  Superfluid helium is an intimate mixture of a viscous normal fluid, with
continuous vorticity, and an inviscid superfluid, where vorticity is
constrained to thin, stable topological defects. One mechanism to generate
turbulence in this system is through the application of a heat flux, so called
thermal counterflow. Of particular interest is how turbulence in the superfluid
responds to both a laminar and turbulent normal fluid in the presence of walls.
We model superfluid vortex lines as reconnecting space curves with fixed
circulation, and consider both laminar (Poiseuille) and turbulent normal fluid
flows in a channel configuration. Using high resolution numerical simulations
we show that turbulence in the normal fluid sustains a notably higher vortex
line density than a laminar flow with the same mean flow rate. We exam Vinen's
relation, $\sqrt{L}=\gamma v_{ns}$, between the steady state vortex line
density $L$ and the counterflow velocity $v_{ns}$. Our results support the
hypothesis that transition to turbulence in the normal fluid is responsible for
the TI to TII transition. We also consider the spectral properties of
fluctuations of the superfluid vortices, which show a good agreement with
previous experimental results.
