  Experiments and numerical simulations of turbulent $^4$He and $^3$He-B have
established that, at hydrodynamic length scales larger than the average
distance between quantum vortices, the energy spectrum obeys the same 5/3
Kolmogorov law which is observed in the homogeneous isotropic turbulence of
ordinary fluids. The importance of the 5/3 law is that it points to the
existence of a Richardson energy cascade from large eddies to small eddies.
However, there is also evidence of quantum turbulent regimes without Kolmogorov
scaling. This raises the important questions of why, in such regimes, the
Kolmogorov spectrum fails to form, what is the physical nature of turbulence
without energy cascade, and whether hydrodynamical models can account for the
unusual behaviour of turbulent superfluid helium. In this work we describe
simple physical mechanisms which prevent the formation of Kolmogorov scaling in
the thermal counterflow, and analyze the conditions necessary for emergence of
quasiclassical regime in quantum turbulence generated by injection of vortex
rings at low temperatures. Our models justify the hydrodynamical description of
quantum turbulence and shed light into an unexpected regime of vortex dynamics.
