A stream function implicit finite difference scheme for 2D incompressible flows of Newtonian fluids

The development of a new algorithm to solve the Navier–Stokes equations by an implicit formulation for the finite difference method is presented, that can be used to solve two‐dimensional incompressible flows by formulating the problem in terms of only one variable, the stream function. Two algebraic equations with 11 unknowns are obtained from the discretized mathematical model through the ADI method. An original algorithm is developed which allows a reduction from the original 11 unknowns to five and the use of the Pentadiagonal Matrix Algorithm (PDMA) in each one of the equations. An iterative cycle of calculations is implemented to assess the accuracy and speed of convergence of the algorithm. The relaxation parameter required is analytically obtained in terms of the size of the grid and the value of the Reynolds number by imposing the diagonal dominancy condition in the resulting pentadiagonal matrixes. The algorithm developed is tested by solving two classical steady fluid mechanics problems: cavity‐driven flow with Re=100, 400 and 1000 and flow in a sudden expansion with expansion ratio H/h=2 and Re=50, 100 and 200. The results obtained for the stream function are compared with values obtained by different available numerical methods, to evaluate the accuracy and the CPU time required by the proposed algorithm. Copyright © 2002 John Wiley & Sons, Ltd.