  We apply the coarse-grained Hall-Vinen-Bekarevich-Khalatnikov (HVBK)
equations to model the statistically steady-state, turbulent, cylindrically
symmetric radial counterflow generated by a moderately large heat flux from the
surface of a cylinder immersed in superfluid $^4$He. We show that a
time-independent solution exists only if a spatial non-uniformity of
temperature and the dependence on temperature of the thermodynamic properties
are accounted for. We demonstrate the formation of a thermal boundary layer
whose thickness grows with temperature of the cylinder's surface, and analyze
the properties of the flow in the radial direction, including the local average
vortex line density.
