  The coupled dynamics of the two-fluid model of superfluid $^4$He is
numerically studied for quantum turbulence of the thermal counterflow in a
square channel. We combine the vortex filament model of the superfluid and the
Navier-Stokes equations of normal fluid. Simulations of the coupled dynamics
show that the velocity profile of the normal fluid is deformed significantly by
superfluid turbulence as the vortices become dense. This result is consistent
with recently performed visualization experiments. We introduce a dimensionless
parameter that characterizes the deformation of the velocity profile.
