An implicit-forcing immersed boundary method for simulating viscous flows in irregular domains

We present a method for solving the incompressible Navier–Stokes equations in irregular domains. These equations are discretized using finite difference method in a uniform Cartesian grid. Stationary rigid boundaries are embedded in the Cartesian grid and singular forces are applied at the rigid boundaries to impose the no-slip conditions. The singular forces are then distributed to the nearby Cartesian grid points using the immersed boundary method. In the present work, the singular forces are computed implicitly by solving a small system of equations at each time step. This system of equations is derived from a second order projection method. The main advantage of this method is that it imposes the no-slip boundary condition exactly and avoids the need for small time step to maintain stability. The ability of the method to simulate viscous flows in irregular domains is demonstrated by applying to 2-dimensional flows past a circular cylinder, multiple rigid obstacles and 3-dimensional flow past a sphere.