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

In this paper we consider the Navier-Stokes-$\alpha$ (NS-$\alpha$) model 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 $\alpha^{2}_{k}$ is based on the mesh spacing, the NS-$\alpha$ 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 $\alpha^{2}_{k}$ is damped in the streak affected region. Results overall 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.