  We present a macro-scale description of quasi-periodically developed flow in
channels, which relies on double volume-averaging. We show that quasi-developed
macro-scale flow is characterized by velocity modes which decay exponentially
in the main flow direction. We prove that the closure force can be represented
by an exact permeability tensor consisting of two parts. The first part, which
is due to the developed macro-scale flow, is uniform everywhere, except in the
side-wall region, where it is affected by the macro-scale velocity profile and
its slip length. The second part expresses the resistance against the velocity
mode, so it decays exponentially as the flow develops. It satisfies a specific
closure problem on a transversal row of the array. From these properties, we
assess the validity of the classical closure problem for the volume-averaged
flow equations. We show that all its underlying assumptions are partly violated
by an exponentially vanishing error during flow development. Furthermore, we
show that it modifies the eigenvalues, modes, and onset point of
quasi-developed flow, when it is applied to reconstruct the macro-scale flow.
The former theoretical aspects are illustrated for high-aspect-ratio channels
with high-porosity arrays of equidistant in-line square cylinders, by means of
direct numerical simulation and explicit filtering of the flow. In particular,
we present extensive solutions of the classical closure problem for Reynolds
numbers up to 600, porosities between 0.2 and 0.95, and flow directions between
0 and 45 degrees, though the channel height has been kept equal to the cylinder
spacing. These closure solutions are compared with the actual closure force in
channels with cylinder arrays of a porosity between 0.75 and 0.94, for Reynolds
numbers up to 300.
