Boundary-velocity error and stability of the accelerated multi-direct-forcing immersed boundary method

Abstract

The multi-direct-forcing immersed boundary method allows for a small velocity error of the no-slip condition in moving-particle problems but suffers from numerical instability if simulation parameters are not carefully chosen. This study investigates the boundary-velocity error and numerical stability of the accelerated multi-direct-forcing immersed boundary method. An analysis of the discretized equations of body motion in moving boundary problems identifies a critical parameter that solely determines the numerical stability for the body motion. Additionally, numerical simulations reveal the optimal acceleration parameter that minimizes the velocity error of the no-slip condition and is independent of details of the boundary discretisation, the boundary shape, and spatial dimensionality. This study provides a guideline for establishing numerically stable simulations of moving boundary problems at optimal boundary-velocity error.

Figures (17)

• Figure 1: Spatial relationship between the boundary point $\bm{X}_k$ and lattice point $\bm{x}$ in the direct-forcing IBM. The support of the weighting function $W$ is given by a few lattice spacings $\Delta{x}$.
• Figure 1: Relationship between $\eta$ and $\lambda_{\rm max} \omega$ for various iteration counts $\ell$.
• Figure 2: Maximum and minimum eigenvalues of the matrix $\mathcal{A}$ with the weighting function $\phi_3$ or $\phi_4$ for a circular cylinder with various values of $\Delta{s}=S/(N\Delta{x})$.
• Figure 3: Local value $a_k$ of the norm for the boundary point $\bm{X}_k$ (whose argument is $\theta = 360^\circ \times k/N$ on the circular cylinder) with the weighting function (a) $\phi_4$ and (b) $\phi_3$ for various values of $\Delta{s}=S/(N\Delta{x})$.
• Figure 4: Maximum boundary-velocity error on the circular cylinder as a function of the iteration count $\ell$ with the weighting function (a) $\phi_4$ and (b) $\phi_3$ for various values of the acceleration parameter $\omega$.