A local and explicit forcing correction for Lagrangian immersed boundary methods

Lagrangian immersed boundary methods (IBM) are widely used to study flows with complex and moving boundaries, such as biological flows. However, it has been noted that in some cases the resulting no-slip condition at the wet tissues is inaccurate. In this work, we propose an improved technique to evaluate the IBM forces with the aim of reducing slip and transpiration velocities at the immersed wet surfaces. In the framework of Moving Least Squares transfer function (MLS-IBM), we first formulate the problem of determining the IB forces in an implicit manner. Although this approach enforces the no-slip condition to machine precision, we demonstrate that this method is unaffordable for moderate/high Reynolds number flows. Nevertheless, insight of the implicit formulation reveals that it is possible to derive analytically a local correction to the IB forcing via an approximate factorisation of the system matrix of the implicit IBM. The resulting correction is fully explicit and it can be easily implemented in existing IB codes and extended to other Lagrangian IBMs. Our approach is then tested and compared against standard IBM-MLS (also in its iterative version) in a series of benchmark flows, including fixed and moving rigid bodies, and finally in the biological flow within a rigid aorta. Importantly, the proposed correction is seen to greatly improve the consistency and convergence properties of the IBM-MLS.