  We consider the evolution of an almost Hermitian metric by the $(1,1)$ part
of its Chern-Ricci form on almost complex manifolds. This is an evolution
equation first studied by Chu and coincides with the Chern-Ricci flow if the
complex structure is integrable and with the K\"ahler-Ricci flow if moreover
the initial metric is K\"ahler. We find the maximal existence time for the flow
in term of the initial data and also give some convergence results. As an
example, we study this flow on the (locally) homogeneous manifolds in more
detail.
