A Penalty Method for the Vorticity–Velocity Formulation

We present a new vorticity–velocity formulation and implementation for the unsteady three-dimensional Navier–Stokes equations, based on a penalty method. It relies on an equivalence theorem that employs exact boundary conditions and the vorticity definition on the domain boundary. This approach is particularly attractive for high-order methods for which the often-used influence matrix method fails to converge for Δt→0. The accuracy and the robustness of the new method is demonstrated in the context of several spectral element simulations of unsteady two- and three-dimensional internal and external flows. In particular, the flow past a finite span cylinder attached to end-plates is studied in some detail in order to evaluate the effects of the aspect ratio on the formation length.