3.1. Torque

Before presenting the full set of simulated cases, it is useful to discuss possible definitions of the Taylor number and to examine whether any of them provides a consistent parametrisation of the torque for arbitrary $\eta$ in the case of a stationary outer cylinder. For example, Di Prima & Swinney (Reference Di Prima and Swinney1981) define the Taylor number as $ \textit{Ta}_d = 4 \, (1 - \eta ) / (1 + \eta ) \, \textit{Re}^2$ , whereas Ostilla-Mónico et al. (Reference Ostilla-Mónico, Verzicco and Lohse2015) use $ \textit{Ta}_o = \sigma (1 / \eta + 1) \textit{Re}^2$ , with the geometric factor $\sigma = (1 + \eta )^4 / (4 \eta )^2$ . Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a ) further propose $ \textit{Ta}_b = \sigma ^2 \textit{Re}_S^2$ . According to all these definitions, the Taylor number typically increases with $\eta$ up to $\eta = 0.5$ , and then either decreases or approaches a plateau for $\eta \gt 0.5$ . In contrast, Chandrasekhar (Reference Chandrasekhar1961, p. 296, (169)) define the Taylor number (for a stationary outer cylinder) as $ \textit{Ta}_c = 4 \textit{Re}^2 \eta ^2 / (1 - \eta )^4 / (1 + \eta )^2$ , which yields a quadratic dependence for small $\eta$ and steeper growth as $\eta \to 1$ .

The key output in TC flow is the non-dimensional torque, defined as

(3.1) \begin{align} \textit{Nu}^\omega = \frac {J^\omega }{J^\omega _0} = \frac {r^3 \left ( \langle v_r' v_\theta ' \rangle - \dfrac {1}{\textit{Re}} \dfrac {\partial (\langle v_\theta \rangle / r)}{\partial r} \right )}{J^\omega _0} = \textit{Nu}_T + \textit{Nu}_V, \hskip 1.em J^\omega _0 = \frac {1}{\textit{Re}} \frac {2 r_i r_o^2}{r_i + r_o}, \end{align}

where $ \textit{Nu}_T$ and $ \textit{Nu}_V$ are the turbulent and viscous contributions, respectively, and $J^\omega _0$ denotes the value in laminar flow. In (3.1), the angle brackets $\langle \, \rangle$ denote averaging in time and over the azimuthal ( $\theta$ ) and axial ( $z$ ) directions. Momentum conservation implies that $ \textit{Nu}^\omega$ must remain exactly constant across the radial direction; therefore, the statistical convergence of the numerical simulations can be assessed from deviations from this constancy. This requirement is particularly stringent for flows with very small $\eta$ , since a long integration time is needed for the inner cylinder to transfer angular momentum to the entire flow field. This constraint helps explain why such configurations have seldom been explored in previous simulations or experiments. Moreover, it has been found that the grid resolution can significantly influence the accuracy of the computed torque, especially in cases with extremely small inner radii.

Eckhardt, Grossmann & Lohse (Reference Eckhardt, Grossmann and Lohse2007) provided a detailed derivation of the non-dimensional torque expression starting from the Navier–Stokes equations, concluding that the proper field to consider as the transported quantity is the angular velocity, $\omega = v_\theta / r$ , rather than the azimuthal velocity or the angular momentum, $L = r v_\theta$ . From a numerical standpoint, it is important to note that radial derivatives appear in the expression for $ \textit{Nu}^\omega$ . Their evaluation on strongly non-uniform grids – required to achieve higher resolution near the walls – may depend on the variable locations and the quantity being differentiated. For instance, we have found that more accurate results near the walls are obtained by expressing

(3.2) \begin{align} {\partial \langle \omega \rangle }/{\partial r} = {\partial \langle v_\theta \rangle / r}/{\partial r} = \left ( \langle \omega _z \rangle - 2 {\langle v_\theta \rangle }/{r} \right ) / r . \end{align}

It is important to recall that in the staggered numerical discretisation that we are using, the velocity components are defined at the centres of the cell faces; hence, the vorticity components are located at the centres of the edges (e.g. $\omega _z(i,j,k+1/2)$ ). Therefore, derivatives as ${\partial \langle \omega \rangle }/{\partial r}$ are evaluated at the grid index $j$ .

For the more general case in which the outer cylinder also rotates independently, Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013 b) decomposed the numerator of (3.1), into a large-scale component – representing the mean variations in the radial and axial directions – and the turbulent fluctuations. The evaluation of this decomposition is rather involved, as it requires axially shifting the instantaneous flow field to keep the Taylor vortices at a fixed height. The results reported by Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013b , figure 4,,) show that, for the special case of a stationary outer cylinder as the present one, the two contributions are nearly identical.

To highlight the pronounced differences in the flow near the inner and outer cylinders, Eckhardt et al. (Reference Eckhardt, Grossmann and Lohse2007) noted that, at steady state,

(3.3) \begin{align} \left . \frac {\partial \langle \omega \rangle }{\partial r} \right |_o = \eta ^3 \left . \frac {\partial \langle \omega \rangle }{\partial r} \right |_i. \end{align}

The time evolution of the two sides of this relation illustrates the approach to steady state and demonstrates that, for small $\eta$ , the simulations must be run for long durations before $\eta ^3 \left . {\partial \langle \omega \rangle }/{\partial r} \right |_i$ converges to $\left . {\partial \langle \omega \rangle }/{\partial r} \right |_o$ . The relevant values are reported in table 2. It is worth noting that, for very small $\eta$ , the wall-normal gradients at the two boundaries differ significantly, with the difference decreasing rapidly as $\eta$ increases. The identity (3.3) is indeed satisfied with good accuracy. Figure 1(a) further illustrates that, for small $\eta$ , (3.3) is satisfied with fewer samples than for larger $\eta$ (see figure 1 b). Table 2 corresponds to $ \textit{Re} = 5000$ , at which a fully turbulent regime is established.

Figure 1.

Time evolution (in the last thousand time units) of the terms in the identity (3.3), namely ${\partial \omega }/{\partial r}|_o$ (solid symbols) and $-\eta ^3 {\partial \omega }/{\partial r}|_i$ (open symbols). Data at $\eta \lt 0.2$ are shown in panel (a) and data at $\eta \geqslant 0.2$ are shown in panel (b). For all cases $ \textit{Re}=5000$ , with values of $\eta$ provided in table 1.

A further verification can be made by evaluating the resolution as a fraction of the Kolmogorov scale, whose local value we estimate as

(3.4) \begin{align} \ell _{\eta } = \big(\nu ^3/\mathcal{D} \big)^{1/4}, \end{align}

where $\mathcal{D}$ is the total dissipation, which will be defined later. The grid spacing in Kolmogorov units is provided in table 1 (the values in parentheses), where $\ell _{\eta }$ is evaluated for each flow case at the peak location of the turbulent kinetic energy. Again, the table confirms that the simulations are properly resolved.

Table 2.

Flow parameters: $\eta$ , maximum and minimum per cent errors of $1-Nu^\omega /\langle Nu^\omega \rangle$ , friction Reynolds number $R_\tau =u_\tau \textit{Re}$ at the inner and outer cylinders, Kolmogorov length scale evaluated with the total dissipation at location of maximum $K^+$ , derivatives of the angular velocity at the inner and outer cylinders.

Table 2 also reports minimum and maximum values of $\varepsilon _{\textit{Nu}} = 1 - Nu^\omega /\langle Nu^\omega \rangle$ , which can be interpreted as deviations from perfect time convergence and satisfactory grid resolution. For $\eta = O(1)$ , the errors are similar to those reported by Ostilla-Mónico et al. (Reference Ostilla-Mónico, Stevens, Grossmann, Verzicco and Lohse2013). Slightly higher errors are found at low $\eta$ , which could be partly due to discretisation errors in the evaluation of the radial derivative of $\langle \omega \rangle$ under such conditions. As reported in table 2, the time convergence error strongly depends on $\eta$ . To identify the regions within the gap which contribute most to the error, in figure 2, we show profiles of $\varepsilon _{\textit{Nu}}$ . It is evident that only in the cases with extremely small $\eta$ does a peak appear near the inner cylinder, which coincides with the region where $ \textit{Nu}_V$ sharply decreases. As previously mentioned, in this region, the evaluation of radial derivatives on a non-uniform grid is sensitive to the numerical scheme used. This peak decreases as $\eta$ increases and the resulting distributions become similar to those reported by Ostilla-Mónico et al. (Reference Ostilla-Mónico, Stevens, Grossmann, Verzicco and Lohse2013), although their results refer to a lower Reynolds number $ \textit{Re} = 1600$ compared with the present case with $ \textit{Re} = 5000$ .

Figure 2.

Profiles of local torque unbalance, $\varepsilon _{\textit{Nu}}=1-Nu^\omega /\langle Nu^\omega \rangle$ , obtained by averaging the last thousand profiles saved every one non-dimensional time unit, for values of $\eta$ indicated in table 1 ( $ \textit{Re}=5000$ for all cases).

As stated in § 1, most previous numerical and experimental studies were conducted for $\eta \gt 0.5$ , for which good collapse of the $ \textit{Nu}^\omega {-}\textit{Ta}$ relationship is obtained regardless of the definition of $ \textit{Ta}$ . As a preliminary step, we have computed two of the test cases carried out by Ostilla-Mónico et al. (Reference Ostilla-Mónico, Verzicco and Lohse2015). The first case correspond to $ \textit{Re}=80$ , $\eta =0.5$ , $\varGamma =2$ , with a $32\times 64\times 64$ grid and a rotational symmetry of order four, for which we find $ \textit{Nu}^\omega =1.136$ , in good agreement with their value $ \textit{Nu}^\omega =1.138$ . The second case corresponds to $ \textit{Re}=1120$ , $\eta =5/7$ , $\varGamma =2\pi$ , which as in Ostilla-Mónico et al. (Reference Ostilla-Mónico, Verzicco and Lohse2015), we have computed with an unresolved and a fully resolved mesh, with rotational symmetry of order 4. Ostilla-Mónico et al. (Reference Ostilla-Mónico, Verzicco and Lohse2015) found that, depending on the initial conditions, four or three pairs of Taylor vortices may form, with associated slight differences in $ \textit{Nu}^\omega$ . In the unresolved simulation, we find $ \textit{Nu}^\omega =4.809$ , as compared with $ \textit{Nu}^\omega =4.835$ . In the well-resolved simulation, we find instead $ \textit{Nu}^\omega =4.688$ , as compared with $ \textit{Nu}^\omega =4.478$ , with a $4.7\,\%$ difference. This difference may be due to the way that $ \textit{Nu}^\omega$ is evaluated. For instance, we estimate $ \textit{Nu}^\omega$ by averaging the total torque along the radial direction.

Representative distributions of $ \textit{Nu}^\omega$ are reported in figure 3, as obtained in a domain with $\varGamma =2$ , without any assumed rotational symmetry. At low Reynolds number ( $ \textit{Re}=800$ ), figure 3(a) shows that for $\eta =0.0244$ , $ \textit{Nu}_T$ is smaller than $ \textit{Nu}_V$ , indicating that the flow is nearly laminar. In contrast, for $\eta =0.5$ , the flow remains turbulent and $ \textit{Nu}^\omega$ decreases with decreasing $ \textit{Re}$ . At $ \textit{Re}=1600$ , the simulation performed on a $192 \times 192 \times 128$ grid (figure 3 b) yields a nearly constant $ \textit{Nu}^\omega$ profile. It is also noteworthy that, for $\eta =0.5$ , the turbulent contribution $ \textit{Nu}_T$ decreases with distance from the inner wall near the centre of the gap, with a corresponding increase in the viscous contribution $ \textit{Nu}_V$ , thereby maintaining a constant total torque. The difference between $ \textit{Nu}_T$ and $ \textit{Nu}_V$ diminishes as $\eta$ and $ \textit{Re}$ decrease. For instance, at $\eta =0.0244$ in figure 3(b), the two contributions are approximately in balance around the gap centre. Figure 3(c) and 3(d) illustrate the influence of grid resolution at higher Reynolds numbers ( $ \textit{Re}=5000$ ). When the resolution is insufficient, as shown in figure 3(c), undulations in the profile of $ \textit{Nu}^\omega$ appear near both walls. These undulations are primarily attributed to variations in the turbulent contribution $ \textit{Nu}_T$ , and their amplitude increases with larger values of $\eta$ . In contrast, the results from higher- resolution simulations, presented in figure 3(d), show a clear reduction in the amplitude of these undulations and a lower overall value of $ \textit{Nu}^\omega$ for $\eta =0.5$ . At $\eta =0.0244$ , the magnitude of the two contributions to $ \textit{Nu}^\omega$ , shown in both figure 3(c) and 3(d), does not significantly change with resolution. However, the more refined simulation yields a more uniform $ \textit{Nu}^\omega$ profile across the entire gap.

Figure 3.

Distributions of non-dimensional total torque $ \textit{Nu}^\omega$ (lines), and its turbulent ( $ \textit{Nu}_T$ , open circles) and viscous ( $ \textit{Nu}_V$ , solid circles) components, versus distance from the inner wall for $\eta =0.0244$ (red) and $\eta =0.5$ (green): (a) $ \textit{Re}=800$ for $\eta =0.0244$ ( $192\times 192 \times 128$ grid) and for $\eta =0.5$ ( $384 \times 192 \times 128$ grid); (b) $ \textit{Re}=1600$ for $\eta =0.0244$ ( $192\times 192 \times 128$ grid) and for $\eta =0.5$ ( $384 \times 192 \times 128$ grid); (c) $ \textit{Re}=5000$ with same resolution as for $ \textit{Re}=1600$ ; (d) $ \textit{Re}=5000$ for $\eta =0.0244$ ( $256\times 256 \times 256$ grid) and for $\eta =0.5$ ( $384 \times 384 \times 256$ grid).

Figure 4.

Total torque coefficient ( $ \textit{Nu}^\omega$ ) as a function of: (a) $ \textit{Re}_S$ ; (b) $ \textit{Ta}_o$ ; (c) $ \textit{Ta}_c$ and (d) compensated $ \textit{Nu}^\omega / \textit{Ta}_c^{1/3}$ versus $ \textit{Ta}_c$ . In panel (a), the inset shows reference data from the literature: Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a ), Wendt (Reference Wendt1933), Froitzheim et al. (Reference Froitzheim, Merbold, Ostilla-Mónico and Egbers2019), Hamede, Merbold & Egbers (Reference Hamede, Merbold and Egbers2023), Ostilla-Mónico et al. (Reference Ostilla-Mónico, Stevens, Grossmann, Verzicco and Lohse2013, table 1), Ostilla-Mónico et al. (Reference Ostilla-Mónico, Van Der, Erwin, Verzicco, Grossmann and Lohse2014, table 1), Lewis & Swinney (Reference Lewis and Swinney1999) and Racina & Kind (Reference Racina and Kind2006). A large part of these data has been extracted by Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a , figure 11a). The values of $\eta$ corresponding to the present simulations are listed in table 1. Solid triangles denote our results for four cases with the same geometry of Ostilla-Mónico et al. (Reference Ostilla-Mónico, Van Der, Erwin, Verzicco, Grossmann and Lohse2014). In panel (a), the green and red lines indicate the trends $ \textit{Re}_S^{0.4}$ , while the black line represents $ \textit{Re}_S^{0.67}$ . In panel (c), the black lines indicate the trend $ \textit{Ta}^{2/9}$ and the short blue line is $27 \textit{Ta}^{1/3}$ .

In figure 4, we consider possible parametrisations of the torque coefficient, using fully resolved simulations with $\varGamma =2$ and without any rotational symmetry for flows with $\eta \leqslant 0.5$ at low and high values of Reynolds numbers. Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a ) conducted numerical simulations of TC flow and compiled both numerical and experimental data, showing in their figure 11(a) a satisfactory collapse of $ \textit{Nu}^\omega$ as a function of the shear Reynolds number $ \textit{Re}_S$ . It is important to note that their dataset corresponds to $\eta \approx 0.7$ . For these cases, at relatively low Reynolds numbers, they reported a scaling law of the form $ \textit{Nu}^\omega = \textit{Re}_S^{0.67}$ . They further noted that this exponent is close to the value $0.69$ obtained by Lewis & Swinney (Reference Lewis and Swinney1999) in experiments conducted at much higher Reynolds numbers and discussed other values of the exponent reported in the literature.

Several previous studies have examined TC flow at different values of $\eta$ . For instance, Hamede et al. (Reference Hamede, Merbold and Egbers2023) carried out experiments at the relatively small $\eta = 0.1$ , close to the present configuration, and their results as a function of $ \textit{Re}_S$ are shown in figure 4(a). These results at high Reynolds number do not align with our simulations at $\eta =0.1$ and low $ \textit{Re}$ (green solid circles in figure 4 a). Numerical results at $\eta = 0.317$ , experimental data from figure 2 of Froitzheim et al. (Reference Froitzheim, Merbold, Ostilla-Mónico and Egbers2019) and the classic measurements of Wendt (Reference Wendt1933) at very high Reynolds numbers for $\eta \gt 0.68$ are also reported in figure 4(a). These comparisons indicate that the present simulations, covering both low and high Reynolds numbers, provide an opportunity to assess whether a transitional value of $ \textit{Re}_S$ can be identified and correlated with $\eta$ . As shown in figure 4(a), for $\eta = 0.5$ (slightly smaller than the value considered by Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a )), simulations with $\varGamma = 2$ at high $ \textit{Re}$ yield the scaling $ \textit{Nu}^\omega \approx \textit{Re}_S^{0.67}$ . This trend is also consistent with the data of Froitzheim et al. (Reference Froitzheim, Merbold, Ostilla-Mónico and Egbers2019) and, despite some scatter, with the low- $ \textit{Re}$ experiments of Hamede et al. (Reference Hamede, Merbold and Egbers2023). Notably, our results for $\eta = 0.5$ align with the classical scaling law $ \textit{Nu}^\omega = \textit{Re}_S^{0.67}$ , which at high $ \textit{Re}_S$ is supported by the experimental measurements of Wendt (Reference Wendt1933) and Lewis & Swinney (Reference Lewis and Swinney1999), as well as by the high- $ \textit{Re}$ simulations of Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a ). The results of Ostilla-Mónico et al. (Reference Ostilla-Mónico, Van Der, Erwin, Verzicco, Grossmann and Lohse2014) at large Reynolds numbers also agree well with those of Wendt (Reference Wendt1933), and up to $ \textit{Re}_S = 10^5$ , they coincide with the measurements of Lewis & Swinney (Reference Lewis and Swinney1999). In the same figure, our simulation at $\eta = 0.714$ and $ \textit{Re} = 37\,400$ falls within the range where the scaling $ \textit{Nu}^\omega \approx \textit{Re}_S^{0.67}$ holds. In this regime, the relation $ \textit{Nu}^\omega \approx \textit{Ta}^{1/3}$ emerges and at higher values, a gradual tendency towards the ultimate regime becomes apparent, as discussed later. At lower Reynolds numbers ( $ \textit{Re} \lt 5000$ ), the scaling becomes $ \textit{Nu}^\omega \approx \textit{Re}_S^{0.40}$ . These latter results, obtained for $800 \lt \textit{Re} \lt 2400$ , correspond to flows where $ \textit{Nu}_T \leqslant Nu_V$ in the central gap region, although both contributions remain non-uniform. In contrast, in cases where $ \textit{Nu}^\omega \approx \textit{Re}_S^{0.67}$ , both the turbulent and viscous contributions are approximately constant across the central region, indicating fully developed turbulence unaffected by the coherent structures that typically form near the inner cylinder. This aspect will be revisited later with the aid of flow visualisations. Additionally, figure 4(a) shows that at $\eta = 0.0244$ , the torque scales as $ \textit{Nu}^\omega \approx \textit{Re}_S^{0.40}$ , suggesting that the flow is in a transitional regime – consistent with the red profiles presented in figure 3. The key conclusion from figure 4(a) is that the shear Reynolds number $ \textit{Re}_S$ does not provide a reliable parameter for collapsing the values of non-dimensional torque across all $\eta$ . However, it appears to perform reasonably well for cases with $\eta \gt 0.5$ .

Considerable effort in previous studies has focused on identifying a scaling relationship between $ \textit{Nu}^\omega$ and the Taylor number. As a first step, we plotted the simulation data for various $\eta$ and Reynolds numbers against $ \textit{Ta}_o$ . Figure 4(b) shows that the data do not collapse: for each value of $\eta$ , the relation $ \textit{Nu}^\omega \approx \textit{Ta}_o^n$ holds with a different proportionality constant and with exponent $n = 2/9$ . The data in table 1 of Ostilla-Mónico et al. (Reference Ostilla-Mónico, Stevens, Grossmann, Verzicco and Lohse2013), together with our results at intermediate $ \textit{Ta}_o$ and $\eta = 0.714$ , also follow the scaling $ \textit{Nu}^\omega \propto \textit{Ta}_o^{2/9}$ , but with a larger prefactor than observed in our simulations at $\eta \leqslant 0.5$ . These observations suggest that the dependence on $\eta$ might, in principle, be absorbed through an $\eta$ -dependent correction factor. Alternatively, using the Taylor number definition proposed by Chandrasekhar (Reference Chandrasekhar1961), figure 4(c) shows that $ \textit{Nu}^\omega$ collapses well for all $\eta$ at $ \textit{Ta}_c \leqslant 10^7$ . A modest scatter is still visible in our data relative to the reference line $0.13,\textit{Ta}_c^{2/9}$ at intermediate $ \textit{Re}$ . A similar degree of scatter appears in the data of Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a ) and Ostilla-Mónico et al. (Reference Ostilla-Mónico, Stevens, Grossmann, Verzicco and Lohse2013), both proportional to $ \textit{Ta}_c^{2/9}$ but with proportionality constants differing from that of our results at $ \textit{Re}=5000$ . The measurements of Hamede et al. (Reference Hamede, Merbold and Egbers2023) lie to the left of our data, whereas those of Wendt (Reference Wendt1933) extend further to the right. This discrepancy may partly arise because the data of Hamede et al. (Reference Hamede, Merbold and Egbers2023) were digitised from their published figures without accounting for error bars, which are substantial owing to the experimental challenges associated with torque measurements (Brauckmann & Eckhardt Reference Brauckmann and Eckhardt2013a ). The measurements of Wendt (Reference Wendt1933) at $\eta = 0.68$ and of Lewis & Swinney (Reference Lewis and Swinney1999) at $\eta =0.724$ agree well with both our results and those of Ostilla-Mónico et al. (Reference Ostilla-Mónico, Van Der, Erwin, Verzicco, Grossmann and Lohse2014) at high Reynolds numbers with $\eta = 0.714$ . In this range, they closely follow the blue line ( $27,\textit{Ta}_c^{1/3}$ ), which characterises the Taylor–Couette regime where the torque is dominated by turbulent stresses (figure 3 d). In this regime, the azimuthal velocity in wall units exhibits a logarithmic profile, with a coefficient that depends on $\eta$ , as discussed later. Figure 4(d) further shows that the normalised torque in figure 4(c) collapses onto a single curve tending towards a constant value (denoted $C_N$ ) when scaled with $ \textit{Ta}_c^{1/3}$ . The inset of figure 4(d) – including data at $\eta = 0.68$ (Wendt Reference Wendt1933), $\eta = 0.724$ (Lewis & Swinney Reference Lewis and Swinney1999) and $\eta = 0.714$ (present data and Ostilla-Mónico et al. Reference Ostilla-Mónico, Van Der, Erwin, Verzicco, Grossmann and Lohse2014) – suggests a consistent value of $C_N = 0.06$ . Finally, it is important to recall again that our conclusions regarding the parametrisation of the torque coefficient may depend critically on the chosen flow configuration, in which the Reynolds number of the inner cylinder is fixed while the radius ratio is varied. Under these conditions, both the shear and rotation Reynolds numbers vary simultaneously (Dubrulle et al. Reference Dubrulle, Dauchot, Daviaud, Longaretti, Richard and Zahn2005).

3.2. Flow visualisations

To our knowledge, TC flow with rotating inner cylinders at small $\eta$ has not been numerically or experimentally investigated. It is thus instructive to investigate the resulting flow structures within the gap. Di Prima & Swinney (Reference Di Prima and Swinney1981) presented flow structures at various Reynolds numbers, obtained by Coles (Reference Coles1965), who introduced small flat flakes into the fluid. These particles were transported by the azimuthal velocity, revealing the formation of azimuthal waves on the laminar rollers, which transition to a fully turbulent regime at higher Reynolds numbers. These visualisations stemmed from a set-up featuring $\eta = 0.88$ . DNS allows to probe the azimuthal fluctuating vorticity component $\omega _\theta '$ , which is a good indicator of the shape of the coherent flow structures, which we present in figure 5. In the case of a relatively large inner cylinder ( $\eta = 0.5$ ), at $ \textit{Re} = 800$ (figure 5 a), $\delta$ -sized rollers are visible which retain symmetry about the $z$ -axis, indicating that the structures do not exhibit azimuthal instability. The vorticity layers are found to be strong near the internal wall than at the outer wall. Additionally, in the central region, two adjacent sheets of opposite-sign vorticity form. At $ \textit{Re} = 1600$ , the azimuthal instability reported by Coles (Reference Coles1965) is clearly evident in figure 5(b), where different orientations of patches of azimuthal vorticity are visible.

Figure 5.

Contours of fluctuating azimuthal vorticity ( $\omega ^\prime _\theta /|\omega ^\prime _\theta |_{\textit{max}}$ ) in transverse ( $r$ – $z$ ) planes, (a,b,e,f) for $\eta =0.5$ ) and (c,d,g,h) $\eta =0.0244$ . (a,c) $ \textit{Re}=800$ , with $|\omega ^\prime _\theta |_{\textit{max}}=6.70$ in panel (a) and $11.45$ in panel (c); (b,d) $ \textit{Re}=1600$ , with $|\omega ^\prime _\theta |_{\textit{max}}=8.76$ in panel (b) and $37.42$ in panel (d); (e,g): $ \textit{Re}=2400$ , with $|\omega ^\prime _\theta |_{\textit{max}}=11.05$ in panel (e) and $65.54$ in panel (g); (f,h): $ \textit{Re}=5000$ , with $|\omega ^\prime _\theta |_{\textit{max}}=16.28$ in panel (f) and $131.40$ in panel (h). Green, blue and cyan denote negative values; yellow, red and magenta denote positive values, with increments $\Delta =0.02$ in panels (a), (b), (e), (f) and $\Delta =0.05$ in panels (c), (d), (g), (h).

In the case of a very small inner cylinder ( $\eta = 0.0244$ ), at $ \textit{Re} = 800$ , the vorticity contours (see figure 5 c) exhibit four weak and thin layers. At $ \textit{Re} = 1600$ , the number of axial instabilities increases and figure 5(d) conveys that these instabilities remain axisymmetric in the azimuthal direction. The vorticity patches do not fill the entire gap, remaining concentrated near the inner cylinder. This change of the flow organisation with the Reynolds number may explain why $ \textit{Nu}_T$ increases less in figure 4(d) than in figure 4(c) for $\eta =0.0244$ . Although the vorticity patches in figure 5(d) appear stronger, they do not significantly affect $ \textit{Nu}^\omega$ , which remains close to its laminar value. For $\eta = 0.5$ the vorticity patches near the inner cylinder become very thin at $ \textit{Re} = 2400$ , as shown in figure 5(e). At $ \textit{Re} = 5000$ , these patches break into small, turbulent-like structures, as shown in figure 5(f), which are concentrated near the inner cylinder. Even for $\eta = 0.024$ at $ \textit{Re} = 2400$ , the vorticity field in figure 5(g) remains axisymmetric, similar to figure 5(d); however, a higher number of instabilities is found along the axial direction. Axi-symmetry is lost at $ \textit{Re} = 5000$ (see figure 5 h), where the non-dimensional torque exceeds unity (see figure 3 b) with $ \textit{Nu}_T \gt Nu_V$ . The formation of azimuthal vorticity patches near the inner cylinder, and not near the outer cylinder, explains the larger difference in the gradients of $ \textit{Nu}_T$ and $ \textit{Nu}_V$ near the inner and the outer cylinder.

3.3. Velocity profiles

In TC flow, scaling in wall units using the friction velocity at the inner cylinder may not always be the most appropriate approach for evaluating flow statistics. This issue was examined by Bilson & Bremhorst (Reference Bilson and Bremhorst2007), who performed direct numerical simulations of Taylor–Couette flow at $\eta = 0.617$ and $ \textit{Re} = 3200$ , with a spatial resolution comparable to that of the present study. In figure 20 of their work, velocity profiles in wall units were shown near both the inner and outer cylinders, each scaled with the respective friction velocity. The authors observed that the velocity departs from the linear profile at $y^+ \approx 2$ , a value much closer to the wall than typically found in canonical wall-bounded flows such as pipes and channels. Our results are consistent with those of Bilson & Bremhorst (Reference Bilson and Bremhorst2007) and further indicate that deviations from the linear region become increasingly pronounced as the inner cylinder radius decreases.

Table 3.

Flow parameters: $\eta$ , $ \textit{Re}_s$ , $ \textit{Ta}_c$ , $ \textit{Nu}^\omega$ , $\kappa _u$ , $B_u$ , $\kappa _o$ , $B_o$ . The log-law constants for $\eta =0.5$ and $\eta =0.714$ and $ \textit{Re}_S=2.952 \times 10^4$ have been used to plot the dashed lines in the inset of figure 6(a) and 6(c).

Ostilla-Mónico et al. (Reference Ostilla-Mónico, Van Der, Erwin, Verzicco, Grossmann and Lohse2014) examined the mean azimuthal velocity profiles for $\eta = 0.714$ , and found that they fitted reasonably well to a logarithmic law, with Kármán constant ( $\kappa _u$ ) and additive constant ( $B_u$ ) depending on the Reynolds number. Their study was inspired by Grossmann, Lohse & Sun (Reference Grossmann, Lohse and Sun2014), who analysed the differences between the azimuthal velocity, $U^+ = \langle V_i - v_\theta \rangle / u_{\tau }|_i$ , and the angular velocity, $\varOmega ^+ = \langle V_i / r_i - v_\theta / r \rangle r_i / u_{\tau }|_i$ . They showed that, for large values of $\rho ^+ = (r - r_i)^+$ , both quantities follow logarithmic profiles reasonably well. The Kármán constant is expected to depend on the flow conditions: at fixed $\eta$ , it may vary with $ \textit{Re}$ and at fixed $ \textit{Re}$ , it may vary with $\eta$ . This is indeed the case when the flow is fully turbulent. The present simulations make it possible to verify this dependence and demonstrate that TC flow differs from the usual behaviour of wall-bounded turbulence, in which at high $ \textit{Re}$ , the constants $\kappa _u$ and $B_u$ are independent of Reynolds number. To determine these constants, the diagnostic function $\varXi = {\rm d}F^+ / {\rm d}\log (\rho ^+)$ was evaluated. The value of $\kappa _u$ was obtained from the minimum of $\varXi$ for $\rho ^+ \gt 15$ and, from the location of this minimum, the corresponding $B_u$ was estimated. For flows that do not exhibit a minimum in $\varXi$ , a logarithmic fit cannot be established, indicating that the Reynolds number must be increased to achieve a fully turbulent azimuthal velocity profile. This condition was indeed met by performing a simulation at $\eta = 0.1$ and $ \textit{Re} = 10\,000$ , which yielded logarithmic profiles whose constants are reported in table 3,

(3.5) \begin{align} U^+ = 1/\kappa _u \log \rho ^+ + B_u, \qquad \varOmega ^+ = 1/\kappa _o \log \rho ^+ + B_o. \end{align}

We focus on the $U^+$ profiles since, as seen in the inset of figure 6(a) and in figure 6(c), they appear straighter than the corresponding $\varOmega ^+$ profiles, consistent with the results of Grossmann et al. (Reference Grossmann, Lohse and Sun2014). The values of $1/\kappa _u$ denoted as NA in table 3 indicate that the profiles bend downward, as shown in figure 6(c) for $\eta \leqslant 0.1$ . For other $\eta$ at the same $ \textit{Re}$ , the profiles are nearly parallel and, consequently, $\kappa _u$ increases slightly with increasing $\eta$ , while a small increase in $B_u$ causes the profiles in figure 6(c) to converge. At $\eta = 0.714$ and $ \textit{Re} = 25000$ , the present profiles agree closely with those of Ostilla-Mónico et al. (Reference Ostilla-Mónico, Van Der, Erwin, Verzicco, Grossmann and Lohse2014) and do not differ significantly from the profiles at $ \textit{Re} = 37\,600$ . In this case, table 3 shows that $\kappa _u$ increases with $ \textit{Re}$ at intermediate Reynolds numbers, indicating a transition regime from $ \textit{Nu} \approx \textit{Ta}_c^{2/9}$ to $ \textit{Nu} \approx \textit{Ta}_c^{1/3}$ . At higher Reynolds numbers, the $U^+$ profiles exhibit $\kappa _u \approx 1$ and further increases in $ \textit{Re}$ merely extend the logarithmic region.

Figure 6.

Profiles of mean relative velocity (a,c) near the inner cylinder $U^+=\langle V_i-v_\theta \rangle ^+$ and (b,d) near the outer cylinder ( $\langle v_\theta \rangle |_o^+$ . Panels (a) and (b) correspond to $\eta =0.714$ , at the Reynolds numbers indicated in the inset, where lines denote the results of Ostilla-Mónico et al. (Reference Ostilla-Mónico, Van Der, Erwin, Verzicco, Grossmann and Lohse2014) and open symbols denote the present results. In panels (c) and (d), we report the velocity profiles at various $\eta$ as in table 1. The solid lines denote DNS data of pipe flow at $ \textit{Re}_\tau =550$ (Pirozzoli et al. Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021). The insets of panels (a) and (c) report profiles of $U^+$ and $\varOmega ^+$ plotted at $ \textit{Re}=25\,000$ in panel (a) and at $\eta =0.5$ in panel (c), the profiles of $U^+$ and $\varOmega ^+$ (symbols), and the fitting logarithmic profiles, with constants given in table 3.

Near the outer cylinder, only at $\eta = 0.5$ does a short, straight layer with a logarithmic $U^+$ profile appear, as shown in figure 6(d). For all other $\eta$ , the profiles remain largely laminar-like. In contrast, at $\eta = 0.714$ , figure 6(b) shows a pronounced $ \textit{Re}$ -dependence at intermediate $ \textit{Re}$ , which vanishes in the $ \textit{Ta}_c^{1/3}$ regime, where the flow becomes independent of $ \textit{Re}$ . At present, we have not been able to determine the log-law constants for $\eta \lt 0.5$ , as reaching sufficiently high Reynolds numbers demands a very large computational effort.

3.4. TKE budgets

In canonical wall-bounded flows, the inner-scaled velocity profiles are typically independent of the Reynolds number. However, the peak value of TKE in the buffer layer increases slightly with $ \textit{Re}$ , and there is still no consensus on whether this growth saturates as $ \textit{Re} \rightarrow \infty$ . Based on the simulations of Pirozzoli et al. (Reference Pirozzoli, Romero, Fatica, Verzicco and Orlandi2021), the $ \textit{Re}$ -dependence of $K^+$ appears to arise primarily from the increase in the streamwise stress. Figure 6 shows that, for Taylor–Couette flow at intermediate Reynolds numbers, the velocity profiles depend strongly on $\eta$ for $\eta \leqslant 0.5$ . At larger $\eta$ , however, the Reynolds-number dependence weakens and tends to vanish at high $ \textit{Re}$ . It is therefore of interest to examine how this behaviour influences the TKE profiles and the corresponding energy budgets. In figure 7(a), the TKE profiles in wall units display a marked increase in the $K^+$ peak with $\eta$ , further amplified by the rise of $u_\tau$ with $\eta$ . Despite this variation, the location of the peak changes only weakly with $\eta$ .

Figure 7.

(a) Inner-scaled turbulent kinetic energy $K$ . The other panels report the turbulent kinetic energy budgets for TC flow: (b) $\eta =0.0244$ ; (c) $\eta =0.049$ ; (d) $\eta =0.1$ ; (e) $\eta =0.5$ ; (f) $\eta =0.714$ . The symbols denote production (green), total dissipation (red), pressure–velocity correlation (blue) and turbulent transfer (magenta).

This behaviour is more clearly reflected in the TKE budgets, reported in the other panels of figure 7. In pipe flow (not shown), the budget is characterised by an approximate balance between production ( $P = -1/r \langle v_r^\prime v^\prime _{\theta } \rangle {\rm d}(\langle v_\theta \gt r)/{\rm d}r$ ) and total dissipation, defined as $\mathcal{D} =\nu u_i {\nabla} ^2 u_i$ . In pressure-driven curved channel flow, Orlandi et al. (Reference Orlandi, Soldati and Pirozzoli2025) found that the TKE budget near the inner wall exhibits a very different behaviour for small $\eta$ , with a positive contribution from the pressure–velocity correlation term ( $\varPi = -1/r\, {\rm d}/{{\rm d}r} ( r \langle v_r^\prime p^\prime \rangle )$ ), which is nearly balanced by total dissipation. In that case, the production term is of minor importance and turbulent diffusion is negative, being smaller than the total dissipation.

The TKE budgets for TC flow at $\eta = 0.0244$ , shown in figure 7(b), exhibit a similar trend to that observed in curved channel flow. However, here the magnitude of the production term is comparable to that of the pressure–velocity correlation. The magnitude of the inner-scaled budget terms decreases with increasing $\eta$ and a universal behaviour is not achieved. Notably, the location of the production peak, in figure 7, remains almost unchanged as $\eta$ varies, coinciding with the position where the viscous stress equals the turbulent stress. As $\eta$ increases, the amplitude of the pressure–velocity correlation term decreases, indicating a change in the pressure distribution near the inner cylinder. At the same time, the turbulent transport term, $T_T = -1{\kern-1pt}/{\kern-1pt}r {\rm d}{\kern-1pt}/{\kern-1pt}{\rm d}r ( r \langle v_r^\prime v_i^\prime v_i^\prime \rangle )$ , increases and compensates for the TKE dissipation. The pronounced increase of $\varPi$ near the rotating inner cylinder can be attributed to the formation of intense, alternating fluctuations of radial velocity that extend azimuthally in short patches, as shown in figure 5(f). For $\eta \gt 0.5$ , these structures exhibit undulations and strong correlation between $v_r^\prime$ and $p^\prime$ .