  In this paper we present a novel hydrodynamic experiment using liquid $^4$He.
The flow is forced inertially by a canonical oscillating grid using either its
normal (He~I) or superfluid (He~II) phase, generating a statistically
stationary turbulence. We characterise the turbulent properties of the flow
using 2D Lagrangian Particle tracking on hollow glass micro-spheres. As
expected for tracer particles, the Vorono\"{i} tessellation on particle
positions does not show a significant departure from a random Poisson process
neither in He~I nor He~II phase. Particles' positions are tracked with high
temporal resolution, allowing to resolve velocity fluctuations at integral and
inertial scales while properly assessing the noise contribution. Additionally,
we differentiate the particles' positions (by convolution with Gaussian
kernels) in order to access small scale quantities like acceleration. Using
these measured quantities and the formalism of classical Homogeneous Isotropic
Turbulence (HIT) to perform an energy budget across scales we extract the
energy injection rate at the large scale, the energy flux cascading through
inertial scales, down to small scales at which it is dissipated. We found that
in such inertially driven turbulence, regardless of the normal or superfluid
state of the fluid, estimates of energy at the different scales are compatible
with each other and consistent with oscillating grid turbulence results
reported for normal fluids in the literature. The largest discrepancy shows up
at small scales where the signal to noise ratio is harder to control and where
the 2D measurement is contaminated by the 3D nature of the flow. This motivates
to focus future experimental projects towards small scales, low noise and 3D
measurements.
