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. 2D experiments are performed, verifying convergence and predicted rates of convergence. It is shown that the α-FE solutions converge at the theoretical limit of O(h) when choosing α = h, in the H norm. Convergence in L is shown to approach O(h), but may plateau below the optimal rate. Furthermore, in the case of flow over a step the NS-α model is shown to resolve vortex separation in the recirculation zone.