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.
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