  We compare the results of Eulerian hydrodynamic simulations of cluster
formation against virial scaling relations between four bulk quantities: the
cluster mass, the dark matter velocity dispersion, the gas temperature and the
cluster luminosity. The comparison is made for a large number of clusters at a
range of redshifts in three different cosmological models (CHDM, CDM and OCDM).
We find that the analytic formulae provide a good description of the relations
between three of the four numerical quantities. The fourth (luminosity) also
agrees once we introduce a procedure to correct for the fixed numerical
resolution. We also compute the normalizations for the virial relations and
compare extensively to the existing literature, finding remarkably good
agreement. The Press-Schechter prescription is calibrated with the simulations,
again finding results consistent with other authors. We also examine related
issues such as the size of the scatter in the virial relations, the effect of
metallicity with a fixed pass-band, and the structure of the halos. All of this
is done in order to establish a firm groundwork for the use of clusters as
cosmological probes. Implications for the models are briefly discussed.
