In computational fluid dynamics, the immersed boundary method originally referred to an approach developed by Charles Peskin in 1972 to simulate fluid-structure (fiber) interactions. Treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for numerical simulations (the elastic boundary changes the flow of the fluid and the fluid moves the elastic boundary simultaneously). In the immersed boundary method the fluid is represented on an Eulerian coordinate and the structure is represented on a Lagrangian coordinate. For Newtonian fluids governed by the incompressible Navier–Stokes equations, the fluid equations are
and in case of incompressible fluids (assuming constant density) we have the condition
The immersed structures are typically represented as a collection of one-dimensional fibers, denoted by
where
where
Since the original development of this method by Peskin, a variety of approaches have been developed to simulate flow over complicated immersed bodies on grids that do not conform to the surface of the body. These include methods such as the immersed interface method, the Cartesian grid method, the ghost fluid method and the cut-cell method. In their review article Mittal and Iaccarino refer to all these (and other related methods) as Immersed Boundary Methods and provide various categorizations of these methods. From the point of view of implementation, they categorize immersed boundary methods into continuous forcing and discrete forcing methods. In the former, a force term is added to the continuous Navier-Stokes equations before discretization, whereas in the latter, the forcing is applied (explicitly or implicitly) to the discretized equations. Under this taxonomy, Peskin's original method is a continuous forcing method whereas Cartesian grid, cut-cell and the ghost-fluid methods are discrete forcing methods.
See the references for more details.