The numerical investigation of the Bondi-Hoyle accretion on to a moving black hole has a long history, both in Newtonian and in general-relativistic physics. By performing new two-dimensional and general-relativistic simulations on to a rotating black hole, we point out a novel feature, namely that quasi-periodic oscillations (QPOs) are naturally produced in the shock cone that develops in the downstream part of the flow. Because the shock cone in the downstream part of the flow acts as a cavity trapping pressure perturbations, modes with frequencies in the integer ratios of 2:1 and 3:1 are easily produced. The frequencies of these modes depend on the black hole spin and on the properties of the flow, and scale linearly with the inverse of the black hole mass. Our results may be relevant for explaining the detection of QPOs in Sagittarius A*, once such detection is confirmed by further observations. Finally, we report on the development of the flip-flop instability, which can affect the shock cone under suitable conditions; such an instability has been discussed before in Newtonian simulations but was never found in a relativistic regime.
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