Immersed Boundary Projection Methods

Immersed boundary methods are an attractive alternative to body-fitted grids for complex geometries and fluid–structure interaction problems. The simplicity of the underlying Cartesian mesh allows for a number of useful conservation and stability properties to be embedded in the numerics, and for the resulting discrete equations to be solved efficiently and scalably. We review the immersed boundary projection method for incompressible flows, which implicitly satisfies the no-slip condition at immersed surfaces by solving a system of algebraic equations for surface traction. We discuss issues related to the smoothness of the surface stresses and solution strategies for strongly coupled fluid–structure interaction. For three-dimensional flows on unbounded domains, we discuss a fast lattice Green's function method that provides for an adaptive domain comprising the vortical flow region and at the same time can be solved efficiently using extensions of the fast multipole method. To illustrate the methods, we present a series of benchmark simulations in two and three dimensions, ranging from inverted flag flutter, flow past spinning and inclined disks, and turbulent flow past a sphere.