Numerical Studies of Counterflow Turbulence, Velocity Distribution of Vortices

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.