  This paper is concerned with the numerical investigation of Quantum
Turbulence (QT) described by the Gross-Pitaevskii (GP) equation. Numerical
simulations are performed using a parallel (MPI-OpenMP) code based on a
pseudo-spectral spatial discretization and second order splitting for the time
integration. We start by revisiting (in the framework of
high-performance/high-accuracy computations) well-known GP-QT settings, based
on the analogy with classical vortical flows: Taylor-Green (TG) vortices and
Arnold-Beltrami-Childress (ABC) flow. Two new settings are suggested to build
the initial condition for the QT simulation. They are based on the direct
manipulation of the wave function by generating a smoothed random phase (SRP)
field, or seeding random vortex rings (RVR) pairs. The new initial conditions
have the advantage to be simpler to implement than the TG and ABC approaches,
while generating statistically equivalent QT fields. Each of these four GP-QT
settings is described in detail by defining corresponding benchmarks that could
be used to validate/calibrate new GP codes. We offer a comprehensive
description of the numerical and physical parameters of each benchmark. We
analyze the results in detail and present values, spectra and structure
functions of main quantities of interest (energy, helicity, etc.) that are
useful to describe the turbulent flow. Some general features of QT are
identified, despite the variety of initial states.
