A central scheme for simulating incompressible fluid-structure interactions

We propose a fully Eulerian model for simulating the interaction of fluid-fluid and fluid- solid structures for incompressible flows and a black-box type numerical scheme for approximating the solutions of the mathematical models that describe them. This mathematical model consists of two Hamilton-Jacobi type PDEs: the fluid-fluid and fluid-solid interfaces are implicitly tracked by a level set equation, and the fluid motion is modeled by the vorticity formulation of Navier-Stokes equation, also of Hamilton-Jacobi type. This approach simplifies the task by allowing us to implement a single numerical scheme for solving both equations. We provide a detail description of the mathematical models and the challenges they pose, and combine several numerical techniques to address those challenges and compute their solutions. Several numerical experiments simulating the coalescence of gas bubbles and the oscillations of an elastic membrane are presented along with our description of the numerical algorithm. These results demonstrate the simplicity and robustness of our proposed approach to solve immersed boundary problems.