  We explore one-point and two-point statistics of the Navier-Stokes-alpha-beta
regularization model at moderate Reynolds number in homogeneous isotropic
turbulence. The results are compared to the limit cases of the
Navier-Stokes-alpha model and the Navier-Stokes-alpha-beta model without
subgrid-scale stress, as well as with high resolution direct numerical
simulation. After reviewing spectra of different energy norms of the
Navier-Stokes-alpha-beta model, the Navier-Stokes-alpha model, and
Navier-Stokes-alpha-beta model without subrid-scale stress, we present
probability density functions and normalized probability density functions of
the filtered and unfiltered velocity increments along with longitudinal
velocity structure functions of the regularization models and direct numerical
simulation results. We highlight differences in the statistical properties of
the unfiltered and filtered velocity fields entering the governing equations of
the Navier-Stokes-alpha and Navier-Stokes-alpha-beta models and discuss the
usability of both velocity fields for realistic flow predictions. The influence
of the modified viscous term in the Navier-Stokes-alpha-beta model is studied
through comparison to the case where the underlying subgrid-scale stress tensor
is neglected. The filtered velocity field is found to have physically more
viable probability density functions and structure functions for the
approximation of direct numerical simulation results, whereas the unfiltered
velocity field is found to have flatness factors close to direct numerical
simulation results.
