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 /nite di1erence method is presented, that can be used to solve two-dimensional incompressible 4ows 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 algorithmis developed which allows a reduction fromthe original 11 unknowns to /ve 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 4uid mechanics problems: cavity-driven 4ow with Re=100; 400 and 1000 and 4ow in a sudden expansion with expansion ratio H=h=2 and Re=50; 100 and 200. The results obtained for the streamfunction are compared with values obtained by di1erent available numerical methods, to evaluate the accuracy and the CPU time required by the proposed algorithm