A parallel high-order compact scheme for the pure streamfunction formulation of the 3D unsteady incompressible Navier–Stokes equation

Abstract

The high-order compact scheme for the pure streamfunction formulation of the Navier-Stokes equation (Yu and Tian, 2019) is extended to unsteady problems. The unsteady term is discretized by the Crank-Nicolson scheme. A new boundary scheme is also established for symmetrical boundaries to reduce the computational cost of symmetrical flow problems. The parallelization of the present scheme is realized through a boundary approximation approach which maintains the accuracy of calculations for the first- and second-order derivatives on the interior boundaries. The multigrid method for solving the governing equations is parallelized as well. These means are essential to feasible large-scale simulations. Test on a problem with an analytical solution proves that the proposed algorithm is fourth-order accurate in space and second-order in time. Several numerical experiments on lid-driven cavity problems with various configurations and Reynolds numbers up to 3200 are carried out. Results show that the present scheme is efficient and robust enough to resolve fine flow structures and temporal characteristics with moderate grid sizes and time steps.