  We determine the velocity correlation function, pairwise peculiar velocity
difference, and root-mean-square pairwise peculiar velocity dispersion of rich
clusters of galaxies, as a function of pair separation, for three cosmological
models: Omega=1 and Omega=0.3 CDM, and Omega=0.3 PBI models (all flat and
COBE-normalized). We find that close cluster pairs, with separation r<10Mpc/h,
exhibit strong attractive peculiar velocities in all models; the cluster
pairwise velocities depend sensitively on the model. The mean pairwise
attractive velocity of clusters on 5Mpc/h scale ranges from 1700 km/s for
Omega=1 CDM, to 1000 km/s for PBI, to 700 km/s for Omega=0.3 CDM. The
small-scale pairwise velocities depend also on cluster mass: richer, more
massive clusters exhibit stronger attractive velocities than less massive
clusters. On large scales, from 20 to 200Mpc/h, the cluster peculiar velocities
are increasingly dominated by bulk and random motions; they are independent of
cluster mass. The cluster velocity correlation function, which reflects the
bulk motion minus the relative motion of pairs, is negative on small scales for
Omega=1 and Omega=0.3 CDM, and positive for PBI; this indicates stronger
pairwise motion than bulk motion on small scales for CDM, and relatively larger
bulk motions for PBI. The cluster velocity correlation function is positive on
very large scales, from 10 to 200Mpc/h, for all models. These positive
correlations, which decrease monotonically with scale, indicate significant
bulk motions of clusters up to 200Mpc/h. The strong dependence of the cluster
velocity functions on models, especially at small separations, makes them
useful tools in constraining cosmological models when compared with
observations.
