Temperature and velocity dependence of superfluid turbulence

Long glass tubes of rectangular cross section (10:1 aspect ratio) are shown to be ideal for the study of superfluid turbulence in He-II thermal counterflow. Analysis of thermal-resistance data from four tubes of different sizes yields definitive results for the velocity and temperature dependence of the equivalent homogeneous vortex line density $L_0$. The velocity dependence is found to be identical with that given by the Schwarz theory of homogeneous turbulence, or by a modified Vinen equation. The temperature dependence of $L_0$ (or of the Vinen parameter ratio $\frac{{\chi }_1}{{\chi }_2}$) is shown to be given approximately by the Schwarz theory. Reanalysis of earlier circular-tube data shows that $\frac{{\chi }_1}{{\chi }_2}$ is geometry dependent, but scales simply with the flow-tube hydraulic diameter. The experimental results are consistent with a simple phenomenological expression for the line density which is applicable to any size or shape of tube and any temperature. The critical heat current is shown to be in approximate agreement with the Vinen formula, although the critical region is found to be poorly defined. A universal and intuitively appealing condition for the critical heat current is found in terms of line density.