A compact streamfunction-velocity scheme for the 2-D unsteady incompressible Navier-Stokes equations in arbitrary curvilinear coordinates

A streamfunction-velocity formulation-based compact difference method is suggested for solving the unsteady incompressible Navier-Stokes equations in the arbitrary curvilinear coordinates, in which the stream function and its first derivatives as the unknown variables are utilized. Numerical examples, involving the boundary layer problem, a constricted channel flow, driven polar cavity flow and trapezoidal cavity flow problem, are solved by the present method. Numerical results demonstrate the accuracy of the proposed scheme and exhibit the numerical capability to simulate the flow problems on geometries beyond rectangular. For driven polar cavity flow problem, the results show that the flow for Re = 5 000 is not steady but time-periodic, and the critical Reynold number (Rec) for the occurrence of a Hopf bifurcation is given.