3.1 Inner-normalised turbulence statistics

Figure 4 shows the inner-normalised mean streamwise velocity profiles above the rough wall. Figure 4(a) presents Case 1 profiles at 21.7 m (for which $U_{{\it\tau}}$ is determined directly from the drag balance data), while figure 4(b) shows single-wire Case 2 data at $U_{\infty }\approx 20~\text{m}~\text{s}^{-1}$ . In both plots, smooth-wall data at ${\it\delta}_{99}^{+}=9830$ are included for comparison, and the location of $z_{I}$ (calculated using (2.4)) for a particular profile is indicated by the symbol with a thick black outline. The latter is true for all remaining figures in this paper. Recall that the Case 1 data are taken at approximately matched ${\it\delta}_{99}/k_{s}$ , with $k_{s}^{+}$ ranging from the transitionally rough ( $k_{s}^{+}=22$ ) to fully rough regime ( $k_{s}^{+}=150$ ) with increasing Reynolds number. Conversely, all Case 2 data are at approximately matched $k_{s}^{+}$ , but ${\it\delta}_{99}/k_{s}$ ranges with streamwise development from relatively small ( ${\it\delta}_{99}/k_{s}=28$ ) to very large ( ${\it\delta}_{99}/k_{s}=198$ ).

Figure 4.

Comparison of streamwise mean velocity between the smooth and rough walls. (a) Shows the rough-wall data for Case 1 at 21.7 m, (b) shows the single-wire rough-wall data for Case 2 at $U_{\infty }\approx 20~\text{m}~\text{s}^{-1}$ with the data at every second streamwise location removed for clarity. Smooth-wall data at ${\it\delta}_{99}^{+}=9830$ is included in both plots. The symbols with thick black outlines show the location of the onset of inertial dynamics according to (2.4). Dashed black lines have a slope of $1/{\it\kappa}=1/0.39$ .

The results in figure 4 show a convincing log-linear region for all profiles, with a slope very similar to the logarithmic slope of the smooth-wall data; the dashed lines in figure 4 show a log-linear slope of $1/{\it\kappa}$ . (Note that the approach used to determine $U_{{\it\tau}}$ for the data shown in figure 4(a) does not assume that there is a logarithmic profile in the mean streamwise velocity.) The results also show the well-known result that the rough wall causes a vertical shift, ${\rm\Delta}U^{+}$ , in the inner-normalised mean streamwise velocity due to the increase in drag above the rough wall (relative to a smooth wall) and the resulting increase in momentum flux towards the wall (Raupach et al. Reference Raupach, Antonia and Rajagopalan1991). In smooth-wall flows the no-slip condition requires that at $z=0$ , $U=0$ , while for rough walls the effective location of the wall varies. Consequently, for the rough wall flows there is no constraint that $U=0$ at $z+{\it\epsilon}=0$ . Nonetheless, there is a location somewhere within the roughness canopy where $U=0$ . Thus, because of the downward shift in the rough-wall profiles ( ${\rm\Delta}U^{+}$ ), the near-wall characteristics of the mean streamwise velocity profile must differ between the smooth and rough walls. Due to this near-wall modification, most profiles in figure 4 for which $7\lesssim {\rm\Delta}U^{+}\lesssim 10$ appear to be approximately log-linear down to the lowest measured wall-normal position. Of course, these profiles may contain error near to the wall due to the choice of ${\it\epsilon}$ described in § 2.4. However, it seems that such behaviour should be observed at some ${\rm\Delta}U^{+}$ for any sensible choice of ${\it\epsilon}$ . Based upon its classically defined attributes, the logarithmic region is herein taken to coincide with the inertial sublayer. Approximate log-linearity of the mean streamwise velocity is thus not sufficient to define the inertial sublayer. That is, although the mean streamwise velocity may exhibit log-linearity external to the roughness crest, the wall-normal location of the beginning of the logarithmic region (or inertial sublayer) may be further out in the boundary layer. Such behaviour occurs in smooth-wall flow, where $U^{+}$ is approximately logarithmic interior to $z_{I}$ (see Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013).

Figure 5.

The friction Reynolds number trends of the streamwise velocity variance above the smooth and rough wall. In (a), the smooth-wall data are presented with some intermediate Reynolds number data removed for clarity. The thin grey line shows the boundary layer DNS of Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013) and the thick grey lines show the two highest Reynolds number smooth-wall profiles, corrected for spatial attenuation using the approach introduced by Smits et al. (Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011). (b) Shows the rough-wall data for Case 1 at 21.7 m, and (c) shows the single-wire rough-wall data for Case 2 at $U_{\infty }\approx 20~\text{m}~\text{s}^{-1}$ with every second streamwise location removed for clarity. The symbols with thick black outlines show the location of the onset of inertial dynamics according to (2.4).

The Reynolds number trends of the streamwise velocity variance are presented in figure 5. Smooth-wall data are shown in figure 5(a), Case 1 data at 21.7 m are shown in figure 5(b) and single-wire Case 2 data at $U_{\infty }\approx$ 20 are presented in figure 5(c). All data show consistent development trends with increasing friction Reynolds number. However, the two highest Reynolds number smooth-wall profiles have significantly poorer spatial resolution ( $l^{+}=$ 61 and 80, respectively) than the other smooth-wall profiles ( $l^{+}<$ 26). Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009) show that in smooth-wall turbulent flows an increase in $l^{+}$ causes attenuation of the streamwise variance that is largest near to the wall, where the contributing scales of motion are relatively small. In this paper, the smooth-wall data are employed to make comparisons with rough-wall data in the outer region of the flow. Such comparisons can lead to specious interpretations when the effect of spatial resolution is not carefully considered. The thick grey lines in figure 5(a) show the two highest Reynolds number smooth-wall profiles, corrected for spatial attenuation affects using the approach of Smits et al. (Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011). The corrected data indicate that for the smooth-wall data there is little effect on $\overline{u^{2}}^{+}$ due to spatial averaging external to $z\approx z_{I}$ (the region of interest in this paper). Although it has not been validated, it is prudent to assume that the correction proposed by Smits et al. (Reference Smits, Monty, Hultmark, Bailey, Hutchins and Marusic2011) provides a reasonable estimate of the effect of spatial attenuation in rough-wall flows, at least away from the near-wall region. It is therefore unlikely that any of the rough-wall $\overline{u^{2}}^{+}$ profiles presented herein (all with $l^{+}<40$ ) are discernibly influenced by spatial attenuation beyond $(z+{\it\epsilon})\approx z_{I}$ .

Comparing figure 5(a–c), the smooth and fully rough variances exhibit very similar development trends with friction Reynolds number. In accord with Townsend’s (Reference Townsend1976) attached eddy hypothesis, both wall conditions appear to exhibit a region of log-linear slope (see § 3.3), that lengthens (in wall units) with increasing Reynolds number. At the lowest Reynolds number in figure 5(b), the rough wall appears to have little effect on the general shape of the streamwise variance profile, with a near-wall peak clearly visible at $z^{+}\approx 15$ . With increasing friction and roughness Reynolds numbers, the magnitude of the near-wall peak diminishes, such that at ${\it\delta}_{99}^{+}=29\,900$ and $k_{s}^{+}=150$ , no near-wall peak is apparent. Note however, that the spatial resolution varies substantially in figure 5(b): from $l^{+}=5.7$ at the lowest friction Reynolds number to $l^{+}=38.5$ at the highest. Therefore, spatial resolution effects are also partially responsible for the reduction in the near-wall peak intensity observed with increasing ${\it\delta}_{99}^{+}$ . However, these effects are not sufficient to account for the complete absence of a near-wall peak at ${\it\delta}_{99}^{+}=29\,900$ ; Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009) suggest from their data that for $l^{+}<20$ the error in the inner-normalised streamwise variance due to spatial averaging should be less than $10\,\%$ for ${\it\delta}_{99}^{+}\gtrsim 3500$ . The lack of a near-wall variance peak is well documented in fully rough flows and is associated with a destruction/disturbance of the near-wall cycle in the immediate vicinity of the roughness elements (Grass Reference Grass1971; Schultz & Flack Reference Schultz and Flack2007). This disturbance of the near-wall cycle appears to be a gradual process with increasing $k_{s}^{+}$ and is likely tied to the corresponding increase in pressure drag at the wall. While a clear near-wall peak is observed in the transitionally rough data in figure 5(b), no such peak is observed in the fully rough data in figure 5(c), even at approximately matched friction Reynolds number. Interestingly, there appears to be little affect in changing ${\it\delta}_{99}/k_{s}$ (which varies from 28 to 198 in figure 5 c) on the general shape of the variance profiles. Note, however, that for the majority of the rough-wall data considered here, the height of the roughness relative to the boundary layer thickness would typically be considered small. This is especially true when compared to some pre-existing measurements in which outer-layer similarity was not observed. For example, Krogstadt & Antonia (Reference Krogstadt and Antonia1999) studied a case with $k_{s}/{\it\delta}_{99}=1/7$ , whereas the largest relative roughness presented here is $k_{s}/{\it\delta}_{99}=1/28$ . It is worth noting that such empirical studies are susceptable to errors in determining the virtual origin (that is, determining ${\it\epsilon}$ ) that can influence the outer region.

3.2 Outer-normalised turbulence statistics

Figure 6.

(a) The velocity defect $(U_{\infty }-U)/U_{{\it\tau}}$ for all smooth- and rough-wall data. The deviation $E$ about the mean defect $\overline{D}^{+}$ as defined in (3.1) is presented for all of the data in (b), and for the smooth wall, rough-wall Case 1 and rough-wall Case 2 data in (c–e), respectively.

Figure 6 compares the smooth- and rough-wall mean velocity profiles in defect form. Examination of such plots is a common approach for assessing the validity of Townsend’s (Reference Townsend1956) wall-similarity hypothesis. If the hypothesis is correct, high Reynolds number data normalised in this way should define a single profile in the outer region regardless of the wall condition. Figure 6(a) shows that, with the exception of a few low Reynolds number smooth-wall profiles, the velocity defect data (smooth and rough wall) follow a single outer-layer profile. To probe these results in greater detail, the deviation of the individual profiles from a mean defect profile is calculated. This is presented in figure 6(c–e) where the deviation, $E$ , is plotted for the smooth-wall data, and Case 1 and Case 2 rough-wall data, respectively. Here,

where $\overline{D}^{+}$ is the mean defect of all of the smooth- and rough-wall profiles. It can be seen that the maximum deviation observed in any of the profiles in the outer layer is ${<}5\,\%$ . Numerous studies have reported an increase in the wake strength, ${\it\Pi}$ , of rough-wall flows relative to that for smooth-wall flows (Krogstad et al. Reference Krogstad, Antonia and Browne1992; Keirsbulck et al. Reference Keirsbulck, Labraga, Mazouz and Tournier2002; Akinlade et al. Reference Akinlade, Bergstrom, Tachie and Castillo2004; Bergstrom, Akinlade & Tachie Reference Bergstrom, Akinlade and Tachie2005; Castro Reference Castro2007). Castro, Segalini & Alfredsson (Reference Castro, Segalini and Alfredsson2013) suggests that ${\it\Pi}$ may be a particularly sensitive measure of whether the outer-region boundary layer structure is truly universal, noting rough-wall studies in which the outer-layer stress profiles collapse well with those for smooth walls, but where ${\it\Pi}$ differs. Based on an examination of extant rough-wall data, Castro et al. (Reference Castro, Segalini and Alfredsson2013) suggest that to ensure outer flow similarity the flow must be fully rough with ${\it\delta}_{99}/k_{s}\gtrsim 11$ (note Castro et al. (Reference Castro, Segalini and Alfredsson2013) actually pose their approximate criterion on ${\it\delta}_{99}/z_{0}$ , where $z_{0}$ is an alternative measure to $k_{s}^{+}$ : using ${\it\kappa}=0.39$ , $k_{s}=27.5z_{0}$ ). In figure 6(a) all data satisfy ${\it\delta}_{99}/k_{s}\gtrsim 11$ . Evidence of invariance, however, is observed even for data that are transitionally rough.

It is important to note that perfect invariance of the mean defect velocity is not expected even for smooth-wall flows. For example, numerous studies (Erm & Joubert Reference Erm and Joubert1991; Schlatter & Örlü Reference Schlatter and Örlü2012; Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015) have shown that ${\it\Pi}$ can be a strong function of $x$ immediately downstream of the trip, with the tripping strength also having an effect. Only at some distance downstream of the trip does ${\it\Pi}$ appear to become asymptotically constant (see Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015). There are also questions regarding whether ${\it\Pi}$ depends on Reynolds number (Coles Reference Coles1962; De Graaff & Eaton Reference De Graaff and Eaton2000; Perry, Marusic & Jones Reference Perry, Marusic and Jones2002). Therefore, the observed agreement between the smooth- and rough-wall defect data provides only ostensible support for Townsend’s wall-similarity hypothesis. Certainly, the deviation $E$ is generally higher for the smooth-wall data (figure 6 c) than for the rough-wall data, particularly at lower speeds. This may be indicative of some dependence on upstream conditions. It is, however, encouraging that the profiles of $E$ are very similar between the smooth and rough walls; suggesting that Townsend’s hypothesis is, at worst, a good approximation for the mean streamwise velocity defect.

Figure 7.

The streamwise velocity variance for all smooth- and rough-wall data. A linear abscissa is used in (a) and a logarithmic abscissa is used in (b). In the inset of (b), the highest Reynolds number smooth-wall profile ( ${\it\delta}_{99}^{+}=21\,430$ ) is compared to the highest Reynolds number rough-wall profile ( ${\it\delta}_{99}^{+}=29\,900$ ).

In figure 7 all smooth- and rough-wall turbulence intensity profiles are presented on an outer-normalised abscissa (linearly and logarithmically spaced in figure 7(a,b), respectively). All rough- and smooth-wall data apparently merge beyond $(z+{\it\epsilon})/{\it\delta}_{99}\approx 0.5$ . However, even at this location the data spreads over a range, $\overline{u^{2}}_{max}^{+}-\overline{u^{2}}_{min}^{+}\approx 0.7$ . In figure 7(c) the highest Reynolds number smooth- and rough-wall data are compared. Here, excellent agreement between these two profiles is observed external to the beginning of the inertial sublayer, providing convincing support for Townsend’s hypothesis, at least for fully rough flows at high friction Reynolds numbers. Comparisons of smooth- and rough-wall streamwise variance profiles across a wide range of friction and roughness Reynolds numbers (such as in figure 7 b) are common in the literature for assessing the efficacy of Townsend’s hypothesis. However, such comparisons can be difficult to interpret since the variance has a strong dependence on ${\it\delta}_{99}^{+}$ . See, for example, figure 8(a) where all inner-normalised smooth-wall variance profiles are plotted on an outer-normalised abscissa. Disregarding any experimental error in the data, and acknowledging that any effects of poor spatial resolution are negligible beyond $z/{\it\delta}_{99}\approx 0.02$ (see § 3.1), the spread of the profiles internal to $z/{\it\delta}_{99}\approx 0.2$ in figure 8(a) is evidently due to the inherent trends with ${\it\delta}_{99}^{+}$ . Returning to figure 7, it appears that data with $k_{s}^{+}\lesssim 70$ (lighter shaded symbols) are attenuated below the fully rough and smooth-wall data. Additionally, the degree of attenuation apparently depends on the magnitude of $k_{s}^{+}$ ; with increasing $k_{s}^{+}$ (darkening symbols) the transitionally rough data lie increasingly closer to the cluster of fully rough and smooth profiles. However, for the present experiments, data with a low $k_{s}^{+}$ typically also have a low ${\it\delta}_{99}^{+}$ . Therefore, in figure 7 it is difficult to isolate trends with $k_{s}^{+}$ from trends with ${\it\delta}_{99}^{+}$ . Note that comparison of rough-wall variance profiles at matched friction Reynolds number but different roughness Reynolds number (such as the lowest friction Reynolds number profiles in figure 8(b,c), respectively) suggest that the trends described above are related to $k_{s}^{+}$ , rather than ${\it\delta}_{99}^{+}$ . This is discussed in further detail in §§ 3.3 and 3.4.

Figure 8.

The friction Reynolds number trends of the streamwise velocity variance above the smooth and rough wall. (a) Shows the smooth-wall data with some intermediate Reynolds number data removed for clarity, (b) shows the rough-wall data for Case 1 at 21.7 m, (c) shows the single-wire rough-wall data for Case 2 at $U_{\infty }\approx 20~\text{m}~\text{s}^{-1}$ with every second streamwise location removed for clarity. The symbols with thick black outlines show the location of the onset of inertial dynamics according to (2.4).

Figure 9.

Streamwise skewness profiles for the smooth and rough walls. (a) Shows the rough-wall data for Case 1 at 21.7 m, (b) shows the single-wire rough-wall data for Case 2 at $U_{\infty }\approx 20~\text{m}~\text{s}^{-1}$ . The highest and lowest Reynolds number smooth-wall profiles are included in each plot for comparison. Note that data beyond $(z+{\it\epsilon})/{\it\delta}_{99}=1.5$ are removed for clarity.

The streamwise velocity skewness profiles for the Case 1 data at 21.7 m and single-wire Case 2 data at $U_{\infty }\approx 20~\text{m}~\text{s}^{-1}$ are presented in figure 9(a,b), respectively. For comparison, the highest and lowest friction Reynolds number smooth-wall skewness profiles are also included in both figures. Note that the skewness is normalised by the standard deviation cubed and is therefore not subject to errors in $U_{{\it\tau}}$ . Interestingly, the smooth- and rough-wall Case 1 data in figure 9(a) apparently merge beyond $(z+{\it\epsilon})/{\it\delta}_{99}\approx 0.02$ , suggesting that $k_{s}^{+}$ has little effect on skewness. Conversely, in figure 7(b) there is a clear trend in the rough-wall data. Inside of $(z+{\it\epsilon})/{\it\delta}_{99}\approx 0.2$ the rough-wall skewness is larger than that of the smooth wall, with a greater difference nearer to the wall. The discrepancy between the smooth- and rough-wall data in this region also appears to decrease with increasing distance from the trip, $x$ (that is, increasing ${\it\delta}_{99}^{+}$ or increasing ${\it\delta}_{99}/k_{s}$ ). Since the range of Reynolds numbers spanned in figure 9(a,b) is approximately the same, it appears that the rough-wall skewness trends observed in figure 9(b) result from the changes in ${\it\delta}_{99}/k_{s}$ rather than the changes in ${\it\delta}_{99}^{+}$ . However, this result is opposite to that observed in figures 5 and 8. Here, changing $k_{s}^{+}$ appeared to influence the general shape of the streamwise velocity variance, whereas changing ${\it\delta}_{99}/k_{s}$ had seemingly little influence (at least over the range of values considered).

3.3 The inertial sublayer in rough-wall flows

In wall-bounded flows, there is now considerable support for the existence of a logarithmic region (or inertial sublayer in physical space). Here, the streamwise velocity is independent of first-order viscous effects and scales with $U_{{\it\tau}}$ regardless of the characteristic length scale for $z$ (an alternative viewpoint is that $z$ is the characteristic length scale). In smooth-wall flows, there is theoretical and experimental evidence that the wall-normal location of the beginning of the inertial sublayer scales with $\sqrt{{\it\delta}^{+}}$ , the geometric mean of the inner and outer length scales of the flow (Klewicki, Fife & Wei Reference Klewicki, Fife and Wei2009; Chin et al. Reference Chin, Philip, Klewicki, Ooi and Marusic2014). In rough-wall flows, Mehdi et al. (Reference Mehdi, Klewicki and White2013) hypothesise and show evidence that, as in smooth-wall flows, the location of the onset of inertial dynamics scales with a geometric ratio of the scales present in the flow, including the length scales imposed by the roughness (see (2.3)).

In what is commonly referred to as the attached eddy hypothesis, Townsend (Reference Townsend1976) theorized that the logarithmic scaling of the wall can be associated with a three-dimensional distribution of self-similar eddying motions whose sizes scale with their distance from the wall. The attached eddy hypothesis is independent of viscosity and consistent with the leading-order mean dynamics in the boundary layer inertial sublayer. Townsend showed that this description leads to logarithmic profiles in the streamwise mean velocities, and in the spanwise and streamwise velocity variances. The latter is shown to have the form:

Note that the constant $A_{1}$ is expected to be universal, but $B_{1}$ is expected to change with external flow geometry. Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013) provide empirical evidence for (3.2) in smooth-wall flows over approximately the same wall-normal domain as the logarithmic region of the mean velocity profile.

In figure 10 the streamwise mean velocity and turbulence intensity are presented for all rough-wall profiles. On the abscissa, the wall-normal location, $z$ , is normalised by the predicted location of the onset of inertial dynamics, $z_{I}$ , aligning $(z+{\it\epsilon})=z_{I}$ for each profile at $(z+{\it\epsilon})/z_{I}=1$ . For clarity, the streamwise mean velocities and turbulence intensities in figure 10 are shifted by their respective values at the location of the beginning of the inertial sublayer. Using (3.2), it is easy to show

That is, figure 10 is independent of the constant $B_{1}$ across the inertial sublayer. The darker points in each profile indicate the region of data contained within the inertial sublayer (defined here for rough-wall flows as $z_{I}^{+}<z^{+}<0.19{\it\delta}_{99}^{+}$ ). The data all show a distinct logarithmic profile in both the streamwise mean velocity and turbulence intensity, with uniformly good agreement across the full range of Reynolds numbers and roughness conditions. A similar observation was also made by Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2013) in high Reynolds number rough-wall pipe flows, but using a different normalisation for $z$ . The dashed black lines that bound the data in figure 10(b) have a slope of $A=1.26$ (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013) and show a range in $\overline{u^{2}}^{+}-\overline{u^{2}}^{+}|_{z=z_{I}}$ of $\pm 0.2$ . Note that in order to obtain $z_{I}$ from the formulation of Mehdi et al. (Reference Mehdi, Klewicki and White2013) a value for the constant, $C_{I}$ , is required. Here, $C_{I}=2.5$ is used, since this yields good collapse of the data in figure 10. This paper, however, is not concerned with determining an exact value for $C_{I}$ . The important result here is that good collapse can be obtained using a fixed value for $C_{I}$ that is $O(1)$ .

Figure 10.

The streamwise mean velocity and turbulence intensity with the abscissa shifted by the estimated location of the onset of inertial dynamics $z_{I}$ , and the ordinate shifted by its respective value at this location. All rough-wall data are presented. The dark points in each profile show the data contained within the inertial sublayer according to the definitions in § 2.4. The dashed black lines that bound the data in 10(b) have a slope of $A=1.26$ (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013) and show a range $\overline{u^{2}}^{+}-\overline{u^{2}}^{+}|_{z=z_{I}}=\pm 0.2$ .

The results in figure 10 suggest that the formulation proposed by Mehdi et al. (Reference Mehdi, Klewicki and White2013) provides, for the current rough surface conditions ( $z_{m}>k_{s}^{+}$ ), a good estimate of the wall-normal scale at which inertial dynamics become dominant. Additionally, the slope of the logarithmic regions of the rough-wall streamwise mean velocities and turbulence intensities agree within experimental uncertainty to those found above smooth walls (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). This suggests similarity between the inertial region dynamics in smooth- and rough-wall flows. The poor collapse of the wake region of the streamwise velocity variance in figure 10 is, to some extent, due to the inherent friction Reynolds number trends associated with this statistic. However, figures 7 and 8 indicate that the outer region of the flow also has a dependence on $k_{s}^{+}$ at least for transitionally rough surfaces at low ${\it\delta}_{99}^{+}$ . In figure 11, the constant $B_{1}$ from (3.2) is presented as a function of friction and roughness Reynolds number. For each profile, $B_{1}$ is determined by minimizing the root-mean-square error between that profile and (3.2) across the inertial region with $A_{1}=1.26$ . For the smooth-wall data, $B_{1}$ appears to become approximately constant for ${\it\delta}_{99}^{+}\gtrsim 4000$ . The hatched regions in figure 11(a,b) show $B_{1}=2.17\pm 0.2$ , where $B_{1}=2.17$ was determined for the highest Reynolds number smooth-wall dataset and the range was chosen to match that shown by the dashed lines in figure 10(b). That is, for any two datasets with $B_{1}$ values that agree to within $0.4$ , $\overline{u^{2}}^{+}$ (when plotted against $(z+{\it\epsilon})/{\it\delta}_{99}$ ) will exhibit equivalent collapse across the entire inertial sublayer to that observed in figure 10(b). For the rough-wall data, $B_{1}$ appears to be a weak function of ${\it\delta}_{99}^{+}$ and a strong function of $k_{s}^{+}$ ; the purple surface in figure 11(c) shows the general trend of the data. At high $k_{s}^{+}$ and ${\it\delta}_{99}^{+}$ , $B_{1}$ appears to plateau to the same approximate value as for the smooth wall. For all flows with $B_{1}\simeq 2.17$ , similarity of the inner-normalised streamwise variance is observed beyond $z_{I}$ regardless of wall condition (accepting, of course, that there is universality in the wake region: see figure 7). Inertial and outer-layer collapse of the inner-normalised streamwise velocity variance is ostensibly observed for the present roughness geometry for ${\it\delta}_{99}^{+}\gtrsim 14\,000$ , or independent of ${\it\delta}_{99}^{+}$ if $k_{s}^{+}\gtrsim 100$ . At low/intermediate friction and roughness Reynolds numbers, however, it appears that for the present roughness $U_{{\it\tau}}$ and ${\it\delta}$ are not sufficient to scale the streamwise velocity in the outer region of the flow, and that some representation of the roughness strength is necessary (e.g. $B_{1}$ in figures 7 and 8). This is discussed in further detail in the following section.

Figure 11.

$B_{1}$ (see (3.2)) as a function of ${\it\delta}_{99}^{+}$ and $k_{s}^{+}$ . The shaded regions in (a) and (b) show $B_{1}=2.17\pm 0.2$ , where $B_{1}=2.17$ was determined for the highest Reynolds number smooth-wall dataset. A two-dimensional exponential surface is fitted to the data in (c) to demonstrate its general trend.

3.4 Matched friction Reynolds number comparisons

Figure 12.

Smooth- and rough-wall streamwise variance and skewness comparisons at approximately matched friction Reynolds numbers ((a) ${\it\delta}_{99}^{+}\approx 4000$ . (b) ${\it\delta}_{99}^{+}\approx 2900$ . (c) ${\it\delta}_{99}^{+}\approx 9800$ . (d) ${\it\delta}_{99}^{+}\approx 5400$ . (e) ${\it\delta}_{99}^{+}\approx 17\,100$ . (f) ${\it\delta}_{99}^{+}\approx 7900$ . (g) ${\it\delta}_{99}^{+}\approx 4700$ . (h) ${\it\delta}_{99}^{+}\approx 22\,000$ ). The inset in each figure shows the $k_{s}^{+}$ and ${\rm\Delta}U^{+}$ values of the rough-wall data in that figure. The figures are ordered from low $k_{s}^{+}$ to high $k_{s}^{+}$ . Dashed black lines mark the wall-normal locations of $3k_{s}/{\it\delta}_{99}$ . The symbols with thick black outlines show the location of the onset of inertial dynamics ( $z_{I}$ ) according to (2.4).

Figure 12 compares the smooth- and rough-wall streamwise variance and skewness at eight approximately matched friction Reynolds numbers in the range $2900\lesssim {\it\delta}_{99}^{+}\lesssim 22\,000$ . The figures are ordered from top to bottom according to the magnitude of $k_{s}^{+}$ . Matched Reynolds number comparisons such as these may avert, to some extent, the inherent friction Reynolds number trends of smooth- and rough-wall statistics that can cloud assessments of outer-layer similarity. However, it is difficult to apply any physical significance to such comparisons, other than that in each case the range of scales (between viscous and outer) present in the flow is approximately the same between the smooth and rough wall. Certainly, from a practical standpoint it may be more informative to compare smooth- and rough-wall boundary layers at matched $Re_{x}$ ( $Re_{x}=xU_{\infty }/{\it\nu}$ , where $x$ is the distance from the laminar/turbulent trip) since such comparisons would demonstrate the effect of roughness in developing engineering flows such as on the hull of a ship or on the wing/fuselage of an aeroplane. The comparisons in figure 12 are presented simply because they reveal interesting trends in rough-wall-bounded flows. The aim here is to reveal any influence of roughness Reynolds number on low order rough-wall streamwise statistics in the absence of known friction Reynolds number trends. Primarily, the focus is on the outer region of the flow, with emphasis on the accuracy of Townsend’s wall-similarity hypothesis.

For all the transitionally rough flows ( $k_{s}^{+}<$ 70) in figure 12, apparent differences in the variance of the streamwise velocity fluctuations exist well into the outer layer when compared to the smooth wall at a matched Reynolds number. Note that this is consistent with the observed variations in $B_{1}$ for these transitionally rough surfaces. Of course, Townsend’s hypothesis is posed for high Reynolds number flows, and all the transitionally rough cases have ${\it\delta}_{99}^{+}\lesssim 10\,000$ . The results, however, indicate that, as the flow becomes fully rough, similarity in the variance of the streamwise velocity fluctuations in the outer layer emerges even for cases in which the friction Reynolds number is relatively low. For example, the ${\it\delta}_{99}^{+}\approx$ 4700, $k_{s}^{+}=$ 121 case (figure 12 g) displays similarity in the outer layer with the smooth-wall case at matching Reynolds number. To the authors’ knowledge, Townsend (Reference Townsend1956) hypothesised wall similarity across all flow conditions, including transitionally rough flows. Certainly, there are numerous instances in the existing literature where comparisons between smooth and transitionally rough boundary layers are made and outer-layer similarity is empirically found to hold (Schultz & Flack Reference Schultz and Flack2003, Reference Schultz and Flack2007; Wu & Christensen Reference Wu and Christensen2007; Allen et al. Reference Allen, Shockling, Kunkel and Smits2007). The present results tentatively suggest, however, that the outer region of the inner-normalised streamwise velocity variance profile has a dependence on $k_{s}^{+}$ across the range of boundary layer and roughness parameters considered here. It is interesting to note that for the rough-wall data in figure 12(a) the drag balance measurements indicate an increase in the wall drag coefficient relative to the smooth wall at matched ${\it\delta}^{+}$ (presumably due to the pressure drag caused by the roughness elements). If, however, the ordinate of figure 12(a) is normalised by $U_{\infty }^{2}$ instead of $U_{{\it\tau}}^{2}$ , excellent collapse between the smooth- and rough-wall variance profiles is observed, indicating that for these data the strength of the streamwise turbulent fluctuations relative to the free stream velocity appears to be relatively unaffected by the rough surface. It is possible that this observation, and indeed the general trends with $k_{s}^{+}$ observed in figure 12, are caused by (or at least augmented by) experimental errors. For reasons discussed in detail in appendix A, it is not believed that this is the case.

Another observation from figure 12 is that the distance from the wall at which similarity begins to be observed – the edge of the roughness sublayer – in flows at high $k_{s}^{+}$ does not seem to scale in any straightforward manner on $k$ , $k_{s}$ or ${\it\delta}_{99}$ . This result stands in contrast to the findings of Raupach, Thom & Edwards (Reference Raupach, Thom and Edwards1980), Flack et al. (Reference Flack, Schultz and Shapiro2005) and Krogstad & Efros (Reference Krogstad and Efros2012) who assert that the region directly influenced by the roughness is confined to within approximately $5k$ of the wall. It should be noted, however, that the conclusions of these studies are based on lower Reynolds number flows in which the scale separation between $k$ and ${\it\delta}_{99}$ was much smaller than in the present case. Furthermore, other studies have documented rough-wall modifications to the Reynolds stresses that extend deep into the outer layer (Krogstadt & Antonia Reference Krogstadt and Antonia1999; Tachie et al. Reference Tachie, Bergstrom and Balachandar2000; Keirsbulck et al. Reference Keirsbulck, Labraga, Mazouz and Tournier2002). Based on the present results, the location at which inertial dynamics emerge, $z_{I}$ , for the rough-wall flows appears to be a better indicator of where similarity in the variance of the streamwise velocity fluctuations is observed, but only when $k_{s}^{+}$ is sufficiently large.

Figure 13.

Smooth- and rough-wall premultiplied streamwise energy spectrograms at approximately matched ${\it\delta}_{99}^{+}$ , and the difference between them. Column 1, the smooth-wall spectrograms; Column 2, the rough-wall spectrograms; Column 3, subtraction of the rough-wall spectrograms from the smooth-wall spectrograms. Rows (a–f) present the spectrograms for most of the comparisons in figure 12 ( ${\it\delta}_{99}^{+}\approx 4000,2900,17\,100,7900,4700,22\,000$ , respectively). Therefore $k_{s}^{+}$ increases from the top to the bottom of the figure. In the third column, contour lines are plotted at $k_{x}{\it\Phi}_{uu}/U_{{\it\tau}}^{2}|_{SW-RW}=\pm [0.15,0.3,0.45,0.6]$ . White lines represent positive contours and black lines represent negative contours. The dashed black vertical lines show the wall-normal location of $k_{s}^{+}$ and the solid black vertical lines show $z_{I}$ .

The comparisons of figure 12 are further scrutinised by considering the inner-normalised premultiplied streamwise turbulent energy spectrograms of selected smooth- and rough-wall datasets in figure 13. The same matched Reynolds number comparisons as those given in figure 12(a,b), (e–h) are presented. The first column in figure 13 shows the smooth-wall spectrograms; the second column shows the rough-wall spectrograms and the third column shows the difference between the smooth- and rough-wall spectrograms. In all comparisons, the Reynolds number is sufficient to reveal in the smooth-wall data wall-normal separation between the near-wall (predominantly small scale) energy peak and the logarithmic/outer-region (large scale) energy peak (Hutchins & Marusic Reference Hutchins and Marusic2007b ). In comparison (a) ( ${\it\delta}_{99}^{+}\approx 4000$ and $k_{s}^{+}=22$ ) the smooth- and rough-wall spectrograms are qualitatively similar, but the magnitude of the rough-wall energy at all scales across the boundary layer is reduced relative to that of the smooth wall. At higher roughness Reynolds numbers, previous studies have shown that the surface roughness causes a reduction in the near-wall energy peak relative to that of the smooth wall (Grass Reference Grass1971; Schultz & Flack Reference Schultz and Flack2007). This observation, however, is not made here due to the lack of resolved energy in this region. Farther into the boundary layer, figure 13(b–f) indicates the differences between the smooth- and rough-wall boundary layers appear to be primarily in the large scales of motion, i.e. motions that are several ${\it\delta}_{99}$ in length (note that a similar observation was also made by Monty et al. (Reference Monty, Allen, Lien and Chong2011) above a braille-type wall roughness). The magnitude of the difference between the smooth- and rough-wall large-scale near-wall motions seems to be related to the friction Reynolds number, with larger differences observed for flows with higher $k_{s}^{+}$ . It also appears that the effect of the rough-wall on the large-scale motions extends farther into the boundary layer (in wall units) with increasing Reynolds number (recall that figure 13 is ordered vertically according to $k_{s}^{+}$ , not ${\it\delta}_{99}^{+}$ ). However, this trend (if real) is weak, and thus difficult to differentiate from experimental uncertainties. Note that in the third column of figure 13, it is apparent that $z_{I}$ (black lines) provides a reliable indicator of the wall-normal location beyond which there is little difference between the smooth- and rough-wall spectrograms. It then follows that this location scales with ${{\it\delta}_{99}^{+}}^{0.64}$ (as per (2.4)), perhaps explaining the apparent trend with ${\it\delta}_{99}^{+}$ .