In computational fluid dynamics, the immersed boundary method originally referred to an approach developed by Charles Peskin in 1972 to simulate fluid-structure interactions. Treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for numerical simulations. 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 we have the condition The immersed structures are typically represented as a collection of one-dimensional fibers, denoted by. Each fiber can be viewed as a parametric curve where is the parameter and is time. Physics of the fiber is represented via the fiber force distribution. Spring forces, bending resistance or any other type of behavior can be built into this term. The force exerted by the structure on the fluid is then interpolated as a source term in the momentum equation using where is the Dirac function. The forcing can be extended to multiple dimensions to model elastic surfaces or three-dimensional solids. Assuming a massless structure, the elastic fiber moves with the localfluid velocity and can be interpolated via the delta function where denotes the entire fluid domain. Discretization of these equations can be done by assuming an Eulerian grid on the fluid and a separate Lagrangian grid on the fiber. Approximations of the Delta distribution by smoother functions will allow us to interpolate between the two grids. Any existing fluid solver can be coupled to a solver for the fiber equations to solve the Immersed Boundary equations. Variants of this basic approach have been applied to simulate a wide variety of mechanical systems involving elastic structures which interact with fluid flows. 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. Mittal and Iaccarino refer to all these 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 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.
