We present a technique to construct a fair sample of simulated galaxy clusters, and build such a sample for a specific cosmological structure formation scenario. Conventionally one extracts such a sample from a single low-resolution large-scale simulation. Here we simulate the clusters individually at high resolution. This is made possible by the method of constrained random fields, in which one can put linear constraints on peaks in the initial smoothed density field. We assume that these peaks are the progenitors of present-day rich clusters, and select clusters for a catalogue by selecting their initial peak parameters. We find that the final cluster mass can be well approximated by a linear function of both the amplitude and the curvature of the initial density peak. Because the probability distributions of these peak parameters are known, we can construct a model catalogue selected on expected final cluster mass. Such a catalogue will not have a well-defined richness limit, because the relation between richness and mass is fairly broad. However, by applying the appropriate completeness corrections, the results for the mass-selected catalogue can be compared with observations for richness- selected cluster catalogues. Each cluster model is evolved from its constrained initial conditions by means of an N-body integrator. This includes an algorithm for galaxy formation, so we produce two-component models consisting of dark matter background particles and galaxies. The latter allow us to obtain directly the observable properties of the cluster models, and match these to the observed properties to define the present time in the models, and thus derive the amplitude of the initial density fluctuation spectrum, σ8. We build a model cluster catalogue for the Ω0 = 1 cold dark matter (CDM) scenario that is designed to mimic the ENACS sample of rich Abell clusters. We use the distribution of richness, corrected for incompleteness, to fix the present epoch. We find σ8 = 0.4–0.5, which is consistent with other determinations. The catalogue is 70 per cent complete for a richness larger than 50, but we do have a complete subsample for richnesses larger than 75. As a first test we compare the cumulative distribution of line-of-sight velocity dispersions to those found for several observational samples, and find that they match best for σ8 ≈ 0.4. This means that we find consistent values for σ8 for the CDM Ω0 = 1 scenario on cluster scales.