  We investigate the nonlinear dynamics of turbulent shear flows, with and
without rotation, in the context of a simple but physically motivated closure
of the equation governing the evolution of the Reynolds stress tensor. We show
that the model naturally accounts for some familiar phenomena in parallel shear
flows, such as the subcritical transition to turbulence at a finite Reynolds
number and the occurrence of a universal velocity profile close to a wall at
large Reynolds number. For rotating shear flows we find that, depending on the
Rayleigh discriminant of the system, the model predicts either linear
instability or nonlinear instability or complete stability as the Reynolds
number is increased to large values. We investigate the properties of
Couette-Taylor flows for varying inner and outer cylinder rotation rates and
identify the region of linear instability (similar to Taylor's), as well as
regions of finite-amplitude instability qualitatively compatible with recent
experiments. We also discuss quantitative predictions of the model in
comparison with a range of experimental torque measurements. Finally, we
consider the relevance of this work to the question of the hydrodynamic
stability of astrophysical accretion discs.
