Convergence of Time-Averaged Statistics of Finite Element Approximations of the Navier-Stokes Equations

When discussing numerical solutions of the Navier-Stokes equations, especially when turbulent flows are concerned, there are at least two questions that can be raised. What is meaningful to compute? How to determine the fidelity of the computed solution with respect to the true solution? This paper takes a step to the answer of these questions for turbulent flows. We consider long time averages of weak solutions of the Navier-Stokes equations, rather than strong solutions. We present time averaged error estimates for the time averaged energy dissipation rate, drag and lift. For shear flows, we address the question of fidelity of the computed solution with respect to the true solution, in view of Kolmogorov’s energy cascade theory.