A compact scheme for the streamfunction-velocity formulation of the 2D steady incompressible Navier-Stokes equations in polar coordinaes

A compact difference scheme is developed for the streamfunction-velocity formulation of the steady incompressible Navier-Stokes equations in polar coordinates, which is of second-order accuracy and carries streamfunction and its first derivatives (velocities) as the unknown variables. Numerical examples, including the biharmonic problem with an analytic solution in the unit circular region, the flow past an impulsively started circular cylinder, the driven polar cavity flow and the wall-driven semi-circular cavity flow problems, are solved by the present method. Compared with the existing values by different available numerical methods and experiments in the literature, numerical results demonstrate the accuracy and efficiency of the currently proposed scheme.