  Here we study the long time behavior of an advection-diffusion equation with
a general time varying (including random) shear flow imposing no-flux boundary
conditions on channel walls. We derive the asymptotic approximation of the
scalar field at long times by using center manifold theory. We carefully
compare it with existing time varying homogenization theory as well as other
existing center manifold based studies, and present conditions on the flows
under which our new approximations give a substantial improvement to these
existing theories. A recent study \cite{ding2020ergodicity} has shown that
Gaussian random shear flows induce a deterministic effective diffusivity at
long times, and explicitly calculated the invariant measure. Here, with our
established asymptotic expansions, we not only concisely demonstrate those
prior conclusions for Gaussian random shear flows, but also generalize the
conclusions regarding determinism to a much broader class of random
(non-Gaussian) shear flows. Such results are important ergodicity-like results
in that they assure an experimentalist need only perform a single realization
of a random flow to observe the ensemble moment predictions at long time.
Monte-Carlo simulations are presented illustrating how the highly random
behavior converges to the deterministic limit at long time. Counterintuitively,
we present a case demonstrating that the random flow may not induce larger
dispersion than its deterministic counterpart, and in turn present rigorous
conditions under which a random renewing flow induces a stronger effective
diffusivity.
