Implementation of a free boundary condition to Navier- Stokes equations

Numerical prediction of flow physics involves the specification of boundary conditions to close the problem. Whether all or part of the boundary needs consideration depends on the nature of the investigated partial differential equations. Taking as an example, closure boundary conditions for Navier-Stokes equations at an incompressible limit take different forms because a time-dependent problem is classified as elliptic-parabolic while it is elliptic in its steady counterpart. Even though we are under the restraint of mathematic classification, in an attempt to obtain a weak-form solution we may impose boundary conditions according to the physics since boundary conditions, no doubt, come from nature. Many flow problems of practical relevance in oceanography and meteorology are defined in a fairly large spatial domain. For some flow problems, like ocean circulation, air pollution, and weather prediction, we need to truncate the physically unbounded domain because of CPU-restrictions, memory-limitations, etc. The ambiguity regarding synthetic boundary conditions poses significant computational difficulties since information is hardly available there. Use of an erroneous outlet boundary condition leads sometimes to numerical instability and very often to appreciable inaccuracy in the interior solution. This has motivated researchers, in the community of computational mechanics, to make efforts towards circumventing this difficulty. We may fabricate outlet boundary by intuition and experience. Inclusion of a far-field perturbation solution or a buffer layer in the analysis is also referred to. In general, these “cures” are problem-dependent and fail to properly represent the local physics in reality. As a consequence, we have felt the need for a treatise on developing a viable outflow boundary condition which can mimic or retain the local real behaviour. We have organized this paper as follows: in the second section, we present Navier-Stokes equations in the form of primitive working variables which are then discretized by a mixed finite element method. In the third section, we discuss some existing outflow boundary conditions and some important issues regarding the consequences of the imposed outlet boundary conditions. We bring forward here a new outflow boundary condition. In the fourth section, we present the computed results of an analytical problem for the validation Received July 1995 Revised April 1996