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A level set‐based immersed interface method for solving incompressible viscous flows with the prescribed velocity at the boundary

A second‐order accurate immersed interface method (IIM) is presented for solving the incompressible Navier–Stokes equations with the prescribed velocity at the boundary, which is an extension of the IIM of Le et al. (J. Comput. Phys. 2006; 220:109–138) to a level set representation of the boundary in place of the Lagrangian representation of the boundary using control points on a uniform Cartesian grid. In order to enforce the prescribed velocity boundary condition, the singular forces at the immersed boundary are applied on the fluid. These forces are related to the jump in pressure and the jumps in the derivatives of both the pressure and velocity, and are approximated via using the local Hermite cubic spline interpolation. The strength of singular forces is determined by solving a small system of equations at each time step. The Navier–Stokes equations are discretized via using finite difference method with the incorporation of jump conditions on a staggered Cartesian grid and solved by a second‐order accurate projection method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate the viscous flows in irregular domains. Copyright © 2009 John Wiley & Sons, Ltd.