Adaptive boundary conditions for exterior stationary flows in three dimensions

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.Comment: 17 pages, 3 tables, 11 figure

Artificial boundary conditions for stationary Navier-Stokes flows past bodies in the half-plane

We discuss artificial boundary conditions for stationary Navier-Stokes flows past bodies in the half-plane, for a range of low Reynolds numbers. When truncating the half-plane to a finite domain for numerical purposes, artificial boundaries appear. We present an explicit Dirichlet condition for the velocity at these boundaries in terms of an asymptotic expansion for the solution to the problem. We show a substantial increase in accuracy of the computed values for drag and lift when compared with results for traditional boundary conditions. We also analyze the qualitative behavior of the solutions in terms of the streamlines of the flow. The new boundary conditions are universal in the sense that they depend on a given body only through one constant, which can be determined in a feed-back loop as part of the solution process

Simulating an exterior domain for drag force computations in the lattice Boltzmann method

The simulation of a stationary fluid flow past an obstacle by means of a lattice Boltzmann method is discussed. The problem of finding appropriate boundary conditions on the boundaries of the truncated numerical domain is addressed by a method recently discussed in the literature, based on a truncated expansion of the solution. The iterative process at the heart of this method is coupled with the iteration steps of a progressive grid refinement technique that allows a rapid convergence towards a well resolved stationary state. It is shown that this combination results in a highly efficient numerical tool which can speed up the resolution process in a substantial manner