A Cartesian Grid Method for Solving the Two-Dimensional Streamfunction-Vorticity Equations in Irregular Regions

Abstract

We describe a method for solving the two-dimensional Navier–Stokes equations in irregular physical domains. Our method is based on an underlying uniform Cartesian grid and second-order finite-difference/finite-volume discretizations of the streamfunction-vorticity equations. Geometry representing stationary solid obstacles in the flow domain is embedded in the Cartesian grid and special discretizations near the embedded boundary ensure the accuracy of the solution in the cut cells. Along the embedded boundary, we determine a distribution of vorticity sources needed to impose the no-slip flow conditions. This distribution appears as a right-hand-side term in the discretized fluid equations, and so we can use fast solvers to solve the linear systems that arise. To handle the advective terms, we use the high-resolution algorithms in CLAWPACK. We show that our Stokes solver is second-order accurate for steady state solutions and that our full Navier–Stokes solver is between first- and second-order accurate and reproduces results from well-studied benchmark problems in viscous fluid flow. Finally, we demonstrate the robustness of our code on flow in a complex domain.