A Sensitivity Study of the Navier-Stokes-$\alpha$ Model

We present a sensitivity study of the of the Navier Stokes-$\alpha$ model (NS$\alpha$) with respect to perturbations of the differential filter length $\alpha$. Parameter-sensitivity is evaluated using the sensitivity equations method. Once formulated, the sensitivity equations are discretized and computed alongside the NS$\alpha$ model using the same finite elements in space, and Crank-Nicolson in time. We provide a complete stability analysis of the scheme, along with the sensitivity results of several benchmark problems in both 2D and 3D. We further demonstrate a practical technique to determine the reliability of the NS$\alpha$ model in problem-specific settings. Lastly, we investigate the sensitivity and reliability of important functionals of the velocity and pressure solutions.