Adaptive boundary conditions for exterior stationary flows in three dimensions

TL;DR

This paper introduces a new numerical method for solving 3D exterior stationary Navier-Stokes flows at low Reynolds numbers, significantly reducing computational domain size and increasing efficiency by using adaptive artificial boundary conditions based on asymptotic behavior.

Contribution

The paper presents a novel approach to define artificial boundary conditions for exterior flows, explicitly depending on drag and lift, improving computational efficiency and accuracy.

Findings

Reduces computational domain size by orders of magnitude.

Achieves high accuracy in calculating hydrodynamic forces.

Provides a self-consistent method for asymptotic boundary conditions.

Abstract

Recently there has been an increasing interest for a better understanding of ultra low Reynolds number flows. In this context we present a new setup which allows to efficiently solve the stationary incompressible Navier-Stokes equations in an exterior domain in three dimensions numerically. The main point is that the necessity to truncate for numerical purposes the exterior domain to a finite sub-domain leads to the problem of finding so called "artificial boundary conditions" to replace the conditions at infinity. To solve this problem we provide a vector filed that describes the leading asymptotic behavior of the solution at large distances. This vector field depends explicitly on drag and lift which are determined in a self-consistent way as part of the solution process. When compared with other numerical schemes the size of the computational domain that is needed to obtain the hydrodynamic forces with a given precision is drastically reduced, which in turn leads to an overall gain in computational efficiency of typically several orders of magnitude.