A fourth order sharp immersed method for the incompressible Navier-Stokes equations with stationary and moving boundaries and interfaces

TL;DR

This paper introduces a high-order immersed interface method for solving incompressible Navier-Stokes equations, achieving fourth order accuracy in both stationary and moving boundary problems, and demonstrating improved efficiency and robustness over existing methods.

Contribution

A novel fourth order immersed interface method with a Runge-Kutta projection scheme for high accuracy in complex flow problems with moving boundaries.

Findings

Demonstrates fourth order convergence in velocity and pressure.

Validates efficiency improvements over second order schemes.

Successfully applies to multiphysics interface problems.

Abstract

We propose a fourth order Navier-Stokes solver based on the immersed interface method (IIM), for flow problems with stationary and one-way coupled moving boundaries and interfaces. Our algorithm employs a Runge-Kutta-based projection method that maintains high-order temporal accuracy in both velocity and pressure for steady and unsteady velocity boundary conditions. Fourth order spatial accuracy is achieved through a novel fifth order IIM discretization scheme for the advection term, as well as existing high-order interface-corrected finite difference schemes for the other differential operators. Using a set of manufactured flow problems with stationary and moving boundaries, we demonstrate fourth order convergence of velocity and pressure in the infinity norm, both inside the domain and on the immersed boundaries. The solver's performance is further validated through a range of practical flow simulations, highlighting its efficiency over a second order scheme. Finally, we showcase the ability of our immersed discretization scheme to handle interface-coupled multiphysics problems by solving a conjugate heat transfer problem with multiple immersed solids. Overall, the proposed approach robustly combines the efficiency of high order discretization schemes with the flexibility of immersed discretizations for flow problems with complex, moving boundaries and interfaces.