On Numerical Boundary Conditions for the Navier-Stokes Equations

Abstract

Part I: We consider the numerical solution of the Navier-Stokes equations governing the unsteady flow of a viscous incompressible fluid. The analysis of numerical approximations to smooth nonlinear problems reduces to the examination of related linearized problems. The well posedness of the linear Navier-Stokes equations and the stability of finite difference approximations are studied by making energy estimates for the initial boundary value problems. Flows with open boundaries (i.e., inflow and outflow) and with solid walls are considered. We analyse boundary conditions of several types involving the velocity components or a combination of the velocity components and the pressure. The properties of these different types of boundary conditions are compared with emphasis on the suppression of undesirable numerical boundary layers for high Reynolds number calculations. The formulation of the Navier-Stokes equations which uses an elliptic equation for the pressure in lieu of the divergence equation for the velocity is shown to be equivalent to the usual formulation if the boundary conditions are treated correctly. The stability of numerical methods which use this formulation is demonstrated. Part II: We consider the numerical solution of the stream function vorticity formulation of the two dimensional incompressible Navier-Stokes equations for unsteady flows on a domain with rigid walls. The no-slip boundary conditions on the velocity components at the rigid walls are prescribed. In the stream function vorticity formulation these become two boundary conditions on the stream function and there is no explicit boundary condition on the vorticity. The accuracy of the numerical approximations to the stream function and the vorticity is investigated.The common approach in calculations is to employ second order accurate finite difference approximations for all the space derivatives and the boundary conditions together with a time marching procedure involving iteration at each time step to satisfy the boundary conditions. With such schemes the vorticity may be only first order accurate. Higher order approximations to the no-slip boundary conditions have frequently been used to overcome this problem. A one dimensional initial boundary value problem containing the salient features is proposed and analysed. With the use of this model, the behaviour observed in calculations is explained.