The application of a parallel finite element scheme to three-dimensional incompressible viscous fluid flow around a circular cylinder is presented. The scheme is based on the Petrov-Galerkin weak formulation with exponential weighting functions. The incompressible Navier-Stokes equations are numerically integrated in time by using a fractional step strategy with second-order accurate Adams-Bashforth scheme for both advection and diffusion terms. Numerical solutions for flow around a circular cylinder are presented. The parallelization and the performance of the present scheme are also demonstrated.
From a practical point of view, the computational simulations of flow around circular cylinders are indispensable in offshore structure engineering fields. In the flow field around a circular cylinder up to high Reynolds number, there are some interesting phenomena such as von Kármán vortex street, the decreasing of the drag coefficients on the cylinder, the transition from laminar flow to turbulence, and so forth. The studies of such flow phenomena have been qualitatively and quantitatively presented by many experimental fluid dynamicists (Wieselsberger, 1921; Roshko, 1954; Achenbach, 1968; West and Apelt, 1982; Schewe, 1983; Cantwell and Coles, 1983; Fey, König and Eckelmann, 1998) and computational fluid ones (Tamura and Kuwahara, 1989; Kashiyama, Tamai, Inomata and Yamaguchi, 1998; Breuer, 2000; Mittal, 2001). Numerical instabilities have been especially experienced in the solution of incompressible Navier-Stokes equations at a high Reynolds number. To stabilize such calculations, various upwind schemes have been successfully presented in finite difference, finite volume and finite element frameworks. Three-dimensional computations of the flow at supercritical Reynolds numbers up to 106have been performed by Tamura and Kuwahara (1989) using the third-order accurate upwind finite difference scheme (Kawamura and Kuwahara, 1984). They found out clearly the mechanism of the drag crisis and the appearance of some longitudinal vortices near separation points.