| We introduce a class of immersed interface methods for incompressible Navier-Stokes equations with singular forces. The algorithms are developed using a new hybrid finite volume/finite difference projection method. This projection method uses the high-resolution finite-volume method from CLAWPACK to advect the cell-centered velocity using the divergence-free edge velocity. Then a finite difference method is used for the diffusion and the projection step. This projection method is second-order accurate and is suitable for flow at high Reynolds numbers or problems containing discontinuous conserved quantities.; The immersed interface method developed in this thesis is motivated by Peskin's immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points. Our method instead incorporates part of this force into jump conditions for the pressure, avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary. This has been implemented by imposing the correct pressure jump in the course of solving the Poisson problem in the projection step. This gives sharp resolution of the pressure across the interface and better volume conservation than the traditional IB method. A fully implicit method that allows us to model immersed elastic boundaries with mass is also presented. |
