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.

中文翻译：

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螺旋 Taylor-Couette 湍流的直接数值模拟

我们对 $400\leqslant Re_{i}\leqslant 1200$ 和 $-2000\leqslant Re_{o}\leqslant -1000$ 的螺旋湍流 Taylor-Couette (TC) 流进行直接数值模拟，即反向旋转。域的纵横比 $\unicode[STIX]{x1D6E4}=\text{height}/\text{gap width}$ 为 $42\leqslant \unicode[STIX]{x1D6E4}\leqslant 125$ ，具有周期性边界条件在轴向，半径比 $\unicode[STIX]{x1D702}=r_{i}/r_{o}=0.91$ 。我们表明，随着 $Re_{i}$ 的减少或 $Re_{o}$ 的减少，从最初的“无特征湍流”流动形成湍流螺旋可以通过金茨堡-朗道方程的现象学来描述，类似正如 Prigent 等人的实验结果所见。(Phys. Rev. Lett., vol. 89, 2002, 014501) 用于 $\unicode[STIX]{x1D702}=0 的 TC 流。98$ 和 $\unicode[STIX]{x1D6E4}=430$ 以及 Rolland & Manneville 对平面 Couette 流中倾斜湍流带的数值模拟（Eur. Phys. J.，vol. 80, 2011, pp. 529–544 ）。因此，我们得出结论，当曲率效应起作用时，Ginzburg-Landau 描述也成立，并且有限波长不稳定性不是实验中上板和下板的无滑移边界条件的结果。我们模拟中最不稳定的轴向波长 $\unicode[STIX]{x1D706}_{z,c}/d\approx 41$ 与 Prigent 等人的发现不同。，其中 $\unicode[STIX]{x1D706}_{z,c}/d\approx 32$ ，因此我们得出结论 $\unicode[STIX]{x1D706}_{z,c}$ 取决于半径比$\unicode[STIX]{x1D702}$ 。此外，我们发现湍流螺旋在间隙中的平均速度的参考系中是静止的，