computational overhead. We also propose a variation of the procedure fo computing hydrodynamic loads within thin lubrication layers, which can arise during the interaction of multiple deformable bodies mediated by a fluid. The method is validated in a series of numerical experiments encompassing analytical flow solutions, separated flows over rigid bodies, and FSI over rigid and deformable bodies. Although the method is here applied to a second-order IBM code, it can be applied to any-order fluid solver, including body-fitted ones.