  The main goal of this paper is to present a comprehensive characterization of
well developed vortex tangles in a turbulent counterflow in quantum fluids
(with a laminar normal fluid component). We analyze extensive numerical
simulations using the vortex filament method, solving the full Biot-Savart
equations for the vortex dynamics in a wide range of temperatures and
counter-flow velocities. In addition to a detailed analysis of traditional
characteristics such as vortex line density, anisotropic and curvature
parameters of the vortex tangle, we stress other dynamical and statistical
characteristics which are either much less studied or even unstudied. The
latter include reconnection rates, mean mutual friction forces, drift
velocities and the probability distribution functions of various tangle
parameters: the loop length, the line curvature, the mean curvature of loops
with a given length, etc. During these studies we compare the three main
reconnection procedures which are widely used in the literature, and identify
which properties are strongly affected by the choice of the reconnection
criteria and which of them are practically insensitive to the reconnection
procedure. The conclusion is that the vortex filament method in the framework
of the Biot-Savart equation sufficiently robust and well suited for the
description of the steady state vortex tangle in a quantum counterflow. The
Local-Induction Approximation to this equation may be successfully used to
analytically establish relationships between mean characteristics of the
stochastic vortex tangle.
