  We continue the investigation of general geometric flows of $G_2$-structures
initiated by the third author in "Flows of $G_2$-structures, I." Specifically,
we determine the possible geometric flows (up to lower order terms) of
$G_2$-structures which are second order quasilinear, by explicitly computing
all independent second order differential invariants of $G_2$-structures which
are $3$-forms. There are four symmetric $2$-tensors and two vector fields. We
do this by deriving explicit computational descriptions of the decompositions
of the curvature and the covariant derivative of the torsion into irreducible
$G_2$-representations, as well as the decomposition of the $G_2$-Bianchi
identity into independent relations. We also show that these six tensors arise
as leading order contributions to the Euler-Lagrange equations for the energy
functionals of the four independent torsion components, and we establish a
$G_2$-analogue of the classical block decomposition of the Riemann curvature
operator on oriented $4$-dimensional Riemannian manifolds.
  Finally, we present a large class of geometric flows of $G_2$-structures
which are directly amenable to a deTurck type trick to establish short-time
existence and uniqueness, with no initial assumption on the torsion, vastly
generalizing an earlier result of Weiss-Witt for the negative gradient flow of
the Dirichlet energy. This result is proved through a careful analysis of the
principal symbols of the linearizations of these operators, establishing
particular linear combinations for which one can prove that the failure of
strict parabolicity is due precisely to the diffeomorphism invariance.
  A detailed introductory section on various foundational results of
$G_2$-structure, several of which are not readily available in the literature,
should be of wider interest and applicability.
