2.1. Parameters for the limit of vanishing curvature

We are interested in the transition from TCF to RPCF, in the limit when the radius ratio ${\it\eta}=r_{i}/r_{o}$ approaches one. This limit is often called the small-gap limit, since it can be obtained for fixed outer radius when the radius of the inner cylinder approaches that of the outer one and the gap width $d=r_{o}-r_{i}$ ultimately vanishes. However, in units of the width $d$ , the radii increase like $r_{i},r_{o}\sim 1/(1-{\it\eta})$ and diverge as ${\it\eta}$ approaches one; this is why this situation was called the ‘limit of large radii’ in Faisst & Eckhardt (Reference Faisst and Eckhardt2000). Moreover, the curvature vanishes ( $R_{C}\rightarrow 0$ ) as ${\it\eta}\rightarrow 1$ , suggesting that curvature effects become less important. However, rotation effects can be maintained.

Figure 2.

Transformation between the ratio of angular velocities ${\it\mu}={\it\omega}_{o}/{\it\omega}_{i}$ and the rotation number $R_{{\it\Omega}}$ for various radius ratios ${\it\eta}$ . For TC systems with vanishing curvature ( ${\it\eta}\rightarrow 1$ ), most ${\it\mu}$ -values result in almost no system rotation ( $R_{{\it\Omega}}\approx 0$ ) and $R_{{\it\Omega}}>0$ is achieved only for ${\it\mu}$ close to one. The symbols mark the values of ${\it\mu}$ and $R_{{\it\Omega}}$ realised by our simulations for $Re_{S}=2\times 10^{4}$ .

For every radius ratio ${\it\eta}$ , the traditional Reynolds numbers can be expressed in terms of the shear Reynolds number and the rotation number as

(2.10a,b ) $$\begin{eqnarray}Re_{i}={\displaystyle \frac{Re_{S}}{2}}+{\displaystyle \frac{{\it\eta}}{2(1-{\it\eta})}}Re_{S}\,R_{{\it\Omega}},\quad Re_{o}=-{\displaystyle \frac{Re_{S}}{2}}+{\displaystyle \frac{1}{2(1-{\it\eta})}}Re_{S}\,R_{{\it\Omega}}\end{eqnarray}$$

and more specifically as $Re_{i}=Re_{S}/2$ and $Re_{o}=-Re_{S}/2$ in the non-rotating case ( $R_{{\it\Omega}}=0$ ) independent of ${\it\eta}$ . In contrast, for finite rotation ( $R_{{\it\Omega}}\neq 0$ ), the cylinder Reynolds numbers $Re_{i}$ and $Re_{o}$ diverge in the vanishing-curvature limit as ${\it\eta}\rightarrow 1$ . As a consequence, the ratio of angular velocities,

(2.11) $$\begin{eqnarray}{\it\mu}={\it\eta}{\displaystyle \frac{Re_{o}}{Re_{i}}}={\displaystyle \frac{-{\it\eta}(1-{\it\eta})+{\it\eta}R_{{\it\Omega}}}{(1-{\it\eta})+{\it\eta}R_{{\it\Omega}}}},\end{eqnarray}$$

converges to ${\it\mu}=-1$ for $R_{{\it\Omega}}=0$ and to ${\it\mu}=1$ for any $R_{{\it\Omega}}\neq 0$ in the vanishing-curvature limit, as shown in figure 2. These limiting values illustrate that the transition between TCF and RPCF is singular in the traditional parameters, $Re_{i}$ , $Re_{o}$ and ${\it\mu}$ , but not in the parameters $Re_{S}$ and $R_{{\it\Omega}}$ that we will generally use here. We note that one exception to this parameter choice occurs in § 3 where the effect of curvature for counter-rotating cylinders can be characterised more adequately by the ratio of angular velocities ${\it\mu}$ . Furthermore, while $R_{C}$ measures the curvature and thus clarifies its disappearance ( $R_{C}\rightarrow 0$ ) for ${\it\eta}\rightarrow 1$ , we will identify the different TC geometries by the radius ratio ${\it\eta}$ as this parameter is more commonly used.

2.2. Momentum transport in Taylor–Couette and rotating plane Couette flow

To study the mean turbulent characteristics of the flow, we calculate wall-normal profiles of the velocity and other flow fields by averaging over surfaces parallel to the wall. The averages are denoted by $\langle \cdots \rangle$ . The surfaces are concentric cylinders in the case of TCF, $\langle \cdots \rangle (r)=\langle \cdots \rangle _{{\it\varphi}z}$ , and planes parallel to the bounding walls in the case of RPCF, $\langle \cdots \rangle (y)=\langle \cdots \rangle _{xz}$ . Equations for the averages follow from the Navier–Stokes equation. The average of the ${\it\varphi}$ -component of the equation of motion (2.5) of TCF results in the continuity equation for the specific angular momentum $\mathscr{L}=ru_{{\it\varphi}}$ ,

(2.12) $$\begin{eqnarray}\partial _{t}\langle \mathscr{L}\rangle +r^{-1}\partial _{r}(rJ^{\mathscr{L}})=0\quad \text{with }J^{\mathscr{L}}=\langle u_{r}\mathscr{L}\rangle -Re_{S}^{-1}r^{2}\partial _{r}\langle {\it\omega}\rangle ,\end{eqnarray}$$

where $J^{\mathscr{L}}$ denotes the angular-momentum flux in dimensionless units (Marcus Reference Marcus1984), ${\it\omega}=u_{{\it\varphi}}/r$ the angular velocity and $r^{-1}\partial _{r}(rJ^{\mathscr{L}})$ the divergence in radial direction. In the statistically stationary state, where $\partial _{t}\langle \mathscr{L}\rangle =0$ , the transverse current $J^{{\it\omega}}=rJ^{\mathscr{L}}$ becomes independent of the radius (Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007b ). In physical units, the torque $T$ needed to drive the cylinders can be calculated from the dimensionless angular-momentum flux $J^{\mathscr{L}}$ as $T={\it\rho}_{f}{\it\nu}^{2}d\,Re_{S}^{2}\,A(r)\,J^{\mathscr{L}}$ , with the dimensionless cylindrical surface area $A(r)=2{\rm\pi}rL_{z}$ and the mass density of the fluid ${\it\rho}_{f}$ . The remaining prefactors compensate for the non-dimensionalization of (2.12). We define the dimensionless torque per axial length,

(2.13) $$\begin{eqnarray}G={\displaystyle \frac{T}{2{\rm\pi}L_{z}{\it\rho}_{f}{\it\nu}^{2}d}}=Re_{S}^{2}\,rJ^{\mathscr{L}}=Re_{S}^{2}\,J^{{\it\omega}},\end{eqnarray}$$

and introduce the Nusselt number,

(2.14) $$\begin{eqnarray}Nu_{{\it\omega}}=G/G_{lam}=J^{{\it\omega}}/J_{lam}^{{\it\omega}},\end{eqnarray}$$

that measures the torque in units of the torque of the laminar profile $G_{lam}$ (Dubrulle & Hersant Reference Dubrulle and Hersant2002; Eckhardt, Grossmann & Lohse Reference Eckhardt, Grossmann and Lohse2007a ).

Equations corresponding to (2.12)–(2.14) can also be derived for RPCF. Averaging the downstream component of the equation of motion (2.1) results in a continuity equation for the specific momentum in the $x$ -direction

(2.15) $$\begin{eqnarray}\partial _{t}\langle u_{x}\rangle +\partial _{y}J^{u}=0\quad \text{with }J^{u}=\langle u_{y}u_{x}\rangle -Re_{S}^{-1}\partial _{y}\langle u_{x}\rangle\end{eqnarray}$$

(see also Pope Reference Pope2000, § 7.1), where $J^{u}$ denotes the momentum flux in dimensionless units. In the statistically stationary state, it is independent of the wall-normal coordinate $y$ . In analogy to the torque $T$ , the force $F$ needed to keep the plane walls at a constant velocity can be calculated from the momentum flux $J^{u}$ multiplied by the dimensionless surface area $A_{xz}=L_{x}L_{z}$ through which the flux passes, $F={\it\rho}_{f}{\it\nu}^{2}Re_{S}^{2}\,A_{xz}\,J^{u}$ . Finally, we define the Nusselt number for the momentum transport in RPCF as in (2.14),

(2.16) $$\begin{eqnarray}Nu_{u}=F/F_{lam}=J^{u}/J_{lam}^{u},\end{eqnarray}$$

that measures the force in units of the value $F_{lam}$ for the laminar flow.

This completes our set of relations between TCF and RPCF that we want to study here. We have identified correspondences between the continuity equations (2.12) and (2.15), the torque and the force ( $T\leftrightarrow F$ ), the angular-momentum flux and the momentum flux ( $J^{\mathscr{L}}\leftrightarrow J^{u}$ ), the Nusselt numbers ( $Nu_{{\it\omega}}\leftrightarrow Nu_{u}$ ) and the mean profiles that enter the diffusive part of the flux in (2.12) and (2.15) ( $\langle {\it\omega}\rangle (r)\leftrightarrow \langle u_{x}\rangle (y)$ ). Since we study the statistically stationary cases only, we also average in time to improve the statistics. Using the dimensionless shear Reynolds number $Re_{S}$ and rotation number $R_{{\it\Omega}}$ (2.2a,b ), we can now study the momentum transport in both systems within the same framework and in a non-singular manner.

We conclude this section by pointing out a peculiar property of the relations that we will take up again in later sections. While the equations of motion (2.1) and (2.5) depend on both driving parameters $Re_{S}$ and $R_{{\it\Omega}}$ , the momentum transport equations (2.12) and (2.15) do not explicitly depend on the rotation number $R_{{\it\Omega}}$ , as noted for RPCF by Salewski & Eckhardt (Reference Salewski and Eckhardt2015). The rotation only indirectly influences the momentum fluxes $J^{\mathscr{L}}$ and $J^{u}$ by changing the turbulent velocity correlations ( $\langle u_{r}\mathscr{L}\rangle$ and $\langle u_{y}u_{x}\rangle$ ) and the mean profiles ( $\langle {\it\omega}\rangle$ and $\langle u_{x}\rangle$ ), as we will discuss for the correlations in §§ 4.1.1 and 5.3 and for the profiles in § 5.2.

2.3. Investigated parameter range

Studies for different ${\it\eta}$ (or $R_{C}$ ) give information on when the properties of RPCF emerge from TCF, and studies of different $Re_{S}$ and $R_{{\it\Omega}}$ give information on the impact of shear and rotation on the turbulence. The range of parameters explored is indicated in figure 3. They were selected as follows.

Figure 3.

Parameter space $(R_{C},Re_{S},R_{{\it\Omega}})$ explored by DNS of TCF and RPCF to study the limit of vanishing curvature, i.e. $R_{C}\rightarrow 0$ equivalent to ${\it\eta}\rightarrow 1$ . (a) For the shear Reynolds numbers $Re_{S}$ and curvature numbers $R_{C}$ indicated here, several simulations with $R_{{\it\Omega}}\in (-0.3,0.9)$ were performed. (b) The range of rotation numbers $R_{{\it\Omega}}$ explored is shown here for $Re_{S}=2\times 10^{4}$ and different curvature numbers $R_{C}=(1-{\it\eta})/\sqrt{{\it\eta}}$ . The curvature numbers correspond to the radius ratios ${\it\eta}=0.99$ , $0.98$ , $0.95$ , $0.90$ , $0.80$ , $0.71$ and $0.5$ . The extra columns for $R_{C}=0$ summarise our RPCF simulations.

For each curvature number $R_{C}$ the evolution of the rotation dependence is analysed with increasing shear, by realising various mean rotation states for three shear Reynolds numbers $Re_{S}=5\times 10^{3}$ , $1\times 10^{4}$ and $2\times 10^{4}$ . Since we adopt part of the results for ${\it\eta}=0.71$ from our previous study (Brauckmann & Eckhardt Reference Brauckmann and Eckhardt2013a ), the simulated values of $Re_{S}=3899$ , $8772$ , $19\,737$ deviate slightly from the target values, as indicated in figure 3(a). In the following, we will skip over these small differences in $Re_{S}$ , as they are not significant within statistical fluctuations. Our highest value of $Re_{S}$ lies in the range where the transition to the fully turbulent state was observed in experiments for ${\it\eta}\sim 0.7$ and stationary outer cylinder (Lathrop, Fineberg & Swinney Reference Lathrop, Fineberg and Swinney1992; Lewis & Swinney Reference Lewis and Swinney1999). We performed simulations for the range of rotation numbers $R_{{\it\Omega}}$ where the occurrence of a torque maximum was observed for medium radius ratios ${\it\eta}$ (Dubrulle et al. Reference Dubrulle, Dauchot, Daviaud, Longaretti, Richard and Zahn2005; Paoletti & Lathrop Reference Paoletti and Lathrop2011; van Gils et al. Reference van Gils, Huisman, Bruggert, Sun and Lohse2011; Paoletti et al. Reference Paoletti, van Gils, Dubrulle, Sun, Lohse and Lathrop2012; Brauckmann & Eckhardt Reference Brauckmann and Eckhardt2013a ; Merbold et al. Reference Merbold, Brauckmann and Egbers2013; Ostilla et al. Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013); this allowed us to extend these studies to ${\it\eta}$ close to one. Moreover, rotation numbers that correspond to strong counter-rotation ${\it\mu}<0$ were selected in order to study the fate of the turbulent intermittency in the outer region (Coughlin & Marcus Reference Coughlin and Marcus1996; van Gils et al. Reference van Gils, Huisman, Grossmann, Sun and Lohse2012; Brauckmann & Eckhardt Reference Brauckmann and Eckhardt2013b ). Thus, the simulated rotation numbers lie mainly in the range $-0.1\leqslant R_{{\it\Omega}}\leqslant 0.6$ as shown in figure 3(b), except for $R_{C}=0.02$ ( ${\it\eta}=0.98$ ) where we focus on a narrow range in $R_{{\it\Omega}}$ only for a study of the intermittency for counter-rotation. The case of no system rotation, i.e. $R_{{\it\Omega}}=0$ , corresponds to counter-rotating cylinders with ${\it\omega}_{o}=-{\it\eta}{\it\omega}_{i}$ . For the highest shear Reynolds number $Re_{S}=2\times 10^{4}$ , the evolution of the rotation dependence is studied for seven curvature numbers $R_{C}>0$ (TCF) and for $R_{C}=0$ (RPCF), as listed in table 1.

Table 1.

Domain size and spatial resolution at the highest $Re_{S}=2\times 10^{4}$ for the different radius ratios ${\it\eta}$ and corresponding curvature numbers $R_{C}$ . The azimuthal periodicity $n_{sym}$ was introduced for the TC simulations to reduce the natural azimuthal length to $L_{x}=2{\rm\pi}r_{a}/n_{sym}$ with the mean radius $r_{a}=(r_{i}+r_{o})/2$ , while the wall-normal (radial) and spanwise (axial) lengths were kept at $L_{y}=1$ and $L_{z}=2$ . $M_{x}$ , $M_{y}$ and $M_{z}$ give the highest order of spectral modes in the corresponding direction. The amplitudes of the highest spectral modes $\widetilde{a}_{x}$ , $\widetilde{a}_{y}$ and $\widetilde{a}_{z}$ estimate the truncation error of the expansion in the corresponding direction. ${\rm\Delta}x^{+}$ , ${\rm\Delta}y^{+}$ and ${\rm\Delta}z^{+}$ denote the grid spacing in inner wall units ${\it\delta}_{{\it\nu},i}$ at the inner cylinder. We listed the maximal amplitude and grid spacing values over all $R_{{\it\Omega}}$ .

2.4. Numerical codes and tests of convergence

For our investigations of TCF and RPCF we perform DNS that approximate a solution to the incompressible Navier–Stokes equation (2.5)–(2.6) and (2.1), respectively. In these simulations we introduce a spanwise (axial) periodicity of length $L_{z}=2$ which is large enough to accommodate one pair of counter-rotating Taylor vortices. Turbulent Taylor vortices are the typical flow structures at and around the torque maximum, and they are suppressed for strongly co- or counter-rotating cylinders (Brauckmann & Eckhardt Reference Brauckmann and Eckhardt2013b ; Huisman et al. Reference Huisman, van der Veen, Sun and Lohse2014; Ostilla-Mónico et al. Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014b ). In RPCF, and as will be shown in large radius ratio TCF, there can exist two peaks in the momentum transport, both of which show turbulent Taylor vortices (Salewski & Eckhardt Reference Salewski and Eckhardt2015). In the cases studied here, the flows near the maxima typically consist of a single vortex pair. Corresponding flow visualizations are shown in Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a ) for TCF and in Salewski & Eckhardt (Reference Salewski and Eckhardt2015) for RPCF with $L_{z}={\rm\pi}$ . In addition, a streamwise periodicity of length $L_{x}=2{\rm\pi}$ was assumed in the RPCF simulations while TCF is naturally periodic in this direction. However, we only simulated a domain of reduced azimuthal length $L_{x}$ that repeats $n_{sym}$ times to fill the entire cylinder circumference. For ${\it\eta}=0.5$ and ${\it\eta}=0.71$ we tested that the reduced azimuthal length does not bias the computed torques for a stationary outer cylinder. The length $L_{x}$ was further increased for larger radius ratios, see table 1.

As discussed in more detail in Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a ), simulating slightly longer or higher domains that are compatible with the natural size of vortices, set by the gap width, i.e. domains that contain more vortices of the same size, does not change the torque. The torques from our previous study (Brauckmann & Eckhardt Reference Brauckmann and Eckhardt2013b ) are compatible with results obtained in a domain that is approximately three times higher and completely winds around the cylinder (Ostilla et al. Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013). Also at higher $Re_{S}\sim 10^{5}$ , Ostilla-Mónico, Verzicco & Lohse (Reference Ostilla-Mónico, Verzicco and Lohse2015) showed that doubling the domain length or height leaves the torque and streamwise velocity profile almost unchanged. However, they noticed small differences in the velocity fluctuations, which would only slightly affect our results. The RPCF simulations with $L_{z}={\rm\pi}$ instead of $L_{z}=2$ reported in Salewski & Eckhardt (Reference Salewski and Eckhardt2015) show the same flow characteristics: two maxima in $Nu_{u}$ , the relevance of Taylor vortices for these maxima and a variation of the mean profile gradient with $R_{{\it\Omega}}$ .

For the TC simulations we employ a numerical code developed by Marc Avila and described in Meseguer et al. (Reference Meseguer, Avila, Mellibovsky and Marques2007). To accomplish large rotation numbers $R_{{\it\Omega}}\sim 0.6$ for radius ratios close to 1, both cylinders have to co-rotate rapidly and reach individual Reynolds numbers $Re_{i}$ and $Re_{o}$ of up to $6\times 10^{5}$ while maintaining the shear $Re_{S}=2\times 10^{4}$ . Since these high absolute velocities introduce numerical instabilities, we perform these simulation cases in a reference frame rotating with the angular velocity ${\it\Omega}_{rf}$ defined in (2.3). For the RPCF simulations we use Channelflow (Gibson, Halcrow & Cvitanović Reference Gibson, Halcrow and Cvitanović2008; Gibson Reference Gibson2012), a pseudospectral code developed by John F. Gibson that was modified to include the Coriolis force, $-R_{{\it\Omega}}\boldsymbol{e}_{z}\times \boldsymbol{u}$ . Both numerical codes use an expansion of the velocity field in Fourier modes in the two periodic directions and in Chebyshev polynomials in the wall-normal direction, and employ a semi-implicit scheme for the time stepping. We denote the highest order of modes employed in the streamwise, wall-normal and spanwise directions by $M_{x}$ , $M_{y}$ and $M_{z}$ , respectively.

The truncation error of the expansions can be assessed by the amplitudes of the highest modes $\widetilde{a}_{x}$ , $\widetilde{a}_{y}$ and $\widetilde{a}_{z}$ which are normalised by the globally strongest mode. Our resolution tests in Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a ) revealed that these amplitudes have to drop to ${\sim}\!10^{-4}$ for the Reynolds numbers $Re_{S}\sim 10^{4}$ simulated here, in order to reach a converged torque computation. All present simulations fulfil this criterion. It turned out that the resolution in the axial direction could be reduced for larger ${\it\eta}$ while maintaining $\widetilde{a}_{z}\lesssim 10^{-4}$ , cf. table 1. This seems plausible since the angular velocity gradients near the inner cylinder are steeper for small ${\it\eta}$ (Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007b ) and thus require a higher resolution. Consequently, the viscous length scale at the inner cylinder  ${\it\delta}_{{\it\nu},i}={\it\nu}/u_{{\it\tau},i}=\sqrt{{\it\eta}/(Re_{S}\,Nu_{{\it\omega}})}$ increases with ${\it\eta}$ so that the axial grid spacing in units of ${\it\delta}_{{\it\nu},i}$ remains at ${\rm\Delta}z^{+}\approx 4$ for the maximal $Nu_{{\it\omega}}$ -value at $Re_{S}=2\times 10^{4}$ . Nevertheless, we did not reduce the resolution in the radial direction for large ${\it\eta}$ so that the wall-normal grid spacing at the inner cylinder ( ${\rm\Delta}y^{+}$ ) decreases. Because of the non-uniform distribution of Chebyshev grid points, the wall-normal grid spacing at the centre increases by a factor of $49$ compared to the grid spacing at the inner cylinder (cf. table 1). The maximal streamwise grid spacing remains at ${\rm\Delta}x^{+}\approx 8$ . Overall, our spacings $({\rm\Delta}x^{+},{\rm\Delta}y^{+},{\rm\Delta}z^{+})$ at the inner cylinder are comparable to the ones used in other DNS of turbulent TCF (Bilson & Bremhorst Reference Bilson and Bremhorst2007; Pirrò & Quadrio Reference Pirrò and Quadrio2008). We furthermore note that the grid spacing measured in outer wall units ${\it\delta}_{{\it\nu},o}={\it\delta}_{{\it\nu},i}/{\it\eta}$ is ${\it\eta}$ -times smaller. Moreover, our simulations satisfy two additional convergence criteria described by Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a ): agreement of the torques computed at the inner and outer cylinder to within $0.5\,\%$ and agreement of the energy dissipation estimated from the torque and from the volume-averaged energy dissipation rate to within $1\,\%$ .

Similar to the difference between inner and outer wall units, the friction Reynolds numbers at the inner wall $Re_{{\it\tau},i}=u_{{\it\tau},i}d/(2{\it\nu})=\sqrt{{\it\eta}^{-1}Re_{S}\,Nu_{{\it\omega}}}/2$ and at the outer wall $Re_{{\it\tau},o}={\it\eta}Re_{{\it\tau},i}$ differ. The simulations for ${\it\eta}=0.5$ maximally reach $Re_{{\it\tau},i}=403$ and $Re_{{\it\tau},o}=201$ , while in our RPCF simulations the maximal friction Reynolds number is $Re_{{\it\tau}}=291$ at both walls.

4.1. Torque due to vortical motion

In this and the following sections, we investigate the momentum transport characteristics that underlie the rotation dependence of the Nusselt number discussed above. Vortical motions, such as Taylor vortices and their turbulent remnants, are known to effectively contribute to this transport by moving fast fluid from the inner cylinder outwards and slow outer fluid inwards. Their effects on the turbulence have been discussed previously (Lathrop et al. Reference Lathrop, Fineberg and Swinney1992; Lewis & Swinney Reference Lewis and Swinney1999; Martínez-Arias et al. Reference Martínez-Arias, Peixinho, Crumeyrolle and Mutabazi2014). Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013b ) quantified the link of the vortical motion to the torque and found that the torque contribution of the mean vortical motion was largest near the torque maximum. Flow visualizations of DNS by Salewski & Eckhardt (Reference Salewski and Eckhardt2015) show distinct vortical states which underlie the narrow and broad maxima for RPCF. Furthermore, using experiments at $Re_{S}\sim 10^{6}$ (Huisman et al. Reference Huisman, van der Veen, Sun and Lohse2014) and DNS at $Re_{S}\sim 10^{5}$ (Ostilla-Mónico et al. Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014b ), these structures have been detected over a range of rotation numbers where their presence is associated with a single torque maximum.

To investigate the effects of vortical structures on the torque, we first extract the mean vortical motion underlying the turbulence in the full simulation and measure its contribution to the momentum transport. The mean vortices consist of temporally and streamwise-averaged turbulent Taylor vortices and are analogous to the ‘large-scale circulation’ found in turbulent Rayleigh–Bénard convection (Ahlers, Grossmann & Lohse Reference Ahlers, Grossmann and Lohse2009). Next, since Taylor vortices do not depend on the streamwise direction, we also consider the torque which results from streamwise-independent flow. We then summarize our results in figure 11.

Figure 11.

Comparison of the total torque $G$ (▪) to the torque that results from the streamwise-invariant flow $G_{2D}$ (●) and to the torque due to the mean flow $\overline{G}$ (▴) for $Re_{S}=2\times 10^{4}$ . All torques are measured in units of the laminar torque $G_{lam}$ giving Nusselt numbers $Nu_{{\it\omega}}$ . The open symbols in ( f) show corresponding results from RPCF simulations.

4.1.1. Mean vortical motion

We follow the procedure introduced in Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013b ) and decompose the flow of the full DNS into a mean contribution $\overline{\boldsymbol{u}}=\langle \boldsymbol{u}\rangle _{{\it\varphi},t}$ (or $\overline{\boldsymbol{u}}=\langle \boldsymbol{u}\rangle _{x,t}$ for RPCF) that includes the mean vortical motion in the $rz$ -plane ( $yz$ -plane) and into the turbulent fluctuations around these mean vortices $\boldsymbol{u}^{\prime \prime }=\boldsymbol{u}-\overline{\boldsymbol{u}}$ , similar to the triple decomposition by Reynolds & Hussain (Reference Reynolds and Hussain1972). Substituting this velocity decomposition into the angular-momentum flux (2.12) results in a separation of the torque into a contribution that is caused by the mean vortical motion $\overline{G}$ and a second contribution that is due to turbulent fluctuations $G^{\prime \prime }$ , i.e. $G=\overline{G}+G^{\prime \prime }$ with

(4.1) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\overline{G}={\displaystyle \frac{Re_{S}^{2}}{r_{2}-r_{1}}}\displaystyle \int _{r_{1}}^{r_{2}}(\langle \overline{u}_{r}\overline{\mathscr{L}}\rangle _{{\it\varphi}z,t}-Re_{S}^{-1}r^{2}\partial _{r}\langle \overline{{\it\omega}}\rangle _{{\it\varphi}z,t})\,r\,\text{d}r,\\ G^{\prime \prime }={\displaystyle \frac{Re_{S}^{2}}{r_{2}-r_{1}}}\displaystyle \int _{r_{1}}^{r_{2}}\langle u_{r}^{\prime \prime }\mathscr{L}^{\prime \prime }\rangle _{{\it\varphi}z,t}\,r\,\text{d}r.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Analogous expressions for the decomposition of the driving force into $F=\overline{F}+F^{\prime \prime }$ can be obtained for RPCF by substituting the velocity decomposition into the momentum flux (2.15). To capture the correct amplitude of the turbulent Taylor vortices, we fix their spanwise position during the temporal average of $\overline{\boldsymbol{u}}$ and in (4.1) by correcting a potential spanwise drift. Such a drift would lead to a cancellation of the large-scale motion over time, even though it might be strong. To this end, for each flow snapshot we calculate an axial profile of the radial velocity $\widetilde{u}_{r}(z)$ (or $\widetilde{u}_{y}(z)$ for RPCF), which was averaged downstream and radially in the central region, and determine the axial shift $z_{0}$ that minimizes the distance of the function $\sqrt{2}\langle \widetilde{u}_{r}\rangle _{rms}\cos [2{\rm\pi}(z-z_{0})/L_{z}]$ to the profile $\widetilde{u}_{r}(z)$ . The flow field is then axially shifted by $z_{0}$ . Note that while the total transport $G$ is constant over $r$ , the individual terms in (4.1) vary with the radius. This motivates the additional radial average in (4.1) in order to quantify the typical strength of the torque contributions. Since we are interested in the mean vortical motion which dominates in the central region and not in the boundary layers, we restrict this average to the range between $(r_{1}-r_{i})/d=1/4$ and $(r_{2}-r_{i})/d=3/4$ .

In figure 11, the torque due to the mean vortical motion $\overline{G}$ is compared to the total torque $G$ . For all cases, $\overline{G}$ exhibits a maximum. For ${\it\eta}\geqslant 0.8$ and RPCF, the maximum of $\overline{G}$ , where the momentum transport by the mean vortices is most effective, nearly coincides with the narrow maximum of the total torque $G$ . Moreover, it is apparent that a distinct narrow peak also occurs in mean-vortex torque $\overline{G}$ for large ${\it\eta}$ . This suggests that the narrow, high- $Re_{S}$ maximum is linked to efficient mean vortices. For lower radius ratios, i.e. ${\it\eta}=0.5$ and $0.71$ , the coincidence between the maxima in $\overline{G}$ and $G$ is not as significant; nevertheless, as the analysis in Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013b ) demonstrates, this coincidence does exist for ${\it\eta}=0.5$ and $0.71$ when the radial average in (4.1) covers the complete radial gap instead of the central region. Finally, the mean-vortex torques $\overline{G}$ in figure 11 reveal that the mean Taylor vortices occur and grow in strength for $R_{{\it\Omega}}>0$ as previously observed in the experiment by Ravelet et al. (Reference Ravelet, Delfos and Westerweel2010).

5.1. Angular momentum profiles

Figure 12.

Profiles of the rescaled specific angular momentum $\widetilde{\mathscr{L}}=\left(\langle \mathscr{L}\rangle -\mathscr{L}_{o}\right)/\left(\mathscr{L}_{i}-\mathscr{L}_{o}\right)$ for $Re_{S}=2\times 10^{4}$ , the rotation numbers $R_{{\it\Omega}}$ specified in the figures and six different radius ratios. The corresponding RPCF profile $\langle u_{x}\rangle -R_{{\it\Omega}}y$ rescaled to the interval $[0,1]$ is shown by the dots. Each figure contains profiles for ${\it\eta}=0.5$ , $0.71$ , $0.8$ , $0.9$ , $0.95$ and $0.99$ as listed in (d). Some of the profiles collapse, especially in panels (c) and (d). In (a) and (b) the profile centre level decreases monotonically with increasing ${\it\eta}\rightarrow 1$ . The dotted line shows the arithmetic mean angular momentum $\widetilde{\mathscr{L}}=0.5$ .

Here, we analyse the mean profiles of the specific angular momentum $\mathscr{L}=ru_{{\it\varphi}}$ which is radially transported between the cylinders. To compensate for the varying mean system rotation, we show rescaled profiles $\widetilde{\mathscr{L}}=\left(\langle \mathscr{L}\rangle -\mathscr{L}_{o}\right)/\left(\mathscr{L}_{i}-\mathscr{L}_{o}\right)$ in figure 12 where $\mathscr{L}_{i}$ and $\mathscr{L}_{o}$ denote the specific angular momentum of the inner and outer cylinder. As a result, the profiles for various radius ratios ${\it\eta}$ at the same rotation number $R_{{\it\Omega}}$ collapse as long as the flow is not subject to the radial partitioning of stability for moderate counter-rotation (cf. § 3): the rotation number $R_{{\it\Omega}}=0.5$ corresponds to co-rotation ( ${\it\mu}\geqslant 0$ ) for all investigated ${\it\eta}$ , and all profiles of $\widetilde{\mathscr{L}}$ in figure 12(d) collapse well. However at $R_{{\it\Omega}}=0.2$ , only the simulation for ${\it\eta}=0.5$ with ${\it\mu}=-0.22$ shows the bursting in the outer region and a deviating profile in figure 12(c). Similarly, at $R_{{\it\Omega}}=0.02$ (figure 12 b), the profiles for ${\it\eta}\leqslant 0.9$ with ${\it\mu}$ -values in the bursting regime clearly deviate from the profiles for ${\it\eta}=0.95$ and $0.99$ with ${\it\mu}>{\it\mu}_{p}$ which are closer together. Finally, no collapse is observed for $R_{{\it\Omega}}=0$ which always corresponds to exact counter-rotation. In addition, figure 12 shows $u_{x}$ profiles from RPCF with the system rotation subtracted, i.e. $\langle u_{x}\rangle -R_{{\it\Omega}}y$ , that agree well with the collapsed $\widetilde{\mathscr{L}}$ profiles from TCF when rescaled to the same interval. The reason for this correspondence between $\langle u_{x}\rangle -R_{{\it\Omega}}y$ in RPCF and the angular momentum profiles in TCF will be discussed in the next section.

The $\widetilde{\mathscr{L}}$ profiles tend toward a universal shape for increasing $R_{{\it\Omega}}$ , as the radial partitioning of stability disappears. The specific angular momentum in the centre becomes nearly flat (with a slightly positive slope) and approaches $\widetilde{\mathscr{L}}=0.5$ which corresponds to $\langle \mathscr{L}\rangle =(\mathscr{L}_{i}+\mathscr{L}_{o})/2$ and indicates that the angular momentum is well mixed. Already Wendt (Reference Wendt1933) noted that in the unstable regime the angular momentum becomes almost constant in the centre which was also repeatedly observed for the case of a stationary outer cylinder (Taylor Reference Taylor1935; Wattendorf Reference Wattendorf1935; Smith & Townsend Reference Smith and Townsend1982; Lewis & Swinney Reference Lewis and Swinney1999; Dong Reference Dong2007). Flat $\mathscr{L}$ profiles resemble the situation in Rayleigh–Bénard convection where the transport of heat results in a well-mixed temperature in the bulk (Tilgner, Belmonte & Libchaber Reference Tilgner, Belmonte and Libchaber1993; Kerr Reference Kerr1996; Brown & Ahlers Reference Brown and Ahlers2007; Ahlers et al. Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012).

5.2. Mixing of angular momentum and neutral stability

The observation that the specific angular momentum $\mathscr{L}=ru_{{\it\varphi}}$ becomes almost flat in the middle is remarkable and calls for an explanation. Moreover, it is of interest whether an analogous phenomenon can be found in RPCF. To address both questions, we will study the stability properties of the mean profile around which the turbulence fluctuates. As long as the mean profile ( $\langle {\it\omega}\rangle (r)$ in TCF and $\langle u_{x}\rangle (y)$ in RPCF) is unstable due to a strong a gradient in the centre, it further enforces turbulence which increases the mixing of angular (or linear) momentum; the mixing flattens and therefore re-stabilises the profile. However, a stable, low-gradient profile decreases turbulent mixing, so that the profile becomes steeper and eventually unstable again. Therefore, we assume that both processes balance in the statistically averaged sense and the mean turbulent profile becomes neutrally stable in the centre.

For TCF with stationary outer cylinder, Wattendorf (Reference Wattendorf1935) and Taylor (Reference Taylor1935) already noted that the mean angular momentum profile in the middle tends towards one of neutral stability according to Rayleigh’s inviscid criterion (Rayleigh Reference Rayleigh1917), which predicts a constant angular momentum profile for this case. Neglecting viscous effects in the central part of the flow becomes justifiable at high Reynolds numbers. The flat $\widetilde{\mathscr{L}}$ profiles in figure 12 indeed realise Rayleigh’s neutral stability criterion in the centre, thus supporting the aforementioned assumptions. More precisely, Rayleigh’s discriminant for neutral stability (Chandrasekhar Reference Chandrasekhar1961) reads

(5.1) $$\begin{eqnarray}\displaystyle 0\equiv {\displaystyle \frac{2\langle {\it\omega}\rangle }{r}}\,\partial _{r}\left(r^{2}\langle {\it\omega}\rangle \right) & \hspace{-2.5pt}=\hspace{-2.5pt} & \displaystyle \left(R_{{\it\Omega}}-1+{\displaystyle \frac{\widetilde{r}^{\,2}}{r^{2}}}\right){\displaystyle \frac{1}{r}}\partial _{r}\langle \mathscr{L}\rangle\end{eqnarray}$$

(5.2) $$\begin{eqnarray}\displaystyle & \hspace{-2.5pt}=\hspace{-2.5pt} & \displaystyle \left(R_{{\it\Omega}}-1+{\displaystyle \frac{\widetilde{r}^{\,2}}{r^{2}}}\right)\left(R_{{\it\Omega}}-1+{\displaystyle \frac{\widetilde{r}^{\,2}}{r^{2}}}+r\partial _{r}\langle {\it\omega}\rangle \right),\end{eqnarray}$$

here expressed so that it serves as a condition for a neutrally stable turbulent mean profile in either $\mathscr{L}$ or ${\it\omega}$ in the middle. For that purpose, we substituted the mean profile $\langle {\it\omega}\rangle$ by the laminar solution ${\it\omega}_{lam}(r)$ from (2.9) because in the centre ${\it\omega}_{lam}$ provides a reasonable approximation to the magnitude of the turbulent mean profile as evidenced by the comparison in Brauckmann & Eckhardt (Reference Brauckmann and Eckhardt2013a ). Since in the middle $r$ is close to the geometric radius $\widetilde{r}=\sqrt{r_{i}r_{o}}/d$ , equation (5.2) implies that $r\partial _{r}\langle {\it\omega}\rangle \approx -R_{{\it\Omega}}$ for $R_{{\it\Omega}}\neq 0$ . Ostilla et al. (Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013) obtained a similar linear relationship by assuming that the Coriolis force term balances the convective term in the ${\it\varphi}$ -component of the equations of motion (2.5). Equations (5.1) and (5.2) then show that both force balance (Ostilla et al. Reference Ostilla, Stevens, Grossmann, Verzicco and Lohse2013) and inviscid stability (Rayleigh Reference Rayleigh1917) predict a profile with flat angular momentum.

The equivalent stability formulation in RPCF was derived from an analogy between rotating shear flows and buoyant flows (Bradshaw Reference Bradshaw1969; Tritton Reference Tritton1992): the stability expressed by a quantity similar to the Richardson number in stratified flows results in the neutral stability condition

(5.3) $$\begin{eqnarray}R_{{\it\Omega}}\left(R_{{\it\Omega}}-\partial _{y}\langle u_{x}\rangle \right)\equiv 0,\end{eqnarray}$$

which is approximately contained in (5.2) since $\widetilde{r}/r\approx 1$ in the middle, and $r\partial _{r}\langle {\it\omega}\rangle$ corresponds to $-\partial _{y}\langle u_{x}\rangle$ . The difference in sign originates from the fact that the radial coordinate in TCF is antiparallel to the wall-normal coordinate in RPCF, cf. figure 1. For RPCF we also assume that the turbulent mean profile in the middle becomes neutrally stable and fulfils (5.3), as proposed previously by Suryadi et al. (Reference Suryadi, Segalini and Alfredsson2014). In conclusion, the negative profile slope $-\partial _{y}\langle u_{x}\rangle$ in RPCF corresponds to the angular velocity gradient $r\partial _{r}\langle {\it\omega}\rangle$  in TCF. Moreover, comparing (5.3) to (5.1) reveals that the combination $(R_{{\it\Omega}}-\partial _{y}\langle u_{x}\rangle )$  in RPCF plays the role of the angular momentum gradient $r^{-1}\partial _{r}\langle \mathscr{L}\rangle$ in TCF, and we therefore compare the modified RPCF profile $\langle u_{x}\rangle -R_{{\it\Omega}}y$ to the angular momentum profiles in figure 12.

Figure 13.

Profile gradients in the middle region versus the rotation number $R_{{\it\Omega}}$ for $Re_{S}=2\times 10^{4}$ and various radius ratios ${\it\eta}$ . (a) Radial gradient of the angular momentum profile $\langle \mathscr{L}\rangle$ in TCF and the corresponding quantity $(R_{{\it\Omega}}-\partial _{y}\langle u_{x}\rangle )$ in RPCF, (b) gradient of the angular velocity profile $\langle {\it\omega}\rangle$ in TCF and the negative downstream velocity gradient $-\partial _{y}\langle u_{x}\rangle$ in RPCF, (c) detail of the maximum in (b). The black dashed line in (b) indicates the estimate $r\partial _{r}\langle {\it\omega}\rangle =-\partial _{y}\langle u_{x}\rangle =-R_{{\it\Omega}}$ from the neutral stability conditions (5.2) and (5.3). All gradients are calculated by averaging the compensated profile derivative $r^{-1}\partial _{r}\langle \mathscr{L}\rangle$ or $r\partial _{r}\langle {\it\omega}\rangle$ over the central region $(r-r_{i})/d\in [0.4,0.6]$ .

To test the assumption of neutrally stable mean profiles, the dependence of the profile gradients on $R_{{\it\Omega}}$ , and the proposed analogies between TCF and RPCF, we evaluate the relevant profile gradients in the central region for $Re_{S}=2\times 10^{4}$ and various radius ratios. Figure 13(a) shows that the specific angular momentum becomes almost flat with a slightly positive slope for $R_{{\it\Omega}}\gtrsim 0.02$ , except for ${\it\eta}=0.5$ where $\mathscr{L}$ is well mixed only for $R_{{\it\Omega}}\gtrsim 0.1$ . Experiments show that the slightly positive angular momentum slope persists for much higher $Re_{S}$ (Smith & Townsend Reference Smith and Townsend1982; Lewis & Swinney Reference Lewis and Swinney1999). In addition, for most $R_{{\it\Omega}}>0$ , the profile slopes measured by $r^{-1}\partial _{r}\langle \mathscr{L}\rangle$ collapse for different ${\it\eta}$ , which demonstrates the universality of the weak $R_{{\it\Omega}}$ -dependence. Furthermore, the quantity $(R_{{\it\Omega}}-\partial _{y}\langle u_{x}\rangle )$ from RPCF also shows this high level of agreement expected from the neutral stability analogy (5.1) and (5.3). In contrast, the profile gradients for $R_{{\it\Omega}}<0$ ( $R_{{\it\Omega}}<0.1$ for ${\it\eta}=0.5$ ) clearly vary with the radius ratio. For these rotation numbers, a pronounced stable region exists which introduces a strong curvature dependence as discussed for the torques in § 4.

The flattening of the specific angular momentum implies that the angular velocity gradients depend on $R_{{\it\Omega}}$ as shown in figure 13(b,c) and in similar figures in Ostilla-Mónico et al. (Reference Ostilla-Mónico, Huisman, Jannink, Van Gils, Verzicco, Grossmann, Sun and Lohse2014a ). In the unstable regime with $R_{{\it\Omega}}>0$ , the profile slopes $r\partial _{r}\langle {\it\omega}\rangle$ approach the neutral stability estimate $r\partial _{r}\langle {\it\omega}\rangle =-\partial _{y}\langle u_{x}\rangle =-R_{{\it\Omega}}$ as ${\it\eta}$ tends to 1. Moreover for low-curvature TCF ( ${\it\eta}\gtrsim 0.9$ ), they coincide with the $u_{x}$ -profile gradients in RPCF which supports the proposed analogy between $\langle {\it\omega}\rangle$ and $\langle u_{x}\rangle$ . The approximate scaling $\partial _{y}\langle u_{x}\rangle \sim R_{{\it\Omega}}$ in RPCF conforms with observations by Bech & Andersson (Reference Bech and Andersson1997) and Suryadi et al. (Reference Suryadi, Segalini and Alfredsson2014). However for smaller ${\it\eta}$ , the $\langle {\it\omega}\rangle$ -gradients depend more strongly on the curvature than the $\langle \mathscr{L}\rangle$ -gradients, consistent with the additional radial dependence in condition (5.2). Furthermore, we note that a slight local super-rotation (corresponding to a shear-inversion in RPCF) occurs in the central region for $R_{{\it\Omega}}\approx 0.03$ as detailed in figure 13(c). This inversion may result from an efficient momentum transport by the large-scale vortices moving fast fluid from the inner wall outwards and slow fluid inwards. The maximum in the mean vortex torque $\overline{G}$ around $R_{{\it\Omega}}\approx 0.03$ indeed supports this picture, cf. figure 11.

For the low-curvature TCF with ${\it\eta}\gtrsim 0.9$ , the flattest ${\it\omega}$ -profile (i.e. maximum in $r\partial _{r}\langle {\it\omega}\rangle$ ) and the narrow torque maximum occur for almost the same rotation number ( $R_{{\it\Omega}}\approx 0.03$ versus $0.02$ ) which extends an observation in Ostilla-Mónico et al. (Reference Ostilla-Mónico, Huisman, Jannink, Van Gils, Verzicco, Grossmann, Sun and Lohse2014a ) for ${\it\eta}\leqslant 0.909$ to larger radius ratios and to the limiting case of RPCF, where the flattest $u_{x}$ -profile is linked to the narrow maximum in $Nu_{u}$ . This connection also exists for ${\it\eta}=0.8$ , thus supporting the independence of the narrow maximum at $R_{{\it\Omega}}=0.02$ from the detachment maximum at $R_{{\it\Omega}}=0.05$ . For ${\it\eta}=0.5$ and ${\it\eta}=0.71$ the $R_{{\it\Omega}}$ -value of the flattest angular velocity profile does not coincide with the location of the torque maximum, which is another evidence that this detachment maximum differs from the narrow torque maximum observed for ${\it\eta}\gtrsim 0.9$ . In contrast, Ostilla-Mónico et al. (Reference Ostilla-Mónico, Huisman, Jannink, Van Gils, Verzicco, Grossmann, Sun and Lohse2014a ) found the flattest ${\it\omega}$ -profile for ${\it\eta}=0.5$ at $R_{{\it\Omega}}=0.25$ and for ${\it\eta}=0.714$ at $R_{{\it\Omega}}=0.12$ which agree with the corresponding torque-maximizing rotation numbers. However, the difference might result from the fact that their values are extrapolated from simulations at $Re_{S}\lesssim 5\times 10^{3}$ lower than the $Re_{S}=2\times 10^{4}$ used here.

The gradients of the ${\it\omega}$ -profile ( $u_{x}$ -profile in RPCF) are relevant when analysing the momentum transport in the centre and the torque (force) needed to drive the walls. They measure the diffusive part of the current $J^{\mathscr{L}}$ ( $J^{u}$ , respectively) which is mediated by viscosity, i.e. the last term in (2.12) and (2.15). A flat profile means that the convective current $J_{c}^{\mathscr{L}}=\langle u_{r}\mathscr{L}\rangle$ ( $J_{c}^{u}=\langle u_{y}u_{x}\rangle$ in RPCF) accomplishes all momentum transport in the middle.

5.3. Turbulent fluctuations

Figure 14.

Fluctuation amplitude of the wall-normal velocity $\langle u_{r}^{\prime }\rangle _{rms}$ ( $\langle u_{y}^{\prime }\rangle _{rms}$ for RPCF) in (a) and of the streamwise velocity $\langle u_{{\it\varphi}}^{\prime }\rangle _{rms}$ ( $\langle u_{x}^{\prime }\rangle _{rms}$ ) in (b) and their cross-correlation coefficient $C_{r{\it\varphi}}$ ( $C_{yx}$ ) in (c) as a function of the rotation number $R_{{\it\Omega}}$ for $Re_{S}=2\times 10^{4}$ and various radius ratios ${\it\eta}$ . All quantities are radially average in the central region $(r-r_{i})/d\in \left[1/3,2/3\right]$ where momentum is mainly transported by the convective current $J_{c}^{\mathscr{L}}=r\langle u_{r}^{\prime }u_{{\it\varphi}}^{\prime }\rangle$ ( $J_{c}^{u}=\langle u_{y}^{\prime }u_{x}^{\prime }\rangle$ ).

Finally, we study the effect of the mean system rotation ( $R_{{\it\Omega}}$ ) on the characteristics of the convective momentum transport by turbulent fluctuations. We demonstrate that not all deviations from the laminar state contribute to the net convective momentum transport but only the turbulent fluctuations around the mean profile. This becomes apparent when introducing the velocity deviation from the mean profile $\boldsymbol{u}^{\prime }=\boldsymbol{u}-\langle \boldsymbol{u}\rangle$ by using the original average $\langle \cdots \rangle =\langle \cdots \rangle _{{\it\varphi}z,t}$ ; note that this differs from $\boldsymbol{u}^{\prime \prime }=\boldsymbol{u}-\langle \boldsymbol{u}\rangle _{{\it\varphi},t}$ , which was introduced in § 4.1.1 to describe turbulent fluctuations around the mean vortical motion, see also Bilson & Bremhorst (Reference Bilson and Bremhorst2007). As a result of incompressibility (2.6) the mean radial flow vanishes, i.e. $\langle u_{r}\rangle =0$ , so that $u_{r}^{\prime }=u_{r}$ . Moreover, by substituting the azimuthal velocity decomposition $u_{{\it\varphi}}=\langle u_{{\it\varphi}}\rangle +u_{{\it\varphi}}^{\prime }$ into the convective angular-momentum flux,

(5.4) $$\begin{eqnarray}J_{c}^{\mathscr{L}}=r\langle u_{r}\,u_{{\it\varphi}}\rangle =r\langle u_{r}\rangle \langle u_{{\it\varphi}}\rangle +r\langle u_{r}\,u_{{\it\varphi}}^{\prime }\rangle =r\langle u_{r}^{\prime }\,u_{{\it\varphi}}^{\prime }\rangle ,\end{eqnarray}$$

we see that only the correlated fluctuations $u_{r}^{\prime }$ and $u_{{\it\varphi}}^{\prime }$ contribute to the net convective transport. Inspired by the analysis of Burin, Schartman & Ji (Reference Burin, Schartman and Ji2010), we measure their amplitude by area- and time-averaged root-mean-square values $\langle u_{r}^{\prime }\rangle _{rms}$ and $\langle u_{{\it\varphi}}^{\prime }\rangle _{rms}$ that enable the definition of a cross-correlation coefficient

(5.5) $$\begin{eqnarray}C_{r{\it\varphi}}={\displaystyle \frac{\langle u_{r}^{\prime }\,u_{{\it\varphi}}^{\prime }\rangle }{\langle u_{r}^{\prime }\rangle _{rms}\langle u_{{\it\varphi}}^{\prime }\rangle _{rms}}}\quad \text{so that }J_{c}^{\mathscr{L}}=r\langle u_{r}^{\prime }\rangle _{rms}\langle u_{{\it\varphi}}^{\prime }\rangle _{rms}C_{r{\it\varphi}}.\end{eqnarray}$$

The corresponding quantities in RPCF are

(5.6) $$\begin{eqnarray}C_{yx}={\displaystyle \frac{\langle u_{y}^{\prime }\,u_{x}^{\prime }\rangle }{\langle u_{y}^{\prime }\rangle _{rms}\langle u_{x}^{\prime }\rangle _{rms}}}\quad \text{so that }J_{c}^{u}=\langle u_{y}^{\prime }\rangle _{rms}\langle u_{x}^{\prime }\rangle _{rms}C_{yx}.\end{eqnarray}$$

Consequently, we may consider increases in the convective (angular) momentum transport as being due to violent fluctuations of the wall-normal or streamwise velocities, or due to a strong cross-correlation between them. All three effects are analysed as functions of $R_{{\it\Omega}}$ for $Re_{S}=2\times 10^{4}$ in figure 14.

Generally, we observe for ${\it\eta}\geqslant 0.8$ and RPCF that the fluctuation amplitudes and correlation coefficients collapse well in the unstable range with $R_{{\it\Omega}}\gtrsim 0$ , which again indicates that low-curvature TCF and RPCF show the same rotation dependence. The wall-normal flow grows in amplitude when rotation sets in ( $R_{{\it\Omega}}>0$ ) and is suppressed by strong rotation forming a maximum around $R_{{\it\Omega}}\approx 0.25$ for ${\it\eta}\geqslant 0.71$ ; this nearly coincides with the broad torque maximum at $R_{{\it\Omega}}=0.2$ for ${\it\eta}\geqslant 0.8$ , and it resembles the maximum observed in the torque $G_{2D}$ from streamwise-invariant simulations (cf. figure 11). The co-occurrence of the maximum wall-normal flow with the broad peak supports the picture that the broad torque maximum is associated with strong vortical motion. While the co-occurrence holds for the plateau next to the actual maximum for ${\it\eta}=0.71$ (figure 8 b), the correlation between the maxima is less clear for ${\it\eta}=0.5$ .

On the other hand, the streamwise velocity fluctuations are strongest without system rotation, forming a sharp maximum around $R_{{\it\Omega}}=0$ , except for the high-curvature TCF with ${\it\eta}=0.5$ (figure 14 b). A corresponding peak in the torques is missing since the narrow torque maximum at $R_{{\it\Omega}}=0.02$ clearly drops off towards $R_{{\it\Omega}}=0$ (cf. figure 9 b).

However, for ${\it\eta}\geqslant 0.9$ the maximum in the cross-correlation coefficients at $R_{{\it\Omega}}\approx 0.03$ (figure 14 c) comes closer to the narrow torque maximum suggesting that the latter is associated with a very efficient (angular) momentum transport by moderately strong but correlated vortices. Moreover, the peak in the cross-correlation coefficients coincides with the maximum in the torque $\overline{G}$ due to the mean vortical flow (cf. figure 11). The maximum in $C_{r{\it\varphi}}$ at $R_{{\it\Omega}}=0.15$ and at $R_{{\it\Omega}}=0.05$ for ${\it\eta}=0.5$ and ${\it\eta}=0.71$ , respectively, does not coincide with the torque maximum for the corresponding radius ratio indicating that the latter differs from the narrow maximum for ${\it\eta}\gtrsim 0.9$ .