4.1 Roughness function

Figure 3. (a,c,e,g) Profiles of the streamwise – azimuthal – velocity $u^{+}$ (solid) and the angular velocity $\unicode[STIX]{x1D714}^{+}$ (dashed) versus the wall-normal distance $y^{+}$ . Black solid lines indicate the viscous sublayer profile $u^{+}=y^{+}$ and the logarithmic law $u^{+}=\unicode[STIX]{x1D705}^{-1}\log y^{+}+B$ , with $\unicode[STIX]{x1D705}=0.4$ and $B=5.0$ . (b,d,f,h) Profiles of the streamwise velocity shift $\unicode[STIX]{x0394}u^{+}$ (solid) and the angular velocity shift $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}$ (dashed). Every row corresponds to a constant Taylor number, (a,b) $Ta=1.0\times 10^{7}$ , (c,d) $Ta=5.0\times 10^{7}$ , (e,f) $Ta=5.0\times 10^{8}$ and (g,h) $Ta=1.0\times 10^{9}$ , see table 1. The grey lines in (g) are logarithmic fits to the smooth profiles for $y^{+}=[150,500]$ .

Figure 3(a,c,e,g) presents the streamwise (i.e. azimuthal) velocity profiles $u^{+}=\langle u(r)-u(r_{i})\rangle _{\unicode[STIX]{x1D703},z,t}/u_{\unicode[STIX]{x1D70F}}$ (solid) and angular velocity profiles $\unicode[STIX]{x1D714}^{+}=\langle u(r_{i})-r_{i}u(r)/r\rangle _{\unicode[STIX]{x1D703},z,t}/u_{\unicode[STIX]{x1D70F}}$ (dashed) versus the wall-normal distance $y^{+}=r^{+}-r_{i}^{+}-h_{m}^{+}$ , where $\langle .\rangle _{\unicode[STIX]{x1D703},z,t}$ indicates averaging over the streamwise and spanwise directions and in time, and $h_{m}^{+}$ is the mean roughness height. Every row corresponds to simulations at constant rotation rate of the inner cylinder (Taylor number), and increasing roughness height.

In line with the previous observations of Huisman et al. (Reference Huisman, Scharnowski, Cierpka, Kähler, Lohse and Sun2013) and Ostilla-Mónico et al. (Reference Ostilla-Mónico, van der Poel, Verzicco, Grossmann and Lohse2014), we also find that the logarithmic profiles of the streamwise velocity $u^{+}$ in smooth-wall TC do not fit the $\unicode[STIX]{x1D705}=0.4$ slope, as found in other wall-bounded flows (e.g. pipe, boundary layer, channel) – for similar values of the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ . However, this asymptotic value is experimentally observed at very high shear rates of $Ta=O(10^{12})$ and $Re_{\unicode[STIX]{x1D70F}}=O(10^{4})$ (Huisman et al. Reference Huisman, Scharnowski, Cierpka, Kähler, Lohse and Sun2013), much higher than can be obtained by the present DNS. The logarithmic profiles of angular velocity $\unicode[STIX]{x1D714}^{+}$ have a slope that is closer to the $\unicode[STIX]{x1D705}=0.4$ asymptote (Ostilla-Mónico et al. Reference Ostilla-Mónico, Verzicco and Lohse2015), especially for the higher $Ta$ figure 3( $g,h$ ). Here we investigate the effects of roughness on both $u^{+}$ and $\unicode[STIX]{x1D714}^{+}$ .

For rough-wall simulations, the logarithmic region shifts downwards – a hallmark effect of a drag increasing surface. Figure 3(b,d,f,h) presents the shifts, where $\unicode[STIX]{x0394}u^{+}=u_{s}^{+}-u_{r}^{+}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}=\unicode[STIX]{x1D714}_{s}^{+}-\unicode[STIX]{x1D714}_{r}^{+}$ . All shifts of the angular and azimuthal profiles are calculated at the centre of the gap, that is at $y^{+}+h_{m}^{+}=Re_{\unicode[STIX]{x1D70F}}$ . As one can see in figure 3(b,d,f,h) of the manuscript, the values of $\unicode[STIX]{x0394}u^{+}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}$ do not depend on $y^{+}$ if one is sufficiently far away from the wall. For lower $Ta$ , there is a small but observable difference between $\unicode[STIX]{x0394}u^{+}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}$ , see figure 3(b), whereas for the higher $Ta$ , this difference diminishes, see figure 3(f,h).

Figure 4. (a) Azimuthal velocity shift (Hama roughness function) $\unicode[STIX]{x0394}u^{+}$ versus the equivalent sand grain roughness height $k_{s}^{+}$ , where $k_{s}^{+}=1.33k^{+}$ . (b) Angular velocity shift $\unicode[STIX]{x1D714}^{+}$ versus $k_{s}^{+}$ . Close overlap with the Nikuradse curve is observed in the transitionally rough regime. The overlap is slightly better for the angular velocity shift, for which we also obtain $k_{s}^{+}=1.33k^{+}$ . The solid blue line represents the fully rough asymptote; $\unicode[STIX]{x0394}u^{+}=2.44\log (k_{s}^{+})+5.2-8.5$ . The green lines represent the fully rough asymptotes obtained from the simulations, with $\unicode[STIX]{x1D705}_{u},\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D714}},A_{u}$ and $A_{\unicode[STIX]{x1D714}}$ extracted from figure 3(g). The spread between statistically similar surfaces, with similar mean and maximum heights, is indicated by the vertical bar. $k_{s}$ is determinded by a best fit between the two data points in the fully rough regime (i.e. cases D4 and D5, see table 1) and the Nikuradse fully rough asymptote.

Figure 5. (a) Azimuthal velocity $u^{+}$ (solid) and the angular velocity $\unicode[STIX]{x1D714}^{+}$ (dashed) versus the wall-normal distance $y/k_{s}$ , where $y=(r-h_{m})$ and $h_{m}$ is the mean roughness height. The three simulations with the highest roughness are plotted (D3, D4 and D5 respectively, see table 1) to convey collapse of the profiles for the fully rough cases. (b) The Nikuradse constant $\tilde{B}$ versus the equivalent sand grain roughness height $k_{s}^{+}$ for both the azimuthal velocity (squares) and the angular velocity (diamonds). Horizontal black line at $\tilde{B}=6.0$ gives the asymptotic value that is observed for fully rough behaviour.

Figure 4(a,b) presents the shift of the streamwise and angular velocity profiles, respectively, versus the equivalent roughness height $k_{s}^{+}$ , for all $Ta$ . Care is taken to ensure overlap for varying $Ta$ and similar $k^{+}$ , to study the $Ta$ dependence of $\unicode[STIX]{x0394}u^{+}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}$ . However, despite the varying $Ta$ numbers, all data collapse onto a single curve, with some scatter. Note that scatter is expected due to the randomness of the surfaces, which are reproduced for every simulation. To obtain an estimate of the expected scatter, we run two simulations with statistically similar surfaces. These are indicated by B3 and BY in table 1 and the velocity profiles are found in figure 3(c,d). We find a difference between the two cases of ${\lesssim}0.2\unicode[STIX]{x0394}u^{+},0.2\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}$ . The measure of this scatter is indicated in figure 4 as a vertical bar ( $=$ spread). We figure 4 as a vertical bar ( $=$ spread). We conclude that the variability in $\unicode[STIX]{x0394}u^{+}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}$ at constant $k^{+}$ and varying $Ta$ falls within the size of that vertical bar, and such conclude that $\unicode[STIX]{x0394}u^{+}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}$ and thus the equivalent roughness height $k_{s}^{+}$ shows little dependence on $Ta$ .

A comparison with the findings of Nikuradse (Reference Nikuradse1933) can be carried out by scaling the fully rough regime to obtain $k_{s}^{+}=Ck^{+}$ , where $C$ is a constant that depends on the surface topology. In figure 5(a) we plot the velocity profiles versus $(r-h_{m})/k_{s}$ for the highest roughness (D3, D4 and D5 respectively, see table 1). Excellent collapse of the D4 and D5 profiles indicates that those simulations are indeed fully rough. In this fully rough regime viscosity can be neglected ( $y\gg k\gg \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ ), whereas the velocity profile is also independent of the system outer length scales ( $y\ll d$ ) i.e. the overlap argument (Pope Reference Pope2000). The gradient of the velocity profile becomes $\text{d}\langle U\rangle /\text{d}y=(u_{\unicode[STIX]{x1D70F}}/y)\unicode[STIX]{x1D6F7}(y/k)$ , where $\unicode[STIX]{x1D6F7}(y/k)$ is a universal function that will go to $1/\unicode[STIX]{x1D705}$ . Integration then gives

(4.1) $$\begin{eqnarray}\displaystyle u_{r}^{+}={\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}\log (y/k)+B={\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}\log (y/k_{s})+\tilde{B}, & & \displaystyle\end{eqnarray}$$

where $B$ is a constant and $y=r-h_{m}$ ; $\tilde{B}$ is the Nikuradse constant. The roughness function in the fully rough regime, (i.e. the fully rough asymptote), is obtained by subtracting (4.1) from the smooth-wall profile $u_{s}^{+}=(1/\unicode[STIX]{x1D705})\log (y^{+})+A$ and rescaling it to overlap with the Nikuradse data,

(4.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}u^{+}={\displaystyle \frac{1}{\unicode[STIX]{x1D705}}}\log (k_{s}^{+})+A-\tilde{B}. & & \displaystyle\end{eqnarray}$$

In figure 4(a), the blue solid line is the fully rough asymptote, with $\unicode[STIX]{x1D705},A,\tilde{B}$ as found in pipe flow (Pope Reference Pope2000). The green solid line is the fully rough asymptote as obtained from our simulations; $\unicode[STIX]{x1D705}_{u}^{-1}=1.22$ and $A_{u}=8.0$ are taken from the fit of the smooth-wall simulation at identical $Ta$ as the fully rough cases (figure 3 g). The fits are in the domain $y^{+}=[150,500]$ , as there the slope becomes approximately constant (figure 3 h). The value of $\tilde{B}$ is plotted in figure 5(b), where we find that $\tilde{B}\approx 6.0$ for the fully rough cases. The mismatch of the slopes in the fully rough regime makes a rescaling to find $k_{s}^{+}$ a priori impossible – a statement that we wish to emphasize. However, to proceed with the comparison of the transitionally rough cases in TC and pipe flow, we choose to rescale the fully rough cases (D4 and D5) with the Nikuradse fully rough asymptote in figure 4. We find that $k_{s}^{+}=1.33k^{+}$ and very close collapse of our data with the Nikuradse data.

In parallel, we analyse the behaviour of $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}$ versus $k_{s}^{+}$ , shown in figure 4(b). Again, the blue solid line represents the fully rough asymptote of Nikuradse. The green solid line is the fully rough asymptote obtained from fits ( $y^{+}=[150,500]$ ) of the smooth-wall angular velocity profile at identical $Ta$ as the fully rough cases, see figure 3(g). We find $\unicode[STIX]{x1D705}_{\unicode[STIX]{x1D714}}^{-1}=2.17$ ( $A_{\unicode[STIX]{x1D714}}=3.7$ ), close to the asymptotic value $\unicode[STIX]{x1D705}^{-1}=2.44$ . Although the differences are marginal, $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}$ fits to the Nikuradse data slightly better than $\unicode[STIX]{x0394}u^{+}$ (note that also here the rescaling is, $k_{s}=1.33k$ ). However, the major difference is the closeness of the fully rough asymptotes.

These results suggest that the near-wall effects of transitionally rough sand grains (and other rough surfaces) in TC flow are similar to the effects of transitionally rough sand grains (and other rough surfaces) in pipe flow (and other canonical systems). Further, we find that these transitionally rough effects are independent of slope of the velocity profile in the logarithmic region, whereas in the fully rough regime, they, a priori, depend on this slope. This is confirmed with similar values of $\unicode[STIX]{x0394}u^{+}$ at similar $k_{s}^{+}$ , for varying $Ta$ , see figure 4. Also the similarity between $\unicode[STIX]{x0394}u^{+}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}^{+}$ in the transitionally rough regime confirms this, whereas the fully rough asymptotes are very dissimilar. We like to point out that simulations (or experiments) at high enough $Ta$ ( $=10^{12}$ (Huisman et al. Reference Huisman, Scharnowski, Cierpka, Kähler, Lohse and Sun2013)), where $\unicode[STIX]{x1D705}=0.4$ , then are expected to also give a collapse to the Nikuradse data in the fully rough regime.

4.2 Global response

Figure 6. Profiles of (a) the friction factor $C_{f}$ , and (b) the Nusselt number $Nu_{\unicode[STIX]{x1D714}}$ versus the roughness height. $k/d$ is the roughness height $k$ relative to the gap width $d$ . (c) Normalized friction coefficient $C_{f}(k_{s}^{+})/C_{f}(k_{s}^{+}=0)$ versus the equivalent sand grain roughness height $k_{s}^{+}$ . (d) Normalized Nusselt number $Nu_{\unicode[STIX]{x1D714}}(k_{s}^{+})/Nu_{\unicode[STIX]{x1D714}}(k_{s}^{+}=0)$ versus the equivalent sand grain roughness height $k_{s}^{+}$ .

Figure 6(a) presents the friction factor $C_{f}$ versus the dimensionless roughness height $k/d$ for varying $Re_{i}$ . For lower $Ta$ numbers, the friction factor $C_{f}$ decreases with increasing $Ta$ , indicating the relevance of viscous drag. For the two highest $Ta$ numbers, representing the higher $k_{s}^{+}$ cases, the friction factor almost collapses onto one line. This tells that $\unicode[STIX]{x1D70F}_{w}\propto u^{2}$ , and thus that pressure drag is dominant over viscous drag, in accordance with the overlap argument presented above. For constant $Ta$ , as expected, the friction factor increases for increasing roughness height. Figure 6(b) presents the global response in terms of $Nu_{\unicode[STIX]{x1D714}}$ . We observe an increase in $Nu_{\unicode[STIX]{x1D714}}$ for increasing $Ta$ , corresponding to the increased transport of the angular velocity that is due to the increased turbulent mixing. Higher roughness leads to increased $J^{\unicode[STIX]{x1D714}}$ as compared to the smooth wall at the same $Ta$ , which also relates to a higher intensity of the turbulent mixing (the $r^{3}(\langle w\unicode[STIX]{x1D714}\rangle _{\unicode[STIX]{x1D703},z,t})$ term of (2.8)) and more plumes ejecting from the boundary layer radially outwards (Zhu et al. Reference Zhu, Verzicco and Lohse2017), on which we will elaborate in § 4.3.

By assuming a logarithmic profile, and integrating this profile over the entire gap (thereby neglecting the contributions of the viscous sublayer), we arrive at an implicit equation for the friction factor $C_{f}$ , namely the celebrated Prandtl’s friction law,

(4.3) $$\begin{eqnarray}\displaystyle (C_{f}/2)^{-1/2}=C_{1}\log ((C_{f}/2)^{1/2}Re_{i})+C_{2}, & & \displaystyle\end{eqnarray}$$

where $C_{1}(Re_{i})$ and $C_{2}(Re_{i})$ for TC at these $Re_{i}$ . Figure 7 shows the friction factor $C_{f}$ versus the inner cylinder Reynolds number $Re_{i}$ , for both smooth-wall (solid) as the rough-wall (symbols) simulations. An upward shift of the friction factor for increasing roughness height is consistent with what is observed for sand grains in pipe flow (Nikuradse Reference Nikuradse1933) and recently also for transverse ribs in Taylor–Couette flow (Zhu et al. Reference Zhu, Verschoof, Bakhuis, Huisman, Verzicco, Sun and Lohse2018b ). Note that this upward shift is directly related to the downward shift of the mean streamwise velocity profile (the roughness function). Since the friction factor and the Nusselt number are related, as expressed in (2.10), we expect the Nusselt number to increase, for increasing roughness height. This is confirmed in figure 7(b), where we plot the Nu number versus the Ta number. The number of simulations with constant $k/d$ is limited, and we vary the Ta number over 2 decades only. However, we observe that the asymptotic, ultimate scaling of $Nu_{\unicode[STIX]{x1D714}}\propto Ta^{0.5}$ , as found for fully rough transverse ribs in (Zhu et al. Reference Zhu, Verschoof, Bakhuis, Huisman, Verzicco, Sun and Lohse2018b ), is not reached. This is expected, as only the inner cylinder is covered with roughness.

Figure 7. (a) Moody representation, showing the friction factor $C_{f}$ as a function of the inner cylinder Reynolds number $Re_{i}$ for varying roughness height $k/d$ . The solid line is the fit of the Prandtl friction law to the smooth-wall simulation data. (b) Compensated plot of the Nusselt number versus the Taylor number for constant $k/d$ . In this regime, $Nu\propto Ta^{0.33}$ . The solid black line indicates the asymptotic scaling of $Nu\propto Ta^{0.50}$ .

4.3 Flow structures

Figure 8. (a–c) Classical turbulent state: contour fields of the instantaneous azimuthal velocity $u(r,\unicode[STIX]{x1D703},z,t)$ for $Ta=5.0\times 10^{7}$ in the meridional plane. (a) Smooth-wall simulation (BS) in which we observe one ejecting plume and one impacting plume. (b) Rough inner cylinder $k/d=0.039$ (B2), with the roughness indicated in grey, exhibiting more plumes ejecting from the inner cylinder radially outwards. (c) Rough inner cylinder $k/d=0.073$ , with the roughness indicated in grey, (B4) leading to a more chaotic flow field, with enhanced mixing and enhanced radial transport of the conserved angular velocity flux. (d–f) Ultimate turbulent state: contour fields of the instantaneous azimuthal velocity $u(r,\unicode[STIX]{x1D703},z,t)$ for $Ta=1.0\times 10^{9}$ in the meridional plane. (d) Smooth inner cylinder (DS) with many plumes ejecting, considerably more chaotic than in (a). (e) Inner cylinder wall roughness (indicated in grey) $k/d=0.035$ (D2) and (f) inner wall roughness $k/d=0.055$ (D5). For the rough cases, we observe more plumes ejecting and more mixing in the bulk, leading to enhanced radial transport of the angular velocity $J^{\unicode[STIX]{x1D714}}$ , expressed in a higher $Nu_{\unicode[STIX]{x1D714}}$ .

To obtain a qualitative understanding of the effect of inner cylinder roughness on the turbulent flow in the gap, we present two series of snapshots of the streamwise azimuthal velocity $u(r,\unicode[STIX]{x1D703},z,t)$ . Figure 8(a–c) exhibits the snapshots for $Ta=5.0\times 10^{7}$ . It is known, and observed here, that for this Taylor number the coherence length of the dominant flow structures becomes smaller than the gap width $d$ , and turbulence develops in the bulk (Grossmann et al. Reference Grossmann, Lohse and Sun2016). On the other hand, the boundary layers remain predominantly laminar and as such the regime is referred to as the ‘classical regime’ of TC turbulence. A divergent colour map is chosen to highlight the turbulent structures in the bulk. A snapshot for the smooth inner cylinder simulation is presented in figure 8(a). At $z/d\approx 0.3$ , one observes an ejecting structure (plume) that detaches from the inner cylinder laminar boundary layer at the location of an adverse pressure gradient. Locally, where this ejecting plume detaches, the flow will be different (i.e. more chaotic) to that in the other parts of the boundary layer. As such, one also expects the local variables (e.g. skin friction, turbulence intensity) to be different. Later we will therefore employ local averaging, to investigate the spatial differences in the flow associated with this structure (van der Poel et al. Reference van der Poel, Ostilla-Mónico, Verzicco, Grossmann and Lohse2015b ; Ostilla-Mónico et al. Reference Ostilla-Mónico, Verzicco, Grossmann and Lohse2016; Zhu et al. Reference Zhu, Mathai, Stevens, Verzicco and Lohse2018a ). The ejecting and impacting (located at $z/d\approx 1.3$ ) plumes have very strong radial velocity components $w$ . From the first term on the right-hand side of (2.8), $r^{3}(\langle w\unicode[STIX]{x1D714}\rangle _{\unicode[STIX]{x1D703},z,t})$ , we then directly see that they strongly contribute to $Nu_{\unicode[STIX]{x1D714}}$ . This brings us to the remaining (figure 8 b,c) snapshots. Many more, small, plumes are seen to eject from the inner cylinder. The roughness there promotes the detachment of ejecting structures and in that way contributes to a higher $Nu_{\unicode[STIX]{x1D714}}$ . An increase in the level of turbulence, as suggested by the increased level of turbulence dissipation, is quantitatively reflected by a decrease in the Kolmogorov scale ( $\unicode[STIX]{x1D702}=(\unicode[STIX]{x1D708}^{3}/\unicode[STIX]{x1D716})^{1/4}$ ), namely $\unicode[STIX]{x1D702}/d=7.1\times 10^{-3}$ for the smooth-wall case BS and $\unicode[STIX]{x1D702}/d=6.5\times 10^{-3}$ for the highest roughness case B5. Note that the decrease in the Kolmogorov scale $\unicode[STIX]{x1D702}$ is only small, since $\unicode[STIX]{x1D702}/d\propto (\unicode[STIX]{x1D716}d^{4}/\unicode[STIX]{x1D708}^{3})^{-1/4}$ . For TC flow, the volume-averaged dissipation rate $\unicode[STIX]{x1D716}$ is related to the angular velocity transport $Nu_{\unicode[STIX]{x1D714}}$ with: $\unicode[STIX]{x1D716}=\unicode[STIX]{x1D708}^{3}d^{-4}\unicode[STIX]{x1D70E}^{-2}(Nu_{\unicode[STIX]{x1D714}}-1)Ta+\unicode[STIX]{x1D716}_{lam}$ , where $\unicode[STIX]{x1D716}_{lam}$ is the laminar volume-averaged dissipation rate, $d$ is the gap width of the set-up and $\unicode[STIX]{x1D70E}=((1+\unicode[STIX]{x1D702})/2/\sqrt{\unicode[STIX]{x1D702}})^{4}$ a geometric parameter (Eckhardt et al. Reference Eckhardt, Grossmann and Lohse2007). As such, we see that $\unicode[STIX]{x1D702}/d\propto Nu_{\unicode[STIX]{x1D714}}^{-1/4}$ only.

Figure 8(d–f) presents snapshots of a flow in the ultimate turbulent state at $Ta=1.0\times 10^{9}$ (Grossmann et al. Reference Grossmann, Lohse and Sun2016). Although less pronounced than for $Ta=5.0\times 10^{7}$ , we still observe distinct ejecting and impacting regions, indicating the survival of the turbulent Taylor rolls. A similar rationale as applied above, to the classical turbulence case, can also be used to explain the enhancement of the Nusselt number for rough inner cylinders in the ultimate turbulent state. In fact, we can also observe more intense plumes for the highest roughness (D5, figure 8 f), in comparison to a lower roughness case (D2, figure 8 e). Note that here we do not observe the stable vortex formation in between roughness elements and the associated ejection of plumes from sharp peaks, as was reported by Zhu et al. (Reference Zhu, Ostilla-Mónico, Verzicco and Lohse2016) for grooved cylinders, for similar Taylor numbers and roughness heights. The increase in the turbulence level is also quantitatively confirmed by a decrease in the Kolmogorov scale here, $\unicode[STIX]{x1D702}/d=2.7\times 10^{-3}$ for the smooth-wall case DS and $\unicode[STIX]{x1D702}/d=2.1\times 10^{-3}$ for the highest roughness case D5.

4.4 Roughness sublayer

The existence of Taylor roll structures is already anticipated in the snapshots of the instantaneous flow in figure 8, from which we observe the ejecting and impacting plume regions. Contour plots of the time and azimuthally averaged azimuthal velocity field, as presented in figure 9, confirm this. Note that the Taylor roll is spatially fixed, allowing for convenient averaging over impacting (solid line), shearing (dashed line) and ejecting (dashed dotted line) regions, a method that we also employed in RB flow (van der Poel et al. Reference van der Poel, Ostilla-Mónico, Verzicco, Grossmann and Lohse2015b ). For an increasing roughness height, the white region (representing $\langle u\rangle _{\unicode[STIX]{x1D703},t}\approx 0.5$ ) shifts radially outwards and the azimuthal velocity in the bulk increases. This process previously has been seen in Zhu et al. (Reference Zhu, Verschoof, Bakhuis, Huisman, Verzicco, Sun and Lohse2018b ), where it is referred to as the bulk velocity ‘getting slaved to’ to the velocity of a cylinder covered with roughness, reflecting the enhanced drag on the rough side.

Figure 9. Contour field of the mean azimuthal velocity $\langle u\rangle _{\unicode[STIX]{x1D703},t}$ in the meridional plane for $Ta=1.0\times 10^{9}$ . (a) Smooth inner cylinder (DS) (b). Inner cylinder wall roughness $k/d=0.0517$ (D2) and (c) inner wall roughness $k/d=0.0818$ (D5). The solid vertical lines indicate the height of the roughness sublayer $h_{r}$ , calculated over the entire cylinder height.

 plume ejection regions,

 sheared regions,

 plume impacting regions.

Figure 10. Contour of the time-averaged azimuthal velocity $\langle u\rangle _{t}^{+}$ , zoomed in on only a few roughness elements, indicated in grey, for $Ta=1.0\times 10^{9}$ (D2). Flow is from left to right. Isolines of $\langle u\rangle _{t}^{+}$ overlay the contour. On the vertical axis, we display the wall-normal coordinate normalized by the gap width $d$ (left) and in viscous units (right). On the horizontal axis, we display the azimuthal coordinate, normalized by the gap width $d$ (below) and in viscous units (above).

Figure 11. (a) Profiles of the root mean square of the form-induced azimuthal velocity fluctuations $\tilde{u} ^{+}=(\langle u\rangle _{t}-\langle u\rangle _{\unicode[STIX]{x1D703},t})/u_{\unicode[STIX]{x1D70F}}$ at incremental heights above the roughness for identical location and conditions as in figure 9. The radial coordinate $r^{+}$ is made dimensionless by the roughness height $k^{+}$ . The horizontal axis represents the azimuthal coordinate in viscous units. (b) Root mean squares of the azimuthally averaged form-induced velocity fluctuations. The profiles are obtained at three heights, namely where the Taylor roll impacts, ejects and in the centre (sheared region) – the extent of the regions are estimated from time-averaged velocity fields. The cyan line gives the mean over the entire height of the cylinder. The solid black lines indicate the maximum roughness height $k_{max}^{+}$ and the height of the roughness sublayer $h_{r}^{+}$ .

A better impression of the local fluid flow disturbances induced by the roughness elements, is obtained from the time-averaged azimuthal velocity $\langle u\rangle _{t}$ , a contour of which we show in figure 10. We zoom in on only a few roughness elements and overlay the contour with isolines of $\langle u\rangle _{t}^{+}$ . We observe that local disturbances are limited to a region of only a few times the roughness height, above which the isolines relax to approximate horizontal lines. We observe small recirculation zones (closed isolines) in open regions behind high roughness elements. To quantify the degree of roughness-induced velocity disturbance, we apply a triple decomposition to the instantaneous azimuthal velocity $u(r,\unicode[STIX]{x1D703},z,t)$ (Pokrajac et al. Reference Pokrajac, Campbell, Nikora, Manes and McEwan2007),

(4.4) $$\begin{eqnarray}\displaystyle u(r,\unicode[STIX]{x1D703},z,t)=\langle u\rangle _{\unicode[STIX]{x1D703},t}(r,z)+\tilde{u} (r,\unicode[STIX]{x1D703},z)+u^{\prime }(r,\unicode[STIX]{x1D703},z,t), & & \displaystyle\end{eqnarray}$$

where $u^{\prime }(r,\unicode[STIX]{x1D703},z,t)=u(r,\unicode[STIX]{x1D703},z,t)-\langle u\rangle _{t}(r,\unicode[STIX]{x1D703},z)$ , the temporal fluctuation and $\tilde{u} (r,\unicode[STIX]{x1D703},z)=\langle u\rangle _{t}(r,\unicode[STIX]{x1D703},z)-\langle u\rangle _{\unicode[STIX]{x1D703},t}(r,z)$ , the component that is strongly related to the roughness induced disturbances and therefore termed the form-induced (or dispersive) velocity fluctuation. Note that by applying the triple decomposition to the full NS equations and then averaging in $\unicode[STIX]{x1D703}$ and $t$ , one will recover the related form-induced stress tensor $\langle \tilde{u} _{i}\tilde{u} _{j}\rangle _{\unicode[STIX]{x1D703},t}$ . However, here we only discuss $\tilde{u} (r,\unicode[STIX]{x1D703},z)$ . The root mean square of the form-induced fluctuations $\sqrt{\tilde{u} (r,\unicode[STIX]{x1D703},z)^{2}}^{+}$ at various heights above the roughness is given in figure 11(a). The horizontal axis corresponds to the roughness elements shown in figure 10. Already for $r/k=4.5$ (with $k$ being the roughness height, i.e. the smallest radius of the ellipsoidal building block) we find it hard to detect spatial fluctuations along $\unicode[STIX]{x1D703}$ . For $r/k=12.0$ , the line is barely distinguishable from the horizontal axis. If we average the lines over the azimuthal direction, we obtain the behaviour of the dispersive fluctations as a function of the wall-normal distance; see figure 11(b). We plot the lines for three respective axial locations, that is the impacting, sheared and ejecting regions, and find little variance in the wall-normal extent of the form-induced fluctuations, for the varying locations. The vertical black line represent the maximum roughness height, below which we average only over fluid regions.

Rough elements on the inner cylinder destroy the streamwise homogeneity of the flow, and thus the root mean square of the dispersive fluctuation is non-zero. Some distance from the wall, the turbulence is well able to ‘mix out’ the form-induced structures and the flow regains streamwise homogeneity. The distance from the wall at which the dispersive fluctuations are zero (or reasonably close to zero), is called the roughness sublayer height $h_{r}$ . In this research, we define $y^{+}=h_{r}^{+}$ where $\sqrt{\langle \tilde{u} ^{+2}\rangle _{\unicode[STIX]{x1D703},z}}(r)=0.01\langle u^{+}\rangle _{\unicode[STIX]{x1D703},z,t}$ and find $h_{r}=3.70k$ ( $h_{r}=2.78k_{s}$ ). This value agrees well with a roughness sublayer height of $2\lesssim k_{s}\lesssim 5$ , as typically found in other canonical flows (Pokrajac et al. Reference Pokrajac, Campbell, Nikora, Manes and McEwan2007).

The existence of a finite height of only a few times the roughness height, at which the dynamical effects of the roughness on the fluid flow vanish, is an important finding in wall-bounded turbulence. As written in Flack & Schultz (Reference Flack and Schultz2014): ‘...the utility of the roughness function itself hinges on similarity of the flow’. This idea (‘wall similarity’) is already heavily tested in other systems (Raupach et al. Reference Raupach, Antonia and Rajagopalan1991; Chung, Monty & Ooi Reference Chung, Monty and Ooi2014; Flack & Schultz Reference Flack and Schultz2014) and in the vast majority of the research found to be correct. However, in a heavily stratified system as TC, where wall-attached structures result from the unstable stratification of centrifugal pressure, the notion of wall similarity is not yet investigated. As such, the findings presented here, that such similarity exists in TC, strengthen our believe that TC could be a valued system to characterize the equivalent sand grain roughness height of a given rough surface.

4.5 Radial profiles

The effects of roughness on a turbulent shear-driven flow, where the shear rate is constant, can be separated into two effects. The first is the change in the wall shear stress $\unicode[STIX]{x1D70F}_{w}$ . The second is the change in the structure of the turbulent flow at a given wall shear stress. Sand grains increase $\unicode[STIX]{x1D70F}_{w}$ and therefore we find an increased momentum transfer to the bulk region with respect to a smooth-wall TC case, at fixed Taylor number. This increased momentum transfer to the bulk (through plumes, very similar to Rayleigh–Bénard flow) leads to a more intense turbulence in the bulk flow, and also to higher velocity fluctuations.

We plot the time and azimuthally averaged azimuthal velocity for $Ta=5.0\times 10^{8}$ in figure 12(a). The roughness covers the inner cylinder, i.e. $(r-r_{i})/d\leqslant 0.07$ . In this section we focus on the behaviour of the statistics above the roughness. Therefore, averaging over both solid and fluid regions is carried out. For increasing roughness height, see the legend for viscous roughness heights, the azimuthal velocity in the bulk increases. Figure 12(b) then presents the corresponding azimuthal velocity profiles for constant $k/d=0.060\pm 0.002$ and varying Taylor number.

Figure 13(a) shows the double-averaged radial profiles $(u)_{rms}^{+}=\langle \langle u^{2}\rangle _{\unicode[STIX]{x1D703},z,t}-\langle u\rangle _{\unicode[STIX]{x1D703},z,t}^{2}\rangle ^{1/2}/u_{\unicode[STIX]{x1D70F}}$ for constant Taylor number $Ta=1.0\times 10^{9}$ . For the smooth-wall case, the peak of $(u^{+})_{rms}$ is located at $y^{+}\approx 10$ . This agrees with the smooth case in (Ostilla-Mónico et al. Reference Ostilla-Mónico, Verzicco, Grossmann and Lohse2016) for TC flow, but is closer to the solid boundary than is found in channel flow; $y^{+}\approx 12$ . We observe a slight decrease in the viscous scaled turbulence intensity $(u)_{rms}^{+}$ for increasing roughness heights, close to the inner cylinder. Further outside the profiles, the profiles collapse.

Figure 12. Profiles of the mean streamwise velocity $\langle u\rangle _{\unicode[STIX]{x1D703},z,t}$ versus the radial coordinate $(r-r_{i})/d$ , where $r_{i}$ is the radius of the inner cylinder and $d$ is the gap width. (a) Constant Taylor number $Ta=5.0\times 10^{8}$ and increasing roughness heights. The profiles exhibit clear ‘slaving’, i.e. the bulk velocity moves towards the velocity of the roughened cylinder. (b) Constant roughness height $k/d=0.060\pm 0.002$ for increasing Taylor number. Increased slaving of the bulk velocity is observed for increasing viscous scaled roughness height $k_{s}^{+}$ , respectively: (blue) $k_{s}^{+}=9$ , (green) $k_{s}^{+}=24$ , (red) $k_{s}^{+}=47$ and (cyan) $k_{s}^{+}=65$ .

Figure 13. (a) Root mean square of the mean streamwise velocity $(u)_{rms}^{+}=\langle \langle u^{2}\rangle _{\unicode[STIX]{x1D703},z,t}-\langle u\rangle _{\unicode[STIX]{x1D703},z,t}^{2}\rangle ^{1/2}/u_{\unicode[STIX]{x1D70F}}$ versus the radius $(r-r_{i})/d$ for $Ta=1.0\times 10^{9}$ . We observe an overall decrease of the viscous scaled turbulence intensity for increasing roughness height $k_{s}^{+}$ . (b) The viscous ${\hat{J}}_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D714}}(y^{+})\equiv -r^{3}\unicode[STIX]{x1D708}\unicode[STIX]{x2202}_{r}\langle \unicode[STIX]{x1D714}\rangle _{\unicode[STIX]{x1D703},z,t}/J_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D714}}(r_{o})$ (solid line) and Reynolds stress ${\hat{J}}_{w\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D714}}(y^{+})\equiv r^{3}\langle w\unicode[STIX]{x1D714}\rangle _{\unicode[STIX]{x1D703},z,t}/J_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D714}}(r_{o})$ (dashed line) terms of the conserved angular velocity flux ${\hat{J}}^{\unicode[STIX]{x1D714}}$ versus the wall-normal distance $y^{+}$ for $Ta=1.0\times 10^{9}$ . The sum of the individual terms represents the total conserved angular velocity flux radially outwards. The black horizontal line at ${\hat{J}}^{\unicode[STIX]{x1D714}}=1.0$ is the sum of the two blue lines for the smooth-wall case. It is observed that ${\hat{J}}^{\unicode[STIX]{x1D714}}$ is conserved above the maximum roughness height (indicated by cross markers).

The reason why we choose to make the variables dimensionless using their friction equivalents is to assess those changes to the flow that go beyond the effects coming from ‘simply’ increasing the momentum input to that region, i.e. to study the structural changes of the turbulent flow for the same momentum input. We can thus infer from figure 13(a) that the bulk of the TC flow is not affected by the roughness elements, other than increasing the momentum input into the bulk would do. A gedankenexperiment (see e.g. Chung et al. (Reference Chung, Monty and Ooi2014)) would make this clear. If one would be placed into the bulk of a rough TC flow, without being able to look into the boundary layer, one could not tell whether the wall is rough or smooth. The reason is that a smooth wall with a higher Taylor number can produce the same $\unicode[STIX]{x1D70F}_{w}$ as a rough wall with a lower Taylor number. This idea is, in fact, Townsend’s outer layer similarity hypothesis (Townsend Reference Townsend1956). (Note: we can only be sure that this holds for the relatively small sand grain surface under scrutiny here, owing to the sufficient scale separation!) As such, we conclude that the influence of the roughness elements pertains only to a region close to the wall.

It has already been mentioned that the angular velocity flux $J^{\unicode[STIX]{x1D714}}(y^{+})$ is conserved in the radial direction, see § 4.2. However, the individual components – i.e. the viscous $J_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D714}}(y^{+})$ and Reynolds stress $J_{w\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D714}}(y^{+})$ terms – are functions of the wall-normal (radial) coordinate. The profiles of these terms are shown in figure 13(b). We normalize all terms with $J_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D714}}(r_{o})$ . The viscous stress terms are presented as solid lines and the Reynolds stress terms as dashed lines. The blue lines represent the smooth-wall case. Very close to the inner cylinder ( $y^{+}=1$ ), $J^{\unicode[STIX]{x1D714}}\approx J_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D714}}$ ; this is expected, since there the gradient of the mean streamwise velocity is maximum and the wall-normal component of the velocity vector goes to zero. On the far right of figure 13(b), the gradient of the mean velocity approaches zero in the bulk, and the correlation function between the wall-normal and angular velocity goes to a maximum; as such $J^{\unicode[STIX]{x1D714}}\approx J_{w\unicode[STIX]{x1D714}}^{\unicode[STIX]{x1D714}}$ . The black line represents the sum of the viscous and Reynold stress terms, for the smooth (blue) case, and is independent of $y^{+}$ . The situation for the rough cylinder cases is more complex. For increasing roughness height, the viscous stress reaches a maximum below the maximum roughness height. This can be explained by the recirculation zones behind the roughness elements. Then, above the roughness, the viscous stress goes to zero in the bulk. For the Reynolds stress terms the increase is monotonic, and similar to the smooth-wall case. However, we observe a steeper increase for the rough cases. Note that under the maximum roughness height, the viscous and Reynolds stress terms do not add up to unity, since the rough surface acts as a radial dependent momentum source term to the NS equations there.