3.1. The $\Omega _z$ cross-section snapshots

The evolution of the most common flow features of the SW bump is shown through a sequence of $\Omega _z$ colour maps in $x$ – $y$ sections at an arbitrary $z$ in figure 3. Past the bump peak, a shear layer emerges in some instances, extending significantly downstream without any immediate rollup of $\Omega _z$ (figure 3 a). Interestingly, subsequent snapshots show that rollup happens simultaneously at different $x$ locations (denoted by R in figure 3 b) – this is atypical of all shear layers where rollup occurs sequentially, not simultaneously. Namely, the shear layer has characteristics of both a spatially developing shear layer and an only time-evolving shear layer. The rollup of the shear layer simultaneously at multiple $x$ locations (although alike in a temporal mixing layer) is indeed rather striking and interesting and may involved coupling with the dynamics of the SB. This deserves careful study. Naturally, this flow features SRs pairing and tearing processes (P and T, respectively, in figure 3 i–n), with associated frequencies discussed later concerning the SB unsteadiness. The shear layer vortices induce near-wall vortices of opposite sign due to the no-slip condition; these two vortices then combine to form a vortex dipole (two antiparallel vortices, henceforth called only dipole; D in figure 3 i,j). The formation of dipoles is characteristic of this flow but is only observable from the instantaneous realization as it is a highly unsteady process and obviously hidden in the mean.

Figure 4. Smooth wall top-view sequence of spanwise vorticity $\Omega _z$ from a low-pass filtered field with filter size $\unicode{x1D6E5} =0.05$ at $Y^+=1$ . The dashed line denotes the $z$ location for figure 3.

Figure 5. (a,c) Iso-surfaces of instantaneous spanwise vorticity $\omega _z$ and (b,d) iso-surfaces of $\lambda _2$ from a low-pass filtered field with filter size $\unicode{x1D6E5} =0.1$ . (a,b) SW and (c,d) GW. The dashed line in both the upper and lower figures indicates the underlying spanwise CS of interest.

3.2. Minibubble

Secondary recirculating regions – named ‘minibubbles’ in HGYS – attached to the wall are found in figure 3 (point M) around $x=4.8$ , identified by having $\Omega _z\gt 0$ . As documented previously, the minibubble forms due to the streamline curvature of the near-wall upstream-moving flow of the SB and is not apparent in the mean flow for SW (note that the mechanism of minibubble formation is similar to that of the upstream flow separation); hence, it is interesting to see the instantaneous realizations and show its presence in SW. Furthermore, more than one minibubble can be observed at a time (figure 3, points M), and in some instances, vorticity from a minibubble can be scooped up by the SRs to form dipoles (figure 3 g,h,n). Note that, while minibubbles can lead to dipole formation, this formation does not require the presence of minibubbles. The motion of a minibubble is due to the combined influences of the SB’s upstream flow near the wall and downstream advection of the minibubble due to its image vortex. The resulting advection is likely small, with no obvious direction of its motion.

Figure 4 shows the top views of $\Omega _z$ at $Y^+=1$ (a constant height above the wall and bump) to reveal the distribution of the minibubble in the $z$ -direction. The minibubble can extend in the whole $z$ domain, but more often is fragmented in $z$ , as seen through the $\Omega _z\gt 0$ regions in figure 4(a,c). The grooves enhance coherence in the flow, always having a minibubble occurring at the grooves. The minibubble’s coherence will be further discussed later through a CS analysis via the phase average of the SRs.

The flow features illustrated in figures 3 and 4 for SW are also observed in GW, thus examination of such snapshots is not repeated. Subsequent discussions will concentrate on the impact of grooves on these flow features, specifically through the incorporation of vortical structures that are directly associated with the grooves at the locations of flow separation.

3.3. The $\Omega _z$ and $\lambda _2$ structures

To better understand the three-dimensionality of the flow structures, figure 5 shows isometric views of $\Omega _z$ and $\lambda _2$ iso-surfaces. The prominent structures past the bump peak are SRs (denoted by dashed lines in each panel of figure 5) with $z$ contortions – similar for both SW and GW. Such three-dimensionality emphasizes the need to carefully align CSs in a phase-average analysis – performed later – as even a single continuous structure varies significantly in $z$ , such that different $x$ – $y$ planes of the same SR realization would reveal different features of th evolving CS.

Figure 6. Sequence from left to right of a zoomed-in view of a subset of $\lambda _2$ -structures coloured by instantaneous streamwise velocity from an unfiltered velocity field: (a–c) SW; (d–f) GW. The dashed line in both the upper and lower figures indicates the underlying spanwise CS of interest. Note that separation occurs at $x=4.4$ for both SW and GW. The SW reattachment is at $x=5.38$ , and GW reattachment is at $x\approx 5.49$ .

The grooves become important near the bump peak, as discussed in HGYS. In particular, small-scale vortical structures with their axis aligned in the streamwise direction and attached to the crest corners of the grooves are observed in GW extending from the bump’s peak into the shear layer. Figure 6 shows a sequence of zoomed-in views of the $\lambda _2$ structures (from the unfiltered field) behind the bump in the shear layer for SW and GW. The streamwise structures in GW weaken due to cross-diffusion as the streamwise vortices for a groove are counter-rotating, although some evidence of the streamwise structures remains on the SRs further downstream, as seen in figure 6(f). Thus, the grooves actively modify the SRs and the downstream flow dynamics. One objective of this study is to explain how the grooves’ corner vortices affect the shear layer and SRs and, hence, flow separation.

Figure 7. Schematic of the typical flow structure of the SB region with grooves. Conceptual elucidation of the organized flow.

3.4. Conceptual elucidation of the flow

For ease of discussion, the sketch in figure 7 highlights the relevant flow structures and dynamics we will analyse in detail. It shows the rollup of the shear layer, leading to the formation of SRs and the streamwise vortices (swirling jets) attached to the crests’ corners. The separated region is below the shear layer up to the reattachment point. In addition, we accentuate the presence of vortex rods (the light blue structures in figure 7) at steep angles in the separated region, which are inferred in the next section to be more prevalent in GW, compared with SW, as well as flow structures that can be found within the SB such as secondary SBs (minibubbles) and vortex dipoles. Quasi-streamwise vortices in the region of accelerating flow upstream of the bump are found to be responsible for negative production and are discussed in § 7.

4.1. Quadrant analysis

In turbulent flows, coherent motions significantly contribute to the generation of the Reynolds shear stress (Lu & Willmarth Reference Lu and Willmarth1973; Jeong et al. Reference Jeong, Hussain, Schoppa and Kim1997; Schoppa & Hussain Reference Schoppa and Hussain2002), which is directly connected to the turbulent momentum transport and, consequently, drag at the wall (Townsend Reference Townsend1976). The Reynolds shear stress can be divided into the contributions of quadrant events in the plane of streamwise and wall-normal fluctuations: ${\rm Q}1(u^{\prime}\gt 0,v^{\prime}\gt 0)$ outward motions, ${\rm Q}2(u^{\prime}\lt 0,v^{\prime}\gt 0)$ ejections, ${\rm Q}3(u^{\prime}\lt 0,v^{\prime}\lt 0)$ inward motions and ${\rm Q}4(u^{\prime}\gt 0,v^{\prime}\lt 0)$ sweeps (Wallace Reference Wallace2016). By understanding which quadrant Reynolds stresses are dominant in the various regions of the flow, one can then formulate the CS responsible.

Figure 8. Colour maps of quadrant Reynolds shear stress in $x$ – $y$ section with ${\mathcal {H}}=3$ . Panels show (a–d) SW, $-\overline {u^{\prime}v^{\prime}}_{Qi}$ ; (e–h) GW at the centre of grooves, $-\langle u^{\prime}v^{\prime}\rangle _{Qi}$ . The dotted line denotes the inflection point of the mean velocity profile. (i) Profiles of ${\rm Q}2$ and ${\rm Q}4$ Reynolds stresses at different streamwise locations for SW, GW at the centre of crests (GW-C), and GW at the centre of groove troughs (GW-T); approximate locations of the CS responsible for Reynolds stresses are shown as dashed circles with the direction of velocity corresponding to the quadrant. For (e–i), the dashed line indicates the crest height position. (j) Sample snapshot of Reynolds shear stress events ${\rm Q}2$ and ${\rm Q}4$ superimposed with streamlines projected into the $x$ – $y$ plane.

Using the quadrant-splitting technique with the concept of the hyperbolic hole, the contribution from each quadrant and hole region can be defined as

(4.1) \begin{align} \langle {u^{\prime}v^{\prime}}\rangle _{Qi}&=\frac {1}{N}\sum u^{\prime}v^{\prime} I_{Qi}; \quad I_{Qi}=\begin{cases} 1, & |u^{\prime}v^{\prime}| \gt {\mathcal {H}}\langle u^{\prime}u^{\prime}\rangle ^{\frac {1}{2}}\langle v^{\prime}v^{\prime}\rangle ^{\frac {1}{2}};\,(u^{\prime},v^{\prime})\,\ {\rm in}\ Qi, \\ 0, & {\rm otherwise,} \end{cases} \end{align}

(4.2) \begin{align} \langle {u^{\prime}v^{\prime}}\rangle _{\mathcal {H}}&=\frac {1}{N}\sum u^{\prime}v^{\prime} I_{\mathcal {H}}; \quad I_{\mathcal {H}}=\begin{cases} 1, & |u^{\prime}v^{\prime}| \leqslant {\mathcal {H}}\langle u^{\prime}u^{\prime}\rangle ^{\frac {1}{2}} \langle v^{\prime}v^{\prime}\rangle ^{\frac {1}{2}},\\ 0, & {\rm otherwise,} \end{cases} \end{align}

where $N$ is the number of realizations, $\mathcal {H}$ is the hole size and $\sum _{i=1}^{4} \langle u^{\prime}v^{\prime}\rangle _{Qi}+\langle u^{\prime}v^{\prime}\rangle _{{\mathcal {H}}}$ (Lu & Willmarth Reference Lu and Willmarth1973). The hole size is chosen to be ${\mathcal {H}}=3$ such that strong events, presumably associated with the large-scale and dynamically relevant CS, are captured. Previous investigations in flow separation (Krogstad & Skåre Reference Krogstad and Skåre1995; Schatzman & Thomas Reference Schatzman and Thomas2017) have shown that with ${\mathcal {H}}=3$ , ${\rm Q}2$ and ${\rm Q}4$ events dominate, while ${\rm Q}1$ and ${\rm Q}3$ make less than $1\,\%$ fractional contribution to $\overline {u^{\prime}v^{\prime}}$ . This dominance also holds for the zero-pressure-gradient TBL (without any separation) (Lu & Willmarth Reference Lu and Willmarth1973); note that, for easy reference, we also use ${\mathcal {H}}=3$ for the quadrant analysis. Moreover, Schatzman & Thomas (Reference Schatzman and Thomas2017) showed that ${\mathcal {H}}=3$ well captures the underlying shear instability and associated strong quadrant motions due to the coherent spanwise vorticity of SRs. A sample snapshot of $-u^{\prime}v^{\prime}$ in figure 8(j) shows that strong ${\rm Q}2$ and ${\rm Q}4$ events result from SRs. The claims below are not significantly affected by altering the threshold $\mathcal {H}$ .

Consistent with that observed by Schatzman & Thomas (Reference Schatzman and Thomas2017), we find that the peaks of $-\langle u^{\prime}v^{\prime}\rangle _{{\rm Q}2}$ and $-\langle u^{\prime}v^{\prime}\rangle _{{\rm Q}4}$ occur above and below, respectively, the inflection point ( $\partial \overline {U}/\partial y=0$ ) of the mean velocity (figure 8 b,d,f,h) – with or without grooves. In agreement with HGYS, the Reynolds shear stress from strong sweep ( ${\rm Q}4$ ) and ejection ( ${\rm Q}2$ ) events decrease in magnitude in GW. Hence, if one considers ${\rm Q}2$ and ${\rm Q}4$ events a direct result of the SRs, the rollers in GW are ‘weaker’ in generating Reynolds shear stress.

The effect of grooves is further detailed through the profiles of quadrant Reynolds shear stress where $-\langle u^{\prime}v^{\prime}\rangle _{{\rm Q}4}$ is significantly suppressed – much more than $-\langle u^{\prime}v^{\prime}\rangle _{{\rm Q}2}$ – near the wall at the onset of the shear layer (figure 8 i at $x=4.5$ ). This suppression is attributed to the fluid ejection promoted by grooves that inhibit the sweep motion. Specifically, a sweep motion below the shear layer – with a $-v^{\prime}$ fluctuation – is countered by the fluid jetting from grooves. Additionally, the profiles in figure 8(i) show a vertical shift of the inflection point (shear layer) away from the wall in GW due to the fluid ejection introduced by grooves at the onset of the shear layer, an aspect previously pointed out in HGYS. The average height of the inflection point is $y=0.143$ for SW and $y=0.158$ for GW with minimal streamwise variation from $x=4.5$ to $5.5$ . These results provide initial insights into the effect of grooves on the SRs, showing that changes in the Reynolds shear stress are due to modification of the SRs. Through the CS analysis later in § 5, we further detail the relation of the Reynolds shear stress uncovered with the quadrant-splitting technique and the SRs.

As a side note, we showed in HGYS that negative Reynolds shear stress occurs on the upstream face of the bump. Figure 8(a,c) emphasizes that this is a consequence of the increased ${\rm Q}1$ events as the flow is accelerated by the bump, narrowing the channel area. On the other hand, the ${\rm Q}3$ events are negligible throughout the channel, for cases with or without grooves (figure 8 e,g).

4.2. Two-point correlations

A consistent swirling motion at a specific position may appear in space as regions of highly correlated velocity fluctuations – providing statistical information about their geometrical characteristics. In this section, we examine the statistical characteristics of CS based on the two-point correlations of velocity and pressure fluctuations.

The generic two-point correlation is defined as

(4.3) \begin{equation} r_{\phi \phi }(x_0,y_0,z_0,\Delta x, \Delta y, \Delta z)=\frac {\langle \phi ^{\prime}(x_0+\Delta x, y_0+\Delta y, z_0+\Delta z,t) \phi ^{\prime}(x_0,y_0,z_0,t)\rangle }{\langle \phi ^{\prime} \phi ^{\prime}\rangle ^{\frac {1}{2}}(x_0,y_0,z_0) \langle \phi ^{\prime}\phi ^{\prime}\rangle ^{\frac {1}{2}}(x,y,z)}, \end{equation}

where coordinate $(x_0,\,y_0,\,z_0)$ is the reference point and $\phi$ can be any independent variable of the flow (i.e. velocity, pressure, etc.). Note that the two-point correlations are averaged in the $z$ -periodic direction over all grooves at the same relative positions, i.e. $z_0$ is a reference point to the span groove’s wavelength ( $\lambda _g$ ), and in SW, the two-point correlations are not a function of $z_0$ .

4.2.1. Velocity correlations

A comparison of the 3-D correlation structure (correlation iso-surface) obtained for the velocity fluctuations for a flat SW, the SW and the GW bumps is provided in figure 9. The $r_{uu}$ iso-surfaces for the flat SW with reference point $y_0=0.12\,(y_0^+=36)$ (figure 9 a) appear to be very elongated in the $x$ -direction with a length of approximately $\Delta x^+=10^3$ – a result due to the near-wall streaks observed in the buffer layer (Jiménez Reference Jiménez2013; Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2014). The $r_{uu}$ iso-surfaces for the SW bump – with the reference point $x_0$ taken above the recirculation bubble where root-mean-square (r.m.s.) of pressure fluctuations ( $p_{rms}$ ) is maximum – are similarly streamwise elongated to that over the flat SW; however, the length is 60 % of that of the flat SW (using $r=+0.2$ ). This is consistent with the findings in HGYS that the streaks are suppressed above the recirculation bubble. Different from the flat SW, the inclination angle of the positive lobe is negative in the $x$ $y$ plane (figure 9 d), which is expected as $r_{uu}$ depicts $u^{\prime}$ -structures in the reattaching shear layer behind the bump. Moreover, the negative $r_{uu}$ lobes are more pronounced and at the same distance from the wall as that for positive $r_{uu}$ lobe – different from the case in the flat wall which are at the wall – indicative of the $r_{uu}$ iso-surfaces describing structures predominantly on the shear layer. For GW, $r_{uu}$ iso-surfaces are no longer elongated in the $x$ -direction, consistent with the observation in HGYS that the flow becomes more isotropic with grooves. The negative $r_{uu}$ lobes in GW (figure 9 g) appear to be more rod-like than in SW (figure 9 d), with a steeper angle in the $x$ $y$ plane, which is linked with the more marked presence of vortex rods at steep angles when analysed together with $r_{ww}$ (detailed below). In GW, the $r_{uu}$ iso-surfaces appear to be disconnected from the bump peak highlighting the disrupted shear layer by the presence of streamwise structures favouring $v^{\prime}$ and $w^{\prime}$ velocity fluctuations.

Figure 9. Three-dimensional iso-surfaces of two-point correlations of the three velocity components for (a–c) the flat SW, (d–f) SW bump and (g–i) GW bump. Panels show (a,d,g) $r_{uu}$ ; (b,e,h) $r_{vv}$ ; (c,f,i) $r_{ww}$ . Iso-surfaces are $r=+0.8$ (red), $r=+0.4$ (green), $r=+0.2$ (blue), $r=-0.1$ (magenta) and $r=-0.2$ (black). The reference point for the flat wall is $y_0=0.12\,(y_0^+=36)$ , for the bump cases are the point of peak r.m.s. $ $ pressure fluctuations: SW at $(x_0,y_0)=(5.2,0.12)$ and GW $(5.47,0.134)$ at the centre of troughs. Quasi-streamwise vortices and SRs responsible for the correlated velocities are sketched in (b,c,e,g,h,i) for reference.

Figure 10. Two-point correlation contours of (a) streamwise velocity fluctuations $r_{uu}$ and (b) spanwise velocity fluctuations $r_{ww}$ in the planar section denoted in figure 9(g,i) with $30^\circ$ in the $x$ $y$ plane with the horizontal. Solid lines indicate positive correlations, and dashed lines negative. (c) Sketch of a vortical structure that would produce the present $r_{uu}$ and $r_{ww}$ , with blue and red corresponding to the contours in (a) and (b), respectively. Note that the green circle in (a) and (b) corresponds to a slice through the CS in (c). (d,e) Instantaneous realizations of $\lambda _2$ -structures for GW coloured by $\omega _x$ , with dashed lines indicating the $\lambda _2$ structures of interest.

The two-point correlations of cross-flow velocity fluctuations ( $r_{vv}$ and $r_{ww}$ ) for the flat SW case (figure 9 b,c) are associated with the quasi-streamwise vortices in the buffer layer (Jiménez Reference Jiménez2018). The effect of the bump modifies this interpretation significantly, particularly for $r_{vv}$ (figure 9 e). The negative $r_{vv}$ lobes appear upstream and downstream of the positive correlation lobe instead of being located on the sides (in the $z$ -direction) as in the flat SW case. Clearly, the presence of SRs is responsible for the distribution of $r_{vv}$ as they induce $+v^{\prime}$ and $-v^{\prime}$ motions in the back and front as they move downstream above the SB, causing $r_{vv}$ to change sign in the $x$ -direction. Similar $r_{vv}$ surfaces are observed for GW (figure 9 h); however, the $r_{vv}=-0.2$ iso-surfaces are larger and overall more symmetric in the $x$ -direction, i.e. the negative lobes are of the same size and same distance with respect $x_0$ – perhaps because of a more matured SR as $x_0$ is slightly further downstream. The $r_{ww}$ iso-surfaces for the bump cases (with or without grooves, figure 9 f,i) are similar to those of the flat SW (figure 9 c); namely, the negative–positive–negative $r_{ww}$ lobes are stacked in the $x$ – $y$ plane. Not surprisingly, the $r_{ww}$ iso-surfaces do not show any indication of SRs, as SRs mostly induce $u^{\prime}$ and $v^{\prime}$ .

An interesting observation is that $r_{uu}$ and $r_{ww}$ in GW, if examined together, can be interpreted to be the result of a vortex rod at a steep angle in the $x$ $y$ plane. For reference, figure 10(a,b) shows the $r_{uu}$ and $r_{ww}$ contours extracted on a plane that is $30^\circ$ with respect to the $-x$ direction for GW (figure 9 g,i). The contours of $r_{uu}$ and $r_{ww}$ support the presence of a vortex rod (indicated by the green-shaded circle), idealized in the sketch in figure 10(c), which would produce such velocity correlations. Therefore, GW is perhaps more notably populated by vortex rods at steep angles past the bump perturbation. Further evidence to elaborate this speculation is provided in figure 10(d,e), where instantaneous $\lambda _2$ -structures at high angles are found within the separated flow region.

Figure 11. Three-dimensional iso-surfaces of two-point correlations for pressure fluctuations. (a) The flat SW, (b) the SW bump and (c) the GW bump. The reference point for the flat wall is $y_0=0.12\,(y_0^+=36)$ , for the bump cases are the point of peak r.m.s. pressure fluctuations: SW at $(x_0,y_0 )=(5.2,0.12)$ and GW $(5.47,0.134)$ at the centre of troughs. (d) Profiles of $r_{pp}$ as a function of streamwise increment $\Delta x$ at $y_0$ for SW and GW.

5.1. Phase-locked ensemble averaging of CS

The eduction of the CS is both sophisticated and elaborate. The interest here is to educe the dominant spanwise CS present in the shear layer, as visualized in Muller & Gyr (Reference Muller and Gyr2020) and conceptually described in Muller & Gyr (Reference Müller and Gyr1986). For this purpose, a visual inspection of the instantaneous structures in an $x$ – $y$ plane is performed using the instantaneous spanwise vorticity ( $\Omega _z$ ) and $-\lambda _2$ . For example, figure 12(a,b) shows the $\Omega _z$ and $-\lambda _2$ colour maps obtained from an arbitrary time and $z$ position for SW. Note that, at this particular instant, $\omega _z$ resembles pairing as described in Hussain & Zaman (Reference Hussain and Zaman1980); however, the $-\lambda _2$ shows a single peak in the centre.

Figure 12. Colour maps of instantaneous (a) spanwise vorticity, $\omega _z$ , and (b) vortex identification criterion, $-\lambda _2$ , in an $x$ $y$ section at an arbitrary spanwise position. (c) Time series of a local circulation, $\gamma$ , computed for the red square in (a,b); the colour maps in (a) and (b) correspond to the time highlighted by the red filled circle in (c). The solid line denotes the mean value of $\gamma$ , the dashed line denotes the standard deviation of $\gamma$ and the grey shaded region corresponds to the values of $\gamma$ considered.

To characterize the dominant spanwise structures, a local time-dependent circulation parameter is computed as

(5.1) \begin{equation} \gamma =\int \Omega _z \mathrm {d}A ,\end{equation}

over a squared region across the shear layer; figure 12(c) shows the evolution of $\gamma$ for the red square box in figure 12(a,b). Note that the side length of the box in figure 12(b) is $0.1$ , which, as mentioned earlier, is approximately the thickness of the mean shear layer. For any selected $x$ location, there is smearing of the educed structure in $y$ if we do not bound in the $y$ direction due to jitter between successive passing structures as well as random spanwise waviness of the SRs. Hence, both $x$ and $y$ locations of the point of eduction (squared box) must be precisely selected a priori; therefore, only a small fraction of the total structures will be used for eduction. The phase average considers only strong events where $\gamma$ has a local negative peak below one standard deviation past the mean $\gamma$ (denoted by the shaded region below the dash line in figure 12 c). The snapshot in figure 12(a,b), which corresponds to the time marked by the filled red circle in figure 12(c), exemplifies that the negative peaks of $\gamma$ correspond to SRs. The criterion for choosing instantaneous realizations based on the time series of $\gamma$ is used for the first estimate ensemble average (zeroth ensemble) with the following two additional conditions: (i) a local peak of $-\lambda _2\gt 0$ must exist in the red square of figure 12(b); and (ii) the vorticity vector at the peak of $-\lambda _2$ must be within a cone of $30^\circ$ along the $-z$ direction. These criteria resulted in only ${\sim}600$ structures being considered for eduction sifted out of millions of structures; this necessarily requires a very long simulation.

Following Hussain & Hayakawa (Reference Hussain and Hayakawa1987) the ensemble average is obtained by phase aligning the structures through a time shift to increase the cross-correlation of $\Omega _z$ in the squared region chosen. A realization is discarded if the correlation coefficient with the ensemble average is lower than a specified value $R_t$ , defined as

(5.2) \begin{equation} R_t=\frac {1}{2N}\sum _{i=1}^{N}R_i. \end{equation}

Here, $R_i$ is the correlation value for the $i$ th realization, and $N$ is the total number of realizations being considered for the new ensemble average. The process is iterated by replacing the zeroth ensemble with the latest ensemble until no more realizations are removed from the considered ensemble. The phase-average procedure is performed at different stations with coordinates corresponding to the centre of the square box at $(x,y)=\{(4.7,0.16),(4.85,0.16),(5,0.16),(5.2,0.12),(5.47,0.13)\}$ . The coordinates are arbitrarily chosen: the first three stations centred at the shear layer, and the second-last and last stations at the peak of $p_{rms}$ for SW and GW, respectively. For both SW and GW, the final phase averages are obtained using ${\sim}250$ structures at each station.

5.2. Coherent structure measures

5.2.1. Educed vortical structures

The phase averaging procedure to educe SRs in the shear layer was performed at five different phases of the evolution of the SRs (we call them stations), moving downstream up to the peak $p_{rms}$ (figure 13). In the case of SW, the peak of $p_{rms}$ is earlier than that in GW, but for completeness, we include for SW the station where the peak $p_{rms}$ occurs for GW. In SW for stations at $x=4.7$ , $4.85$ and $5$ , we see that $\widetilde {\omega _z}$ (for the threshold chosen) depicts a shear layer and not closed circular loops, although a well-defined SR structure is obtained from $\lambda _2$ of the phase-averaged field (figure 13 a–c,f–d). In the early stages of the rollup process, the shear layer is constant and steady and, therefore, emerges in the phase average. As the roller matures moving downstream, the contours of $\widetilde {\omega _z}$ become circular – like $\lambda _2$ – and clearly identify the SR as a separate structure from the shear layer. The effect of groove spinning jets is evident in altering $\widetilde {\omega _z}$ by breaking the coherence of the shear layer in the early stages of the formation of SRs. The fact that $\widetilde {\omega _z}$ does not have a well-established shear layer in GW may indicate that the rollup process is being delayed, hence this is evidence of the delayed peak of turbulence and the changes to $-\langle u^{\prime}u^{\prime}\rangle \partial \langle U \rangle /\partial x$ seen by HGYS.

Figure 13. Contours of $-\lambda _2(\partial [ U_i]/\partial x_j)=0.5,3,5.5,8$ (red dotted line) at different stations from top to bottom superimposed with the coherent streamlines $(\tilde {u},\tilde {v})$ , contours of $\widetilde {\omega _z}=-1.3,-2.3$ (blue lines), the phase-average $([ U ],[V])$ dividing streamline (magenta solid line) and time-averaged dividing streamline (magenta dotted line) for (a–e) SW, and (f–j) GW at the centre of grooves. The eduction locations (the red contour centres) are arbitrarily chosen – the same for SW and GW. The red patches identify the educed SRs, the yellow patches denote nearby SRs and the green patches the near-wall minibubbles. Both the yellow and green are unavoidably smeared due to the inherent jitter with respect to the educed red structures.

Interestingly, in the coherent velocity field, minibubbles are found embedded within the large SB on the downstream side of the bump. The fact that the eduction of SRs always captures minibubbles implies spatial coherence between the minibubble and SRs. Such spatial coherence is enhanced by the induction of the image vortices. The minibubble is influenced by the shear layer rollup but is also subjected to the induction of the image, the combined effect would be an interesting study. Here, the mean field is indirectly represented by the phase-average dividing streamline (purple lines in figure 13 a–j) with no evidence of the minibubble for SW (figure 13 a–e) in contrast to GW, which shows the minibubble within grooves at $x\approx 4.8$ in figure 13(h–j). The effect of grooves is to enhance the extent and prevalence of the minibubble in the coherent velocity within the SB (figure 13 f–j). Here, we understand why the minibubble also appears in the mean flow for GW as it is more dominant and frequent in the coherent flow, always having a footprint at $x=4.8$ inside the grooves. Note that in GW, for the early stages of the SR, there are three simultaneous minibubbles versus only one in SW. In HGYS, we observed lower skin friction in GW compared with SW near the downstream end of the bump (around $x=5$ ), where near-wall flow is moving upstream. The lower skin friction could be attributed to the minibubbles deflecting the upstream-moving flow away from the wall, leading to a reduced time-average skin friction there. Moreover, the minibubbles contribute to skin friction because the velocity underneath them is downstream – further compounded by its image vortex. These three effects – deflection of the SB flow by the minibubble away from the wall, the minibubble bottom flow downstream and the effect of the image vortex – combine together to alter the skin friction and form drag in this region.

Figure 14. Coherent Reynolds shear stress $-\tilde {u} \tilde {v}$ at different $x$ -stations superimposed with contours of $\widetilde {\omega _z}$ at levels $-2.3$ , $-1.3$ (red dotted lines) for (a) SW, and (b) GW at the centre of grooves. Here, $x_c$ and $y_c$ denote the $(x,y)$ locations of the centres for eduction in each panel. Panel (c) shows the $-\tilde {u} \tilde {v}$ area average over the dotted squares in (a,b), $\alpha _{\tilde {u}\tilde {v}}=A_{{square}}^{-1}\iint -\tilde {u}\tilde {v} \text {d}x\text {d}y$ ; solid line SW and dashed line GW. (d) Sketch comparing the coherent Reynolds shear stress distributions of CS of different cross-sections: a circular SR, an elliptical SR inclined upstream and downstream (solid line positive values and dashed lines negative values). In (a,b), a $+$ sign identifies the location for eduction alignment and is very close to the location of $\widetilde {\omega _z}$ peak; ${\rm Q}1$ , ${\rm Q}2$ , ${\rm Q}3$ , and ${\rm Q}4$ identify the quadrants, and the $x$ -range of each panel is $0.2$ .

5.2.2. Coherent Reynolds shear stress

In the early stage of the SR in SW (figure 14 a at $x=4.7$ ), the coherent Reynolds shear stress is dominated by ${\rm Q}4$ sweep events and ${\rm Q}3$ inward motions, while the ${\rm Q}2$ ejections and ${\rm Q}1$ outward motions are inhibited, perhaps expected because the upstream half of the SR is closer to the wall and bump peak, i.e. the SR is not yet well developed and the upstream half is very close to the onset of the shear layer. A cloverleaf-like pattern in $-\tilde {u}\tilde {v}$ is formed at $x=4.85$ (figure 14 a), similar to SRs found in mixing layers (Hussain & Zaman Reference Hussain and Zaman1980; Hussain & Hayakawa Reference Hussain and Hayakawa1987). In GW, the cloverleaf pattern $-\tilde {u}\tilde {v}$ is delayed, indicating the disruption of the rollup of $-\Omega _z$ due to the presence of grooves inducing a series of swirling jets at the point of separation near the bump peak. The cloverleaf patterns for both SW and GW are not exactly axisymmetric; therefore, on average, they make some (positive) contribution to the total (time-averaged) Reynolds shear stress. If the structure cross-section is circular (in the $x$ – $y$ plane), the four lobes will be equal (figure 14 d). For an elliptic SR where the lobes from each axis are distinct from each other, the spatial distribution of $-\tilde {u}\tilde {v}$ has regions of both positive and negative values, and which sign dominates in the time average depends on the orientation of the SR. The sketch in figure 14(d) shows the effect of the inclination of an elliptic SR; the peak of $-\tilde {u}\tilde {v}$ is positive if the SR is inclined toward the wall, and otherwise, if it inclines away from the wall. Here, we see that $-\tilde {u}\tilde {v}$ always makes a positive contribution to $-\langle u^{\prime}v^{\prime}\rangle$ throughout the evolution of SRs for both SW and GW as the educed structure is always elliptic and with an inclination toward the wall – perhaps characteristic of SRs in the shear layer above a SB. The peak of $-\tilde {u}\tilde {v}$ oscillates between ${\rm Q}2$ and ${\rm Q}4$ events for the five stations analysed. The limited thickness of the bubble and the limited length of the shear layer together prevent the typical tumbling of the SRs. This contrasts a typical mixing layer in which the SRs would tumble, varying their geometry among the three shapes in figure 14(d) occurring randomly in space and time – resulting in a net zero contribution to the time-averaged Reynolds shear stress; the net zero Reynolds stress results not only from the symmetric but also from the tilted structures occurring randomly in space and time (Zaman & Hussain Reference Zaman and Hussain1980).

Figure 15. Coherent production, ${\mathcal {P}}_c=-\tilde {u}_i \tilde {u}_j\partial \langle U_i\rangle /\partial x_j$ . Panel (a) is for SW, and (b) is for GW at the groove centre. Here, $x_c$ and $y_c$ denote the $(x,y)$ locations of the point of alignment for eduction in each panel. All panels are superimposed with corresponding contours of $\widetilde {\omega _z}$ at levels $-1.3$ and $-2.3$ (dotted lines). A $+$ sign identifies the location for eduction alignment and is very close to the location of the $\widetilde {\omega _z}$ peak. The $x$ -range of each panel is $0.2$ . Panel (c) shows the ${\mathcal {P}}_c$ area average over the dotted squares in (a,b), $\alpha _{{\mathcal {P}}_c}=A_{{square}}^{-1} \iint {\mathcal {P}}_c \text {d}x\text {d}y$ ; solid line SW and dashed line GW.

Figure 16. (a,b) Incoherent turbulence intensity, $[u^{\prime\prime} u^{\prime\prime}]+[v^{\prime\prime} v^{\prime\prime}]+[w^{\prime\prime} w^{\prime\prime}]$ . (c, d) Incoherent production, ${\mathcal {P}}_r=-[u^{\prime\prime}_{\!\!\!i} u^{\prime\prime}_j][\partial U_i/\partial x_j]$ . Panels (a,c) are for SW, and (b,d) for GW at the groove centre. Here, $x_c$ and $y_c$ denote the $(x,y)$ locations of the point of alignment for eduction in each panel. All panels are superimposed with corresponding contours of $\widetilde {\omega _z}$ at levels $-1.3$ and $-2.3$ (dotted lines). A $+$ sign identifies the location for eduction alignment and is very close to the location of the $\widetilde {\omega _z}$ peak. The $x$ -range of each panel is $0.2$ . Panel (e) shows the ${\mathcal {P}}_r$ area average over the dotted squares in (c,d), $\alpha _{{\mathcal {P}}_r}=A_{{square}}^{-1} \iint {\mathcal {P}}_{r} \text {d}x\text {d}y$ ; solid line SW and dashed line GW.

5.2.3. Coherent and incoherent productions

When examining the coherent production (production of coherent motion by mean shear), ${\mathcal {P}}_c=-\tilde {u}_i \tilde {u}_j \langle S_{ij}\rangle$ , where $S_{ij}=(\partial U_i/\partial x_j + \partial U_j/\partial x_i)/2$ (figure 15 a,b), we obtain a cloverleaf pattern similar to that in $-\tilde {u}\tilde {v}$ , indicating that $-\tilde {u}\tilde {v}\langle S_{12} \rangle$ is the dominant contribution to ${\mathcal {P}}_c$ – but of course with different peak values. The coherent Reynolds stress for both SW and GW are increasing up to the last station measured (figure 13 c), while the maximum ${\mathcal {P}}_c$ values occur at $x_c =5$ (SW) and $x_c=5.2$ (GW). The peak values of ${\mathcal {P}}_c$ occur before the peaks of production ( $-\overline {u_i u_j} \overline {S_{ij}}$ in SW; $-\langle u^{\prime}_i u^{\prime}_j\rangle \langle S_{ij}\rangle$ in GW) at $x_c=5.2$ and $x_c=5.47$ for SW and GW, respectively (see HGYS). The shear action in producing coherent motion is maximum prior to the point of breakup of the SRs (on average), which is where the peak of production and dissipation is to be expected.

The incoherent (random) turbulence intensity, $ [u^{\prime\prime}_{\!\!\!i} u^{\prime\prime}_{\!\!\!i} ]$ , is similar to the (time average) turbulence intensity, $\langle u^{\prime\prime}_{\!\!i} u^{\prime\prime}_{\!\!i}\rangle$ , reported by HGYS – increasing with $x$ past the bump peak and with peak values at the shear layer. Throughout all five stations, lower $ [u^{\prime\prime}_{\!\!\!i} u^{\prime\prime}_{\!\!\!i} ]$ is observed for GW with respect to SW (figure 16 a,b). The incoherent production (production of random motion by coherent motion), ${\mathcal {P}}_r=- [u^{\prime\prime}_{\!\!\!i} u^{\prime\prime}_j ] [S_{ij} ]$ , corroborates the lowered production, particularly in the core of SRs in GW, but also a reduction of the peaks in front and behind (figure 16 c,d,e) – note that the peaks of ${\mathcal {P}}_r$ are expected to be in front and behind of the SR where strain rate is maximum (Hussain Reference Hussain1983). The non-zero ${\mathcal {P}}_r$ in the core of SRs emphasizes that the SRs are turbulent and not exactly uniform in $z$ . The lower ${\mathcal {P}}_r$ (particularly at the core) in GW can be attributed to the swirling jets suppressing incoherent production by organizing the flow in the form of swirling streamwise jets. The fact that there is lower ${\mathcal {P}}_r$ in GW than in SW suggests that flow should be more organized (coherent); therefore, coherent measures such as $-\tilde {u}\tilde {v}$ can be expected to be higher in magnitude in GW. From figure 14(a,b), it is not evident that this is the case. The streamwise evolution trend of $- \tilde {u}\tilde {v}$ via spatial averages (figure 14 c) supports this expectation that coherent measures would be higher with lower incoherent turbulence production, particularly evident for stations at $x_c=5.2$ and $5.47$ .

Figure 17. Three-dimensional iso-surfaces of $-\lambda _2(\partial [U_i ]/\partial x_j)$ for (a) SW and (b) GW of the educed structures at different $x$ locations (shifted in $z$ to illustrate the time evolution). Colour maps of $\partial \overline {U}/\partial Y$ for (c,f) SW and $\partial \langle U\rangle /\partial Y$ for (d,g) GW with superimposed contours of $\langle \Omega _x\rangle$ . (e) Profiles of $\partial \overline {U}/\partial Y$ for SW and $\partial \langle U\rangle /\partial Y$ for GW at the centre crests (GW-C), at the centre of grooves (GW-T) and GW spanwise averaged (GW $z$ -avg) at different streamwise locations. The vertical bars in (e) and (h) represent the width of the profiles in these two panels; the width is measured where the value is half of the peak value of each profile.

The results of the coherent and incoherent measures show that SRs are more coherent in GW but generate less intense random fluctuations (turbulence intensity), seemingly by the spinning jets generated by the grooves. This confirms the inferences made from the quadrant Reynolds shear stress analysis and two-point correlations of pressure fluctuations – more coherent but weaker SRs. So far, we have focused on coherent measures in $x$ – $y$ planes and found evidence of the suppression of turbulence for the GW due to spinning jets but have not yet shown the spinning jets emerging from the phase average.

6.1. Propagation velocity of the SRs

An unanswered question is why the SB in GW is longer than that in SW. One explanation is that the fluid channelled through the grooves and ejected behind the bump pushes the SRs – resulting in a higher propagation velocity and a longer travelling distance than those in the SW case. To confirm this, the convective velocity of the SRs is investigated. Note that the computation of propagation velocity is a non-trivial task, and estimation without taking into account the size of structures can sometimes lead to errors, particularly when using Taylor’s frozen hypothesis (Zaman & Hussain Reference Zaman and Hussain1981; Del Álamo & Jiménez Reference lamo, Juan and Jiménez2009). Here, the propagation velocity is considered from the signal propagation velocity of instantaneous spanwise vorticity fluctuations ( $\omega^{\prime}_z$ ) measured based on the peak locations of space–time correlations (Hussain & Clark Reference Hussain and Clark1981). The two-point correlation (4.3) is extended to construct space–time correlation maps by considering a time delay between the two-point variable fluctuations, defined as

(6.1) \begin{equation} r_{\omega _z \omega _z}(x_0,y_0,z_0,\Delta x,\Delta t)=\frac { \langle \omega^{\prime}_z (x_0+\Delta x, y_0, z_0,t+ \Delta t)\omega^{\prime}_z (x_0,y_0,z_0,t)\rangle }{\langle \omega^{\prime}_z \omega^{\prime}_z \rangle ^{\frac {1}{2}}(x_0,y_0,z_0) \langle \omega^{\prime}_z \omega^{\prime}_z\rangle ^{\frac {1}{2}}(x,y,z)}. \end{equation}

Figure 18. Space–time correlations of spanwise vorticity fluctuations $r_{\omega _z \omega _z}$ for (a) the SW, (b) the GW at the centre of grooves and (c) GW at the centre of crests; the reference point coordinates are the point of peak r.m.s. pressure fluctuations: SW at $(x_0,y_0 )=(5.2,0.12)$ and GW $(5.47,0.134)$ . (d) Line fit connecting the coordinate points with $\partial r_{\omega _z \omega _z}/\partial \Delta x=0$ at every $\Delta t$ from panels (a,b,c); solid line is SW, dashed is GW at the centre of crests (GW-C) and dotted line, GW at the centre of grooves (GW-T).

Space–time correlation contour maps of $r_{\omega _z \omega _z }$ are shown in figure 18(a–c) for SW, GW at the grooves and GW at the crests, respectively. Note that the reference points $(x_0)$ for $r_{\omega _z \omega _z }$ (given in the caption of figure 18) are based on the location of peak pressure fluctuations – which coincide with the location of the shear layer and SRs. The propagation velocity is obtained by fitting a line connecting coordinates points with $\partial r_{\omega _z \omega _z }/\partial \Delta x=0$ at every $\Delta t$ (Hussain & Clark Reference Hussain and Clark1981), i.e. the local maximum correlation at every $\Delta t$ . The propagation velocities are $u_c=0.23$ for SW, $0.31$ at the centre of grooves and $0.30$ at the centre of crests for GW, confirming that indeed the SRs are travelling downstream faster in GW. A lower propagation velocity at crests indicates that perhaps SRs are stretched between the crests and grooves, which would suggest waviness in the spanwise direction (see figure 17 c) with a forced wavelength matching the grooves. However, the difference in propagation velocity between crests and grooves is rather small, and, as discussed before, the SRs realizations (at this $x$ ) are expected to be relatively spanwise uniform. The inferred slight waviness is just a history effect of the flow ejection at the grooves and streamwise vortices at the crest corners earlier in the shear layer.

Perhaps an additional reason for having a higher propagation velocity for the SR in GW is the image vortex and the fact that the SRs are forming from a shear layer slightly farther from the wall (compare the peaks in $\partial \langle U\rangle /\partial Y$ in figure 17(h) between SW and GW, from $y^+\approx 18$ – $24$ ). The increased separation distance between the SR and image vortex within the wall would reduce the upstream induced velocity (SRs have naturally $-\omega^{\prime}_z$ , and the induced motion with their image would be in the $-x$ direction), hence increasing the SR downstream velocity.

6.2. Unsteadiness of the separation bubble

We use the point of zero wall shear stress ( $\tau _w=0$ ) to identify the point of reattachment for both SW and GW – this criterion for GW was shown by HGYS to be in agreement with various other criteria. The instantaneous detachment/reattachment lines are approximately uniform in the spanwise direction (figure 19 a,b); the uniform detachment/reattachment lines have also been observed for the backward-facing step (Le, Moin & Kim Reference Le, Moin and Kim1997) and flow separation due to APG (Na & Moin Reference Na and Moin1998; Wu & Piomelli Reference Wu and Piomelli2018). Therefore, as in Le et al. (Reference Le, Moin and Kim1997) and Na & Moin (Reference Na and Moin1998), we track the location of the reattachment point ( $x_r$ ) based on the spanwise-averaged wall shear stress ( $\tau _{w,zavg}$ ) (for the GW case, spanwise averaging is performed at the same relative positions of the grooves). Specifically, the reattachment point is defined where $\tau _{w,zavg}$ switches from negative to positive (figure 19 c–e).

Figure 19. Instantaneous colour maps of wall shear stress, $\tau _w$ , for (a) SW and (b) GW. The thick black solid line denotes the location with $\tau _w=0$ continuously connected through the span of the wall. Locations of spanwise-averaged reattachment point $x_r$ for (c) SW, (d) GW at the centre of crests and (e) GW at the centre of grooves. In (c–e), the red solid line has a slope equal to the convective velocity obtained in the previous section for each configuration, respectively, as a reference.

The instantaneous detachment/reattachment lines and overall SB motions, not surprisingly, are linked with the SRs that develop in the shear layer after flow separation (rollup frequency) and shedding of vorticity of the SB (Eaton & Johnston Reference Eaton and Johnston1980; Neto et al. Reference Neto, Grand, Métais and Lesieur1993). Since the grooves change the SRs, we further investigate how the GW modifies the unsteadiness of the $x_r$ .

Figure 20. Histogram of downstream travelled length ( $\Delta _{x_r}$ ) and the slope ( $m_{x_r}$ ) of the rate of increase in the downstream travelled length from the time series of $x_r$ for (a) SW, (b) GW at the centre of grooves and (c) GW at the centre of crests.

Figure 21. Pre-multiplied frequency spectra of (a) spanwise vorticity vertically integrated across the shear layer at $x=4.9$ , $\Gamma$ , (b) the reattachment position $x_r$ and (c) the SB area ( $A$ ). The solid line is SW, the dashed line is GW at the centre of crests (GW-C) and the dash-dot line is GW at the centre of grooves (GW-T). (d) An instantaneous snapshot as reference for the computations of $\Gamma$ , $A$ and $x_r$ , as suggested by a referee.

Figure 20 shows histograms of the downstream travelled distance $\Delta _{x_r}$ and time rate of increase of the streamwise travel distance $m_{x_r}$ ; the latter illustrating the speed of $x_r$ . The travelled distances $(\Delta _{x_r})$ are obtained by starting at the instant when the reattachment position changes direction from moving upstream to downstream and ending at the opposite situation, i.e. from a local minimum to a local maximum in figure 19(c–e). The $\Delta _{x_r}$ probability distribution for GW shows many occurrences for larger lengths of the SB compared with SW, particularly at the centre of grooves. Such increased probability of large $\Delta _{x_r}$ suggests that the grooves somewhat stabilize the growth of the SB, allowing for longer travelled distances of $x_r$ and modifying the shedding dynamics of the SB.

The histogram of $m_{x_r}$ establishes that the travelling speed of $x_r$ for the GW is higher than the SW case, with a more noticeable increase at troughs (figure 20 b). More importantly, the mean values of $m_{x_r}$ coincide with the propagation velocity obtained based on $r_{\omega _z \omega _z}$ at the shear layer in § 6.1. Therefore, the histogram of the slopes confirms that the movement of the SRs at the shear layer indeed leads the movement of the reattachment point, which travels at a higher speed for the GW case. Note that more occurrences with low $m_{x_r}$ are observed at crests, which agrees with the slightly slower propagation velocity at crests.

The sudden retraction of $x_r$ is due to the shedding of vorticity from the SB, which clearly shows a quasi-periodic behaviour in the signal shown in figure 19(c–e). This periodic behaviour is not necessarily connected with the rollup frequency of SRs – as we will see later – but with the saturation of the growth of the SB and the drag that it is subjected to, having a characteristic shedding frequency.

The unsteadiness of the reattachment point can be quantified based on the standard deviation of $x_r$ ( $0.071$ for the SW and $0.131$ and $0.101$ at the grooves and crests for GW, respectively), notably increasing for the GW. The higher standard deviation of $x_r$ in GW is attributed to the more prevalent minibubbles that can reach further downstream, as figure 13(f–j) shows, which is expected to alter the shedding of vorticity from the SB.

The rollup frequency of SRs, quantified by the spectra of the signal

(6.2) \begin{equation} \Gamma (t)=\int _{0.11}^{0.21} \Omega _{z,zavg}(x=4.9,y,t)\mathrm {d}y ,\end{equation}

(figure 21 a; the shear layer is located between $y=0.11$ and $0.21$ , as seen in figure 13) shows that the peak frequency in GW is lower than SW, consistent with pressure correlations in § 4.2. Perhaps this behaviour is expected since the shear layer in GW is more diffused in $y$ and with a lower peak in $\partial \langle U\rangle /\partial y$ , so the rollup process takes longer.

On the other hand, the peak frequency in the spectra of $x_r$ is found to be lower – for both SW and GW – than the rollup of SRs (figure 21 b). As mentioned before, the peak frequencies of $x_r$ are found to characterize the shedding of vorticity from the SB, verified via a movie visualization (available as supplementary material at https://doi.org/10.1017/jfm.2025.4) of the SB synchronized with the evolution of the reattachment position. Interestingly, the grooves do not affect the peak frequency of $x_r$ with respect to SW. However, the magnitude of $\phi _{x_r}$ for the peak frequency is significantly different between GW and SW, and different from that observed for $\phi _\Gamma$ . This means that, while the peak frequency of $x_r$ remains the same between SW and GW, in GW, the amplitude of the oscillations is larger, as previously pointed out through the histogram of $\Delta x_r$ in figure 20(a–c). Perhaps the grooves are facilitating the movement of the reattachment point by inducing an effective slip velocity at the wall.

We also examine the spectra of the SB area, $A$ (computed similarly to that in Fang et al. (Reference Fang, Tachie, Bergstrom, Yang and Wang2021)), as follows:

(6.3) \begin{equation} \begin{aligned} A(t)=\int _0^{H} \int _{4.25}^{12} I_A(\psi )\mathrm {d}x \mathrm {d}y; \quad I_A(\psi )=\begin{cases} 1, & \psi \lt 0,\\ 0, & {\rm otherwise,} \end{cases}\\ \psi (x,y,t)=\int _{y=wall}^{y} U(x,y^{\prime})_{z-avg} \mathrm {d}y^{\prime}, \end{aligned} \end{equation}

and as shown in figure 21(c) (integrated from x = 4.25 where the peak of the bump is to the end of the channel at x = 12). Unlike $\Gamma$ or $x_r$ , which are localized measurements, $A$ provides information about the instantaneous size of the SB by integrating large-scale and small-scale phenomena. The $A$ spectrum for SW exhibits a distinct peak at a low frequency $f=0.12$ . Several researchers have reported such low-frequency oscillations (Hudy & Naguib Reference Hudy and Naguib2007; Wu et al. Reference Wu, Meneveau and Mittal2020; Fang et al. Reference Fang, Tachie, Bergstrom, Yang and Wang2021), which are frequently referred to as the ‘breathing’ motions of the SB; however, no consensus has been reached regarding their origin. This low-frequency peak also agrees with the results obtained by Eaton & Johnston (Reference Eaton and Johnston1980); Hudy & Naguib (Reference Hudy and Naguib2007) for a backward-facing step separating flow. The $A$ spectra in SW also show a higher peak frequency – perhaps not so clear as the magnitude of $\phi _A$ is very low – which is very close to the shedding frequency of vorticity from the SB as quantified through $\phi _{x_r}$ . The rollup frequency, as quantified through $\phi _\Gamma$ , does not clearly surface in the spectra of $A$ . This is not surprising since the rollup process only distorts the upper surface of the SB, changing the area minimally (see movie in the supplementary material).

In GW, at the centre of the grooves, the peak frequency of the $A$ is the shedding frequency of the SB, which completely dominates over the low-frequency peak; while at crests, the low frequency is dominant, and the shedding frequency peak becomes more noticeable. In addition, like $\phi _{x_r}$ , the corresponding peak magnitude in GW at the centre of grooves significantly increases.

Figure 22. Colour maps of turbulent kinetic energy (TKE) production for (a) SW and (b) GW at the centre of grooves. Wall-normal profiles of production and the separate contributions from Reynolds normal stresses and Reynolds shear stress at (c,d) $x=4.05$ where flow accelerates and at (e,f) $x=5.625$ inside the SB; (c,e) SW and (d,f) GW.

8.1. Coherent structure dynamics and its quadrant analysis

The quadrant analysis demonstrates that the time-averaged Reynolds shear stress, which results from strong sweep ( ${\rm Q}4$ ) and ejections ( ${\rm Q}2$ ) (associated with the SRs), decreases in GW. From the CS analysis, SR generates sweep downstream of the structure and ejection upstream for both SW and GW. The grooves generate a series of streamwise jets with ejections and sweeps, the former stronger than the latter, in $y$ – $z$ planes, which interact and modify the SR ejections and sweeps. The quadrant analysis shows that the swirling jet ejections are effective in countering the SR sweeps, weakening the ${\rm Q}4$ Reynolds shear stress. This effect is more prominent at the initiation of the shear layer close to the bump peak because the swirling jets impact there first. The peaks in Reynolds shear stress from strong events (sweeps and ejections) for GW shift in $y$ farther from the wall, induced by the jetting effect at the groove centres.

8.2. Coherent structure characterization via two-point correlations

Two-point velocity correlations, as expected, also identify SRs behind the bump peak, particularly that from wall-normal velocity, for both SW and GW. The grooves have a significant effect on the streamwise velocity correlations, which decrease faster in $x$ than in SW. Moreover, this decrease suggests that near-wall vortex rods at steep angles are more prevalent in GW. Interestingly, while the two-point correlations of pressure highlight the SRs, the correlation extends more in the $z$ -direction for GW than SW, suggesting that grooves increase $z$ coherence. Since the spanwise row of streamwise swirling jets all start at the same $x$ and have virtually the same circulation, their evolution will be identical, and hence their interaction at the point of their maturity makes SRs aligned in $z$ . Also, the streamwise correlation of pressure fluctuations in the shear layer reveals that SRs are farther apart on average in GW, suggesting a lower frequency of rollup of SRs.

8.3. Minibubbles, their role in form and skin-friction drag

Analysis via phase-average eduction of SRs reveals several new features. Surprisingly, the minibubble is persistently present in the coherent velocity field, suggesting a direct link between SRs and minibubbles – for both SW and GW. Although the importance of minibubbles is not well established, its omnipresence tells us that it is fundamental to perhaps most separated flows and likely will help us to understand flow phenomena (e.g. mass transport) over mountains, dunes, etc. In the case of GW, the minibubbles are more numerous, larger and stronger (higher circulation) compared with SW, each of which has a notable effect on both skin-friction and form drag.

8.4. Spanwise rollers and their persistent contribution to positive production

The rollup of a plane shear layer, whether initially laminar or turbulent, starts with the formation of an elliptic cross-sectioned SR tilted downward on its downstream side, transitions into a circular cross-section and then tilts its downstream side upward – as in a tumbling manner. The locations of these orientations are random as the SR advects downstream, with the time average reflecting no clear footprint of any particular SR geometry, and hence, production is always positive. In the case of the shear layer enveloping the SB, the upstream inclination of the initial elliptical SR is repeated and sustained in space (with minimized tumbling). As a consequence, the production is always positive, even instantaneously, leading to a higher positive production than in a typical plane shear layer. That is, the SRs always generate a higher time-averaged coherent Reynolds shear stress compared with a plane shear layer, where the time-averaged coherent Reynolds shear stress is much lower.

8.5. Coherent and incoherent productions

Coherent production, ${\mathcal {P}}_c$ , of SRs becomes maximum upstream of the peak total turbulence production for both SW and GW. The subsequent decay in ${\mathcal {P}}_c$ is expected as the mean shear decreases faster than the increase of $-\tilde {u}\tilde {v}$ . These results naturally imply that the control of the SRs in the shear layer needs to occur upstream of the peak of turbulence in the uncontrolled scenario.

The incoherent production ${\mathcal {P}}_r$ indeed shows its peak value at the same location as the total turbulence production. The value of ${\mathcal {P}}_r$ is lower in GW than in SW by an average of 25 % (figure 16 c,d) due to the reduced $\partial \langle U\rangle /\partial y$ across the shear layer. The shear layer commences as wavy as well as thicker in GW compared with SW, and the streamwise jets accentuate the waviness, with the net effect that, in GW, $\partial \langle U\rangle /\partial y$ is smaller (concomitantly SRs are larger); hence, the reduced incoherent production and turbulence intensity. Based on the educed 3-D structure, we speculate that the streamwise structures induced by the grooves prevent breakup, pairing and the core dynamics of the SRs.

8.6. Spanwise roller rollup, their propagation velocity and SB reattachment

In HGYS, we found that the SB is longer (approximately 20%) in GW than SW and attributed this to the jetting induced by the grooves. Here, we articulated that it is the jetting of fluid from grooves at the bump peak that pushes the SRs, inducing a higher propagation velocity of the SRs, hence their longer travel before impingement on the wall for reattachment. Space–time correlation maps of spanwise vorticity fluctuations verify that SRs have higher propagation velocity (20 %) in GW.

8.7. Time-dependent dynamics of SB

We identify two main flow features of SB unsteadiness: high-frequency vorticity shedding from the SB and low-frequency ‘breathing’ of SB. In SW, the low-frequency area fluctuations are stronger, while in GW, the high-frequency fluctuations are stronger. The spanwise organization by the swirling jets does not seem to be quite in consonance with this behaviour of the SB, and this begs a rigorous study of the SB dynamics with and without grooves.

8.8. Negative production

Inspection of instantaneous vortical structures using the $\lambda _2$ -criterion, along with instantaneous regions of negative production, elucidated the vortex organization that leads to the average negative production reported in HGYS. The average negative production obtained on the upstream side of the bump where flow accelerates due to streamwise stretching of fluid elements occurs every time two counter-rotating vortices appear in the accelerating region, inducing a strong ejection (Q2) event. This is somewhat counter-intuitive as this is the typical situation in flat wall turbulence, but if it occurs in a region of $+\partial \langle U \rangle /\partial x$ , it leads to a significant transfer of energy from the turbulent field to the mean field. In GW, the additional region of negative production due to counter-gradient Reynolds shear stress is due to vortical structures entering the grooves, possibly connecting groove size and CS diameter as a way to modify negative production around the bump.