On kernel functions for the Immersed Boundary Method in the Lattice Boltzmann framework

The Immersed Boundary Method (IBM) is a renowned approach for numerical fluid-structure interactions, whose methodology and numerical techniques have been refined and commonly integrated into classical Navier-Stokes solvers such as Finite Difference, Finite Element, and Finite Volume. However, the effects of kernel functions and their implementation are seldom discussed in the framework of the Lattice Boltzmann Method (LBM). The theoretical foundation of LBM has distinctions from classical Navier-Stokes solvers, which require a conscientious use of IBM in the LBM regime. Because LBM uses discrete velocities that connect only neighboring nodes, mass and momentum exchange cannot occur beyond the nearest neighbors within a single time step. For direct-forcing IBM-LBM, commonly used kernel functions cover more than a single fluid node, resulting in unstable force fluctuations. In this work, we investigate issues related to force fluctuations caused by existing Dirac delta functions. We propose more compact kernel functions, such as linear shape functions, Gaussian kernels, and an error function, to overcome observed fluctuations. We validate the proposed kernels through 2D benchmark tests.