A coarse-mesh semi-analytical framework for incompressible flows: Extending the Nodal Integral-Immersed Boundary Method

This work extends the Nodal Integral–Immersed Boundary Method (NIM–IBM) to the solution of steady incompressible Navier–Stokes equations in complex geometries. The NIM provides a coarse-mesh, semi-analytical discretization that maintains second-order spatial accuracy, while the sharp-interface IBM enforces boundary conditions on non-body-fitted Cartesian grids. To address the challenges of pressure–velocity coupling and mass conservation near immersed boundaries, the formulation integrates a hybrid MAC–SOLA (Marker and Cell-Solution Algorithm) pressure–correction scheme, preserving the analytical structure of NIM and avoiding complex matrix couplings at cut cells. The proposed framework is validated against multiple benchmark problems involving internal and external flows. Results show that the method accurately captures key flow features and benchmark quantities, even on coarse meshes, with good agreement to experimental and high-resolution numerical data. The approach offers a computationally efficient and geometrically flexible alternative for incompressible flow simulations, with potential for extension to unsteady and high-Reynolds-number regimes.