A derivation of the NS-α model and preliminary application to plane channel flow

Abstract

In this paper the Navier–Stokes-α (NS-α) model is considered within a large-eddy simulation framework. An investigation is carried out using fully developed turbulent channel flow at a fairly low Reynolds number. This is a flow where diffusion plays a prominent role, and presents a challenge to the nonlinear model investigated here. It is found that when α² _k is based on the mesh spacing, the NS-α model has a tendency to tilt spanwise vorticity in the streamwise direction, leading to high skin friction. This is due to interaction between the spanwise vorticity, the model, and the streamwise streaks. To overcome this problem α² _k is damped in the streak-affected region. The overall results demonstrate the potential of the model to reproduce some features of the DNS (helicity statistics and small-scale features), but more work is required before the full potential of the model can be achieved. In addition to the channel flow investigation, a derivation of the governing using Hamilton's principle is given. The derivation is intended to be clear and accessible to a wide audience, and contains a new interpretation of the model parameter.

Keywords:

• Navier-Stokes-α
• large-eddy simulation
• plane channel flow
• regularization model
• subgrid-scale model

Acknowledgements

This work has been supported by the Natural Science and Engineering Research Council of Canada (NSERC) and Mathematics of Information Technology and Complex Systems (MITACS), and was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET) and Western Canada Research Grid (WESTGRID). We would like to thank Kevin Lamb for his useful suggestions regarding this work, and the reviewers whose valuable comments greatly improved the organization of the manuscript.

Notes

1. It is not clear at this point why this is the case. One possibility is, given that the streak spacing is believed to emerge from a secondary instability of the Tollmein–Schlicting wave (Jimenez [48], p. 219), the spacing we see here may be related to possible differences that would arise through a stability analysis of the NS-α equation as compared to the same analysis for the Navier–Stokes equation. For example, it has been shown that the model lowers the critical wavenumber for baroclinic instability in a two-layer quasi-geostrophic model. Although the initialization here was not representative of a true transition process, there were significant differences observed in how the flow became turbulent from the perturbed laminar state when the NS-α model was used, as compared to without its use.