2.1. Problem definition

The dynamics of the flow is established by the incompressible Navier–Stokes equations,

where $p$ is the fluctuating pressure, ${\it\rho}$ is the density of the fluid, ${\it\nu}$ is the kinematic viscosity, $\boldsymbol{g}$ is a forcing vector and $\boldsymbol{N}(\boldsymbol{u})$ represents the nonlinear advection terms. Since a wavy-wall pipe is axisymmetric, it is convenient to examine the problem in a cylindrical-coordinate system, and thus we denote the axial, radial and azimuthal components of velocity by $\boldsymbol{u}(z,r,{\it\theta})(t)=(u,v,w)(t)$ . For a fully developed turbulent pipe flow the driving force $\boldsymbol{g}=(g,0,0)$ corresponds to the mean pressure gradient in the streamwise $z$ direction, and, as is common in simulations of this type, is used in order to allow both the pressure and the velocity to be axially periodic. No-slip boundary conditions are applied to the velocity field along the walls of the domain.

The fundamental length scale is the diameter $D$ of the corresponding smooth pipe. The highest Kármán number or friction Reynolds number is $\mathit{Re}_{{\it\tau}}=u_{{\it\tau}}D/2{\it\nu}=314$ . Here, the friction velocity is defined by $u_{{\it\tau}}=({\it\tau}_{o}/{\it\rho})^{1/2}$ , where ${\it\tau}_{o}$ is the mean wall shear stress. The axial mean pressure gradient per unit mass $g_{o}=4{\it\tau}_{o}/{\it\rho}D={\it\lambda}{\it\rho}U_{b}^{2}/2D$ (where $U_{b}$ is the bulk-flow velocity) is required to drive the flow in a smooth pipe, and the friction factor ${\it\lambda}$ in smooth pipes is given by

Here, $U_{b}$ is used to define the bulk-flow Reynolds number as $\mathit{Re}_{D}=U_{b}D/{\it\nu}=4\langle Q\rangle /{\rm\pi}D{\it\nu}$ , where $\langle Q\rangle$ is the average volumetric flow rate. From the definitions above one can also determine that

The methodology pursued in the present work produces sets of simulations, each with constant $\mathit{Re}_{{\it\tau}}$ , from the maximum value $\mathit{Re}_{{\it\tau}}=314$ down to the laminar regime, typically with the driving force per unit mass $g_{o}$ halved between successive sets. Since

the friction Reynolds number typically varies by a factor of $(2)^{-1/2}$ between successive sets of simulations. It should be noted that the Blasius friction factor correlation for smooth pipes is given by ${\it\lambda}=0.3164\mathit{Re}_{D}^{-1/4}$ (Nikuradse Reference Nikuradse1933); for moderate turbulent Reynolds numbers this yields the relationship $\mathit{Re}_{{\it\tau}}=0.09944\mathit{Re}_{D}^{7/8}$ . Consequently, the maximum value of the bulk-flow Reynolds number in the smooth pipe for the present work, with $\mathit{Re}_{{\it\tau}}=314$ , is $\mathit{Re}_{D}\approx 10\,000$ .

All the domains have the same mean radius $\bar{R}=D/2$ , and for turbulent flow calculations the same axial length $L=2{\rm\pi}D$ . The length to diameter ratio was chosen based on the previous pipe length convergence studies by Chin et al. (Reference Chin, Ooi, Marusic and Blackburn2010). They showed that a pipe length of at least $2{\rm\pi}D$ is required to achieve converged turbulent-flow statistics for $\mathit{Re}_{{\it\tau}}\approx 500$ .

For the wavy-wall flows, 15 corrugation wavelengths were chosen within the domain length of $2{\rm\pi}D$ . This number is large enough to ensure that, even at transitional Reynolds numbers, the number of wavelengths is sufficient to reduce the streamwise correlation of near-wall structures to acceptably low values at half the domain length (Chin et al. Reference Chin, Ooi, Marusic and Blackburn2010). With a corrugation amplitude of $a=h/2$ , the radius of the wavy pipe, $R(z)$ , is given by

as illustrated in figure 1. For laminar-flow calculations, only a single module of the axial wave is represented, i.e. the domain length is reduced to $L_{m}=2{\rm\pi}D/15\approx 0.41888D$ .

When attempting to define both the bulk-flow and friction Reynolds numbers for a wavy-wall (or any non-uniform) pipe, one needs an equivalent diameter. For simplicity, we have adopted the mean diameter $D=2\bar{R}$ for this measure. However, for a constant mean radius the volume of the domain increases as the corrugation height increases, and thus we reduce the driving force $g$ as $h$ increases in order to maintain the total body force constant at each $\mathit{Re}_{{\it\tau}}$ .

From Pappus’ second theorem (Kern & Bland Reference Kern and Bland1948), the domain volume can be found in closed form as ${\rm\pi}(\bar{R}^{2}+h^{2}/8)L$ , provided that the length comprises an integral number of wavelengths. Using the equivalent diameter, an equivalent mean wall shear stress is found by equating the mean wall tractive force to the mean body force on the domain, i.e. 

from which

Thus, if one wishes to keep the friction velocity and hence $\mathit{Re}_{{\it\tau}}$ constant as $h$ increases, the driving force per unit mass $g$ must be reduced as $h$ increases according to

Here, $V_{o}$ is the volume of the smooth-wall pipe and $g_{o}$ is the corresponding axial driving force.

By employing $u_{{\it\tau}}$ and $D$ for normalization, equation (2.1) takes on the dimensionless form

where $\boldsymbol{F}^{+}=(4V_{o}/V,0,0)$ and the ‘ $+$ ’ superscript indicates normalization by ${\it\nu}$ and $u_{{\it\tau}}$ . The use of (2.9) is well-suited for numerical simulation, as one only needs to assign the value of $\mathit{Re}_{{\it\tau}}$ . The resultant value of $\mathit{Re}_{D}$ can be easily evaluated after the flow field becomes statistically stationary.

Napoli, Armenio & De Marchis (Reference Napoli, Armenio and De Marchis2008) introduced an important roughness parameter called the ‘effective slope’ (ES). The effective slope ES accounts for the roughness corrugation shape, and is defined by

where, in the present case, $L$ is an integral number of corrugations. This function will allow us to investigate the influence of a rough wall on the roughness function as well as friction and pressure drag.

Table 1 summarizes the main parameters of the wavy-wall geometries. The maximum peak-to-peak corrugation amplitude of $h/D=0.07952$ was chosen to be 50 wall units at $\mathit{Re}_{{\it\tau}}=314$ , i.e.  $h^{+}=50$ . Direct numerical simulations were performed for a range of Reynolds numbers at this corrugation height (case G). The post-transitional turbulent regime simulations were at $\mathit{Re}_{{\it\tau}}=180$ and 250, as also listed in table 1. Simulations were also carried out for corrugation heights of $h^{+}=40$ , 30, 20, 10, 5 and 0 (all at $\mathit{Re}_{{\it\tau}}=314$ ). It can be seen that the fractional increase in volume, $V/V_{o}$ , and hence the fractional reduction in driving force, is less than half a per cent even at the largest corrugation height, case G. The corrugation wavelength, $L_{m}=2{\rm\pi}D/15$ , corresponds to 263 wall units at $\mathit{Re}_{{\it\tau}}=314$ .

2.2. Mean momentum equation

The mean momentum equation for flow inside a wavy-walled pipe is developed and discussed in this section according to the analysis of Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005a ,Reference Wei, McMurtry, Klewicki and Fife b ). By applying the Reynolds decomposition, time averaging and simplifying for statistically stationary axisymmetric flow, the streamwise component of (2.1) becomes

where $U$ is the mean velocity component in the $z$ direction, $P$ is the mean pressure and $\langle u^{\prime }v^{\prime }\rangle$ is the Reynolds shear stress. The expression for the mean pressure gradient (2.8) allows (2.11) to be written as

Equation (2.12) contains two unknown functions, $U$ and $\langle u^{\prime }v^{\prime }\rangle$ , and thus is unclosed. The boundary conditions at the pipe centreline, $r=0$ , are

Integrating (2.12) with respect to $r$ and making use of (2.13) yields

Using (2.14), (2.12) then becomes

It is convenient to rewrite (2.15) using $y=\bar{R}-r$ , where it is understood that $y$ is the average wall position. With this, one obtains

The friction velocity, $u_{{\it\tau}}$ , inner length scale, ${\it\nu}/u_{{\it\tau}}$ , and outer length scale, $\bar{R}$ , constitute the basic normalization parameters. Hence, the inner-normalized mean momentum equation can be obtained from (2.16) as

The small parameter ${\it\varepsilon}$ is defined by

where ${\it\varepsilon}\rightarrow 0$ as $\mathit{Re}_{{\it\tau}}\rightarrow \infty$ , $y^{+}=yu_{{\it\tau}}/{\it\nu}$ is the inner-wall-normalized distance, $U^{+}=U/u_{{\it\tau}}$ is the inner-normalized streamwise-mean velocity and $T_{u}^{+}=-\langle u^{\prime }v^{\prime }\rangle /u_{{\it\tau}}^{2}$ is the inner-normalized Reynolds shear stress. It is important to note that ${\it\varepsilon}$ explicitly contains the roughness effect, thus indicating the combined roughness and Reynolds number nature of the problem. Equation (2.17) indicates a balance of three terms, namely $\mathit{VF}=\text{the mean viscous force}$ (viscous stress gradient), $\mathit{TI}=\text{the mean effect of turbulent inertia}$ (Reynolds shear-stress gradient) and $\mathit{PG}=\text{the mean pressure gradient}$ . The relative magnitudes of these three terms determine a distinct layer structure characterized by the leading-order balances in (2.17).

The outer form of the equation uses the mean pipe radius $\bar{R}$ to normalize the wall-normal distance ${\it\eta}=y/\bar{R}$ . This gives

and the boundary conditions at ${\it\eta}=1$ are

Equations (2.17)–(2.21) are considered in § 5 to help to clarify the scaling behaviours associated with the mean dynamics.

3.1. Discretization

A cylindrical-coordinate spectral element/Fourier spatial discretization is employed (Blackburn & Sherwin Reference Blackburn and Sherwin2004). Nodal spectral elements are deployed to discretize the meridional semi-plane, and Fourier expansions are used in the azimuthal direction. This is possible because the domain is axisymmetric. The velocity, $\boldsymbol{u}^{+}$ , and pressure, $p^{+}$ , can be projected onto a set of two-dimensional complex Fourier modes,

where $k_{o}$ is the azimuthal wavenumber. Only a finite number of these modes are represented in the calculation.

The nonlinear advection terms $\boldsymbol{N}(\boldsymbol{u}^{+})$ are computed in skew-symmetric form $\boldsymbol{N}(\boldsymbol{u}^{+})=(\boldsymbol{u}^{+}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{u}^{+}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{u}^{+}\boldsymbol{u}^{+})/2$ for robustness, but are not explicitly dealiased. Time integration is carried out via a second-order mixed implicit–explicit pseudo-spectral velocity correction scheme (Karniadakis, Israeli & Orszag Reference Karniadakis, Israeli and Orszag1991; Guermond & Shen Reference Guermond and Shen2003). More detail about the numerical method is presented by Blackburn & Sherwin (Reference Blackburn and Sherwin2004), who demonstrate that the method attains spectral convergence for non-axisymmetric flows and provide a full explanation of how geometric singularities at the axis are overcome.

The computational meshes retain a quadrilateral spectral element strategy that has been successfully implemented in our previous DNS and wall-resolving large-eddy simulations for smooth-wall geometries (Schmidt et al. Reference Schmidt, McIver, Blackburn, Rudman and Nathan2001; Blackburn & Schmidt Reference Blackburn and Schmidt2003; Chin et al. Reference Chin, Ooi, Marusic and Blackburn2010; Saha et al. Reference Saha, Chin, Blackburn and Ooi2011). The viscous length scale, $\ell _{v}={\it\nu}/u_{{\it\tau}}$ , is first determined for the maximum Reynolds number to be attempted. In the radial direction, the distance of the first element boundary from the wall is set at $10\ell _{v}$ . This resolves the viscous sublayer. The distance from the wall to the second element boundary is then set near the height of maximum turbulent energy production (here, we have used $25\ell _{v}$ ). The remaining element heights to the pipe centreline form a geometric progression, where the number of elements is chosen on the basis of experience. The result is then checked to ensure that features near the centreline of the pipe are adequately resolved.

To estimate the remaining mesh parameters, rules of thumb for resolving wall-flow DNS are adopted from Piomelli (Reference Piomelli, Liu and Liu1997). Near the wall, ${\it\Delta}_{y}^{+}<1$ , ${\it\Delta}_{z}^{+}\approx 15$ and ${\it\Delta}_{{\it\theta}}^{+}\approx 6$ in the wall-normal, streamwise and cross-flow directions respectively. Given the wall-normal height of the first element, we employ 10th-order nodal shape functions (i.e. with 11 points along the edge of an element) in order to satisfy ${\it\Delta}_{y}^{+}<1$ . Because the elements employ equal-order tensor-product shape functions, there is the same number of points within each element in the streamwise direction as in the wall-normal direction, and we use ${\it\Delta}_{z}^{+}\approx 15$ for the streamwise length of a near-wall element. In the present case, this requires 26 elements to reach the streamwise domain length of $2{\rm\pi}D$ .

We therefore adopt 30 elements in the streamwise direction for convenience, allowing two elements per geometric wavelength. For the present problem, the added complexity of a hierarchical mesh design to resolve fine-scale near-wall geometric features is not required or justified, and thus we have a simple (logically rectangular) $30\times 8$ array of elements to cover the meridional semi-plane. These elements are distorted isoparametrically to accommodate the wavy-wall shapes, as shown in figure 2. In order to establish the number of planes of data required in the azimuthal direction, we find from ${\it\Delta}_{{\it\theta}}^{+}\approx 6$ that 320 planes (160 Fourier modes) are adequate at $\mathit{Re}_{{\it\tau}}=314$ . Overall, the number of independent mesh nodes in the meridional semi-plane is 7020, and the total number of nodes is then approximately 2.25 million for the simulations conducted at $\mathit{Re}_{{\it\tau}}=314$ . Fewer mesh nodes are required at lower values of $\mathit{Re}_{{\it\tau}}$ . The same spectral element outlines are retained, but the orders of the element shape functions and Fourier azimuthal interpolants are reduced as appropriate.

Figure 2. Spectral element meshes, each with 240 elements in the meridional semi-plane. Here, $h$ is the peak-to-peak wave height and $a=h/2$ is the corrugation amplitude. The labels match those used in table 1: (a) smooth-wall pipe; (b)  $h_{314}^{+}=20$ ; (c)  $h_{314}^{+}=50$ .

We note that since the corrugated domains are two-dimensional, it follows that the associated time-mean flows and local turbulent statistics are too. When we present one-dimensional profile data for turbulence statistics below, the values have also been averaged in the streamwise direction. This implies that profiles representing averages of product terms, such as Reynolds stresses, contain contributions from streamwise averages of products of the deviations of the time-mean flow velocity components from the streamwise-mean profile as well as from the streamwise averages of the local fluctuation products. Contributions from the time-mean or ‘coherent’ streamwise fluctuations from the mean velocity profile are typically only significant within one corrugation height of the mean radius.

3.2. Validation for smooth-pipe flow

Due to the lack of availability of wavy-wall pipe flow data, the veracity of the smooth-wall turbulent-flow calculations was tested by comparing statistical profiles with laser Doppler velocimetry (LDV) measurements obtained at $\mathit{Re}_{{\it\tau}}=314.5$ by den Toonder & Nieuwstadt (Reference den Toonder and Nieuwstadt1997). Comparisons were presented in figure 7 of Blackburn et al. (Reference Blackburn, Ooi and Chong2007). It should be noted that statistical data were only calculated after all the initial transients had convected out of the computational domain and the flow field had reached a statistically stationary state. The mean velocity showed excellent agreement with the measurements. The comparison of the second-order statistics between the DNS and measurements was also very good, except in the near-wall region where measurement inaccuracies became apparent. The r.m.s. profile of the radial fluctuations was also slightly but consistently lower than the experimental measurements. Such deviations are, however, common, e.g. as observed by Westerweel (Reference Westerweel1993) and den Toonder & Nieuwstadt (Reference den Toonder and Nieuwstadt1997). The reasons behind this underestimation of the radial velocity r.m.s. apparently remains an open question.

5.1. Mean velocity profiles

The effects of the corrugation height and Reynolds number are evident in the behaviour of the mean velocity profile. Figure 5 presents mean velocity profiles in both inner and velocity defect forms. The results for the smooth-wall pipe (case A) are also included for comparison. The mean statistics in the corrugated pipe were computed by averaging over $z{-}{\it\theta}$ planes uniformly up to the minimum sensible radius $R_{min}$ . The velocity field around the wavy surface is strongly influenced by the surface profile. This is clearly seen by comparing the near-wall flow with the smooth-wall case. With an increase in corrugation height, the maximum centreline velocity decreases in comparison to the smooth pipe. The offset in the maximum mean velocity profiles depends on the magnitude of the corrugation amplitude, as suggested by Blackburn et al. (Reference Blackburn, Ooi and Chong2007). By plotting the mean velocity profiles for case G for $\mathit{Re}_{{\it\tau}}=180\,250$ and 314, figure 5(a) shows an increasing downward shift of the inner-normalized profiles with increasing Reynolds number. The near-wall velocity profiles also remain substantially different from the smooth-wall pipe flow. The existence of the no-slip boundary condition causes the mean velocity profiles to vary sharply within the near-wall corrugated region. Hence, the overall velocity field (averaged over $z{-}{\it\theta}$ planes) exhibits a larger momentum deficit when compared with the smooth-wall flow. In contrast, the mean velocity profiles in defect form (figure 5 b) exhibit agreement for all cases in the region ${\it\eta}\geqslant 0.2$ for the present conditions. This is in accord with Townsend’s wall-similarity hypothesis (Townsend Reference Townsend1956). Observations similar to these were made by Shockling et al. (Reference Shockling, Allen and Smits2006) for distributed roughness pipe flows in the transitionally and fully rough regimes. The mean profiles systematically vary as a function of corrugation amplitude within the corrugated sublayer. This is consistent with the study of Wu & Christensen (Reference Wu and Christensen2007).

Figure 5. Mean velocity profile for different corrugation heights at $\mathit{Re}_{{\it\tau}}=314$ and for different values of $\mathit{Re}_{{\it\tau}}$ at the highest corrugation height ( $h/D=0.46024$ ): (a) inner scaling and (b) outer scaling (velocity defect law). The symbol shapes for the DNS data are given in table 1.

The effect of the corrugation height is characterized here using a corrugation amplitude, $a=h/2$ . This has similarity to an equivalent sand-grain roughness height, $k_{s}$ (Nikuradse Reference Nikuradse1933; Schlichting Reference Schlichting1936). When $a$ increases to become comparable to ${\it\nu}/u_{{\it\tau}}$ , the inner-scaled profiles exhibit a downward shift relative to the smooth-wall profile. The wall-normal extent of this deficit, which can be interpreted as an internal layer within which the length scales imposed by the corrugated surface directly impact the dynamics, is surface-dependent, with the highest corrugation height showing the largest wall-normal extent of velocity deficit ( $y^{+}\geqslant 50$ ). Consistent with enhanced drag, this deficit increases with increasing corrugation height, and is largest for case G. For the set of conditions explored, the mean velocity profile can be expressed as

where ${\rm\Delta}U^{+}$ is denoted as the corrugation function. Analogous to the roughness function of Hama (Reference Hama1954), the corrugation function depends on $a^{+}$ or $h^{+}$ . The relationship between ${\rm\Delta}U^{+}$ and $a^{+}$ follows from the concept of equivalent sand-grain roughness. For the profiles of figure 6(a), ${\rm\Delta}U^{+}=4.45$ and 8.0 for $h_{314}^{+}=20$ and 50 respectively. According to the analogy with equivalent sand-grain roughness, for sufficiently large $a^{+}$ there should be a loss of dependence on viscosity, allowing the log-law velocity profile to be written as

where $A^{\prime }$ is the effective corrugation function, analogous to the Nikuradse roughness function.

Figure 6. Mean velocity profile plotted (a) against $y^{+}$ showing the logarithmic region and (b) against $y/a$ . The data sets are as for figure 5.

Figure 6(b) plots $U^{+}$ versus $y/a$ . In the fully rough regime, the viscous sublayer no longer exists. Consistently, the effective corrugation function appears to approach an approximately constant value of $A^{\prime }\simeq 5.6$ , which is smaller than the Nikuradse roughness function value of approximately 8.5 in the fully rough regime. We also observe that for the conditions explored herein the increasing $a^{+}$ profiles at fixed $\mathit{Re}_{{\it\tau}}$ continue to exhibit a small but apparently persistent variation. This probably indicates that the largest $a^{+}$ condition may still not be in the fully rough (fully corrugated) regime. For completeness we note that from (5.1) and (5.2), the corrugation function in the fully rough regime is given by

In the present work, the corrugation function ${\rm\Delta}U^{+}$ is plotted against the ES as shown in figure 7. For comparison, data from Napoli et al. (Reference Napoli, Armenio and De Marchis2008), Schultz & Flack (Reference Schultz and Flack2009) and De Marchis & Napoli (Reference De Marchis and Napoli2012) are included in the plot. The results are presented for $0\leqslant \mathit{ES}\leqslant 0.38$ at the highest Reynolds number and are found to show a trend similar to other observations. Napoli et al. (Reference Napoli, Armenio and De Marchis2008) observed linear variation of ${\rm\Delta}U^{+}$ for $\mathit{ES}\leqslant 0.15$ , whereas the corrugation function for sinusoidally corrugated pipes increases linearly up to approximately $\mathit{ES}\sim 0.24$ .

Figure 7. Dependence of the roughness function ${\rm\Delta}U^{+}$ on the ES of the wall corrugations at $\mathit{Re}_{{\it\tau}}=314$ . Data for different types of wall corrugation and $\mathit{Re}_{{\it\tau}}$ from Napoli et al. (Reference Napoli, Armenio and De Marchis2008), Schultz & Flack (Reference Schultz and Flack2009) and De Marchis & Napoli (Reference De Marchis and Napoli2012) are included here for comparison.

5.2. Velocity fluctuation r.m.s. profiles

The effects of roughness in the inner and outer regions of the flow are examined in the context of Townsend’s outer similarity hypothesis by comparing the turbulence intensity profiles from the wavy- and smooth-wall pipes. Figure 8(a–f) shows the turbulence intensities in the streamwise, wall-normal and azimuthal directions as normalized by the friction velocity, and for both inner- and outer-normalized corrugation heights. The root-mean-square (r.m.s.) profiles of the streamwise velocity fluctuations in figure 8(a) do not vary with the presence of corrugation for $y^{+}\geqslant 60$ , and agree very well with the results for the smooth-pipe flow in the outer region of the flow. As the wall is approached, however, the wavy-wall data fall below the smooth-wall data. This is generically consistent with the findings for other rough-wall pipe flows, e.g. Allen et al. (Reference Allen, Shockling, Kunkel and Smits2007).

Figure 8. Root mean square profiles of turbulent intensities for different corrugation heights at $\mathit{Re}_{{\it\tau}}=314$ and for different values of $\mathit{Re}_{{\it\tau}}$ at the largest corrugation height ( $h/D=0.46024$ ): (a,b) streamwise velocity fluctuation; (c,d) radial velocity fluctuation; (e,f) azimuthal velocity fluctuation; (a,c,e) profiles representing inner normalization; (b,d,f) profiles in outer-scaled variables. The data sets are as for figure 5.

The profiles of radial and tangential velocity fluctuation (see figure 8 c–f) also merge outside the corrugated sublayer. These findings are consistent with the notion of outer-layer similarity. Relative to ${u^{\prime }}_{rms}^{+}$ , the effect of the roughness is more subtly realized for the ${v^{\prime }}_{rms}^{+}$ and ${w^{\prime }}_{rms}^{+}$ profiles, whose near-wall peak values and locations show little change from the smooth-wall profile. Overall, the main influence on the ${u^{\prime }}_{rms}^{+}$ profile is that its peak value attenuates near the wavy wall and shifts away from the wall with increasing $h^{+}$ . On the other hand, the primary apparent effect on the ${v^{\prime }}_{rms}^{+}$ and ${w^{\prime }}_{rms}^{+}$ profiles is that the locations of their peaks shift slightly towards the pipe centre with increasing $h^{+}$ .

By considering increasing $\mathit{Re}_{{\it\tau}}$ for a fixed corrugation height, the traditional description leads to the expectation of encountering a dynamically smooth wall, followed by a transitionally rough wall and culminating with a fully rough-wall flow. The Reynolds number range of the present study ( $\mathit{Re}_{{\it\tau}}=180,250,314$ ), however, apparently only resides in the transitional rough-wall regime. This is even so for the case of the largest corrugation height ( $h/D=0.07952$ ), which corresponds to $h^{+}\approx 30,40$ and 50 respectively. Figure 8 also displays the inner- and outer-normalized turbulence intensities for the cases G180 and G250. These data indicate an increasing trend in the peak values of ${v^{\prime }}_{rms}^{+}$ and ${w^{\prime }}_{rms}^{+}$ with $\mathit{Re}_{{\it\tau}}$ , while the peak in the ${u^{\prime }}_{rms}^{+}$ profile decreases. Continued profile variations of the type displayed for increasing $h^{+}$ provide evidence of transitional rough-wall flow. The data in the outer region are in imperfect agreement with Townsend’s similarity hypothesis, indicating results similar to those presented by Flack et al. (Reference Flack, Schultz and Shapiro2005). Likely reasons for this include low Reynolds number and insufficiently small ${\it\delta}/h$ .

5.3. Reynolds shear-stress profile

Profiles of the Reynolds shear stress are presented in figure 9. Over a considerable outer-region extent, and for inner- and outer-normalized distance from the wall, the wavy-wall profiles $T_{u}^{+}=-\langle uv\rangle ^{+}$ convincingly merge with the smooth-wall profile. The effect of corrugation becomes stronger only at high corrugation height, and thus the profiles near the wavy-wall fall below the smooth-wall data. Under outer normalization the corrugated- and smooth-wall profiles consistently merge at a wall-normal position just beyond the peak of $T_{u}^{+}$ . As discussed in § 7, this behaviour correlates with an important feature of the mean dynamical balance. The influence of $\mathit{Re}_{{\it\tau}}$ is also apparent. The inner-normalized profiles of figure 9(a) exhibit an expected shift with increasing $\mathit{Re}_{{\it\tau}}$ . The outer-normalized profiles of figure 9(b), however, show excellent agreement beyond the $T_{u}^{+}$ peak, suggesting robust support for outer similarity in the domain where the mean dynamics is inertially dominated (see § 7).

Figure 9. Reynolds shear-stress profiles for different corrugation heights at $\mathit{Re}_{{\it\tau}}=314$ and for different values of $\mathit{Re}_{{\it\tau}}$ at the largest corrugation height ( $h/D=0.46024$ ): (a) inner scaling and (b) outer scaling. The data sets are as for figure 5.

Figure 10. Average kinetic energy in non-axisymmetric modes as a function of the azimuthal wavenumber $k_{o}$ for turbulent flow at $\mathit{Re}_{{\it\tau}}=314$ . Values are normalized by the average axisymmetric energy, $\langle E_{0}\rangle$ . Inset: data replotted in linear coordinates, emphasizing the emergence with increasing corrugation height of a peak in energy at low wavenumbers centred around $k_{o}=3$ .

7.1. The four-layer structure

A sketch representing the smooth-wall four-layer structure at a fixed Reynolds number is illustrated in figure 13. Not all of the terms in (2.17) are of leading order throughout, and this leads to a different magnitude ordering of terms in each of the four layers: layer I, $|\mathit{PG}|\cong |\mathit{VF}|\gg |\mathit{TI}|$ ; layer II, $|\mathit{VF}|\cong |\mathit{TI}|\gg |\mathit{PG}|$ ; layer III, $|\mathit{VF}|\cong |\mathit{TI}|\cong |\mathit{PG}|$ ; layer IV, $|\mathit{TI}|\cong |\mathit{PG}|\gg |\mathit{VF}|$ . For smooth-wall pipe flow, the magnitude of the ratio ( $\mathit{VF}/\mathit{TI}$ ) exceeds unity in layer I, indicating a nominal balance between the pressure gradient and the mean viscous force. Thus, layer I coincides with the viscous sublayer, which is obliterated in the presence of dynamically significant roughness. In layer II, the dominant balance is between the viscous stress gradient and the Reynolds stress gradient. Conventional notions lead to the expectation that for transitionally or fully rough flows, layer II is obliterated as well, since roughness is typically viewed as generating inertially dominated flow from the roughness crests outward. Consistent with the results of Mehdi et al. (Reference Mehdi, Klewicki and White2013), the results presented below reveal that this is distinctly not the case. Namely, layer II continues to exist. Across layer III, all three terms are of equal order, but by layer IV the $\mathit{VF}$ term loses leading-order importance and there exists a dominant balance between the $\mathit{PG}$ and $\mathit{TI}$ terms.

Figure 13. Sketch of the four layers of turbulent flow in a smooth-wall pipe at fixed Reynolds number (Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005a ). It should be noted that across layer III the $\mathit{TI}$ term changes sign and there is an exchange of dominant balance. From the outer edge of layer III to the pipe centreline the leading-order mean dynamics is governed by the inertial terms, $\mathit{TI}$ and $\mathit{PG}$ .

Roughness imposes scales of motion in the range between $O({\it\nu}/u_{{\it\tau}})$ and $O({\it\delta})$ . These scales modify the distribution of dynamically relevant length scales with distance from the wall. The net result of this is that the layer structure that describes the leading-order mean dynamics, and, in particular, the position at which the $\mathit{VF}$ term loses leading order, now becomes a function of the roughness and Reynolds number. Apart from this, the analysis and evidence presented by Mehdi et al. (Reference Mehdi, Klewicki and White2013) indicate that the overall scale separation in rough-wall flows can be segregated into contributions associated with the scale separation between the inner and roughness scale and the roughness and outer scale. By adopting a natural extension of the smooth-wall theory, they further show that a unifying element in describing the structure of both smooth- and rough-wall flows is that the onset of inertially dominated mean dynamics is described by an intermediate length scale. For the smooth-wall flow, this length scale is analytically shown to be solely a function of the overall scale separation (as reflected in the value of $\mathit{Re}_{{\it\tau}}$ ), while in the rough-wall case, this intermediate scale is a function of the relative scale separations between the inner, roughness and outer scales, and the Reynolds number (overall scale separation).

Due to the geometric complexity of the roughnesses employed in the experiments of Mehdi et al. (Reference Mehdi, Klewicki and White2013), the wall-normal location of the peak in $T_{u}$ was used as a surrogate for the intermediate length scale, i.e. a surrogate for the position from the wall to the outer edge of layer III – the position where the leading-order mean dynamics becomes wholly inertial. A remarkable feature of the present wavy-wall roughness is that it allows one to consider normalizations of (2.17) in which the driving mean pressure gradient is given by an explicit analytical expression that includes the effect of corrugation height. Thus, for example, the definition of ${\it\varepsilon}$ in (2.19) allows one to recover the exact inner-normalized mean momentum equation for smooth-wall pipe flow, i.e. as $h^{+}\rightarrow 0$ . More importantly, this representation also allows the description of mean dynamics given by Mehdi et al. (Reference Mehdi, Klewicki and White2013) to be formally tested by replacing the surrogate intermediate length with an exact analytical expression.

The ratio of the gradient of the viscous stress to the gradient of the Reynolds stress is plotted against the wall-normal distance $y^{+}$ in figure 14(a). As generically observed by Mehdi et al. (Reference Mehdi, Klewicki and White2013), the ratio ( $\mathit{VF}/\mathit{TI}$ ) attains a value of approximately $-1$ interior to where the $\mathit{TI}$ term crosses zero (i.e. where the ratio asymptotes to $\pm \infty$ ). In this region of the flow, there is also more data spread than farther from the wall. The profile variability here is a direct effect of the roughness, while the scatter within each profile is a consequence of taking the ratio of derivative quantities in a region where the statistical convergence below the roughness elements is not as good as away from the wall. For all corrugation heights and Reynolds numbers explored, the data of figure 14 unambiguously reveal the existence of layer II. The physical significance of this is that the $\mathit{VF}$ term retains leading order. The highest corrugation height (case G) exhibits a noticeable $y^{+}$ shift in the position of layer II as the Reynolds number increases from 180 to 314. Moreover, the combined effect of the Reynolds number and corrugation height does not noticeably influence the existence of the other two distinct layers (layers III and IV), albeit their $y^{+}$ position is clearly influenced by $\mathit{Re}_{{\it\tau}}$ and $h^{+}$ . As anticipated, for all values of $h^{+}$ greater than 5, layer I is completely destroyed.

Figure 14. Ratio of the mean viscous force to the mean effect of turbulent inertia: (a) versus $y^{+}$ ; (b) versus ${\hat{y}}$ , the intermediate normalization that yields the invariant form (7.6). Inset: the data replotted showing the shift in layer II with increasing Reynolds number. The data sets are the same as in figure 5.

7.2. Scaling via invariant representation of the mean dynamics

We now determine an invariant form of the relevant mean momentum equation that is applicable over most of the flow domain and that inherently accounts for the combined effects of roughness and Reynolds number. We begin by noting that the inner and outer variables $y^{+}$ and ${\it\eta}$ are appropriate to the inner and outer domains respectively. Layer III in figure 13 is of particular interest, since it has been shown (for smooth-wall flow) that its structure is replicated as a function of $y$ over a hierarchy of scaling layers that span from $O({\it\nu}/u_{{\it\tau}})$ to $O({\it\delta})$ (Fife et al. Reference Fife, Klewicki and Wei2009). The distinctive feature of each layer in this hierarchy, and layer III, is that all of the terms in (2.17) have the same order of magnitude.

According to the approach used for channel flow, rescaling is most easily accomplished for the differentials $\text{d}y^{+}$ and $\text{d}T_{u}^{+}$ (Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005a ). The simplest successful rescaling takes the form

where $\hat{U}$ and $\hat{T}_{u}$ are the rescaled $O(1)$ functions of ${\hat{y}}$ and ${\it\varepsilon}$ , and ${\rm\pi}_{1}$ and ${\rm\pi}_{2}$ are scaling parameters, functions of ${\it\varepsilon}$ , to be determined. The terms in (2.17) then transform as

Invariance is satisfied when the derivatives on the right of (7.2), namely $\text{d}^{2}\hat{U} /\text{d}{\hat{y}}^{2}$ and $\text{d}\hat{T}_{u}/\text{d}{\hat{y}}$ , are rescaled to formally become $O(1)$ quantities for all Reynolds numbers and roughness heights. By the requirement established above, the orders of magnitude of both terms on the right, namely $1/{\rm\pi}_{1}^{2}$ and ${\rm\pi}_{2}/{\rm\pi}_{1}$ , must match (in order of magnitude) the third term in (2.17), namely ${\it\varepsilon}^{2}:{\rm\pi}_{1}^{-2}={\rm\pi}_{2}/{\rm\pi}_{1}={\it\varepsilon}^{2}$ . This is only possible if ${\rm\pi}_{2}={\it\varepsilon},{\rm\pi}_{1}={\it\varepsilon}^{-1}$ .

Thus, from (7.1)

Integrating (7.3) gives two integration constants which are chosen to be $y_{m}^{+}$ and $T_{um}^{+}$ ; they are the values of $y^{+}$ and $T_{u}^{+}$ where ${\hat{y}}=0$ and $\hat{T}_{u}=0$ . The result is

where the quantities with subscript $m$ are the values of those variables at the maximum point $y_{m}^{+}$ of $T_{u}^{+}$ . The rescaled intermediate variables now become

where $y_{m}^{+}$ and $T_{um}^{+}$ are the maximum Reynolds shear-stress location and value respectively and $U_{m}^{+}$ is the value of the mean streamwise velocity at the maximum point $y_{m}^{+}$ .

Normalization of the mean momentum equation according to these variables results in

and thus provides the desired parameter-free representation in which all scaled terms are formally represented as being $O(1)$ . The rescaled functions satisfy $\hat{T}_{u}(0)=\text{d}\hat{T}_{u}/\text{d}{\hat{y}}(0)=0$ , $\text{d}^{2}\hat{U} /\text{d}{\hat{y}}^{2}(0)=-1$ . It should be noted that the present rescaling differs from that used for smooth-wall flows. In the smooth-wall case, the intermediate variable is equal to the geometric mean between the inner and outer length scales, i.e.  ${\hat{y}}=y/({\it\delta}({\it\nu}/u_{{\it\tau}}))^{1/2}$ . Due to (2.19), the new meso-variable takes the form ${\hat{y}}=y((V_{o}/V)/({\it\delta}{\it\nu}/u_{{\it\tau}}))^{1/2}$ , where the factor $V_{o}/V=1+h^{2}/(8\bar{R}^{2})$ analytically accounts for the influence of varying corrugation height. While the present analyses support and illuminate the notions associated with the dynamical relevance of the intermediate length scale in rough-wall flows, we stress that this analysis still requires knowledge of the location of the $T_{u}^{+}$ peak, which cannot be obtained without first carrying out the simulations. Hence, we still do not have an a priori description of the effect of a particular roughness.

The ratio of $\mathit{VF}/\mathit{TI}$ profiles is plotted versus ${\hat{y}}$ in figure 14(b). As is apparent, from layer III outward, the agreement between the different $h^{+}$ and $\mathit{Re}_{{\it\tau}}$ profiles is much improved. Furthermore, a good portion of the scatter in layer II is rationally attributable to the aforementioned uncertainty associated with statistical convergence, while the remaining profile variations in this region are probably associated with genuine dependences on $h^{+}$ and $\mathit{Re}_{{\it\tau}}$ . These findings are consistent with the smooth-wall analysis, indicating that, with increasing $\mathit{Re}_{{\it\tau}}$ , self-similar mean dynamics is most rapidly developed in a domain beginning near the outer edge of layer III (i.e. the log layer), while over a region interior to that point, and extending to the lower portion of layer II, Reynolds number invariance is established more slowly (Klewicki Reference Klewicki2013b ). These findings are also in accord with the rough-wall observations of Mehdi et al. (Reference Mehdi, Klewicki and White2013), indicating that interior to the outer edge of layer III the interactions involving the viscous and inertial forces are, in general, significantly influenced by the scales of motion imposed by the roughness. Conversely, beyond layer III they surmise that for roughnesses that sufficiently three-dimensionalize and/or reduce the characteristic scales of the layer II vorticity field the flow properties beyond layer III (i.e. in the domain where the leading-order dynamics is wholly inertial) become independent of the roughness details, and thus satisfy Townsend’s outer similarity hypotheses. The results of §§ 5.2 and 5.3 generally support these findings in that outer similarity is empirically observed to hold near to, or slightly beyond, the wall-normal location of the peak in $T_{u}^{+}$ .

The veracity of the scaling depicted in figure 14(b) is more directly tested by examining whether the solutions of (7.6) are indeed invariant for variations in $h^{+}$ and $\mathit{Re}_{{\it\tau}}$ . The mean velocity and Reynolds shear-stress profiles ranging from smooth to the maximum $h^{+}$ explored are plotted in figure 15(a,b). The result is encouraging, as these profiles exhibit convincing invariance, except possibly for the $h^{+}=50$ case, where the roughness height is approximately $1/6$ of the mean pipe radius. This is likely to be a case where the vorticity field in layer II does not sufficiently three-dimensionalize at small scales, and outer similarity breaks down. Here, it is also relevant to note that the invariance approximated in figure 15 should improve with increasing $\mathit{Re}_{{\it\tau}}$ .

Figure 15. Meso-normalized profiles of the mean velocity (a) and Reynolds shear stress (b) for turbulent flow in smooth and wavy-walled pipes. The data sets are the same as in figure 5.

Evidence in support of this interpretation is given by figure 16, which shows the normalized magnitude of the mean vorticity, ${\it\Omega}_{z}=-\text{d}U/\text{d}y$ . Analysis of the mean momentum equation reveals that the mean vorticity will decay from its wall value of $|{\it\Omega}_{z}^{+}|=1$ to a value that is $O({\it\varepsilon})$ by $y^{+}=O({\it\varepsilon}^{-1})$ , and then decay from there to $O({\it\varepsilon}^{2})$ by $y^{+}=O({\it\varepsilon}^{-2})$ , e.g. Klewicki (Reference Klewicki2013b ). These features predicted by the analysis can be directly tested by plotting the data according to the normalization of figure 16. Smooth-wall pipe data in the range $1800\leqslant \mathit{Re}_{{\it\tau}}\leqslant 530\,000$ (not shown in figure 16) and boundary layer data in the range $1000\leqslant \mathit{Re}_{{\it\tau}}\leqslant 17\,000$ confirm that, for all $\mathit{Re}_{{\it\tau}}$ , ${\it\varepsilon}^{-1}|{\it\Omega}_{z}^{+}|=1$ at the outer edge of layer III, i.e.  ${\it\varepsilon}y^{+}\simeq 2.6$ (Klewicki Reference Klewicki2013b ). Furthermore, by using the position of the peak in $T_{u}^{+}$ as a surrogate for the intermediate length scale, Mehdi et al. (Reference Mehdi, Klewicki and White2013) showed that, except for large organized roughness (i.e. large two-dimensional transverse bars), this same condition is also satisfied for a range of roughnesses covering over a decade in $\mathit{Re}_{{\it\tau}}$ and over two decades in $k_{s}^{+}$ . Physically, this observed decay rate stems from the reduction in scale and three-dimensionalization of the near-wall vorticity field, caused by vorticity stretching, and that results in an exchange of mean enstrophy to fluctuating enstrophy (Klewicki Reference Klewicki2013a ). In accord with this interpretation, the $h^{+}=50$ profile in figure 16 does not quite adhere to the predicted decay rate, presumably due to the roughness-induced motions in layer II being too large and organized. Again, the distinct (and remarkable) advantage of the present analysis is that its explicit representation of the pressure drop as a function of the corrugation height allows the present scaling theory to be analytically tested.

Figure 16. Meso-normalized mean velocity gradient ( $-{\it\Omega}_{z}$ ) profiles for smooth and wavy-walled pipes. The horizontal line denotes ${\it\varepsilon}^{-1}|{\it\Omega}_{z}^{+}|=1$ and the vertical line denotes ${\it\varepsilon}y^{+}=2.6$ . The expression for ${\it\varepsilon}$ is given by (2.19).