The immersed interface method for simulating prescribed motion of rigid objects in an incompressible viscous flow

In the immersed interface method, a boundary immersed in a fluid is represented as a singular force in the Navier–Stokes equations. This paper presents an explicit approach for computing the singular force to enforce prescribed motion of a rigid boundary in an incompressible viscous flow. The tangential component of the singular force is related to the surface vorticity and is calculated from the normal derivative of the velocity. The normal component of the singular force is determined from a predictor and a corrector. The predictor uses the normal derivative of the vorticity. The corrector superposes a homogeneous solution to the pressure Poisson equation to achieve the desired normal derivative of the pressure. In the current immersed interface method, the velocity and the pressure are solved using the MAC scheme with the incorporation of jump conditions induced by the singular force and a discontinuous finite body force. The body force is applied to obtain the rigid motion of the fluid enclosed by the boundary. Circular Couette flow, flow past a cylinder, and flow around flappers are simulated to test the accuracy, stability, and efficiency of the method as well as the effect of the corrector. With no stiff springs to model rigid boundaries, the method is stable at relatively high Reynolds numbers.