The Regularized Dia Closure for Two-Dimensional Turbulence

Abstract

A regularized version of the direct interaction approximation closure (RDIA) is compared with ensemble averaged direct numerical simulations (DNS) for decaying two-dimensional turbulence at large-scale Reynolds numbers ranging between low (≈ 50) and high (≈ 4000). The regularization localizes transfer by removing the interaction between large-scale and small-scale eddies depending on a specified cut-off ratio α. It thus eliminates spurious convection effects of small-scale eddies by large-scale eddies in the Eulerian direct interaction approximation (DIA) that causes the underestimation of small-scale kinetic energy by the DIA. Cumulant update versions of the RDIA closure that have comparable performance but are much more efficient computationally have also been analyzed. Both the closures and DNS use discrete wavenumber representations relevant to flows on a doubly periodic domain. This means that any differences between them are intrinsic and not partly due to using continuous wavenumber formulation for the closures.

Comparisons between the regularized closures and DNS have focused on evolved kinetic energy and palinstrophy spectra and as well on enstrophy flux spectra and on the evolution of skewness which depends sensitively on small-scale differences. All of these diagnostics compare quite well when α = 6. And this is the case for runs started from each of three initial spectra, for the range of evolved large-scale Reynolds numbers ranging from ≈ 50 to ≈ 4000 and for regularized DIA closures with, and particularly without, cumulant update restarts. The performance of the RDIA compared with quasi-Lagrangian closure models is discussed.

Keywords:

• Turbulence
• Direct interaction
• Closures
• Regularization

Notes

For both spectrums A and B, the ordinate in Figs. 1 and 4 of FD should be , not P(k) as stated there.