A streamfunction-velocity approach for 2D transient incompressible viscous flows

We recently proposed (J. Comput. Phys. 2005, 207(1):52-68) a new paradigm for solving the steady-state two-dimensional (2D) Navier-Stokes (N-S) equations using a streamfunction-velocity (Psi-nu) formulation. This formulation was shown to avoid the difficulties associated with the traditional formulations (primitive variables and streamfunction-vorticity formulations). The new formulation was found to be second-order accurate and was found to yield accurate solutions of a number of fluid flow problems. In this paper, we extend the ideas and propose a second-order implicit, unconditionally stable Psi-nu formulation for the unsteady incompressible N-S equations. The method is used to solve several 2D time-dependent fluid flow problems, including the flow decayed by viscosity problem with analytical solution, the lid-driven square cavity problem, the backward-facing step problem and the flow past a square prism problem. For the problems with known exact solutions, our coarse grid transient solutions are extremely close to the analytical ones even for high Reynolds numbers (Re). For the driven cavity problem, our time-marching steady-state solutions up to Re=7500 provide excellent matches with established numerical results, and for Re=10000, our study concludes that the asymptotic stable solution is periodic as has been found by other authors in recent studies. For the backward step problem, our numerical results are in excellent agreement with established numerical and experimental results. Finally, for the flow past a square prism, we have very successfully simulated the von Karman vortex street for Re=200. Copyright (C) 2009 John Wiley & Sons, Ltd.