This paper compares the results of Eulerian hydrodynamic simulations of cluster formation with virial scaling relations between four bulk quantities: cluster mass, dark matter velocity dispersion, gas temperature, and cluster luminosity. The comparison is made for a large number of clusters at various redshifts in three cosmological models (CHDM, CDM, and OCDM). 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. The normalizations for the virial relations are computed and compared with existing literature, finding good agreement. The Press-Schechter prescription is calibrated with the simulations, again finding results consistent with other authors. The paper also examines 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. These comparisons help establish a firm groundwork for using clusters as cosmological probes. The results show that the virial relations describe clusters fairly well, although the normalization may need slight modification. The paper also discusses the implications for the models. The results indicate that the temperature is an excellent virial mass indicator. The paper also addresses the effects of metallicity on X-ray emissivity, finding that the ratio of line to continuum emission is small, making the accurate determination of a cluster's metallicity difficult. However, this makes the results largely insensitive to the assumed metallicity. The paper concludes that the simulations reproduce the observed scatter in the virial relations.This paper compares the results of Eulerian hydrodynamic simulations of cluster formation with virial scaling relations between four bulk quantities: cluster mass, dark matter velocity dispersion, gas temperature, and cluster luminosity. The comparison is made for a large number of clusters at various redshifts in three cosmological models (CHDM, CDM, and OCDM). 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. The normalizations for the virial relations are computed and compared with existing literature, finding good agreement. The Press-Schechter prescription is calibrated with the simulations, again finding results consistent with other authors. The paper also examines 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. These comparisons help establish a firm groundwork for using clusters as cosmological probes. The results show that the virial relations describe clusters fairly well, although the normalization may need slight modification. The paper also discusses the implications for the models. The results indicate that the temperature is an excellent virial mass indicator. The paper also addresses the effects of metallicity on X-ray emissivity, finding that the ratio of line to continuum emission is small, making the accurate determination of a cluster's metallicity difficult. However, this makes the results largely insensitive to the assumed metallicity. The paper concludes that the simulations reproduce the observed scatter in the virial relations.