Immersed boundary method

One of the most difficult tasks in the numerical simulation of fluid flow is the generation of a grid around the object being modeled. This has always involved a large amount of time-consuming user interaction. If the object is moving and/or deforming, then the necessary regridding is an even greater and computationally expensive problem. After regridding, the solution variables must be transferred (i.e. interpolated) from the old to the new grid. This interpolation is a source of error and may also destroy the conservative properties of numerical schemes. The immersed boundary method has the potential to simplify these problems associated with the grid. In particular, the computation of the grid around the object being modeled can be completely automated. In the immersed boundary method, the Navier–Stokes equations are generally solved on a Cartesian grid, which removes the effort needed to generate a body-fitted grid and enables the use of efficient numerical methods that can be parallelized in a relatively easy manner. The influence of objects on the flow is simulated by the addition of a force density (which represents the force of the surface of the object on the fluid) to the Navier–Stokes equations. This force density, if chosen properly, should result in a solution to the Navier–Stokes equation which satisfies the boundary conditions on the surface of the object. This is in contrast to other methods such as body-fitted curvilinear or unstructured grids, which require the grid to be built around or inside the objects being modeled.

<math>
\frac{u^{n+1}-u^n}{\delta t}=-\nabla p +\Gamma ^n + F^{n+1}
</math>

'''Basic Interpolation Method'''

If the immersed boundary coincides with the velocity grid points then <math>F^{n+1}</math> can be calculated as

<math>F^{n+1} =\frac{u^{n+1}_{IB}-u^n }{\Delta t} -(-\nabla p^{n+1}+\Gamma ^n)</math>

If not, then an interpolation operator will be necessary to find <math>u^{n+1}</math> at the grid points(near the immersed boundary), where the force density is added.
<math>\frac{u^{n+1}-u^n}{\delta t}=-\nabla p +\Gamma ^n +(\frac{ S_{NB}(u^{n+1}_{NB})+S_{IB}(u^{n+1}_{IB})-u^n}{\Delta t}-(-\nabla p^{n+1}+\Gamma^n))</math>

The above equation simplifies to Equation 

<math>u^{n+1}=S_{NB}(u^{n+1}_{NB})+S_{IB}(u^{n+1}_{IB})</math>

which replaces the discretized momentum equation at grid points adjacent to the immersed boundary at which the force density <math>F^{n+1}</math> is added.