A high-order compact scheme for the pure streamfunction (vector potential) formulation of the 3D steady incompressible Navier–Stokes equations

In this paper, a high-order compact finite difference algorithm is proposed for the pure streamfunction (vector potential) formulation of the three dimensional steady incompressible Navier–Stokes equations, in which the grid values of the streamfunction (vector potential), its first-order and second-order derivatives are carried as the unknown variables. The numerical boundary schemes are also established for a general set of flow problems with no normal speed on the boundaries. Numerical examples, including a test problem with an analytical solution and three types of lid-driven cubic cavity flow problems, are solved numerically by the newly proposed scheme. The results obtained prove that the present numerical method has the ability to solve the three dimensional incompressible flow with high accuracy.