3.1. Friction coefficient and flow transition

The friction coefficient is a key parameter for characterising the transition from laminar to turbulent flow in curved channels. According to the various definitions of friction velocity given in (2.1), we define local friction coefficients at the inner and at the outer wall, as well as a global friction coefficient,

(3.1) \begin{equation} C_{f,i}=2\left (\frac {u_{\tau ,i}}{u_b}\right )^2,\quad C_{f,o}=2\left (\frac {u_{\tau ,o}}{u_b}\right )^2,\quad C_{f,g}=2\left (\frac {u_{\tau ,g}}{u_b}\right )^2, \end{equation}

shown in figure 3 as a function of the bulk Reynolds number. The global friction (in the inset) is related to the mean-pressure gradient (B3), hence to the power expenditure to drive the flow. For comparison with the plane channel, we also report the friction law for laminar flow, $C_f=12/Re_b$ (dashed line in the inset), and the logarithmic friction relation,

(3.2) \begin{equation} \sqrt {\frac {2}{C_f}}=\frac {1}{k}\left [\ln \left (\frac {Re_b}{2} \sqrt {\frac {C_f}{2}}\right )-1\right ]+A \end{equation}

(solid line), obtained by Zanoun, Nagib & Durst (Reference Zanoun, Nagib and Durst2009) from the logarithmic law of the wall, with log-law constants $k=0.41$ , $A=5.17$ .

Figure 3. Friction coefficient as a function of the bulk Reynolds number for the R40 flow cases (a) and R1 flow cases (b). Red circles denote the local friction coefficient at the inner wall ( $C_{f,i}$ ), blue circles at the outer wall ( $C_{f,o}$ ), black circles in the insets denote the global friction coefficient ( $C_{f,g}$ ); dashed lines denote the analytical friction law for laminar flow (red for the convex wall, blue for the concave wall, black for the plane channel), black solid lines the logarithmic friction relation for plane channel flow (3.2). A magnified view of the friction trend for the R40 flow cases is shown in panel (c).

In the turbulent regime, the friction at the outer wall is consistently higher than at the inner wall due to increased turbulence intensity. For the R40 flow cases (figure 3 a), the friction coefficient of a plane channel at the same flow rate (black solid line) lies approximately midway between the values at the two walls of the curved channel and is comparable to the global friction. In contrast, for the R1 flow cases (figure 3 b), the inner-wall friction is significantly lower than that of an equivalent plane channel. However, the global friction is higher and aligns with the plane channel only for $Re_b\ge 40\,000$ .

Figure 4. Mean streamwise velocity ( $U/u_b$ ) at various Reynolds numbers for the R40 flow cases (a) and R1 flow cases (b). Black dashed lines refer to the Poiseuille profile, black circles to the velocity profile for laminar curved channel flow (3.3).

To comment more clearly on the friction trend in the laminar and transitional regime, it is useful to consider the mean velocity profiles at representative Reynolds numbers, as reported in figure 4. At low Reynolds number, the flow in a curved channel is laminar and the velocity profile can be derived analytically as

(3.3) \begin{equation} U(r)=\alpha r-\frac {\beta }{r}-r\ln r, \end{equation}

where

(3.4) \begin{equation} \alpha =\frac {r_o^2\ln r_o - r_i^2\ln r_i}{r_o^2-r_i^2},\quad \beta =\frac {r_i^2 r_o^2 \ln (r_o/r_i)}{r_o^2-r_i^2}. \end{equation}

In the R40 flow cases, the laminar velocity profile is nearly identical to the parabolic profile of plane channel flow, meaning that the shear stress distribution is almost symmetrical. As the curvature increases, the pressure gradient in the radial direction becomes stronger and the location of maximum velocity shifts towards the inner wall. As a result, the shear stress is larger at the inner wall than at the outer wall. In the laminar regime, the friction coefficient at the two walls can be determined analytically from (3.3), thus obtaining

(3.5) \begin{equation} \frac {\mathrm {d} U(r)}{\mathrm {d} r}=\alpha +\frac {\beta }{r^2}-\ln r-1. \end{equation}

For the R40 flow cases, we find $C_{f,i} \approx 12.03/Re_b$ , hence nearly identical to plane Poiseuille flow, and $C_{f,o} \approx 11.83/Re_b$ . In contrast, in the R1 flow cases the friction coefficient at the inner wall, $C_{f,i} \approx 18.34/Re_b$ , is approximately twice as at the outer wall, $C_{f,o} \approx 9.04/Re_b$ . Flow transition occurs when the Reynolds number attains a critical value at which centrifugal instability sets in and Dean vortices form, similarly to the Taylor vortices in flow between two rotating cylinders (Taylor Reference Taylor1923). Here, the critical Reynolds number is defined as the point where the friction coefficient deviates from the laminar trend by at least $2\, \%$ , an amount higher than the uncertainty of time-averaged measurements. For the R40 flow cases the critical Reynolds number is $Re_b\approx 250$ , which is in good agreement with the value of $228.5$ found by Finlay et al. (Reference Finlay, Keller and Ferziger1988), whereas for the R1 flow cases transition starts earlier, at $Re_b\approx 50$ . The transitional regime is greatly affected by curvature, with convex curvature (inner wall) tending to stabilise the flow, and concave curvature (outer wall) having the opposite effect. This is particularly evident in the R40 flow cases: at $Re_b=1000$ , the flow near the inner wall is laminar, as visible from the mean velocity profile in figure 4(a). In contrast, secondary motions are generated near the outer wall, which results in stronger momentum exchange. The peak velocity then shifts towards the outer wall, resulting in higher friction. At $Re_b = 4000$ , the flow near the outer wall is fully turbulent (as will be shown later through flow visualisations), whereas the velocity profile near the inner wall remains close to the laminar state. Hence, mild convex curvature tends to delay transition, as plane channel flow becomes turbulent at $Re_b \approx 2600$ (Yimprasert et al. Reference Yimprasert, Kvick, Alfredsson and Matsubara2021). Due to this stabilising effect, the transitional flow regime near the inner wall extends well into the range of Reynolds numbers for which fully turbulent flow is expected in plane channels. As a result, the global friction in the range $3000\le Re_b\le 10\,000$ is less than predicted from (3.2), as clearly shown from the magnified view in figure 3(c). This deviation reaches its maximum at $Re_b=4000$ , with a drag reduction of $12.2\, \%$ with respect to the plane channel data (Orlandi, Bernardini & Pirozzoli Reference Orlandi, Bernardini and Pirozzoli2015). By increasing the Reynolds number to $5000$ , the flow near the inner wall undergoes abrupt transition to a fully turbulent state, as indicated by the jump in the friction trend indicated by red circles in figure 3(a). This type of transition from laminar to turbulent regime is typical of canonical wall flows (see e.g. Patel & Head Reference Patel and Head1969), however, it quite different than at the outer wall, where transition is facilitated by centrifugal instabilities and occurs more smoothly.

Figure 5. Mean shear rate ( $\mathcal {S}\delta /u_b$ ) at various Reynolds numbers for the R40 flow cases (a) and R1 flow cases (b). Circles denote the analytical profile for the laminar case.

For the R1 flow cases, friction in the laminar regime is higher at the inner wall. By increasing the Reynolds number, the velocity profiles tend to flatten at the outer wall, where friction increases. Turbulence is inhibited near the inner wall, and friction deviates less markedly from the laminar trend. An inversion occurs past $Re_b \approx 1000$ , whereby friction at the outer wall becomes higher that at the inner wall. Flow transition is facilitated by strong channel curvature, and it occurs smoothly at both walls. At the outer wall, the different transition from canonical wall flows is due to centrifugal (primary) instabilities. As we will show, in the R1 flow cases secondary motions associated with centrifugal instabilities do not reach the inner wall, and transition is affected by streamwise (secondary) instabilities, which lead to the formation of large-scale cross-stream structures.

In figure 5 we show the mean shear rate, $\mathcal {S}=\mathrm {d} U/\mathrm {d} r-U/r$ . Similar to the case of the plane channel, the location where the mean shear vanishes can be used to define a bound for the regions of influence of the two walls. In the R40 flow cases (figure 5 a) the shear rate profile is nearly symmetrical, hence this location coincides roughly with the channel centreline, with exception of the case $Re_b=1000$ , for which it occurs at $y/\delta \approx 0.71$ . In contrast, for the R1 flow cases (figure 5 b) the shear rate profile is strongly asymmetrical, the zero crossing shifting towards the inner wall and approaching it as the Reynolds number increases. Hence, the region of influence of the inner wall is much narrower than for the outer wall. In case of strong curvature, the shear rate profile exhibits a local maximum near the outer wall and a local minimum near the inner wall, corresponding to inflectional points of the mean velocity profiles. This showcases the presence of shear layers that can support inflectional instabilities.

3.2. Flow visualisations

The previous discussion on flow transition can be visualised through the instantaneous fields of streamwise velocity in cross-stream and wall-parallel planes, which we show in figures 6 and 7 for the R40 flow cases and in figures 8 and 9 for R1 flow cases. As for the R40 flow cases, the cross-stream and longitudinal planes at $Re_b=1000$ , shown in figures 6(a) and 7(a), highlight the onset of two symmetric large-scale ejections carrying low-speed fluid from the outer wall towards the channel core. These secondary motions are generated by two pairs of counter-rotating roll cells, which resemble closely Dean vortices. The inner-wall flow region is almost unaffected by these vortices, as one can see from figure 7(b). Figures 6(b) and 7(c), depicting the cross-stream and longitudinal planes at $Re_b=4000$ , show that fine-scale ejections and organised structures of momentum streaks characterise the outer wall, from which one can infer that the flow is fully turbulent. Traces of the streamwise vortices appear as streamwise-aligned bands of high-speed fluid near the inner wall, as visible in figure 7(d). Being centrifugally stable, the inner-wall flow region is still transitional, as no clear signs of turbulent activity appear. The cross-stream planes at $Re_b=20\,000$ and $87\,000$ (figure 6 c,d) show clear signs of streaky patterns at both walls. Overlaying the small scales of turbulence, streamwise large-scale structures are found near the outer wall, whose footprint appears in the wall-parallel planes of figures 7(e) and 7(g) as wide regions of alternating high and low speed fluid. These large-scale regions fill the entire streamwise extent of the domain, and their spanwise width is comparable to the channel height, hence two pairs of roll cells are present. The influence of the latter is found to extend to the inner wall at high Reynolds numbers. In fact, low-speed regions at the outer wall, corresponding to large-scale ejections, are found at the same spanwise location as high-speed regions at the inner wall, corresponding to large-scale sweeps (figure 7 f,h).

Figure 6. Instantaneous streamwise velocity fields in a cross-stream plane for the R40 flow cases at $Re_b=1000$ (a), $4000$ (b), $20\,000$ (c), $87\,000$ (d). The flooded contours range from $0.2\,u_b$ (blue) to $1.2\,u_b$ (red).

Figure 7. Instantaneous fields of streamwise velocity fluctuations for the R40 flow cases in wall-parallel planes near the outer (a,c,e,g) and inner wall (b,d,f,h) at $y^+\approx 12$ . The panels correspond to $Re_b=1000$ (a,b), $4000$ (c,d), $20\,000$ (e,f), $87\,000$ (g,h). The flooded contours range from $-0.3\,u_b$ (blue) to $0.3\,u_b$ (red). Streamwise and spanwise coordinates are shown in both outer units ( $r_c\theta /\delta$ , $z/\delta$ ), and local wall units ( $r_c\theta ^+$ , $z^+$ ), where the streamwise domain extent is measured at the centreline. Mean flow goes from left to right.

Figure 8. Instantaneous streamwise velocity fields in a cross-stream plane for the R1 flow cases at $Re_b=1000$ (a), $4000$ (b), $20\,000$ (c), $87\,000$ (d). The flooded contours range from $0.4\,u_b$ (blue) to $1.5\,u_b$ (red). Only half of the domain is shown.

Figure 9. Instantaneous fields of streamwise velocity fluctuations for the R1 flow cases in wall-parallel planes near the outer (a,c,e,g) and inner wall (b,d,f,h) at $y^+\approx 12$ . The panels correspond to $Re_b=1000$ (a,b), $4000$ (c,d), $20\,000$ (e,f), $87\,000$ (g,h). The flooded contours range from $-0.3\,u_b$ (blue) to $0.3\,u_b$ (red). Streamwise and spanwise coordinates are shown in both outer units ( $r_c\theta /\delta$ , $z/\delta$ ), and local wall units ( $r_c^+\theta$ , $z^+$ ), where the streamwise domain extent is measured at the centreline. Only half of the domain is shown.

Cross-stream visualisations show clearly how the region with higher momentum shifts towards the inner wall for the R1 flow cases (figure 8), as expected from the mean velocity profiles (figure 4). As well as for the R40 flow cases, signatures of streamwise roll cells are visible as large-scale ejections from the outer wall, although they are now more numerous and chaotic. As displayed in the wall-parallel plane of figure 9(a), streaky structures appear to be quite distinct at the outer wall at $Re_b=1000$ , confirming that increasing curvature promotes the transition to turbulence. Figure 9(c,e,g) show that momentum streaks at the outer wall exhibit a wavy pattern with a streamwise wavelength comparable to the domain size. This pattern resembles the ‘undulating Dean vortex flow’ found by Finlay et al. (Reference Finlay, Keller and Ferziger1988), which is likely associated with inflectional instabilities of the wall-normal velocity profile (Le Cunff & Bottaro Reference Le Cunff and Bottaro1993), as evidenced by figures 4(b) and 5(b). The streaks are wider for the R1 flow cases than for R40, whereas streamwise large-scale structures are narrower, making the distinction between fine-scale turbulence and large-scale structures less clear cut. In fact, streamwise roll cells do not clearly show up in the wall-parallel planes of fluctuating streamwise velocity. This, however, does not convey that they are absent: as we shall see, in case of strong curvature these secondary motions are organised differently and mainly impact the wall-normal velocity. As visible from the wall-parallel planes shown in figure 9(b,d,f), the inner-wall flow region is devoid of turbulent structures, or nearly so for the highest Reynolds number (figure 9 h), whereas it is characterised by the presence of large-scale structures elongated in the spanwise direction. The latter are visible in the form of alternating regions of high and low speed fluid aligned along the spanwise with length comparable to $\delta$ . The streamwise extent of each pair of spanwise structures is approximately $\pi \delta$ , meaning that the computational domain can fit two pairs, with exception of the case $Re_b=4000$ , at which the streamwise wavelength seems slightly shorter.

3.3. Velocity spectra

A clearer understanding of the energetic relevance of the various scales of motions can be achieved from inspecting the spectra of the velocity fluctuations. Spectra of both streamwise and wall-normal velocity fluctuations are presented, the former to detect large-scale structures typical of plane channel flows (Lee & Moser Reference Lee and Moser2015), whereas the latter are instrumental in revealing the presence of streamwise roll cells (Dai, Huang & Xu Reference Dai, Huang and Xu2016). Figures 10 and 11 display the premultiplied spectra of streamwise velocity fluctuations in the spanwise ( $k_z^* E^*_{uu}$ ) and streamwise ( $k_\theta ^* E^*_{uu}$ ) directions as a function of the distance from the wall for R40 and R1 flow cases, respectively. For clarity, we refer to the distance from the inner wall as $y_i=y$ , and from the outer wall as $y_o=\delta -y$ ), both displayed on the left-hand vertical axis. The distance from the inner and outer wall is also reported on the right-hand vertical axis in local wall units, $y^+_i$ and $y^+_o$ , respectively.

Figure 10. Premultiplied spectra of streamwise fluctuating velocity as a function of spanwise wavelength, $k_z^* E^*_{uu}$ , (a–c ) and of streamwise wavelength, $k_\theta ^* E^*_{uu}$ , (d–f) as the wall distance varies for the R40 flow cases. The panels correspond to $Re_b=4000$ (a,d), $20\,000$ (b,e), $87\,000$ (c,f). Wall distance from the inner and outer wall, spanwise and streamwise wavelengths are reported in outer units ( $y_i/\delta$ , $y_o/\delta$ , $\lambda _z/\delta$ , $\lambda _\theta /\delta$ ), and in local wall units ( $y_i^+$ , $y_o^+$ , $\lambda _z^+$ , $\lambda _\theta ^+$ ), respectively. Dashed lines mark wavelength and wall distance of the energy peak associated with streamwise large-scale structures.

In the R40 flow cases a primary energy peak appears in the spanwise spectra (figures 10 a–c and 11 a–c) near both walls at $y_i^+\approx y_o^+\approx 12$ , $\lambda _z^+\approx 100$ , which is the typical signature of the regeneration cycle of momentum streaks (Hamilton, Kim & Waleffe Reference Hamilton, Kim and Waleffe1995). An exception is the case at $Re_b=4000$ : the inner-wall primary peak is nearly absent, confirming the inferences from flow visualisations. From the streamwise spectra (figures 10 d–f and 11 d–f) we note that the streaks-related peak is at $\lambda _\theta ^+\approx 800$ , meaning that the mean wavelength of momentum streaks is slightly shorter than the typical value of plane channel flows, namely $\lambda _x^+\approx 1000$ (Monty et al. Reference Monty, Hutchins, Ng, Marusic and Chong2009). A secondary peak appears in the spanwise spectra farther from walls, at $y_i\approx y_o\approx 0.1 \delta$ at $Re_b=4000$ (figures 10 a and 11 a), and $y_i\approx y_o\approx 0.2\delta$ at $Re_b=20\,000$ (figures 10 b and 11 b) and $87\,000$ (figures 10 c and 11 c). As seen from the intersection of the dashed lines, the energy of the peaks is concentrated at $\lambda _z/\delta \approx 2$ . If the domain contains $n$ pairs of roll cells, we expect to find a peak in the spanwise spectra at $\lambda _z=L_z/n$ , where $L_z=4\delta$ for the R40 flow cases. Flow visualisations indeed revealed the presence $n=2$ roll cells pairs, hence we interpret these peaks as the footprint of streamwise large-scale structures. A third peak at $\lambda _z/\delta \approx 1$ appears faintly at $Re_b=20\,000$ , and sharply emerges at $Re_b=87\,000$ . This peak corresponds to $n=4$ pairs of streamwise cells, indicating that roll cells with various sizes could be present, which we will investigate below. The streamwise spectra reveal that the scales of motion with the maximum streamwise wavelength, $\lambda _\theta =L_\theta =2\pi$ , are quite energetic, suggesting that the streamwise vortices span the whole domain along the streamwise direction.

Figure 11. Premultiplied spectra of streamwise fluctuating velocity as a function of spanwise wavelength , $k_z^* E^*_{uu}$ , (a–c) and of streamwise wavelength, $k_\theta ^* E^*_{uu}$ , (d–f) as the wall distance varies for the R1 flow cases. FThe panels correspond to $Re_b=4000$ (a,d), $20\,000$ (b,e), $87\,000$ (c,f). Wall distance from the inner and outer wall, spanwise and streamwise wavelengths are reported in outer units ( $y_i/\delta$ , $y_o/\delta$ , $\lambda _z/\delta$ , $\lambda _\theta /\delta$ ), and in local wall units ( $y_i^+$ , $y_o^+$ , $\lambda _z^+$ , $\lambda _\theta ^+$ ), respectively. Dashed lines mark wavelength and wall distance of the energy peak associated with spanwise large-scale structures.

As for the R1 flow cases (figure 11) the primary peak in the spanwise spectra due to momentum streaks appears on the outer wall only, suggesting that the near-wall turbulence regeneration cycle still takes place at the outer wall (figure 11 a–c). As noted in flow visualisations, streaky structures have a greater spanwise wavelength than the R40 flow cases, which is approximately $\lambda _z^+\approx 120$ . Their streamwise wavelength is instead shorter, approximately $\lambda _\theta ^+\approx 500$ , as visible from the streamwise spectra (figure 11 d–f). The absence of a primary peak near the inner wall in the spanwise spectra highlights the stabilising effect of convex curvature. More specifically, the near-wall peak is absent at $Re_b=4000$ (figure 11 d), starts to form at $Re_b=20\,000$ (figure 11 e), and only becomes visible at $Re_b=87\,000$ (figure 11 f). This trend, which is also found in the streamwise spectra, suggests that the inhibiting effect of convex curvature on turbulence becomes gradually less intense as the Reynolds number increases. In contrast to the R40 flow cases, never does a secondary peak emerge in the spanwise spectra near the outer wall. This implies that the streamwise roll cells, if present, have a negligible effect on the streamwise velocity fluctuations. A secondary peak is instead present near the inner wall at $\lambda _z/\delta \approx 2$ , which moves closer to the wall (in $\delta$ units) as $Re_b$ increases. This peak cannot be related to streamwise roll cells for two reasons: (i) they are most intense at the outer wall, yet no secondary peak is detected there, and (ii) the spanwise wavelength of streamwise vortices must be shorter, as one can infer from figure 8 and as we will show more clearly below. Rather, we can ascribe this peak to the presence of spanwise large-scale structures. To substantiate this statement, we consider the streamwise energy spectra, which highlight energetic modes near the inner wall at long wavelengths: $\lambda _\theta /\delta \approx 2\pi /3$ at $Re_b=4000$ (figure 11 d) and $\lambda _\theta /\delta \approx \pi$ at $20\,000$ (figure 11 e) and $87\,000$ (figure 11 f), as marked by the dashed lines. These energy peaks cannot be related either to turbulence structures, which are absent or strongly inhibited near the inner wall, or to streamwise roll cells, as we have just seen through the spanwise spectra that they do not affect streamwise velocity fluctuations. However, flow visualisations in wall-parallel planes near the inner wall (figure 9 b,d,f) revealed the presence of spanwise structures, organised in pairs with a streamwise length of approximately $\pi \delta$ . The wavelength of these peaks corresponds to $n=L_\theta /\lambda _\theta =3$ pairs of eddies at $Re_b=4000$ and $n=L_\theta /\lambda _\theta =2$ pairs at $20\,000$ and $87\,000$ . Hence, we interpret the energy peaks in the streamwise spectra as the signature of cross-flow structures. Moreover, the spanwise size of these structures is comparable to $\delta$ , explaining the peaks in the spanwise spectra at $\lambda _z/\delta \approx 2$ .

Figure 12. Premultiplied spectra of fluctuating wall-normal velocity ( $k_z^* E^*_{vv}$ ) as a function of the spanwise wavelength and wall distance for the R40 flow cases (a–c) and R1 flow cases (d–f ). The panels correspond to $Re_b=4000$ (a,d), $20\,000$ (b,e), $87\,000$ (c,f). Wall distance from the inner and outer wall and spanwise wavelength are reported in outer units ( $y_i/\delta$ , $y_o/\delta$ , $\lambda _z/\delta$ ) and local wall units ( $y_i^+$ , $y_o^+$ , $\lambda _z^+$ ), respectively. Dashed lines mark the wavelength and wall-distance of the energy peak related to streamwise large-scale structures.

In figure 12 we show the premultiplied spectral densities of wall-normal velocity fluctuations in the spanwise direction ( $k_z^* E^*_{vv}$ ). In the R40 flow cases (figure 12 a–c) energy peaks related to near-wall turbulent structures form at $y_i^+\approx y_o^+\approx 50$ , $\lambda _z^+\approx 100$ , except for the inner wall at $Re_b=4000$ , in agreement with the flow visualisations. Two secondary peaks in the channel core, more prominent at $Re_b=87\,000$ , highlight the presence of energetic modes with $\lambda _z/\delta \approx 1$ and $\lambda _z/\delta \approx 2$ at $y_o/\delta \approx 0.3$ . Following the same reasoning about the streamwise velocity spectra, we trace these energetic modes to streamwise roll cells, which generate large-scale radial sweeps and ejections.

The effect of strong curvature on the energy distribution of the wall-normal velocity fluctuations (figure 12 d–f ) is rather striking. Contrary to the streamwise velocity spectra, the energy content is now much higher in the case of strong curvature, which points to substantial structural changes. No distinct peak is observed in the inner part of the channel, which conveys that the spanwise large-scale structures that we noted above have no impact (on average) on the wall-normal velocity fluctuations. A prominent energy peak is located at $y_o\approx 0.3\delta$ , $\lambda _z\approx 0.8\delta$ , marked by the intersection of the dashed lines. This energetic mode is not present in plane channel flow (see e.g. Cho, Hwang & Choi Reference Cho, Hwang and Choi2018) and cannot be related to cross-flow structures, which are more intense near the inner wall. Hence, we interpret them as the signatures of streamwise roll cells. This would mean that the typical configuration of streamwise large-scale structures consists of $n=L_z/\lambda _z=10$ pairs of roll cells (with $L_z=8$ , $\lambda _z=0.8\delta$ ), which we will ascertain in the following.

3.4. Streamwise large-scale structures

Flow visualisations and energy spectra reveal the presence of streamwise large-scale structures, which are akin to the Dean vortices observed in laminar flow (Dean Reference Dean1928). To understand how these structures depend on curvature and Reynolds number and how they affect the flow field, we separate the coherent contribution from the underlying turbulence, where coherence is intended in the sense given by Hussain (Reference Hussain1986). For that purpose we use the triple decomposition of Hussain & Reynolds (Reference Hussain and Reynolds1970), whereby a generic field variable, $\varphi (\theta ,r,z,t)$ , is decomposed as

(3.6) \begin{equation} \varphi (\theta ,r,z,t)={\Phi }(r)+\tilde {\varphi }(r,z,t)+\varphi ''(\theta ,r,z,t), \end{equation}

where $\Phi$ is the space average over the homogeneous directions ( $\theta$ , $z$ ) and time ( $t$ ),

(3.7) \begin{equation} \tilde {\varphi }(r,z,t)=\langle \varphi (\theta ,r,z,t)\rangle _\theta -{\Phi }(r), \end{equation}

is the coherent contribution from the streamwise vortices, $\langle \varphi (\theta ,r,z,t)\rangle _\theta$ is the average over the streamwise direction, and

(3.8) \begin{equation} \varphi ''(\theta ,r,z,t)=\varphi (\theta ,r,z,t)-\langle \varphi (\theta ,r,z,t)\rangle _\theta \end{equation}

is the instantaneous turbulent fluctuation. The total fluctuation is then $\varphi '=\tilde {\varphi }+\varphi ''$ . The instantaneous streamwise-averaged fields ( $\tilde {\varphi }$ ) are then averaged over time to extract the coherent contribution, namely $\overline {\tilde {\varphi }}$ . One should note that, in principle, the averages defined in (3.7) would be zero in infinitely long domains, and their long-time averages would also tend to zero. Hence, the coherent motions herein considered refer to a finite streamwise length associated with the extent of the computational domain (here, $L_{\theta } = 2 \pi \delta$ ), and to a finite time horizon, here taken to be $600 \delta / u_b$ . Whereas the intensity of those coherent motions is inevitably affected by these choices (see appendix D), the conclusions regarding the influence of Reynolds number and wall curvature still stand.

Figure 13. Mean Stokes stream function ( $\overline {\tilde {\psi }}$ ) for streamwise-coherent disturbances in an $(r,z)$ -plane overlaid to flooded contours of coherent streamwise velocity ( $\overline {\tilde {u}}$ ), for the R40 flow cases (a–c) and for the R1 flow cases (d–f). The panels correspond to $Re_b=4000$ (a,d), $20\,000$ (b,e), $87\,000$ (c,f). Positive values of $\tilde {\psi }$ (solid lines) indicate a clockwise-rotating roll cell, and vice versa for negative values (dashed lines). The flooded contours range from $-1.5\,u_{\tau ,g}$ (blue) to $1.5\,u_{\tau ,g}$ (red). Each panel shows the spanwise distribution of the mean shear stress at the two walls, $\overline {\tau }_{w,i}^+(z)$ and $\overline {\tau }_{w,o}^+(z)$ , defined in (3.9). Only half of the domain is shown for the R1 flow cases.

In figure 13 we show the Stokes stream function for streamwise-coherent motions ( $\overline {\tilde {\psi }}$ ), defined such that $\tilde {w}= {\partial \tilde \psi }/{\partial r}$ , $\tilde {v}=-{\partial \tilde \psi }/{\partial z}$ . The overlaid flooded contours represent the mean coherent streamwise velocity ( $\overline {\tilde {u}}$ ). In addition, we report the spanwise distribution of the mean shear at the inner and outer wall,

(3.9) \begin{equation} \overline {\tau }_{w,i}^+(z)=\frac {\nu }{u_{\tau ,i}^2}\frac {\partial \overline {\langle u\rangle _{\theta }}}{\partial r}\bigg |_{r_i},\qquad \overline {\tau }_{w,o}^+(z)=\frac {\nu }{u_{\tau ,o}^2}\frac {\partial \overline {\langle u\rangle _{\theta }}}{\partial r}\bigg |_{r_o}. \end{equation}

In the R40 flow cases (figure 13 a–c) two pairs of counter-rotating vortices appear which nearly span the whole channel thickness and with a spanwise wavelength $\lambda _z/\delta \approx 2$ , which is in agreement with the energy spectra in figure 10. Streamwise roll cells with the same size were detected by Moser & Moin (Reference Moser and Moin1987). The spanwise inhomogeneity due to these secondary eddies has a strong impact on the distribution of the wall shear stress. Indeed, large-scale ejections are generated between any pair of counter-rotating vortices (blue contours), where low-speed fluid is diverted away from the wall, and local friction attains a minimum. Correspondingly, large-scale sweeps generate at the opposite wall whereby high-speed fluids is pushed towards the wall (red contours), causing local increase of wall friction. This tendency is clearer as the Reynolds number increases, as one can infer from the spanwise distribution of the local wall shear, which shows excursions of approximately $10\, \%$ at $Re_b=4000$ (figure 13 a) and $Re_b=20\,000$ (figure 13 b), and reaching up to $20\, \%$ at $Re_b=87\,000$ (figure 13 c).

As for the R1 flow cases (figure 13 d–f), the number of pairs of counter-rotating vortices increases to approximately 10 (only five are visible as half of the domain is shown). Hence, the spanwise wavelength of the roll cells is $\lambda _z/\delta \approx 0.8$ , which again corresponds to what found from the spectral analysis in figure 12. Spanwise shortening of the streamwise roll cells is also observed in rotating channel flows, in which similar vortices develop on account of Coriolis forces (Matsson & Alfredsson Reference Matsson and Alfredsson1990). Through DNS of rotating channel flow, Brethouwer (Reference Brethouwer2017) found that the size of the streamwise roll cells is smaller at higher rotation numbers (the rotation number in rotating channel flow is the counterpart of the curvature ratio in curved channels), in agreement with previous numerical studies (Kristoffersen & Andersson Reference Kristoffersen and Andersson1993; Yang & Wu Reference Yang and Wu2012). In addition, Brethouwer (Reference Brethouwer2017) found that the size of the roll cells is independent of the Reynolds number, which is also the case here. Due to strong channel curvature, the roll cells are pushed towards the outer wall and cannot fill the entire channel thickness. Hence, the flow region near the inner wall (say, $y/\delta \lt 0.2$ ) is not affected, as seen in the spanwise distribution of the wall shear, which is nearly flat. As for the shear at the outer wall, spanwise excursions of $\tau _{w,o}^+(z)$ have a maximum amplitude of approximately $4\, \%$ at $Re_b=4000$ (figure 13 d) and $Re_b=20\,000$ (figure 13 e), whereas they are very small at $Re_b=87\,000$ (figure 13 f), at which the effect of the streamwise vortices is outweighed by turbulence. The reason why the outer-wall shear stress is barely affected by streamwise vortices is their unsteadiness. The coherent stream function shows indeed that roll cells are less organised in the R1 flow cases than in R40, and on average they do not affect the streamwise velocity. Nonetheless, the impact of those eddies on the instantaneous flow is certainly not negligible in the case of strong curvature. The strength of streamwise large-scale structures can be quantified in terms of the maximum amplitude of the coherent wall-normal velocity (Canton et al. Reference Canton, Örlü, Chin, Hutchins, Monty and Schlatter2016), which results in $\text {max}|\tilde {v}|/u_b\approx 6\, \%$ for the R40 flow cases, nearly independent of the Reynolds number. Streamwise vortices in the R1 flow cases are much stronger, their strength being $\text {max}|\tilde {v}|/u_b\approx 20\, \%$ at $Re_b=4000$ and $12\, \%$ at $87\,000$ .

Figure 14. Time history of the premultiplied spanwise energy spectra of fluctuating streamwise velocity ( $k_z^*E^*_{uu}$ ) for the R40 flow case at $Re_b=87\,000$ , at $y/\delta =0.8$ . The black dashed lines correspond to $\lambda _z/\delta =2$ and $\lambda _z/\delta =1$ , at which the expected number of streamwise vortices are indicated on the right vertical axis.

Figure 15. Coherent stream function ( $\tilde {\psi }$ ) at the time instants marked by the red dashed lines in figure 14, for the R40 flow case at $Re_b=87\,000$ .

3.5. Splitting and merging events

The eduction procedure based on triple decomposition allowed to quantify the statistical organisation of streamwise large-scale structures. However, the energy spectra highlighted the presence of multiple peaks at large wavelengths, which are particularly evident for the R40 flow case at $Re_b=87\,000$ (figure 10). Multiple peaks suggest either the coexistence of large-scale structures of different sizes, or the occurrence of splitting and merging events. To clarify this point, we inspect the time history of the spanwise energy spectra of streamwise velocity fluctuations, shown in figure 14 for the R40 flow case at $Re_b=87\,000$ , at the wall distance where the peak of the energy associated with the streamwise vortices occurs. A time window from $tu_b/\delta =1000$ to $1120$ was selected for the analysis and we took six subsequent snapshots of $\tilde {\psi }$ , shown in figure 15, at intervals of $20\delta /u_b$ marked by the red dashed lines in figure 14. From $tu_b/\delta \approx 1000$ to $1040$ most energy is clustered around $\lambda _z/\delta =1$ , hence $n=4$ pairs of vortices would be expected (the number of vortices pairs is indicated on the right-hand vertical axis). However, the snapshots of $\tilde {\psi }$ at $tu_b/\delta \approx 1010$ (figure 15 a) and $1030$ (figure15 b) reveal the occurrence of $n=3$ pairs. In this respect, we note that the roll cells are not uniform in size, as the pair centred at $z/\delta \approx 0.5$ has a wavelength $\lambda _z/\delta \approx 2$ , whereas the two remaining pairs have a wavelength $\lambda _z/\delta \approx 1$ . A possible explanation is that the two pairs of small-size vortices are stronger than the large-size one, hence more energy is concentrated at $\lambda _z/\delta =1$ . A vortex-merging event is observed between $tu_b/\delta \approx 1050$ and $1090$ . Figures 15(c) and 15(d) depict that the pair of smallest vortices located at $z/\delta \approx 3$ decrease in size and strength, as the isolines of the stream function get sparser. This process continues until the smallest pair is embedded within the adjacent clockwise rotating vortex (figure 15 e), which appears more regular and strong at $tu_b/\delta \approx 1110$ (figure 15 f). From $tu_b/\delta \approx 1080$ on, the energy peak is clustered around $\lambda _z/\delta =2$ and, as expected, $n=2$ pairs of roll cells are found.

Figure 16. Time history of the premultiplied spanwise energy spectra of fluctuating wall-normal velocity ( $k_z^*E^*_{vv}$ ) for the R1 flow case at $Re_b=87\,000$ , at $y/\delta =0.7$ . The black dashed lines correspond to $\lambda _z/\delta =0.73$ , $\lambda _z/\delta =0.80$ and $\lambda _z/\delta =0.89$ , at which the expected number of streamwise vortices are indicated on the right-hand vertical axis.

Figure 17. Coherent stream function ( $\tilde {\psi }$ ) at the time instants marked by the red dashed lines in figure 16 for the R1 flow case at $Re_b=87\,000$ . Only half of the domain is shown.

The results obtained for the R1 flow case at $Re_b=87\,000$ are presented in figure 16, which reports the time evolution of the spanwise spectra of the wall-normal velocity at the peak position of energy associated with the streamwise vortices. The time window under scrutiny is half as for the R40 flow case, and the six subsequent snapshots of $\tilde {\psi }$ , shown in figure 17, are sampled at intervals of $10\delta /u_b$ instead of $20$ on account of stronger unsteadiness in cases with strong curvature. The time evolution of the spectra shows that the energy peak is clustered around $\lambda _z/\delta \approx 0.89$ from $tu_b/\delta =1055$ to $1065$ . Consistently, figure 17(a) displays nine vortices in the region under scrutiny ( $n=9$ pairs are present in the whole domain). This configuration seems to be unstable because of the lack of organisation of the vortex pair located at $z/\delta \approx 1$ , and because two anticlockwise vortices are adjacent straddling $z/\delta \approx 2$ . At $tu_b/\delta =1070$ the energy peak shifts at $\lambda _z/\delta \approx 0.8$ , indeed the related figure 17(b) shows that a new clockwise rotating vortex emerged at $z/\delta \approx 2.2$ fitting in between the two anticlockwise rotating vortices and increasing the number of vortices to $10$ (i.e. $n=10$ ). The number of vortex pairs increase further to $n=11$ at $tu_b/\delta \approx 1080$ , as visible in figure 17(c), and the energy peak decreases accordingly to $\lambda _z/\delta \approx 0.73$ . From $tu_b/\delta \approx 1085$ to $1105$ the energy peak settles at $\lambda _z/\delta \approx 0.8$ , and figures 17(d) and 17(e) highlight that the vortex configuration consists of $n=10$ pairs. The energy peak shifts again to $\lambda _z/\delta \approx 0.73$ at $tu_b/\delta =1110$ . Figure 17(f) shows that the main vortices are still 10, however, small secondary vortices tend to split off from the primary ones, bringing more energy to smaller scales. In the case of strong curvature, the transitions from one vortex configuration to the other can be attributed to the unsteady dynamics of the vortices, which can be inferred from their spanwise motions, distorted shapes and different sizes, rather than to splitting and merging phenomena, which could not be clearly identified.

3.6. Role of streamwise vortices on velocity fluctuations

In figure 18 we show the root-mean-square (r.m.s.) of the streamwise (figure 18 a,b) and wall-normal (figure 18 c,d) velocity fluctuations, as well as the turbulent shear stress (figure 18 e,f). Total fluctuations are reported along with the contributions due to the streamwise vortices, which we have determined by taking the r.m.s. of the coherent contribution (3.7) along the spanwise direction and in time. For the R40 flow cases, the streamwise velocity fluctuations (figure 18 a) are higher at the outer than at the inner wall. As made clear from the increased intensity of the coherent fluctuations, this asymmetry is mainly due to the streamwise vortices, which are stronger near the outer wall. An exception is the case at $Re_b=4000$ (green line), in which the coherent contribution at the outer wall is comparable to that at the inner wall. In general, the coherent contribution is approximately half of the total. The peak of the coherent contribution occurs between $y/\delta \approx 0.8$ and $y/\delta \approx 0.9$ , corresponding to a bump in the profile of the total fluctuations. As for the R1 flow cases, the profiles of streamwise velocity fluctuations (figure 18 b) show a near-wall peak on the concave side comparable to that of the corresponding R40 flow cases, and little contribution from coherent fluctuations. The profile of the velocity fluctuations is almost flat in the channel core, and the near-wall peak on the convex side is very different from the case with mild curvature, extending much farther from the wall. Wall-normal velocity fluctuations (figure 18 c) are only relevant in the channel core, with a peak at $y/\delta \approx 0.7$ . For the R1 flow cases (figure 18 d) fluctuations of wall-normal velocity are up to two limes larger than the streamwise velocity. Substantial contribution is found to be provided by the coherent fluctuations, which also attain a peak at $y/\delta \approx 0.7$ , as observed in the velocity spectra.

Figure 18. Profiles of r.m.s. streamwise (a,b) and wall-normal (c,d) velocity fluctuations and mean turbulent shear stress (e,f) at various Reynolds numbers for the R40 flow cases (a,c,e) and R1 flow cases (b,d,f). Solid lines refer to total fluctuations and dashed lines refer to coherent fluctuations due to streamwise vortices.

As for the turbulent shear stress, it is nearly symmetrical in fully turbulent R40 flow cases (figure 18 e), resembling the case of a plane channel (see e.g. Kim et al. Reference Kim, Moin and Moser1987). However, two differences should be noted: (i) the point of zero crossing is shifted towards the inner wall at $y/\delta \approx 0.4$ , and (ii) the peak value increases near the outer wall and decreases near the inner wall. The effect of convex curvature is particularly evident at $Re_b=4000$ , at which the zero crossing is even closer to the inner wall. The coherent turbulent stress is approximately half of the total at $y/\delta \approx 0.8$ , pointing to significant contribution of streamwise vortices to momentum transport. Similar conclusions were also reached by Moser & Moin (Reference Moser and Moin1987) and Brethouwer (Reference Brethouwer2022). The effects of strong curvature are shown in figure 18(f). In the channel core, where viscous effects are negligible, the turbulent stress profile becomes quadratic rather than linear, as after the analytical distribution of the total shear stress (C2). As seen in the inset of figure 18(f), the peak value is greatly reduced near the inner wall, which aligns with the findings of So & Mellor (Reference So and Mellor1973) in experiments of turbulent boundary layers over a convex surface. The reduction in turbulent shear stress at the inner wall is not only due to the stabilising effect of convex curvature but also results from the influence of spanwise large-scale structures, as will be discussed below. The coherent Reynolds stress is negligible and vanishes entirely near the inner wall, as the streamwise vortices are displaced towards the outer wall (see § 3.4). This analysis demonstrates that strong curvature significantly alters the role of streamwise-aligned coherent structures in turbulence. Specifically, their contribution to wall-normal fluctuations becomes substantially larger than to streamwise fluctuations, while their contribution to momentum transport is suppressed.

3.7. Spanwise large-scale structures

Visualisations of the flow near the inner wall of strongly curved channels (figure 9) revealed the presence of alternating regions with positive/negative velocity fluctuations elongated along the spanwise direction. Those wavy patters are the footprint of spanwise large-scale structures, which are originated from streamwise instabilities (Finlay et al. Reference Finlay, Keller and Ferziger1988) and which are convected at the mean flow speed (Matsson & Alfredsson Reference Matsson and Alfredsson1992). As we did for the streamwise vortices, we exploit triple decomposition to separate the effects of the cross-stream structures from those of turbulence. A generic field variable, $\varphi (\theta ,r,z,t)$ , is decomposed as

(3.10) \begin{equation} \varphi (\theta ,r,z,t)={\Phi }(r)+\hat {\varphi }(\theta ,r,t)+\varphi '''(\theta ,r,z,t), \end{equation}

where

(3.11) \begin{equation} \hat {\varphi }(\theta -tu_c/r,r,t)=\langle \varphi (\theta ,r,z,t)\rangle _z-{\Phi }(r), \end{equation}

is the spanwise-coherent contribution, $\langle \varphi (\theta ,r,z,t)\rangle _z$ is the average along the spanwise direction, and $\varphi '''(\theta ,r,z,t)$ is the instantaneous turbulent fluctuation. Since spanwise-aligned structures are advected with the flow, phase alignment is required to educe the associated coherent contribution. Hence, the time averages are evaluated by shifting consecutive spanwise-averaged field by an angle $\Delta t_s u_c/r_c$ , where $\Delta t_s$ is the sampling interval, set equal to a time unit, and $u_c$ is the convection velocity of coherent disturbances. The convection velocity was estimated from the analysis of the wavenumber-frequency spectra of the streamwise velocity fluctuations for the R1 flow case at $Re_b=4000$ , resulting in $u_c\approx 0.67u_b$ (see appendix E), which is comparable to the value of $u_c\approx 0.80u_b$ found in experiments (Matsson & Alfredsson Reference Matsson and Alfredsson1992). Moreover, the convection velocity in inner units is $u_c^+\approx 11$ , which is similar to the mean speed of the near-wall energy-containing eddies (Kim & Hussain Reference Kim and Hussain1993; Jimenez et al. Reference Jimenez, Uhlmann, Pinelli and Kawahara2001). This value of $u_c$ was also considered for the flow cases at higher Reynolds number, as it was found to ensure maximum coherence in time.

Figure 19. Mean stokes stream function ( $\overline {\hat {\psi }}$ ) for spanwise-coherent disturbances, overlaid to flooded contours of the mean coherent pressure ( $\overline {\hat {p}}$ ) in a $(\theta ,r)$ -plane for the R1 flow cases at $Re_b=4000$ (a), $20\,000$ (b), $87\,000$ (c). The flooded contours range from $-0.3u_{\tau ,g}^2$ (blue) to $0.3u_{\tau ,g}^2$ (red). Positive values of $\hat {\psi }$ (solid lines), indicate a clockwise-rotating roll cell, associated with negative coherent pressure (blue contours), whereas negative (dashed lines) correspond to anticlockwise rolls and positive coherent pressure (red contours). The mean flow is clockwise.

In figure 19 we show the mean Stokes stream function for spanwise-coherent motions ( $\overline {\hat {\psi }}$ ), defined such that $\hat {v}= ({\partial \hat \psi }/{\partial \theta })/r$ , $\hat {u}=-{\partial \hat \psi }/{\partial r}$ , overlaid to flooded contours of the mean coherent pressure ( $\overline {\hat {p}}$ ) in the $(\theta ,r)$ -plane. Alternating high- and low-pressure regions are observed, marking the presence of spanwise large-scale structures. Those are organised into three pairs of roll cells at $Re_b=4000$ (figure 19 a), whereas only two pairs are found at $Re_b=20\,000$ (figure 19 b) and $87\,000$ (figure 19 c). Although the radial extent of the spanwise structures is comparable to $\delta$ , they are most intense near the inner wall and weaker towards the outer wall. The stream function shows that the cross-stream structures have an irregular shape and tend to split up at high Reynolds number. This more chaotic organisation yields reduced strength of the spanwise structures, as measured in terms of the magnitude of the coherent streamwise velocity, $\mathrm {max}|\hat {u}|$ . Indeed, we found $\mathrm {max}|\hat {u}|/u_b\approx 9\, \%$ at $Re_b=4000$ and $\mathrm {max}|\hat {u}|/u_b\approx 7\, \%$ at $20\,000$ and $87\,000$ , hence spanwise large-scale structures are approximately half as strong as the streamwise ones (see § 3.4). The centres of the roll cells are approximately at the same location as the local minima of the mean shear rate (figure 5), supporting the idea that the spanwise structures may originate from a shear-layer instability (Finlay et al. Reference Finlay, Keller and Ferziger1988; Yu & Liu Reference Yu and Liu1991).

Figure 20. Profiles of r.m.s. pressure fluctuations (a) and mean turbulent stress near the inner wall (b) at various Reynolds numbers, for the R1 flow cases. Solid lines refer to total fluctuations and dashed lines refer to coherent fluctuations due to spanwise large-scale structures (the latter are denoted with the hat symbol).

Figure 20(a) displays the distributions of the r.m.s. pressure fluctuations for the R1 flow cases, and includes the contribution of coherent fluctuations due to the spanwise structures, which are obtained from the spanwise-coherent contribution according to (3.11). Near the inner wall, pressure fluctuations attain a peak which is twice as high as the outer wall at $Re_b=4000$ , and almost four times as high at $Re_b=87\,000$ . The impact of the spanwise-aligned structures on pressure fluctuations is substantial, as the peak value of the coherent contribution near the inner wall is approximately one third of the total. Another effect of spanwise structures, which is related to the increase of pressure fluctuations, is the suppression of the turbulent shear stress. This can be ascertained in figure 20(b), where we show the total turbulent stress (solid lines) and the coherent turbulent stress due to spanwise structures (dashed lines). The coherent turbulent shear stress yields a negative contribution, conveying that the spanwise structures tend to suppress ejections and sweeps near the inner wall. A similar result was reported by Kuwata (Reference Kuwata2022) simulating a turbulent flow over high-aspect-ratio streamwise ribs, in which case the spanwise large-scale structures (originated by a Kelvin–Helmholtz instability) were found to increase locally pressure fluctuations and suppress the turbulent shear stress.

3.8. Role of spanwise structures on wall shear and pressure

Suppression of the turbulent shear stress associated with spanwise-coherent structures was found to yield frictional drag reduction by many authors (Koumoutsakos Reference Koumoutsakos1999; Fukagata, Kasagi & Sugiyama Reference Fukagata, Kasagi and Sugiyama2005; Mamori & Fukagata Reference Mamori and Fukagata2011). In addition, experiments on turbulent boundary layers revealed that characteristic and identifiable variation of the wall pressure accompanies the advection of large organised structures, which are responsible for variation of the wall shear stress (Thomas & Bull Reference Thomas and Bull1983). To understand whether spanwise large-scale structures play a role in reducing drag at the inner wall, we then investigate if strong pressure fluctuations due to the spanwise-coherent structures have a footprint on friction at the inner wall. For that purpose, we preliminarily verify whether those quantities are correlated by inspecting the joint probability density function (JPDF) of the wall shear stress and of the fluctuating pressure at the inner wall, which we report in figure 21. Strong positive correlation emerges at $Re_b=4000$ (figure 21 a) and $Re_b=20\,000$ (figure 21 b), at which flow near the inner wall is dominated by spanwise-coherent structures. This correlation becomes less distinct at $Re_b = 87\,000$ (figure 21 c), at which turbulent fluctuations start reach down to the near-wall region. Hence, the effect of spanwise-coherent structures at the inner wall is to enhance pressure fluctuations, which are strongly correlated with shear stress fluctuations.

Figure 21. The JPDF of wall shear stress and pressure fluctuations at the inner wall, $P(\tau _w', p_w')$ , for the R1 flow cases at $Re_b=4000$ (a), $20\,000$ (b) and $87\,000$ (c).

Figure 22. Probability density function (PDF) of the wall shear stress, $P(\tau _w^+)$ , at various Reynolds numbers for flow cases R40 (a) and R1 flow cases (b). Dashed lines refer to the inner wall and solid lines refer to the outer wall.

Strong fluctuations of the wall shear can contribute to friction reduction at the inner wall by increasing the number of backflow events. This insight is supported by the PDF of the wall shear stress reported in figure 22. The mildly curved cases, in which spanwise-coherent structures are not detected, can serve as a comparison. In addition, in table 2 we list the probability of backflow events. In the R40 flow cases (figure 22 a) the PDFs of the wall shear stress at the inner and the outer wall are nearly identical, with increasing probability of large values of positive shear as the Reynolds number increases. Backflow events are very rare, with probability not exceeding $0.3\, \%$ . The picture is quite different for the R1 flow cases (figure 22 b). At the outer wall (solid lines), the probability of negative shear is lower than in the R40 flow cases. As evidenced by the increased probability of sweep motions on the concave wall (see § 3.9), we ascribe this backflow inhibition to the strong secondary flow pushing high-speed fluid towards the wall. Hence, the flow on the concave side is less prone to separate, as backflow probability decreases with flow acceleration (Zaripov et al. Reference Zaripov, Li, Lukyanov, Skrypnik, Ivashchenko, Mullyadzhanov and Markovich2023). A similar reduction in backflow events was observed in the outer bend of a toroidal pipe by Chin et al. (Reference Chin, Vinuesa, Örlü, Cardesa, Noorani, Chong and Schlatter2020), who attributed it to a similar mechanism, suggesting an analogy between the effects of concave curvature and favourable pressure gradient. In contrast, the tails of the PDF widen at the inner wall (dashed lines), showcasing the amplification of wall-shear fluctuations induced by spanwise-coherent structures. The widening of the negative tail corresponds to an increase in backflow events at the inner wall, where their probability exceeds $4\, \%$ .

Further insights into the relationship between wall pressure and wall shear are provided in figure 24, where we show the streamwise distribution of the coherent pressure and of the coherent shear stress at the inner wall, both normalised by their maximum value. Streamwise inhomogeneity induced by spanwise-coherent structures imposesa clear imprint on the wall pressure (blue lines), which features peaks and troughs corresponding to the high- and low-pressure regions observed in figure 19. The coherent wall shear stress (red lines) also features peaks and troughs, which we explain as follows. The coherent streamwise velocity of a clockwise-rotating roll cell (positive values of the stream function in figure 19) opposes the mean flow near the inner wall, reducing locally the streamwise velocity and hence the wall shear. Anticlockwise rotating roll cells act in the opposite way. The small-scale oscillations of the wall shear stress superposed to the large-scale ones at $Re_b=87\,000$ (figure 22 c) are due to turbulent activity, which explains reduced correlation between wall shear and wall pressure (see figure 21). The effects of spanwise-coherent structures on the flow field can be further characterised by analysing the phase shift between the wall shear stress and the wall pressure. The wavy pressure distribution generates local pressure gradients in the streamwise direction, which tend to accelerate and decelerate the fluid and result in alternating regions with high and low wall shear stress. Looking back at figure 19, one can see that high-pressure regions correspond to anticlockwise rotating eddies, and vice versa for the low-pressure regions. Hence, between any pair of counter-rotating eddies where the flow is locally subjected to an adverse pressure gradient ( $u'\lt 0$ ) high-speed fluid is pushed towards the inner wall ( $v'\lt 0$ ). Between any neighbouring pair, there is a favourable pressure gradient ( $u'\gt 0$ ) and simultaneously low-speed fluid is diverted away from the inner wall ( $v'\gt 0$ ). In both cases, the combination of these motions yields to $-u'v'\lt 0$ , explaining why spanwise structures make a negative contribution to the production of Reynolds shear stress, which we have highlighted in figure 20(b).

Table 2. Probability of backflow events at the two walls at various Reynolds numbers, for the R40 ( $r_c/\delta =40.5$ ) and R1 ( $r_c/\delta =1$ ) flow cases.

Figure 23. Streamwise distribution of mean coherent shear stress ( $\overline {\hat {\tau }}_{w,i}$ , red lines), and mean coherent pressure ( $\overline {\hat {p}}_{w,i}$ , blue lines) at the inner wall, for the R1 flow cases at $Re_b=4000$ (a), $20\,000$ (b), $87\,000$ (c). Both quantities are normalised by their maximum value.

3.9. Quadrant analysis

To provide a quantitative basis for the above analysis, we consider the JPDF of streamwise and wall-normal fluctuations, $P(u',v')$ , defined such that

(3.12) \begin{equation} -\overline {u'v'}=\int _{-\infty }^{+\infty } u'v' P(u',v') \mathrm {d}u'\mathrm {d}v', \end{equation}

where the covariance integrand, $u'v' P(u',v')$ , is a measure of the contribution of each pair of $u'$ and $v'$ to the turbulent shear stress (Wallace & Brodkey Reference Wallace and Brodkey1977). Each quadrant of the $(u',v')$ plane corresponds to a class of motions, specifically Q2 and Q4 quadrant events correspond to ‘ejections’ ( $u'\lt 0$ and $v'\gt 0$ ) and ‘sweeps’ ( $u'\gt 0$ and $v'\lt 0$ ), yielding positive contribution to the overall turbulent shear stress, whereas Q1 and Q3 quadrants correspond to ‘outward interactions’ ( $u'\gt 0$ and $v'\gt 0$ ), and ‘inward interactions’ ( $u'\lt 0$ and $v'\lt 0$ ), which yield negative contribution to it.

Figure 24. The JPDF of streamwise and wall-normal velocity fluctuations, superimposed to flooded contours of the covariance integrand, near the outer wall (a–c) and the inner wall (d–f) at $y^+\approx 12$ , for the R1 flow cases. The panels correspond to $Re_b=4000$ (a,d), $20\,000$ (b,e), $87\,000$ (c,f).

In figures 23 and 25 we show flooded contours of $P(u',v')$ superimposed to isolines of $u'v'P(u',v')$ for the R40 and R1 flow cases, respectively. The JPDF is evaluated in wall-parallel planes near the outer and the inner wall at $y^+\approx 12$ . In plane channel flow, this location is the ‘balance point’ where the contributions of ejections and sweeps are equal (Kim et al. Reference Kim, Moin and Moser1987). In the R40 flow cases, the JPDF at the outer wall (figures 23 a–c and 25 a–c) has a roughly elliptical shape with major axis inclined along the Q2 and Q4 quadrants, pointing to high probability of sweeps and ejections. Similar observations apply to the fully turbulent R40 flow cases near the inner wall (figure 25 e,f), and also for the R1 flow cases near the outer wall (figure 23 a–c). However, strong curvature seems to increase the probability of sweeps more than ejections, which can be inferred from the shift of the JPDF peak towards the Q4 quadrant. As previously noted, this tendency is associated with strong secondary motions that transport high-velocity fluid from the channel core towards the outer wall, increasing near-wall velocity and thereby suppressing the formation of regions with negative velocity (see figure 22). The scenario is different at the inner wall. In the R40 flow case at $Re_b=4000$ , as depicted in figure 25(d), the peak probability is mainly concentrated in the Q2 and Q3 quadrants, and to a lesser extent in the Q4 quadrant, on account of stronger, less probable values of $u'$ and $v'$ . The roughly circular shape of the JPDF at the inner wall in the R1 flow cases (figure 23 d–f) points to the suppression of the negative correlation between $u'$ and $v'$ which is typical of near-wall turbulent flow. In addition, the covariance integrand highlights that the contribution to the turbulent stress is not dominated by the Q2 and Q4 motions, but rather strong contributions also come from Q1 and Q3 motions.

Figure 25. The JPDF of streamwise and wall-normal velocity fluctuations, superimposed to flooded contours of the covariance integrand, near the outer wall (a–c) and near the inner wall (d–f ) at $y^+\approx 12$ , for the R40 flow cases. Thehe panels correspond to $Re_b=4000$ (a,d), $20\,000$ (b,e), $87\,000$ (c,f).

The same results are quantified in table 3, where we list the integrated contributions to the turbulent shear stress from each quadrant, at $y^+\approx 12$ . For the R40 fully turbulent flow cases the results are comparable to experimental results for plane channel flow (Wallace, Eckelmann & Brodkey Reference Wallace, Eckelmann and Brodkey1972), namely the contribution of both ejections and sweeps is approximately $70\, \%$ , whereas inward and outward interactions each contribute negatively by approximately $20\, \%$ . At $Re_b=4000$ , the Q2 contribution near the inner wall reduces to $30\, \%$ , which is comparable to the Q1 contribution but opposite in sign, whereas the Q4 contribution exceeds $110\, \%$ of the shear stress. Combining this value with the shape of the JPDF in figure 25(d), one can infer that larger, energetic but infrequent motions are the main contributors to the turbulent shear stress. These strong sweeps can be attributed to the streamwise vortices pushing the high-speed mean flow from the channel core towards the inner wall. As for the R1 flow cases, the quadrant contributions at the outer wall are similar to those of the R40 flow cases, except for greater contribution of ejections as compared with sweeps. Hence, the most probable motions (i.e. sweeps, as visible in figure 23) do not contribute as much to the turbulent shear stress as the ejections, see table 3. At the inner wall, the fractional contributions of the outward (Q1) and inward (Q3) interactions exceeds half the contributions of ejections (Q2) and sweeps (Q4). This result confirms that the spanwise large-scale structures, which were found to generate Q1 and Q3 motions in § 3.8, dampen the turbulent shear stress on the convex side of strongly curved channels.

Table 3. Percent contribution of quadrants to turbulent shear stress ( $\overline {u'v'}_{Qi}/\overline {u'v'}$ ), for the R40 and R1 flow cases near the inner and outer wall, at $y^+\approx 12$ .

Figure 26. (a) The TKE production ( $\mathcal {P}^+$ ) near the outer wall (solid lines) and near the inner wall (dashed lines). (b) Turbulent shear stress ( $-\overline {u'v'}^+$ , solid lines) and viscous shear stress ( $\nu \mathcal {S}^+$ , dashed lines) near the inner wall. All quantities are reported in local wall units for the R1 flow cases.

3.10. Energy production reversal

The Q1 and Q3 motions contribute negatively to turbulent shear stress, hence they yield negative contribution to the production of TKE, $\mathcal {P}=-\overline {u'v'}\mathcal {S}$ , where $\mathcal {S}=\mathrm {d}U/\mathrm {d}r-U/r$ is the mean shear rate. Based on the analysis in § 3.9, we expect that TKE production be reduced near the inner wall in flow cases with strong curvature. In figure 26(a) we then show the TKE production for the R1 flow cases. Near the outer wall (solid lines) all distributions tend to collapse, especially at high Reynolds number, revealing that the flow similarity is preserved near the outer wall. The peak production is located at $y^+\approx 12$ (except at $Re_b=4000$ , for which the peak occurs at $y^+\approx 9$ ) and the peak value is $\mathcal {P}^+\approx 0.25$ , similar to plane channel flow (Kim et al. Reference Kim, Moin and Moser1987; Laadhari Reference Laadhari2002). In contrast, TKE production near the inner wall (dashed lines) depends heavily on the Reynolds number, implying that classical wall scaling no longer holds near highly convex surfaces. Notably, a region appears where the production is negative, with an extent of $0.12\delta$ at $Re_b=4000$ and gradually reducing at higher Reynolds numbers. For the production to be positive everywhere, the turbulent shear stress and the viscous shear stress must have the same sign and vanish at the same location. This is not the case for the strongly curved channel, as clearly illustrated in figure 26(b), which shows profiles of the turbulent shear stress, $-\overline {u'v'}^+$ (solid lines), and of the viscous shear stress, $\nu \mathcal {S}^+$ (dashed lines), near the inner wall in the R1 flow cases. The displacement between the zero-crossings of the turbulent shear stress and of the viscous shear is related to asymmetry of the mean velocity profile (Beguier et al. Reference Beguier, Giralt, Fulachier and Keffer2005) and leads to a region of opposing shear where $-\overline {u'v'}^+\lt 0$ and $\nu \mathcal {S}^+\gt 0$ , hence $\mathcal {P}^+\lt 0$ .

As pointed out in § 3.1, the zero crossing of the mean shear marks the interface between the two distinct flow structures developing at each wall. Straddling this interface there is a diffusive transfer of shear stress from the outer- to the inner-wall region that has a sign opposite to the locally produced shear stress, leading to negative net TKE production (Hanjalić & Launder Reference Hanjalić and Launder1972). In the region with negative production local energy reversal takes place, whereby energy is transferred from turbulent fluctuations to the mean flow (Eskinazi & Erian Reference Eskinazi and Erian1969). Besides its physical significance, this result provides useful caveat for use of RANS models for simulations of turbulent flows over convex walls. In fact, standard eddy-viscosity models would clearly fail if turbulent shear stress and mean shear do not go to zero at the same location.