An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies

TL;DR

This paper introduces an immersed interface method for 2D Navier-Stokes equations that effectively handles complex geometries, ensuring high accuracy and efficient computation for flows with multiple obstacles.

Contribution

The paper presents a novel polynomial extrapolation approach for immersed interface methods, improving simplicity and accuracy in simulating multiply connected domains.

Findings

Achieves second order spatial accuracy.

Attains third order temporal accuracy.

Validated on diverse 2D flow scenarios.

Abstract

We present an immersed interface method for the vorticity-velocity form of the 2D Navier Stokes equations that directly addresses challenges posed by multiply connected domains, nonconvex obstacles, and the calculation of force distributions on immersed surfaces. The immersed interface method is re-interpreted as a polynomial extrapolation of flow quantities and boundary conditions into the obstacle, reducing its computational and implementation complexity. In the flow, the vorticity transport equation is discretized using a conservative finite difference scheme and explicit Runge-Kutta time integration. The velocity reconstruction problem is transformed to a scalar Poisson equation that is discretized with conservative finite differences, and solved using an FFT-accelerated iterative algorithm. The use of conservative differencing throughout leads to exact enforcement of a discrete Kelvin's theorem, which provides the key to simulations with multiply connected domains and outflow boundaries. The method achieves second order spatial accuracy and third order temporal accuracy, and is validated on a variety of 2D flows in internal and free-space domains.