Adaptive Boundary Conditions for Exterior Stationary Flows in Three Dimensions

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.