Extending the immersed boundary projection method for Robin-type and other mixed-type boundary conditions

We extend the immersed boundary projection method (IBPM) to accommodate Robin-type boundary conditions, with a focus on the slip-wall condition in incompressible viscous flows. Our approach employs the jump discontinuity relations as the basis of forming the algebraic constraints for the IBPM. The resulting formulation yields a slightly larger system of differential-algebraic equations compared to the original IBPM. We first examine one- and two-dimensional linear problems before extending the method to the Navier–Stokes equation. Results demonstrate that the technique accurately captures discontinuities without requiring special treatment of the differential operators near the boundary. As a comprehensive validation, we simulate flow past a two-dimensional circular cylinder with slip-wall conditions. Additionally, we apply the method to a heat transfer problem over an array of cylinders to illustrate its versatility.