  In this paper we argue that differential rotation can possibly sustain
hydrodynamic turbulence in the absence of magnetic field. We explain why the
non-linearities of the hydrodynamic equations (i.e. turbulent diffusion) should
not be neglected, either as a simplifying approximation or based on boundary
counditions. The consequences of lifting this hypothesis are studied for the
flow stability and the enhanced turbulent transport. We develop a simple
general model for the energetics of turbulent fluctuations in differentially
rotating flows. By taking into account the non-linearities of the equations of
motions, we give constraints on the mean flow properties for the possible
development of shear instability. The results from recent laboratory
experiments on rotating flows show -- in agreement with the model -- that the
pertinent parameter for stability appears to be the Rossby number Ro. The
laboratory experiments seem to be compatible with Ro<-1/2 and Ro > 1 in the
inviscid or high rotation rates limit. Our results, taken in the inviscid
limit, are coherent with the classical linear stability analysis, in the sense
that the critical perturbation equals zero on the marginal linear stability
curve. We also propose a prescription for turbulent viscosity which generalize
the beta-prescription derived in Richard & Zahn 1999.
