An immersed interface method for simulating the interaction of a fluid with moving boundaries

Abstract

In the immersed interface method, boundaries are represented as singular force in the Navier–Stokes equations, which enters a numerical scheme as jump conditions. Recently, we systematically derived all the necessary spatial and temporal jump conditions for simulating incompressible viscous flows subject to moving boundaries in 3D with second-order spatial and temporal accuracy near the boundaries [Sheng Xu, Z. Jane Wang, Systematic derivation of jump conditions for the immersed interface method in three-dimensional flow simulation, SIAM J. Sci. Comput., 2006, in press]. In this paper we implement the immersed interface method to incorporate these jump conditions in a 2D numerical scheme. We study the accuracy, efficiency and robustness of our method by simulating Taylor–Couette flow, flow induced by a relaxing balloon, flow past single and multiple cylinders, and flow around a flapping wing. Our results show that: (1) our code has second-order accuracy in the infinity norm for both the velocity and the pressure; (2) the addition of an object introduces relatively insignificant computational cost; (3) the method is equally effective in computing flow subject to boundaries with prescribed force or boundaries with prescribed motion.