  Taylor--Couette (TC) flow is the shear-driven flow between two coaxial
independently rotating cylinders. In recent years, high-fidelity simulations
and experiments revealed the shape of the streamwise and angular velocity
profiles up to very high Reynolds numbers. However, due to curvature effects,
so far no theory has been able to correctly describe the turbulent streamwise
velocity profile for given radius ratio, as the classical Prandtl--von
K\'arm\'an logarithmic law for turbulent boundary layers over a flat surface at
most fits in a limited spatial region.
  Here we address this deficiency by applying the idea of a Monin--Obukhov
curvature length to turbulent TC flow. This length separates the flow regions
where the production of turbulent kinetic energy is governed by pure shear from
that where it acts in combination with the curvature of the streamlines. We
demonstrate that for all Reynolds numbers and radius ratios, the mean
streamwise and angular velocity profiles collapse according to this separation.
We then derive the functional form of the velocity profile. Finally, we match
the newly derived angular velocity profile with the constant angular momentum
profile at the height of the boundary layer, to obtain the dependence of the
torque on the Reynolds number, or, in other words, of the generalized Nusselt
number (i.e., the dimensionless angular velocity transport) on the Taylor
number.
