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We obtain the von K\'arm\'an-Howarth relation for the stochastically forcedthree-dimensional Hall-Vinen-Bekharvich-Khalatnikov (3D HVBK) model ofsuperfluid turbulence in Helium ($^4$He) by using the generating-functionalapproach. We combine direct numerical simulations (DNSs) and analyitcal studiesto show that, in the statistically steady state of homogeneous and isotropicsuperfluid turbulence, in the 3D HVBK model, the probability distributionfunction (PDF) $P(\gamma)$, of the ratio $\gamma$ of the magnitude of thenormal fluid velocity and superfluid velocity, has power-law tails that scaleas $P(\gamma) \sim \gamma^3$, for $\gamma \ll 1$, and $P(\gamma) \sim\gamma^{-3}$, for $\gamma \gg 1$. Furthermore, we show that the PDF$P(\theta)$, of the angle $\theta$ between the normal-fluid velocity andsuperfluid velocity exhibits the following power-law behaviors: $P(\theta)\sim\theta$ for $\theta \ll \theta_*$ and $P(\theta)\sim \theta^{-4}$ for $\theta_*\ll \theta \ll 1$, where $\theta_*$ is a crossover angle that we estimate. Fromour DNSs we obtain energy, energy-flux, and mutual-friction-transfer spectra,and the longitudinal-structure-function exponents for the normal fluid and thesuperfluid, as a function of the temperature $T$, by using the experimentallydetermined mutual-friction coefficients for superfluid Helium $^4$He, so ourresults are of direct relevance to superfluid turbulence in this system.

The Statistical Properties of Superfluid Turbulence in $^4$He from the Hall-Vinen-Bekharevich-Khalatnikov Model