3.1. Flow structures near the surface of the cube

The main vortical structures around one of the cubes for $\lambda =2.5a$ (widest gap) is visualized by an iso-surface of $\varLambda _2$ (Jeong & Hussain Reference Jeong and Hussain1995) in figure 3(a). Corresponding sample vortex lines with the vorticity magnitude colour coded are presented in figure 3(b). They are selected to intersect with the peak vorticity magnitude on the cube mid-$x$–$y$ plane above and behind the cube as well as with $y=0.12a$ far upstream. Distributions of streamlines in wall-normal, spanwise and axial planes coinciding with the middle of the cube are shown in figures 3(c), 3(d) and 3(e), respectively. Contour lines of $\varLambda _2$ at the same level as in figure 3(a) are also presented to facilitate a comparison between them. The vortex located upstream of the cube close to the channel wall is part of the horseshoe vortex. It forms as the boundary layer separates and rolls up (figure 3d), presumably owing to the local cube-induced adverse pressure gradient (Simpson Reference Simpson2001). As the horseshoe vortex wraps around the cube, its legs become aligned in the streamwise direction. In the inner side (the one facing the other cube), the swirling motion induced by the horseshoe leg can be seen at $z/a_{3z}=-0.75$ in figure 3(e). In the outer side, the leg signature appears as the curved streamlines centred around $z/a_{3z}=-3$. Along with these legs, there are a series of other streamwise structures whose origin and interactions among them are discussed later.

Figure 3. Characteristic primary flow structure around one of the cubes for $\lambda =2.5a$. (a) Iso-surface of $\varLambda _2=-$0.1$U^2_c/a^2$; and (b) vortex lines colour coded with the vorticity magnitude. (c–e) Selected in-plane streamlines: (c) $x$–$z$ plane at $y/a_{3y}=0.5$, (d) $x$–$y$ plane at $z/a_{3z}=-1.79$ and (e) $y$–$z$ plane at $x/a_{3x}=0.5$. Blue lines: contour of $\varLambda _2=-0.1U^2_c/a^2$ showing the intersections with the vortical canopy, arch-like and horseshoe vortices.

A vortex canopy covers the cube and the separated flow regions on its top and side surfaces. The vortex lines indicate that this canopy is dominated by wall-normal vorticity along the inner and outer sides, and by spanwise vorticity above the cube. Consistent with the orientation of the vortex lines, as quantified later, the magnitude of the vorticity associated with the canopy is substantially higher than that of the secondary streamwise structures. The arch-like vortex behind the cube (figure 3a), as referred to by Hussein & Martinuzzi (Reference Hussein and Martinuzzi1996), is located within the separated region there. Its vertical legs and spanwise top are also evident in the sample $x$–$z$ and $x$–$y$ planes, respectively. Note that the streamlines in figure 3(d) indicate that this separated zone is not a closed ‘bubble’, with the flow passing through its centre exiting above the arch-like vortex. As elucidated later, all the separated zones around the cube are open and involve complex 3-D flow structures. Even for the widest gap, the presence of the neighbouring cube causes significant flow asymmetry. Figure 3(c) demonstrates that the separation regions along the sidewall and behind the cube in the inner side are smaller than those at the outer side. Furthermore, figure 3(e) shows that the flow induced by the horseshoe vortex and other secondary streamwise structures are not symmetric. There is also a net spanwise flow towards the outer side over most of the area surrounding these vortices, which affects the interaction among them further downstream.

The extent of asymmetry induced by the neighbouring cube and its dependence on the spacing are illustrated in figure 4 by a series of $x$–$z$ sections showing the streamlines and distributions of $\varOmega _y$. Note that since the cube sizes and locations in these plots are matched, the displayed FOVs differ owing to the slight differences in cube size (table 1), magnification and location of the sample area for the three cases. Several trends are evident. First, at all elevations, the wall-normal vorticity components associated with the canopy and the arch-type vortex are parts of the same pair of continuous vorticial layers. They start near the front surface of the cube, have high peaks outside of the separated regions along its sides and extend well downstream of the arch, where they spread and decay. The magnitude of $\varOmega _y$ increases with distance from the channel floor, peaks at $y/a_{iy}=0.5\text {--}0.75$ and then decreases near the top of the cube. The trends at low $y/a_{iy}$ can be explained by near-wall interactions between the canopy and the secondary streamwise vortices (details follow). Near the top, the wall-normal vorticity transitions to a layer with high spanwise vorticity above the cubes (see figure 5). The flow at the outer side appears to be less affected by the cube spacing. Here, the length and width of the separated regions along the sidewall as well as distribution of $\varOmega _y$ appear to be similar for all cases. Near the inner sidewall, the separated region is thinner, and the width of the layer with high $\varOmega _y$ decreases with the spacing. Further details on the separated flows over the cube surfaces obtained using thinner interpolation volumes are discussed in the following section and presented in figure 6. The asymmetry is also evident in the relative size of the two legs of the arch-like vortex.

Figure 4. In-plane streamlines in a series of $x$–$z$ planes superimposed on colour contour of $\varOmega _ya/U_c$ for (a,d,g,j,m) $\lambda =1.0a$; (b,e,h,k,n) $\lambda =1.5a$; and (c,f,i,l,o) $\lambda =2.5a$. The planes are located at (a–c) $y/a=0.025$; (d–f) $y/a_{iy}=0.15$; (g–i) $y/a_{iy}=0.50$; (j–l) $y/a_{iy}=0.75$; (m–o) $y/a_{iy}=1.02$. The size of FOV in each column, represented by individual panel size, has been adjusted such that the cubes in each row are aligned and of the same size, to aid comparison among the three cases.

Figure 5. Streamlines in the $x$–$y$ mid-planes superimposed on the colour contour of (a,c,e) $\varOmega _za/U_c$ and (b,d, f) $U/U_c$. (a,b) $\lambda =1.0a$; (c,d) $\lambda =1.5a$; and (e, f) $\lambda =2.5a$. These mid-planes are located at (a,b) $z/a_{1z}=-0.96$; (c,d) $z/a_{2z}=-1.27$; (e, f) $z/a_{3z}=-1.79$. The size of FOV in each row, represented by individual panel size, has been adjusted such that the cubes in each column are aligned and of the same size, to aid comparison among the three cases.

The streamlines near the wall within the separated regions reveal a series of singular points. The separated flows in front of the cubes visible at $y/a=0.025$ and $y/a_{iy}=0.15$ (figure 4a–f) and in the $x$–$y$ sample data (figure 5) are associated with the horseshoe vortex rollup. The swirling flows and foci associated with the arch-like vortex behind the cube extend from the channel floor to approximately $y/a_{iy}=0.8$ (0.75 is shown), while decreasing in area. A series of saddle points appear on the boundaries and inside the separated regions in front of, behind and on top of the cube. Their locations along with that of the stagnation point on the front surface are listed in table 2. All the positions are presented in their scaled coordinates. The locations of the centres of relevant surfaces are also provided for convenience. As the spacing decreases, the saddle points in front of ($S^B_1$) and behind the cube ($S^B_2$ and $S^B_3$) shift towards the inner side, while those on the top ($S^T_1$ and $S^T_2$) shift towards the outer side. Here, superscripts $B$ and $T$ refer to the vicinity of the bottom and the top surface, respectively. These shifts are associated with changes to the mean flow direction. Near the bottom the incoming flow upstream of cube is deflected towards the outer side, presumably by cube-induced blockage, as is particularly evident for the narrowest spacing (figure 4a). It should be noted that saddle points $S^B_1$, $S^B_2$ and $S^B_3$ in the near-wall region have been seen in the DNS results for a single cube by Yakhot et al. (Reference Yakhot, Liu and Nikitin2006). The asymmetry is also noticeable in the near-wall streamlines along the outer perimeter to the sides and behind the cubes, e.g. in the converging streamlines which are associated with interactions of the horseshoe vortices with the outer flow. These patterns along the outer periphery behind a single cube have been reported by Martinuzzi & Tropea (Reference Martinuzzi and Tropea1993), based on experimental surface visualizations, and by Yakhot et al. (Reference Yakhot, Liu and Nikitin2006).

Table 2. The scaled 3-D coordinates ($x/a_{ix}$, $y/a_{iy}$, $z/a_{iz}$) of the front surface stagnation point (ST), the centres of the cube surfaces ($C_{s}^{F}$, $C_{s}^{I}$, $C_{s}^{O}$ and $C_{s}^{T}$), the singular points marked in figures 4–6 as well as the streamwise extent of the separation regions in front of ($X_{sep}^{F}$) and behind ($X_{sep}^{A}$) the cube. The superscripts $F$, $I$, $O$, $T$, $A$ and $B$ refer to the front, inner, outer and top surfaces as well as the arch vortex, and the vicinity of the bottom wall, respectively.

Spanwise vorticity and streamwise velocity distributions in $x$–$y$ cross-sections aligned with the mid-plane of the cube showing the horseshoe vortex, vortical canopy, and the arch-like vortex are presented in figure 5. Several trends are evident: (i) it appears that in this plane, the different gaps between the cubes have limited influence on the spanwise vorticity distribution; (ii) the spanwise vorticity has peaks in the frontal head of the horseshoe vortex, and at the top of the vortical canopy, the latter being consistent with the wall-normal vorticity along the sides of the cube. Further downstream, including in the separated region behind the cube, $\varOmega _z$ decreases and appears to be diffused over a broad area, also in accordance with the trends of $\varOmega _y$ in top views (figure 4); (iii) upstream, it appears that a substantial fraction of the boundary layer vorticity is entrained into the horseshoe vortex and the much thinner structure in front of it. The height of the latter is 6$\delta _{\nu \infty }$, i.e. slightly higher than the viscous sublayer far upstream. While the rollup of a horseshoe vortex ahead of obstacles has been seen before in numerous studies, including those involving large cubes (Hearst et al. Reference Hearst, Gomit and Ganapathisubramani2016; Diaz-Daniel et al. Reference Diaz-Daniel, Laizet and Vassilicos2017), the formation of a second structure farther upstream have mostly been seen under laminar upstream conditions (Diaz-Daniel et al. Reference Diaz-Daniel, Laizet and Vassilicos2017; Schröder et al. Reference Schröder, Willert, Schanz, Geisler, Jahn, Gallas and Leclaire2020). Considering the distance from the wall, laminar-like behaviour should not be surprising; (iv) only a small fraction of the upstream vorticity is located above the streamline leading to the stagnation point on the front surface, hence contributes to the vorticial canopy above the cube. The rapid increase in $\varOmega _z$ occurs close to the top front corner of the cube, suggesting that the origin of most of this vorticity, as quantified later, is located along the front surface; and (v) while the reverse flow inside the separated region above the cube can be as high as $-$0.15$U_c$, owing to blockage effect the velocity at $y\sim 1.3a$ is already 0.7$U_c$, i.e. the shear strain there is of the order of $10^4$ s$^{-1}$. This level is comparable to the wall shear strain rate far upstream ($U^2_{\tau \infty }/\nu$). In several places, such as under the horseshoe vortex, as well as the bottom parts of the forward face and behind the cube, the fast reverse flow, which can be as high as $-$0.2$U_c$, generates opposite sign vorticity. The locations of the marked singular points are included in table 2. The height of the stagnation point, ranging from 0.65 to 0.67 cube heights for different spacings, is in good agreement with those reported in Yakhot et al. (Reference Yakhot, Liu and Nikitin2006), Hearst et al. (Reference Hearst, Gomit and Ganapathisubramani2016) and Diaz-Daniel et al. (Reference Diaz-Daniel, Laizet and Vassilicos2017). The heights of the horseshoe head ($F^B_1$) and the top of the arch-like vortex ($F^A_3$) are similar (differences less than 0.05$a$) to those reported based on DNS in Diaz-Daniel et al. (Reference Diaz-Daniel, Laizet and Vassilicos2017).

Figure 6. The flow structure very close to the surfaces of the cube: streamlines in surface-parallel planes and velocity vectors in surface-normal mid-planes superimposed on colour contour of the streamwise velocity; (a–c) 15 $\mathrm {\mu }$m from the outer surface, (d–f) 15 $\mathrm {\mu }$m from the inner surface and (g–i) 20 $\mathrm {\mu }$m from the top surface. For (a,d,g) $\lambda =1.0a$, (b,e,h) $\lambda =1.5a$ and (c,f,i) $\lambda =2.5a$. The dash-dot lines correspond to $U=0$. The spanwise coordinates of the outer and inner surfaces ($z^{O}_i/a_{iz}$ and $z^{I}_i/a_{iz}$, respectively) are provided in table 2.

The next discussion focuses on the flow structure very near the cubes’ surfaces. As noted in § 2.2, to resolve these flows, the interpolation ellipsoid is 20 $\mathrm {\mu }$m thick in the surface-normal direction. Figure 6 provides two views for each surface and cube spacing. The wall-parallel views correspond to planes located 15 $\mathrm {\mu }$m away from sidewalls and 20 $\mathrm {\mu }$m above the top surface. The wall-normal views show the mid-$x$–$z$ planes for the sidewalls (i.e. $y/a_{iy}=0.5$) and $x$–$y$ plane crossing the centre of the cubes for the top. Each wall-parallel streamline pattern contains at least one, but in most cases four singular points whose locations are also listed in table 2. The separation lines in the forward parts, where streamlines converge from both sides, have a saddle point in the middle. The latter is labelled as $S^O_1$, $S^I_1$ and $S^T_1$, where the superscripts $O$, $I$ and $T$ refer to the outer, inner and top surfaces, respectively. Attachment lines, where the streamlines diverge, with a saddle point in their centres ($S^O_2$, $S^I_2$ and $S^T_2$) are also evident further downstream. The flow patterns along the outer (figure 6a–c) and inner side (figure 6d–f) appear to be qualitatively similar (except for the narrowest spacing), each featuring a major focus close to the top ($F^O_1$, $F^I_1$) and a minor one near the bottom ($F^O_2$, $F^I_2$), in addition to the separation and attachment lines. However, in the inner side, the foci are sources, i.e. the radial velocity component near the centre is positive, and in the outer side, the foci are sinks. The top surfaces have separation lines bounded in both sides by sink-like foci ($F^T_1$ and $F^T_2$), and what appears to be curved attachment lines that start parallel to sidewalls and end close to the back of the cubes. In all cases and surfaces the separated regions do not form closed ‘bubbles’, i.e. part of the flow circumvents the separation lines and penetrates the separated zones somewhere along the forward side. This observation implies that outlets should also exist somewhere, presumably out of plane. For example, along the inner sides, the flow enters the separated regions from the upstream bottom corner. On the top, the upstream flows penetrate from both sides between the foci and the peripheral attachment line.

To elucidate the complex phenomena involved, figure 7 presents selected 3-D streamlines, with their distance from the relevant surfaces colour coded. Starting from the inner wall (figure 7a,d,g), streamlines originating from the forward surface, turn around the bottom corner and enter the domain located downstream of the separation line. They then turn upward (owing to the flow induced by the streamwise vortices) and subsequently upstream near the middle of the surface, as part of the reverse flow seen in the planar views. These streamlines then spiral away from the wall around the centre of the focus located close to the top of the cube in the planar view, and then leave the separated zones close to the top surface. Other streamlines (not shown) spiral around the bottom focus. Along the outer side (figure 7b,e,h), the trajectories have similar general features although magnitudes and locations differ. One of the selected sample streamlines enters the separated region near the bottom, spirals around the bottom focus away from the wall and then get entrained into the separated zone behind the cube, which is also open. The other enters the vicinity of the surface from the downstream end of the cube, and then circumvent the upper focus before turning downstream. On the top of the cube (figure 7c,f,i), one of the shown streamlines enters the separated region from the side, rotates around the focus, lifts upward upon reaching the front separation line and then turns downstream. The other streamline has a 3-D spiralling trajectory that starts near the focus. These trajectories elucidate the flow phenomena that generate the complex streamline patterns observed in the wall-parallel planes. Subsequent discussions on the evolution of streamwise structures will elucidated some of the causes for these phenomena. Finally, the corresponding surface-normal planes demonstrate the width of the open separated regions, which increase from 20 $\mathrm {\mu }$m to 70 $\mathrm {\mu }$m with increasing gap for the inner surface but remain at approximately 120 $\mathrm {\mu }$m for the mid-outer and mid-top planes.

Figure 7. Selected 3-D streamlines, which are entrained into the open-type separation regions for $\lambda =2.5a$ along the cube's (a,d,g) inner side, (b,e,h) outer side (note the change in orientation) and (c,f,i) top. For (a–c) 3-D perspectives looking at the surface, (d–f) spanwise view and (g–i) top view. The streamlines are colour coded with the distance from the relevant surface.

As mentioned before and demonstrated by selected streamlines presented in figure 8, the separated region behind the cube is also open. The flow enters this region from the lower sides, spirals around the centre of the arch-like vortex while changing its orientation from $x$–$z$ to $x$–$y$ planes and then leaves at an elevation of about the cube height. Accordingly, the $x$–$z$ planar views for low elevations (figure 4a–f) show streamlines from both sides of the cube being entrained into the arch legs. Such entrainment does not occur at higher elevations. Similarly, the $x$–$y$ planes (figure 5a,c,e) show the flow leaving the separated region above the top of the arch. Before concluding this section, it should be noted that occurrence of open separation on all the surfaces is consistent with topologically based kinematic theorems by Hunt et al. (Reference Hunt, Abell, Peterka and Woo1978). They argue that separation regions formed around any 3-D surface-mounted obstacle must be open.

Figure 8. Selected 3-D streamlines, which are entrained into the open-type separation region behind the cube for the $\lambda =2.5a$. (a) A 3-D perspective, (b) top view and (c) spanwise view. The streamlines are colour coded with the distance from the bottom wall.

3.2. Evolution and impact of streamwise vortices

The next discussion follows the evolution and interactions of streamwise vortices. A series of $y$–$z$ sections showing the distributions of $\varOmega _x$ and vectors of ($V$, $W$) aimed at elucidating the evolution of these structures around and behind the cube are presented in figures 9 and 10, respectively. We have also tried to present the evolution of streamwise structures in terms of swirling strength (not shown), but trends appear to be quite similar to those depicted using the vorticity. As is evident, the cubes are surrounded by multiple vortices, which are labelled for clarity as $A$–$D$ with superscripts $I$ and $O$ referring to the inner and outer surfaces, respectively (figure 9l). The counter-rotating legs of the horseshoe vortex that rolls up upstream of the cube (focus $F^B_1$ in figure 5a) and persist, at least for a while downstream of it, are labelled as $A^{I}$ and $A^{O}$. The asymmetry of this legs increases with decreasing cube spacing, with the inner leg being located closer to the side surface and having a higher peak vorticity magnitude than those of the outer leg. Consequently, in the narrowest case, the horseshoe vortex leg associated with the neighbouring cube is also visible in the left-most corner of the plots. Upstream of the front surface, the horseshoe vortices and thin layers with counter-rotating vorticity under them are the only visible structures. As the flow impinges on the front surface, it creates a source-like radial outflow pattern originating from the central stagnation point. The spanwise location of this point is biased towards the inner side and its elevation is approximately 2/3 of the cube's height. Both coordinates are not affected significantly by the spacing (figure 9(d–f) and point ST in table 2). Early signs of counter-rotating secondary vortices, which form between the horseshoe legs and the sidewalls ($B^{I}$ and $B^{O}$), appear at $x/a_{ix}=-0.06$, become distinct downstream the leading edge (figure 9g–i) and reach maximum size at $x/a_{ix}=0.24$. With decreasing cube spacing, their peak vorticity magnitude and area (hence presumably their strength) decrease along the inner sides. Additional counter-rotating pairs ($C^{I}$/$D^{I}$ and $C^{O}$/$D^{O}$) appear near the top corners immediately downstream of the leading edge (figure 9g–i). The origin and mechanisms affecting the generation and evolution of each of these structures is discussed in the following section.

Figure 9. A series of $y$–$z$ planes showing the distributions of $\varOmega _xa/U_c$ superimposed on $V$–$W$ vectors for (a,d,g,j,m,p,s) $\lambda =1.0a$, (b,e,h,k,n,q,t) $\lambda =1.5a$ and (c,f,i,l,o,r,u) $\lambda =2.5a$. The plane locations are $x/a_{ix}=$ (a–c) $-$0.24, (d–f) $-$0.06, (g–i) 0.04, (j–l) 0.24, (m–o) 0.48, (p–r) 0.90 and (s–u) 1.08. The increment between contour lines is 0.2. The 0-contour line is omitted for clarity. The size of FOV in each column, represented by individual panel size, has been adjusted such that the cubes in each row are aligned and of the same size, to aid comparison among the three cases.

Figure 10. A series of $y$–$z$ planes showing the distributions of $\varOmega _xa/U_c$ and in-plane velocity vectors highlighting the evolution of streamwise vortices behind the cubes for (a,d,g,j,m,p) $\lambda =1.0a$, (b,e,h,k,n,q) $\lambda =1.5a$ and (c,f,i,l,o,r) $\lambda =2.5a$. The planes are located at $x/a_{ix}=$ (a–c) 1.38, (d–f) 1.74, (g–i) 2.16, (j–l) 2.58 and (m–o) 3.78. Increment between contour lines is 0.1. The 0-contour line is omitted for clarity. (p–r) Corresponding smoothed iso-surfaces of $\varOmega _xa/U_c=\pm 0.16$ between and downstream of the cubes. The size of FOV in each column, represented by individual panel size, has been adjusted such that the cubes in each row are aligned and of the same size, to aid comparison among the three cases.

Around $x/a_{ix}=0.5$, in all cases, the pairs of vortices around the top edges start shifting towards the outer side with the mean lateral flow. In the process, vortex $C^{I}$ migrates to the top of the cube (figure 9m–r), and $B^{I}$ to the upper inner corner. Furthermore, vortices $D^{I}$ and $C^{O}$ diffuse (expand and become weaker), and vortex $D^{O}$ shifts to the outer top corner and starts merging with $B^{O}$ while pushing $C^{O}$ away from the surface. The merging process is particularly evident at $x/a_{ix}=1.08$, where the combined structure is labelled as $E^{O}$. An additional pair of vortices ($G^{I}$ and $G^{O}$) forms near the bottom wall within the separated region. Subsequent developments behind the cube are illustrated in figure 10 (note the difference in colour scale). While still visible at $x/a_{ix}=1.38$, the positive vortices $B^{I}$, $D^{I}$ and $C^{O}$ disappear, and $G^{I}$ diminishes by $x/a_{ix}$=1.74. The negative vortices, namely $C^{I}$ and $E^{O}$ ($B^{O}$ and $D^{O}$) roll around each other and merge to form a larger structure ($H$, figure 10d–f), which is centred behind the cube. This vortex entrains part of the positive vorticity originating from the outer leg of the horseshoe vortex ($A^{O}$) towards the bottom. The extent of interactions with the inner leg depends on the cube spacing. For $\lambda =1.0a$, the inner leg ($A^{I}$) is already merged with $H$ at $x/a_{ix}=2.16$ to create a large streamwise vortex that occupies most of the space behind the cube (figure 10g). At $x/a_{ix}$=2.58, the shown FOV covers both cubes (FOV 1 in figure 2) to demonstrate that a mirror image of this process evolves around the neighbouring cube as well with the vortex rotating in the opposite direction. The resulting 3-D wake structure is also depicted in figure 10(p) using iso-surfaces of $\varOmega _x$. For $\lambda =1.5a$, the merging with $A^{I}$ only begins at $x/a_{ix}\sim 2.16$ and is almost completed at $x/a_{ix}=3.78$ (figures 10(n) and 10(q)). For $\lambda =2.5a$ one can still find a distinct inner horseshoe vortex legs at $x/a_{ix}=3.78$, although there are signs of interaction with other structures at this location. Note that the direction of the large streamwise vortex behind each cube is the same as that of the inner leg of the horseshoe vortex. The vorticity from the other/outer legs is entrained into narrow near-bottom layers. Interestingly, if the two cubes are replaced with a single obstacle with a similar total width, the near wake would have a pair of horseshoe legs rotating in the opposite direction. Possible reasons for the dominance of one of the vorticity components (negative in figures 9 and 10) might be related to asymmetry in the peak vorticity of the horseshoe legs, with that of the inner one being higher. Specific reasons for this asymmetry, involving realignment and stretching of vortices, is discussed in the following section. Another source of asymmetry involves the stronger positive wall-normal flow between the cubes, which initiates the migration of vortex $C^{I}$ around the top corner. Between the large vortices dominating the near wake, the induced flow points away from the wall. As discussed later, this induced flow affects the distributions of wall shear stresses and streamwise velocity between the cubes.

3.3. Origin and evolution of vortical structures

This section identifies the origin of the vortical structures on the various surfaces along with key phenomena affecting their evolution by stretching and turning. The analysis focuses on the role of the mean flow features, deferring discussions on the impact of the turbulent terms, such as gradients of vorticity–velocity correlations, to subsequent papers. As discussed before, the near-surface velocity is resolved using first-order SVD involving an interpolation volume shaped as an oblate ellipsoid with a base diameter of 200 $\mathrm {\mu }$m in the surface-parallel directions, and a surface-normal size of 20 $\mathrm {\mu }$m. The grid spacing is 10 and 60 $\mathrm {\mu }$m in the surface-normal and -parallel directions, respectively. The corresponding velocity gradients as well as the vorticity and wall shear stress components are calculated by re-applying SVD with the same oblate interpolation volume but using the structured velocity distributions. Application of SVD here serves as a convenient low-pass filter. The figures correspond to the intermediate spacing ($\lambda =1.5a$), but the discussions include comments on the effects of spacing with reference to results presented before.

The following discussion shows that most of the vortical structures discussed before have origins either upstream of the cube or on its front surface. Figure 5 already shows that head of the horseshoe vortex upstream of the cube contains rolled-up boundary layer vorticity. The next question involves the origin of the vortical canopy engulfing the cube. The vorticity distributions along with the streamlines 30 $\mathrm {\mu }$m upstream of the front surface are presented figure 11(a–c). As is evident, the radial flow away from the stagnation point generates $\varOmega _y>0$ on the inner half and $\varOmega _y<0$ on the outer half of the surface, as well as $\varOmega _z>0$ below the stagnation point and $\varOmega _z<0$ above it. The vorticity signs and directions are consistent with those of the canopy (figures 4 and 5). In contrast, and as one would expect, $\varOmega _x$ is very small. We have also tried to calculate the surface-normal gradients of $\varOmega _y$ and $\varOmega _z$ to estimate the vorticity diffusion from the wall. While the results give the correct signs of vorticity flux, questions about the uncertainty in the vorticity gradients compared to their magnitude have led to a decision not to present them. Unlike the front of the cube, the side and top surfaces have little impact on the vortical canopy, and in fact, figures 4 and 5 show that the canopy is separated from these surfaces. As for the streamwise vortices, they are not generated by viscous diffusion, but rather by realignment of $\varOmega _y$ and $\varOmega _z$ along the edges of the front surface. This process is demonstrated in figure 11(d–f), which shows the distributions of vortex straining terms that affect $\varOmega _x$ in the vorticity transport equations. Summing $\varOmega _y\partial U/\partial y$ (figure 11e), $\varOmega _z\partial U/\partial z$ (figure 11f) and the streamwise straining term, $\varOmega _x\partial U/\partial x$ (which is negligible here), results in a series of peaks with alternating signs distributed along the edges of the front surface (figure 11d). The signs and location of these peaks are consistent with those of vortices $B^{I}$, $C^{I}$, $D^{I}$, $D^{O}$, $C^{O}$ and $B^{O}$ evident in figure 9(g–l). This agreement confirms that, except for the horseshoe legs, the rest of the streamwise structures originate from the realignment of $\varOmega _y$ and $\varOmega _z$ at the edges of the front surface. Specifically, the upper pairs ($C$ and $D$) are results of opposing effects with spatially varying magnitudes of $\varOmega _y$ and $\varOmega _z$, while both contribute constructively to the near-bottom secondary vortex ($B$).

Figure 11. The structure of vorticity at $x/a_{2x}=-0.03$ for $\lambda =1.5a$. (a–c) Colour contours of (a) $\varOmega _x$, (b) $\varOmega _y$ and (c) $\varOmega _z$ superimposed on the in-plane streamlines. (d–f) Colour contours of vortex straining terms affecting the streamwise vorticity: (d) $\Omega_x \partial U/\partial x + \Omega_y \partial U/ \partial y + \Omega_z \partial U/ \partial z$, (e) $\Omega_y \partial U/ \partial y$ and ( f) $\Omega_z \partial U/ \partial z$.

Vortex stretching dominates in the evolution of the vortical canopy. A sample comparison between the stretching term for $\varOmega _z$ ($\varOmega _z\partial W/\partial z$) and the sum of all the spanwise vorticity straining terms ($\varOmega _j\partial W/\partial x_j$) in the cube mid-plane is presented in figure 12. There is little difference between them, with both showing that the spanwise stretching peaks at the head of the horseshoe vortex and above the forward part of the cube, both consistent with the location of the $\varOmega _z$ maxima (figure 5). In both cases, the spanwise stretching is associated with the flow circumventing an obstacle, namely the bottom of the cube near the horseshoe head (figures 4 and 9a–f), and the separation line above the cube (figures 6(g–i) and 9(g-i)). Further downstream the magnitude of $\varOmega _z$ above the cube is reduced by flow contraction that continues up to $x/a_{2x}=2.5$. Phenomena affecting the evolution of $\varOmega _y$ along the sides are demonstrated in figures 13(a) and 13(b). After being produced on the front surface, $\varOmega _y$ is enhanced by vertical stretching as the flow circumvents the forward part of the cube (figure 9g–l) and the separated regions along the side surfaces (figure 6a–f). This stretching occurs along most of the sides (figure 13c), but not at their bottom corners, where the flow contracts vertically under the influence of the axial secondary vortices $B^{I}$ and $B^{O}$. The resulting reduction in the magnitude of $\varOmega _y$ near the bottom corners is evident in figure 4(a–f). With decreasing spacing, the area with vertical stretching weakens along the inner side (not shown), hence the thickness of the canopy along this surface decreases (figure 4g–i). Near the trailing edges and behind the cube, $\varOmega _y\partial V/\partial y$ changes sign owing to vertical contraction induced by several effects, most prominently, the flow induced by vortices $B^{I}$ and $B^{O}$ and other structures that they interact with (figure 9m–r). In the near field behind the cube, the vertical contraction is associated with an upward flow induced by the arch vortex and downward flow behind the cube (figure 9s–u). Note that this discussion focuses on the mean flow effects, but the evolution of vorticity is also affected by turbulent dispersion, which is beyond the present scope.

Figure 12. The $x$–$y$ mid-plane for $\lambda =1.5a$, showing colour contours of vortex straining terms affecting $\varOmega _z$: (a) $\varOmega _x {\partial W}/{\partial x}+\varOmega _y{\partial W}/{\partial y}+\varOmega _z {\partial W}/{\partial z}$ and (b) $\varOmega _z {\partial W}/{\partial z}$, both superimposed on contour lines of $\varOmega _z$. Incremental difference between lines is $\varOmega _za/U_c$=0.5.

Figure 13. The $x$–$z$ mid-plane for $\lambda =1.5a$, showing colour contours of vortex straining terms affecting $\varOmega _y$: (a) $\varOmega _x {\partial V}/{\partial x}+\varOmega _y {\partial V}/{\partial y}+\varOmega _z {\partial V}/{\partial z}$ and (b) $\varOmega _y {\partial V}/{\partial y}$ both superimposed on contour lines of $\varOmega _y$. Incremental difference between lines is $\varOmega _ya/U_c=0.5$; solid lines: $\varOmega _y>0$; and dashed lines: $\varOmega _y<0$. (c) Contours of $\varOmega _y{\partial V}/{\partial y}$ superimposed on in-plane velocity vectors in a $y$–$z$ plane located at $x/a_{2x}=0.24$.

Finally, mechanisms affecting the evolution of horseshoe legs and the secondary structures between them and the sidewall are highlighted in figure 14. The discussion focuses on the $x$–$z$ plane that intersects the centre of the horseshoe legs, i.e. the magnitude of $\varOmega _x$ there is maximum (figure 14a). The sum of straining terms affecting $\varOmega _x$ (figure 14b), and the distributions of the individual terms (figure 14c–e) demonstrate, as expected, that the legs of the horseshoe vortex originate from realignment of $\varOmega _z$ in front of the cube (figure 14e). Subsequently, around the corner and forward part of the sides, $\varOmega _x$ is enhanced by axial stretching (figure 14c). The maximum of $\varOmega _x$ coincides with these regions, with the inner leg subjected to higher straining, hence have a higher maximum vorticity compared to the outer leg. This latter trend increases with decreasing cube spacing (figure 9j–l). Further downstream, the total straining term along the paths of the legs changes sign, consistent with the reduction in vorticity there, but its magnitude is smaller than that near the leading edge. As discussed before, in the aft part and downstream of the cube, the legs interact with other axial vortices. The outer leg is entrained under the large structure forming behind the cube, which, depending on spacing, entrains the inner leg (figure 10d–i). As for the secondary structures ($B^{I}$ and $B^{O}$), as discussed before, they are originated from realignment of $\varOmega _y$ and $\varOmega _z$ at the bottom corners of the front surface (figure 11d–f). Consistent effects are also evident from figures 14(d) and 14(e). Along the sides, $\varOmega _z\partial U/\partial z$ continues to contribute constructively to the axial vorticity, but $\varOmega _y\partial U/\partial y$ changes sign, and together with streamwise contraction (figure 14c) reduces $\varOmega _x$. Overall, straining increases $\varOmega _x$ magnitude in the $B^{I}$ and $B^{O}$ structures in the forward part of the cube, and decreases it in the aft part. These trends are consistent with those observed in figure 9. To elucidate the contribution of $\varOmega _z$ in this region, note that a layer with $\varOmega _z>0$ forms on both sides of the cube (see line contours in figure 14e) after being fed from the lower part of the front surface (figure 11c) and enhanced by spanwise stretching and realignment of $\varOmega _y$.

Figure 14. The structure of streamwise vorticity in the $x$–$z$ plane located at $y/a_{2y}=0.18$ for $\lambda =1.5a$: (a) colour contour of $\varOmega _x$ superimposed on the in-plane streamlines; and (b–e) the corresponding colour contours of vortex straining terms affecting $\varOmega _x$: (b) $\varOmega _x{\partial U}/{\partial x}+\varOmega _y{\partial U}/{\partial y}+\varOmega _z{\partial U}/{\partial z}$ superimposed on contour line of $\varOmega _x$; (c) $\varOmega _x{\partial U}/{\partial x}$ superimposed on contour line of $\varOmega _x$; (d) $\varOmega _y{\partial U}/{\partial y}$ superimposed on contour line of $\varOmega _y$; and (e) $\varOmega _z{\partial U}/{\partial z}$ superimposed on contour line of $\varOmega _z$.

3.4. Wall shear stress distributions

The flow structure around the cubes causes significant variations in the wall shear stress distributions, as depicted in figures 15 and 16. The results are normalized by the wall shear sufficiently far upstream, $\rho U^2_{\tau \infty }$. Focusing first on the vicinity of the cube (figure 15), the streamwise shear stress, $\tau ^+_{xy}$, peaks along the sides of the cube, in the space between the horseshoe legs and vortices $B^{I}$ and $B^{O}$, which are marked by red and blue lines, respectively. The downward flow induced by these vortices brings high-momentum fluid towards the wall, increasing the wall-normal velocity gradients there. The peak magnitude is 2.4 in the outer side and increases from 2.1 to 2.7 with increasing spacing on the inner side. In front of the cube, $\tau ^+_{xy}<0$, and its magnitude peaks under the horseshoe heads, which are marked by black lines. Another region with $\tau ^+_{xy}<0$ is located behind the cube, peaking in magnitude under the central part of the arch-like vortex. The extents of the separation regions upstream and downstream of the cube, which are defined by the lines where $\tau ^+_{xy}=0$ and marked as $X^F_{sep}$ and $X^A_{sep}$ in figure 15(b), are listed in table 2. In front of the cube, the length of the separation region is about 1.0$a$, close to that measured in Martinuzzi & Tropea (Reference Martinuzzi and Tropea1993), but smaller than the 1.2$a$ and 1.4$a$ reported by Yakhot et al. (Reference Yakhot, Liu and Nikitin2006) and Diaz-Daniel et al. (Reference Diaz-Daniel, Laizet and Vassilicos2017), respectively, for larger cubes. Behind the cube, the separation region extends to approximately 1.35 cube heights downstream from the back surface. This value is slightly smaller than the 1.4a–1.5$a$ range found in Martinuzzi & Tropea (Reference Martinuzzi and Tropea1993), Yakhot et al. (Reference Yakhot, Liu and Nikitin2006), Hearst et al. (Reference Hearst, Gomit and Ganapathisubramani2016), Diaz-Daniel et al. (Reference Diaz-Daniel, Laizet and Vassilicos2017) and Schröder et al. (Reference Schröder, Willert, Schanz, Geisler, Jahn, Gallas and Leclaire2020). This discrepancy might be attributable to the differences in $h/a$, presumably owing to the lower momentum, but higher turbulence levels in the region where the present cube is located. The magnitude of $\tau ^+_{yz}$ (figure 15d–f) peaks between the region where the horseshoe legs turn downstream and the front corners but remains high under the legs along the sides of the cubes. The peak magnitudes on both sides appear to increase with cube spacing, with the inner values being lower than the outer ones. Behind the cube the distributions of $\tau ^+_{yz}$ have local maxima downstream of the foci corresponding to the arch-like vortex near the wall (figure 4a–c). Elevated $\tau ^+_{yz}$ can also be seen under the large streamwise vortex forming behind the cube, especially for $\lambda =1.0a$, where it develops closer to the back surface (figure 10g–i).

Figure 15. Colour contours and contour lines of the distributions of (a–c) $\tau ^+_{xy}$, and (d–f) $\tau ^+_{yz}$ on the bottom wall. For (a,d) $\lambda =1.0a$, (b,e) $\lambda =1.5a$ and (c, f) $\lambda =2.5a$. Increment between lines is 0.3. Thick lines show the locations of maximum vorticity magnitude for black – $\varOmega _z$ in the horseshoe head, red – $\varOmega _x$ of the horseshoe legs and blue – $\varOmega _x$ of secondary streamwise vortices $B^{I}$ and $B^{O}$ in figure 9.

Figure 16. Colour contours and contour lines of $\tau ^+_{xy}$ on the bottom wall: (a–c) upstream, and (d–f) in and downstream of the gap between the cubes. For (a,d) $\lambda =1.0a$, (b,e) $\lambda =1.5a$ and (c, f) $\lambda =2.5a$. Increment between lines is 0.3. Purple line in (d) is the location of maximum magnitude of $\varOmega _x$ in the merged large pair of streamwise vortices.

The streamwise shear stress distributions further upstream and downstream of the cubes are plotted in panels (a,b,c) and (d,e, f) of figure 16, respectively, showing data obtained from different fields of view. In some cases, the magnitudes of the peaks differ by approximately 10 %, which, as discussed in the Appendix, is consistent with the uncertainty in the wall shear stress measurements. In general, the shear stress distribution around both cubes appear to be quite similar (within the uncertainty limit), confirming the symmetric alignment of the cubes relative to the flow. As far as 3$a$ upstream of the cubes, $\tau ^+_{xy}$ is already significantly smaller than 1, implying the cube-induced adverse pressure gradients already influence the flow there. This observation is also demonstrated by the mean velocity profiles at $x=-3a$ for two spanwise locations and all the spacings, which are presented in figure 17(a). There is already a momentum deficit in the viscous sublayer and the covered range of the log layer compared to the fully developed turbulent channel flow measured in the same facility and location without the roughness cubes (Zhang et al. Reference Zhang, Wang, Blake and Katz2017). The corresponding wall-normal velocity profiles presented in figure 17(b) clearly show that $\partial V/\partial y>0$ for all cases, implying that $\partial U/\partial x<0$, i.e. the flow is being slowed down by the adverse pressure gradients. The magnitude of $\partial W/\partial z$ in this region is negligible. Linear fitting to the local $\partial U/\partial x$ and the magnitude of the velocity deficit suggest that the cube effects could extend to about $x=-5a$. The magnitude of $\partial V/\partial y$ decreases with increasing cube spacing, both at $z=0$, i.e. the line of symmetry between cubes, and at the spanwise location of the cube centre. Following the streamwise development of the wall stress along $z=0$, $\tau ^+_{xy}$ decreases upstream the cube, then increase rapidly between the cubes, and then decreases again further downstream. The variations in stress increase with decreasing spacing, characterized by the minimum ($\tau ^+_{xy}\sim 0.2$) at $x/a\sim -1$, the maximum at $x/a=0.5$ associated with flow channelling and streamwise vortices, and the broad area with $\tau ^+_{xy}<0.3$ at $x/a>2.3$. The latter is associated with the positive wall-normal flow induced by the large pair of merged streamwise vortices that form behind the cubes (figure 10). It should be noted that the instantaneous realizations (not shown) indicate that the near-wall flow direction in this area becomes negative intermittently.

Figure 17. Profiles of (a) $U^+$, and (b) $V^+$ at $x=-3a$, both at $z=0$ and in spanwise planes aligned with the cube centre.

It would be of interest to demonstrate how the evolution of shear stresses is related to the development of the entire mean velocity profiles along the $z=0$ line and the effect of spacing on them. The streamwise and wall-normal velocity profiles are presented in figures 18(a) and 18(b), respectively. In each series, the first row provides data for $x\leq 0$, and the second row, for $x\geq 0$. Starting from $x/a=-2.52$, there is already an increasing streamwise momentum deficit with decreasing spacing. The difference between profiles peaks at $x/a=-0.54$, at about the same planes of the horseshoe head, and then starts decreasing as the flow starts accelerating into the gap between the cubes. There, the very near-wall flow ($y/a<0.1$) accelerates, but a spacing-dependent momentum deficit develops at $0.1<y/a<0.6$. Both phenomena are associated with the horseshoe legs and to some extent, the secondary axial vortex $B^{I}$. At $y/a<0.1$, high-momentum flow is driven towards the wall between these vortices, as discussed before, and some of it makes it to the other side of the horseshoe leg under this vortex (figure 9j–r). This effect is intensified for the narrow spacing owing to the close proximity of both inner legs. At $0.1<y/a<0.6$, the horseshoe legs induce a positive wall-normal velocity along the centreline, which drives low-momentum flow away from the wall. At $x/a>1$, the persistent decrease in streamwise momentum is driven initially by the horseshoe legs, and then further downstream by the large vortical structures developing behind the cubes (figure 10g–o). As one would expect, the momentum deficit increases with decreasing spacing. All the vertical velocity profiles (figure 18b) have positive slopes at $x/a\leq -0.54$ that increase with decreasing spacing, owing to the adverse pressure gradients there. At $-0.54<x/a<1.5$, the near-wall profiles have maxima induced by the horseshoe legs. Further downstream, the maxima shift upward under the influence of the large scale merged streamwise vortices.

Figure 18. Evolution of (a) $U/U_c$, and (b) $V/U_c$ profiles along $z=0$.

3.5. Gap-dependent blockage and channelling

The blockage to the channel, the flow channelling between cubes and the asymmetry in the flow structure around each cube can be evaluated by analysing the evolution of bulk velocity, $U_b$. It is defined as the streamwise velocity averaged over an area with height $a_{iy}$, and width $L$ (figure 19a) upstream and downstream of the cubes, and $L-a_{iz}/2$ at $0\leq x/a\leq 1$. For $\lambda _i=1.0a$ and 1.5$a$, $L=(\lambda _i+a_{iz})/2$, i.e. the sample area extends from the centre of the cube to $z$=0 on the inner side, and by the same distance on the outer side. For $\lambda =2.5a$, $L=(\lambda +a_{3z})/2\text {--}150\ \mathrm {\mu }$m since for this case, the FOV ends in the outer side. The values of $U_b$ are normalized both by $U_c$ and $U^{\prime }_b$, the latter being the bulk velocity over the same height far upstream of the cubes based on the streamwise velocity profiles in the same channel flow available in Zhang et al. (Reference Zhang, Wang, Blake and Katz2017). The implications of cube-induced flow blockage are further evaluated by examining the evolution of the vertical and spanwise velocity components along the edges of the sample area. The vertical velocity component, $V_b$ (figure 19c), is averaged along upper edges of these areas (blue lines in figure 19a), and the spanwise velocity, $W_b$ (figure 19d), along their outer boundaries (red lines in figure 19a). Note that based on the continuity equation $a\cdot \partial U_b/\partial x$ is balanced by $V_b$ and $a\cdot W_b/L$. In this discussion, the flow asymmetry induced by the neighbouring cube can be readily observed by comparing the results for the inner and outer sides. Several trends are evident. First, in all parts of the flow field, the differences between bulk velocity components in the inner and outer sides increase with decreasing cube spacing. They are substantial for $\lambda =1.0a$, especially between and downstream of the cube, and decrease to a few per cent for $\lambda =2.5a$. Second, at $x<0$, $U_b$ decreases and the other components increase in magnitude with decreasing distance from the front surface, as the axial blockage accelerates the flow vertically and laterally outward. As discussed before, even at $x/a=-3$, $U_b/U'_b \sim 0.9$, i.e. the flow is already slowed down and begins to circumvent the cubes. The decrease in $U_b$ and the corresponding increases in $V_b$ and magnitude of $W_b$ intensify with decreasing spacing. In contrast, the lateral flow along the plane of symmetry between cubes (at $z=0$) remains persistently very small ($<$0.01$U_c$), confirming that the flow around the two cubes is symmetric, and that the observed phenomena are not caused by cubes’ misalignments. Third, at $0<x/a<1$, $U_b$ increases abruptly on both sides owing to flow channelling, but then continues to decrease at a mild rate along the inner side while remaining higher than that on the outer side. While $V_b$ decreases on both sides, it remains positive along the inner side, but becomes negative at $x/a>0.5$ in the outer side. The persistent upward flow between cubes, and downward flow along the outer side contribute to the previously discussed circulation that causes the migration and merging of axial vortices around the cube. Finally, downstream of the cube, the bulk flow appears to be influenced by the gap-dependent development of large streamwise vortices. Hence, $V_b$ remains negative and $U_b$ increases monotonically in the outer side, while in the inner side, $U_b$ does not change significantly with $x/a$, and its magnitude decreases with decreasing spacing. The latter is consistent with trends of the wall shear stress in this region.

Figure 19. (a) The areas in the inner and outer sides of the cube used for evaluating the blockage. At $x<0$ and $x/a>1$, this area extends from the centre of the cube (black solid line), and at 0$\leq x/a\leq 1$, it extends outward from the side surface of the cube (black dashed line). (b–d) Axial profiles of (b) the streamwise bulk velocity, (c) wall-normal velocity averaged over the upper edge of the sample area (blue line in a) and (d) spanwise velocity averaged over the lateral edge of the sample area (red line in a). Here, $U'_b$ (${=}0.53U_c$) is the bulk velocity in the fully developed channel flow far upstream of the cubes calculated for the same area from data available in Zhang et al. (Reference Zhang, Wang, Blake and Katz2017).