Contact. Comparing horizon-scale observations of Sgr A* and M 87* with numerical simulations has provided considerable insight into their interpretation. Most of these simulations are variations of the same physical scenario consisting of a rotation-supported torus seeded with poloidal magnetic fields. However, this approach has several well-known limitations such as secular decreasing trends in mass-accretion rates that render long-term variability studies difficult; a lack of connection with the large-scale accretion flow, which is replaced by an artificial medium emulating vacuum; and significant differences with respect to the predictions of models of accretion onto Sgr A* fed by stellar winds. Aims. We aim to study the flow patterns that arise on horizon scales in more general accretion scenarios that have a clearer connection with the large-scale flow, and are at the same time controlled by a reduced set of parameters. Methods. As a first step in this direction, we performed three-dimensional general relativistic hydrodynamic simulations of rotating transonic flows with velocity perturbations injected from a spherical boundary located far away from the central object (1000 gravitational radii). We studied the general properties of these flows with varying perturbation amplitudes and angular momentum. We analyzed time series of mass and angular-momentum radial fluxes, angle- and time-averaged profiles, and synthetic bremsstrahlung light curves, as well as the three-dimensional structure of the flow, and quantified shock and sonic transitions in the solutions. Results. We observe a rich phenomenology in accretion patterns, which includes smooth Bondi-like flows, turbulent torus-like structures, shocks, filaments, and complex sonic structures. For sufficiently large perturbations and angular momentum, radial profiles deviate from the constant entropy and constant angular-momentum profiles used for initialization and resemble those of advection-dominated accretion flows, showing evidence of entropy generation and angular-momentum redistribution not mediated by magnetic fields. Time series do not show the secular decreasing trend and are suitable for long-term variability studies. We see that the fluctuations are amplified and extend further in frequency than the injected spectrum, producing a red noise spectrum both for the mass-accretion rate and the synthetic light curves. Conclusions. We present a simulation setup that can produce a wide variety of flow patterns at horizon scales and incorporate information from large scale accretion models. The future inclusion of magnetic fields and radiative cooling could make this type of simulation a viable alternative for the numerical modeling of general low-luminosity active galactic nuclei (AGNs).