We perform direct numerical simulations of spiral turbulent Taylor–Couette (TC) flow for
$400\leqslant Re_{i}\leqslant 1200$
and
$-2000\leqslant Re_{o}\leqslant -1000$
, i.e. counter-rotation. The aspect ratio
$\unicode[STIX]{x1D6E4}=\text{height}/\text{gap width}$
of the domain is
$42\leqslant \unicode[STIX]{x1D6E4}\leqslant 125$
, with periodic boundary conditions in the axial direction, and the radius ratio
$\unicode[STIX]{x1D702}=r_{i}/r_{o}=0.91$
. We show that, with decreasing
$Re_{i}$
or with decreasing
$Re_{o}$
, the formation of a turbulent spiral from an initially ‘featureless turbulent’ flow can be described by the phenomenology of the Ginzburg–Landau equations, similar as seen in the experimental findings of Prigent et al. (Phys. Rev. Lett., vol. 89, 2002, 014501) for TC flow at
$\unicode[STIX]{x1D702}=0.98$
an
$\unicode[STIX]{x1D6E4}=430$
and in numerical simulations of oblique turbulent bands in plane Couette flow by Rolland & Manneville (Eur. Phys. J., vol. 80, 2011, pp. 529–544). We therefore conclude that the Ginzburg–Landau description also holds when curvature effects play a role, and that the finite-wavelength instability is not a consequence of the no-slip boundary conditions at the upper and lower plates in the experiments. The most unstable axial wavelength
$\unicode[STIX]{x1D706}_{z,c}/d\approx 41$
in our simulations differs from findings in Prigent et al., where
$\unicode[STIX]{x1D706}_{z,c}/d\approx 32$
, and so we conclude that
$\unicode[STIX]{x1D706}_{z,c}$
depends on the radius ratio
$\unicode[STIX]{x1D702}$
. Furthermore, we find that the turbulent spiral is stationary in the reference frame of the mean velocity in the gap, rather than the mean velocity of the two rotating cylinders.