Accretion onto supermassive black holes at close to the Eddington rate is expected to drive powerful winds, which have the potential to majorly influence the properties of the host galaxy. Theoretical models of such winds can simultaneously explain observational correlations between supermassive black holes and their host galaxies, such as the $M- relation, and the powerful multi-phase outflows that are observed in a number of active galaxies. Analytic models developed to understand these processes usually assume simple galaxy properties, namely spherical symmetry and a smooth gas distribution with an adiabatic equation of state. However, the interstellar medium in real galaxies is clumpy and cooling is important, complicating the analysis. We wish to determine how gas turbulence, uneven density distribution, and cooling influence the development of active galactic nucleus (AGN) wind-driven outflows and their global properties on kiloparsec scales. We calculated a suite of idealised hydrodynamical simulations of AGN outflows designed to isolate the effects of turbulence and cooling, both separately and in combination. All simulations initially consisted of a 1 kpc gas shell with an AGN in the centre. We measured the main outflow parameters -- the velocity, the mass outflow rate ($ M out $), and the momentum ($ p out c/L_ AGN $) and energy ($ E out /L_ AGN $) loading factors -- as the system evolves over 1.2 Myr and estimated plausible observationally derived values. We find that adiabatic simulations approximately reproduce the analytical estimates of outflow properties independently of the presence or absence of turbulence and clumpiness. Cooling, on the other hand, has a significant effect, reducing the outflow energy rate by one to two orders of magnitude in the smooth simulations and by up to one order of magnitude in the turbulent ones. The interplay between cooling and turbulence depends on AGN luminosity: in Eddington-limited AGN, turbulence enhances the coupling between the AGN wind and the gas, while in lower-luminosity simulations, the opposite is true. This mainly occurs because dense gas clumps are resilient to low-luminosity AGN feedback but get driven away by high-luminosity AGN feedback. The overall properties of multi-phase outflowing gas in our simulations qualitatively agree with observations of multi-phase outflows, although there are some quantitative differences. We also find that using `observable' outflow properties leads to their parameters being underestimated by a factor of a few compared with real values. We conclude that the AGN wind-driven outflow model is capable of reproducing realistic outflow properties in close-to-realistic galaxy setups and that the $M- relation can be established without efficient cooling of the shocked AGN wind. Furthermore, we suggest ways to improve large-scale numerical simulations by accounting for the effects of AGN wind.
