Cool cores of galaxy clusters are thought to be heated by low-power active galactic nuclei (AGN), whose accretion is regulated by feedback. However, the interaction between the hot gas ejected by the AGN and the ambient intracluster medium is extremely difficult to simulate, as it involves a wide range of spatial scales and gas that is Rayleigh-Taylor (RT) unstable. Here we use a subgrid model for RT-driven turbulence to overcome these problems and present the first observationallyconsistent hydrodynamical simulations of AGN self-regulation in galaxy clusters. For a wide range of parameter choices the cluster in our three-dimensional simulations regulates itself for at least several 109 years. Heating balances cooling through a string of outbreaks with a typical recurrence time of ≈ 80 Myrs, a timescale that depends only on the global cluster properties.