4.1. Turbulent structures

The turbulent structures populating the flow upstream of the bump tip are observed via the four non-zero components of the structure function tensor, plotted in figure 5 in the $(X,r_z,Y)$ space with $r_x=r_y=0$. Figure 6 additionally shows the streamwise evolution of the intensity of local maxima of the diagonal components, $\left \langle {\delta u_i \delta u_j} \right \rangle _m$ with $i=j$, and of their wall-normal position $Y_m$ and spanwise scale $r_{z,m}$. In fact, in this $r_x=r_y=0$ space a positive/negative peak of the structure function $ \left \langle {\delta u_i \delta u_j} \right \rangle $ corresponds to a negative/positive peak of the correlation function ${\mathsf{R}}_{ij}$ (see (2.3)), and provides structural information about the flow (Jiménez Reference Jiménez2018).

Figure 5.

Structure functions $ \left \langle {\delta u \delta u} \right \rangle $, $ \left \langle {\delta v \delta v} \right \rangle $, $ \left \langle {\delta w \delta w} \right \rangle $ and $\left \langle {\delta u \delta v} \right \rangle $ (from a–d). Contour lines are drawn at 99 %, 95 %, 90 %, 75 %, 50 %, 20 % of the maximum in each plane, and the isosurfaces are computed for a value of 85 % of the maximum in the volume. The thick black line in panel (d) is the zero contour level.

Figure 6.

Streamwise evolution of the intensity and the $(Y,r_z)$ location of the maxima of the diagonal components of the structure function tensor. The intensity of $ \left \langle {\delta v \delta v} \right \rangle $ and $\left \langle {\delta w \delta w} \right \rangle $ in panel (a) is multiplied by a factor of 10 for visualisation. The black line shows the bump (arbitrary vertical scale). Regions identified by the dashed lines are defined in figure 4.

In the plane channel flow, streaks of HSS and LSS and QSVs are at the core of the near-wall turbulence cycle. For $r_x=r_y=0$, HSS/LSS and QSVs produce peaks of $ \left \langle {\delta u \delta u} \right \rangle $ and $ \left \langle {\delta v \delta v} \right \rangle $, respectively; the corresponding $r_z$ and $Y$ indicate their characteristic spanwise spacing/size, and wall-normal distance. The streaks, indeed, induce negative $R_{uu}$ at their characteristic $r_z$ spacing, and QSVs induce negative $R_{vv}$ at their lateral sides. (We refer the reader to figure 7 of Gatti et al. (Reference Gatti, Chiarini, Cimarelli and Quadrio2020), where the AGKE has been computed from a velocity field induced by the ensemble-averaged quasi-streamwise vortex.) This is what happens in the upstream periodic portion of the computational domain before the inlet I, where peaks of $ \left \langle {\delta u \delta u} \right \rangle $ and $ \left \langle {\delta v \delta v} \right \rangle $ are found at $(r_z,Y)=(0.28,0.07)$ and $(r_z,Y)=(0.3,0.21)$, or $(r_z^+,Y^+)=(55,14)$ and $(r_z^+,Y^+)=(59,42)$. The classical spacing $r_z^+ \approx 100$ between LSS (Kim, Moin & Moser Reference Kim, Moin and Moser1987; Robinson Reference Robinson1991) is twice the $r_z^+\approx 50$ separation of maximum negative $R_{uu}$ correlation between the low- and high-speed streaks. In this region, $ \left \langle {\delta u \delta u} \right \rangle $ is larger than $ \left \langle {\delta v \delta v} \right \rangle $ and $ \left \langle {\delta w \delta w} \right \rangle $, indicating that streamwise velocity fluctuations are dominant over the cross-stream ones.

In region R1, the mild adverse pressure gradient determined by the bump curvature intensifies the streamwise fluctuations, with $ \left \langle {\delta u \delta u} \right \rangle _m$ increasing by 15 % from $X=15.5$ to $X=16$. Wall-normal fluctuations are affected too, but only marginally, with $ \left \langle {} \right \rangle _m$ increasing by 3 %. Such intensification of turbulent fluctuations confirms the result of Wu & Squires (Reference Wu and Squires1998), who found a local peak of the turbulent kinetic energy $k$ just upstream of the bump. In region R2, where the pressure gradient becomes favourable again, the streaks weaken as the magnitude of $ \left \langle {\delta u \delta u} \right \rangle $ decreases, and their spanwise spacing increases by 14 % from $r_{z,m}=0.28$ at $X=16.37$ (at zero curvature) to $r_{z,m}=0.32$ at $X=16.7$ (at the bump tip). This differs from the results of Marquillie et al. (Reference Marquillie, Laval and Dolganov2008), who observed for a higher bump a slight decrease of the spanwise spacing of the streaks, measured in outer units. If the spacing is expressed in viscous units computed with the local $u_\tau$, its increase becomes even more evident: $r_z^+=81$ at $X=16.37$ goes up to $r_z^+=112$ at $X=16.55$, the position of maximum $Re_\tau$, to decrease again to $r_z^+=97$ at the tip. The lack of viscous scaling indicates that the streaks do not immediately respond to the changing wall friction (Marquillie et al. Reference Marquillie, Laval and Dolganov2008). Moreover, $\Delta Y_m$ decreases along the bump, meaning that the streaks are pushed towards the wall.

Unlike the streaks, QSVs are marginally affected by the bump, as seen from the map of $ \left \langle {\delta v \delta v} \right \rangle $; only a small increase of $ \left \langle {\delta v \delta v} \right \rangle _m$ and a decrease of $r_{z,m}$ close to the bump tip can be noticed. This difference is explained by the different typical wall distance at which the two types of structures reside. The bump height $h_b=0.0837$, or $h_b^+=25$ ($h_b^+=17$ when computed with the friction velocity of the planar channel), is in fact comparable to the average position of the streaks, but lower than that of the QSVs, which exist at $Y \approx 0.2$ or $Y^+ \approx 40$ in the planar channel region (Jeong et al. Reference Jeong, Hussain, Schoppa and Kim1997).

In R2, the presence of the bump leads to a new statistical feature, related to a type of turbulent structure which does not exist over a plane wall. Figure 7 shows a cross-section at $X=16.44$, i.e. near the end of the concave portion of the bump, and highlights how structures of spanwise velocity are generated close to the bump surface. Here (and in the whole range $16.28 \le X \le 16.7$), the map of $ \left \langle {\delta w \delta w} \right \rangle $ shows a distinct near-wall positive peak at $r_{z,m} \approx 0.3-0.4$ and $\Delta Y_m \approx 0.03-0.04$, indicating structures with negatively correlated spanwise velocity for $r_z \neq 0$. The peak cannot derive from the QSVs, which produce a negative $R_{ww}$ only when $r_y \neq 0$ (Schoppa & Hussain Reference Schoppa and Hussain2002; Gatti et al. Reference Gatti, Chiarini, Cimarelli and Quadrio2020). Instead, the observed maximum is consistent with the incipient development of TGVs. They are pairs of near-wall mushroom-shaped counter-rotating vortices, which originate from an inviscid instability mechanism over concave surfaces, because of the imbalance between the centrifugal force and the radial pressure gradient (Görtler Reference Görtler1941; Floryan Reference Floryan1991; Saric Reference Saric1994; Xu et al. Reference Xu, Liu and Wu2020). In laminar boundary layers, TGVs are known to advect upwards/downwards low/high momentum, and to induce streamwise velocity fluctuations larger than spanwise ones (Xu, Zhang & Wu Reference Xu, Zhang and Wu2017; Xu et al. Reference Xu, Liu and Wu2020). However, the contribution of the TGVs to $\left \langle {\delta u \delta u} \right \rangle $ cannot be distinguished from that of the streaks as the latter structures induce much larger fluctuations at a similar spanwise scale (see figure 7).

Figure 7.

Structure functions on a plane at $X=16.44$. Contour lines as in figure 5.

The presence of these vortices is usually assessed using the Görtler number $G$, which is very large in the concave portion of the bump; using $h$ and $U_b$ as reference length and velocity scales, the Görtler number is approximately $G = Re_b (h/R)^{1/2} \approx 3000$ at $x=16.6$ where $R$ is minimum (see figure 2a). According to the linear stability theory, in a laminar boundary layer the first onset of this instability is experienced at $G=(U_\infty \delta /\nu )(\delta /R)^{1/2}=0.4638$, where $\delta =(\nu x_0/U_\infty )^{1/2}$ is the local boundary layer thickness at $x=x_0$ (Floryan & Saric Reference Floryan and Saric1982). Note that, in the turbulent regime, if present, TGVs are immersed in a random background that makes their identification more difficult. Although their presence causes a near-wall peak in the profile of the single-point spanwise Reynolds stress $ \left \langle {ww} \right \rangle $ (result not shown), the absence of the scale information prevents their certain identification. Hence, the scale-space information of double-point statistics is crucial for their detection.

The peak of $ \left \langle {\delta w \delta w} \right \rangle $ moves towards larger $r_z$ along the bump, while its wall distance increases but only slightly; it is located at $(r_{z,m},\Delta Y_m)=(0.29,0.03)$ for $X=16.37$, and at $(r_{z,m},\Delta Y_m)=(0.39,0.04)$ for $X=16.7$. This is again consistent with the evolution of the TGVs, which progressively move apart in the spanwise direction and increase in size (Xu et al. Reference Xu, Liu and Wu2020).

As the streamwise evolution of the magnitude of $ \left \langle {\delta w \delta w} \right \rangle _m$ confirms, these structures are sustained only over the concave portion of the bump, whereas the convex wall inhibits their growth (Benmalek & Saric Reference Benmalek and Saric1994): here the TGVs are progressively damped by the redistribution via pressure–strain (see § 4.3). When studying the nonlinear development of TGVs over walls with variable curvature, indeed, Benmalek & Saric (Reference Benmalek and Saric1994) found that a convex wall reduces their growth more than a flat wall (Peerhossaini & Wesfreid Reference Peerhossaini and Wesfreid1988), and eliminates the three-dimensional structures of streamwise velocity.

A detailed description of the sustaining mechanism of TGVs is provided later in § 4.3, where the scale-space energy redistribution is addressed.

4.2. Production of turbulent kinetic energy

The components of the turbulent kinetic energy production tensor are shown in figure 8. Due to the spanwise homogeneity, $W$ and its derivatives are zero, so ${\mathsf{P}}_{33}=0$ everywhere.

Figure 8.

Production terms ${\mathsf{P}}_{11}$ (a) and ${\mathsf{P}}_{22}$ (b). See caption of figure 5 for contour lines and isosurfaces.

In R1, production resembles that in a plane channel flow; ${\mathsf{P}}_{11}$ is positive at all scales and positions and it is one order of magnitude larger than ${\mathsf{P}}_{22}$. Hence, energy is transferred from the mean field mainly towards the streamwise fluctuating field, feeding the streaks. In a plane channel, indeed, QSVs (responsible for cross-stream fluctuations) are known to be energised by redistribution only (Gatti et al. Reference Gatti, Chiarini, Cimarelli and Quadrio2020).

Over a curved wall, production is known to differ from the plane wall (Smits & Wood Reference Smits and Wood1985). In R2 the amount of streamwise energy drained from the mean flow to feed the streaks decreases, in agreement with their loss of coherency observed above in § 4.1. The maximum production ${\mathsf{P}}_{11,m}$ decreases along the streamwise coordinate; its evolution resembles that of the scale shear stresses $ \left \langle {\delta u \delta v} \right \rangle $ ($- 2 \left \langle {\delta u \delta v} \right \rangle \partial U/ \partial y$ is the dominant contribution here), i.e. moves towards smaller $r_z$ and $\Delta Y$.

Careful inspection of figure 8 reveals a second region of positive ${\mathsf{P}}_{11}$ close to the wall, just after the concave–convex curvature change; see the peak of ${\mathsf{P}}_{11}$ at $(X,r_z, \Delta Y) \approx (16.5, 0.14, 0.016)$ (see also figure 3). This production sustains the streamwise fluctuations in the upstream internal layer, and is responsible for the positive production of turbulent kinetic energy downstream of the bump, shown for example in figure 12 of Banchetti et al. (Reference Banchetti, Luchini and Quadrio2020). Above this layer, ${\mathsf{P}}_{11}$ is negative for $16.4 \le X \le 16.6$ and $r_z>0.4$, due to the positive $\partial U / \partial x$ (not shown), with a local minimum that identifies the spanwise scale $r_z \approx 0.42$. For these scales and positions, the interaction between the near-wall cycle and the changing mean velocity field leads to a sink for the streamwise fluctuations; here, the turbulent activity is reduced as the mean flow accelerates. This is in agreement with the relaminarisation of the boundary layer over a bump observed by other authors in the same region (see for example Webster et al. Reference Webster, Degraaff and Eaton1996a).

The term ${\mathsf{P}}_{22}$, i.e. the production of $ \left \langle {\delta v \delta v} \right \rangle $, is one order of magnitude smaller than ${\mathsf{P}}_{11}$ at all scales and positions; this is to be expected, owing to the small height of the bump considered in this work. In fact, the velocity derivatives $\partial V / \partial x$ and $\partial V / \partial y$, and thus ${\mathsf{P}}_{22}$, are determined by the geometry of the bump, as already shown for example by Bradshaw (Reference Bradshaw1973), who used $\partial V/\partial x$ as a proxy for the curvature of the wall. In R1 and in the part of R2 upstream of the inflection point, the concave wall induces ${\mathsf{P}}_{22}>0$ for $Y \gtrapprox 0.1$, and ${\mathsf{P}}_{22}<0$ close to the wall. When ${\mathsf{P}}_{22}>0$, the mean flow feeds not only the streamwise velocity streaks (as in the canonical channel flow) but the QSVs too. When ${\mathsf{P}}_{22}<0$, instead, energy is drained from the $v$ fluctuations to feed the mean field. After the concave–convex change of curvature, the region with ${\mathsf{P}}_{22}<0$ shrinks and then grows again over the bump tip, where it extends for all $r_z$ and $\Delta Y \gtrapprox 0.02$, with a clear peak associated with the QSVs at $(r_z,\Delta Y) \approx (0.22,0.05)$, or $Y \approx 0.13$. This negative ${\mathsf{P}}_{22}$ resembles what was observed slightly upstream for ${\mathsf{P}}_{11}$, indicating a net sink for both the streamwise and vertical velocity fluctuations, consistently with the above-mentioned tendency of the flow to laminarise near the bump tip.

In the internal layer that originates at the inflection point (see § 3 and figure 3), a small region of positive ${\mathsf{P}}_{22}$ arises close to the surface, and shrinks at the bump tip (it then grows again in regions R3 and R4; see later § 5.2). In the internal layer, therefore, both streamwise and vertical fluctuations are sustained by positive production. However, the ${\mathsf{P}}_{22}>0$ region extends for a lower $\Delta Y$ range and is weaker compared with ${\mathsf{P}}_{11}$: overall, the production of vertical fluctuations is less affected by the internal layer. This is again consistent with the results of Wu & Squires (Reference Wu and Squires1998).

4.3. Redistribution of turbulent kinetic energy

Lastly, we consider the energy redistribution with figure 9, which shows the pressure–strain terms for the three normal components over the upstream half of the bump.

Figure 9.

Pressure strain terms $\varPi _{11}$ (a), $\varPi _{22}$ (b) and $\varPi _{33}$ (c). See caption of figure 5 for contour lines and isosurfaces.

In the plane channel and in R1, two redistribution mechanisms take place (Gatti et al. Reference Gatti, Chiarini, Cimarelli and Quadrio2020). For $\Delta Y \gtrapprox 0.1$, part of the energy drained by the mean flow to feed the streamwise fluctuations is redistributed towards the cross-stream components to feed the QSVs. This is seen here in the $X=15.9$ plane, via the negative peak of $\varPi _{11}$ at $(r_z,\Delta Y) \approx (0.23,0.09)$ and the positive peaks of $\varPi _{22}$ and $\varPi _{33}$ at $(r_z,\Delta Y)=(0.17,0.13)$ and $(r_z,\Delta Y)=(0.23,0.07)$. Close to the wall, however, the vertical fluctuations are redistributed towards the in-plane ones owing to the so-called splatting (Mansour, Kim & Moin Reference Mansour, Kim and Moin1988); here, $\varPi _{11}>0$, $\varPi _{22}<0$ and $\varPi _{33}>0$ (see the colour maps in the $Y \rightarrow 0$ plane in figure 9).

In R2, an additional and intense redistribution takes place close to the wall. In the concave part, streamwise and vertical contributions to the fluctuating energy are redistributed towards the spanwise one; this is shown at $X=16.3$ by the negative peaks of $\varPi _{11}$ and $\varPi _{22}$ very near the wall at $r_z \approx 0.3$ and $r_z \approx 0.2$, respectively, accompanied by the positive peak of $\varPi _{33}$ at $(r_z,\Delta Y) \approx (0.24,0.02)$. The scale $r_z \approx 0.2-0.3$ and the wall distance $\Delta Y \approx 0.02$ are consistent with the position of local maximum of $ \left \langle {\delta w \delta w} \right \rangle $ discussed above in § 4.1 and associated with the TGVs. Hence, the generation of TGVs in the concave part of the bump is accompanied, in the vicinity of the wall, by an intense reorientation of both streamwise and vertical fluctuations into spanwise ones. In particular, the main contribution derives from the streamwise fluctuations, since $|\varPi _{11}|>|\varPi _{22}|$ there. This is consistent with the decrease of $ \left \langle {\delta u \delta u} \right \rangle _m$ and the increase of $ \left \langle {\delta w \delta w} \right \rangle _m$ observed in figure 6 at these streamwise coordinates.

After the inflection point, the redistribution changes nature; near the bump tip streamwise and spanwise fluctuations are reoriented into vertical ones. Indeed, $\varPi _{11}<0$, $\varPi _{22}>0$ and $\varPi _{33}<0$ for $0.1 \le Y \le 0.15$; at $X=16.6$ for example, their peak is found at $(r_z,\Delta Y) \approx (0.3,0.02-0.08)$. This is consistent with the observation that here $ \left \langle {\delta w \delta w} \right \rangle _m$ decreases with $X$, while $ \left \langle {\delta v \delta v} \right \rangle _m$ increases (see figure 5). Therefore, along the convex portion of the bump the TGVs progressively vanish; they are not sustained anymore, and their energy feeds mostly the vertical fluctuations.

5.1. Turbulent structures

In analogy with figure 5, which dealt with the upstream part of the bump, figure 10 plots the components of the structure function tensor for the downstream part, which includes regions R3 and R4.

Figure 10.

Structure functions $ \left \langle {\delta u \delta u} \right \rangle $, $ \left \langle {\delta v \delta v} \right \rangle $, $ \left \langle {\delta w \delta w} \right \rangle $ and $ \left \langle {\delta u \delta v} \right \rangle $ (from a–d). Contour lines are drawn at 99 %, 95 %, 90 %, 75 %, 50 %, 20 % of the maximum in each plane, and isosurfaces correspond to 85 % of the maximum in the volume.

The upstream internal layer, generated upstream in R2, grows rapidly in R3, owing to the adverse pressure gradient, and influences the spatial organisation of the turbulent fluctuations (Webster et al. Reference Webster, Degraaff and Eaton1996b; Wu & Squires Reference Wu and Squires1998). The velocity streaks become more vigorous and tend to decrease their characteristic spanwise spacing. In fact, the maximum $ \left \langle {\delta u \delta u} \right \rangle _m$ increases with $X$, and its spanwise scale $r_{z,m}$ shrinks, especially if quantified in local viscous units (see also figure 6). This should be contrasted with the opposite behaviour observed in R2, where the pressure gradient is favourable.

The QSVs are weakly affected in R3 (like in R2): $ \left \langle {\delta v \delta v} \right \rangle _m$ and its scale $r_{z,m}$ and wall distance $Y_m$ remain almost constant with $X$. The map of $ \left \langle {\delta w \delta w} \right \rangle $, moreover, shows that the TGVs generated upstream quickly vanish after the bump tip; the local maximum of $ \left \langle {\delta w \delta w} \right \rangle $ close to the wall at $r_z \approx 0.2$, indeed, disappears for $X \ge 16.9$.

Close to the downstream convex–concave change of curvature at $X \approx 17.1$, a second internal layer is formed, in agreement with previous observations (see for example Webster et al. Reference Webster, Degraaff and Eaton1996b). Unlike the first internal layer, however, this one has a minor effect on the velocity fluctuations, because it develops across a smaller pressure gradient and for a shorter spatial extent. This is consistent with the wall-curvature parameter $\Delta \kappa ^+$ being smaller here than upstream (see § 3).

Further downstream, in the (small) recirculating region for $16.3 \le X \le 17.6$ the near-wall contribution of $\left \langle {\delta w \delta w} \right \rangle $ to the turbulent kinetic energy increases. This is due to the impinging flow upon reattachment, which reorients vertical and streamwise fluctuations into spanwise ones (see for example Chiarini et al. Reference Chiarini, Gatti, Cimarelli and Quadrio2022a, and section § 5.3). The recirculation affects streaks and QSVs only marginally. Figures 10 and 6 confirm that there is no downstream evolution of the maxima $\left \langle {\delta u_i \delta u_j} \right \rangle _m$ for $16.3 \le X \le 17.6$. Clearly, this is due to the marginal recirculation created by the tiny bump. Mollicone et al. (Reference Mollicone, Battista, Gualtieri and Casciola2018), for example, found that a higher bump with a larger recirculation leads to a streamwise and wall-normal stretching of the flow structures.

Lastly, in R4 the flow is attached again, and gradually recovers towards the canonical plane channel flow. However, as visible in figure 6, during the recovery $\left \langle {\delta u \delta u} \right \rangle $, $\left \langle {\delta v \delta v} \right \rangle $ and $\left \langle {\delta w \delta w} \right \rangle $ peak at different streamwise positions, i.e. $X=17.8$, $X=21$ and $X=20$ respectively. The energy associated with the streamwise velocity streaks is maximum just after the reattachment, where streaks have been intensified by the adverse pressure gradient in R3. The peak position of $\left \langle {\delta v \delta v} \right \rangle _m$ and $\left \langle {\delta w \delta w} \right \rangle _m$, instead, are found more downstream. This is consistent with the results of Marquillie et al. (Reference Marquillie, Ehrenstein and Laval2011). They observed that, near the peak of the turbulent kinetic energy $k$ just downstream the bump, which coincides with the peak of its largest contributor $\left \langle {\delta u \delta u} \right \rangle $, the streaks become unstable and break down into smaller structures, which further downstream evolve into hairpin-like streamwise vortices, so that the reorganisation of the fluctuations in the vertical and spanwise components occurs later than the streamwise one.

5.2. Production of turbulent kinetic energy

Figure 11 plots the production terms for $\left \langle {\delta u \delta u} \right \rangle $ and $\left \langle {\delta v \delta v} \right \rangle $ in regions R3 and R4. The map of ${\mathsf{P}}_{11}$ shows that streamwise energy is drained from the mean flow to feed the streamwise fluctuating field almost everywhere, except in a thin near-wall region close to the mean-flow reattachment point (see figure 12). The scales and wall-normal distances identified by the positive local maxima of ${\mathsf{P}}_{11}$, i.e. $r_z \approx 0.2$ and $\Delta Y \approx 0.02$, indicate that the streamwise velocity fluctuations are again mainly sustained by the interaction of the wall cycle with the mean flow. In R3, ${\mathsf{P}}_{11}$ increases with $X$, consistently with the strengthening of the streaks observed above in § 5.1; this is also shown by the downstream evolution of the local maximum of ${\mathsf{P}}_{11}$, which is ${\mathsf{P}}_{11,m}=0.096$ at $X=17$ and increases to ${\mathsf{P}}_{11,m}=0.12$ at $X=17.5$. The global maximum of ${\mathsf{P}}_{11}$ is found close to the maximum of $\left \langle {\delta u \delta u} \right \rangle $, just after the flow reattachment at $(X,r_z,Y) =(17.68,0.2,0.09)$, in agreement with the results of Marquillie et al. (Reference Marquillie, Ehrenstein and Laval2011).

Figure 11.

Production terms ${\mathsf{P}}_{11}$ (a) and ${\mathsf{P}}_{22}$ (b). See caption of figure 10 for contour lines and isosurfaces.

Figure 12.

Zoom of the production term ${\mathsf{P}}_{11}$ in the recirculating bubble at $X=17.75$. The black thick line is for ${\mathsf{P}}_{11}=0$. The black thin lines indicate isovalues of ${\mathsf{P}}_{11}$, with an increase of $0.0002$.

In the near-wall region close to the flow reattachment point, i.e. in the range $17.45 \le X \le 18$ at $\Delta Y \le 0.02$, streamwise energy is drained from the fluctuating field to feed the mean flow at all scales: figure 11 shows a very thin region with slightly negative ${\mathsf{P}}_{11}$ (see also the near-wall zoom of figure 12). This negative region results from the positive $\partial U / \partial x$, which corresponds to an acceleration of the reverse/forward near-wall mean flow in the region downstream/upstream of the reattachment point. Interestingly, the same negative production due to the mean-flow acceleration has been observed in the separating and reattaching flow past an elongated rectangular cylinder (Chiarini et al. Reference Chiarini, Gatti, Cimarelli and Quadrio2022a); this suggests that a near-wall region with negative production is a general feature of reattaching flows.

The production term for $\left \langle {\delta v \delta v} \right \rangle $ is almost two orders of magnitude smaller than ${\mathsf{P}}_{11}$. In R3, ${\mathsf{P}}_{22}$ is negative, with its local minima related to QSVs. Hence, QSVs are energised by the mean flow in the upstream side of the bump, but release energy back to the mean flow in the downstream side. The negative peak of ${\mathsf{P}}_{22}$ moves away from the wall as $X$ increases. At the same time, a positive ${\mathsf{P}}_{22}$ appears near the wall, close to the convex–concave change of curvature. This layer with ${\mathsf{P}}_{22}>0$ thickens, and its intensity increases, until in R4 (for $X \gtrapprox 18$) it extends for all $Y$. The maximum intensity of ${\mathsf{P}}_{22}$ occurs at $(X,r_z,Y) \approx (18,0.18,0.07)$. Further downstream, it gradually decreases to become eventually zero as for the plane channel flow.

5.3. Redistribution of turbulent kinetic energy

Figure 13 plots the pressure–strain terms for the three normal components of the structure function tensor. Four different redistribution mechanisms take place in regions R3 and R4; two are localised near the wall, and the other two exist at larger $Y$.

Figure 13.

Pressure–strain terms $\varPi _{11}$ (a), $\varPi _{22}$ (b) and $\varPi _{33}$ (c). See caption of figure 10 for contour lines and isosurfaces.

Along the convex portion of the bump, the TGVs progressively vanish under the action of the pressure strain, which, close to the wall, reorients the spanwise fluctuations into vertical fluctuations. This explains the negative peak of $\varPi _{33}$ at $(X,r_z,\Delta Y)=(16.7,0.37,0.02)$, and the positive peak of $\varPi _{22}$ at the same scales and positions. Moving downstream, this redistribution mechanism is absent already at $X \approx 17$, as the TGVs do not survive for long (see § 5.1).

The second redistribution mechanism occurs at larger wall distances ($Y \approx 0.2$) and can only be seen for $X \le 17.2$. Here, the energy of the streaks advected from the bump tip is partially redistributed towards the cross-stream components and feeds the QSVs. The local negative minimum of $\varPi _{11}$ at $(r_z, \Delta Y) \approx (0.24, 0.1)$ and the local positive maximum of $\varPi _{22}$ and $\varPi _{33}$ at $(r_z, \Delta Y) \approx (0.62, 0.07)$ and $(r_z,\Delta Y) \approx (0.32,0.13)$ at $X=16.9$ support this scenario. For all the diagonal components, the quantity $\varPi _{ij}+{\mathsf{P}}_{ij}$ is positive at these scales and positions, implying that the interaction of the wall cycle with the mean flow is a net source for the velocity fluctuations in the three directions, with the streamwise ones being sustained by production, and the cross-stream ones by redistribution.

In R3 a third redistribution mechanism originates near the wall at $X \approx 16.9$, and acts again to partially reorient streamwise fluctuations into cross-stream ones; see the negative and positive peaks of $\varPi _{11}$ and $\varPi _{33}$ at $X \approx 16.9$, $\Delta Y \approx 0$ and $r_z \approx 0.2$ in figure 13(a,c). It is associated with the inner layer generated after the convex–concave change of curvature; its intensity and vertical extent grow with the streamwise coordinate (note the streamwise evolution of the corresponding peaks of $\varPi _{ij}$), and at $X \gtrapprox 17.4$ becomes the main redistribution process active in R4.

Lastly, the fourth redistribution mechanism is at work close to the mean-flow reattachment point and inside the recirculating region. Here, $\varPi _{11}$ and $\varPi _{33}$ are positive, with $\varPi _{22}<0$: this is the statistical signature of the flow impinging (see the figure 14(a) for a zoom of $\varPi _{33}$ at $X=17.42$). In this region, $\varPi _{33}>\varPi _{11}$, and the local peaks of $\varPi _{22}$ and $\varPi _{33}$ identify a small spanwise scale of $r_z \approx 0.22$. Hence, the pressure–strain preferentially organises the velocity fluctuations in small-scale $w$-structures, similarly to the findings of Chiarini et al. (Reference Chiarini, Gatti, Cimarelli and Quadrio2022a). This mechanism is expected to become more significant when the recirculation is larger and more intense. To prove that these $w$-structures are associated with the recirculating bubble, we consider an additional bump that does not induce separation after the bump tip. This second bump has the same upstream part, but an horizontal expansion factor of $2.5$ is applied to the rear part; for further details see bump $G_2$ in Banchetti et al. (Reference Banchetti, Luchini and Quadrio2020). Figure 14(b), plots $\varPi _{33}$ for this bump, at the position of minimum $C_f$ (that in this case is positive as there is no separation). The absence of a local peak of $\varPi _{33}$ for $\Delta Y \rightarrow 0$, confirms that when the flow remains fully attached and does not impinge on the wall, this redistribution mechanism does not occur, and the fluctuations do not organise in $w$-structures.

Figure 14.

Pressure strain term $\varPi _{33}$ is the $Y\unicode{x2013}r_z$ plane with minimum $C_f$. Panel (a) is for the bump that induces separation after the tip (considered in this work). Panel (b) is for the bump $G_2$ of Banchetti et al. (Reference Banchetti, Luchini and Quadrio2020) that does not induce flow separation. The cross in panel (a) highlights the near-wall local peak of $\varPi _{33}$. See caption of figure 10 for contour lines.