  A counter-rotating Taylor-Couette flow with relatively small radius ratios of
$\eta$ = 0.2--0.5 was investigated over a wide range of the Reynolds number,
from laminar to turbulent regime, by means of three-dimensional direct
numerical simulations. We investigated the $\eta$ dependence of the flow
structure and determined a critical value between $\eta$ = 0.2 and 0.3, below
which, the stable outer cylinder side exhibited a modal structure that was
different from the Taylor-vortex flow on the inner side. At $\eta \geq$ 0.3,
the Taylor-vortex on the unstable inner side dominated the entire flow field
between the cylinders, whose footprints were observed in the vicinity of the
outer cylinder wall. However, for $\eta$ = 0.2, the influence from the inner
side was limited up to the centre of the cylinder gap. Moreover, on the stable
outer cylinder side, there appeared a modal structure that was axially
homogeneous, azimuthally periodic, and similar to the Tollmien-Schlichting (TS)
instability wave. As the Reynolds number increased with a fixed $\eta$ = 0.2,
the modal structure changed its azimuthal wavenumber and thickened radially in
the wall unit. Although the Reynolds shear stress on the outer side remained
approximately zero, the intensity of the velocity fluctuations was comparable
to the Taylor-vortex flows in the central part.
