An Immersed Interface Method for Incompressible Flows and Near Contact

We present an enhanced immersed interface method for simulating incompressible fluid flows in thin gaps between closely spaced immersed boundaries. This regime, common in engineered structures such as tribological interfaces and bearing assemblies, poses significant computational challenges because of limitations in grid resolution and the prohibitive cost of mesh refinement near contact. The immersed interface method imposes jump conditions that capture stress discontinuities generated by forces that are concentrated along immersed boundaries. Our approach introduces a bilinear velocity interpolation operator in two spatial dimensions that incorporates jump conditions from multiple nearby interfaces if they occupy the same interpolation stencil. Furthermore, we show that the methodology developed for resolving distinct interfaces in near contact also improves accuracy for single-interface geometries with intrinsically sharp features, in which multiple interface segments intersect a single grid cell. Numerical results demonstrate substantial improvements in both interface and Eulerian velocity accuracy compared with a lubrication-corrected immersed boundary method, even for interfaces whose separation is one-fiftieth of a computational grid cell. The results also show that this formulation effectively resolves flows near sharp geometric features. The proposed method improves upon previous interpolation schemes and eliminates the need for prior knowledge of interface orientation or geometry. This makes it broadly applicable to a wide range of fluid-structure interaction problems involving near-contact dynamics.