Implicit Algorithm for Numerical Solution of the Laminar Incompressible Two-Dimensional Navier-Stokes Equations

摘要

A numerical procedure is presented for solution of the laminar incompressible two dimensional Navier Stokes equations written in terms of the stream function and vorticity. The full implicit unified method is capable of treating both steady and unsteady flows. The governing equations are solved in conservation form, incorporating a general transformation of the independent variables so that arbitrary geometrical configurations may be considered. In the steady case, the algorithm corresponds identically to Newton's method as applied to the solution of nonlinear partial differential equations. For unsteady flows, the procedure is second order accurate temporally as well as spatially. An analysis of a model system of equations is provided which indicates numerical stability. Utility of the method is verified by calculation of representative examples of both steady and unsteady flow fields. These include the classic driven cavity problem and the unsteady flow about a circular cylinder. Results of these computations are compared with previous numerical solutions as well as with experimental data. Keywords; Navier Stokes equations; Computational fluid dynamics; Newton's method.