Eigensolution analysis of immersed boundary method based on volume penalization: applications to high-order schemes

TL;DR

This paper analyzes the eigensolution and stability of immersed boundary methods based on volume penalization, proposing improvements with second-order damping to enhance accuracy and reduce time step restrictions in high-order schemes.

Contribution

It introduces a novel analysis of volume penalization IBMs using eigensolution and non-modal methods, and proposes a second-order damping term to improve accuracy and stability.

Findings

Volume penalization adds dissipation without affecting dispersion.

Optimal penalization parameters can be guided by derived relationships.

Adding second-order damping improves accuracy and relaxes time step restrictions.

Abstract

This paper presents eigensolution and non-modal analyses for immersed boundary methods (IBMs) based on volume penalization for the linear advection equation. This approach is used to analyze the behavior of flux reconstruction (FR) discretization, including the influence of polynomial order and penalization parameter on numerical errors and stability. Through a semi-discrete analysis, we find that the inclusion of IBM adds additional dissipation without changing significantly the dispersion of the original numerical discretization. This agrees with the physical intuition that in this type of approach, the solid wall is modelled as a porous medium with vanishing viscosity. From a stability point view, the selection of penalty parameter can be analyzed based on a fully-discrete analysis, which leads to practical guidelines on the selection of penalization parameter. Numerical experiments indicate that the penalization term needs to be increased to damp oscillations inside the solid (i.e. porous region), which leads to undesirable time step restrictions. As an alternative, we propose to include a second-order term in the solid for the no-slip wall boundary condition. Results show that by adding a second-order term we improve the overall accuracy with relaxed time step restriction. This indicates that the optimal value of the penalization parameter and the second-order damping can be carefully chosen to obtain a more accurate scheme. Finally, the approximated relationship between these two parameters is obtained and used as a guideline to select the optimum penalty terms in a Navier-Stokes solver, to simulate flow past a cylinder.