Using classical ensemble model, we investigate the sensitivity of double ionization in circularly polarized laser fields to the initial ensembles that are generated for Ar atom with a definite total angular momentum.The numerical results show that the double-ionization probability of the counter-rotating case where the electrons initially rotate against the laser field can be much higher than that of the co-rotating case where the electrons rotate with the laser field.By analyzing the energy distributions of double-ionized electrons versus time, we find that, for the counter-rotating case, the ionization of second electron occurs much earlier than that of that co-rotating case, and the first ionization can leave the Ar+ ion in a higher energy state, which can increase the second ionization probability.These results indicate that the nonadiabatic effects can play an important role in double-ionization dynamics.