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Several processes may limit the accretion rate onto a super-massive blackhole (SMBH). Two processes that are commonly considered (e.g., for sub-gridprescriptions) are Bondi-Hoyle-Lyttleton accretion and the Eddington limit. Athird one is AGN wind feedback. It has been long suggested that such a windfeedback regulates the final SMBH mass, however, it has been shown recentlythat AGN winds can also regulate the average accretion rate at a levelconsistent with observations of high redshift AGNs. In this paper we study theeffect of wind feedback on the accretion rate using 2D, high resolutionhydrodynamic simulations, that incorporate a self-consistent wind injectionscheme and resolves the SMBH sphere of influence. Two different cases areexplored and compared: one in which the initial gas density is uniform, and onein which it has an isothermal sphere profile. We also compare simulations withand without cooling. Our main finding is that for reasonable parameters, AGNfeedback always limits the accretion rate to be far below theBondi-Hoyle-Lyttleton limit. For typical wind parameters and a uniform ISMdensities of the order $1 {cm}^{-3}$, the accretion rate is found to be severalorders of magnitudes smaller than that inferred in large samples of highredshift AGNs. On the other hand, the accretion rate obtained for initiallyisothermal density profile is found to be consistent with the observations,particularly when cooling is included. Furthermore, it roughly scales as$\sigma^{5}$ with the velocity dispersion of the bulge, in accord with the$M-\sigma$ relation.

Numerical simulations of AGN wind feedback on black hole accretion: probing down to scales within the sphere of influence