An Immersed Interface Method for the Incompressible Navier-Stokes Equations

Abstract

We present an immersed interface algorithm for the incompressible Navier Stokes equations. The interface is represented by cubic splines which are interpolated through a set of Lagrangian control points. The position of the control points is implicitly updated using the fluid velocity. The forces that the interface exerts on the fluid are computed from the constitutive relation of the interface and are applied to the fluid through jumps in the pressure and jumps in the derivatives of pressure and velocity. A projection method is used to time advance the Navier-Stokes equations on a uniform cartesian mesh. The Poisson-like equations required for the implicit solution of the diffusive and pressure terms are solved using a fast Fourier transform algorithm. The position of the interface is updated implicitly using a quasi-Newton method (BFGS) within each timestep. Several examples are presented to illustrate the flexibility of the presented approach.