4.1. Smooth-wall validation

The accuracy of the code CDP in simulating pipe flows is first validated by comparing our results against other published DNS results for the smooth-wall case at similar Reynolds numbers. The mean velocity profile and the turbulence intensities are shown in figures 3(a) and 3(b) respectively. It can be seen that the present mean velocity profile has excellent agreement with other existing literature (both second-order and spectral codes). The velocity profiles do not fall on the high- $\mathit{Re}$ log law owing to the low Reynolds number of the flow. Comparing the turbulence intensities, we also obtain good agreement with data in the existing literature. The largest discrepancy (2.8 %) occurs in the comparison of the peak values of the streamwise and azimuthal turbulence intensities with those of Eggels et al. (Reference Eggels, Unger, Weiss, Westerweel, Adrian, Friedrich and Nieuwstadt1994). An explanation for this is offered by Wu & Moin (Reference Wu and Moin2008) who argue that this is due to the relatively coarse mesh and shorter domain used by Eggels et al. (Reference Eggels, Unger, Weiss, Westerweel, Adrian, Friedrich and Nieuwstadt1994). The variation from Loulou et al. (Reference Loulou, Moser, Mansour and Cantwell1997) and Chin et al. (Reference Chin, Ooi, Marusic and Blackburn2010) is likely to be due to the slightly different Reynolds numbers, $\mathit{Re}_{{\it\tau}}=170$ and $\mathit{Re}_{{\it\tau}}=190$ , respectively, and possibly also because of the spectral code. The Reynolds shear stress $\langle \overline{u_{r}^{\prime }u_{x}^{\prime }}\rangle ^{+}$ is shown in figure 3(c) as a function of wall-normal distance. The agreement with Eggels et al. (Reference Eggels, Unger, Weiss, Westerweel, Adrian, Friedrich and Nieuwstadt1994), Fukagata & Kasagi (Reference Fukagata and Kasagi2002) and Wu & Moin (Reference Wu and Moin2008) is good and the differences observed with Loulou et al. (Reference Loulou, Moser, Mansour and Cantwell1997) and Chin et al. (Reference Chin, Ooi, Marusic and Blackburn2010) are due to the different Reynolds number. Also shown in figure 3(c) are the viscous stress and the total shear stress for the current simulation. It can be seen that the total shear stress is a linear function, which gives confidence that all relevant statistics have converged. There is some variation of ${p^{\prime }}_{rms}^{+}$ , present in the literature. The peak ${p^{\prime }}_{rms}^{+}$ is found to be fairly consistently located at $y^{+}=30$ ; however the value ranges from 1.82 to 2.01. The present DNS results have a peak ${p^{\prime }}_{rms}^{+}$ of 1.95 (figure 3 d), which is closer to the values obtained by Loulou et al. (Reference Loulou, Moser, Mansour and Cantwell1997) and Wu & Moin (Reference Wu and Moin2008). Generally, the agreement with other similar simulations from the literature is good. Even with ${p^{\prime }}_{rms}^{+}$ , for which the literature exhibits substantial variation, data from the current simulation are within the reported scatter.

4.2. Investigation of the virtual origin

Defining the virtual origin ( $y=0$ ) for a rough wall is not straightforward. For a smooth surface $y=0$ is easily defined as the point at which the no-slip condition acts. For a rough surface, the virtual origin depends on roughness geometry, and is typically located somewhere between the peak and trough of the rough surface. Knowledge of the virtual origin is critical for the ability to compare statistics from different surfaces. This is especially so for the current simulations where the ratio of the reference radius of the pipe $R_{0}$ to the roughness height $h$ is low. However, for flows with large $R_{0}/h$ ratios, the virtual origin offset ${\it\epsilon}$ , which lies somewhere between the peak and trough of the roughness, only occupies a very small fraction of the boundary layer and therefore has less influence on the profiles of turbulent statistics. Typically, the only method to determine the virtual origin offset experimentally is via the modified Clauser chart method. For example, Perry & Li (Reference Perry and Li1990) link the virtual origin with the expected log-region collapse. For the present low Reynolds number DNS data, the logarithmic region is not sufficiently defined to reliably use this technique. However, we do have the option of employing the mean momentum absorption plane method to determine the virtual offset. This method was introduced by Thom (Reference Thom1971) to determine the virtual origin of a vegetation canopy. The method essentially considers the virtual origin ${\it\epsilon}_{1}$ to be the point at which the integrated resultant force acts, and can be expressed as

(4.1) $$\begin{eqnarray}{\it\epsilon}_{1}=\left.\int _{-h}^{h}yF_{tot}(y)\text{d}y\right/\!\int _{-h}^{h}F_{tot}(y)\text{d}y\end{eqnarray}$$

where $F_{tot}\;(=\langle \,\overline{f}_{p}+\overline{f}_{{\it\nu}}\rangle )$ is the time and spatial (azimuthal and streamwise) average of the total drag force acting on the roughness elements due to the viscous (  $\overline{f}_{{\it\nu}}$ ) and pressure forces (  $\overline{f}_{p}$ ). Data on the spatial distribution of the drag and the ratio between the pressure and viscous drag on the surface of rough walls are difficult to obtain experimentally but can be easily assessed when simulated numerically. Figures 4(a) and 4(b) respectively show the normalised time-averaged pressure and viscous drag acting on the roughness elements for case 20\_141 simulated at $\mathit{Re}_{{\it\tau}}=180$ . The maximum viscous drag is located at the crest of the roughness element due to the large streamwise velocity gradient in the wall-normal direction. The location of the maximum pressure drag is located on the slope of the forward face of the roughness element. Calculating the virtual origin using this method results in a positive virtual origin offset for all roughnesses tested here ( ${\it\epsilon}_{1}>0$ ).

Figure 4. Time-averaged (a) pressure $\overline{f}_{p}$ and (b) viscous $\overline{f}_{{\it\nu}}$ drag on the surface of the roughness for case 20\_141 normalised by the total drag across the pipe $F_{drag}$ , where $F_{drag}=\sum _{y=-h}^{h}F_{tot}(y)$ . Black contour lines denote regions with positive values while white contour lines denote regions with negative values.

An alternative method which could be used to determine the virtual origin is to locate the position where the mean streamwise velocity is zero ( $U^{+}=0$ ), a condition which is satisfied by the no-slip condition at the surface of the smooth wall. The difficulty of this method is that this location is not always uniquely defined since larger roughness heights will often exhibit flow reversal in the roughness troughs, in which case multiple points satisfy this condition. The virtual origin offset obtained using this method (taking the outermost point that satisfies the condition) differs from the results from the mean momentum absorption methods, yielding a value, ${\it\epsilon}_{2}$ , which is always negative. A variation of this method is to find the location where $U^{+}=1$ and then shift it by $y^{+}=-1$ to obtain the offset of the virtual origin ( ${\it\epsilon}_{3}$ ). The offset obtained using this method is fairly consistent for all roughness cases with values fluctuating around 0, from $-2.6<{\it\epsilon}_{3}^{+}<2.2$ (except for case 80\_565).

A final suggested technique to determine the virtual origin is to collapse the total stress profile outside the roughness layer regardless of whether the pipe is smooth or not. The total stress ${\it\tau}(r)$ across the pipe is the sum of the viscous stress $-{\it\nu}\text{d}U/\text{d}r$ and the Reynolds stress $\langle \overline{u_{r}^{\prime }u_{x}^{\prime }}\rangle$ and can be expressed as

(4.2) $$\begin{eqnarray}{\it\tau}(r)=-{\it\nu}\frac{\text{d}U}{\text{d}r}+\langle \overline{u_{r}^{\prime }u_{x}^{\prime }}\rangle =\frac{1}{2}F_{x}r\end{eqnarray}$$

where $F_{x}$ is defined as the streamwise driving pressure gradient, as in (2.2), and $r$ is taken above the roughness elements. Rearranging (4.2) and dividing both sides with the radial location of the virtual origin of the pipe $R_{v}$ (currently arbitrary), we obtain

(4.3) $$\begin{eqnarray}\frac{{\it\tau}(r)}{R_{v}F_{x}/2}=\frac{{\it\tau}(r)}{{\it\tau}_{w}}=\frac{r}{R_{v}}\end{eqnarray}$$

where ${\it\tau}_{w}=F_{x}R_{v}/2$ , identified as total wall stress which is balanced by the pressure drop in the pipe. Equation (4.3) shows that $R_{v}$ can be chosen arbitrarily to ensure collapse, i.e. there is no unique solution. However, if, in analogy with a smooth wall, one insists that the wall friction is defined as the drag per plan area, then a unique definition for $R_{v}$ emerges. To see this, consider the volume integral of the streamwise momentum equation:

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\rho}\frac{\text{d}U_{b}}{\text{d}t}={\it\rho}F_{x}V+\int _{\partial S}(-pn_{x}+{\it\nu}\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}u)\text{d}S={\it\rho}F_{x}V-F_{drag}=0 & \displaystyle\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\displaystyle & \Rightarrow F_{x}=F_{drag}/({\it\rho}V) & \displaystyle\end{eqnarray}$$

$n$

$V$

$\partial S$

$R_{h}=\sqrt{V/({\rm\pi}L_{x})}$

$F_{x}$

${\it\tau}_{w}$

(4.6) $$\begin{eqnarray}{\it\tau}_{w}=\frac{R_{v}F_{x}}{2}=\frac{R_{v}F_{drag}}{2{\it\rho}V}=\frac{R_{v}F_{drag}}{2{\it\rho}{\rm\pi}{R_{h}}^{2}L_{x}}=\frac{\overbrace{(R_{v}/R_{h})^{2}}^{\text{Constant}}F_{drag}}{{\it\rho}\underbrace{(2{\rm\pi}R_{v}L_{x})}_{\begin{array}{@{}c@{}}\text{Plan area based}\\ \text{on virtual radius}\end{array}}}.\end{eqnarray}$$

Here, to ensure that ${\it\tau}_{w}$ physically represents the drag acting on the wall-plan area, the virtual radius has to be equal to the hydraulic radius $R_{v}=R_{h}$ such that the value of the constant is 1. It can be shown that, even for the largest roughness case simulated here ( $R_{0}/h=9$ ), the reference radius of the pipe is approximately equal to the hydraulic radius of the pipe $R_{0}\approx R_{h}$ with 0.15 % relative difference. Therefore using the reference radius $R_{0}$ as the location of the virtual origin is also valid ( ${\it\epsilon}_{4}^{+}=0$ ).

The different methods used to obtain the virtual origin offset are tested by applying ${\it\epsilon}$ to the total stress, velocity defect and streamwise turbulence intensity profiles. The virtual origin offset is applied to case 20\_141 using values of ${\it\epsilon}_{1}^{+}=11.7$ , ${\it\epsilon}_{2}^{+}=-4.1$ and ${\it\epsilon}_{4}^{+}=0$ as illustrated in figure 5. Figure 5(a) is a plot of the total stress of the fluid. Unlike a smooth wall where the normalised total stress approaches 1 at the wall, the total shear stress of the fluid for a rough wall within the roughness elements reduces and is balanced by the pressure and viscous stress acting on the elements of the rough surface (the sum of the pressure and viscous stress acting on the elements is shown by the grey solid line on figure 5 a). Setting the virtual radius to the reference radius of the pipe, we obtain an unsurprising collapse in the total stress of the rough-wall cases with the smooth-wall case. In addition, a good collapse in the radial turbulence intensity with the smooth wall is observed almost immediately above the roughness elements for ${\it\epsilon}_{4}$ (figure 5 b). While a better collapse of the velocity defect is observed for ${\it\epsilon}_{2}$ compared to ${\it\epsilon}_{4}$ (figure 5 c), using ${\it\epsilon}_{1}$ as the virtual origin offset does not result in a better collapse in any of the three statistics considered. Scotti (Reference Scotti2006) observed that a virtual origin determined from the mean momentum absorption plane provided a good collapse in mean statistics. It is possible that the small differences reported here between ${\it\epsilon}_{4}$ and ${\it\epsilon}_{1}$ would not be easily discernible in the higher Reynolds number results presented by Scotti (Reference Scotti2006) (where the Reynolds shear stress or the total stress were also not shown).

Overall, for the cases considered here, the virtual origin offset of the pipe ${\it\epsilon}_{4}$ seems to offer the most consistently reliable estimate of the virtual origin for the roughness cases simulated. Good collapse in the turbulent statistics, total-stress profile and velocity defect are also obtained for all the roughness cases in the outer region of the flow when ${\it\epsilon}_{4}=0$ . It should, however, be noted that the analysis presented above in (4.2)–(4.6) that leads to ${\it\epsilon}=0$ would not be true for an external boundary layer where the flow is unbounded in the wall-normal direction and the boundary layer thickness of a rough wall differs greatly from the smooth-wall case. This would suggest that the same roughness tested at the same Reynolds number in a pipe and an external boundary layer would probably exhibit different virtual origins as well as different flow close to the roughness elements. Effectively, the constrained pipe geometry would lead to more flow within the roughness canopy, compared to the boundary layer, where the flow is effectively able to ride over the roughness elements through changes in the virtual origin. For all subsequent plots, the virtual origin of the wall is defined as the reference radius of the rough-wall pipe.

Figure 5. Plot of (a) total stress normalised by the wall stress, (b) velocity defect and (c) streamwise turbulence intensity, for the smooth wall and case 20\_141 at $\mathit{Re}_{{\it\tau}}=180$ with virtual origin offset ${\it\epsilon}_{1}^{+}=11.7$ , ${\it\epsilon}_{2}^{+}=-4.1$ and ${\it\epsilon}_{4}^{+}=0$ . Thick grey line in (a) is the total wall stress (viscous and pressure) of the rough surface with ${\it\epsilon}_{4}^{+}=0$ . Inset: zoomed in view of the boxed region.

4.3. Investigation of low Reynolds number and low ratio of reference radius to roughness height $R_{0}/h$

In this section, we investigate the feasibility of simulating the three-dimensional sinusoidal roughness at low Reynolds number using case $\mathit{RE}$ . Roughness causes a downward shift in the viscous-scaled mean velocity profile, which is typically expressed as a modified logarithmic law,

(4.7) $$\begin{eqnarray}U^{+}=\frac{1}{{\it\kappa}}\ln (y^{+})+C-{\rm\Delta}U^{+},\end{eqnarray}$$

where ${\rm\Delta}U^{+}$ is known as the Hama roughness function (Hama Reference Hama1954). To determine if the roughness function ${\rm\Delta}U^{+}$ obtained in the low Reynolds number simulations is applicable to flows at higher Reynolds number, case 20\_141 (the case with the largest roughness) is simulated at three different Reynolds numbers ( $\mathit{Re}_{{\it\tau}}=180,360$ and 540). In this study, where Townsend’s outer-layer similarity (Townsend Reference Townsend1980) seems to be conserved (see §§ 4.5 and 4.4.5), ${\rm\Delta}U^{+}$ is determined 50 wall units above the crest of the roughness elements for all of the cases simulated since the log layer of the flow at low and moderate Reynolds numbers is poorly defined. As the roughness elements have the same viscous scale, the physical geometrical size of the roughness element relative to the pipe radius reduces with increasing Reynolds number. The sketches of the rough pipe in figure 6 clearly illustrates this where $R_{0}/h$ increases from 9 at $\mathit{Re}_{{\it\tau}}=180$ to 27 when $\mathit{Re}_{{\it\tau}}=540$ . Hence, these experiments also address the influence of $R_{0}/h$ for this particular roughness.

Figure 6. Streamwise velocity profile for smooth wall and case 20\_141 at $\mathit{Re}_{{\it\tau}}=180,360$ and 540. Dash-dotted lines show $U^{+}=y^{+}$ and $U^{+}=(1/{\it\kappa})\log (y^{+})+C$ , ${\it\kappa}=0.40$ and $C=5.3$ . Inset: roughness function ${\rm\Delta}U^{+}$ against $y^{+}$ for the different Reynolds number simulation. Note that $R_{0}/h$ ranges from 9 for the $\mathit{Re}_{{\it\tau}}=180$ simulation to 27 for the $\mathit{Re}_{{\it\tau}}=540$ simulation. Refer to table 1 for symbol key.

Examining the mean velocity profiles for the smooth wall in figure 6, the low Reynolds number effect at $\mathit{Re}_{{\it\tau}}=180$ is apparent, causing an upward shift from the logarithmic region (shown by the dot–dashed line for ${\it\kappa}=0.4$ and $C=5.3$ ). This is the well-documented pressure gradient effect of low Reynolds number turbulent pipe flow (see for example Nickels Reference Nickels2004). However, for the rough cases, a closer collapse is obtained for the mean velocity profile simulated at all three Reynolds numbers. Although the low Reynolds number effect is still observable for case 20\_141 at $\mathit{Re}_{{\it\tau}}=180$ , it is not as significant as for the smooth wall simulated at the same Reynolds number. Calculating the difference in the mean velocity profile between the smooth and rough cases at the same Reynolds number (inset of figure 6), we observe that the profiles remain fairly constant above the roughness elements, particularly so above $y^{+}=70$ , indicating that the shape of the mean velocity profile in the outer part of the flow for both the smooth and rough cases is similar. Due to the low Reynolds number effect of the smooth pipe at $\mathit{Re}_{{\it\tau}}=180$ , the roughness function ${\rm\Delta}U^{+}$ is slightly overestimated. However, the maximum difference obtained in ${\rm\Delta}U^{+}$ between the $\mathit{Re}_{{\it\tau}}=180$ and 540 simulation is only approximately 0.7. This error can be relatively insignificant in high Reynolds number flows which are of engineering interest.

An overestimation of ${\rm\Delta}U^{+}$ by some amount ${\it\varepsilon}$ , causes an overestimation of $k_{s}$ for a surface by approximately $\exp ({\it\kappa}{\it\varepsilon})$ , which is approximately 33 % when ${\it\varepsilon}=0.7$ . For full-scale moderate to high Reynolds number flows in the fully rough regime, this translates to an overestimation of the total drag coefficient by only approximately 5 %. Such an error is rather insignificant when one considers the traditional error associated with determining $C_{f}$ experimentally, and also that a fully rough surface can lead to $O(100\,\%)$ increases in drag coefficient. More important for the current simulations is that the low Reynolds number simulations are able to capture the correct trends in ${\rm\Delta}U^{+}$ with varying roughness geometry. This is confirmed for the case of varying effective slope in figure 10 below. An improved method which could be used to obtain a more accurate value of ${\rm\Delta}U^{+}$ for large roughnesses simulated at low $\mathit{Re}_{{\it\tau}}$ would be to calculate the shift in the log-law from a higher Reynolds number smooth-wall simulation.

For the transitional roughness cases which have been simulated, the observance of outer-layer similarity in the outer part of the flow appears to be insensitive to changes in the $R_{0}/h$ ratio. The total stress profiles of the smooth and rough cases are shown in figure 7(a), and as expected show an excellent collapse for all the cases when the virtual origin is set to be the reference radius of the pipe. In addition, very good collapse is observed in the radial turbulence intensity between the smooth and rough cases as illustrated in figure 7(c). Reasonable collapse is also obtained when the mean velocity profile is plotted in defect form in figure 7(b), with no pronounced difference in this collapse with varying $R_{0}/h$ . Overall, the adherence to Townsend’s outer-layer similarity of the mean statistics for the rough cases is invariant with Reynolds number and hence also with the ratio $R_{0}/h$ .

Figure 7. (a) Total stress normalised by the wall stress; (b) velocity defect; and (c) radial turbulence intensity for the smooth wall and case 20\_141 at $\mathit{Re}_{{\it\tau}}=180,360$ and 540.

Based on these observations, the decision was made to run the majority of our simulations at $\mathit{Re}_{{\it\tau}}=180$ . Although $\mathit{Re}_{{\it\tau}}=360$ would give a slightly more accurate representation of ${\rm\Delta}U^{+}$ (by avoiding the low Reynolds number pressure gradient effect), these simulations are approximately eight times more expensive. In this instance, we believe that the contribution from this study would be maximised by testing an increased number of surface geometries and thus we choose to deploy our computational resources in this manner. However, throughout this study, certain surfaces will also be computed at $\mathit{Re}_{{\it\tau}}=360$ , to continually verify that the important results and trends observed in this study are not artifacts of low $\mathit{Re}$ or low $R_{0}/h$ .

4.4. Mean velocity profiles

4.4.1. Case C1: geometrically scaled roughness (fixed ${\it\lambda}/h$ )

The mean streamwise velocity profile for the geometrically scaled roughness cases (C1) are plotted in figure 8 for the cases simulated at $\mathit{Re}_{{\it\tau}}=180$ and figure 9 for the cases simulated at $\mathit{Re}_{{\it\tau}}=540$ . Geometrically increasing the size of the roughness increases the roughness average height, with proportionate increases in ${\it\lambda}_{x}$ and ${\it\lambda}_{s}$ . The aspect ratio and average slope of the surface (and hence $\mathit{ES}$ ) remains unchanged. An increase in the geometric scaling factor yields an increase in the roughness function as observed in the insets of figures 8 and 9. It is important to highlight that for cases 02\_018 and 05\_035, there are only four and eight grid points per roughness wavelength and hence the topological features of the sinusoidal surface are not fully resolved (a more faceted version of the roughness is effectively simulated). Thus, this may result in some minor discrepancies in the obtained ${\rm\Delta}U^{+}$ (which can be confirmed by conducting a grid sensitivity analysis).

Figure 8. Streamwise velocity profile for geometrically scaled roughness cases at $\mathit{Re}_{{\it\tau}}=180$ . Dash-dotted lines show $U^{+}=y^{+}$ and $U^{+}=(1/{\it\kappa})\log (y^{+})+C$ , ${\it\kappa}=0.40$ and $C=5.3$ . Inset: roughness function ${\rm\Delta}U^{+}$ against roughness height $h^{+}$ . Dash-dotted line of the inset flow demarcates the fully rough asymptote where $k_{s}^{+}=4.1h^{+}$ .

Figure 9. Streamwise velocity profile for geometrically scaled roughness cases at $\mathit{Re}_{{\it\tau}}=540$ . Dash-dotted lines show $U^{+}=y^{+}$ and $U^{+}=(1/{\it\kappa})\log (y^{+})+C$ , ${\it\kappa}=0.40$ and $C=5.3$ . Inset: roughness function ${\rm\Delta}U^{+}$ against roughness height $h^{+}$ . Dash-dotted line of the inset flow demarcates the fully rough asymptote where $k_{s}^{+}=4.1h^{+}$ . Refer to table 1 for symbol key.

It appears that the flow reaches the fully rough regime when $h^{+}\approx 60$ which corresponds to an equivalent sand grain roughness of $k_{s}^{+}=4.1h^{+}$ . Hence this dataset spans the full transitional regime, from smooth up to fully rough. To the best of the authors’ knowledge this is the first time that $k_{s}^{+}$ verses ${\rm\Delta}U^{+}$ has been mapped from smooth up to the fully rough regime using DNS with a body-conforming grid.

4.4.2. Case C2: decreasing roughness wavelength ${\it\lambda}^{+}$ (fixed $h^{+}=10$ )

Figure 10 shows mean streamwise velocity profiles for the cases where the wavelength of the roughness elements is varied while maintaining a constant roughness height (case C2). Decreasing the wavelength of the roughness elements increases the $\mathit{ES}$ of the surface and allows us to analyse the effects of $\mathit{ES}$ on the roughness function independently of roughness height. In addition, the roughness elements become ‘more dense’ with decreasing wavelength (increased solidity ${\it\Lambda}$ ). Despite reducing the wavelength of the roughness elements by about a quarter (four times the number of roughness elements), there was only an increase of ${\approx}1.56$ in ${\rm\Delta}U^{+}$ between case 10\_283 and 10\_070. The very gradual increase in the roughness function observed in the inset of figure 10 when the roughness wavelength decreases suggests that $\mathit{ES}$ has only a secondary importance in the flow compared to the average roughness height for the current surface geometry in the transitionally rough regime. It is interesting to consider the asymptotic limits that we might expect for variations in  ${\it\lambda}$ . When ${\it\lambda}\rightarrow \infty$ we would expect the surface to approach the smooth case. We might expect a similar behaviour as ${\it\lambda}\rightarrow 0$ (although with this case, the pipe radius will be altered). Selected duplicated cases were simulated at $\mathit{Re}_{{\it\tau}}=360$ (circled symbols) and a similar trend is observed with varying wavelength, further assuaging any doubts that the low $\mathit{Re}$ (and low $R_{0}/h$ ) of these simulations is influencing observed trends.

Figure 10. Streamwise velocity profile for cases with decreasing roughness wavelength ${\it\lambda}^{+}$ . Dash-dotted lines show $U^{+}=y^{+}$ and $U^{+}=(1/{\it\kappa})\log \!(y^{+})+C$ , where ${\it\kappa}=0.40$ and $C=5.3$ . Inset: roughness function ${\rm\Delta}U^{+}$ against roughness wavelength ${\it\lambda}^{+}$ . Circled symbols in shades of red are simulated at $\mathit{Re}_{{\it\tau}}=360$ . Refer to table 1 for symbol key.

4.4.3. Case C3: increasing roughness height $h^{+}$ (fixed ${\it\lambda}^{+}=141$ )

Figure 11 plots the mean streamwise velocity profile for the cases where roughness height is increased while the wavelength remains fixed (case C3). This yields increasing aspect ratio (and increasing $\mathit{ES}$ ) as $h$ increases. Not surprisingly, increasing $h^{+}$ leads to increments in ${\rm\Delta}U^{+}$ . Comparing the inset of figure 11 with figure 9 shows that the geometrically scaled surfaces (case C1) yield a much higher ${\rm\Delta}U^{+}$ at low $h^{+}$ than the fixed wavelength case C3. This is an expected result from figure 10, since the geometrically scaled cases have a higher value of effective slope, and therefore a higher ${\rm\Delta}U^{+}$ , than the C3 cases at low $h^{+}$ .

Figure 11. Streamwise velocity profile for cases with increasing roughness height $h^{+}$ . Dash-dotted lines show $U^{+}=y^{+}$ and $U^{+}=(1/{\it\kappa})\log (y^{+})+C$ , where ${\it\kappa}=0.40$ and $C=5.3$ . Inset: roughness function ${\rm\Delta}U^{+}$ against roughness height $h^{+}$ . Circled symbols and dotted symbols show the same cases simulated at $\mathit{Re}_{{\it\tau}}=360$ and 540 respectively (refer to table 1 for symbol key).

4.4.4. Consolidated mean velocity results

Figures 10 and 11 show that increasing the roughness height $h^{+}$ (with fixed ${\it\lambda}$ ) or decreasing the wavelength of the roughness ${\it\lambda}^{+}$ (with fixed $h$ ) both lead to an increase in ${\rm\Delta}U^{+}$ . Both of these experiments are varying the $\mathit{ES}$ of the surface. On figure 12 we further investigate these data, showing ${\rm\Delta}U^{+}$ as a function of $\mathit{ES}$ in plot (a) and as a function of $k_{a}^{+}$ (which is proportional to $h^{+}$ ) in plot (b). Additional data from Blackburn et al. (Reference Blackburn, Ooi and Chong2007) (corrugated pipe, $\mathit{Re}_{{\it\tau}}=314$ ), De Marchis & Napoli (Reference De Marchis and Napoli2012) (2D and 3D inhomogeneous rough channel, $\mathit{Re}_{{\it\tau}}=395$ ), Napoli et al. (Reference Napoli, Armenio and De Marchis2008) (2D inhomogeneous rough channel, $\mathit{Re}_{{\it\tau}}=395$ ), Bhaganagar, Kim & Coleman (Reference Bhaganagar, Kim and Coleman2004) (‘egg-carton’ asymmetric rough channel, $\mathit{Re}_{{\it\tau}}=400$ ) and Schultz & Flack (Reference Schultz and Flack2009) (close-packed pyramids) are also included in the plots for a more comprehensive comparison. A horizontal line at ${\rm\Delta}U^{+}=9$ is added to mark the approximate start of the fully rough regime.

Figure 12. ${\rm\Delta}U^{+}$ as a function of (a) $\mathit{ES}$ ; (b) $k_{a}^{+}$ . The horizontal dashed line at ${\rm\Delta}U^{+}=9$ demarcates the start of the fully rough regime. Solid coloured symbols show results of the current study (symbols as defined in table 1). Light shaded symbols show results from the literature $\otimes$ : $\oplus$ , $\odot$ , close-packed pyramids of Schultz & Flack (Reference Schultz and Flack2009) (for $\mathit{ES}=0.19$ , 0.4 and 1 respectively); $\times$ , ‘egg-carton’ roughness of Bhaganagar et al. (Reference Bhaganagar, Kim and Coleman2004);  $\ast$ , 2D and 3D inhomogeneous roughness of De Marchis & Napoli (Reference De Marchis and Napoli2012);  $\boxdot$ , corrugated pipe of Blackburn et al. (Reference Blackburn, Ooi and Chong2007); ♢, 2D inhomogeneous roughness of Napoli et al. (Reference Napoli, Armenio and De Marchis2008).

In figure 12(a), it is clearly observed that ${\rm\Delta}U^{+}$ does not scale solely with $\mathit{ES}$ . This is especially true for data obtained by Schultz & Flack (Reference Schultz and Flack2009) where the same rough surfaces were tested at various Reynolds numbers, thus changing the value of $k_{a}^{+}$ while maintaining constant $\mathit{ES}$ . It is clearly visible in their data that ${\rm\Delta}U^{+}$ increases as $k_{a}^{+}$ increases for a given $\mathit{ES}$ . Our current simulations, which are mostly in the transitionally rough regime, also do not collapse onto the trend reported by Napoli et al. (Reference Napoli, Armenio and De Marchis2008) (♢ symbols). Yuan & Piomelli (Reference Yuan and Piomelli2014) also did not find a collapse onto the curve given by Napoli et al. (Reference Napoli, Armenio and De Marchis2008), observing that the plateau in $\mathit{ES}$ versus ${\rm\Delta}U^{+}$ can occur at larger values of $\mathit{ES}$ for different surfaces. In figure 12(b), we plot ${\rm\Delta}U^{+}$ against $k_{a}^{+}$ , which in §§ 4.4.1 and 4.4.3 were found to be strongly correlated. When scaled in this manner, a reasonable collapse in the data is observed in the fully rough regime as the roughness function follows a log function of ${\rm\Delta}U^{+}=(1/{\it\kappa})\log (C_{eq}k_{a}^{+})+5.3-8.5$ , where $C_{eq}$ is the multiplicative factor for the roughness function to fall on to the fully rough asymptote of the sand grain roughness. However, there is still considerable scatter in the data, notably in the transitionally rough regime. Cases with low effective slope ( $\mathit{ES}<0.2$ ), do not appear to approach the fully rough regime and contribute to the scatter. Obvious outliers are the pyramid data from Schultz & Flack (Reference Schultz and Flack2009) for the cases when $\mathit{ES}=0.19$ ( $\otimes$ symbols). The small gradients of these close-packed pyramids resemble more a gently undulating (‘wavy’) surface rather than a sawtooth roughness, thus indicating that $\mathit{ES}$ has a significant role in determining the asymptotic behaviour of the rough surface (e.g. if it approaches the fully rough regime). This is also observed in the experiments conducted by Acharya et al. (Reference Acharya, Bornstein and Escudier1986) where the ${\rm\Delta}U^{+}$ profile does not have the same slope as the fully rough sand grain roughnesses for the tested surface with the largest ${\it\lambda}/k_{a}$ ratio, which might be in the ‘wavy’ regime. Results from their experimental data also appear to suggest that the multiplicative factor $C_{eq}$ might depend on $\mathit{ES}$ . In addition, the study of Napoli et al. (Reference Napoli, Armenio and De Marchis2008) clearly demonstrates that $\mathit{ES}$ is an important parameter. They simulated six different roughnesses with the same $k_{a}^{+}=19.75$ , but varying $\mathit{ES}$ , reporting a large variation in the measured roughness function ( $0.78<{\rm\Delta}U^{+}<10.29$ ). To a lesser extent, we find a similar (yet weaker) dependence on $\mathit{ES}$ for our three-dimensional roughness (as shown in figures 10 and 12 a). Despite highlighting the fact that both $k_{a}^{+}$ and $\mathit{ES}$ are important roughness parameters, it would appear that neither $\mathit{ES}$ nor $k_{a}^{+}$ alone is suitable for quantifying the roughness function.

Figure 13. (a) Surface plot showing ${\rm\Delta}U^{+}$ as a function of $\mathit{ES}$ and $k_{a}^{+}$ . (b–d) Comparison of measured ${\rm\Delta}U^{+}$ and prediction from (4.8) with (b) numerical simulations of Bhaganagar et al. (Reference Bhaganagar, Kim and Coleman2004), Blackburn et al. (Reference Blackburn, Ooi and Chong2007), Napoli et al. (Reference Napoli, Armenio and De Marchis2008) and De Marchis & Napoli (Reference De Marchis and Napoli2012); (c) experimental data of Schultz & Flack (Reference Schultz and Flack2009); (d) numerical simulations of realistic roughness of Yuan & Piomelli (Reference Yuan and Piomelli2014). See Yuan & Piomelli (Reference Yuan and Piomelli2014) for definition of symbols. Other symbols are as in figure 12. Also shown are sketches of: (i) corrugated roughness of Blackburn et al. (Reference Blackburn, Ooi and Chong2007), (ii) close-packed right-angle pyramids and (iii) sand grain roughness of Yuan & Piomelli (Reference Yuan and Piomelli2014).

To better understand the relationship of ${\rm\Delta}U^{+}$ with $k_{a}^{+}$ and $\mathit{ES}$ , a three-dimensional plot is drawn in figure 13(a). Though there is some scatter, the data seem to lie approximately on the surface

(4.8) $$\begin{eqnarray}{\rm\Delta}U_{est}^{+}={\it\alpha}\log k_{a}^{+}+{\it\beta}\log \mathit{ES}+{\it\gamma}\end{eqnarray}$$

where ${\rm\Delta}U_{est}^{+}$ is the predicted roughness function for known values of $k_{a}^{+}$ and $\mathit{ES}$ and ${\it\alpha}=1/{\it\kappa}$ , ${\it\beta}=1.12$ and ${\it\gamma}=1.47$ are the coefficients obtained from the curve fit. This fit at least captures the basic behaviour observed from figures 10–12 (namely that ${\rm\Delta}U^{+}$ increases with increasing $h^{+}$ or increasing $\mathit{ES}$ ). However, the fit is simplistic, and lacks the correct behaviour in the limits. For large $k_{a}^{+}$ , we would expect ${\rm\Delta}U^{+}$ to increase as the log of $k_{a}^{+}$ , which is captured. However, at very large or very small $\mathit{ES}$ , we might expect ${\rm\Delta}U^{+}$ to tend to zero and this is not reflected in the fit. Despite these limitations, the fit embodies the basic behaviour and will be used to demonstrate that ${\rm\Delta}U^{+}$ is highly dependent on both $k_{a}^{+}$ and $\mathit{ES}$ .

In figure 13(b) we compare the actual roughness function ${\rm\Delta}U^{+}$ of numerically simulated turbulent flows (Bhaganagar et al. Reference Bhaganagar, Kim and Coleman2004; Blackburn et al. Reference Blackburn, Ooi and Chong2007; Napoli et al. Reference Napoli, Armenio and De Marchis2008; De Marchis & Napoli Reference De Marchis and Napoli2012) with the roughness function ${\rm\Delta}U_{est}^{+}$ predicted from (4.8) based on $k_{a}^{+}$ and $\mathit{ES}$ . Even with this simple linear–linear surface fit, we observe a clearer trend than using either $k_{a}^{+}$ or $\mathit{ES}$ alone (as shown in figure 12). More surprisingly, this surface fit also seems to predict the behaviour of the roughness function within the transitionally rough regime. The roughness function for the asymmetric channel of Bhaganagar et al. (Reference Bhaganagar, Kim and Coleman2004) ( $\times$ symbols) is consistently underpredicted by the surface fit but still follows the same approximate trend. A further notable outlier is the data of Napoli et al. (Reference Napoli, Armenio and De Marchis2008) for the two-dimensional roughness case which has very low $\mathit{ES}$ but high $k_{a}^{+}$ (♢ symbol). Despite this, the correlation coefficient of determination of the curve fit with the results of Napoli et al. (Reference Napoli, Armenio and De Marchis2008) is good with $r^{2}=0.763$ as compared to $r^{2}=0.850$ for current simulations. A strong correlation is also observed for the experimental data of Schultz & Flack (Reference Schultz and Flack2009) as illustrated in figure 13(c) where $r^{2}=0.819$ . One would also wonder if the curve fit model which is based on data (mostly) from three-dimensional regular and homogeneous roughness would be applicable to more realistic looking irregular surfaces. In figure 13(d), the roughness function model is applied to the roughness data of Yuan & Piomelli (Reference Yuan and Piomelli2014) who simulated a variety of irregular roughnesses. Again, a good correlation is observed with $r^{2}=0.827$ . This clearly indicates that the roughness function for the rough surfaces tested, either homogeneous or inhomogeneous, depends on both $\mathit{ES}$ and $k_{a}^{+}$ . It is important to note that while the roughness function ${\rm\Delta}U^{+}=f(\mathit{ES},k_{a}^{+})$ is applicable to the data which we have tested, this model will not be able to quantify the alignment (sheltering) of the roughness elements, since neither $k_{a}^{+}$ which is a statistical parameter nor $\mathit{ES}$ which is a gradient, contain any information on the organisation of the surface points. (Note: $\mathit{ES}$ only has compressed (mean) information on the location of the downstream surface point relative to the current point.) There are certainly additional roughness parameters that need to be considered in order to improve functions such as (4.8).

4.4.5. Summary of mean statistics

The mean flow properties for the various roughness cases are summarised in table 3. Figure 14 shows the velocity defect for all tested surfaces at (a,b) $\mathit{Re}_{{\it\tau}}=180$ and (c,d) $\mathit{Re}_{{\it\tau}}=540$ , on both (a,c) linear and (b,d) semi-logarithmic axes. For all cases tested, which cover the transitionally rough to the fully rough regimes, the mean streamwise velocity profiles, in velocity defect form, collapse in the outer layer regardless of the roughness height and wavelength. Even for the highest roughness amplitude ( $h^{+}=80$ ), simulated at $\mathit{Re}_{{\it\tau}}=540$ , the velocity defect profile collapses when $y/R_{0}>0.2$ ( $y^{+}>108$ at $\mathit{Re}_{{\it\tau}}=540$ ), offering support (for this particular roughness) for Townsend’s outer-layer similarity hypothesis (Townsend Reference Townsend1980).

Table 3. Bulk flow properties for the different cases simulated. $\mathit{Re}_{{\it\tau}}$ , $\mathit{Re}_{cl}$ and $\mathit{Re}_{D}$ are the viscous scaled, centreline and bulk Reynolds number which are calculated using the reference radius of the pipe $R_{0}$ .  $U_{cl}^{+}$ is the normalised centreline mean velocity, $U_{b}^{+}$ is the normalised bulk velocity and ${\rm\Delta}U^{+}$ is the Hama roughness function.

Figure 14. Mean velocity defect profiles for the simulated rough and smooth pipes at (a,b) $\mathit{Re}_{{\it\tau}}=180$ and (c,d) $\mathit{Re}_{{\it\tau}}=540$ . (a,c) Linear $y^{+}$ axis; (b,d) semi-logarithmic. Symbols are as given in table 1.

4.5. Second-order statistics

For this section, the turbulence intensities in the radial, azimuthal and streamwise directions are analysed. The velocity fluctuations are calculated about the global average velocity of the flow (temporally and spatially averaged). The turbulence intensity in the radial or wall-normal direction is important in understanding the physics of the flow as it plays a major role in the momentum transfer of high-speed fluid from the outer layer to the wall and low-speed fluid from the wall to the outer layer. Orlandi (Reference Orlandi2013) found a proportionality between the wall-normal turbulence intensity at the top of the roughness and the roughness function ${\rm\Delta}U^{+}$ for simulated two- and three-dimensional cube roughness. This observation formed the basis of an alternative Moody’s diagram where the equivalent sand grain height is replaced by the wall-normal turbulence intensity. This finding is interesting since it suggests that roughness can be modelled at the wall by introducing some magnitude of wall-normal turbulence intensity. The wall-normal intensities for the current pipe flow simulations are shown in figure 15. It can be seen that the conclusion of Orlandi (Reference Orlandi2013) approximately holds for the current data. However, for the group of simulations in the transitionally rough regime where the wavelength is altered (while $h^{+}$ remains constant) there is a noticeable change in the roughness function (see figure 10) while the radial turbulence intensities at the peak of the roughness elements ( $y^{+}=10$ ) remain virtually unchanged (see figure 15 b). However, in general there is a pronounced increase in the radial turbulence intensities at the peak of the roughness elements with increasing roughness size and increasing ${\rm\Delta}U^{+}$ . There is also a pronounced variation in the radial turbulence intensities within the roughness elements (for $y<h$ ); however the profiles collapse on to the smooth-wall case very quickly and convincingly above the roughness elements (for $y^{+}\gtrsim 2h^{+}$ ). This is contrary to the experiments conducted by Krogstad & Antonia (Reference Krogstad and Antonia1999) and George & Simpson (Reference George and Simpson2000) for a rough-wall boundary layer where the authors found that the wall-normal fluctuation increases in the inner region of the rough wall compared to a smooth-wall boundary layer. Krogstad & Antonia (Reference Krogstad and Antonia1999) also found that the wall-normal fluctuation in the outer region is significantly different from the smooth-wall fluctuation for matched Reynolds numbers. It should be noted that the collapse, or lack thereof, of statistics in the near-wall region can be extremely sensitive to the choice of virtual origin, which in the case of the roughened pipe is relatively well defined (see § 4.2), but for external boundary layers is very challenging to accurately determine experimentally.

The azimuthal turbulence intensities are shown in figure 16. The collapse in azimuthal intensities is very convincing for both Reynolds numbers beyond $y^{+}\approx 60$ . For an external flow, the height of the boundary layer increases in the presence of the rough surface. However, for a pipe, this is not an option as the boundary layer cannot thicken and the fixed virtual origin (see § 4.2) ensures that the flow is channelled between the roughness elements. At the front of the roughness elements, flow is diverted in either the positive of negative spanwise direction, which gives rise to strong azimuthal fluctuations. This is clearly visible in the contour plot of the azimuthal turbulence intensity in the streamwise rough plane ( $A$ ) which is shown in figure 17(a) for case 20\_141.  This fluctuation causes the appearance of a secondary peak observed below the roughness crest in figure 16(a,b) which seems to move further outwards as $h^{+}$ increases. Along the ‘smooth’ streamwise plane ( $B$ ), the flow near the wall is channeled and this is reflected in the regions with weak azimuthal fluctuations which periodically occurs every half of the roughness wavelength (see figure 17(b)).

Figure 15. Radial turbulence intensities for various rough-wall pipes at (a,b) $\mathit{Re}_{{\it\tau}}=180$ and (c,d) $\mathit{Re}_{{\it\tau}}=540$ . (a,c) Linear $y^{+}$ axis; (b,d) semi-logarithmic. Symbols are as given in table 1.

Figure 16. Azimuthal turbulence intensity for various rough-wall pipes at (a,b) $\mathit{Re}_{{\it\tau}}=180$ and (c,d) $\mathit{Re}_{{\it\tau}}=540$ . (a,c) Linear $y^{+}$ axis; (b,d) semi-logarithmic. Symbols are as given in table 1.

Figure 17. Contours of time-averaged azimuthal fluctuation intensities, $u_{{\it\theta},rms}^{\prime +}$ , for case 20\_141 at (a) plane $A$ and (b) plane $B$ . The occurrence of the secondary peak is due to the high fluctuations at the leading edge of the roughness elements. Flow is from left to right.

There is significant variation in the streamwise turbulence intensity for the inner region of the flow (figure 18). However, for $y/R_{0}\gtrsim 0.33$ there is good collapse in the outer region. The maximum streamwise turbulence intensity reflects the radial momentum transfer, which pulls high-speed fluid from the centre of the pipe towards the wall, and ejects low-speed fluid radially outwards from the near-wall region. For a smooth wall, the maximum streamwise turbulence intensity is located at $y^{+}=15$ , but in a rough wall there appears to be two competing factors. For the transitionally rough cases, there are signs that a peak or local maximum in streamwise intensity persists at $y^{+}=15$ , albeit of diminishing strength as $h^{+}$ increases. There are also signs of a further local maximum or bump in the streamwise intensity occurring close to, or just beneath, the roughness crest height $h^{+}$ . This is difficult to discern when $h^{+}$ is close to 15, but is subtly visible for cases with $h^{+}<10$ , becoming very pronounced for the higher Reynolds number cases where $h^{+}$ becomes large (figure 18 d).  Figure 19 shows planar views of the streamwise intensity on planes $A$ and $B$ for case 20\_141. The maximum streamwise turbulence intensity occurs above the roughness crest at $y^{+}=27$ in plane $A$ and at $y^{+}=12$ in plane $B$ , yielding a spatially averaged value that lies very close to the crest of the roughness elements.

Figure 18. Streamwise turbulence intensity for various rough-wall pipes at (a,b) $\mathit{Re}_{{\it\tau}}=180$ and (c,d) $\mathit{Re}_{{\it\tau}}=540$ . (a,c) Linear $y^{+}$ axis; (b,d) semi-logarithmic. Symbols are as given in table 1.

Figure 19. Contours of time-averaged streamwise velocity fluctuation intensities, $u_{x,rms}^{\prime +}$ , for case 20\_141 at (a) plane $A$ and (b) plane $B$ . Dash-dotted lines: location of maximum turbulence intensity ( $y^{+}=27$ and $y^{+}=12$ ). Flow is from left to right.

The Reynolds shear stresses are plotted in figure 20. The profiles for the rough surfaces in the transitionally and fully rough regime collapse onto the smooth-wall case almost immediately above the peak of the roughness elements. It is surprising how convincingly the Reynolds stress of the rough surfaces collapses onto the smooth-wall pipe data for the cases tested, and how close to the roughness crest this collapse occurs. This collapse occurs within the region that is classically considered to be the roughness sublayer.

Figure 20. Reynolds shear stress for various rough-wall pipes at (a,b) $\mathit{Re}_{{\it\tau}}=180$ and (c,d) $\mathit{Re}_{{\it\tau}}=540$ . (a,c) Linear $y/R_{0}$ axis; (b,d) logarithmic $y^{+}$ axis. Symbols are as given in table 1.

Figures 15–20 demonstrate that the mean statistics for all three fluctuating components, along with the Reynolds shear stress, collapse in the outer region, providing apparent evidence in support of Townsend’s outer-layer similarity hypothesis. Various other numerical and experimental studies have provided similar support for Townsend’s outer-layer hypothesis (Flack, Schultz & Connelly Reference Flack, Schultz and Connelly2007; Wu & Christensen Reference Wu and Christensen2007; Leonardi & Castro Reference Leonardi and Castro2010; Efros & Krogstad Reference Efros and Krogstad2011). For a boundary layer, outer-layer similarity is typically believed to depend on the blockage ratio ${\it\delta}/k$ ( ${\it\delta}$ is the boundary layer thickness), which should usually be large for any similarity to be present (i.e. ${>}40$ , see Jimenez Reference Jimenez2004). In an open rough-wall channel where a free-slip boundary condition is imposed on the upper surface, outer-layer similarity was observed by Coceal et al. (Reference Coceal, Thomas, Castro and Belcher2006) and Leonardi & Castro (Reference Leonardi and Castro2010) for ${\it\delta}/k=8$ and ${\it\delta}/k=4$ respectively. These values are significantly smaller than the values proposed for a boundary layer. The present rough-pipe simulations offer support for outer-layer similarity in the mean statistics even for our largest roughness cases which have ${\it\delta}/k=6.75$ . It should be highlighted that, throughout this study, we have only considered notions of outer-layer similarity in terms of mean statistics (and only up to second-order statistics). This type of similarity is important for the prediction of rough-wall flows at application-type Reynolds numbers. However, there are certain notable instances in the literature, where more subtle differences have been observed in the instantaneous coherent structure between rough and smooth flows (see for example Hong, Katz & Schultz Reference Hong, Katz and Schultz2011).

It is notable that our sinusoidal pipe roughness, which is geometrically quite similar to the ‘egg carton’ roughness elements used by Bhaganagar et al. (Reference Bhaganagar, Kim and Coleman2004) in their one-sided rough-channel simulations ( $h^{+}=5.4,10.8,21.6$ and ${\it\lambda}^{+}=100$ ), produced such different results. Bhaganagar et al. (Reference Bhaganagar, Kim and Coleman2004) did not obtain a collapse in the outer layer of the flow for any intensities and found that the location of the peak turbulence intensity moves further away from the rough wall as the roughness height is increased. It should be noted that the asymmetry in their channel gives rise to a number of differences with respect to standard symmetrical internal flows, that could influence the perception of outer-layer similarity. (i) The boundary layer at the rough side grows at the expense of the smooth side. As a result of this growth, it is possible that the virtual origin may differ for an asymmetric channel. (ii) The total wall drag on the rough surface is higher than that on the smooth surface. At the edge of the layer, this precludes the possibility of true outer-layer similarity, since the same variance values are normalised by different $U_{{\it\tau}}$ values for the smooth and rough wall. Therefore, in this asymmetric case, there can be no collapse obtained in the outer layer of the flow when the statistics are viscously scaled. (iii) In comparing rough-wall statistics from one side of the channel, to smooth-wall statistics from the other, one is not comparing flows at matched Reynolds number. (iv) In the light of the interaction that is known to take place between the two walls of turbulent channel flow (Sabot & Comte-Bellot Reference Sabot and Comte-Bellot1976; Antonia et al. Reference Antonia, Teitel, Kim and Browne1992; Kwon et al. Reference Kwon, Philip, de Silva, Hutchins and Monty2014), it is doubtful whether the ‘smooth’ wall in such asymmetric simulations is really representative of the smooth-wall canonical turbulent channel.

Pressure statistics are examined in figure 21. Bhaganagar, Coleman & Kim (Reference Bhaganagar, Coleman and Kim2007) noted that for their ‘egg carton’ roughness, the increased pressure drag due to the roughness elements is associated with the intensity of the pressure fluctuations, where the intensity of the pressure fluctuations of the rough wall were higher than of the smooth wall of their asymmetric channel. However, when the pressure fluctuations of the rough wall were scaled locally with the friction velocity of the rough side, the pressure intensity of the rough wall was lower than the pressure fluctuations of the smooth wall. The results in figure 21 contradict the primary observation of Bhaganagar et al. (Reference Bhaganagar, Coleman and Kim2007), where the current simulations exhibit significant increases in pressure fluctuations for the rough pipes above the smooth-wall level when scaled with local $U_{{\it\tau}}$ . The outer region of the pipe is not affected by the intense pressure fluctuations occurring in the vicinity of the roughness elements and collapses for all cases for $y/R_{0}\approx 0.33$ .

Figure 21. Root-mean-square pressure fluctuation profile for various rough-wall pipes at (a,b) $\mathit{Re}_{{\it\tau}}=180$ and (c,d) $\mathit{Re}_{{\it\tau}}=540$ . (a,c) Linear $y^{+}$ axis and (b,d) semi-logarithmic. Symbols are as given in table 1.