We simulate high-velocity flow in a self-affine channel with a
constant perpendicular
opening by solving numerically the Navier–Stokes equations, and analyse
the resulting
flow qualitatively and quantitatively. At low velocity, i.e. vanishing
inertia, the effective
permeability is dominated by the narrowest constrictions measured perpendicular
to
the local flow direction and the flow field tends to fill the channel due
to the diffusion
generated by the viscous term in the Stokes equation. At high velocity
(strong inertia),
the high-velocity zones of the flow field resemble a narrow tube of essentially
constant
thickness in the direction of flow, since the transversal diffusion is
weak compared to
the longitudinal convection. The thickness of the flow tube decreases with
Reynolds
number. This narrowing in combination with mass balance results in an average
velocity in the flow tube which increases faster with Reynolds number than
the
average velocity in the fracture. In the low-velocity zones, recirculation
zones appear
and the pressure is almost constant.The flow tube consists of straight sections. This is due to inertia.
The local curvature
of the main stream reflects the flow-tube/channel-wall interaction.
A boundary layer
is formed where the curvature is large. This boundary layer is highly dissipative
and
governs the large pressure loss (inertial resistance) of the medium. Quantitatively,
vanishing, weak and strong inertial flow regimes can be described by the
Darcy, weak
inertia and Forchheimer flow equations, respectively. We observe a cross-over
flow
regime from the weak to strong inertia, which extends over a relatively
large range
of Reynolds numbers.
