  We present the results of a numerical investigation of three-dimensional
homogeneous and isotropic turbulence, stirred by a random forcing with a power
law spectrum, $E_f(k)\sim k^{3-y}$. Numerical simulations are performed at
different resolutions up to $512^3$. We show that at varying the spectrum slope
$y$, small-scale turbulent fluctuations change from a {\it forcing independent}
to a {\it forcing dominated} statistics. We argue that the critical value
separating the two behaviours, in three dimensions, is $y_c=4$. When the
statistics is forcing dominated, for $y<y_c$, we find dimensional scaling, i.e.
intermittency is vanishingly small. On the other hand, for $y>y_c$, we find the
same anomalous scaling measured in flows forced only at large scales. We
connect these results with the issue of {\it universality} in turbulent flows.
