We investigate the utility of the spectrally hyperviscous Navier–Stokes equations (SHNSE) in providing a viable computational model for three-dimensional fluid turbulence, in particular, at high Reynolds numbers. In the present paper, the SHNSE model is implemented for a periodic box by using pseudo-spectral methods, and numerical results obtained from simulating decaying turbulence are first compared with those obtained by direct numerical simulation. Next we perform numerical experiments with very high Reynolds numbers and for experimentally selected parameter choices obtain close agreement with the Kolmogorov energy dissipation power law. Our numerical experiments also validate some theoretical properties of the SHNSE. Overall our results indicate that the SHNSE model has potential to be a robust platform for studying turbulence which can retain spectral accuracy while significantly reducing the number of degrees of freedom needed for accurate simulation.