3.1 Mean velocity

Figure 2.

Present rough wall mean velocity profiles at several Reynolds numbers, $630\leqslant \unicode[STIX]{x1D6FF}_{99}^{+}\leqslant 7200$ . See table 1 for symbols. The dot-dashed line represents the slope of the log region in a smooth wall turbulent boundary layer. Inset: variation of $U_{\infty }^{+}$ with $\unicode[STIX]{x1D6FF}_{99}^{+}$ (open symbols represent the corresponding $\unicode[STIX]{x1D6FF}^{+}$ values).

Figure 2 shows mean velocity profiles normalised using inner scaling parameters, i.e. $U^{+}$ versus $y^{+}$ , at a fixed $x$ location as $Re_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D6FF}_{99}^{+}$ increases from 630 to 7200. It is clearly evident that the mean velocity profile shifts downward and towards larger $y^{+}$ as $Re_{\unicode[STIX]{x1D70F}}$ increases, which is due to $U_{\unicode[STIX]{x1D70F}}$ increasing with $Re_{\unicode[STIX]{x1D70F}}$ . However, the rate of the downward shift decreases and eventually becomes zero, i.e. the mean velocity profile ceases to move downwards once $Re_{\unicode[STIX]{x1D70F}}$ reaches a value of approximately 4000. Beyond this value, the profile simply shifts to larger $y^{+}$ , while its shape remains unchanged. The inset plot in figure 2 shows the variation of $U_{\infty }^{+}$ as a function of $Re_{\unicode[STIX]{x1D70F}}$ (or $\unicode[STIX]{x1D6FF}_{99}^{+}$ ). The trend is clear: as $Re_{\unicode[STIX]{x1D70F}}$ increases, $U_{\infty }^{+}(\equiv [\sqrt{C_{f}/2}]^{-1})$ first decreases before reaching a constant value for $Re_{\unicode[STIX]{x1D70F}}\geqslant 4000$ . This indicates that the downward shift ceases at the same time as the coefficient of friction becomes constant and the TBL can be considered fully rough; in this fully rough regime, the drag is solely made of the form drag of the roughness elements. The continuous shift of the entire profile to larger values of $y^{+}$ suggests that the length scale $\unicode[STIX]{x1D708}/U_{\unicode[STIX]{x1D70F}}$ is not an appropriate length scale. The profile continues to shift horizontally because $U_{\unicode[STIX]{x1D70F}}$ increases with $Re_{\unicode[STIX]{x1D70F}}$ (see table 1), thus resulting in $y^{+}$ increasing for a fixed value of $y$ , but maintaining its shape.

To verify the fixed form of the shape, we plot in figure 3 the three velocity profiles in figure 2 for which $U_{\infty }^{+}$ is constant as a function of $y/k$ . There is a clear collapse of the profiles across the entire boundary layer onto a single curve of the form:

(3.1) $$\begin{eqnarray}U^{+}=F(y/k),\end{eqnarray}$$

where $F$ is a function, which appears to be independent of Reynolds number. A similar behaviour ( $U_{\infty }^{+}=\text{const}$ ) can be observed in the sand-grain rough wall profiles of Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016) ( $3970\leqslant Re_{\unicode[STIX]{x1D70F}}\leqslant 29\,900$ ). However, it is achieved at a much higher $Re_{\unicode[STIX]{x1D70F}}$ than in the present rough wall, illustrating the fact that a sand-grain rough wall TBL requires a larger $Re_{\unicode[STIX]{x1D70F}}$ than a 2-D rod rough wall TBL before it can become fully rough. For comparison, we also report in figure 3, the rough wall mean velocity profile of Krogstad & Efros (Reference Krogstad and Efros2012) (2-D transverse square bars with $p=8k$ ). Considering that $\unicode[STIX]{x1D6FF}^{+}=13\,300$ and $k_{s}\simeq 325$ , one expects the profile to be in the fully rough regime. Despite the large difference in $Re_{\unicode[STIX]{x1D70F}}$ between our data and those of Krogstad & Efros (Reference Krogstad and Efros2012), the latter profile is not too different from the present distributions, although its value of $U_{max}^{+}$ is slightly larger, reflecting a lower $C_{f}$ in comparison to the present study; $C_{f}\simeq 0.0072$ , and 0.0068, for the present 2-D rough wall and that of Krogstad & Efros (Reference Krogstad and Efros2012), respectively.

Figure 3.

Mean velocity profiles in the present rough wall (black symbols) at several Reynolds numbers, $3940\leqslant Re_{\unicode[STIX]{x1D70F}}\leqslant 7200$ , sand-grain roughness (red symbols, $Re_{\unicode[STIX]{x1D70F}}=20\,160$ , 25 020 and 29 900, Squire et al. Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016) and square bar roughness (blue symbols, $Re_{\unicode[STIX]{x1D70F}}=13\,300$ , Krogstad & Efros Reference Krogstad and Efros2012) as a function of $y/k$ .

Figure 4 shows a clear collapse of the mean velocity profiles over the present rough wall, when plotted as a function of $y/\unicode[STIX]{x1D6FF}_{99}$ . The same trend is observed in the sand-grain roughness data of Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016), however, the latter collapse onto a different curve. This difference is due to the difference in $C_{f}$ between the two rough wall TBLs. It is well known that there is no such collapse for smooth wall TBL profiles because of $Re_{\unicode[STIX]{x1D70F}}$ -dependence of $C_{f}$ . This is well illustrated in figure 4, where we plot the high Reynolds number smooth wall data of Marusic et al. (Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015) ( $3370\leqslant Re_{\unicode[STIX]{x1D70F}}\leqslant 9830$ ) and the direct numerical simulation (DNS) data of Schlatter & Örlü (Reference Schlatter and Örlü2010) ( $Re_{\unicode[STIX]{x1D70F}}=674$ ). Of particular interest, is the rough wall velocity profile of Krogstad & Efros (Reference Krogstad and Efros2012), also reported in the figure for comparison. This profile collapses relatively well onto the present 2-D rough wall profiles, as it could have been anticipated from figure 3 and the similar values of $C_{f}$ , suggesting that both TBLs share similarities. This may not be too surprising considering the rough walls share almost the same roughness geometry, cylindrical rods and square bars with the same $k$ and $p$ , although $\unicode[STIX]{x1D6FF}/k$ for the square bar rough wall is approximately twice the present value.

Figure 4.

Mean velocity profiles over the present 2-D rough wall (back symbols) as a function of $y/\unicode[STIX]{x1D6FF}$ . Symbols as in table 1. Red symbols: (Squire et al. Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016,   $Re_{\unicode[STIX]{x1D70F}}=20\,160$ , 25 020, and 29 900); blue symbols: (Krogstad & Efros Reference Krogstad and Efros2012,   $Re_{\unicode[STIX]{x1D70F}}=13\,300$ ); dashed lines: smooth wall data (Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015,   $Re_{\unicode[STIX]{x1D70F}}=3370$ , 4760, 6450 and 9830) and solid line: smooth wall DNS data (Schlatter & Örlü Reference Schlatter and Örlü2010,   $Re_{\unicode[STIX]{x1D70F}}=674$ ). Only the profiles for which $U_{\infty }/U_{\unicode[STIX]{x1D70F}}$ is constant in the present rough wall and sand-grain roughness (Squire et al. Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016) TBLs are shown.

Altogether, the results presented in figures 2–4 are consistent with a $Re$ independence of the normalised mean velocity profile, which would suggest that the Reynolds number similarity is well achieved in a rough wall TBL, at least in the context of the mean velocity. Note that unlike a smooth wall TBL, where one must exclude the viscous-dominated near-wall region if such similarity is to be observed, the Reynolds number similarity is observed across practically the entire boundary layer for a fully rough TBL. This is possible because the spatially averaged viscosity effects in the near-wall region are either zero or negligible. The sand-grain rough wall data of Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016) also show that when $Re$ is large enough the normalised velocity profiles collapse onto a single distribution. However, that distribution differs from that of the present data due to a difference in the value of the form drag coefficient. To illustrate this, we employ the so-called diagnostic plot (Alfredsson Reference Alfredsson and Örlü2010; Alfredsson, Segalini & Örlü Reference Alfredsson, Segalini and Örlü2011; Örlü et al. Reference Örlü, Segalini, Klewicki and Alfredsson2016), which consists of plotting $u^{\prime }$ as a function of $U$ , thus removing any ambiguity associated with knowing $y$ and $U_{\unicode[STIX]{x1D70F}}$ accurately. Figure 5 shows such plots, where $u^{\prime }$ and $U$ are normalised by $U$ and $U_{\infty }$ , respectively. While for each surface there is a clear collapse of the distributions across the entire boundary layer, the collapse is different between the two surfaces, in conformity with two distinct drag coefficients ( $C_{f}\simeq$ 0.0072 and 0.0032 for the present data and the sand-grain roughness, respectively). Further, the trend shown by the data indicates that the distributions for the two rough walls will remain distinct, regardless of the Reynolds number, hence suggesting two distinct $Re$ -independent profiles. It appears thus that a $Re$ -independent normalised mean velocity profile for the complete TBL is possible for fully rough wall TBLs. However, the profile cannot be universal since it is controlled by the rough wall drag coefficient which varies between roughness geometries as evidenced in figure 4.

Figure 5.

Diagnostic plot for the present rough wall (black symbols), sand-grain roughness (red symbols, $Re_{\unicode[STIX]{x1D70F}}=20\,160$ , 25 020, and 29 900, Squire et al. Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016) and the square bar roughness (blue symbols, $Re_{\unicode[STIX]{x1D70F}}=13\,300$ , Krogstad & Efros Reference Krogstad and Efros2012).

Another parameter that is also likely to control the $Re$ -independent profile of a rough wall TBL, for a given roughness geometry, is the ratio $\unicode[STIX]{x1D6FF}/k$ . It is expected that for a given roughness geometry, $C_{f}$ will change with $\unicode[STIX]{x1D6FF}/k$ ; $\unicode[STIX]{x1D6FF}/k=63$ and 200 for the present profiles and those of Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016), respectively, shown in figures 4 and 5. This raises the following question: can the $Re$ -independent profiles for both rough walls collapse if one varies $\unicode[STIX]{x1D6FF}/k$ ? (Such collapse would result in both rough walls yielding the same diagnostic distribution.) The above results show that simply invoking Townsend’s Reynolds number similarity to address the question is not enough. The answer requires one to carry out a series of parametric measurements for both rough wall TBLs in the fully rough regime by systematically changing the value of $\unicode[STIX]{x1D6FF}/k$ . However, one expects to observe the same result as presented here. Indeed, consider for example the 2-D rough wall where one changes $k$ , while maintaining $p/k$ constant and equal to 8. Regardless of $k$ and as long as $\unicode[STIX]{x1D6FF}/k$ is large enough (at least larger than 40 (Jiménez Reference Jiménez2004)), the fully rough regime will yield the same $C_{f}$ as for the present case, and thus the same normalised velocity profiles as shown in figure 2 will be obtained. This appears to be confirmed by the data of Krogstad & Efros (Reference Krogstad and Efros2012) reported in figure 5. Despite the difference in the cross-sectional shape between the present roughness and that of Krogstad & Efros (Reference Krogstad and Efros2012) (circular rods versus square bars), the distributions are practically identical. Of course, the critical $Re$ at which the fully rough regime will be achieved will vary with $k$ . This implies that two distinct roughness geometries, as for example in the present work, will lead to two distinct $Re$ -independent normalised mean velocity profiles in the fully rough regime. These differences can be further elucidated by comparing the turbulence intensity profiles in the outer region of the smooth and rough wall TBLs based on a modified diagnostic plot proposed by Castro, Segalini & Alfredsson (Reference Castro, Segalini and Alfredsson2013) and reported in figure 6. This modified diagnostic plot produces a better collapse of all curves than the original one. It seems that by introducing the velocity deficit ( $\unicode[STIX]{x0394}U$ ), the difference between smooth, and different types of rough walls is dramatically reduced. Note though that the collapse is not perfect. While the non-collapse for $U^{\prime }/U_{\infty }^{\prime }\leqslant 0.7$ can be explained in terms of the differences in turbulence levels in the near-wall region of different types of TBLs, the reason for the variation in the region $U^{\prime }/U_{\infty }^{\prime }\geqslant 0.8$ between the 2-D bars and the sand-grain roughness is not clear; this may reflect a real difference associated with the roughness geometry or the possibility that the sand-grain paper data of Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016) are not fully rough (albeit $C_{f}$ appears to be constant in the experimental data of Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016)).

Figure 6.

Comparison of turbulence intensities in the outer layer of smooth (dotted line, $Re_{\unicode[STIX]{x1D70F}}=9830$ , Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015) and rough wall turbulent boundary layers (data from figure 5) using the modified diagnostic plot (Castro et al. Reference Castro, Segalini and Alfredsson2013). Here, $U^{\prime }=U+\unicode[STIX]{x0394}U$ and $U_{\infty }^{\prime }=U_{\infty }+\unicode[STIX]{x0394}U$ and $\unicode[STIX]{x0394}U$ is the velocity defect. The solid line is the smooth wall outer layer straight line from Alfredsson, Örlü & Segalini (Reference Alfredsson, Örlü and Segalini2012).

Figure 7.

Comparison of velocity defect profiles in smooth wall (black dashed lines, $Re_{\unicode[STIX]{x1D70F}}=3370$ , 4760, 6450 and 9830, Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015), sand-grain roughness (red symbols, $Re_{\unicode[STIX]{x1D70F}}=20\,160$ , 25 020 and 29 900, Squire et al. Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016), square bar (blue symbols, $Re_{\unicode[STIX]{x1D70F}}=13\,300$ , Krogstad & Efros Reference Krogstad and Efros2012) and the present rough wall data (black symbols) normalised by (a) $U_{\unicode[STIX]{x1D70F}}$ and (b) $U_{\infty }$ . Refer to caption of figure 4 for symbols used for smooth wall, sand-grain and square bar rough wall TBLs.

It is common to test the universality of TBL velocity profiles by plotting the velocity profiles in the deficit form, (Townsend Reference Townsend1956):

(3.2) $$\begin{eqnarray}\frac{U_{\infty }-U}{u_{o}}=f(y/\unicode[STIX]{x1D702}),\end{eqnarray}$$

where $u_{o}$ and $\unicode[STIX]{x1D702}$ are velocity and length scales, respectively, and have been the subject of many investigations (i.e. Townsend (Reference Townsend1976), George & Castillo (Reference George and Castillo1997), Zagarola & Smits (Reference Zagarola and Smits1998), Jones, Nickels & Marusic (Reference Jones, Nickels and Marusic2008), Talluru et al. (Reference Talluru, Djenidi, Kamruzzaman and Antonia2016)). Here, we use $U_{\unicode[STIX]{x1D70F}}$ , $U_{\infty }$ and $\unicode[STIX]{x1D6FF}$ because Talluru et al. (Reference Talluru, Djenidi, Kamruzzaman and Antonia2016) showed that these are correct scaling variables for rough wall TBLs (use of the length scale $\unicode[STIX]{x1D6FF}^{\ast }[2/C_{f}]^{1/2}$ (Rotta Reference Rotta1962), where $\unicode[STIX]{x1D6FF}^{\ast }$ is the displacement thickness, does not change the results). Figure 7(a,b) shows the velocity defect profiles for the rough wall data and the same smooth wall data as reported in figure 4 normalised by $U_{\unicode[STIX]{x1D70F}}$ and $U_{\infty }$ , respectively. Universality is thought to be reflected in the collapse of the normalised velocity defect profiles. For the rough wall data, the use of either $U_{\unicode[STIX]{x1D70F}}$ or $U_{\infty }$ as a scaling velocity results in a good collapse. In particular, the collapse is perfect for the profiles for which $U_{\infty }^{+}$ is constant, confirming that a $Re$ -independent normalised velocity distribution is achieved. The smooth wall data present an interesting paradox: the velocity defect profiles collapse when normalised with $U_{\unicode[STIX]{x1D70F}}$ , but not when normalised with $U_{\infty }$ . In order to solve this paradox we write the left-hand side of (3.2) as follows,

(3.3) $$\begin{eqnarray}\frac{U_{\infty }-U}{U_{\unicode[STIX]{x1D70F}}}=\left[\frac{2}{C_{f}}\right]^{1/2}\left(\frac{U_{\infty }-U}{U_{\infty }}\right).\end{eqnarray}$$

For the smooth wall, $C_{f}$ is function of $Re$ ; $C_{f}$ decreases with increasing $Re$ , thus compensating for the $Re$ dependence of $(U_{\infty }-U)/U_{\infty }$ which leads to the collapse of the profiles. We note though that they collapse relatively well with the rough wall profiles when normalised with $U_{\unicode[STIX]{x1D70F}}$ . The collapse of $(U_{\infty }-U)/U_{\unicode[STIX]{x1D70F}}$ between the rough wall and the smooth wall data is often used to demonstrate the universality of the velocity profiles in TBLs and validate Townsend’s Reynolds number similarity. Interestingly, Townsend (Reference Townsend1956) states that ‘if, as must be assumed, Reynolds number similarity of self-preserving flow exists, the form of the self-preserving functions is universal for any one type of flow’. This implies that $(U_{\infty }-U)/U_{\unicode[STIX]{x1D70F}}$ and $(U_{\infty }-U)/U_{\infty }$ must be similar in form for any $Re$ , if both $U_{\unicode[STIX]{x1D70F}}$ and $U_{\infty }$ are proper scaling velocities compliant with self-preservation analysis (Talluru et al. Reference Talluru, Djenidi, Kamruzzaman and Antonia2016).

For the rough wall, $C_{f}$ is constant in the fully rough regime. Thus, using either $U_{\unicode[STIX]{x1D70F}}$ or $U_{\infty }$ leads to the same collapse which is consistent with the Reynolds number similarity. The data of Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016) show the same result. Regardless of the choice of $u_{o}$ , the normalised velocity defect profiles collapse. However, while the sand-grain rough wall data collapse with the present velocity defect profiles when normalised by $U_{\unicode[STIX]{x1D70F}}$ , they collapse onto a different curve than the present rough wall one when normalised by $U_{\infty }$ . The data of Krogstad & Efros (Reference Krogstad and Efros2012) differ slightly from the present data when normalised by $U_{\infty }$ , certainly reflecting the slightly smaller value of $C_{f}$ .

For the smooth wall, equation (3.3) indicates that $(U_{\infty }-U)/U_{\unicode[STIX]{x1D70F}}$ and $(U_{\infty }-U)/U_{\infty }$ cannot both be $Re$ -independent, as shown in figure 7(a,b), since $C_{f}$ is $Re$ -dependent. George & Castillo (Reference George and Castillo1997) argued, based on the asymptotic invariance principle, that the velocity deficit in the outer layer scales as $(U_{\infty }-U)/U_{\infty }$ in the $Re$ -independent regime. Note that even if the smooth wall profiles $(U_{\infty }-U)/U_{\infty }$ collapse (in the outer part of the layer); figure 7(b) shows that they will certainly collapse onto a different curve than that for the rough walls. Notice how the trend shown by the distributions altogether in figure 7(b) reflects that of the profiles in figure 4; as the distributions in figure 4 shift downwards, they shift upwards in figure 7(b). This can be explained if we write $1-U/U_{\infty }=1-(U/U_{\unicode[STIX]{x1D70F}})(C_{f}/2)^{1/2}$ . It appears then that $(U/U_{\unicode[STIX]{x1D70F}})(C_{f}/2)^{1/2}$ is larger for a smooth wall TBL than a rough wall TBL; further, while it is constant (i.e. independent of $Re$ ) for a given fully rough wall TBL, it increases with $Re$ for a smooth wall TBL. This result shows that the distribution ( $1-U/U_{\infty }$ ) or simply ( $U/U_{\infty }$ ) should be used for assessing a $Re$ -independence state. Indeed, when such state is reached, the profiles, $U/U_{\infty }$ , must collapse onto a single curve; this curve varies with the roughness geometry or the coefficient of drag.

It is remarkable that while the profiles $(U_{\infty }-U)/U_{\infty }$ differ between smooth and rough walls, the profiles for $(U_{\infty }-U)/U_{\unicode[STIX]{x1D70F}}$ collapse relatively well onto each other. We saw above that the compensational effect of $C_{f}$ on $(U_{\infty }-U)/U_{\infty }$ leads to a collapse of the profiles $(U_{\infty }-U)/U_{\unicode[STIX]{x1D70F}}$ for the smooth wall. However, the collapse of $(U_{\infty }-U)/U_{\unicode[STIX]{x1D70F}}$ between the rough and smooth wall data may not in fact be too surprising. To see this, let us first note that the downward shift of the rough wall mean velocity profiles, $\unicode[STIX]{x0394}U^{+}$ , can be expressed across most of the boundary layer, as

(3.4) $$\begin{eqnarray}\unicode[STIX]{x0394}U^{+}=U_{\infty }^{+}|_{smooth}-U_{\infty }^{+}|_{rough}=\left(\frac{2}{C_{f}}\right)_{smooth}^{1/2}-\left(\frac{2}{C_{f}}\right)_{rough}^{1/2}.\end{eqnarray}$$

We can then write

(3.5) $$\begin{eqnarray}U_{smooth}^{+}=U_{rough}^{+}+\unicode[STIX]{x0394}U^{+}=U_{rough}^{+}+\left(\frac{2}{C_{f}}\right)_{smooth}^{1/2}-\left(\frac{2}{C_{f}}\right)_{rough}^{1/2}.\end{eqnarray}$$

Using (3.5), we can express the normalised defect profile $(U_{\infty }-U)/U_{\unicode[STIX]{x1D70F}}$ as,

(3.6) $$\begin{eqnarray}\left(\frac{U_{\infty }-U}{U_{\unicode[STIX]{x1D70F}}}\right)_{smooth}=\left(\frac{2}{C_{f}}\right)_{smooth}^{1/2}-U_{rough}^{+}-\left(\frac{2}{C_{f}}\right)_{smooth}^{1/2}+\left(\frac{2}{C_{f}}\right)_{rough}^{1/2}.\end{eqnarray}$$

Thus, we have

(3.7) $$\begin{eqnarray}\left(\frac{U_{\infty }-U}{U_{\unicode[STIX]{x1D70F}}}\right)_{smooth}=\left(\frac{U_{\infty }-U}{U_{\unicode[STIX]{x1D70F}}}\right)_{rough}.\end{eqnarray}$$

This explains the collapse of $U_{\unicode[STIX]{x1D70F}}$ -normalised velocity defect profiles between rough wall and smooth wall TBLs. A similar equality can be established for velocity profiles over two different rough wall TBLs. Accordingly, one can expect that all smooth and rough wall (ZPG) TBL velocity defect profiles normalised by $U_{\unicode[STIX]{x1D70F}}$ will collapse onto a single curve. Of course, equation (3.7) implies that the mean velocity ( $U^{+}$ ) profiles in the rough wall TBLs are shifted downward by the same amount, $\unicode[STIX]{x0394}U^{+}$ , across most of the layer (e.g. $y/\unicode[STIX]{x1D6FF}_{99}\geqslant 0.1$ ) with respect to the smooth wall TBL profiles when $\unicode[STIX]{x1D6FF}_{99}^{+}$ is the same for both TBLs. This seems a reasonable assumption according to figure 4 and the agreement observed in figure 7, which means that the rough wall mean velocity profiles are, apart from a downward shift, similar to the smooth wall mean velocity profiles. These results show that while the normalised mean velocity profiles do not collapse between different walls because of the change in drag coefficient, the shape of the distributions is the same. This is precisely Townsend’s Reynolds number similarity applied to the outer region of TBLs. It is important to remember though that $U_{\unicode[STIX]{x1D70F}}$ is not a scaling velocity compliant with self-preservation for the smooth wall TBL, at least for the outer part of the layer. This last remark should be discussed in the context of scaling of a ZPG TBL at a moderate Reynolds number. It can be shown (see, e.g. Townsend Reference Townsend1976; Talluru et al. Reference Talluru, Djenidi, Kamruzzaman and Antonia2016) for a ZPG TBL that the scaling velocity, $u_{o}$ , which complies with self-preservation, must satisfy the following condition

(3.8) $$\begin{eqnarray}\frac{u_{o}}{U_{\infty }}=C,\end{eqnarray}$$

where $C$ is a constant. We can write (3.8) as

(3.9) $$\begin{eqnarray}u_{o}=CU_{\infty }=CU_{\unicode[STIX]{x1D70F}}\left[\frac{2}{C_{f}}\right]^{1/2}.\end{eqnarray}$$

For a fully rough ZPG TBL, where $C_{f}$ is constant (independent of $Re$ and $x$ ), both $U_{\infty }$ and $U_{\unicode[STIX]{x1D70F}}$ can be used as appropriate scaling velocities leading to normalised velocity profiles, to conform with self-preservation and $Re$ -independence. For a smooth wall ZPG TBL, the use of $u_{o}=U_{\infty }$ (or $U_{\unicode[STIX]{x1D70F}}[2/C_{f}]^{1/2}$ ) appears correct for a given $U_{\infty }$ , which implies that $U_{\unicode[STIX]{x1D70F}}$ alone cannot be a scaling velocity since $C_{f}$ is not constant (either with $x$ or $Re$ ). Further, for a smooth wall, self-preservation (see Talluru et al. Reference Talluru, Djenidi, Kamruzzaman and Antonia2016) requires

(3.10) $$\begin{eqnarray}\unicode[STIX]{x1D702}\sim \frac{\unicode[STIX]{x1D708}}{u_{o}},\end{eqnarray}$$

while $u_{o}$ still obeys (3.9). Accordingly, we have

(3.11) $$\begin{eqnarray}\unicode[STIX]{x1D702}\sim \frac{\unicode[STIX]{x1D708}}{U_{\infty }}=\frac{\unicode[STIX]{x1D708}}{U_{\unicode[STIX]{x1D70F}}\left[{\displaystyle \frac{2}{C_{f}}}\right]^{1/2}}.\end{eqnarray}$$

In the case of a fully rough wall TBL, this constraint does not exist since the viscous term in the mean equation of motion is removed (Talluru et al. Reference Talluru, Djenidi, Kamruzzaman and Antonia2016), thus discarding both $\unicode[STIX]{x1D708}/U_{\infty }$ and $\unicode[STIX]{x1D708}/U_{\unicode[STIX]{x1D70F}}$ as appropriate length scales. We already saw (figure 2) that $\unicode[STIX]{x1D708}/U_{\unicode[STIX]{x1D70F}}$ does not scale the velocity profiles. For a rough wall, since $U_{\infty }\sim U_{\unicode[STIX]{x1D70F}}$ then $\unicode[STIX]{x1D708}/U_{\infty }$ cannot scale the velocity profiles either.

3.2 Streamwise Reynolds normal stress

The previous section strongly suggests that a rough wall TBL reaches a $Re$ -independent state at a moderate $Re$ in a fully rough regime in which the mean velocity profile remains unchanged when normalised by length and velocity which comply with self-preservation. It is then of interest to assess how the normalised Reynolds stresses, and in particular the streamwise Reynolds stress, behave as the rough wall TBL approaches its $Re$ -independent state.

Figure 8.

Profiles of $\overline{{u^{+}}^{2}}$ in smooth and rough wall TBLs at several $Re_{\unicode[STIX]{x1D70F}}$ as a function of (a) $y^{+}$ ; (b) $y/k$ ; and (c) $y/\unicode[STIX]{x1D6FF}$ . Smooth wall DNS (solid black line, $Re_{\unicode[STIX]{x1D70F}}=674$ Schlatter & Örlü Reference Schlatter and Örlü2010), smooth wall TBL (black dashed lines, $Re_{\unicode[STIX]{x1D70F}}=3370$ , 4760, 6450 and 9830, Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015), sand-grain roughness (red symbols, $Re_{\unicode[STIX]{x1D70F}}=20\,160$ , 25 020 and 29 900, Squire et al. Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016), square bar (blue symbols, $Re_{\unicode[STIX]{x1D70F}}=13\,300$ , Krogstad & Efros Reference Krogstad and Efros2012) and the present rough wall data (black symbols). Refer to caption of figure 4 for symbols used for smooth wall, sand-grain and square bar rough wall TBLs.

There is strong experimental evidence in the literature showing that the profile of $\overline{u^{2}}$ in a smooth wall TBL is considerably altered when $Re$ increases (see Metzger et al. Reference Metzger, Klewicki, Bradshaw and Sadr2001; Hutchins & Marusic Reference Hutchins and Marusic2007; Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010; Smits, McKeon & Marusic Reference Smits, McKeon and Marusic2011, for example). This behaviour is illustrated in figure 8(a), which shows viscous-scale-normalised distributions of $\overline{u^{2}}$ in a smooth wall TBL as $Re$ increases (DNS data of Schlatter & Örlü (Reference Schlatter and Örlü2010), $Re_{\unicode[STIX]{x1D70F}}$ = 674; wind tunnel measurements of Marusic et al. (Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015), $3370\leqslant Re_{\unicode[STIX]{x1D70F}}\leqslant 9830$ ). At low Reynolds number, the turbulence intensity profile presents a characteristic peak at $y^{+}\simeq 15$ . Some relatively recent studies show that the near-wall peak exhibits a weak $Re$ dependence when scaled on $U_{\unicode[STIX]{x1D70F}}$ (see, e.g. Klewicki & Falco Reference Klewicki and Falco1990; De Graaff & Eaton Reference De Graaff and Eaton2000; Metzger et al. Reference Metzger, Klewicki, Bradshaw and Sadr2001; Hoyas & Jiménez Reference Hoyas and Jiménez2006; Hutchins & Marusic Reference Hutchins and Marusic2007; Vincenti et al. Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013). However, Monkewitz & Nagib (Reference Monkewitz and Nagib2015) argue that this inner peak cannot increase indefinitely with $Re$ . In any case, the inner peak does not disappear with increasing $Re$ . The data of Marusic et al. (Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015) on a smooth wall turbulent boundary layer show convincingly that the inner peak remains even though the magnitude of $\overline{u^{+2}}$ increases in the outer layer, possibly leading to an outer peak which may eventually exceed the near-wall peak value. A similar behaviour is observed for the high $Re$ ZPG TBL data of Fernholz & Finley (Reference Fernholz and Finley1996), Vincenti et al. (Reference Vincenti, Klewicki, Morrill-Winter, White and Wosnik2013) and the high Reynolds number smooth wall pipe data of Morrison et al. (Reference Morrison, McKeon, Jiang and Smits2004) and Hultmark et al. (Reference Hultmark, Vallikivi, Bailey and Smits2012). These latter authors attributed the occurrence of the outer peak to a shift in the production away from the wall when $Re$ increases, supposedly resulting from an increasing separation between large and small scales.

The continuous evolution (in shape and magnitude) of the $\overline{{u^{+}}^{2}}$ distribution with an ever increasing Reynolds number may be interpreted as reflecting the transient nature of the smooth wall TBL, and contrasts with that of a rough wall TBL. This can be seen in figure 8(a), where we have reported profiles of $\overline{{u^{+}}^{2}}$ for the present 2-D rough wall at different $Re_{\unicode[STIX]{x1D70F}}$ ranging from 620 to 7200, smooth wall data of Marusic et al. (Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015) and the rough wall data of Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016). The data of Krogstad & Efros (Reference Krogstad and Efros2012) at $Re_{\unicode[STIX]{x1D70F}}=13\,300$ have also been included. Our measurements are consistent with those of Krogstad & Efros (Reference Krogstad and Efros2012) and Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016).

At low $Re$ , profiles of $\overline{{u^{+}}^{2}}$ on both the 2-D rod and sand-grain rough wall TBLs present a near-wall peak. Lee & Sung (Reference Lee and Sung2007), Lee et al. (Reference Lee, Lee, Kim and Sung2009) and Volino, Schultz & Flack (Reference Volino, Schultz and Flack2011) respectively reported a similar inner peak in the low Reynolds number DNS, particle image velocimetry (PIV) and LDV measurements of a turbulent boundary layer over 2-D rough walls with a similar spacing $p=8k$ as in the present case. As $Re$ increases, this peak subsides while an outer layer peak appears. A similar behaviour of a decreasing inner peak was reported by Monty et al. (Reference Monty, Chong, Mathis, Hutchins, Marusic and Allen2010), who compared the turbulence statistics (obtained with matched $l^{+}$ values) over a range of $Re$ in a turbulent boundary layer over a regular braille-type roughness. This behaviour is clearly evident in the Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016) data. These results confirm that the near-wall peak attenuation is not due to spatial resolution issues usually associated with hot-wire anemometry (see also the study of Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009)). The first five ( $Re_{\unicode[STIX]{x1D70F}}\leqslant 4000$ ) of our 2-D rod rough wall turbulence intensity profiles have $l^{+}$ values smaller than 20 (see table 1 for details). Looking at those five profiles, one can easily discern that the reduction in the magnitude of the inner peak as $Re$ increases is due to roughness and not due to insufficient spatial resolution in our hot-wire measurements. Nonetheless, we realise that there is some spatial attenuation in our results particularly at high $Re$ but emphasise that this does not affect the main conclusions of the paper. Squire et al. (Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016) reached similar conclusions on the influence of $l^{+}$ on the inner peak. As the inner peak decreases, it also shifts towards larger $y^{+}$ . Whilst it is not clear whether or not the inner peak is ‘removed’ or simply ‘hidden’ due to the large values of $\overline{{u^{+}}^{2}}$ in the outer layer, the former scenario is expected because the near-wall region of the flow is entirely altered by the roughness at least when $Re$ is sufficiently large and $4\leqslant p/k\leqslant 16$ (e.g. Wang, Hejcik & Sunden Reference Wang, Hejcik and Sunden2007; Djenidi et al. Reference Djenidi, Antonia, Amielh and Anselmet2008; Lee et al. Reference Lee, Lee, Kim and Sung2009).

The horizontal shift is not limited to the inner peak. There is a clear systematic shift of the entire distribution of $\overline{{u^{+}}^{2}}$ , similar to that observed in the $U^{+}$ profiles, as $Re_{\unicode[STIX]{x1D70F}}$ increases. However, the shape of the distributions seems to suggest a possible independence on $Re$ of the $\overline{{u^{+}}^{2}}$ profiles with increasing $Re$ . This indicates that, like the mean velocity field, the fluctuating velocity field on the rough walls may comply with the Reynolds number similarity hypothesis. To ascertain this, we report in figure 8(b,c) some of the rough wall distributions of $\overline{{u^{+}}^{2}}$ as functions of $y/k$ and $y/\unicode[STIX]{x1D6FF}$ , respectively; these distributions correspond to the mean velocity profiles reported in figure 3. There is generally a good collapse for both sand-grain and 2-D rough wall boundary layers. For the present 2-D rough wall, the distribution for $Re_{\unicode[STIX]{x1D70F}}=3945$ deviates from the other two in the region $y/k\leqslant 10$ , possibly reflecting a persistence of the transient state, which would indicate that the fluctuating velocity field is not yet ‘fully’ independent of $Re$ . Just as in the case of normalised mean velocity profiles, the 2-D rough wall and the sand-grain rough wall have distinct $Re$ -independent distributions of $\overline{{u^{+}}^{2}}$ .