3.1. Friction coefficient

The friction coefficient is a key parameter for characterising the influence of the curvature radius on flows in concentric annuli and curved channels. Based on the friction velocities (2.6), we define friction coefficients at the inner and outer wall as

(3.1) \begin{align} 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, \end{align}

which are listed in table 1 and shown in figure 2, as a function of the inner radius. For axial flows (figure 2 a), there is a significant decrease in friction at the inner wall (green dots) as $r_i$ increases, accompanied with moderate increase at the outer wall (red dots). This can be explained by looking at the mean axial velocity profiles (not shown here), whose maxima shift towards the inner wall as $r_i$ decreases. The friction difference between two walls decreases as the inner radius increases, vanishing for $r_i \gt 2.5$ . Quadrio & Luchini (Reference Quadrio and Luchini2002) evaluated the overall resistance of the two walls by weighting their friction coefficients with the corresponding wetted areas, and noticed that the overall friction is essentially unchanged from its planar value, for $r_i \geqslant 1$ . The data in table 1 do confirm this behaviour, at least for $r_i \geqslant 1$ . However, the overall drag increases for smaller $r_i$ . This suggests that when wall curvature is transverse to the mean flow, the effects of concave and convex walls are significant only when the inner radius of the annulus is small compared to the distance between the two walls.

Figure 2. Friction coefficient at the inner wall ( $C_{f,i}$ , green) and at the outer wall ( $C_{f,o}$ , red) versus the inner radius for (a) axial flows and (b) azimuthal flows.

For azimuthal flows (figure 2 b), the friction trend at the two walls is more complex. In particular, $C_{f,i}$ has a minimum at $r_i \approx 14.5$ , $C_{f,o}$ has a maximum at $r_i \approx 1.5$ , and there is a crossover of friction at the two walls at $r_i \approx 0.25$ . At very large values of $r_i$ , $C_{f,i}$ is slightly less than $C_{f,o}$ , with a difference of approximately $10\,\%$ . This behaviour allows us to identify four flow regimes, namely: (i) mild curvature for $r_i \gt 14.5$ ; (ii) moderate curvature for $1.5 \lt r_i \leqslant 14.5$ ; (iii) strong curvature for $0.25 \lt r_i \leqslant 1.5$ ; (iv) extreme curvature for $r_i \leqslant 0.25$ .

In these simulations, the bulk Reynolds number is set to $R_b = 5000$ , which is sufficiently high to result in fully turbulent flow conditions. Soldati et al. (Reference Soldati, Orlandi and Pirozzoli2025) found that for mildly curved channels ( $r_i = 40$ ), the friction at the inner wall is permanently lower than at the outer wall as the Reynolds number increases. In contrast, for strongly curved channels ( $r_i = 0.5$ ), the friction at the inner wall exceeds that at the outer wall for $R_b \lt 1000$ , and becomes lower for higher $R_b$ . Figure 2(b) displays a similar behaviour, with crossing point at $r_i \approx 0.25$ . The maximum difference between the friction at the two walls is obtained for $r_i=2.5$ , at which $C_{f,o}$ is more than twice $C_{f,i}$ .

This complex behaviour arises in azimuthal flows because of two competing effects. First, as the curvature radius is reduced, destabilisation at the outer wall and stabilisation at the inner wall yield, respectively, increase and decrease of the friction coefficient. By further reducing the inner radius, the wetted area of the inner wall becomes very small, such that the maximum velocity (not shown here) approaches it, with consequent tendency for the friction coefficient to increase at the inner wall. The inversion of friction at the two walls occurring in azimuthal flows will be further characterised through the analysis of the TKE transport equation, which reveals the dominance of the pressure–velocity correlation term and a reduction in production.

3.2. Total stress and turbulent energy production

The expressions for the total stress in the concentric annulus and curved channel configurations are given by

(3.2) \begin{equation} \tau _{T,z}=\frac {1}{Re}\frac {\partial \overline {v_z}}{\partial r}-\overline {v_z'v_r'}, \qquad \tau _{T,\theta }=\frac {1}{Re}\frac {r \, \partial (\overline {v_{\theta }}/r)}{\partial r}-\overline {v_\theta 'v_r'}, \end{equation}

respectively, where the fluctuating velocities are defined as $v_i' = v_i - \overline {v_i}$ . Given the large number of configurations investigated, only a selection of flow cases is displayed in figure 3, specifically those with $r_i = 0.025$ , $0.125$ , $1$ , $5.5$ , $14.5$ and $79.5$ , for both axial (figure 3 a) and azimuthal (figure 3 b) flows. The achievement of smooth profiles indicates that a good statistical steady state has been reached. For azimuthal flows, Brethouwer (Reference Brethouwer2022) derived the following analytical expression for the total stress profile:

(3.3) \begin{equation} \tau _{T,\theta }(r) = \frac {r_i^2 \tau _{w,i} (r_o^2 - r^2) + r_o^2 \tau _{w,o} (r_i^2 - r^2)}{2 r^2 r_c \delta }. \end{equation}

The azimuthal flow cases with extreme value of curvature, namely $r_i = 0.025$ (red) and $r_i = 0.125$ (green), exhibit a total stress profile that deviates more significantly from a linear trend. In figure 3(b), we report the analytical profile (3.3) for these two cases (open circles), which shows excellent agreement with the computed profiles (dots), further indicating well-converged statistics.

Figure 3. Total stress profiles scaled by bulk velocity versus distance from inner wall for (a) axial flows and (b) azimuthal flows. The values of $r_i$ are reported in the legend. The open circles denote the analytical profiles of $\tau _{T,\theta }(r)/u_b^2$ from (3.2) for the azimuthal flow cases with $r_i=0.025$ and $r_i=0.125$ .

Figure 4. (a) Minimum of negative production ( $-P_{\theta }|_{min}$ ) versus the inner radius; in the inset, the thickness $\Delta P_N$ of the layer with $P_\theta \lt 0$ is plotted versus $r_i$ . Red dots denote results obtained with the $rv_{\theta }$ -formulation, red circles with the $v_{\theta }$ -formulation, and green triangles with the GPU-version of the $v_{\theta }$ -formulation. (b) Profiles of TKE production scaled by bulk velocity versus distance from the inner wall; the inset shows a close-up view of the production profiles for flow cases with $r_i\gt 1$ . The values of $r_i$ are reported in the legend. Both plots refer to azimuthal flows.

In wall-bounded flows, TKE production typically scales well with wall friction, especially as the Reynolds number increases. For smooth, flat walls, this production reaches a maximum value of approximately $0.25$ in wall units, located at a fixed distance from the wall (Pope Reference Pope2000). However, the stabilising effect of the convex wall leads to a reduction in TKE production, which may even become negative, implying a transfer of energy from turbulent fluctuations to the mean flow. As mentioned in the Introduction, there has been a debate regarding the existence of a layer with negative TKE production in concentric annulus flows. The last column of table 1 reports the mean extent in the radial direction of the flow region where production is negative, which we define as $\Delta P_N$ . For axial flows, $\Delta P_N$ is zero for all values of the inner radius, confirming the conclusions drawn by Boersma & Breugem (Reference Boersma and Breugem2001). In contrast, a completely different behaviour is observed for azimuthal flows, for which a negative production layer exists at each value of $r_i$ .

This point is further investigated in figure 4(a), which focuses on azimuthal flows and illustrates the minimum of $P_\theta = -\overline {v_\theta^{\prime}v_r^{\prime}} \, {r \, \partial (U_\theta /r)}/{\partial r}$ in absolute value as a function of the inner radius. The figure highlights the large values of negative production for small $r_i$ , and its sharp reduction as $r_i$ increases. A local maximum is observed for $r_i=24.5$ ; however, the semi-logarithmic scale amplifies values that are actually extremely lower than those obtained for small $r_i$ . The inset of figure 4(a) shows the extent of the layer with negative production $\Delta P_N$ , also listed in table 1. Notably, $\Delta P_N$ increases for large values of the inner radius, specifically between $r_i=9.5$ and $r_i=40$ . However, these relatively high values of $\Delta P_N$ are not as significant as those observed for small $r_i$ . This can be inferred from figure 4(b), where the profiles of $P_\theta$ are shown for the same flow cases as in figure 3(b): the broad layer with negative production observed for large $r_i$ corresponds to negative values of $P_\theta$ that are nearly zero.

The profiles of $P_\theta$ in figure 4(b) further characterise the occurrence of negative TKE production in azimuthal flows. Near the outer wall, $P_\theta$ is always positive, though the amplitude of its peak varies in a complex manner. For $r_i = 1$ , at which the peak amplitude is maximum, the high turbulence activity near the outer wall influences the channel core, but does not affect significantly the layer near the inner wall, where the stabilising effect of convex curvature tends to suppress $P_\theta$ . For flow cases with $r_i\gt 1$ , the peak production near the outer wall decreases in amplitude with $r_i$ , whereas it increases near the inner wall, without reaching the value near the outer wall. For values $r_i\lt 1$ , production increases in the channel core, and decreases to large negative values near the inner wall. Since the production is inherently related to the flow structures, the complex profiles of $P_\theta$ indicate a modification of the flow structures for the azimuthal flow cases with $r_i \lt 1$ , which will be stressed in the following through a detailed analysis of the budget terms in the TKE transport equation.

3.3. Velocity and pressure correlations

As can be inferred from the results presented so far, identifying a suitable scaling for both the mean velocity and the normal stresses with respect to the friction velocities at the two walls is quite challenging. Thus we consider the normalised anisotropy tensor $b_{ij}$ , which is typically defined as $b_{ij} = \overline {v_i^{\prime}v_j^{\prime}}/\overline {K}-1/3\delta _{ij}$ , where $\overline {K}=2\overline {K'}$ is twice the mean TKE. Here, we adopt an alternative definition, normalising each component of the anisotropy tensor by its maximum value, namely $b_{ij} = \overline {v_i^{\prime}v_j^{\prime}}/\overline {v_i^{\prime}v_j^{\prime}}|_{max}$ , in such a way that quantities range between $0$ and $1$ . This definition allows for the fluctuating pressure correlations, which we define as $b_{pp} = \overline {p'p'}/\overline {p'p'}|_{max}$ consistently with $b_{ij}$ , to be visualised on the same plot.

Examining the profiles of the diagonal components $b_{ii}$ in the present configurations can reveal whether wall curvature enhances or diminishes flow anisotropy with respect to flat walls, for which $b_{ii}$ exhibit strong inhomogeneity near the wall, and a fair isotropy farther from the wall. To facilitate comparison between flow over curved and flat walls, we provide in figure 5 the profiles of $b_{ii}$ obtained from the data by Lee & Moser (Reference Lee and Moser2015) for plane channel flow at various friction Reynolds numbers ( $R_\tau$ ), specifically $R_\tau =180,550,1000,2000,5200$ . These data reveal that in the buffer region, the streamwise (red dots) and spanwise (orange dots) components exhibit a maximum, whose location is independent of the friction Reynolds number for $R_\tau \gt 500$ . In fact, the only profiles that do not collapse on the others in the viscous wall region are those at $R_\tau =180$ (solid lines). Regarding the pressure fluctuations, the values of $b_{pp}$ (black) show a Reynolds number dependence, but the peak location remains invariant.

Figure 5. Diagonal components of the anisotropy tensor (streamwise in red, wall-normal in green, and spanwise in orange) and fluctuating pressure correlations (in black) scaled by their maximum values. Profiles obtained by Lee & Moser (Reference Lee and Moser2015) for plane channel flow at friction Reynolds numbers $R_\tau =180$ (lines), and $R_\tau =550, 1000, 2000, 5200$ (dots).

Figure 6. Diagonal components of the anisotropy tensor (streamwise in red, wall-normal in green, and spanwise in orange) and fluctuating pressure correlations (in black) scaled by their maximum value versus distance from the inner radius. Left-hand panels correspond to axial flows, and right-hand panels to azimuthal flows, with (a,b) $r_i=0.125$ , (c,d) $r_i=1$ and (e,f) $r_i=79.5$ .

For curved wall configurations, we report in figure 6 the profiles of $b_{ii}$ and $b_{pp}$ for axial flows (left-hand panels) and azimuthal flows (right-hand panels) with $r_i = 0.125$ (figures 6 a,b), $r_i = 1$ (figures 6 c,d), and $r_i = 79.5$ (figures 6 e,f). To facilitate a quick comparison between the two flow configurations, the components along the flow direction are plotted in red, namely $b_{\theta \theta }$ for flow in the $\theta$ -direction, and $b_{zz}$ for flow along the $z$ -direction. The three panels on the left-hand side for axial flows show similar trends with minor differences. For $r_i = 79.5$ (figure 6 e), a fairly good symmetry is observed for all quantities. Comparing figure 6(e) and figure 5, some differences in amplitude are observed. These differences are due to the Reynolds number, which is $R_\tau \approx 160$ in the present simulations for $r_i=79.5$ . For $r_i = 1$ (figure 6 c), the symmetry of the flow breaks down, as is visible from the minimum of $b_{zz}, b_{pp}$ and the maximum of $b_{rr}, b_{\theta \theta }$ in the central region shifting slightly towards the inner wall. This trend is even more pronounced for $r_i = 0.125$ (figure 6 a). For axial flows, the maximum of $b_{pp}$ remains fixed near the outer wall, whereas its peak near the inner wall diminishes in amplitude with $r_i$ , decreasing to half the absolute maximum for $r_i=0.125$ (figure 6 a). The minimum of $b_{pp}$ is located at the same position where $b_{rr}$ is maximum.

For the azimuthal flow case with $r_i = 79.5$ , the profiles of $b_{ii}$ in figure 6(f) do not differ significantly from those in figure 6(e) for the axial flow in the same geometry. However, a slight asymmetry is present, which is particularly evident in the profile of $b_{pp}$ . This suggests that even a small curvature in the longitudinal direction influences the normal stresses and especially the fluctuating pressure. As for axial flows, the maximum of $b_{pp}$ is located near the outer wall for large values of the inner radius. All the profiles change drastically for azimuthal flows with small values of the inner radius. At $r_i=1$ (figure 6 d), a significant increase in $b_{rr}$ is observed with respect to the corresponding axial flow case in figure 6(c). This increase in radial velocity fluctuations, more pronounced towards the outer wall, is explained by the presence of large ejections that influence the flow structures near the outer wall, as will be shown in the following through flow visualisations. These large-scale ejections are also responsible for the large values near the outer wall of total stress and TKE production, both reaching their peak for $r_i=1$ (cyan dots in figures 3 b and 4 b). In a small layer around $r - r_i \approx 0.9$ , the flow tends to be isotropic due to a strong reduction in $b_{\theta \theta }$ . This small layer of isotropic flow near the outer wall is also characterised by the absence of a distinct peak in $b_{pp}$ . The reduction of $b_{\theta \theta }$ is even stronger near the inner wall, explaining the sharp reduction of $P_\theta$ in that region (cyan dots in figure 4 b). Together with the formation of a prominent peak in $b_{pp}$ near the inner wall, this suggests a complete rearrangement of the budgets in the TKE transport equation. For the azimuthal flow case with $r_i=0.125$ , what emerges from figure 6(b) is a large peak of $b_{rr}$ near the inner wall, in correspondence with the maximum of $b_{pp}$ . The presence of this peak in radial velocity fluctuations suggests that the rearrangement of the TKE budgets is even more marked for this flow case with extreme curvature, as discussed in the next subsection.

The results obtained so far indicate that transverse curvature does not alter substantially the structure of axial flows with respect to canonical wall-bounded flows, whereas longitudinal curvature induces major modifications in azimuthal flows. Therefore, the remainder of this paper will focus on azimuthal flows only.

3.4. Budgets of TKE

In the previous discussion, greater emphasis was placed on azimuthal flows, with a special focus on how flow statistics change near the inner wall as the curvature increases. Important insights into these flow modifications can be gained through the analysis of the TKE budgets. For azimuthal flows, the transport equation for the TKE is

(3.4) \begin{align} & 0= -C_\theta \underbrace {{} -\frac {\overline {{v_r^\prime }{v_z^\prime }}}{r} \frac {{\rm d} (\overline {v_z} r) }{{\rm d} r} -\frac {\overline {{v_r^\prime }{v_{\theta }^\prime }}}{r} \frac {{\rm d} (\overline {v_\theta } r) }{{\rm d} r} }_{P_{\theta S}} \underbrace {{} -\frac {\overline {{v_{\theta }^\prime }{v_r^\prime }}}{r} \overline {v_{\theta }} +\frac {\overline {{v_{\theta }^\prime }{v_{\theta }^\prime }}}{r} \overline {v_r} }_{P_{\theta C}} \underbrace {{} -\frac {1}{r}{\frac { {\rm d} \overline {{v_r^\prime } K^\prime } r }{{\rm d} r}} }_{T_\theta } \underbrace {{} -\frac {1}{r}{\frac {{\rm d} \overline {{v_r^\prime } p^\prime } r }{{\rm d} r}} }_{\Pi _\theta } \nonumber \\ & \textstyle\underbrace {{} +\textstyle\frac {1}{Re} \!\left \lbrack \overline {{v_z^\prime }\, \nabla ^2 {v_z^\prime } } +\overline {{v_{\theta }^\prime }\, \nabla ^2 {v_{\theta }^\prime } }+ \overline {{v_r^\prime }\, \nabla ^2 {v_r^\prime } }\! \right \rbrack +\textstyle\frac {1}{Re} \!\left \lbrack \textstyle\frac {1}{r^2} \left (\! -\overline {{v_{\theta }^\prime } {v_{\theta }^\prime }}-\overline {{v_r^\prime } {v_r^\prime }} +2 \!\left (\overline {{v_{\theta }^\prime }\textstyle\frac {\partial {v_r^\prime }}{\partial \theta }} -\overline {{v_r^\prime }\textstyle\frac {\partial {v_{\theta }^\prime }}{\partial \theta }}\right )\! \right )\! \right \rbrack }_{D_\theta }, \nonumber\\[-8pt] \end{align}

where $C_\theta$ , the convective contribution, is zero as $\overline {v_r}=0$ , $P_\theta$ is the production, $D_\theta$ is the total dissipation, $T_\theta$ is the transfer due to triple velocity correlations, and $\Pi _\theta$ is the transfer due to pressure–velocity correlations. The production term $P_\theta$ can be decomposed into two components: $P_{\theta S}$ , which is the product of the Reynolds stress and the shear strain rate, and $P_{\theta C}$ , the curvature-related term. Generally, $P_{\theta C}$ provides a positive contribution to the total production. As shown by Liu et al. (Reference Liu, Zhu, Bao, Zhou and Han2022), $P_{\theta C}$ may be the critical factor driving relaminarisation in turbulent pipe flow with streamwise-varying wall rotation, as this term facilitates the transfer of energy from turbulence to the circumferential mean flow. In axial flows, $P_{\theta C}$ is relatively large at very small values of $r_i$ , but becomes negligible when $r_i\geqslant 1$ . The inset of figure 7(a) illustrates that $P_{\theta C}$ exceeds $P_{\theta }$ near the wall, and it is insufficient to offset the negative values of $P_{\theta S}$ . This imbalance leads to a negative $P_{\theta }$ , as shown in figure 4(b), although this effect diminishes for larger $r_i$ . The profiles of each term are depicted in figure 7 for the azimuthal flow cases with $r_i=0.025$ , $0.125$ , $1$ and $79.5$ . To elucidate the budgets of $\overline {K}$ in the region near the inner wall, a logarithmic scale for $r - r_i$ is employed.

Figure 7. Profiles of each term of the TKE transport equation (3.5) versus the distance from inner radius for azimuthal flow cases with (a) $r_i=0.025$ , (b) $r_i=0.125$ , (c) $r_i=1$ , and (d) $r_i=79.5$ . The symbols used for each term are reported in the legend. In the inset of (a) are plotted $P_\theta$ (red symbols), $P_{\theta S}$ (green lines) and $P_{\theta C}$ (orange lines).

Figure 8. Profiles of mean TKE versus the distance from the inner wall for (a) azimuthal flows and (b) axial flows. The values of the inner radius are reported in the legend.

For $r_i = 79.5$ (figure 7 d), the TKE budgets exhibit a distribution typical of plane channels, featuring a balance between production $P_\theta$ (red dots), and total dissipation $D_\theta$ (green dots). The peak of total dissipation is located slightly closer to the inner wall compared to the production peak. The triple velocity transfer term $T_\theta$ (black dots) indicates an energy transfer from the near-wall region, dominated by ribbon-like structures, towards the adjacent layer, dominated by rod-like structures (Orlandi Reference Orlandi2019).

For $r_i = 1$ (figure 7 c), the contribution of the pressure–velocity correlation $\Pi _\theta$ (purple dots) is still relatively small, albeit greater than for $r_i = 79.5$ , and comparable in magnitude to the other terms. Although the semi-logarithmic scale does not illustrate clearly the budgets close to the outer wall, it is evident that the amplitude of each term near the inner wall is markedly lower than at the outer wall. This substantial reduction of all budgets near the inner wall is reflected in the profiles of $\overline {K}$ , shown in figure 8 as a function of the distance from the inner wall. For $r_i = 1$ (cyan dots), a near-wall bump is absent near the inner wall, in contrast to the peak observed near the outer wall.

For $r_i=0.125$ and $0.025$ (respectively figures 7 b,a), the budgets of TKE change dramatically. The pressure–velocity correlation term increases to an order of magnitude higher than the production term observed for $r_i=79.5$ . (Note that the vertical axis ranges are different.) This increase is likely associated with the negative production, which draws fluctuations from small to large scales of motion. The large-scale fluctuating pressure interacts with large ejections of radial velocity from the inner wall, leading to a significant gain of TKE. This strong energy gain is subsequently dissipated and partially transferred away from the inner wall region via the triple velocity transfer term. This increase in energy gained and lost results in higher peaks of $\overline {K}$ near the inner wall compared to the outer wall, as is clearly visible in figure 8(a) for $r_i = 0.125$ (green dots) and even more markedly for $r_i = 0.025$ (red dots).

The non-monotonic trend of $\overline {K}$ with $r_i$ is analogous to the $C_f$ trend shown in figure 2(b). A comparison of the TKE profiles for azimuthal flows in figure 8(a) and for axial flows in figure 8(b) reveals that for axial flows, the two peaks of $\overline {K}$ have similar magnitudes; however, the minimum shifts from the centreline towards the inner wall as $r_i$ decreases. The relatively minor changes in the TKE profiles for axial flows near both walls are replicated in the budget profiles, which are not shown here as they do not differ significantly from those observed in canonical wall-bounded flows.

3.5. Flow visualisations

As mentioned previously, major changes in the statistics of azimuthal flows, which are particularly pronounced for small values of the inner radius, indicate the presence of distinctive flow structures developing near the inner wall. These structures can be characterised by inspecting flow visualisations after a careful selection of the relevant quantities, essential for obtaining meaningful insights into the flow organisation. According to Orlandi (Reference Orlandi2013), the wall-normal velocity stress is a crucial quantity in flows over both smooth and rough walls, making the correlation between $v_r^\prime$ and any pertinent quantity a preferred subject of scrutiny. The atypical distribution of the pressure–velocity correlation term in the TKE transport equation, which deviates from the canonical behaviour observed in wall-bounded flows, points to the significant role of pressure fluctuations in azimuthal flows. Therefore, special attention should be paid to the correlations between $p^{\prime }$ and $v_r^\prime$ .

Due to the complexity of three-dimensional surface contours, the flow structures from a single realisation are visualised via planar contours. As for the transverse ( $r$ – $z$ ) and longitudinal ( $\theta$ – $r$ ) planes, flow contours are obtained through space averages. Specifically, the three-dimensional field of a generic quantity $q_{ijk}$ is averaged along the $\theta$ -direction, $\widetilde {q} = \sum_i {q_{ijk}}/{N_\theta }$ , and along the $z$ -direction, $\widehat {q} = \sum_k {q_{ijk}}/{N_z}$ , to obtain visualisations in the $r$ – $z$ plane and in the $\theta$ – $r$ plane, respectively. Contours of $\widetilde {q}$ and $\widehat {q}$ are scaled by their respective maximum values to emphasise the shape of flow structures. As for the wall-parallel ( $\theta$ – $z$ ) planes, flow contours are displayed both at the centreline and at the radius where $\Pi _\theta$ attains its maximum near the inner wall (see figure 7). In this case, the three-dimensional quantities $v_r^\prime$ and $p^{\prime }$ are normalised by $\overline {K}^{1/2}$ and $\overline {p^{\prime } p^{\prime }}^{1/2}$ , respectively. The comparison between the $\theta$ – $z$ planes at these two locations provides insights into the increase in the pressure–velocity correlation $\Pi _\theta$ (see (3.5)) as the inner radius decreases. Since wall-parallel planes have a different extent in the $\theta$ -direction depending on $r_i$ , the streamwise coordinate has been scaled so that it ranges between $0$ and $2\pi$ .

Figure 9. Contours in transverse ( $r$ – $z$ ) planes of azimuthal flow cases with (a,c) $r_i=0.025$ and (b,d) $r_i=1$ . Fluctuating radial velocity $\widetilde {{v_r^\prime }}$ scaled with the maxima (a) $\widetilde {{v_r^\prime }}_{max}=-0.25$ and (b) $\widetilde {{v_r^\prime }}_{max}=-0.128$ . Pressure $\widetilde {p^{\prime }}$ scaled with the maxima (c) $\widetilde {p^{\prime }}_{max}=0.26$ and (d) $\widetilde {p^{\prime }}_{max}=-0.0137$ . The outer wall is at the top, and the inner wall is at the bottom of each plot. Blue, green, cyan represent negative values, and yellow, red, magenta represent positive values, with increments $\varDelta =0.1$ . Profiles of $\widetilde {p^{\prime }}$ on the inner wall (red) and outer wall (black) of (e) $r_i=0.025$ and (f) $r_i=1$ .

Figure 10. Contours in wall-parallel ( $\theta$ – $z$ ) planes of azimuthal flow cases with (a,b,e,f) $r_i=0.025$ and (c,d,g,h) $r_i=1$ : fluctuating radial velocity ${v_r^\prime }/\overline {K}^{1/2}$ (top panels) and pressure $p^{\prime }/\overline {p^{\prime } p^{\prime }}^{1/2}$ (bottom panels). Wall-parallel planes are displayed at (a,c,e,g) the radius of maximum $\Pi _\theta$ and (b,d,f,h) the centreline. Blue, green, cyan represent negative values, and yellow, red, magenta represent positive values, with increments (a) $\varDelta =0.005$ , (c) $\varDelta =0.025$ , (b,d) $\varDelta =0.06$ , (e,g) $\varDelta =0.04$ , (f,h) $\varDelta =0.08$ . Mean flow is from left to right.

For $r_i = 0.025$ , the shape of the contours in figure 9(a) indicates that $\widetilde {{v_r^{\prime}}}$ is concentrated towards the outer top wall (solid black line) and tends to vanish near the inner bottom wall (solid green line). However, when scaled by $\overline {K}^{1/2}$ (figure 10 a), contours of $v_r$ reveal that alternating positive and negative structures with spanwise coherence are present near the inner wall. For pressure, an almost opposite behaviour is observed. Flow structures with large values of $\widetilde {p^{\prime }}$ form near the inner wall (figure 9 c), while they are absent in the outer half of the channel. This difference is quantified in figure 9(e), which shows that for $r_i=0.025$ , pressure oscillations at the inner wall (red line) have significantly higher amplitude than at the outer wall (black line). Conversely, for $r_i=1$ , figure 9(f) shows that pressure oscillations at the two walls are similar, although their overall magnitude is smaller than for $r_i=0.025$ .

The spanwise coherence of $p^{\prime }$ in figure 10(e) is less evident than that of $v_r$ in figure 10(a). Nonetheless, high positive/negative values of $p^{\prime }$ tend to align with high negative/positive values of $v_r$ , suggesting that a negative correlation exists between the two quantities. Streamwise-coherent structures with alternating positive and negative amplitudes of radial velocity are visible in the wall-parallel plane at the centreline (figure 10 b), where the spanwise coherence is reduced. Accordingly, numerous weak patches of $\widehat {{v_r^\prime }}$ appear in the $\theta$ – $r$ plane of figure 11(a) far away from the inner wall, where instead two strong structures are present. A spanwise coherence of $p^{\prime }$ is undetectable in the $\theta$ – $z$ plane at the centreline (figure 10 f). In fact, contour lines of $\widehat {p^{\prime }}$ disappear completely in the outer part of the $\theta$ – $r$ plane (figure 11 c). Two distinct patches of $\widehat {p^{\prime }}$ are present near the inner wall, shifted by approximately $\pi /2$ with respect to the two $\widehat {{v_r^\prime }}$ patches.

Figure 11. Contours in longitudinal ( $\theta$ – $r$ ) planes of azimuthal flow cases with (a,c) $r_i=0.025$ and (b,d) $r_i=1$ , and fluctuating (a,b) radial velocity $\widehat {{v_r^\prime }}$ and (c,d) pressure $\widehat {p^{\prime }}$ , both averaged along the $z$ -direction, scaled by their maximum values: (a) $\widehat {{v_r^\prime }}_{max}=0.053$ , (b) $\widehat {{v_r^\prime }}_{max}=0.026$ , (c) $\widehat {p^{\prime }}_{max}=0.092$ , (d) $\widehat {p^{\prime }}_{max}=0.006$ . Blue, green, cyan represents negative values, and yellow, red, magenta represent positive values, with increments $\varDelta =0.05$ . Mean flow is clockwise.

For the flow case with $r_i = 1$ , a similarity in the $\widetilde {{v_r^\prime }}$ contours (figure 9 b) is observed, with those of the case with $r_i = 0.025$ (figure 9 a) near the inner wall. Farther from the wall, the radial velocity is organised in streamwise-coherent structures, which are pushed towards the outer wall and are larger than those at $r_i = 0.025$ . These structures are also visible in the wall-parallel plane at the centreline (figure 10 d), and are caused by longitudinal vortices that originate from centrifugal instabilities (Soldati et al. Reference Soldati, Orlandi and Pirozzoli2025). Both spanwise and streamwise coherence of $v_r$ are reduced near the inner wall, as is visible in figure 10(c). This suggests a tendency of the flow towards more isotropic conditions near the inner wall, which we already noticed from the profiles of the anisotropy tensor diagonal components and from the reduction of all terms in the TKE budgets equation; see figures 6(d) and 7(c). From the contours in the $\theta$ – $z$ plane (figure 11 b), numerous weak patches of $\widehat {{v_r^\prime }}$ appear, increasing from a single pair at $r_i = 0.025$ (figure 11 a) to a higher number, which is challenging to evaluate exactly due to the azimuthal convection of these flow patterns. From the pressure contours in the transverse and wall-parallel planes of figures 9(d) and 10(g), it appears that pressure is characterised by a strong spanwise coherence near the inner wall, which reduces at the centreline (figure 10 h). This behaviour is reflected in the longitudinal plane (figure 11 d), where large flow patterns are visible near the inner wall, and tend to break up towards the outer wall.

3.6. Correlation between radial velocity and pressure

Figure 12. Correlation between fluctuating radial velocity and pressure versus distance from the inner wall for azimuthal flows. The values of $r_i$ are reported in the legend.

The increase in the pressure–velocity correlation $\Pi _\theta$ (see (3.5)), identified as a key feature of azimuthal flows with small inner radius, can be explained by analysing the correlations between the fluctuating radial velocity and pressure. Additionally, these correlations can provide valuable insights for quantifying the number of spanwise-coherent flow patterns.

Figure 12 illustrates the trend of $\overline {{v_r^\prime } p^{\prime }}$ as a function of the distance from the inner wall, using a semi-logarithmic scale to magnify the region near the inner wall. In the case of smallest inner radius $r_i = 0.025$ (red circles), the prominent peak of $\overline {{v_r^\prime } p^{\prime }}$ indicates a strong negative correlation between fluctuating pressure and radial velocity near the inner wall, consistent with flow visualisations. As $r_i$ increases, the peak amplitude decreases sharply, and the peak location shifts further from the inner wall.

Figure 13. Contours in longitudinal ( $\theta$ – $r$ ) planes of azimuthal flow cases with $r_i$ values (a) 0.025, (b) 0.075, (c) 0.125, (d) 0.5, and (e) 1; $z$ -averaged correlation between fluctuating radial velocity and pressure ( $\widehat {{v_r^\prime } p^{\prime }}$ ). Blue, green, cyan represent negative values, and yellow, red, magenta represent positive values, with increments (a) $\varDelta =0.0001$ , (b,c,d,e) $\varDelta =0.00002$ .

This trend can be visualised in figure 13, where the contours of the $z$ -averaged correlations between fluctuating pressure and radial velocity, $\widehat {{v_r^\prime } p^{\prime }}$ (scaled by their maximum value), are shown in longitudinal planes. Consistent with the profiles of $\overline {{v_r^\prime } p^{\prime }}$ , negative (blue) contour lines of $\widehat {{v_r^\prime } p^{\prime }}$ are predominant near the inner wall for $r_i=0.025$ (figure 13 a), as well as for $r_i=0.075$ and $r_i=0.125$ (figures 13 b,c). Further from the inner wall, alternating regions of positive and negative contour lines appear, which are due to the phase shift between the patches of radial velocity (figure 11 a) and pressure (figure 11 c). These patches are the footprints of spanwise-coherent flow structures, which are organised in two pairs for $r_i=0.025$ (figure 13 a), and three pairs for $r_i=0.075$ and $0.125$ (figures 13 b,c).

For $r_i=0.5$ and $r_i=1$ (figures 13 d,e), there is still some negative correlation between $\widehat {{v_r^\prime }}$ and $\widehat {p^{\prime }}$ near the inner wall, which tends to become positive towards the outer wall. Qualitatively, the number of flow patches increases with $r_i$ , though their precise quantification from an instantaneous flow field is quite challenging due to the reduced spanwise coherence of these structures for larger values of inner radius. In addition, these patches are convected by the mean flow, and rotate in time around the inner wall, making it tricky to use time averages. A detailed analysis of the spanwise structures can be found in Soldati et al. (Reference Soldati, Orlandi and Pirozzoli2025), who provided a more accurate evaluation of their number using velocity spectra and phase averages.