An Immersed Interface Method for Incompressible Flows and Geometries with Sharp Features

TL;DR

This paper develops an immersed interface method that employs discontinuous Galerkin representations to accurately model incompressible flows around geometries with sharp features, overcoming errors associated with traditional continuous methods.

Contribution

The study introduces a DG-based immersed interface method that improves accuracy near sharp geometrical features in fluid-structure interaction simulations.

Findings

DG approach achieves accuracy comparable to CG for smooth interfaces

DG method's time step restrictions are insensitive to sharp features

CG method's time step size is limited by interface sharpness

Abstract

The immersed interface method (IIM) for models of fluid flow and fluid-structure interaction imposes jump conditions that capture stress discontinuities generated by forces that are concentrated along immersed boundaries. Most prior work using the IIM for fluid dynamic applications has focused on smooth interfaces, but boundaries with sharp features such as corners and edges can appear in practical analyses, particularly on engineered structures. The present study builds on our work to integrate finite element-type representations of interface geometries with the IIM. Initial realizations of this approach used a continuous Galerkin (CG) finite element discretization for the boundary, but as we show herein, these approaches generate large errors near sharp geometrical features. To overcome this difficulty, this study introduces an IIM approach using discontinuous Galerkin (DG) representation of the jump conditions. Numerical examples explore the impacts of different interface representations on accuracy for both smooth and sharp boundaries, particularly flows interacting with fixed interface configurations. We demonstrate that using a DG approach provides accuracy that is comparable to the CG method for smooth cases. Further, we identify a time step size restriction for the CG representation that is directly related to the sharpness of the geometry. In contrast, time step size restrictions imposed by DG representations are demonstrated to be insensitive to the presence of sharp features.