  We performed the numerical simulation of quantum turbulence produced by
thermal counterflow in superfluid $^{4}${\rm He} by using the vortex filament
model. The pioneering work was made by Schwarz, which has two defects. One is
neglecting non-local terms of the Biot-Savart integral (localized induction
approximation, LIA), and the other is the unphysical mixing procedure in order
to sustain the statistically steady state of turbulence. We succeeded in making
the statistically steady state without the LIA and the mixing. This state shows
the characteristic relation $L=\gamma^2 v_{ns}^2$ between the
line-length-density $L$ and the counterflow relative velocity $v_{ns}$ with the
quantitative agreement of the coefficient $\gamma$ with some typical
observations. We compare our numerical results to the observation of experiment
by Paoletti {\it et al}, where thermal couterflow was visualized by solid
hydrogen particles.
