  We perform a numerical analysis of superfluid turbulence produced by thermal
counterflow in He II by using the vortex filament model. Counterflow in a low
aspect ratio channel is known to show the transition from laminar flow to the
two turbulent states TI and TII. The present understanding is that the TI has
the turbulent superfluid and the laminar normal fluid but both fluids are
turbulent in the TII state. This work studies the vortex tangle in the TI
state. Solid boundary condition is applied to walls of a square channel, and
the velocity field of the normal fluid is prescribed to be a laminar Poiseuille
profile. An inhomogeneous vortex tangle, which concentrates near the solid
boundaries, is obtained as the statistically steady state. It is sustained by
its characteristic space-time oscillation. The inhomogeneity of the vortex
tangle shows the characteristic dependence on temperature, which is caused by
two competitive effects, namely the profile of the counterflow velocity and the
mutual friction.
