Fourth- and tenth-order compact finite difference solutions of perturbed circular vortex flows

In this study, high-order compact finite difference calculations are reported for 2D unsteady incompressible circular vortex flow in primitive variable formulation. The fourth-order Runge-Kutta temporal discretization is used together with fourth- or tenth-order compact spatial discretization. Dependent on the perturbation initially imposed, the solutions display a tripole, triangular or square vortex. The comparison of the predictions with the detailed spectral calculations of Kloosterziel and Carnevale (J. Fluid Mech. 1999; 388:217-257) shows that the vorticity fields are very well captured. The spectral resolution of the present method was quantified from the decomposition of the vorticity distribution in its azimuthal components and compared with reported spectral results. Using identical grid resolution to the reference results yields negligible differences in the main features of the flow. The perturbation amplitude and its first harmonic are virtually identical to the reference results for both fourth- or tenth-order spatial discretization, as theoretically expected but seldom a posteriori verified. The differences between the two spatial discretizations appear only for coarser grids, favouring the tenth-order discretization. Copyright (c) 2005 John Wiley & Sons, Ltd.