5.1 Effective wall-normal distance

We first discuss the determination of an effective distance from a rough surface before we discuss statistical results because defining the wall-normal distance from a rough surface is not straightforward. Conventionally, in order to define the distance from a rough surface, a virtual origin for the rough surface has been introduced not only for discussing flow physics (Perry & Li Reference Perry and Li1990) but also for modelling rough-wall turbulence (Durbin et al. Reference Durbin, Medic, Seo, Eaton and Song2001; Aupoix & Spalart Reference Aupoix and Spalart2003). When the roughness height is negligibly lower than the boundary layer thickness, as in the case of experiments in a boundary layer (Schultz & Flack Reference Schultz and Flack2009; Flack et al. Reference Flack, Schultz, Barros and Kim2016), determination of the effective wall-normal distance is not an essential issue. However, it is quite important for the current DNS study because the mean roughness height occupies a relatively large fraction of the boundary layer thickness: $h_{m}/\unicode[STIX]{x1D6FF}=0.022{-}0.078$ .

One of the most reasonable and comprehensive approaches is shifting the origin to fit a mean velocity profile to a log-law profile (Perry & Li Reference Perry and Li1990). This idea has also been used in flows over porous walls or vegetated and urban canopies, and the shifted thickness is usually referred to as the displacement thickness (Raupach Reference Raupach1994a ). The other approach, suggested by Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015), is to collapse the total shear stress profile over the rough surface regardless of the presence of the wall roughness, and they reported that the virtual origin coincided with the position of the mean roughness height.

The present study attempts to analytically derive the effective distance starting from the double-averaged Navier–stokes equation for incompressible flows (Whitaker Reference Whitaker1986):

(5.5) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\langle \overline{u_{i}}\rangle ^{f}}{\unicode[STIX]{x2202}t}+\langle \overline{u}_{j}\rangle ^{f}\frac{\unicode[STIX]{x2202}\langle \overline{u}_{i}\rangle ^{f}}{\unicode[STIX]{x2202}x_{j}} & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\langle \overline{p}\rangle ^{f}}{\unicode[STIX]{x2202}x_{i}}+\frac{1}{\unicode[STIX]{x1D711}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}}\left(\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}\langle \overline{u}_{i}\rangle ^{f}}{\unicode[STIX]{x2202}x_{j}}\right)-\frac{1}{\unicode[STIX]{x1D711}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}}\left(\underbrace{\unicode[STIX]{x1D711}\langle \tilde{\overline{u}}_{i}\tilde{\overline{u}}_{j}\rangle ^{f}}_{{\mathcal{T}}_{ij}}+\underbrace{\unicode[STIX]{x1D711}\langle \overline{u_{i}^{\prime }u_{j}^{\prime }}\rangle ^{f}}_{R_{ij}}\right)\nonumber\\ \displaystyle & & \displaystyle -\,\underbrace{\frac{\unicode[STIX]{x1D708}}{\unicode[STIX]{x1D711}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}x_{j}}\frac{\unicode[STIX]{x2202}\langle \overline{u}_{i}\rangle ^{f}}{\unicode[STIX]{x2202}x_{j}}}_{\overline{g}_{i}^{\unicode[STIX]{x1D711}}}-\underbrace{\left(\frac{1}{\unicode[STIX]{x1D70C}S_{f}}\int _{L}\overline{\tilde{p}}n_{i}\,\text{d}\ell -\frac{\unicode[STIX]{x1D708}}{S_{f}}\int _{L}n_{k}\frac{\unicode[STIX]{x2202}\tilde{\overline{u}}_{i}}{\unicode[STIX]{x2202}x_{k}}\,\text{d}\ell \right)}_{\overline{f}_{i}}.\end{eqnarray}$$

Here $L$ represents the obstacle perimeter within an averaging $x$ – $z$ plane, $\ell$ represents the circumference length of solid obstacles, and $n_{k}$ is its unit normal vector pointing outwards from the fluid to the solid phase. The second moments ${\mathcal{T}}_{ij}$ and $R_{ij}$ are the plane-dispersive covariance and the plane-averaged Reynolds stress, respectively. The dispersive covariance arises because of the spatial inhomogeneity of the mean velocity field. The inhomogeneous correction term $\overline{g}_{i}^{\unicode[STIX]{x1D711}}$ arises because of the spatial inhomogeneity of the plane porosity $\unicode[STIX]{x1D711}$ . In the present study, the $x$ – $z$ plane is considered as the averaging plane, and thus $\unicode[STIX]{x1D711}$ varies with respect to $y$ below the roughness peaks. Hence, $\overline{g}_{i}^{\unicode[STIX]{x1D711}}$ has a non-zero contribution to the momentum transfer below the roughness peak ( $\unicode[STIX]{x1D711}<1$ ). The plane-averaged drag force $f_{i}$ consists of two integration terms. The first term is expressed as the line integration of the pressure dispersion representing the form drag, while the second term consists of the dispersive mean velocity gradient representing the viscous drag. By integrating equation (5.5) over the wall-normal direction from $0$ to $y$ , the shear stress balance for the present flow system can be derived after some manipulation (Kuwata & Kawaguchi Reference Kuwata and Kawaguchi2017):

(5.6) $$\begin{eqnarray}\displaystyle -\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\langle \overline{p}\rangle ^{f}}{\unicode[STIX]{x2202}x}\unicode[STIX]{x1D6FF}_{e}\left(1-\frac{1}{\unicode[STIX]{x1D6FF}_{e}}\int _{0}^{y}\unicode[STIX]{x1D711}\,\text{d}y\right) & = & \displaystyle \unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}\langle \overline{u}\rangle ^{f}}{\unicode[STIX]{x2202}y}-R_{12}-{\mathcal{T}}_{12}-\left(\int _{0}^{y}\unicode[STIX]{x1D711}\overline{g}_{x}^{\unicode[STIX]{x1D711}}\,\text{d}y-\int _{0}^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D711}\overline{g}_{x}^{\unicode[STIX]{x1D711}}\,\text{d}y\right)\nonumber\\ \displaystyle & & \displaystyle -\,\left(\int _{0}^{y}\unicode[STIX]{x1D711}\overline{f}_{x}\,\text{d}y-\int _{0}^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D711}\overline{f}_{x}\,\text{d}y\right).\end{eqnarray}$$

With the help of a definition of the friction velocity (3.3), the total stress on the left-hand side of (5.6) can be further written as

(5.7) $$\begin{eqnarray}-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\langle \overline{p}\rangle ^{f}}{\unicode[STIX]{x2202}x}\unicode[STIX]{x1D6FF}_{e}\left(1-y_{e}/\unicode[STIX]{x1D6FF}_{e}\right)=u_{\unicode[STIX]{x1D70F}}^{2}\left(1-y_{e}/\unicode[STIX]{x1D6FF}_{e}\right),\end{eqnarray}$$

where the effective wall-normal distance from the rough surface is introduced as

(5.8) $$\begin{eqnarray}y_{e}=\int _{0}^{y}\unicode[STIX]{x1D711}\,\text{d}y.\end{eqnarray}$$

The expression (5.7) for the total shear stress suggests that the total shear stress is expressed as a function of the normalized effective distance $y_{e}/\unicode[STIX]{x1D6FF}_{e}$ . To get more insight from (5.7), we transform the effective channel height $\unicode[STIX]{x1D6FF}_{e}$ as

(5.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{e}=\int _{0}^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D711}\,\text{d}y=\unicode[STIX]{x1D6FF}-\int _{0}^{y_{p}}\left(1-\unicode[STIX]{x1D711}\right)\,\text{d}y=\unicode[STIX]{x1D6FF}-\frac{1}{L_{x}L_{z}}\underbrace{\int _{0}^{y_{p}}L_{x}L_{z}\left(1-\unicode[STIX]{x1D711}\right)\,\text{d}y}_{Vs},\end{eqnarray}$$

where $y_{p}$ is the wall-normal location of the roughness peak. It is interesting to note that the integral in the second term on the right-hand side can be interpreted as the total volume of the solid phase (roughness elements) $Vs$ . The total volume $Vs$ can also be written as an integral of the roughness height $h$ with infinitesimal area $\text{d}x\,\text{d}z$ as

(5.10) $$\begin{eqnarray}Vs=\iint h\,\text{d}x\,\text{d}z.\end{eqnarray}$$

By inserting the definition of the mean roughness height $h_{m}$ of (4.1) into (5.10), a simple relation can be derived from (5.9) and (5.10):

(5.11) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{e}=\unicode[STIX]{x1D6FF}-h_{m}.\end{eqnarray}$$

Additionally, since the plane porosity is unity outside the rough wall $y>y_{p}$ , the effective distance outside the rough wall becomes

(5.12) $$\begin{eqnarray}\displaystyle y_{e}=\int _{0}^{y}\unicode[STIX]{x1D711}\,\text{d}y & = & \displaystyle y-\int _{0}^{y_{p}}(1-\unicode[STIX]{x1D711})\,\text{d}y=y-h_{m}.\end{eqnarray}$$

It is readily understood from (5.9) and (5.12) that the normalized effective distance $y_{e}/\unicode[STIX]{x1D6FF}_{e}$ outside the rough wall coincides with the expression proposed by Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015), who considered the virtual origin as $h_{m}$ to define the wall-normal distance $(y-h_{m})/(\unicode[STIX]{x1D6FF}-h_{m})$ . However, it should be noted that, unlike the expression of Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015), as the origin exactly indicates the points where there is no fluid phase, $\unicode[STIX]{x1D711}=0$ , the streamwise mean velocity strictly goes to zero at the origin $y_{e}/\unicode[STIX]{x1D6FF}_{e}=0$ in the present expression.

Figure 3.

Sum of the plane-averaged Reynolds shear stress and the plane-dispersive shear stress profiles with different scalings: profiles with (a)  $y/\unicode[STIX]{x1D6FF}$ , (b)  $y_{e}/\unicode[STIX]{x1D6FF}_{e}$ and (c)  $y^{\ast }=(y-h_{m})/(\unicode[STIX]{x1D6FF}-h_{m})$ . The dashed thin line indicates the smooth-wall result by Iwamoto, Suzuki & Kasagi (Reference Iwamoto, Suzuki and Kasagi2002).

Figure 4.

Streamwise mean velocity profiles with defect form: profiles with (a)  $y/\unicode[STIX]{x1D6FF}$ , (b)  $y_{e}/\unicode[STIX]{x1D6FF}_{e}$ and (c)  $y^{\ast }=(y-h_{m})/(\unicode[STIX]{x1D6FF}-h_{m})$ . The dashed thin line indicates the smooth-wall result by Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002).

To demonstrate the advantage of employing the effective distance $y_{e}$ , profiles of the sum of the plane-averaged Reynolds shear stress $-R_{12}^{+}$ and the plane-dispersive shear stress $-{\mathcal{T}}_{12}^{+}$ for all the samples are compared between the scalings of $y/\unicode[STIX]{x1D6FF}$ , $y_{e}/\unicode[STIX]{x1D6FF}_{e}$ and $y^{\ast }=(y-h_{m})/(\unicode[STIX]{x1D6FF}-h_{m})$ in figure 3. For comparison, DNS results in smooth-wall turbulent channel flow at friction Reynolds number of 650 (Iwamoto et al. Reference Iwamoto, Suzuki and Kasagi2002) are also plotted. Also, figure 4 depicts profiles of $U^{+}$ by defect form with reference to $U_{e}^{+}$ . Here, $U^{+}=\langle \overline{u}\rangle ^{f}$ and $U_{e}^{+}$ denote $U^{+}$ at the top slip boundary at $y=\unicode[STIX]{x1D6FF}$ . It is clear that the profiles of $-(R_{12}^{+}+{\mathcal{T}}_{12}^{+})$ as a function of $y_{e}/\unicode[STIX]{x1D6FF}_{e}>0.15$ and $y^{\ast }>0.15$ in figures 3(b) and 3(c) collapse very well, whereas the profiles as a function of $y/\unicode[STIX]{x1D6FF}$ in figure 3(a) do not. Jiménez (Reference Jiménez2004) pointed out that when the roughness height $h$ occupied a relatively large fraction of the boundary layer thickness, $\unicode[STIX]{x1D6FF}/h<40$ , turbulence in the whole region was modified by the roughness and then the similarity of the profiles outside the roughness sublayer was not preserved. However, almost perfect similarity can be found in the mean velocity defect with the scaling of $y_{e}/\unicode[STIX]{x1D6FF}_{e}$ as shown in figure 4(b) in the region of $y_{e}/\unicode[STIX]{x1D6FF}_{e}>0.15$ , which substantiates that the outer layer similarity may be maintained in the present rough surfaces with relatively high roughness height, $\unicode[STIX]{x1D6FF}/h_{m}=13{-}45$ . The observation of the similarity for the rough surfaces whose roughness height is a significant fraction of the boundary layer thickness is in accordance with the observations by another experimental study (Flack, Schultz & Connelly Reference Flack, Schultz and Connelly2007) and DNS studies (Chan et al. Reference Chan, MacDonald, Chung, Hutchins and Ooi2015; Forooghi et al. Reference Forooghi, Stroh, Schlatter and Frohnapfel2018a ). Moreover, the onset of the region $y_{e}/\unicode[STIX]{x1D6FF}_{e}=0.15$ where the similarity of the streamwise mean velocity can be clearly seen corresponds to $y_{e}/h_{m}=2{-}7$ . This supports the previous finding that similarity outside a roughness sublayer extends $5h$ from the wall (Raupach, Antonia & Rajagopalan Reference Raupach, Antonia and Rajagopalan1991). The fact that good collapse can be observed outside the roughness sublayer implies that $h_{m}$ can also be regarded as the displacement thickness, which is usually computed to fit the mean velocity profile to the log-law profile. Although good collapse outside the roughness sublayer can be confirmed with both scalings of $y_{e}$ and $y^{\ast }$ , a difference is obvious near the origin of the rough walls $y_{e}\simeq 0$ and $y^{\ast }\simeq 0$ . As mentioned above, as the origin of the effective distance is located at a position where there is no fluid phase, $\unicode[STIX]{x1D711}=0$ , the fluid velocity always goes to zero at $y_{e}=0$ . In contrast to that of $y_{e}$ , the scaling of $y^{\ast }$ assumes the origin to be $y=h_{m}$ , resulting in non-zero velocity at the origin. It is noted that the perfect collapse can be confirmed among the rough-wall results in figure 4(b,c) while the profile for a smooth wall does not perfectly agree with the rough-wall results. This discrepancy is considered to be due to the difference in the flow configurations. The present DNS is performed in the open-channel flow system while the smooth-wall result is obtained in the channel flow simulation, which may cause the difference in the mean velocity profile in the core region.

To discuss the profiles in the immediate vicinity of the origin, figure 5 compares profiles of $-(R_{12}^{+}+{\mathcal{T}}_{12}^{+})$ between the different scalings of $y^{+}$ and $y_{e}^{+}$ together with the smooth-wall result. The limiting behaviour of the total shear stress towards the origin with the scaling of $y^{+}$ is not consistent, whereas a consistent trend can be found in the profiles with $y_{e}^{+}$ despite the fact that the characteristics of the rough-surface geometry are very different. This suggests that the introduction of the effective distance $y_{e}$ enables us to compare statistics from different rough surfaces, not only outside the roughness sublayer but also in the immediate vicinity of the origin. This observation also suggests that the use of the effective distance is promising for modelling turbulence near rough surfaces. Noticeably, although the total shear stress for the rough wall is rapidly damped compared with the smooth-wall result in the region of $y_{e}^{+}\simeq 10$ , the limiting behaviour of the total shear stress towards the bottom wall ( $1<y_{e}^{+}<5$ ) is found to be consistent, albeit there exists the wall roughness. This observation suggests the overwhelming advantage of employing the effective distance for modelling turbulence in the vicinity of a rough wall.

Figure 5.

Sum of the plane-averaged Reynolds shear stress and the plane-dispersive shear stress profiles in the vicinity of the origin: profiles with (a)  $y^{+}$ and (b)  $y_{e}^{+}$ .

Figure 6.

Plane-averaged streamwise mean velocity profiles with semi-logarithmic wall-scaling $y_{e}^{+}$ : (a) cases R1–R3 and (b) cases S1–S5. The dashed thin line indicates the smooth-wall result by Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002).

5.2 Mean velocity

Figure 6 presents the $x$ – $z$ plane-averaged streamwise mean velocity $U^{+}=\langle \overline{u}\rangle ^{f+}$ with the logarithmic scaling of the effective wall-normal distance $y_{e}^{+}$ together with the smooth-wall results. The profiles for the rough surfaces are significantly lower than the smooth-wall result, while the profiles seem to maintain the logarithmic profile away from the rough walls of $y_{e}^{+}>100$ . The mean velocity profile near the channel centre at $y=\unicode[STIX]{x1D6FF}$ for the smooth-wall case, however, exhibits a different trend from that observed in the rough-wall results, which is primarily due to the use of the slip boundary condition to the channel centre instead of simulating the channel flow. The downward shift of the streamwise mean velocity increases with $h_{rms}^{+}$ or $Sk$ . This indicates that rough-wall skin friction increases with increasing $h_{rms}^{+}$ or $Sk$ . The values of the downward shifts of $U^{+}$ referred to as roughness function $\unicode[STIX]{x0394}U^{+}$ are calculated by comparing $U^{+}$ at $y_{e}^{+}\simeq 100$ ( $y_{e}/\unicode[STIX]{x1D6FF}_{e}\simeq 0.167$ ) with the standard log law. Note that since the streamwise mean velocity defects in the region of $y_{e}/\unicode[STIX]{x1D6FF}_{e}>0.15$ are perfectly overlapped, as discussed in figure 4, the arbitrary choice of $y_{e}^{+}\sim 100$ does not affect the computed roughness function. (The procedure for obtaining $\unicode[STIX]{x0394}U^{+}$ is the same as that used by Forooghi et al. (Reference Forooghi, Stroh, Magagnato, Jakirlić and Frohnapfel2017).) The equivalent roughness $k_{s}^{+}$ can be uniquely determined based on the relation between $\unicode[STIX]{x0394}U^{+}$ and $k_{s}^{+}$ in the fully rough regime (Flack & Schultz Reference Flack and Schultz2010):

(5.13) $$\begin{eqnarray}B-\unicode[STIX]{x0394}U^{+}+\frac{1}{\unicode[STIX]{x1D705}}\text{ln}(k_{s}^{+})=8.5.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D705}$ and $B$ respectively stand for von Kármán constant and log-law intercept for a smooth wall, and the values of $\unicode[STIX]{x1D705}=0.4$ and $B=5.0$ are chosen in the present study. The computed roughness functions $\unicode[STIX]{x0394}U^{+}$ and $k_{s}^{+}$ are listed in table 5. The roughness function ranges from $5.9$ to $10.4$ , and the corresponding inner-scaled equivalent roughness ranges from $47$ to $248$ . The lowest value of $k_{s}^{+}=47$ in case S1 roughly corresponds to the onset of the fully rough regime according to an experimental report on turbulent boundary layers over a packed sphere bed (Ligrani & Moffat Reference Ligrani and Moffat1986) in which the transitionally rough regime spanned $15<k_{s}^{+}<50$ . Hence, the present simulations are all expected to be in the fully rough regime. It is confirmed from table 5 that $k_{s}^{+}$ increases with increasing $h_{rms}^{+}$ or $Sk$ . Although the increases in $h_{rms}^{+}$ and $Sk$ both give rise in $k_{s}^{+}$ , the effects of the $h_{rms}^{+}$ and $Sk$ on $U^{+}$ profiles near the rough surface $y_{e}^{+}<50$ are very different. In figure 6(a), as $h_{rms}^{+}$ increases, the blockage effect on the $U^{+}$ profiles near the rough surface $y_{e}^{+}<50$ is more significant. In contrast, figure 6(b) confirms that the blockage effect is relaxed in the positively skewed rough wall of case S5, resulting in an increase in $U^{+}$ in the vicinity of the rough wall $y_{e}^{+}<20$ . These observations suggest that, although the increases in $h_{rms}^{+}$ and $Sk$ have the same effects on $\unicode[STIX]{x0394}U^{+}$ , the driving mechanism differs.

Table 5.

Roughness function $\unicode[STIX]{x0394}U^{+}$ and equivalent roughness $k_{s}^{+}$ .

5.3 Reynolds stress and dispersive covariance

Profiles of the $x$ – $z$ plane-averaged Reynolds normal stresses $R_{ij}^{+}$ , which appear in the double-averaged momentum equation (5.5), are shown in figure 7. The location of the roughness peak is also indicated in the figure. In figure 7(a,c,e), as $h_{rms}^{+}$ increases, the Reynolds stresses are damped below the roughness peak. Reduction in the streamwise and spanwise components is pronounced in figures 7(a) and 7(e) whereas $R_{22}^{+}$ in figure 7(c) is insensitive to the change in the rough-surface geometry, indicating that turbulence tends to approach an isotropic state with increasing $h_{rms}^{+}$ . These trends are consistent with other experimental and DNS observations (Ligrani & Moffat Reference Ligrani and Moffat1986; Krogstad et al. Reference Krogstad, Andersson, Bakken and Ashrafian2005; Flack et al. Reference Flack, Schultz and Connelly2007; Kuwata & Suga Reference Kuwata and Suga2016b ,Reference Kuwata and Suga c ; Forooghi et al. Reference Forooghi, Stroh, Schlatter and Frohnapfel2018a ). In figure 7(a,c,e), $R_{11}^{+}$ and $R_{33}^{+}$ take the maximum peak just above the roughness peak, and then they are damped below the roughness peak. The significance of the damping effect depends on $h_{rms}^{+}$ , but the limiting behaviour of $R_{ij}^{+}$ towards the bottom wall is found to be similar. In figure 7(b,d,f), as $Sk$ increases, damping of $R_{11}^{+}$ and $R_{33}^{+}$ just below the roughness peak is more significant whereas the modification of $R_{22}^{+}$ is marginal, which is the same trend as in cases $\text{R}1{-}\text{R}3$ . However, in contrast to cases $\text{R}1{-}\text{R}3$ , the trends of $R_{11}^{+}$ and $R_{33}^{+}$ in the vicinity of the bottom wall $y_{e}/\unicode[STIX]{x1D6FF}_{e}<0.025$ are different. As $Sk$ increases, $R_{11}^{+}$ and $R_{33}^{+}$ gain strength in the vicinity of the bottom wall. This is because the number of hemispheres decreases with increasing $Sk$ , as seen in figure 1, and the blockage effect is consequently relaxed. Another important observation in the figure is that, although the behaviour of the Reynolds normal stresses near the rough surfaces of $y_{e}/\unicode[STIX]{x1D6FF}_{e}<0.1$ strongly depends on the roughness geometry, almost perfect collapse with the smooth-wall result can be found in $R_{22}^{+}$ and $R_{33}^{+}$ at $y_{e}/\unicode[STIX]{x1D6FF}_{e}=0.15$ ; however, a slight discrepancy can still be found in $R_{11}^{+}$ at the same location.

Figure 7.

Plane-averaged Reynolds normal stress profiles: (a) streamwise component in cases R1–R3, (b) streamwise component in cases S1–S5, (c) wall-normal component in cases R1–R3, (d) wall-normal component in cases S1–S5, (e) spanwise component in cases R1–R3, and (f) spanwise component in cases S1–S5. The location of the roughness peak is also indicated as the thin lines. The dashed thin line indicates the smooth-wall result by Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002).

Unlike in flows over a smooth wall, mean velocity becomes spatially inhomogeneous near rough surfaces owing to the presence of roughness elements. Hence, mean velocity dispersion defined as $\tilde{\overline{u}}(x,y,z)=\overline{u}(x,y,z)-\langle \overline{u}\rangle ^{f}(y)$ exists, and the plane-dispersive covariance, ${\mathcal{T}}_{ij}=\langle \tilde{\overline{u}}_{i}\tilde{\overline{u}}_{j}\rangle$ , accordingly arises in the double-averaged momentum equation as well as $R_{ij}$ in (5.5). To compare the intensities of the mean velocity dispersion and the velocity fluctuation, the plane-dispersive kinetic energy, ${\mathcal{K}}^{+}={\mathcal{T}}_{kk}^{+}/2$ , and the turbulent kinetic energy, $k^{+}=R_{kk}^{+}/2$ , are compared in figure 8. For comparison, the turbulent kinetic energy profile for the smooth-wall case is also plotted. The figure confirms that ${\mathcal{K}}^{+}$ in figure 8(c,d) has a maximum below the roughness peak and it steeply decreases as it separates from the wall while $k^{+}$ in figure 8(a,b) has a maximum above the roughness peak. The maximum peak value of ${\mathcal{K}}^{+}$ is found to be far larger than that of $k^{+}$ in all cases, indicating that the magnitude of the mean velocity dispersion overwhelms the velocity fluctuation below the roughness peak. Although ${\mathcal{K}}^{+}$ steeply decreases outside the rough surfaces, it still has a meaningful value even away from the rough wall at $y_{e}/\unicode[STIX]{x1D6FF}_{e}=0.15$ . This implies that a spatial inhomogeneity of the mean velocity exists far away from the rough wall. The maximum peak of ${\mathcal{K}}^{+}$ slightly increases as $h_{rms}^{+}$ increases in figure 8(c), whereas the consistent trend between the maximum peak of ${\mathcal{K}}^{+}$ and $Sk$ cannot be confirmed in figure 8(d). The pronounced peak of ${\mathcal{K}}^{+}$ can be seen in the positively skewed rough-surface case S5, which may be due to an increase in the streamwise mean velocity in the vicinity of the rough surface in figure 6.

Figure 8.

Plane-averaged turbulent kinetic energy and plane-dispersive kinetic energy profiles: (a) turbulent kinetic energy in cases R1–R3, (b) turbulent kinetic energy in cases S1–S5, (c) plane-dispersive kinetic energy in cases R1–R3, and (d) plane-dispersive kinetic energy in cases S1–S5. The location of the roughness peak is also indicated as the thin lines. The dashed thin line indicates the smooth-wall result by Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002).

To investigate the influence of the mean velocity dispersion on the momentum transfer, figure 9 shows the contributions by the plane-averaged Reynolds shear stress $-R_{12}^{+}$ and the plane-dispersive covariance $-{\mathcal{T}}_{12}^{+}$ in the double-averaged momentum equation (5.5). The plane-dispersive covariance $-{\mathcal{T}}_{12}^{+}$ in figure 9(c,d) takes the maximum peak just above the roughness peak, which is similar to the trend for the plane-dispersive kinetic energy ${\mathcal{K}}^{+}$ in figure 8. However, the maximum peak value of $-{\mathcal{T}}_{12}^{+}$ is found to be far smaller than that of $-R_{12}^{+}$ despite the fact that the peak value of ${\mathcal{K}}^{+}$ overwhelms $k^{+}$ in figure 8. The same observation was reported in DNS studies (Busse et al. Reference Busse, Lützner and Sandham2015; Kuwata & Kawaguchi Reference Kuwata and Kawaguchi2018). Busse et al. (Reference Busse, Lützner and Sandham2015) reported that the wall-normal velocity dispersion was approximately one order smaller than the streamwise velocity dispersion, and the plane-dispersive shear stress accordingly showed a smaller value compared with the Reynolds shear stress. In figure 9(a,c), the plane-dispersive shear stress in cases R2 and R3 has an appreciable value away from the rough surface of $y_{e}/\unicode[STIX]{x1D6FF}_{e}=0.15$ , whereas $-R_{12}^{+}$ in cases R2 and R3 shows a lower value in that region compared with its value in the smooth-wall case. However, it should be recalled that the sum of the two shear stresses, $-(R_{12}^{+}+{\mathcal{T}}_{12}^{+})$ , collapses irrespective of the rough geometry as seen in figure 3. The collapse of $-(R_{12}^{+}+{\mathcal{T}}_{12}^{+})$ indicates the collapse of the viscous shear stress, which results in the collapse of the streamwise mean velocity defect. Therefore, it can be said that the spatial inhomogeneity of the mean velocity distribution does not alter the plane-averaged streamwise mean velocity profile but alters the plane-averaged Reynolds shear stress profile.

Figure 9.

Plane-averaged shear stress profiles: (a) plane-averaged Reynolds shear stress in cases R1–R3, (b) plane-averaged Reynolds shear stress in cases S1–S5, (c) plane-dispersive shear stress in cases R1–R3, and (d) plane-dispersive shear stress in cases S1–S5. The location of the roughness peak is also indicated as the thin lines. The dashed thin line indicates the smooth-wall result by Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002).

Figure 10.

Turbulence production terms in case R3: (a) production terms for the plane-averaged Reynolds normal stresses and (b) production terms for the plane-averaged Reynolds shear stress.

For a better understanding of the influence of the mean velocity dispersion on the Reynolds stress, production terms for $R_{ij}^{+}$ are discussed. In the double-averaged system, two production terms, namely the mean shear production $P_{ij}^{s}$ and the mean dispersion-related production $P_{ij}^{d}$ ,

(5.14) $$\begin{eqnarray}\displaystyle P_{ij}^{s} & = & \displaystyle -R_{ik}\frac{\unicode[STIX]{x2202}\langle \overline{u_{j}}\rangle ^{f}}{\unicode[STIX]{x2202}x_{k}}-R_{jk}\frac{\unicode[STIX]{x2202}\langle \overline{u_{i}}\rangle ^{f}}{\unicode[STIX]{x2202}x_{k}},\end{eqnarray}$$

(5.15) $$\begin{eqnarray}\displaystyle P_{ij}^{d} & = & \displaystyle -\unicode[STIX]{x1D711}\left\langle \left(\overline{\tilde{u} _{j}^{\prime }\tilde{u} _{k}^{\prime }}+\overline{\langle u_{k}^{\prime }\rangle ^{f}\tilde{u} _{j}^{\prime }}+\overline{\tilde{u} _{k}^{\prime }\langle u_{j}^{\prime }\rangle ^{f}}\right)\frac{\unicode[STIX]{x2202}\tilde{\overline{u}}_{i}}{\unicode[STIX]{x2202}x_{k}}\right\rangle ^{f}\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D711}\left\langle \left(\overline{\tilde{u} _{i}^{\prime }\tilde{u} _{k}^{\prime }}+\overline{\langle u_{k}^{\prime }\rangle ^{f}\tilde{u} _{i}^{\prime }}+\overline{\tilde{u} _{k}^{\prime }\langle u_{i}^{\prime }\rangle ^{f}}\right)\frac{\unicode[STIX]{x2202}\tilde{\overline{u}}_{j}}{\unicode[STIX]{x2202}x_{k}}\right\rangle ^{f},\end{eqnarray}$$

can be defined (Kuwata & Suga Reference Kuwata and Suga2016c ). The production term of $P_{ij}^{s}$ is the turbulence generation due to the mean shear of the plane-averaged mean velocity, and the components of $P_{11}^{s}$ and $P_{22}^{s}$ remain in the present flow system because $\unicode[STIX]{x2202}\langle \overline{u}\rangle ^{f}/\unicode[STIX]{x2202}y$ exists. On the other hand, the production of $P_{ij}^{d}$ termed as the mean dispersion-related production is the production term due to the dispersive mean velocity gradients, and all components $P_{11}^{d},P_{22}^{d},P_{33}^{d}$ and $P_{12}^{d}$ have the possibility to appear near the rough surfaces. It should be mentioned that $P_{ij}^{d}$ also appears in the transport equation of ${\mathcal{T}}_{ij}$ with an opposite sign (Kuwata & Suga Reference Kuwata and Suga2016c ), and thus $P_{ij}^{d}$ is interpreted as an energy transfer term between ${\mathcal{T}}_{ij}$ and $R_{ij}$ . The contributions of those productions ( $P_{ij}^{s}$ and $P_{ij}^{d}$ ) normalized by $u_{\unicode[STIX]{x1D70F}}$ and $\unicode[STIX]{x1D708}$ in case R3, in which the mean velocity dispersion intensity is the most significant as shown in figure 8, are compared in figure 10. The productions due to the mean shear $P_{11}^{s}$ and $P_{12}^{s}$ are found to be the dominant contributors to the generation of $R_{11}$ and $R_{22}$ . In terms of the production terms for $R_{11}$ and $R_{22}$ , the sign of the mean dispersion-related production $P_{ij}^{d}$ is the opposite sign of $P_{ij}^{s}$ near the roughness peak $0.075<y_{e}/\unicode[STIX]{x1D6FF}_{e}<0.13$ . This indicates that $P_{ij}^{d}$ acts as a sink term for $R_{ij}$ near the roughness peak. The magnitude of the peak values of $P_{11}^{d+}$ and $P_{12}^{d+}$ near the roughness peak reach 44 % of $P_{11}^{s+}$ and 50 % of $P_{12}^{s+}$ , respectively. Although the terms $P_{11}^{d}$ and $P_{12}^{d}$ substantially reduce $R_{11}$ and $R_{22}$ near the roughness peak region of $0.075<y_{e}/\unicode[STIX]{x1D6FF}_{e}<0.13$ , it is observed that they work as a source in the other region. In contrast to $P_{11}^{d}$ and $P_{12}^{d}$ , the other components $P_{22}^{d}$ and $P_{33}^{d}$ do not work as sink terms, but their contributions are found to be small. The significance of $P_{11}^{d}$ and $P_{12}^{d}$ partly explains the noticeable reduction in the streamwise Reynolds stress and the Reynolds shear stress near the roughness peak as observed in figure 7. Furthermore, this also explains the discrepancies in $R_{11}^{+}$ and $R_{12}^{+}$ even away from the rough wall of $y_{e}/\unicode[STIX]{x1D6FF}_{e}=0.15$ in figures 7 and 9 because the significance of $P_{11}^{d}$ and $P_{12}^{d}$ , which are generated due to the dispersive mean velocity gradients, strongly depends on the roughness topology. Additionally, considering that $P_{12}^{d}$ appears in the transport equation of ${\mathcal{T}}_{12}$ with an opposite sign, it can be said that a reduction in $R_{12}$ leads to the generation of ${\mathcal{T}}_{12}$ . This may also explain the observation that the sum of $-(R_{12}^{+}+{\mathcal{T}}_{12}^{+})$ collapses in all cases irrespective of the significance of ${\mathcal{T}}_{12}^{+}$ as shown in figures 3 and 9. However, to give a complete answer, a full budget term analysis for ${\mathcal{T}}_{ij}$ and $R_{ij}$ is essential.

Figure 11.

Plane-averaged drag force contribution (DFC) and the inhomogeneous correction contribution (ICC): (a) DFC in cases R1–R3, (b) DFC in cases S1–S5, (c) ICC in cases R1–R3, and (d) ICC in cases S1–S5.

5.4 Drag force and inhomogeneous correction terms

The plane-averaged Reynolds shear stress and the plane-dispersive covariance are damped on approaching the bottom wall at $y_{e}/\unicode[STIX]{x1D6FF}_{e}=0$ , while the inhomogeneous correction contribution (ICC) and the drag force contribution (DFC) in (5.5), defined as

(5.16) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle ICC=-\left(\int _{0}^{y}\unicode[STIX]{x1D711}\overline{g}_{x}^{\unicode[STIX]{x1D711}}\,\text{d}y-\int _{0}^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D711}\overline{g}_{x}^{\unicode[STIX]{x1D711}}\,\text{d}y\right),\\ \displaystyle DFC=-\left(\int _{0}^{y}\unicode[STIX]{x1D711}\overline{f}_{x}\,\text{d}y-\displaystyle \int _{0}^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D711}\overline{f}_{x}\,\text{d}y\right),\end{array}\right\}\end{eqnarray}$$

are generated below the roughness peak instead. To investigate the contribution of ICC and DFC defined in (5.16), the terms (DFC and ICC) are normalized by $u_{\unicode[STIX]{x1D70F}}$  (figure 11). The drag force term DFC in figure 11(a) and (b) increases towards the bottom wall of $y_{e}/\unicode[STIX]{x1D6FF}_{e}=0$ and it reaches almost unity because the other stress terms become negligibly small at the bottom wall. However, in case S5, as a slight viscous stress exists at the bottom wall (not shown here), DFC shows a lower value at $y_{e}/\unicode[STIX]{x1D6FF}_{e}=0$ compared with those in the other cases. In contrast to DFC, the contribution of ICC in figure 11(c) and (d) is negligibly smaller compared with DFC. As $h_{rms}^{+}$ or $Sk$ increases, the location of the roughness peak scaled with the effective distance moves away from the bottom wall, and then the influence of DFC and ICC extends away from the bottom wall. The term ICC at the bottom wall is found to consistently decrease with $Sk$ or $h_{rms}^{+}$ .

5.5 Skin friction coefficient

This subsection discusses the contribution of the stress terms in (5.6) to the skin friction coefficient $C_{f}$ defined as $C_{f}=\unicode[STIX]{x1D70F}_{w}/(0.5\unicode[STIX]{x1D70C}U_{b}^{2})$ . Here, the bulk mean velocity is computed with the mass flow rate $Q$ divided by the effective cross-sectional area $S_{yz}$ as $U_{b}=Q/S_{yz}$ . Applying further double integration over the wall-normal direction to (5.6) with normalization of $\unicode[STIX]{x1D6FF}$ and $2U_{b}$ , the FIK identity (Fukagata, Iwamoto & Kasagi Reference Fukagata, Iwamoto and Kasagi2002) for the double-averaged system can be derived as

(5.17) $$\begin{eqnarray}\displaystyle C_{f} & = & \displaystyle \underbrace{\frac{\unicode[STIX]{x1D6FF}_{e}}{\unicode[STIX]{x1D6FF}C^{\prime }}\frac{4}{Re_{b}}}_{laminar}\underbrace{+\,\frac{8}{C^{\prime }}\int _{0}^{1}\left(1-y\right)\left(-R_{12}\right)\,\text{d}y}_{turbulence}\underbrace{+\frac{8}{C^{\prime }}\int _{0}^{1}\left(1-y\right)\left(-{\mathcal{T}}_{12}\right)\,\text{d}y}_{dispersion}\nonumber\\ \displaystyle & & \displaystyle \underbrace{+\,\frac{8}{C^{\prime }}\int _{0}^{1}\left(y-\frac{1}{2}y^{2}\right)\left(\unicode[STIX]{x1D711}\overline{f}_{x}\right)\,\text{d}y}_{drag}\underbrace{+\,\frac{8}{C^{\prime }}\int _{0}^{1}\left(y-\frac{1}{2}y^{2}\right)\left(\unicode[STIX]{x1D711}\overline{g^{\unicode[STIX]{x1D711}}}_{x}\right)\,\text{d}y}_{inhomogeneous~correction},\end{eqnarray}$$

where $Re_{b}$ is the bulk mean Reynolds number, and constant $C^{\prime }$ is expressed as

(5.18) $$\begin{eqnarray}C^{\prime }=\frac{\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D6FF}_{e}}\int _{0}^{1}\left(y-\frac{1}{2}y^{2}\right)\unicode[STIX]{x1D711}\,\text{d}y.\end{eqnarray}$$

In addition to the laminar and turbulence contributions, the dispersion, drag and inhomogeneous correction contribution terms appear as contributors to $C_{f}$ owing to the presence of a rough surface. (See appendix A for the detailed derivation of (5.17).) As well as the turbulence contribution, the dispersion contribution is expressed as an integration of the weighted plane-dispersive covariance $(1-y){\mathcal{T}}_{12}$ , which suggests that ${\mathcal{T}}_{12}$ near the bottom wall contributes more to the skin friction coefficient. Unlike $R_{12}$ and ${\mathcal{T}}_{12}$ , the weighting function for $\overline{f}_{x}$ and $\overline{g}_{x}^{\unicode[STIX]{x1D711}}$ , $(y-\frac{1}{2}y^{2})$ , increases with distance from the bottom wall, suggesting that $\overline{f}_{x}$ and $\overline{g}_{x}^{\unicode[STIX]{x1D711}}$ away from the bottom wall contribute more to the skin friction coefficient. Additionally, unlike the FIK identity for the turbulent channel or pipe flows as discussed by Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002), the constant $C^{\prime }$ appears in the denominator of each contribution. Hence, all contributions are consistently affected depending on the value of $C^{\prime }$ . When we consider the smooth channel flow (i.e. $\unicode[STIX]{x1D711}=1$ for the entire region and $\unicode[STIX]{x1D6FF}_{e}=\unicode[STIX]{x1D6FF}$ ), $C^{\prime }$ takes value $1/3$ and (5.17) consequently reduces to the FIK identity for the fully developed turbulent channel flow. For the presently tested rough surfaces, however, variation in $C^{\prime }$ is not significant; $C^{\prime }$ varies in the range from 0.34 (case S5) to 0.35 (case R3). Hence, the value of $C^{\prime }$ is not a key parameter that determines $C_{f}$ in the presently tested cases.

Figure 12.

Weighted drag force term of (5.19): (a) cases R1–R3 and (b) cases S1–S5.

The contribution terms in (5.17) are presented in figure 13. It is revealed that the most dominant contribution to $C_{f}$ is turbulence, and the second contribution is the drag force in all cases. The laminar contribution is almost independent of the change in $h_{rms}^{+}$ or $Sk$ , and the inhomogeneous correction term is found to hardly contribute to an increase in $C_{f}$ . The dispersion contribution increases with increasing $h_{rms}^{+}$ in figure 13(a) and its contribution in case R3 reaches 37 % of the turbulence contribution. In contrast, as shown in figure 13(b), the dispersion contribution is far smaller compared with the turbulence contribution, and the dispersion contribution takes almost an constant value irrespective of the change in $Sk$ . In terms of the turbulence and drag force contribution terms, the turbulence and drag force contribution terms in figure 13(a) increase with $h_{rms}^{+}$ . In particular, the increase in the drag contribution is found to be prominent (the drag contribution in case R3 is $3.3$ times larger than that in case R1). In contrast, it is observed in figure 13(b) that an increase in $Sk$ dose not significantly affect an increase in the drag contribution while the turbulence contribution increases with $Sk$ . Hence, it is understood that the impacts of $h_{rms}^{+}$ and $Sk$ on the drag force contribution are different.

To identify the reason for these observations, figure 12 shows the integral function for the drag force contribution defined as follows:

(5.19) $$\begin{eqnarray}F_{d}(y)=\frac{8}{C^{\prime }}\left(y-\frac{1}{2}y^{2}\right)\left(\unicode[STIX]{x1D711}\overline{f}_{x}\right).\end{eqnarray}$$

Corresponding to an increase in the roughness peak height scaled with the effective distance with $h_{rms}^{+}$ or $Sk$ , the influence of $F_{d}$ is extended away from the bottom wall with $h_{rms}^{+}$ or $Sk$ . However, it is observed that the maximum peak of $F_{d}$ in figure 12(b) steeply decreases with increasing $Sk$ while the maximum peak of $F_{d}$ in figure 12(a) gradually increases with $h_{rms}^{+}$ . The primary reason for the decrease in the maximum peak $F_{d}$ with $Sk$ is a decrease in the surface area of the roughness element. Since the pressure and viscous drag terms are expressed as the sum of the surface integration of the viscous stress and the pressure per a certain $x$ – $z$ plane as in (5.5), the decrease in the surface area directly results in a decrease in the drag force. As $Sk$ increases, the number of hemispheres decreases, keeping the hemisphere diameter almost constant, which obviously decreases the surface area, as can be seen in figure 1, resulting in the decrease in $F_{d}$ .

Figure 13.

Contribution terms to the skin friction coefficient in (5.17): (a) cases R1–R3 and (b) cases S1–S5.

Figure 14.

Variation of $k_{s}^{+}$ with $h_{rms}^{+}$ ; data of De Marchis et al. (Reference De Marchis, Napoli and Armenio2010) are for two-dimensional irregular wavy walls and data of Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015) are for three-dimensional sinusoidal rough walls.

5.6 Improvement of correlation function for the equivalent roughness

This subsection concentrates on the relation between the equivalent roughness and the statistical moments, namely $h_{rms}^{+}$ and $Sk$ . However, it should be cautioned that equivalent roughness is not a physical property but a flow property, and the equivalent roughness is not necessarily guaranteed to have a relation with the roughness topological parameters. Figure 14 shows the variation in $k_{s}^{+}$ with $h_{rms}^{+}$ in case R1–R3. For comparison, we also plot data for the rough surfaces of $Sk\simeq 0$ and $ES>0.35$ in the fully rough regime, namely two-dimensional irregular wavy walls (De Marchis et al. Reference De Marchis, Napoli and Armenio2010) and three-dimensional sinusoidal rough walls (Chan et al. Reference Chan, MacDonald, Chung, Hutchins and Ooi2015). Furthermore, linear lines with slopes of 3.0, 5.0 and 7.0 are shown. It is observed that $k_{s}^{+}$ in the present data linearly increases with $h_{rms}^{+}$ , and the factor of the proportionality is approximately $5.0$ . The results in two-dimensional irregular wavy walls by De Marchis et al. (Reference De Marchis, Napoli and Armenio2010) are close to the present results of cases R1 and R2 whereas the data of Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015) are substantially larger than the present results. The discrepancy can be partly explained by the difference in the roughness arrangement. The rough surfaces in this study and the rough surfaces of De Marchis et al. (Reference De Marchis, Napoli and Armenio2010) are irregular while the roughness corrugations of Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015) are regularly aligned with a staggered arrangement. Forooghi et al. (Reference Forooghi, Stroh, Magagnato, Jakirlić and Frohnapfel2017) reported that the staggered arrangement of the roughness elements resulted in higher equivalent roughness compared with randomly distributed roughness elements. Another implication in terms of the discrepancy is the presence of more suitable roughness height parameters. Indeed, although the present study chooses $h_{rms}$ as the roughness height parameter, there are different possible choices for the roughness height parameter (e.g. mean peak height, peak-to-valley height, or roughness height amplitude). However, as the data are not sufficient to explore alternative parameters to $h_{rms}$ , we do not discuss it here. (See Flack & Schultz (Reference Flack and Schultz2010), Flack, Schultz & Rose (Reference Flack, Schultz and Rose2012), Forooghi et al. (Reference Forooghi, Stroh, Magagnato, Jakirlić and Frohnapfel2017) and Thakkar et al. (Reference Thakkar, Busse and Sandham2017) for detailed discussions on the peak-to-valley height, and see De Marchis et al. (Reference De Marchis, Napoli and Armenio2010) and Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015) for the roughness height amplitude.) In terms of the relation between $h_{rms}$ and $k_{s}$ , Flack & Schultz (Reference Flack and Schultz2010) derived a correlation by fitting a large number of experimental data, including several kinds of real rough surfaces:

(5.20) $$\begin{eqnarray}k_{s}=4.43h_{rms}(1+Sk)^{1.37}.\end{eqnarray}$$

The correlation suggests that the proportionality constant becomes $4.43$ for $Sk=0$ , which is found to be close to the value obtained by the present DNS. The other reports on the relation between $k_{s}$ and $h_{rms}$ state that $k_{s}\sim 3h_{rms}$ for honed surfaces (Shockling, Allen & Smits Reference Shockling, Allen and Smits2006) and $k_{s}\sim 1.6h_{rms}$ for a commercial steel pipe (Langelandsvik et al. Reference Langelandsvik, Kunkel and Smits2008), whose proportionality constants are significantly lower than those obtained in this study. However, it should be cautioned that those proportionality constants were not for rough surfaces of $Sk=0$ .

Figure 15.

Variation $k_{s}/h_{rms}$ against $Sk$ ; data of Schultz & Flack (Reference Schultz and Flack2009) are for pyramid roughness, data of Yuan & Piomelli (Reference Yuan and Piomelli2014) are for numerically generated sand grains, data of Busse et al. (Reference Busse, Thakkar and Sandham2017) are for graphite roughness and grit-blasted roughness, data of Forooghi et al. (Reference Forooghi, Stroh, Schlatter and Frohnapfel2018a ) are for randomly distributed hemispheres/cones, and data of Forooghi et al. (Reference Forooghi, Weidenlener, Magagnato, Böhm, Kubach, Koch and Frohnapfel2018b ) are for combustion chamber deposits. The dashed line indicates the correlation of (5.20) (Flack & Schultz Reference Flack and Schultz2010), and the solid line indicates the fitted curve of (5.21).

To see the influence of $Sk$ on $k_{s}^{+}$ , figure 15 shows $k_{s}/h_{rms}$ versus $Sk$ . For comparison, we also plot experimental data for pyramid roughness (Schultz & Flack Reference Schultz and Flack2009), and other DNS data for randomly distributed hemispheres/cones (Forooghi et al. Reference Forooghi, Stroh, Schlatter and Frohnapfel2018a ), graphite roughness (Busse et al. Reference Busse, Thakkar and Sandham2017), grit-blasted roughness (Busse et al. Reference Busse, Thakkar and Sandham2017), numerically generated sand grains (Yuan & Piomelli Reference Yuan and Piomelli2014) and combustion chamber deposits (Forooghi et al. Reference Forooghi, Weidenlener, Magagnato, Böhm, Kubach, Koch and Frohnapfel2018b ) with $ES>0.35$ . The dashed line indicates the correlation function of (5.20) by Flack & Schultz (Reference Flack and Schultz2010), and the solid line displays the best-fitted curve

(5.21) $$\begin{eqnarray}k_{s}/h_{rms}=4.0\left(1+0.17Sk\right)^{4}.\end{eqnarray}$$

Clearly, all the results suggest that $k_{s}^{+}$ monotonically increases with $Sk$ in the range $-1<Sk<3$ . This observation is in line with the correlation of Flack & Schultz (Reference Flack and Schultz2010) whereas it is in contrast to the findings by Flack et al. (Reference Flack, Schultz, Barros and Kim2016) that an increase in $Sk$ slightly reduces $k_{s}$ . It should be noted that the finding by Flack et al. (Reference Flack, Schultz, Barros and Kim2016) was, however, based on experimental data for grit-blasted surfaces with negative skewness, and is not the implication for a generic rough surface. The correlation function by Flack & Schultz (Reference Flack and Schultz2010) reasonably works in the range of $-0.5<Sk<0.5$ ; however, the correlation function is found to overestimate the influence of $Sk$ in the rough-wall case with significantly positive/negative skewness $Sk<-0.5,~Sk>0.5$ . One possible reason for the discrepancy is the influence of small short-wavelength undulation. Besides the numerically generated rough surfaces in Chan et al. (Reference Chan, MacDonald, Chung, Hutchins and Ooi2015) and Forooghi et al. (Reference Forooghi, Stroh, Magagnato, Jakirlić and Frohnapfel2017) and the presently used samples, real rough surfaces usually include short- and long-wavelength undulations. Howell & Behrends (Reference Howell and Behrends2006) reported in terms of measurements on real rough surfaces that sampling interval and sampling length had significant effects on roughness statistics. The rough-surface samples used in the DNS (Busse et al. Reference Busse, Thakkar and Sandham2017; Forooghi et al. Reference Forooghi, Weidenlener, Magagnato, Böhm, Kubach, Koch and Frohnapfel2018b ) were filtered to remove uninfluential undulation by the low-pass filter. Additionally, it should be noted that DNS can completely reflect the scanned rough-surface geometry even though they contain some measurement uncertainty originating from insufficiency of resolution, sampling interval or sampling length. On the other hand, as the samples used in Flack & Schultz (Reference Flack and Schultz2010) were not filtered, the influence of the short-wavelength undulation or the measurement condition may emerge. Another possible reason for the underprediction of $k_{s}/h_{rms}$ for the rough surfaces with $Sk<-0.5$ by Flack & Schultz (Reference Flack and Schultz2010) is that only two samples were used in Flack & Schultz (Reference Flack and Schultz2010) which had negative skewness, and both were relatively mild ( $Sk=-0.455,-0.19$ ). The values $k_{s}/h_{rms}$ are inconsistent even though $Sk$ are the same, which is clearly seen in the results by Schultz & Flack (Reference Schultz and Flack2009) and Forooghi et al. (Reference Forooghi, Weidenlener, Magagnato, Böhm, Kubach, Koch and Frohnapfel2018b ). This points out the fact that $k_{s}/h_{rms}$ is not simply expressed as a function of $Sk$ , but other roughness parameters should be included to take account of the other roughness characteristics such as the slope or the solidity. Indeed, Forooghi et al. (Reference Forooghi, Stroh, Magagnato, Jakirlić and Frohnapfel2017) demonstrated that peak height distribution independently affected the equivalent roughness as well as the skewness and the effective slope, and Thakkar et al. (Reference Thakkar, Busse and Sandham2017) presented the significant impact of the streamwise correlation length on the equivalent roughness. Since the presently used data are limited, it is unknown whether the correlation function of (5.21) successfully works for another type of rough surface or for rough surfaces with significantly higher equivalent roughness. Further examination is essentially required to derive a universal correlation for the equivalent roughness. Another important finding from figures 14 and 15 is that, even though the presently considered rough surfaces have inconsistent kurtosis as seen in table 4, the consistent trend of $k_{s}^{+}$ can be clearly confirmed in the present DNS results. This observation supports the statement by Flack & Schultz (Reference Flack and Schultz2010) that the kurtosis of the surface elevation distribution does not have strong correlation with $k_{s}^{+}$ compared with $h_{rms}^{+}$ and $Sk$ .