Immersed boundary simulations of flows driven by moving thin membranes

Immersed boundary methods are extensively used for simulations of dynamic solid objects interacting with fluids due to their computational efficiency and modeling flexibility compared to body-fitted grid methods. However, thin geometries, such as shells and membranes, cause a violation of the boundary conditions across the surface for many immersed boundary projection algorithms. Using a one-dimensional analytical derivation and multi-dimensional numerical simulations, this manuscript shows that adjustment of the Poisson matrix itself is require to avoid large velocity, pressure, and force prediction errors when the pressure jump across the interface is substantial and that these errors increase with Reynolds number. A new minimal thickness modification is developed for the Boundary Data Immersion Method (BDIM-σ), which avoids these issues while still enabling the use of efficient projection algorithms for high-speed immersed surface simulations.