  The velocity dispersion of galaxies on small scales ($r\sim1h^{-1}$ Mpc),
$\sigma_{12}(r)$, can be estimated from the anisotropy of the galaxy-galaxy
correlation function in redshift space. We apply this technique to
``mock-catalogs'' extracted from N-body simulations of several different
variants of Cold Dark Matter dominated cosmological models to obtain results
which may be consistently compared to similar results from observations. We
find a large variation in the value of $\sigma_{12}(1 h^{-1} Mpc)$ in different
regions of the same simulation. We conclude that this statistic should not be
considered to conclusively rule out any of the cosmological models we have
studied. We attempt to make the statistic more robust by removing clusters from
the simulations using an automated cluster-removing routine, but this appears
to reduce the discriminatory power of the statistic. However, studying
$\sigma_{12}$ as clusters with different internal velocity dispersions are
removed leads to interesting information about the amount of power on cluster
and subcluster scales. We also compute the pairwise velocity dispersion
directly and compare this to the values obtained using the Davis-Peebles
method, and find that the agreement is fairly good. We evaluate the models used
for the mean streaming velocity and the pairwise peculiar velocity distribution
in the original Davis-Peebles method by comparing the models with the results
from the simulations.
