Convergence analysis and computational testing of the finite element discretization of the Navier–Stokes alpha model

This report performs a complete analysis of convergence and rates of convergence of finite element approximations of the Navier–Stokes‐α (NS‐α) regularization of the NSE, under a zero‐divergence constraint on the velocity, to the true solution of the NSE. Convergence of the discrete NS‐α approximate velocity to the true Navier–Stokes velocity is proved and rates of convergence derived, under no‐slip boundary conditions. Generalization of the results herein to periodic boundary conditions is evident. Two‐dimensional experiments are performed, verifying convergence and predicted rates of convergence. It is shown that the NS‐α‐FE solutions converge at the theoretical limit of O(h²) when choosing α = h, in the H¹ norm. Furthermore, in the case of flow over a step the NS‐α model is shown to resolve vortex separation in the recirculation zone. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010