Immersed boundary method

The immersed boundary method (IBM) enables flow simulations over complex obstacles using stationary computational meshs, that encompasses the solid bodies. The flexibility comes at a price of the imposition of interpolated boundary conditions on locations in-between the fluid grid nodes.

Note

IBM goes under other names, such as Cartesian grid method, the immersed interface method, the embedded boundary method, the fictitious domain method, etc.

Basic Principles

An IBM Lagrangian mesh is superposed on an LBM Eulerian grid. The computational points from Lagrangian mesh are referred as “solid” nodes. In return, the “fluid” nodes are those located in Eulerian grid.

To assure a no-slip boundary condition, the IBM uses an extra forcing function that’s distributed over a thin band of neighbooring fluid nodes to take effect (diffuse interface), as illustrated below:

where the black grid nodes represent the thin band over which a correction force \(f_{\alpha}\) calculated on Lagrangian mesh (blue nodes) is distributed. The force field accounts for the presence of a immersed solid body.

The formulation of the correction force requires an interpolated fluid velocity over the solid node. This information is usually obtained via a combine operation from fluid nodes near the local solid node. While the correction force has to be spread to the near fluid nodes after it has been calculated.

Note

The combine operation is commonly referred as interpolation. However, since many interpolative processes were adopted during the whole algorithm development, we opted to use the combine nomenclature for IBM.

The figure below illustrates the combine and spread operations. The interpolations are obtained by a smoothed delta function.

Note

The diffuse interface is defined by the fluid nodes that are affected by the spread operation.

Note

The IBM operations enter the main solution after the macroscopics calculation and before LBM collision.