# A comparative study of immersed boundary method and interpolated bounce-back scheme for no-slip boundary treatment in the lattice Boltzmann method: Part I, laminar flows

## TL;DR

This study systematically compares the immersed boundary method and interpolated bounce-back scheme for no-slip boundary treatment in lattice Boltzmann simulations of laminar flows, highlighting their accuracy, limitations, and practical guidelines.

## Contribution

It provides a comprehensive analysis of the accuracy and performance differences between the two boundary treatment methods through theoretical and numerical investigations.

## Key findings

- Immersed boundary methods typically show first-order accuracy with velocity and force/torque.
- Interpolated bounce-back schemes generally achieve second-order accuracy.
- Interpolated bounce-back has higher fluctuations in force calculations during particle movement.

## Abstract

The interpolated bounce-back scheme and the immersed boundary method are the two most popular algorithms in treating a no-slip boundary on curved surfaces in the lattice Boltzmann method. While those algorithms are frequently implemented in the numerical simulations involving complex geometries, such as particle-laden flows, their performances are seldom compared systematically over the same local quantities within the same context. In this paper, we present a systematic comparative investigation on some frequently used and most state-of-the-art interpolated bounce-back schemes and immersed boundary methods, based on both theoretical analyses and numerical simulations of four selected 2D and 3D laminar flow problems. Our analyses show that immersed boundary methods (IBM) typically yield a first-order accuracy when the regularized delta-function is employed to interpolate velocity from the Eulerian to Lagrangian mesh, and the resulting boundary force back to the Eulerian mesh. This first order in accuracy for IBM is observed for both the local velocity and hydrodynamic force/torque, apparently different from the second-order accuracy sometimes claimed in the literature. Another serious problem of immersed boundary methods is that the local stress within the diffused fluid-solid interface tends to be significantly underestimated. On the other hand, the interpolated bounce-back generally possesses a second-order accuracy for velocity, hydrodynamic force/torque, and local stress field. The main disadvantage of the interpolated bounce-back schemes is its higher level of fluctuations in the calculated hydrodynamic force/torque when a solid object moves across the grid lines. General guidelines are also provided for the necessary grid resolutions in the two approaches in order to accurately simulate flows over a solid particle.