  Poincar\`e gauge theory (PGT) is an alternative gravity theory, which
attempts to bring the gravity into the gauge-theoretic frame, where the
Lagrangian is quadratic in torsion and curvature. Recently, the cosmological
models with torsion based on this theory have drawn many attentions, which try
to explain the cosmic acceleration in a new way. Among these PGT cosmological
models, the one with only even parity dynamical modes -- SNY model, for its
realistic meaning, is very attractive. In this paper, we first analyze the
past-time cosmic evolution of SNY model analytically. And based on these
results we fit this model to the most comprehensive SNeIa data (Union 2) and
thus find the best-fit values of model parameters and initial conditions, whose
related $\chi^{2}$ value is consistent with the one from $\Lambda$CMD at the
1$\sigma$ level. Also by the $\chi^{2}$ estimate, we provide certain
constraints on these parameters. Using these best-fit values for the Union 2
SNeIa dataset, we are able to predict the evolution of our real universe over
the late time. From this prediction, we know the fate of our universe that it
would expand forever, slowly asymptotically to a halt, which is in accordance
with the earlier works.
