A sharp immersed method for 2D flow-body interactions using the vorticity-velocity Navier-Stokes equations

TL;DR

This paper introduces a sharp-interface immersed method for 2D flow-body interactions using vorticity-velocity Navier-Stokes equations, achieving second-order accuracy and improved efficiency over existing first-order methods.

Contribution

The paper presents a novel moving boundary treatment, a two-way coupling methodology without pressure dependence, and demonstrates second-order accuracy for flow-body interaction simulations.

Findings

Achieves second-order spatial accuracy near boundaries.

Provides significant efficiency improvements over first-order methods.

Validates the approach through extensive testing and validation.

Abstract

Immersed methods discretize boundary conditions for complex geometries on background Cartesian grids. This makes such methods especially suitable for two-way coupled flow-body problems, where the body mechanics are partially driven by hydrodynamic forces. However, for the vorticity-velocity form of the Navier-Stokes equations, existing immersed geometry discretizations for two-way coupled problems only achieve first order spatial accuracy near solid boundaries. Here we introduce a sharp-interface approach based on the immersed interface method to handle the one- and two-way coupling between an incompressible flow and one or more rigid bodies using the 2D vorticity-velocity Navier-Stokes equations. Our main contributions are three-fold. First, we develop and analyze a moving boundary treatment for sharp immersed methods that can be applied to PDEs with implicitly defined boundary conditions, such as those commonly imposed on the vorticity field. Second, we develop a two-way coupling methodology for the vorticity-velocity Navier-Stokes equations based on control-volume momentum balance that does not require the pressure field. Third, we show through extensive testing and validation that our resulting flow-body solver reaches second-order accuracy for most practical scenarios, and provides significant efficiency benefits compared to a representative first-order approach.