  Direct numerical simulations of the Taylor-Couette (TC) problem, the flow
between two coaxial and independently rotating cylinders, have been performed.
The study focuses on TC flow with mild curvature (small gap) with a radius
ratio of $\eta=r_i/r_o=0.909$, an aspect ratio of $\Gamma=L/d=2\pi/3$, and a
stationary outer cylinder. Three inner cylinder Reynolds of $1\cdot10^5$,
$2\cdot10^5$ and $3\cdot 10^5$ were simulated, corresponding to frictional
Reynolds numbers between $Re_\tau\approx 1400$ and $Re_\tau \approx 4000$. An
additional case with a large gap, $\eta=0.5$ and driving of $Re=2\cdot10^5$ was
also performed. Small-gap TC was found to be dominated by spatially-fixed
large-scale structures, known as Taylor rolls (TRs). TRs are attached to the
boundary layer, and are active, i.e. they transport angular velocity through
Reynolds stresses. An additional simulation with inner cylinder Reynolds number
of $Re=1\cdot10^5$ and fixed outer cylinder with an externally imposed axial
flow of comparable strength as the wind of the TRs was also conducted. The
axial flow was found to convect the TRs without any weakening effect. For
small-gap TC, evidence for the existence of logarithmic velocity fluctuations,
and of an overlap layer, in which the velocity fluctuations collapse in outer
units, was found. Profiles consistent with a logarithmic dependence were also
found for the angular velocity in large-gap TC, albeit in a very reduced range
of scales. Finally, the behaviour of both small- and large-gap TC was compared
to other canonical flows. Small-gap TC has similar behaviour in the near-wall
region to other canonical flows, while large-gap TC displays very different
behaviour.
