  We perform a numerical analysis of counterflow quantum turbulence of
superfluid 4He with nonuni- form flows by using the vortex filament model. In
recent visualization experiments nonuniform laminar flows of the normal fluid,
namely, Hagen-Poiseuille flow and tail-flattened flow, have been observed.
Tail-flattened flow is a novel laminar flow in which the outer part of the
Hagen-Poiseuille flow becomes flat. In our simulation, the velocity field of
the normal fluid is prescribed to be two nonuniform profiles. This work
addresses a square channel to obtain important physics not revealed in the
preceding numerical works. In the studies of the two profiles we analyze the
statistics of the physical quantities. Under Hagen-Poiseuille flow,
inhomogeneous quantum turbulence appears as a statistically steady state. The
vortex tangle shows a characteristic space-time oscillation. Under
tail-flattened flow, the nature of the quantum turbulence depends strongly on
that flatness. Vortex line density increases significantly as the profile
becomes flatter, being saturated above a certain flatness. The inhomogeneity is
significantly reduced in comparison to the case of Hagen-Poiseuille flow.
Investigating the behavior of quantized vortices reveals that tail-flattened
flow is an intermedi- ate state between Hagen-Poiseuille flow and uniform flow.
In both profiles we obtain a characteristic inhomogeneity in the physical
quantities, which suggests that a boundary layer of superfluid appears near a
solid boundary. The vortex tangle produces a velocity field opposite to the
applied superfluid flow, and, consequently, the superfluid flow becomes smaller
than the applied one.
