On the Use of Compact Streamfunction-Velocity Formulation of Steady Navier-Stokes Equations on Geometries beyond Rectangular

• Authors: Swapan K. Pandit
• Venue: Journal of Scientific Computing
• Year: 2008

In this paper, a new methodology has been proposed to solve two-dimensional (2D) Navier-Stokes (N-S) equations representing incompressible viscous fluid flows on irregular geometries. It is based on second order compact finite difference discretization of the fourth order streamfunction equation on computational plane. The important advantage of this formulation is not only to overcome the difficulties existing in the velocity-pressure and streamfunction-vorticity formulations, but also for being applicable to complex geometries beyond rectangular. We first apply the proposed scheme to a problem having analytical solution and then to the well-studied benchmark problem (problem of lid-driven cavity flow) in viscous fluid flow. Finally, we demonstrate the robustness of our proposed scheme on flow in a complex domain (e.g. constricted channel and dilated channel). It is seen to efficiently capture steady state solutions of the N-S equations with Dirichlet as well as Neumann boundary conditions. In addition to this, it captures viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variations. Estimates of the error incurred show that the results are very accurate on a coarser grid. The results obtained using this scheme are in excellent agreement with analytical and numerical results whenever available and they clearly demonstrate the superior scale resolution of the proposed scheme.