An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries
We present an immersed interface method for the incompressible NavierStokes equations capable of handling both rigid and flexible boundaries. The immersed boundaries are represented by a number of Lagrangian control points. In order to ensure that the no-slip condition on the rigid boundary is satisfied, singular forces are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep. For flexible boundaries, the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions. The position of the flexible boundary is updated implicitly using a quasi-Newton method (BFGS) within each timestep. The NavierStokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate.
immersed interface method navierstokes equations cartesian grid method finite difference fast poisson solvers irregular domains fluidmembrane interaction