We perform numerical studies of the Navier--Stokes-$\alpha\beta$ equations, which are based on a 
general framework for fluid-dynamical theories with gradient dependencies. In particular, we examine 
the effect of the length scales $\alpha$ and $\beta$ on the energy spectrum in three-dimensional 
homogeneous and isotropic turbulent flows in a periodic cubic domain. The limiting cases of the 
Navier--Stokes-$\alpha$ and Navier--Stokes equations are included as special cases. A significant 
increase in the accuracy of the energy spectrum at large wave numbers arises for $\beta<\alpha$. 
We also examine the alignment between the vorticity and the eigenvalues of the stretching tensor. 
The alignment predicted by the Navier--Stokes-$\alpha\beta$ equations improves as $\beta$ decreases 
away from $\alpha$. The observed accuracy increases are, however, limited by the grid resolution. 
Optimal choices for $\alpha$ and $\beta$ are thus likely to depend not only on the problem of interest 
but also on the grid resolution.