A study of the Navier–Stokes-α model for two-dimensional turbulence

Abstract

The Navier–Stokes-α model of turbulence is a mollification of the Navier–Stokes equations, in which the vorticity is advected and stretched by a smoothed velocity field. The smoothing is performed by filtering the velocity field over spatial scales of size smaller than α. This is achieved by convolution with a kernel associated with Green's function of the Helmholtz operator scaled by a parameter α. The statistical properties of the smoothed velocity field are expected to match those of Navier–Stokes turbulence for scales larger than α, thus providing a more computable model for those scales. For wavenumbers k such that kα ≫ 1, corresponding to spatial scales smaller than α, there are three candidate power laws for the energy spectrum, corresponding to three possible characteristic time scales in the model equations: one from the smoothed field, the second from the rough field and the third from a special combination of the two. In two dimensions, the second time scale may be understood to characterize the dynamics of the conserved enstrophy. We measure the scaling of the energy spectra from high-resolution simulations of the two-dimensional Navier–Stokes-α model, in the limit as α→∞. The energy spectrum of the smoothed velocity field scales as k ⁻⁷ in the direct enstrophy cascade regime, consistent with dynamics dominated by the timescale associated with the rough velocity field. We are thus able to deduce that the dynamics of the dominant cascading conserved quantity, namely the enstrophy of the rough velocity, governs the scaling of all derived statistical quantities.

Keywords:

• Turbulence models
• Sub-grid scale models
• Large-eddy simulations
• Two-dimensional Navier–Stokes-α model

2000 Mathematics Subject Classifications:

• 76F55
• 76F65

Acknowledgments

We are grateful to B. Nadiga and B. Wingate for useful discussions and helpful comments on this study. This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under contract no. DE-AC52-06NA25396, partially supported by the Laboratory Directed Research and Development Program and the DOE Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research. This work was also supported in part by the NSF grant no. DMS-0504619, the ISF grant no. 120/06, the BSF grant no. 2004271 and the US Civilian Research and Development Foundation, grant no. RUM1-2654-M0-05.

Notes

^aIn units of Δ x.