Open boundary conditions in numerical simulations of unsteady incompressible flow

In numerical simulations of unsteady incompressible flow, mass conservation can be enforced using a fractional step method in which the momentum equations are solved in combination with a pressure or pressure correction equation. While there is an obvious way to implement most spatial boundary conditions commonly applied to these equations, the situation for so-called open boundaries is less clear. Open boundaries must allow fluid to leave the domain with minimal effect on the upstream flow and, at the same time, ensure global conservation of mass. Previous work found that the open boundary condition implementations commonly used with lower order fractional step methods, such as the projection-1 and projection-2 methods, can cause the higher order projection-3 method to become unstable. We discuss the implementation of open boundary conditions in combination with fractional step methods, and investigate three possible approaches. References M. P. Kirkpatrick and S. W. Armfield. On the stability and performance of the projection-3 method for the time integration of the Navier--Stokes equations. ANZIAM J., 49:C559--C575, 2008. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/307. M. P. Kirkpatrick, S. W. Armfield, and J. H. Kent. A representation of curved boundaries for the solution of the Navier--Stokes equations on a staggered three-dimensional Cartesian grid. J. Comp. Phys., 184:1--36, 2003. doi:10.1016/S0021-9991(02)00013-X. M. P. Kirkpatrick, S. W. Armfield, J. H. Kent, and T. Dixon. Simulation of vortex-shedding flows using high-order fractional step methods. ANZIAM J., 42(E):856--876, 2000. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/625. J. Shen. A remark on the projection-3 method. Int. J. Numer. Meth. Fluids, 16:249--253, 1993. doi:10.1002/fld.1650160308. S. W. Armfield and R. Street. An analysis and comparison of the time accuracy of fractional-step methods for the Navier--Stokes equations on staggered grids. Int. J. Numer. Meth. Fluids, 38:255--282, 2002. doi:10.1002/fld.217. S. W. Armfield and R. Street. Modified fractional-step methods for the Navier--Stokes equations. ANZIAM J., 45:C364--C377, 2004. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/894. P. Gresho. On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via finite element method that also introduces a nearly consistent mass matrix. Part 1: Theory. Int. J. Numer. Meth. Fluids, pages 587--620, 1990. doi:10.1002/fld.1650110509. M. P. Kirkpatrick. A large eddy simulation code for industrial and environmental flows. PhD thesis, The University of Sydney, 2002