We present results from a comprehensive number of relativistic, time-dependent, axisymmetric simulations of the runaway instability of non-constant angular momentum thick discs around black holes. This second paper extends earlier results where only constant angular momentum discs were considered. All relevant aspects of the theory of stationary thick discs around rotating black holes, necessary to build the initial state in our simulations, are presented in great detail. The angular momentum of the discs is assumed to increase outwards with the radial distance according to a power law. The main simplifying assumptions of our approach are not to include magnetic fields and self-gravity in the discs. Furthermore, the dynamics of the spacetime is accounted for by computing the transfer of mass and angular momentum from the disc to the black hole through the event horizon : the evolution of the central black hole is assumed to follow a sequence of Kerr black holes of increasing mass and spin. In agreement with previous results based on stationary models we find that by allowing the mass and the spin of the black hole to grow, constant angular momentum discs rapidly become unstable on a dynamical timescale. The comparison with the results of paper I shows that the effect of the angular momentum transfer from the torus to the black hole is to make constant angular momentum discs less unstable, increasing the timescale of the instability. However, we find that non-constant angular momentum discs are dramatically stabilized for very small values of the angular momentum slope. Our time-dependent simulations confirm, thus, the predictions of stationary studies concerning the stabilizing effect of non-constant angular momentum distributions.
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