3.1 Time-averaged secondary flows

Secondary flows are large regions of recirculation within the boundary layer that appear as large vortical structures in the time-averaged flow fields. These structures are clearly visible in the time-averaged velocity fields in both the experimental measurements and numerical simulations presented in figure 2. The plots show colour contours of the streamwise velocity with velocity vectors representing the in-plane velocity components over one period of the surface pattern. The mean streamwise velocity exhibits large spanwise variations that extend the full thickness of the boundary layer but are most pronounced near the wall. Low-momentum pathways (LMPs) form around the elevated ridges of the surfaces and high-momentum pathways (HMPs) occur in the valleys. Between these zones, large secondary flows form. In both the experiments and the numerical simulations, these vortices extend well beyond the roughness sublayer to approximately $0.5\unicode[STIX]{x1D6FF}$ .

Figure 2.

Time-averaged velocity and signed swirling strength.

Despite the fact that the numerical simulations correspond to a Reynolds number approximately ten times weaker than the experiments, we find a very similar flow topology, demonstrating the robustness of the secondary flows. This is different from the mean secondary flow in ducts, whose shape changes as a function of Reynolds number (Kawahara et al. Reference Kawahara, Uhlmann and Van Veen2012).

The mean signed swirling strength maps exhibit the secondary flows very clearly and are also presented in figure 2. These maps are computed as the time average of the instantaneous swirling strength multiplied by the sign of the local instantaneous vorticity. The map from the experimental measurements was shown previously by Vanderwel & Ganapathisubramani (Reference Vanderwel and Ganapathisubramani2015) where the large pair of counter-rotating vortices are the secondary flows and the smaller pair of vortices located just above each protrusion were labelled as ‘primary’ flows. When the spacing of the ridges was small, the primary flows dominated and secondary flows did not appear, but as the spacing became $O(\unicode[STIX]{x1D6FF})$ the secondary flows manifested. The map from the numerical simulations produces the same vortex topology. The magnitude of the swirling strength is the same order of magnitude as those in the experiment.

The map from the numerical simulations now provides access to what is happening below the roughness height. It illustrates how the ‘primary’ vortices emanate from the corners of the large surface protrusions, and that the secondary vortices appear detached and engulf the bulk of the boundary layer. Smaller vortex pairs are also produced by each of the small bumps in the valleys; however, proximity to the larger secondary flows mitigates the strength of one side of this smaller vortex pair for many of the bumps. Notice that in the middle of the valley ( $z/\unicode[STIX]{x1D6FF}=0,1$ ) both sides of this small vortex pair appear with nearly equal strength over the roughness bumps. The vortices associated with the small roughness bumps are not expected to influence the secondary flows significantly. To test this, the numerical simulations were repeated with all the connecting pins of the LEGO removed. These results are shown in the Appendix and demonstrate that the same structure of primary and secondary flows persists around a square wall protrusion.

3.2 Instantaneous velocity fields

It is clear that secondary flows are a dominant feature of the time-averaged flow fields, however, what is still unclear is how these secondary flows feature in the instantaneous snapshots of the flow. Example snapshots (taken from the supplementary movies) of the instantaneous velocity field are presented in figure 3. It is immediately apparent that the structure of the instantaneous flow is completely different and more complex than the time-averaged flow field. The vector fields of the in-plane motion do not exhibit large zones of recirculation but rather much finer-scale motions consistent with the scale of the wrinkles in the interfaces of the low- and high-momentum pathways. The bases of the low-speed zones remain fixed above the elevated regions of the surface; however, the low-momentum pathways themselves extend far away from the wall in narrow wisps that wave from side to side in the flow in a way that brings to mind the motion of a candle flame. Similarly, high-momentum pathways penetrate the boundary layer and reach close to the wall in the regions above the valleys of the surface roughness.

Figure 3.

Snapshots of the instantaneous flow field from the experiment and the simulation (videos of the experimental measurements and the simulation results are available online in mp4 format on the Cambridge Core website, please see supplementary movies available at https://doi.org/10.1017/jfm.2018.955).

In the following sections we apply several analyses to the instantaneous snapshots in order to better understand the structure of the instantaneous flow features. We investigate how the presence of the secondary motions is reflected in dispersive and random contributions to the Reynolds stresses and which of the discrete structures present in the instantaneous maps are relevant to make up the large secondary flows that are observed in the time-averaged flow fields.

3.3 Reynolds stress distribution

The spanwise inhomogeneity of the Reynolds stresses is known to sustain secondary motions of Prandtl’s second kind. Based on the mean momentum budget or streamwise vorticity transport one can identify particular gradients of the Reynolds stress tensor compensating for the presence of the secondary motion (Anderson et al. Reference Anderson, Barros, Christensen and Awasthi2015; Dai et al. Reference Dai, Huang, Xu and Cui2015; Stroh et al. Reference Stroh, Hasegawa, Kriegseis and Frohnapfel2016). Due to the periodically varying surface inhomogeneity in the spanwise direction we expect a similar periodicity in the statistical features of the flow field. A measure for the additional momentum transport due to surface inhomogeneities in rough-wall boundary layers is given by the so-called dispersive stresses (Finnigan Reference Finnigan2000; Cheng & Castro Reference Cheng and Castro2002; Finnigan & Shaw Reference Finnigan and Shaw2008). The definition of dispersive stresses is based on a distinction between spatial and temporal averaging in the definition of the velocity fluctuations.

In the present set-up, a triple decomposition, as suggested by Reynolds & Hussain (Reference Reynolds and Hussain1972), can be employed. The triple decomposition of the velocity component $u_{i}$ in the $y$ – $z$ cross-plane is given by

(3.1) $$\begin{eqnarray}u_{i}(y,z,t)=\langle \bar{u}_{i}\rangle (y)+\tilde{u} _{i}(y,z)+u_{i}^{\prime \prime }(y,z,t),\end{eqnarray}$$

where $\langle \bar{u}_{i}\rangle$ is the time-averaged and spanwise-averaged field, $\tilde{u} _{i}$ and $u_{i}^{\prime \prime }$ denote dispersive and random fluctuations, respectively. Combination of the second and third terms from (3.1) gives the total fluctuation

(3.2) $$\begin{eqnarray}u_{i}^{\prime }(y,z,t)=\tilde{u} _{i}(y,z)+u_{i}^{\prime \prime }(y,z,t).\end{eqnarray}$$

This approach is explained in detail in Türk et al. (Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014). With this framework, the spatially averaged stress tensor becomes

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{ij}=\underbrace{\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}\langle \bar{u}_{i}\rangle }{\unicode[STIX]{x2202}x_{j}}}_{viscous\,stress}\underbrace{-\langle \overline{u_{i}^{\prime \prime }u_{j}^{\prime \prime }}\rangle }_{\,random\,stress}\,\,\underbrace{-\langle \tilde{u} _{i}\tilde{u} _{j}\rangle }_{dispersive\,stress}.\end{eqnarray}$$

In addition to the usual viscous stress and Reynolds random stress, $\unicode[STIX]{x1D70F}_{ij}$ also contains the dispersive stress $\langle \tilde{u} _{i}\tilde{u} _{j}\rangle$ , which represents the transport of momentum due to the spanwise heterogeneity of the mean flow. The dispersive stresses are a measure of the spatial inhomogeneity of the flow, as a time-averaged turbulent flow field without any spanwise variations would exhibit zero dispersive stresses.

Figure 4.

Viscous, random and dispersive components of the total shear stress $\unicode[STIX]{x1D70F}_{12}=\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}\langle \bar{u}\rangle /\unicode[STIX]{x2202}y)-\langle \overline{u^{\prime \prime }v^{\prime \prime }}\rangle -\langle \tilde{u} \tilde{v}\rangle$ evaluated from the experimental measurement (a,c) runtime statistics of the numerical simulation. Corresponding statistical quantities computed from the velocity fields reconstructed with 6 POD modes are shows in (b,d).

Figure 4 demonstrates the contributions of the viscous, random and dispersive stresses to the total shear stress $\unicode[STIX]{x1D70F}_{12}$ in the experiment (a,b) and the numerical simulation (c,d). The profiles are normalized by the total shear stress extrapolated to the location of average surface height $\bar{k}=0.017\unicode[STIX]{x1D6FF}$ (in the experiment) and $\bar{k}=0.019\unicode[STIX]{x1D6FF}$ (in the DNS). Note, that the POD-filtered results as shown in figure 4(b,d) will be discussed in § 3.4, but appear next to figure 4(a,c) for optimal comparability.

The relative contributions of dispersive stress and turbulent stress to the total stress are very similar in both cases, which demonstrates that both the experiment and simulation generate very similar secondary flows. We find a much smaller contribution of the viscous stress to the total stress in the experiment; however, this difference is related to the difference in Reynolds number. At the larger experimental Reynolds number the region where the viscous stress prevails would have been much smaller and is not captured here.

It is evident that the dispersive stress is not only present in the vicinity of the surface roughness but also strongly augments the random stress in the region $0.1<y/\unicode[STIX]{x1D6FF}<0.7$ , where it constitutes up to $28\,\%$ of the total stress for numerical and experimental data. These findings differ from turbulent flows over irregular rough surfaces where large secondary motions do not occur. For irregular rough surfaces the dispersive stress is restricted to the vicinity of the rough surface and its maximum contribution to the total shear stress is significantly weaker, e.g. approximately $16\,\%$ – $17\,\%$ as reported by De Marchis, Napoli & Armenio (Reference De Marchis, Napoli and Armenio2010) or Forooghi et al. (Reference Forooghi, Stroh, Schlatter and Frohnapfel2018). Also for regular roughness structures without secondary motions, the reported dispersive stresses are significantly smaller (Coceal et al. Reference Coceal, Thomas, Castro and Belcher2006; Xie & Fuka Reference Xie and Fuka2018). The present results confirm the observation reported by Vermaas, Uijttewaal & Hoitink (Reference Vermaas, Uijttewaal and Hoitink2011) that the momentum exchange due to secondary flow can be the same order of magnitude as turbulent momentum exchange.

Figure 5.

Distribution of dispersive fluctuations $\widetilde{u_{i}}$ and their products $\widetilde{u_{i}}\widetilde{u_{j}}$ for experimental measurement and numerical simulation.

In order to visualize the origin of the dispersive stresses, maps of the dispersive fluctuations and their products are presented in figure 5. The fluctuating components $\tilde{v}$ and $\tilde{w}$ directly correspond to the time-averaged secondary motion indicated by the arrows in figure 3, while $\tilde{u}$ is related to the bulging of the mean streamwise velocity profile visible in the same figure. The spatially averaged covariance of the dispersive fluctuations gives the dispersive stresses from (3.1). The comparison between experimental and numerical data reveals good qualitative and quantitative agreement. The only visible difference is in the wall-normal location at which $\tilde{w}$ changes its sign. This is located around $y/\unicode[STIX]{x1D6FF}=0.4$ for the experiment whereas it is found below $y/\unicode[STIX]{x1D6FF}=0.3$ in the simulation data. From the DNS results it can be seen that the near-wall region in between the raised surface elements does not contribute substantially to the dispersive stress, which suggests that the large-scale secondary flows are the main source of the observed large contribution of the dispersive stress $-\langle \tilde{u} \tilde{v}\rangle$ to the total shear stress. The secondary flow basically generates large-scale regions with up-drafts of low-speed momentum and downwashes of high-speed momentum occurring above the peaks and valleys of the surface, respectively.

Figure 6.

Time-averaged products of random fluctuations for experimental measurement and numerical simulation.

Figure 6 shows the spatial distributions of the random stresses. For these data, we also see very good agreement between experimental and numerical results. Note that due to the nature of the applied averaging, the top row of figure 6 represents $\overline{u_{i}^{\prime \prime }u_{i}^{\prime \prime }}$ whereas the top row of figure 5 contains $\widetilde{u_{i}}$ . By definition $\overline{u_{i}^{\prime \prime }u_{i}^{\prime \prime }}$ can only assume positive values.

It can be seen that $\overline{v^{\prime \prime }v^{\prime \prime }}$ , and to a lesser extent also $\overline{w^{\prime \prime }w^{\prime \prime }}$ , is mainly concentrated in a large blob around the elevated region of the surface. This suggests that the up-draft of the secondary motion is accompanied by strong random fluctuations while the region of downwash appears to be much calmer. Hence, the secondary flow does not only generate significant dispersive stress, but also appears to reorganize the turbulent stress such that higher turbulence intensity is found above the elevated ridges. This will be further investigated in § 3.6.

With respect to the contributions to the total stress to $\unicode[STIX]{x1D70F}_{12}$ , it can be observed that in the outer flow region ( $y>0.15\unicode[STIX]{x1D6FF}$ ), $-\langle \overline{u^{\prime \prime }v^{\prime \prime }}\rangle$ is strongly governed by the enhanced wall-normal fluctuations above the elevated ridges, while the contribution in the near-wall region in between the elements seems to stem from the classical near-wall turbulence behaviour. In contrast, the main contribution to $-\langle \tilde{u} _{i}\tilde{u} _{j}\rangle$ is located just above the edges of the elevated ridges where strong upward motions occur simultaneously with low streamwise velocities. It is also worth noting the scales of each of these maps which again demonstrate that the random stresses are comparable in magnitude to that of the dispersive stresses.

3.4 Proper orthogonal decomposition

In order to identify the highest energy-containing flow structures and to gain insight into the impact of large-scale flow patterns on the overall secondary flows, the sets of instantaneous flow fields were decomposed with POD using snapshot POD as introduced by Sirovich (Reference Sirovich1987). Accordingly, the convolution of eigenvectors and underlying velocity basis, in contrast to the direct POD, provides the spatial velocity patterns, i.e. the POD modes. A straightforward implementation is described by Meyer, Pedersen & Özcan (Reference Meyer, Pedersen and Özcan2007), for instance.

The relative and cumulative energy distributions of the POD modes are presented in figure 7, which shows that roughly $20$ modes are required to capture $50\,\%$ of the turbulent kinetic energy of the flow. However, investigation of the corresponding flow patterns indicates that the first six modes are associated with large-scale ( $\unicode[STIX]{x1D6FF}$ -scale) motion. Therefore, a basis of these six characteristic patterns has been chosen to test the hypothesis that this class of patterns predominates the secondary motions.

Figure 7.

The relative and cumulative energy distributions across the modes.

These first six modes are presented in figure 8 for both the numerical and experimental data. In both cases, the data are cropped to a single period of the surface pattern. Both the experiment and the simulations reveal nearly identical flow patterns in these first few modes. The dominant modes (modes 1 and 2) are large vortical motions that fill the boundary layer depth and are essentially lateral shifts of the main secondary flow patterns. The later modes (3–6) divide the flow up into smaller vortices but still are quite active right out to the edge of the boundary layer. As the mode number increases further, the size of the vortices decreases, and the range of vortex scales corresponds with the cascade of scales in turbulence.

The reduced basis of six modes captures roughly one third of the turbulent kinetic energy (see figure 7). Interestingly, a reconstruction of the instantaneous snapshots using only the first six modes provides $66\,\%$ of the total magnitude of the total Reynolds stress $\overline{u^{\prime }v^{\prime }}$ , illustrating that these large-scale features are responsible for the majority of the Reynolds stress. We note that the dispersive contribution to the Reynolds shear stress is determined by the spanwise variation of the mean flow field, therefore it is not surprising that the POD-based model captures a large percentage of $\overline{u^{\prime }v^{\prime }}$ . The suitability of the reduced basis of models is further emphasized in figure 4, where the direct comparison of full and filtered data sets demonstrates that the dispersive stresses are preserved with the six-mode basis.

The fact that the simulations and the experiments produce similar modes despite the fact that the experimental data are missing at the wall indicates that the near-wall motions are not very important – at least in terms of energy contribution. Repeating the analysis using a wider domain confirmed that the largest structures are centred on the raised ridges of the surface, as shown in modes 1 and 2. The fact that they do not exceed the dimensions of the single period domain retroactively verified the choice of this particular domain, which consequently was used for all subsequent analyses.

Even though the shape of the energy distributions of the POD modes are consistent between the experiment and the simulation, there is a slight variation in the magnitudes of the lower modes, as can be identified from figure 7. This can be attributed to the different spatial resolution of the data as it is well established that a coarsely resolved data set will lead to a larger fraction of energy in lower modes (Placidi & Ganapathisubramani Reference Placidi and Ganapathisubramani2018) and the resolution of the present experiments is more coarse than the DNS.

Figure 8.

The highest energy modes present in the flow over the roughness.

Figure 9.

Complete instantaneous flow field with its POD-reconstructed counterpart (a,b) and the corresponding velocity fluctuations (c,d). In the upper row, the colour code corresponds to the streamwise velocity component and the arrows indicate the in-plane motion, whereas in the bottom row, the colour code represents the instantaneous deviation of the streamwise velocity component from $\langle \bar{u}\rangle (y)$ . A video of the temporal flow evolution based on the DNS data and the reduced-order reconstruction is available online in mp4 format on the Cambridge Core website.

While the POD modes do not represent actual realizations of the flow, they do indicate how the energy in the velocity fluctuations is distributed. For the present case, it has been shown that the highest energy flow patterns correspond to large $\unicode[STIX]{x1D6FF}$ -scale vortical motions. The fact that the dominant features of the secondary motions are captured by the six most energetic modes in turn means that it is possible to use a POD filter to focus the analysis of the instantaneous data on the large-scale motions. In the following sections, we use the results of the POD to reconstruct ‘filtered’ instantaneous velocity fields using the reduced basis of six modes. A snapshot of the reconstructed flow field in comparison to the full DNS velocity field is shown in figure 9. The lateral shift related to the first two modes of figure 8 is clearly visible in the corresponding movie. This large-scale lateral motion appears to be a characteristic dynamic behaviour of the secondary motion which is consistently observed in experiments and DNS.

In general, it can clearly be observed that the reduced-order reconstruction removes the effects of the finer fluctuations (which have relatively less energy) and maintains only the large-scale motions (which contribute the majority of the total Reynolds stress $\overline{u^{\prime }v^{\prime }}$ ). In the following we use the ‘filtered’ flow field in comparison with the full flow field in order to analyse whether characteristic instantaneous features of the secondary flow can be captured with the most energetic modes only.

3.5 Vortex identification

The time-averaged results in § 3.1 show that the time-averaged secondary flow consists of a large counter-rotating vortex pair, with many smaller vortices located near the wall. However, we would like to identify if the instantaneous fields also contain such large counter-rotating vortical structures or if they are simply the result of a inhomogeneous distribution of smaller structures. If the latter, what then are the characteristics of the instantaneous vortices and their distribution?

Figure 10.

Instantaneous structure of the flow visualized using isosurfaces of $\unicode[STIX]{x1D706}_{2}$ -criterion ( $\unicode[STIX]{x1D706}_{2}=-0.005$ ) and coloured by their rotational direction (video of the simulation visualization is available online in mp4 format on the Cambridge Core website).

Figure 10 (and the associated movie) show the instantaneous three-dimensional flow structure from the DNS. This looks qualitatively similar to what one would expect to see when visualizing a smooth turbulent boundary layer (e.g. Wu & Moin Reference Wu and Moin2009); however, in the present case, the flow structures seem to predominantly appear in the regions along the ridges, which is in agreement with the observed enhanced turbulence intensity in the region above the ridges. Similar observations in visualizations of the near-wall turbulent structures have been previously reported in flows with spanwise heterogeneity of boundary conditions resulting in formation of secondary flows (Benschop & Breugem Reference Benschop and Breugem2017; Ni et al. Reference Ni, Lu, Fang, Moulinec and Yao2018) or flows with directly induced large-scale secondary motion (Canton et al. Reference Canton, Örlü, Chin, Hutchins, Monty and Schlatter2016). Additionally, one can also observe a local clustering of the detected structures mainly located along the sides of the ridges. These small-scale vortex groups are rather compact, exhibiting a streamwise extent of approximately $2{-}3\unicode[STIX]{x1D6FF}$ and appear to be convected downstream along the ridges, slowly changing their composition and shape.

In order to further characterize the vortices we applied a vortex identification algorithm to identify the vortices in the instantaneous snapshots from the experiments and the simulations. The algorithm used peaks in swirling strength above a fixed threshold to identify the locations of vortices and then applied an area-growing algorithm to determine the extent of each of the vortices (Vanderwel & Tavoularis Reference Vanderwel and Tavoularis2011). The circulation of each vortex was then determined by integrating the vorticity over the area of the vortex. The swirling strength maps were determined for the full flow field and for the reduced-order instantaneous velocity maps reconstructed from six POD modes as outlined in the previous section.

Figure 11.

Flow diagram showing an example of using a reduced-order model to effectively filter the instantaneous snapshots while preserving the highest energy structures. (a,e) The original instantaneous velocity map. (b,f) The reconstructed map using only 6 modes. (c,d,g,h) The vortices identified from peaks in swirling strength.

Figure 11 provides an example of the process, where filtered and unfiltered velocity fields and the corresponding swirling strength of random snapshots from both data sets are shown. We found that the typical snapshot has over 100 discrete vortices identified but that only 2–3 of them are dominant, with total circulations two orders of magnitude larger than the rest. Furthermore, in 91 % of instances, the two largest vortices had opposite rotation, indicating the predominance of vortex pairs. However, the average diameter of these dominant vortices is considerably smaller than the time-average secondary vortices, as demonstrated in figure 11(c,g). The swirling strength distributions of the reduced basis emphasize the dominant (counter-rotating) vortices and eliminate the small-scale fluctuations (see figure 11 d,h). The validity of this elimination process is shown in figure 12, where scatter plots summarize all the locations of all the vortices and provide the resulting probability distribution of the locations of the vortices. Interestingly, at first glance, the unfiltered data indicate an even distribution of vortex locations across the domain (figure 12 a,e). The corresponding probability density functions (PDF) as plotted in figure 12(c,g), however, indicate that clockwise vortices tend to occur to the right of the elevated ridge, while counter-clockwise vortices occur to the left, consistent with the pattern of the secondary motions. The filtered data clearly emphasize this trend, since the randomly oriented small-scale fluctuations are eliminated from the scatter plots (figure 12 c,g), which in turn also results in more salient spatial separation of the counter-oriented events (see figure 12 d,h). Here, the immediate comparison of filtered and unfiltered PDFs particularly shows that the six-mode basis is sufficient to capture a major property of the instantaneous flow behaviour.

Overall, this leads us to conclude that, in the instantaneous flow fields, the secondary flows manifest as smaller yet stronger counter-rotating vortex pairs, that can appear in a range of orientations but average out to form these large secondary flow structures in the mean. These findings are consistent with the results of Kevin et al. (Reference Kevin, Monty, Bai, Pathikonda, Nugroho, Barros, Christensen and Hutchins2017), who investigated how the secondary flow patterns are organized in instantaneous snapshots of the flow over converging–diverging riblets. They also filtered their data and found that the diameters of the instantaneous motions were on average smaller than the time-averaged secondary flows and that the centre of these motions meandered in space. They also concluded that the large $\unicode[STIX]{x1D6FF}$ -scale secondary flows were ‘the superposition of stronger individual vortices’.

Figure 12.

Scatter plot of vortex locations and difference of PDF distribution between clockwise and counter-clockwise rotating vortices.

3.6 Instantaneous stresses

In a similar way to our analysis of the instantaneous vortex distribution, we can also investigate the instantaneous structure of the Reynolds random stress field. The time-averaged maps of Reynolds random stress $\overline{u^{\prime \prime }v^{\prime \prime }}$ shown in figure 6 indicate that a high magnitude of stress is concentrated above the elevated surfaces and extends beyond $y/\unicode[STIX]{x1D6FF}=0.5$ . By investigating the instantaneous maps of this quantity, we can assess whether this is the result of the superposition of many smaller ejection/sweep events or whether the instantaneous stress events also extend to most of the depth of the boundary layer.

Figure 13.

Instantaneous distribution of the Reynolds random stresses $u^{\prime \prime }v^{\prime \prime }$ , coloured by their quadrant based on the original velocity data and on the reduced-order filtered velocity data.

Figure 13 presents typical instantaneous maps of the Reynolds random stress $u^{\prime \prime }v^{\prime \prime }$ from both the experiment and the simulations. The unfiltered stress maps are shown alongside maps determined from the reduced-order instantaneous velocity maps reconstructed using six POD modes, in order to focus on the large-scale motions in the flow as was done when analysing the vortices. In this case, the POD filtering is crucial to identify the appropriate extent of the dominant structures, which would otherwise be obscured by the fine-scale turbulence. Events are categorized (and coloured) via quadrant analysis based on the sign of the local vertical and streamwise velocity fluctuations, such that the Q2 events ( $u^{\prime \prime }<0$ , $v^{\prime \prime }>0$ ) are associated with ejections and Q4 events ( $u^{\prime \prime }>0$ , $v^{\prime \prime }<0$ ) are associated with sweeps, following previous studies which also employed quadrant analysis (Finnigan Reference Finnigan1979; Antonia Reference Antonia1981; Vanderwel & Tavoularis Reference Vanderwel and Tavoularis2016).

We find that the magnitudes of the instantaneous stresses are similar to the magnitudes of the time-averaged Reynolds and dispersive stresses (i.e. $u^{\prime \prime }v^{\prime \prime }/U_{\infty }^{2}\sim 4\times 10^{-3}$ ). The Q2 and Q4 events are dominant and often appear to alternate along the span with a period similar to the roughness spacing. Between these events the vector maps show the signatures of vortices. The dominant instantaneous Q2 and Q4 events appear to have an average vertical extent of $0.4\unicode[STIX]{x1D6FF}$ , indicating that these important momentum transfer events extend over a large portion of the boundary layer thickness.

Q4 events (sweeps) appear to occur closer to the surface whereas Q2 events (ejections) appear to occur further away from the surface, which is consistent with the fact that sweeps tend to dominate momentum transfer near the wall for flows over rough surfaces (Raupach Reference Raupach1981). Following Raupach’s analysis, this difference is highlighted by plotting a map of the difference between the fractional stress contributions by sweeps and ejections as

(3.4) $$\begin{eqnarray}\unicode[STIX]{x0394}S=S_{4}-S_{2},\quad \text{where }S_{i}=\langle u^{\prime \prime }v^{\prime \prime }\rangle _{i}/\overline{u^{\prime \prime }v^{\prime \prime }}.\end{eqnarray}$$

Maps of $\unicode[STIX]{x0394}S$ are presented in figure 14 from the experimental and numerical data, with and without the POD filter. In all the cases, $\unicode[STIX]{x0394}S$ becomes positive with values of approximately 0.2–0.9 in the proximity of the elevated surface element, which is only slightly larger than the range of values between 0.1 and 0.5 observed by Raupach in a rough-wall turbulent boundary layer with $Re_{\unicode[STIX]{x1D70F}}\sim 400$ . The main difference observed between the present results and those from a rough boundary layer without secondary flows is that $\unicode[STIX]{x0394}S$ is not uniform across the span and the region of positive $\unicode[STIX]{x0394}S$ (where sweeps dominate) extends much farther into the outer boundary layer, due to the presence of the secondary flows. This extended region of positive $\unicode[STIX]{x0394}S$ coincides with the location of upwash in the mean velocity fields; so, although the mean dispersive velocity component is upwards, the dominant fluctuations are downwards sweeps. This makes sense as the largest stresses would occur when the velocity fluctuation is the most different from the mean, and this would be consistent with a lateral instability of the large-scale low-momentum regions, which was observed by Kevin et al. (Reference Kevin, Monty, Bai, Pathikonda, Nugroho, Barros, Christensen and Hutchins2017).

Figure 14.

Maps of the fractional difference between the stress contributions by sweeps and ejections, $\unicode[STIX]{x0394}S=S_{4}-S_{2}$ for full velocity fields and velocity field reconstructed with 6 POD modes.

It is interesting to see that this extended region of positive $\unicode[STIX]{x0394}S$ is least accentuated with the unfiltered DNS results and most accentuated when the six-mode POD reconstruction is applied. This suggests that the scale resolution of the data affects this property of the turbulent flow. The unfiltered DNS is effectively the most finely resolved data set capturing all the fine-scale structures, whereas the experimental data are effectively filtered at the PIV resolution of approximately 35 wall units, and of course the six-mode reconstructions isolate the contributions of the large-scale dominant flow structures. This leads to the conclusion that, away from the wall in the upwash created by the secondary flows, the large-scale motions comprise predominantly sweeps (which lead to positive $\unicode[STIX]{x0394}S$ ) whereas the smaller-scale motions appear to be mostly ejections (which, when included, tend to decrease $\unicode[STIX]{x0394}S$ ). In other words, when the data are limited to only the dominant larger-scale structures, the extended region of positive $\unicode[STIX]{x0394}S$ is enhanced, highlighting the inhomogeneity of the flow due to secondary flows.

This analysis demonstrates that the Reynolds random stress appears to be the superposition of strong individual stress events, that meander somewhat around the region of the elevated surfaces and often extend far into the outer boundary layer. These stress events appear superimposed on the mean dispersive secondary flow structure and can take the opposite direction of the mean flow. The spatial organization of the turbulent flow structures is attributed to the secondary flows that manifest in these flows, which leads to greater turbulent Reynolds stress in the regions above the elevated surfaces, composed of mainly Q4 events (sweeps) typical of near-wall behaviour in homogeneous rough walls.