  Several processes may limit the accretion rate onto a super-massive black
hole (SMBH). Two processes that are commonly considered (e.g., for sub-grid
prescriptions) are Bondi-Hoyle-Lyttleton accretion and the Eddington limit. A
third one is AGN wind feedback. It has been long suggested that such a wind
feedback regulates the final SMBH mass, however, it has been shown recently
that AGN winds can also regulate the average accretion rate at a level
consistent with observations of high redshift AGNs. In this paper we study the
effect of wind feedback on the accretion rate using 2D, high resolution
hydrodynamic simulations, that incorporate a self-consistent wind injection
scheme and resolves the SMBH sphere of influence. Two different cases are
explored and compared: one in which the initial gas density is uniform, and one
in which it has an isothermal sphere profile. We also compare simulations with
and without cooling. Our main finding is that for reasonable parameters, AGN
feedback always limits the accretion rate to be far below the
Bondi-Hoyle-Lyttleton limit. For typical wind parameters and a uniform ISM
densities of the order $1 {cm}^{-3}$, the accretion rate is found to be several
orders of magnitudes smaller than that inferred in large samples of high
redshift AGNs. On the other hand, the accretion rate obtained for initially
isothermal 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.
