2.1 Effect of channel width

Figure 2.

(a) Mean velocity profile for smooth-wall (solid) and rough-wall (dashed) minimal channels with ${\mathcal{W}}^{+}\approx 100$ , $k^{+}\approx 300$ . Darker grey refers to increasing channel width (table 1). (b) Difference in smooth-wall and rough-wall velocity. The origin in $z^{+}$ is at the roughness crest.

The flow of the minimal-span channel is intrinsically related to the spanwise width, with the critical height scaling as $z_{c}=0.4L_{y}$ . Previous studies support the view that the roughness crest must be submerged in healthy turbulence, requiring $z_{c}^{+}>k^{+}\Rightarrow L_{y}^{+}>k^{+}/0.4$ (Chung et al. Reference Chung, Chan, MacDonald, Hutchins and Ooi2015). This condition ensures that the roughness sublayer (the region of flow directly affected by the roughness) is fully captured, the height of which scales with the roughness height for conventional $k$ -type roughness (Raupach, Antonia & Rajagopalan Reference Raupach, Antonia and Rajagopalan1991; Flack, Schultz & Connelly Reference Flack, Schultz and Connelly2007). For the present roughness, however, the flow in the troughs would become essentially stationary as the bars become increasingly tall. This means that the roughness sublayer would be a region of altered turbulence close to the bar crests, with a region of quiescent flow below this. The roughness sublayer would instead likely scale on the spacing between the bars, ${\mathcal{W}}$ , with $k$ no longer altering the dynamics of the flow. We will therefore re-examine the relationship needed to capture the roughness sublayer for the present deep bars.

Figure 2(a) shows the effect of increasing channel width on the mean velocity profile for bar roughness with ${\mathcal{W}}^{+}\approx 100$ , $k^{+}\approx 300$ . As observed in previous minimal-span channels, increasing the channel width $L_{y}^{+}$ , and hence critical height $z_{c}$ , reduces the centreline velocity in the altered outer layer. A larger proportion of energy-containing eddies is captured by the wider channels and the results tend to match more closely with conventional full-span channel flow. The rough-wall flow with the smallest width of $L_{y}^{+}=153\approx 1.5{\mathcal{W}}^{+}$ (light grey dashed line) is seen to result in a slightly increased mean velocity below the critical height of $z_{c}^{+}\approx 0.4L_{y}^{+}\approx 61$ , when compared to the larger width cases. This is not observed in the smooth-wall flow (solid lines) so that the velocity difference between smooth- and rough-wall flows is nearly $2U_{\unicode[STIX]{x1D70F}}$ lower for $L_{y}^{+}=153$ (figure 2 b). When $L_{y}^{+}\geqslant 306\approx 3{\mathcal{W}}^{+}$ , little difference is seen with increasing channel width. That is, we have obtained a $\unicode[STIX]{x0394}U^{+}$ that is insensitive to the minimal span and the roughness sublayer is completely captured. In our simulations, scales larger than $L_{y}$ are, from the point of view of the roughness, much larger and perceived as effectively infinitely long. In terms of the critical height, $z_{c}$ , the above results suggest that for spanwise bar roughness we require $z_{c}\gtrsim 1.2{\mathcal{W}}$ , or that the height of the healthy turbulence region above the bar crests must be greater than the fluid width between bars, ${\mathcal{W}}$ . These results can also be interpreted as indicating that a large-scale separation between the largest turbulent length scale, $z_{c}$ , and the roughness length scale, ${\mathcal{W}}$ , is not necessary to ensure that the roughness effects are fully captured.

The classical description of $d$ -type roughness is that the roughness function scales on the largest turbulent length scale, which is conventionally the channel half-height, $h$ . In minimal-span channels the critical height $z_{c}$ can be regarded as the largest captured turbulent length scale in the flow as opposed to $h$ . The above results, however, show that increasing $z_{c}$ above $1.2{\mathcal{W}}$ has diminishing influence on the flow. This suggests that the largest turbulent length scales of the flow are independent of the present bar roughness geometry.

4.1 Parameter space

Figure 5.

Parameter space of the current study, showing the different series of spanwise bars. Fixed ${\mathcal{W}}^{+}$ shown by vertical solid lines, fixed $k/{\mathcal{W}}$ shown by horizontal dashed and dotted lines. Lines of constant $k^{+}$ are shown by diagonal lines in this log–log plot. A selection of roughness cross-sections are indicated by the arrows.

Figure 5 shows the parameter space of different ${\mathcal{W}}^{+}$ values used in this study. The following figures will all use the same symbols, where a data series is grouped by ${\mathcal{W}}^{+}$ (vertical lines in figure 5). In some situations, it is useful to consider fixed $k/{\mathcal{W}}$ , with this study considering two different values of $k/{\mathcal{W}}=3$ and $k/{\mathcal{W}}=6$ (horizontal dashed and dotted lines, respectively). A fixed $k/{\mathcal{W}}$ is similar to how laboratory experiments are conducted, in that a single surface (of fixed $k/{\mathcal{W}}$ ) is studied at different flow speeds, or ${\mathcal{W}}^{+}$ . In the current simulations at fixed $Re_{\unicode[STIX]{x1D70F}}=395$ , the physical size of the bars is changed so that ${\mathcal{W}}^{+}$ changes. Also shown in figure 5 are lines of constant $k^{+}$ , shown by the diagonal lines in this logarithmic plot. Studies investigating the effect of solidity, $\unicode[STIX]{x1D6EC}=k/(2{\mathcal{W}})$ , are typically done at constant $k^{+}$ , whereby roughness elements with fixed height are studied with different spacings between the elements, ${\mathcal{W}}^{+}$ (e.g. Leonardi et al. Reference Leonardi, Orlandi and Antonia2007; MacDonald et al. Reference MacDonald, Chan, Chung, Hutchins and Ooi2016).

4.2 Mean flow

Figure 6.

Streamlines of time-averaged velocity for (a) ${\mathcal{W}}^{+}=10$ with $5\leqslant k^{+}\leqslant 60$ and (b) $k/{\mathcal{W}}=6$ with $10\leqslant {\mathcal{W}}^{+}\leqslant 200$ . Streamlines are defined such that the mass flux between each streamline is approximately constant. The contour shows the vertical (wall-normal) turbulence intensity $w^{\prime +2}$ with a logarithmic colour bar.

Streamlines of time-averaged velocity are shown in figure 6 over a colour contour of the vertical (wall-normal) turbulence intensity, $w^{\prime +2}$ where the colour bar has a logarithmic scale. The intensity is defined using the triple decomposition of $w=W+\widetilde{w}+w^{\prime }$ (Reynolds & Hussain Reference Reynolds and Hussain1972; Finnigan Reference Finnigan2000) where the time-independent (spatially dependent) component, $\widetilde{w}(x,z)$ , of the fluctuation is subtracted off and $w^{\prime }$ is the purely turbulent fluctuation. Firstly, we will just consider the effect of increasing $k^{+}$ for a fixed ${\mathcal{W}}^{+}=10$ (figure 6 a). Streamlines are defined such that the mass flow between two streamlines in the roughness cavity is approximately constant, $\int _{z_{a}}^{z_{b}}(U^{+}+\widetilde{u}(x_{m},z))\,\text{d}z=A_{s}{\mathcal{W}}\approx 0.04{\mathcal{W}}$ , where $x_{m}$ is the streamwise mid-point of the roughness cavity. A clear difference for $k^{+}=5$ is seen, with the small roughness height restricting the recirculating ‘bubble’. There is a subtle difference with $k^{+}=10$ , but once $k^{+}\gtrsim 20$ little effect is seen with increasing $k^{+}$ . This suggests that once $k/{\mathcal{W}}\gtrsim 2$ there is very little effect in changing $k$ . At this point, there is a single recirculating bubble rotating in the clockwise direction. There is a cascade of alternately rotating recirculating regions below this primary bubble that reaches the roughness trough; however, the strength of these regions is negligible compared to the turbulent flow above, with the local velocities being less than one per cent of $U_{\unicode[STIX]{x1D70F}}$ . The contour of $w^{\prime +2}$ shows that the turbulent motions do not penetrate far into the roughness canopy, tending to zero within 10–20 viscous units.

Increasing the width of the fluid gap whilst retaining a fixed aspect ratio $k/{\mathcal{W}}$ (figure 6 b) results in a strengthening of the fluid recirculation region, as well as a greater penetration of the turbulence into the roughness canopy in terms of $z^{+}$ . However, it still appears that the turbulent fluctuations do not penetrate much past $z/{\mathcal{W}}\approx -2$ . When ${\mathcal{W}}^{+}\geqslant 100$ , a secondary recirculating bubble emerges below the primary one based on the constant mass flux definition of the streamlines defined above. This secondary recirculating bubble is nearly the same strength as the primary recirculating bubble seen when ${\mathcal{W}}^{+}=10$ in figure 6(a). This secondary bubble is still present for ${\mathcal{W}}^{+}<100$ , but its strength is substantially diminished. The streamlines in the final panel of figure 6(b) appear particularly dense, but are spaced approximately one grid cell apart.

Figure 7.

Profiles of (a) vertical turbulence intensity $w^{\prime +2}$ and (b) streamwise velocity $U^{+}+\widetilde{u}^{+}$ at the streamwise mid-point of the trough, for $k/{\mathcal{W}}=6$ . Darker grey lines and arrows indicate increasing ${\mathcal{W}}^{+}$ .

To better describe the flow within the cavity shown in figure 6(b), vertical profiles are shown in figure 7 of the vertical turbulence intensity (a) and streamwise velocity (b). Note that these are not spatially averaged but are instead profiles at the streamwise mid-point of the troughs. The vertical turbulence intensity shows a clear peak forming at $z/{\mathcal{W}}=-1$ which is especially evident for the largest bar spacing of ${\mathcal{W}}^{+}=200$ . This is associated with the bottom edge of the primary recirculating bubble observed in figure 6(b). Below $z/{\mathcal{W}}=-1$ , the turbulence intensity quickly decreases to become negligible below $z/{\mathcal{W}}\approx -2$ . The streamwise velocity, $U^{+}+\widetilde{u}^{+}$ (figure 7 b), is negative near the bottom of the primary recirculating bubble near $z/{\mathcal{W}}=-1$ , with increasing ${\mathcal{W}}^{+}$ resulting in stronger recirculation. The wider bar spacings of ${\mathcal{W}}^{+}=100$ and ${\mathcal{W}}^{+}=200$ also show the bottom of the secondary recirculating bubble at $z/{\mathcal{W}}\approx -2$ , where a positive streamwise velocity is observed from this anticlockwise rotating vortex. These results suggest that the strength of these recirculating bubbles continues to increase with increasing ${\mathcal{W}}^{+}$ . While later results will suggest the turbulent flow above the roughness is close to an asymptotic fully rough state, with the mean streamwise velocity profile collapsing when plotted against $z/{\mathcal{W}}$ , this does not appear to extend to the flow within the roughness cavity itself.

The streamlines of figure 6 suggest that, in the mean, the flow above the roughness sees an alternating slip/no-slip interface at the roughness crest. This scenario is often used in the superhydrophobic surface literature to describe the flow over textured surfaces similar to the present ones, in which bubbles of gas are trapped inside the roughness canopy with a liquid flowing over the top. The solid posts present a no-slip interface to the liquid, while the gas presents a zero shear interface which can promote drag reduction. This is similar to the present flow, with the notable exception of zero permeability in the region between the bars. A key parameter in the study of superhydrophobic surfaces is the slip length, $L^{+}$ , which relates the slip velocity at the roughness crest, $U_{k}^{+}$ (averaged over both slip and no-slip surfaces) to the average velocity gradient at the crest, $U_{k}^{+}=L^{+}\text{d}U_{k}^{+}/\text{d}z^{+}$ .

Figure 8.

(a) Slip velocity at the roughness crest $U_{k}^{+}$ and (b) slip length $L^{+}$ for varying fluid widths ${\mathcal{W}}^{+}$ . Data are for fixed $k/{\mathcal{W}}=6$ . The solid line in (b) indicates the Stokes flow limit $L^{+}=2C_{b}{\mathcal{W}}^{+}$ for alternating slip/no-slip surfaces (Lauga & Stone Reference Lauga and Stone2003), with $C_{b}=0.055$ for ${\mathcal{W}}/\unicode[STIX]{x1D706}=0.5$ . The dashed line indicates a fit of $C_{b}\approx 0.018$ for the present data.

The slip velocity is shown in figure 8(a) for $k/{\mathcal{W}}=6$ and can be seen to increase with ${\mathcal{W}}^{+}$ . The slip length in figure 8(b) also increases with ${\mathcal{W}}^{+}$ , in agreement with previous studies of turbulent flow over superhydrophobic surfaces comprising square posts (Seo & Mani Reference Seo and Mani2016). Here, Seo & Mani (Reference Seo and Mani2016) suggested that in the limit of ${\mathcal{W}}^{+}\rightarrow 0$ the slip length approaches the Stokes flow solution for flow over a slip/no-slip wall. Lauga & Stone (Reference Lauga and Stone2003) determined that the asymptotic limit for Stokes flow over spanwise-aligned textures with alternating slip/no-slip boundary conditions was $L^{+}=2C_{b}(\unicode[STIX]{x1D70E}){\mathcal{W}}^{+}$ , where $\unicode[STIX]{x1D70E}={\mathcal{W}}/\unicode[STIX]{x1D706}$ is the fraction of fluid area at the crest and the coefficient $C_{b}=-\log (\cos (\unicode[STIX]{x1D70E}\unicode[STIX]{x03C0}/2))/(2\unicode[STIX]{x03C0})$ . This is displayed in figure 8(b) with $C_{b}(\unicode[STIX]{x1D70E}=0.5)=0.055$ , where the present turbulent flow data show a smaller slip length to that predicted by this Stokes flow solution with the constant estimated as $C_{b}\approx 0.018$ . This is expected as the gaps between the spanwise bars are permeable and are not a pure slip boundary, as assumed in the Stokes flow case. However, these similarities to superhydrophobic surfaces (which neglect any flow within the roughness canopy) may suggest there is little interaction between the roughness canopy flow and the flow above the roughness.

Figure 9.

(a,b) Wall shear stress against streamwise position and (c,d) total pressure difference across a single roughness element against vertical position ( $z^{+}=0$ is located at the roughness crest). Distance between bars of (a,c) ${\mathcal{W}}^{+}\approx 10$ and (b,d) ${\mathcal{W}}^{+}\approx 100$ . Lighter grey denotes larger $k^{+}$ (table 1).

Figure 9 shows the time-averaged viscous stress and pressure distribution acting on a single roughness element for cases with ${\mathcal{W}}^{+}\approx 10$ and ${\mathcal{W}}^{+}\approx 100$ . The viscous stress, $\unicode[STIX]{x1D70F}_{visc}^{+}(x)$ , only acts on the horizontal roughness crests and troughs and can be related to the viscous drag force via $F_{\unicode[STIX]{x1D708}}=\int _{0}^{\unicode[STIX]{x1D706}}\unicode[STIX]{x1D70F}_{visc}\,\text{d}x\cdot L_{y}L_{x}/\unicode[STIX]{x1D706}$ . The viscous stress distribution along the roughness crest does not vary with $k$ for either ${\mathcal{W}}^{+}\approx 10$ (a) or ${\mathcal{W}}^{+}\approx 100$ (b). There is some variation in the viscous stress at the trough, whereby the shortest roughness heights have a negative stress due to recirculation within the roughness cavity. However, once $k/{\mathcal{W}}\gtrsim 1$ the trough is sufficiently removed from this recirculation element such that the viscous stress is approximately zero.

The total pressure distribution, $\unicode[STIX]{x0394}p^{+}(z)$ , is shown for ${\mathcal{W}}^{+}\approx 10$ and ${\mathcal{W}}^{+}\approx 100$ in figure 9(c,d). As discussed in § 3, the pressure across a single element, $\unicode[STIX]{x0394}p^{+}$ , is the combination of the driving (mean) pressure $\unicode[STIX]{x0394}P^{+}$ and the periodic component $\unicode[STIX]{x0394}\tilde{p}^{+}$ . The total pressure drag force in the channel is then $F_{p}=\int _{-kt}^{0}\unicode[STIX]{x0394}p\,\text{d}z\cdot L_{y}L_{x}/\unicode[STIX]{x1D706}$ . This pressure difference can be seen to tend to zero when $k/{\mathcal{W}}\gtrsim 1$ for ${\mathcal{W}}^{+}\approx 10$ (c). However, for the roughness with ${\mathcal{W}}^{+}\approx 100$ (d) there is a slight negative pressure difference below $z^{+}\approx -30$ , before tending to zero at $z^{+}\approx -150$ . This is likely due to the greater strength of the secondary vortex seen over $-200\lesssim z^{+}\lesssim -100$ for ${\mathcal{W}}^{+}=100$ (figure 6 b).

Figure 10.

Ratio of total pressure drag force, $F_{p}$ , to the total drag force. Symbols: +, ${\mathcal{W}}^{+}=10$ ; ○, ${\mathcal{W}}^{+}=20$ ; ♢, ${\mathcal{W}}^{+}=50$ ; $\times$ , ${\mathcal{W}}^{+}=100$ ; ▵, ${\mathcal{W}}^{+}=200$ .

Figure 10 shows the ratio of the total pressure drag force $F_{p}$ to the overall drag force, $D_{1}=F_{p}+F_{\unicode[STIX]{x1D708}}$ . For the three narrowest gaps of ${\mathcal{W}}^{+}\lesssim 50$ , the pressure drag is approximately 32 % of the total drag and is almost independent of $k$ . However, the larger widths have larger pressure drag contributions, of 42 % of the total drag for ${\mathcal{W}}^{+}=100$ and 57 % for ${\mathcal{W}}^{+}=200$ . These magnitudes are similar to those observed in three-dimensional sinusoidal roughness which exhibit $k$ -type behaviour (Chan et al. Reference Chan, MacDonald, Chung, Hutchins and Ooi2015; MacDonald et al. Reference MacDonald, Chan, Chung, Hutchins and Ooi2016). This increase in pressure drag contribution with increasing ${\mathcal{W}}^{+}$ is due to the increased turbulence penetration and strengthening of the recirculating bubbles that was observed in figure 6. In the seminal $d$ -type work of Perry et al. (Reference Perry, Schofield and Joubert1969), the authors assumed that the viscous drag was negligible so that the pressure drag, which could be experimentally measured using pressure tappings, was the only contribution to $U_{\unicode[STIX]{x1D70F}}$ . While this assumption was for bars with ${\mathcal{W}}^{+}\gtrsim 250$ , the present data for ${\mathcal{W}}^{+}\approx 200$ show that the viscous drag remains at approximately 40 % of the total drag, suggesting that the assumption of negligible viscous drag may not be ideal.

Figure 11.

Reynolds stress decomposed into dispersive and turbulent components for $k/{\mathcal{W}}=6$ , with (a) ${\mathcal{W}}^{+}=10$ and (b) ${\mathcal{W}}^{+}=100$ . Line styles: ——, total Reynolds stress (dispersive $+$ turbulent); – –, dispersive stress $\langle \widetilde{u}\widetilde{w}\rangle ^{+}$ ; – - –, turbulent stress $\langle \overline{u^{\prime }w^{\prime }}\rangle ^{+}$ ; - - -, smooth-wall total Reynolds stress. The cross denotes virtual origin derived from the force-centroid method (Jackson Reference Jackson1981). The origin in $z^{+}$ is the roughness crest.

The Reynolds stress for rough-wall flows can be decomposed into two components based on the triple decomposition defined above (Reynolds & Hussain Reference Reynolds and Hussain1972; Finnigan Reference Finnigan2000). These are the time-independent (spatially dependent) component, known as the dispersive stress $-\langle \widetilde{u}\widetilde{w}\rangle ^{+}$ and the purely turbulent (time-dependent) stress $-\langle \overline{u^{\prime }w^{\prime }}\rangle$ . Here, the overbar denotes temporal averaging, with $\overline{\widetilde{u}}=\widetilde{u}$ , while the angle brackets denote the superficial spatial average over a wall-parallel volume which is thin in the vertical direction. This superficial average includes both fluid and solid regions, where the solid region has zero velocity everywhere. This can be related to the intrinsic spatial average (averaging over just the fluid regions) by multiplying the superficially averaged quantity by the ratio of total volume to fluid-only volume, in this case $\unicode[STIX]{x1D706}/{\mathcal{W}}=2$ . The distinction between superficial and intrinsic averaging becomes irrelevant above the roughness crests ( $z>0$ ) as there is only fluid region in the wall-parallel plane. The cross-terms involving $u_{i}^{\prime }$ and $\widetilde{u}_{i}$ will be zero, because $\overline{u_{i}^{\prime }}=0$ and $\langle \widetilde{u}_{i}\rangle =0$ . Figure 11 shows these two components, along with the total Reynolds stress (sum of dispersive and turbulent stresses). For the narrow bars with ${\mathcal{W}}^{+}=10$ (figure 11 a), the dispersive stress is negligible between the bars, and zero above the roughness crest. The total Reynolds stress is therefore essentially the same as the turbulent stress. However, compared to the smooth-wall Reynolds stress (dotted line), the rough-wall total Reynolds stress appears to have a different virtual origin that is within the roughness canopy.

Figure 11(b) shows that the dispersive stress (dashed line) for the wider bars ( ${\mathcal{W}}^{+}=100$ ) is dominant and counter-gradient within the roughness canopy, while the turbulent stress is near zero in this region. This strong dispersive stress (relative to the total) would be due to the recirculating bubbles within the canopy, which due to the time-independent nature of the dispersive stress suggest that these bubbles do not vary much with time. Moreover, the near-zero turbulent Reynolds stress suggests that even though vertical turbulent fluctuations penetrate into some of the canopy (contours in figure 6), this does not result in significant momentum flux through the roughness canopy. Above the roughness crest, the dispersive stress quickly tends to zero as the flow retains little spatially dependent information of the bars. Note that this does not imply the flow has lost all information regarding the bars; rather it could have temporally dependent eddies that would not appear as a time-independent dispersive stress. As with the narrower bars (figure 11 a), the total Reynolds stress appears to have a virtual origin that is within the roughness canopy.

The dispersive stress has often been assumed to be negligible relative to the total turbulent Reynolds stress. This is based on laboratory experiments such as those of Poggi, Katul & Albertson (Reference Poggi, Katul and Albertson2004) and Poggi & Katul (Reference Poggi and Katul2008), who determined that the dispersive stress became negligible for three-dimensional vertical cylinders when $\unicode[STIX]{x1D6EC}>0.1$ . Similarly, Böhm et al. (Reference Böhm, Finnigan, Raupach and Hughes2013) showed that the dispersive stress is smaller than the turbulent stress for a three-dimensional array of bulbs with $\unicode[STIX]{x1D6EC}=0.38$ . However, a recent analysis by Castro (Reference Castro2017) on the numerical studies of Leonardi & Castro (Reference Leonardi and Castro2010) and Castro et al. (Reference Castro, Xie, Fuka, Robins, Carpentieri, Hayden, Hertwig and Coceal2017) showed that the dispersive stress is significant for cuboid roughnesses, supporting the present results.

4.3 Virtual origin

A common issue in roughness studies is determining the location of the origin in the vertical coordinate $z$ , often termed the zero-plane displacement height or the virtual origin, $\unicode[STIX]{x1D716}$ . The zero-plane displacement height, often denoted by $d$ , is a dynamic parameter associated with the origin of the logarithmic region of the flow, given by $U^{+}=\unicode[STIX]{x1D705}^{-1}\log (z-d)+B_{s}-\unicode[STIX]{x0394}U^{+}$ . This is therefore found by fitting the log law to the mean velocity profile, although this can be problematic as the roughness function $\unicode[STIX]{x0394}U^{+}$ is also unknown. Furthermore, for the present simulations, $Re_{\unicode[STIX]{x1D70F}}$ is relatively small and the use of the minimal channel means that only the beginning of the logarithmic region is captured. This makes it difficult to fit a log law to the velocity profile so that the zero-plane displacement height cannot be easily measured in this way. The term virtual origin is often used interchangeably with the zero-plane displacement height, but typically refers to methods that estimate the origin based on flow parameters other than the logarithmic velocity profile.

In the limit of tightly packed bars ( ${\mathcal{W}}^{+}\ll 1$ ), the roughness is submerged in a viscous-dominated flow while the turbulent flow above appears as a time-dependent shear. The small-scale roughness alters the near-wall flow by promoting a slip velocity at the crests, but otherwise the outer shear is unaffected apart from the slip velocity. This idea was successfully used by Luchini, Manzo & Pozzi (Reference Luchini, Manzo and Pozzi1991) for riblets, where the difference in streamwise and spanwise shear flows helps explain the drag-reducing properties of these small-scale protrusions. However, using the slip velocity ( $U_{k}^{+}$ , shown in figure 8 a) to define the origin breaks down for larger ${\mathcal{W}}^{+}$ as inertial effects begin to dominate and the assumption of viscous-dominated flow can no longer apply.

Figure 12.

Virtual origin, calculated using (a) the force-centroid method (Jackson Reference Jackson1981) and (b) the location of zero Reynolds stress. Symbols are the same as figure 5.

A physically appealing method that is often employed is to define $\unicode[STIX]{x1D716}$ as the centroid of the moment of the drag forces acting on the rough wall (Jackson Reference Jackson1981). Figure 12(a) shows the virtual origin, calculated using Jackson’s method. The origin for the three smallest ${\mathcal{W}}^{+}$ values is almost invariant with $k$ , with $\unicode[STIX]{x1D716}\approx 0.11{\mathcal{W}}$ . Note that this means $\unicode[STIX]{x1D716}^{+}$ increases with ${\mathcal{W}}^{+}$ , but the collapse with the scaling of $\unicode[STIX]{x1D716}/{\mathcal{W}}$ suggests ${\mathcal{W}}$ is a crucial parameter for the present roughness. For the larger values of ${\mathcal{W}}^{+}\geqslant 100$ , $\unicode[STIX]{x1D716}$ is substantially larger than the narrower cases, but still appears to asymptote to constant values for $k/{\mathcal{W}}\geqslant 3$ . These constants are approximately $\unicode[STIX]{x1D716}\approx 0.36{\mathcal{W}}$ for ${\mathcal{W}}^{+}=100$ , and $\unicode[STIX]{x1D716}\approx 0.55{\mathcal{W}}$ for ${\mathcal{W}}^{+}=200$ . This increase with ${\mathcal{W}}^{+}$ is most likely due to the secondary recirculating bubble observed in figure 6(b), which exists over $-200\lesssim z^{+}\lesssim -100$ . This recirculating bubble causes a non-negligible pressure difference to act across the roughness elements (figure 9 d), which in turn causes $\unicode[STIX]{x1D716}^{+}$ to increase. Given that this secondary recirculating bubble extends down to $z^{+}\approx 200$ , then it follows that we require at least $k^{+}>200\Rightarrow k/{\mathcal{W}}>2$ to reach this asymptote. These results are in qualitative agreement with the high aspect ratio model of Sadique et al. (Reference Sadique, Yang, Meneveau and Mittal2017), based on flow over three-dimensional rectangular prisms, which predicts $\unicode[STIX]{x1D716}/{\mathcal{W}}$ to asymptote to a constant for $k/{\mathcal{W}}\gtrsim 5$ . It is likely that the weaker sheltering behaviour of three-dimensional roughness (Yang et al. Reference Yang, Sadique, Mittal and Meneveau2016) is what requires a larger $k/{\mathcal{W}}$ ratio to reach the asymptotic limit, compared to $k/{\mathcal{W}}\gtrsim 2$ observed for the present two-dimensional roughness.

This virtual origin using Jackson’s method was shown in figure 11 as a cross symbol on the total Reynolds shear stress. This is seen to be located close to the position of the strongest dispersive stress, with the other ${\mathcal{W}}^{+}$ cases (not shown) supporting this observation. This supports the view that the drag centroid virtual origin is related to the recirculating bubbles. However, the turbulent flow above the roughness is unlikely to depend on these temporally independent recirculating bubbles. Instead, perhaps a better location to use as the origin is where the Reynolds shear stress reaches zero (Nepf et al. Reference Nepf, Ghisalberti, White and Murphy2007; Nepf Reference Nepf2012). This would lead to a better collapse of the smooth- and rough-wall total Reynolds shear stress in the near-wall region as they will both reach zero at the same location. In the present roughness, both the turbulent and dispersive Reynolds shear stress reach zero near the same location (see figure 11 b for an example with ${\mathcal{W}}^{+}=100$ ). Figure 12(b) shows the location of the zero crossing of the turbulent Reynolds stress, $\langle \overline{u^{\prime }w^{\prime }}\rangle ^{+}$ . This shows a much better collapse for the larger ${\mathcal{W}}^{+}$ values, with $\unicode[STIX]{x1D716}\approx 0.11{\mathcal{W}}$ for most of the cases with ${\mathcal{W}}^{+}\geqslant 20$ . It is only the narrowest case with ${\mathcal{W}}^{+}=10$ that the Reynolds stress reaches zero slightly deeper into the troughs with $\unicode[STIX]{x1D716}\approx 0.15{\mathcal{W}}$ , although given ${\mathcal{W}}^{+}$ is so small then this is a minor difference in terms of viscous units. This Reynolds stress zero crossing will be used to define the origin when computing the roughness function in § 4.4, although using a different origin only changes the numerical values of $\unicode[STIX]{x0394}U^{+}$ and not the conclusions of this paper.

4.4 Mean velocity and roughness function

Figure 13.

(a,c) Mean velocity profile and (b,d) difference in smooth- and rough-wall velocities for fluid width (a,b) ${\mathcal{W}}^{+}=10$ and (c,d) ${\mathcal{W}}^{+}=100$ . Line styles: black, smooth wall; grey, rough wall. Lighter grey denotes larger $k^{+}$ (table 1). The arrow shows increasing $k^{+}$ . Roughness crest denoted by + symbol, origin $\unicode[STIX]{x1D716}^{+}$ defined using the Reynolds stress zero crossing (figure 12 b). The vertical dashed line shows the minimal-span critical height $z_{c}^{+}\approx 0.4L_{y}^{+}$ .

The mean velocity profile is shown in figure 13 for the cases with ${\mathcal{W}}^{+}=10$ and ${\mathcal{W}}^{+}=100$ . The velocity difference between smooth- and rough-wall flows for ${\mathcal{W}}^{+}=10$ (figure 13 b) has a dependence on $k$ for $k^{+}\lesssim 20$ , wherein increasing $k^{+}$ increases the velocity difference but when $k$ is sufficiently large ( $k^{+}\gtrsim 30\Rightarrow k/{\mathcal{W}}\gtrsim 3$ ) little difference is observed. Similarly, for ${\mathcal{W}}^{+}=100$ a collapse is observed for $k/{\mathcal{W}}\gtrsim 3$ (figure 13 c,d), where the cases with the largest peak-to-trough heights ( $k^{+}=300$ , $600$ and $1200$ ) are almost indistinguishable from one another. Here, the case with the smallest roughness height of $k^{+}=50$ has a velocity difference slightly greater then the larger $k^{+}$ values, but this is likely due to differences in the virtual origin (figure 12 b).

Figure 14.

Roughness function $\unicode[STIX]{x0394}U^{+}$ against (a) roughness trough-to-peak height $k^{+}$ and (b) solidity $\unicode[STIX]{x1D6EC}=k/(2{\mathcal{W}})$ . Symbols are the same as figure 5. The vertical black dashed line at $\unicode[STIX]{x1D6EC}=0.15$ in (b) denotes the dense regime. The vertical grey dashed and dotted lines show fixed $\unicode[STIX]{x1D6EC}=1.5$ and $\unicode[STIX]{x1D6EC}=3.0$ and are plotted in figure 15 as a function of ${\mathcal{W}}^{+}$ .

The roughness function $\unicode[STIX]{x0394}U^{+}$ is computed by taking the value of $U_{s}^{+}-U_{r}^{+}$ at the vertical critical height $z_{c}^{+}\approx 0.4L_{y}^{+}$ , shown by the vertical dashed line in figure 13(b,d). Figure 14(a) shows the roughness function against the peak-to-trough height, $k^{+}$ . It is clear that increasing $k^{+}$ for fixed values of ${\mathcal{W}}^{+}$ does not tend towards the fully rough asymptote of $k$ -type roughness, that is $\unicode[STIX]{x1D705}^{-1}\log (k^{+})$ . Instead, for ${\mathcal{W}}^{+}\approx 10$ , it seems to asymptote to a constant roughness function that is independent of roughness height. The roughness functions for the larger spacings ( ${\mathcal{W}}^{+}\gtrsim 20$ ) all appear to be entirely within this asymptotic regime, with $\unicode[STIX]{x0394}U^{+}$ being approximately equal to a constant that only depends on ${\mathcal{W}}^{+}$ . However, consider fixed values of $k/{\mathcal{W}}$ , shown by the dashed and the dotted grey lines for $k/{\mathcal{W}}=3$ and $k/{\mathcal{W}}=6$ , respectively. This is more representative of a laboratory study in which a single geometry (with fixed $k/{\mathcal{W}}$ ) is studied at various flow speeds which leads to the roughness Reynolds number varying. By considering fixed $k/{\mathcal{W}}$ , the data now appear to behave more like a $k$ -type roughness in that the larger values of $k^{+}$ are tending towards the fully rough asymptote. This is despite the previous results showing very little change in the flow with such large aspect ratios. It would seem that the offset $B$ in $\unicode[STIX]{x1D705}^{-1}\log (k^{+})+B$ is dependent on $k/{\mathcal{W}}$ , with increasing $k/{\mathcal{W}}$ leading to reduced $B$ .

The roughness function as a function of solidity, $\unicode[STIX]{x1D6EC}=k/(2{\mathcal{W}})$ (figure 14 b), shows how the roughness function for a particular ${\mathcal{W}}^{+}$ tends towards a constant value for large solidity (large $k/{\mathcal{W}}$ ratios). This value depends on the fluid width ${\mathcal{W}}^{+}$ , with increasing ${\mathcal{W}}^{+}$ leading to an increasing roughness function. It is important to emphasise that here we are varying $k^{+}$ for a variety of ${\mathcal{W}}^{+}$ values. Previous studies that examine the roughness function for varying solidity typically do so by keeping $k^{+}$ fixed and varying ${\mathcal{W}}^{+}$ . This results in the roughness function decreasing with solidity in the so-called dense regime when $\unicode[STIX]{x1D6EC}\gtrsim 0.15$ (Jiménez Reference Jiménez2004; Flack & Schultz Reference Flack and Schultz2014; MacDonald et al. Reference MacDonald, Chan, Chung, Hutchins and Ooi2016), as observed in figure 1(a). If we were to consider a fixed $k^{+}$ value for the present bar roughness and increase the wavelength, we would still obtain a reducing roughness function in the dense regime ( $\unicode[STIX]{x1D6EC}\gtrsim 0.15$ , shown by the vertical black dashed line in figure 14 b). To see this, consider the first data point for ${\mathcal{W}}^{+}\approx 100$ (red cross, $k^{+}\approx 50$ , $\unicode[STIX]{x1D6EC}=0.25$ ) and final data point for ${\mathcal{W}}^{+}\approx 10$ (black plus, $k^{+}\approx 60$ , $\unicode[STIX]{x1D6EC}=3$ ). Even though the roughness heights are slightly different, it is clear that the roughness function is reducing with solidity for these approximately matched $k^{+}$ values.

Figure 15.

(a) Roughness function $\unicode[STIX]{x0394}U^{+}$ against fluid width ${\mathcal{W}}^{+}$ , and (b) mean velocity profiles against vertical position normalised on ${\mathcal{W}}^{+}$ , for solidity values – –, $\unicode[STIX]{x1D6EC}=1.5$ , and - - - -, $\unicode[STIX]{x1D6EC}=3.0$ (figure 14 b). Data only shown for the region of healthy turbulence, $z<z_{c}$ . Symbols: +, ${\mathcal{W}}^{+}=10$ ; ○, ${\mathcal{W}}^{+}=20$ ; ♢, ${\mathcal{W}}^{+}=50$ ; $\times$ , ${\mathcal{W}}^{+}=100$ ; ▵, ${\mathcal{W}}^{+}=200$ . The solid purple line shows the fully rough asymptote, $\unicode[STIX]{x0394}U^{+}=\unicode[STIX]{x1D705}^{-1}\log ({\mathcal{W}}^{+})+C$ in (a) and $U^{+}=\unicode[STIX]{x1D705}^{-1}\log (z/{\mathcal{W}})+B_{s}-C$ in (b), with $B_{s}\approx 5.0$ and $C\approx -7.4$ .

To better show the trend with ${\mathcal{W}}^{+}$ , the roughness function is considered in figure 15(a) for surfaces with fixed aspect ratios of $k/{\mathcal{W}}=3$ and $k/{\mathcal{W}}=6$ (indicated by the vertical dashed and dotted lines in figure 14 a). For these fixed aspect ratio bars, the roughness function is seen to collapse. This suggests that for these high aspect ratio (tall and narrow) rough surfaces, it is ${\mathcal{W}}$ , and not $k$ that is the relevant length scale. The $k$ -type asymptote, $\unicode[STIX]{x1D705}^{-1}\log (k^{+})+B$ for fixed $k/{\mathcal{W}}$ would then become $\unicode[STIX]{x1D705}^{-1}\log ({\mathcal{W}}^{+})+C$ , where $C=B+\unicode[STIX]{x1D705}^{-1}\log (k/{\mathcal{W}})$ . If we assume the roughness function for ${\mathcal{W}}^{+}=200$ is in the fully rough regime, then this offset $C$ can be estimated as $C\approx -7.4$ from the present data and is independent of $k/{\mathcal{W}}$ (solid line in figure 15). We can also evaluate the equivalent sand grain roughness, $k/k_{s}\equiv \exp (-\unicode[STIX]{x1D705}(3.5+B))=(k/{\mathcal{W}})\exp (-\unicode[STIX]{x1D705}(3.5+C))$ . With $C\approx -7.4$ for $k/{\mathcal{W}}\geqslant 3$ we have $k_{s}\approx {\mathcal{W}}\exp (-3.9\unicode[STIX]{x1D705})\approx 0.21{\mathcal{W}}$ , indicating that $k_{s}$ is solely a function of ${\mathcal{W}}$ . The ratio $k_{s}/k$ can therefore be given as $k_{s}/k=0.11\unicode[STIX]{x1D6EC}^{-1}$ , where $\unicode[STIX]{x1D6EC}=k/(2{\mathcal{W}})$ is the solidity. This implies that for any bar roughness in this extremely dense regime of $\unicode[STIX]{x1D6EC}\geqslant 1.5$ , the equivalent sand grain roughness scales with $\unicode[STIX]{x1D6EC}^{-1}$ .

Conventional $k$ -type roughness in the fully rough regime exhibits a collapse in the mean velocity profile when plotted against the vertical position normalised on roughness height, $k$ . For the present roughness, the analogous scaling is to use the vertical position normalised on the bar spacing, ${\mathcal{W}}$ . This is shown in figure 15(b) for both $k/{\mathcal{W}}=3$ and $k/{\mathcal{W}}=6$ . Here, data are only shown for the region of healthy turbulence, $z<z_{c}=0.4L_{y}$ , as the spanwise width of the minimal domain varies across different ${\mathcal{W}}^{+}$ cases. In the near-wall region, there is a slight difference between the two cases with the largest spacings ( ${\mathcal{W}}^{+}=100$ and ${\mathcal{W}}^{+}=200$ ), although this is likely because the flow has not quite reached the asymptotic fully rough state. Moreover, the logarithmic scaling of this plot exaggerates small differences close to the wall. However, there is a clear collapse in the logarithmic layer for ${\mathcal{W}}^{+}=100$ and ${\mathcal{W}}^{+}=200$ , where the fully rough asymptote in mean velocity in the logarithmic layer is $U^{+}=\unicode[STIX]{x1D705}^{-1}\log (z/{\mathcal{W}})+B_{s}-C\approx \unicode[STIX]{x1D705}^{-1}\log (z/{\mathcal{W}})+12.4$ , where $B_{s}\approx 5.0$ is the smooth-wall log-law constant. This is simply an alternate representation of figure 15(a) and shows that the flow is becoming fully rough, where the dominant roughness length scale is the spacing between the bars, ${\mathcal{W}}$ .

The results of figure 15 suggest that the flow has reached the asymptotic fully rough regime. However, as discussed by Busse, Thakkar & Sandham (Reference Busse, Thakkar and Sandham2017), even though bulk flow properties like $\unicode[STIX]{x0394}U^{+}$ and $U_{b}^{+}$ may suggest we have obtained this universal regime, other flow properties such as the non-zero viscous drag (figure 10) and near-wall flow may still exhibit a Reynolds number dependence. We must therefore be careful and instead note that the flow appears to be tending towards this fully rough state, although it has not yet been comprehensively obtained.

Figure 16.

Equivalent sand grain roughness of several roughness geometries, adapted from figure 1 of Jiménez (Reference Jiménez2004). The drag coefficient $C_{D}$ is to normalise for different roughness geometries; here we use $C_{D}\approx 1.25$ as in Jiménez (Reference Jiménez2004). The fully rough asymptote determined for the present data (figure 15) is shown by the thick dashed line for $\unicode[STIX]{x1D6EC}\geqslant 1.5$ , where $k_{s}/k\approx 0.11\unicode[STIX]{x1D6EC}^{-1}$ .

Figure 16 shows the equivalent sand grain roughness against solidity for several different roughness geometries, adapted from figure 1 of Jiménez (Reference Jiménez2004). The sparse regime ( $\unicode[STIX]{x1D6EC}\leqslant 0.15$ ), which has been extensively studied, is seen to scale with $\unicode[STIX]{x1D6EC}$ . Most authors term any roughness with $\unicode[STIX]{x1D6EC}>0.15$ to be dense; however, no qualitative scaling for this regime has been proposed, and few experimental studies have been able to examine this regime. The present study has demonstrated that the equivalent sand grain roughness for spanwise-aligned rectangular bars scales with $\unicode[STIX]{x1D6EC}^{-1}$ for very dense rough surfaces with $\unicode[STIX]{x1D6EC}\gtrsim 1.5$ (see figure 15 a and associated discussion). This is shown in figure 16 by the thick dashed line, where we have used a drag coefficient of $C_{D}=1.25$ , as recommended in Jiménez (Reference Jiménez2004) for spanwise bar roughness. Note that if we define the drag coefficient with the slip velocity at the roughness crests, $C_{D}=\unicode[STIX]{x1D70F}_{w}/(1/2)\unicode[STIX]{x1D70C}U_{k}^{2}=2/U_{k}^{+2}$ , then from figure 8 we have $C_{D}\approx 0.9$ –1.3. This minor variation in $C_{D}$ only shifts the thick black line in figure 16 up or down slightly and does not affect the main result that $k_{s}/k$ is scaling with $\unicode[STIX]{x1D6EC}^{-1}$ .

This leads to a new interpretation of figure 16, where the sparse ( $\unicode[STIX]{x1D6EC}\leqslant 0.15$ ) and very dense ( $\unicode[STIX]{x1D6EC}\gtrsim 1.5$ ) regions are separated by the dense region. In the well-studied sparse regime, the equivalent sand grain roughness increases with $\unicode[STIX]{x1D6EC}$ , and the roughness height $k$ is the dominant roughness length scale. In the very dense regime, the equivalent sand grain roughness scales with $\unicode[STIX]{x1D6EC}^{-1}$ and we have shown that the spacing between the bars ${\mathcal{W}}$ is an important roughness length scale. This suggests that in the intermediary dense region ( $0.15\leqslant \unicode[STIX]{x1D6EC}\lesssim 1.5$ ) both $k$ and ${\mathcal{W}}$ are competing roughness length scales, which may help in understanding why no scaling for $\unicode[STIX]{x1D6EC}\gtrsim 0.15$ has received widespread support. It is likely that this dense regime is highly dependent on the roughness geometry. Moreover, the ratio of the bar width to wavelength, $b/\unicode[STIX]{x1D706}$ , is an important parameter, which has been kept constant at $b/{\mathcal{W}}=1$ in the present study.

4.5 Near-wall flow

Figure 17.

Two-dimensional pre-multiplied energy spectrum of vertical (wall-normal) velocity for $k/{\mathcal{W}}=6$ (shaded contour) for (a) ${\mathcal{W}}^{+}=10$ at $z^{+}+\unicode[STIX]{x1D716}^{+}\approx 6.4$ and (b) ${\mathcal{W}}^{+}=100$ at $z^{+}+\unicode[STIX]{x1D716}^{+}\approx 16.1$ . This corresponds to a location approximately 5 viscous units above the roughness crests. The thick vertical line at the bottom of the plots marks the bar wavelength $\unicode[STIX]{x1D706}^{+}=b^{+}+{\mathcal{W}}^{+}=2{\mathcal{W}}^{+}$ . The line contour shows the smooth-wall spectrum at matched $z^{+}+\unicode[STIX]{x1D716}^{+}$ locations. Contour spacing is (a) $0.002U_{\unicode[STIX]{x1D70F}}^{2}$ and (b) $0.005U_{\unicode[STIX]{x1D70F}}^{2}$ .

In order to analyse the changes to the flow structures that occur as we tend towards the fully rough regime of this extremely dense roughness, figure 17 shows the two-dimensional pre-multiplied energy spectra of vertical (wall-normal) velocity, $k_{x}k_{y}E_{w^{\prime }w^{\prime }}$ . This is done for two cases with $k/{\mathcal{W}}=6$ , at a horizontal (wall-parallel) location approximately 5 viscous units above the roughness crest. This spectrum is related to the vertical root-mean-square velocity fluctuations as $w_{rms}^{\prime 2}=\int _{0}^{\infty }\int _{0}^{\infty }k_{x}k_{y}E_{w^{\prime }w^{\prime }}\,\text{d}\log (k_{x})\text{d}\log (k_{y})$ , where the single prime denotes purely turbulent fluctuations, with the temporally independent component $\widetilde{w}$ subtracted off. The case with the narrower bars ( ${\mathcal{W}}^{+}=10$ , figure 17 a) has moderate agreement with the smooth-wall flow, in that the peak is located close to $\unicode[STIX]{x1D706}_{x}^{+}\approx 250$ and $\unicode[STIX]{x1D706}_{y}^{+}\approx 50$ . This corresponds to the quasi-streamwise vortices (Jeong et al. Reference Jeong, Hussain, Schoppa and Kim1997) that accompany the familiar near-wall streaks (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967). The strength of the energy spectrum in the rough-wall flow is much greater, as the roughness enables the vertical velocity to penetrate into the roughness canopy while the smooth-wall flow has the impermeability condition at $z^{+}=0$ . However, when the roughness spacing is increased to ${\mathcal{W}}^{+}=100$ (figure 17 b), we see that there is significant energy located at long spanwise wavelengths at $\unicode[STIX]{x1D706}_{x}^{+}\approx 100$ that is absent in the smooth-wall flow. This resembles the identification of long spanwise rollers over streamwise-aligned riblets (García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011), that leads to the degradation of the drag-reducing regime. These rollers develop due to an inflectional point in the mean velocity profile, which leads to a two-dimensional Kelvin–Helmholtz-like instability. These rollers have long been associated with canopy flows over high aspect ratio roughness (e.g. Ghisalberti & Nepf Reference Ghisalberti and Nepf2004; Nepf Reference Nepf2012), where Raupach, Finnigan & Brunet (Reference Raupach, Finnigan and Brunet1996) demonstrated a similarity between the canopy shear flow and mixing layers.

Figure 18.

Pre-multiplied streamwise cospectra of the Reynolds stress, $k_{x}E_{uw}^{+}$ , where the cospectra have been integrated over $\unicode[STIX]{x1D706}_{y}^{+}\geqslant 153$ , for (a) smooth wall, (b) ${\mathcal{W}}^{+}=10$ , (c) ${\mathcal{W}}^{+}=50$ and (d) ${\mathcal{W}}^{+}=100$ . All bar roughness cases are for $k/{\mathcal{W}}=6$ . The thick vertical line marks the bar wavelength $\unicode[STIX]{x1D706}^{+}=2{\mathcal{W}}^{+}$ Contour spacing is $3\times 10^{-3}$ , with the shaded area denoting positive values.

To better identify these long spanwise rollers, figure 18 shows the cospectra of the Reynolds stress, where the spectra have been integrated only over $\unicode[STIX]{x1D706}_{y}^{+}\geqslant 153$ . This is similar to figure 11 of García-Mayoral & Jiménez (Reference García-Mayoral and Jiménez2011). The smooth-wall flow (figure 18 a) is mostly positive (counter-gradient), but as the spacing between the bars increases the Reynolds stress becomes negative. It forms a local maximum in which the shear stress is concentrated around $z^{+}\approx 4$ , $\unicode[STIX]{x1D706}_{x}^{+}\approx 130$ for ${\mathcal{W}}^{+}=100$ (figure 18 d). Although this agrees well with the streamwise-aligned bars of García-Mayoral & Jiménez (Reference García-Mayoral and Jiménez2011) in the drag-increasing regime, where the maxima was at $z^{+}\approx 4$ , $\unicode[STIX]{x1D706}_{x}^{+}\approx 150$ , the present structures could originate from different mechanisms as these structures have a streamwise wavelength that is of the same order as the wavelength of the roughness. In any case, the existence of spanwise coherence in these spectrograms suggests that a Kelvin–Helmholtz instability drives the shear layer that forms between each spanwise-aligned bar of the present roughness.

Figure 19.

Contours of instantaneous vertical (wall-normal) turbulent fluctuations (with the time-independent dispersive component $\widetilde{w}$ removed) in a horizontal (wall-parallel) plane at $z^{+}\approx 2$ , for $k/{\mathcal{W}}=6$ and (a) smooth wall, (b) ${\mathcal{W}}^{+}=10$ , (c)  ${\mathcal{W}}^{+}=20$ , (d) ${\mathcal{W}}^{+}=50$ , (e) ${\mathcal{W}}^{+}=100$ and (f) ${\mathcal{W}}^{+}=200$ . Vertical dotted lines show the leading edge of the bars.

An instantaneous snapshot of vertical turbulent fluctuations (with the time-independent dispersive component $\widetilde{w}$ removed) at $z^{+}\approx 2$ is shown in figure 19, for $k/{\mathcal{W}}=6$ bars. The smooth-wall flow (figure 19 a) has near-zero vertical velocity due to the impermeability constraint, although structures that are elongated in the streamwise direction can be seen faintly. The two narrower bars ( ${\mathcal{W}}^{+}=10$ and ${\mathcal{W}}^{+}=20$ , a and b) more clearly show this streaky structure. There is a definite coherence across the spanwise bars, even though there are strong fluctuations occurring on the leading edges of the bars. For ${\mathcal{W}}^{+}\geqslant 50$ , little coherence between adjacent bars remains to be seen at this scale, with the strong vertical velocity saturating across much of the fluid region between the bars. The above analyses suggests that the near-wall streaky structure has been replaced by spanwise rollers with $\unicode[STIX]{x1D706}_{x}^{+}\approx 150$ , although it is difficult to observe this in instantaneous snapshots and with such a narrow spanwise domain.