We design an artificial boundary condition for the steady incompressible Navier-Stokes equations in streamfhction-vorticity formulation in a flat channel with slip boundary conditions on the wall. The new boundary condition is derived fiom the Oseen equations and the method of lines. A numerical experiment for the non-linear Navier-Stokes equations is presented. The artificial boundary condition is compared with Dirichlet and Neumann boundary conditions for the flow past a rectangular cylinder in a flat channel. The numerical results show that our boundary condition is more accurate. KEY WORDS: Navier-Stokes equations; O m equations; method of lines; artificial boundary condition 1. INTRODUCTION
Many numerical simulations of viscous flow problems in physically unbounded domains are carried out in 'truncated' bounded computational domains with an artificial boundary. Artificial boundary conditions such as Neumann or Dirichlet boundary conditions are then prescribed at the artificial boundary. In general the above artificial boundary conditions are only very rough approximations of the exact boundary condition at the artificial boundary. Hence the bounded computational domain must be quite large when high accuracy is required, so the cost of the computation is increased. In order to limit the computational cost, the artificial boundary is often chosen not too far from the domain of interest. The proper specification of the boundary condition at a given artificial boundary for solving partial differential equations on an unbounded domain has been studied. For example, Goldstein' and Feng2 studied Helmholtz-type equations and designed asymptotic radiation conditions at the given circle artificial boundary. Han and W d 4 presented a sequence of artificial boundary conditions with high accuracy for the Laplace equation and the linear elasticity system. Hagstrom and Kelle? obtained the exact boundary condtion and artificial boundary conditions at an artificial boundary for partial differential equations in a cylinder, which were used to solve the non-linear problem.6 Halpern' and Halpern and Schatzman' developed a family of artificial boundary conditions for the unsteady Oseen equations, which was then applied to the unsteady Navier-Stokes (N-S) equations. Nataf designed an open boundary condition for the steady Oseen equations in a flat channel with slip boundary conditions on the wall. Hagstrom'o*ll proposed asymptotic boundary conditions at an artificial boundary for the simulation of timedependent fluid flows. Recently Han el al." designed a discrete artificial boundary condition for a system of linear N-S equations in a flat channel with no-slip boundary conditions on the wall.