Accurate computation of surface stresses and forces with immersed boundary methods

TL;DR

This paper presents a post-processing filtering technique for immersed boundary methods that reduces spurious oscillations in surface stresses, leading to more accurate and smooth force calculations without affecting velocity convergence.

Contribution

It introduces a novel filtering approach for surface stresses in immersed boundary methods, improving physical accuracy and smoothness of stress distributions.

Findings

Filtered stresses converge to physical stresses in test problems.

The method reduces oscillations without impacting velocity convergence.

Surface force calculations become more reliable and physically meaningful.

Abstract

Many immersed boundary methods solve for surface stresses that impose the velocity boundary conditions on an immersed body. These surface stresses may contain spurious oscillations that make them ill-suited for representing the physical surface stresses on the body. Moreover, these inaccurate stresses often lead to unphysical oscillations in the history of integrated surface forces such as the coefficient of lift. While the errors in the surface stresses and forces do not necessarily affect the convergence of the velocity field, it is desirable, especially in fluid-structure interaction problems, to obtain smooth and convergent stress distributions on the surface. To this end, we show that the equation for the surface stresses is an integral equation of the first kind whose ill-posedness is the source of spurious oscillations in the stresses. We also demonstrate that for sufficiently smooth delta functions, the oscillations may be filtered out to obtain physically accurate surface stresses. The filtering is applied as a post-processing procedure, so that the convergence of the velocity field is unaffected. We demonstrate the efficacy of the method by computing stresses and forces that converge to the physical stresses and forces for several test problems.