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Newtonian theory predicts that the velocity V of free test particles on circular orbits around a spherical gravity center is a decreasing function of the orbital radius r, dV/dr<0. Only very recently, Aschenbach [B. Aschenbach, Astronomy and Astrophysics, 425, 1075 (2004)] has shown that, unexpectedly, the same is not true for particles orbiting black holes: for Kerr black holes with the spin parameter a>0.9953, the velocity has a positive radial gradient for geodesic, stable, circular orbits in a small radial range close to the black-hole horizon. We show here that the Aschenbach effect occurs also for nongeodesic circular orbits with constant specific angular momentum l=l(0)=const. In Newtonian theory it is V=l(0)/R, with R being the cylindrical radius. The equivelocity surfaces coincide with the R=const surfaces which, of course, are just coaxial cylinders. It was previously known that in the black-hole case this simple topology changes because one of the "cylinders" self-crosses. The results indicate that the Aschenbach effect is connected to a second topology change that for the l=const tori occurs only for very highly spinning black holes, a>0.99979.