Investigations of the NS-alpha model using a lid-driven cavity flow

In this paper we investigate a subgrid model based on an anisotropic version of the NS-$\alpha$ model using a lid-driven cavity flow at a Reynolds number of 10,000. Previously the NS-$\alpha$ model has only been used numerically in the isotropic form. The subgrid model is developed from the Eulerian-averaged anisotropic equations [Holm, \textit{Physica D}, v.133, pp 215-269, 1999]. It was found that when $\alpha^{2}$ was based on the mesh numerical oscillations developed which manifested themselves in the appearance of streamwise vortices and a `mixing out' of the velocity profile. This is analogous to the Craik-Leibovich mechanism, with the difference being that the oscillations here are not physical but numerical. The problem could be traced back to the discontinuity in $\alpha^{2}$ encountered when $\alpha^{2}=0$ on the endwalls. An alternative definition of $\alpha^{2}$ based on velocity gradients, rather than mesh spacing, is proposed and tested. Using this definition the results with the model shown a significant improvement. The splitting of the downstream wall jet, rms and shear stress profiles are correctly captured a coarse mesh. The model is shown to predict both positive and negative energy transfer in the jet impingement region, in qualitative agreement with DNS results.