  To understand the expansion dynamics of the universe from galaxy cluster
scales, using the angular diameter distance (ADD) data from two different
galaxy cluster surveys, we constrain four cosmological models to explore the
underlying value of $H_0$ and employ the model-independent Gaussian Processes
to investigate the evolution of the equation of state of dark energy. The ADD
data in the X-ray bands consists of two samples covering the redshift ranges
[0.023, 0.784] and [0.14, 0.89], respectively. We find that: (i) For these two
samples, the obtained values of $H_0$ are more consistent with the recent local
observation by Riess et al. than the global measurement by the Plank
Collaboration, and the $\Lambda$CDM model is still preferred utilizing the
information criterions; (ii) For the first sample, there is no evidence of
dynamical dark energy (DDE) at the $2\sigma$ confidence level (CL); (iii) For
the second one, the reconstructed equation of state of dark energy exhibits a
phantom-crossing behavior in the relatively low redshift range over the
$2\sigma$ CL, which gives a hint that the late-time universe may be actually
dominated by the DDE from galaxy cluster scales; (iv) By adding a combination
of Type Ia Supernovae, cosmic chronometers and Planck-2015 shift parameter and
HII galaxy measurements into both ADD samples, the DDE exists evidently over
the $2\sigma$ CL.
