  We perform numerical simulations of finite temperature quantum turbulence
produced through thermal counterflow in superfluid $^4$He, using the vortex
filament model. We investigate the effects of solid boundaries along one of the
Cartesian directions, assuming a laminar normal fluid with a Poiseuille
velocity profile, whilst varying the temperature and the normal fluid velocity.
We analyze the distribution of the quantized vortices, reconnection rates and
quantized-vorticity production as a function of the wall-normal direction. We
find that the quantized vortex lines tend to concentrate close to the solid
boundaries with their position depending only on temperature and not on the
counterflow velocity. We offer an explanation of this phenomenon by considering
the balance of two competing effects, namely the rate of turbulent diffusion of
an isotropic tangle near the boundaries and the rate of quantized-vorticity
production at the centre. Moreover, this yields the observed scaling of the
position of the peak vortex line density with the mutual friction parameter.
Finally we provide evidence that upon the transition from laminar to turbulent
normal fluid flow, there is a dramatic increase in the homogeneity of the
tangle, which could be used as an indirect measure of the transition to
turbulence in the normal fluid component for experiments.
