4.1. Skin-friction coefficient

The mean skin-friction coefficient is $C_f={\tau }_{w}^{b}/({\rho } U_b^{2}/2)$, where $U_b$ is the bulk velocity. Figure 9 compares the computed values of $C_f$ at the two $Re_{\tau }$ to the experimental results of Flack et al. (Reference Flack, Schultz and Barros2020). Note that the rough surface is nearly hydraulically smooth at the lower Reynolds number. The skin-friction coefficient decreases as $Re$ increases and the flow becomes transitionally rough. Both the viscous drag and form drag contribute to the overall skin friction in this flow regime. As $Re$ further increases, the rough surface exhibits fully rough behaviour where the skin friction becomes independent of $Re$. The $C_f$ values of Case R400 and Case R600 are shown in table 4. The errors of the simulation relative to the experiment at the two $Re_{\tau }$ are $0.8\,\%$ and $0.3\,\%$, respectively, showing acceptable agreement.

Figure 9.

Skin-friction coefficient of cases R400 and R600, compared with experimental results (Flack et al. Reference Flack, Schultz and Barros2020). The rough surfaces correspond to the surface with $k_{rms}\approx 88\ \mathrm {\mu }\textrm {m}$ and $Sk=-0.07$ in Flack et al. (Reference Flack, Schultz and Barros2020).

Table 4.

Skin-friction coefficient from experiments.

4.2. Double-averaged velocity

The DA velocity profiles for cases R400 and R600 are discussed below. Figure 10$(a)$ shows the streamwise mean velocity profile in outer coordinates. Parameter $U$ is normalized by the average friction velocity $u_{\tau }$ and $y$ is shifted by subtracting the reference plane $y_0$, and then normalized by $\delta$. The smooth cases at $Re_{\tau }=395$ and $Re_{\tau }=590$ from Moser et al. (Reference Moser, Kim and Mansour1999) are also presented for comparison. A velocity deficit can be observed for the rough cases and the velocity decrease is more significant in the lower half-channel. Compared with the smooth case, the peak of the mean velocity profile at $Re_{\tau }=400$ decreases by $1.6\,\%$ and shifts away from the rough wall by $5\,\%$. For the higher $Re_{\tau }$, a more significant velocity deficit is observed. The peak mean velocity is reduced by $4.4\,\%$ and shifted by $8.4\,\%$ compared with the smooth case at the same $Re_{\tau }$.

Figure 10.

Mean velocity profile normalized by $(a)$ average friction velocity $u_{\tau }$ and $(b)$ rough-wall friction velocity $u_{\tau }^b$, compared with DNS of smooth channel flow of Moser et al. (Reference Moser, Kim and Mansour1999). The vertical dotted lines in $(b)$ denote the top of the roughness layer, $k_c^+$.

Figure 10$(b)$ provides a closer view of the viscous wall region for the rough wall. The mean velocity and the wall-normal distance of the smooth cases are normalized by the average friction velocity $u_{\tau }$ and $\nu /u_{\tau }$, respectively. The smooth-wall curves at different $Re_{\tau }$ collapse in inner coordinates. Scaling with the bottom-wall friction velocity $u_{\tau }^{b}$ provides a more accurate representation of slip and drag effects (e.g. Alamé & Mahesh Reference Alamé and Mahesh2019). Hence, for the rough cases, the mean velocity profile of the lower half-channel is normalized by $u_{\tau }^{b}$. Case R400 shows a slip velocity at the wall and the profile is lowered by $6.7\,\%$ relative to the smooth case. The slip velocity is increased further for Case R600 and the profile is shifted downwards by $13.2\,\%$. The two profiles intersect at $y^{+}=5.7$. This indicates that a larger slip velocity exists at the wall and an overall increase of drag is exhibited in the viscous wall region as $Re_{\tau }$ increases.

The mean velocity profile is shown in figure 11$(a)$ using the same scaling as figure 10$(b)$, but in semi-log coordinates. Due to the slip effect of the roughness, the velocity profile shows a gradual increase away from the wall. This trend appears to end at $y^{+}=5$ which is the transition from the viscous sublayer to the buffer layer. In the buffer layer, the velocity for Case R600 increases more slowly and displays a more significant velocity deficit than Case R400. A similar trend was observed by Busse et al. (Reference Busse, Thakkar and Sandham2017). In the logarithmic-law region, the smooth-wall profiles follow the logarithmic law:

(4.1)\begin{equation} U_{s}^{+}=\frac{1}{\kappa} \ln{y^{+}}+ B, \end{equation}

where $\kappa$ is the von Kármán constant and $B$ is the intercept for a smooth wall. The rough-wall profiles conform to the logarithmic law but display an offset from the smooth-wall profiles, where the roughness effect on mean velocity can be evaluated from this difference. Nikuradse (Reference Nikuradse1933) found that the logarithmic velocity distribution for the mean velocity profile still held for rough walls, with the same value of $\kappa$ as

(4.2)\begin{equation} U_{r}^{+}=\frac{1}{\kappa} \ln({y/k_{s}})+ 8.5. \end{equation}

The roughness function is obtained by taking the difference of mean velocities in wall units between smooth and rough walls within the logarithmic layer:

(4.3)\begin{equation} B-\Delta U^{+}+\frac{1}{\kappa} \ln{k_{s}^{+}}=8.5. \end{equation}

Flack & Schultz (Reference Flack and Schultz2010) show good collapse in the fully rough regime for different roughness types when using $k_{s}$ as the roughness scale. However, in the transitional regime, the roughness function depends on the roughness type, and the onset of the fully rough regime is unknown for most surfaces. Difference $\Delta U^{+}$ from the simulations is plotted in figure 11$(b)$ as a function of the corresponding roughness Reynolds number $k_{s}^{+}$ obtained from experiment. The results show that cases R400 and R600 are in the transitionally rough regime, and match the experimental results of Flack et al. (Reference Flack, Schultz and Barros2020).

Figure 11.

$(a)$ Mean velocity profile in the bottom half of the channel in inner coordinates. The legend is the same as figure 10. $(b)$ Roughness function compared with the experimental results of Flack et al. (Reference Flack, Schultz and Barros2020) and the fully rough asymptote.

4.3. Reynolds stresses

The Reynolds stresses scaled by $u_{\tau }^{2}$ are shown in outer coordinates in figure 12. The location of the peak of $\langle u'u' \rangle ^+$ is dependent on $Re$, as shown in figure 12$(a)$, hence both smooth and rough cases at $Re_{\tau }=600$ are closer to the wall than at $Re_{\tau }=400$. The peak of $\langle u'u' \rangle ^+$ in the lower half-channel for Case R400 is decreased by $2.8\,\%$ compared with the smooth wall of Moser et al. (Reference Moser, Kim and Mansour1999). This behaviour is consistent with Busse et al. (Reference Busse, Lützner and Sandham2015). The minimum value is shifted to the top wall by $7.9\,\%$ due to the asymmetry induced by the roughness. For $Re_{\tau }=600$, the peak value of $\langle u'u' \rangle ^+$ is further decreased by $9.1\,\%$ and the minimum value is shifted to the top wall by $9.2\,\%$; i.e. the roughness has a more significant effect on the streamwise velocity fluctuations as $Re_{\tau }$ increases.

Figure 12.

Reynolds stress profiles normalized by $u_{\tau }$: $(a)$ streamwise, $(b)$ wall-normal and $(c)$ spanwise components and $(d)$ Reynolds shear stress. The legend is the same as figure 10.

The wall-normal Reynolds stress is shown in figure 12$(b)$. Compared with the smooth channel, $\langle v'v' \rangle ^+$ in the lower half-channel is increased, while that in the upper half-channel is decreased. The peak value of $\langle v'v' \rangle ^+$ in the lower half-channel is increased by $5.3\,\%$ at $Re_{\tau }=400$ and $8.9\,\%$ at $Re_{\tau }=600$. The spanwise velocity fluctuations also increase in the lower half-channel (figure 12$c$), and they decrease in the upper half-channel compared with the smooth channel. The Reynolds shear stress increases in magnitude at the rough wall, (figure 12$d$), consistent with the increase in wall-normal velocity fluctuations. The profile of $\langle u'v' \rangle ^+$ shifts downwards by $4.9\,\%$ at $Re_{\tau }=400$ and by $8.4\,\%$ at $Re_{\tau }=600$.

The Reynolds stresses are normalized by the local friction velocity at each wall and plotted separately in figure 13. This scaling yields better collapse in the logarithmic regions of the smooth and rough walls. In the near-wall region of the rough wall, the peak value of the streamwise component of the rough-wall side decreases. The spanwise component shows a higher peak value while the wall-normal component and Reynolds shear stress show relatively weaker variation in their peak values. These trends become more pronounced with increasing $Re_{\tau }$.

Figure 13.

Near-wall Reynolds stresses scaled by local friction velocity $u_{\tau }^l=u_{\tau }^b$ for the bottom wall and $u_{\tau }^t$ for the top wall. $(a)$ Case R400 and $(b)$ Case R600. The vertical dashed line denotes the top of the roughness layer $k_c^+$.

Busse et al. (Reference Busse, Lützner and Sandham2015) studied a series of surfaces with different levels of filtering (a low-pass filter was used to remove the high-wavenumber contributions to the surface data). For their smallest roughness height ($k_{rms}^+=4.6$) and largest skewness ($Sk=1.15$, where $Sk>0$ means the surface is peak-dominant), they found an increase in the peak of $\langle v'v'\rangle ^+$, and a slight increase in both $\langle v'v'\rangle ^+$ and $\langle u'v'\rangle ^+$ in the roughness layer. Compared with their surfaces, the present rough surfaces have smaller roughness heights in wall units for cases R400 and R600, and zero skewness. That is, our surfaces have fewer, and smaller, asperities to increase $\langle v'v'\rangle ^+$ and $\langle u'v'\rangle ^+$.

4.4. Dispersive flux and correlations

The dispersive stresses obtained from DA are examined in the near-wall region of the rough wall in figure 14. Compared with the Reynolds stresses, the dispersive stresses are mostly confined to the roughness layer, consistent with the observations of Yuan & Jouybari (Reference Yuan and Jouybari2018) and Jelly & Busse (Reference Jelly and Busse2019). For Case R400, $\langle \tilde {u} \tilde {u} \rangle ^+$ is maximum at $(y-y_0)/\delta =0.015$ and drops to zero by $(y-y_0)/\delta =0.05$, underlining the importance of spatial inhomogeneity in the roughness layer. The profiles of $\langle \tilde {v} \tilde {v} \rangle ^+$ and $\langle \tilde {w} \tilde {w} \rangle ^+$ peak at $(y-y_0)/\delta =0.011$ and $0.016$, respectively. The dispersive shear stress $\langle \tilde {u} \tilde {v} \rangle ^+$ shows the same peak location as $\langle \tilde {v} \tilde {v} \rangle ^+$ and is comparatively small. The level of dispersive stresses is enhanced for higher $Re_{\tau }$, especially in the roughness layer, and the peak values occur closer to the wall.

Figure 14.

Dispersive stress profiles scaled by $u_{\tau }^b$ over the bottom half of the channel. The vertical dotted line denotes the top of the roughness layer $k_c$. The legend of $(b)$ is same as $(a)$.

Dispersive stresses arise from regions where the local temporal average is different from the average computed over time and the entire rough wall. The form-induced velocity and pressure are examined at $(y-y_0)/\delta =0.0075$ and $z/\delta =1.57$ in figure 15 to illustrate how the roughness topography contributes to the local inhomogeneity in the roughness layer. Figure 15$(a)$ shows that large positive $\tilde {u}$ (red region) occurs in the large trough regions between roughness asperities, while large negative $\tilde {u}$ (blue region) occurs in the wake regions behind the roughness elements. Both these high- and low-momentum regions contribute to the streamwise dispersive stress. In the areas where the roughness protrusions are relatively peaky and dense, corresponding to the $x$–$y$ plane probed in figure 15$(b)$, regions with negative $\tilde {u}$ are observed near the roughness protrusions.

Figure 15.

Form-induced quantities for Case R400 at $(y-y_0)/\delta =0.0075$ ($y^+=3$): ($a$) $\tilde {u}^+$, ($c$) $\tilde {v}^+$, ($e$) $\tilde {w}^+$, ($g$) $\tilde {p}^+$. A slice at $z/\delta =1.57$: ($b$) $\tilde {u}^+$, ($d$) $\tilde {v}^+$, ($\,f$) $\tilde {w}^+$, ($h$) $\tilde {p}^+$. The velocities are normalized by $u_{\tau }^b$ and pressure is normalized by $(u_{\tau }^b)^2$. The rectangular regions labelled by $A$ and $B$ are two representative roughness geometries discussed in the text.

Figures 15$(c)$ and 15$(d)$ show that impulsive upward velocities, shown by the high-magnitude positive $\tilde {v}$ (red regions), occur mostly in front of the roughness protrusions. Downward velocities, shown by the negative $\tilde {v}$ (blue regions), occur in the wake of the protrusions. The strength of these upward and downward motions is positively correlated with the roughness height.

Figures 15$(e)$ and 15$(\,f)$ show pairs of large positive $\tilde {w}$ (red region) and negative $\tilde {w}$ (blue region) in front of the roughness crests. This suggests that the impulsive upward velocity produces a pair of streamwise vortices in front of the roughness elements. Similar behaviour was observed by Muppidi & Mahesh (Reference Muppidi and Mahesh2012) in their investigation of an idealized rough-wall supersonic boundary layer. Also, $\tilde {v}$ displays larger wall-normal length scales than $\tilde {w}$, which explains why the peak location of $\langle \tilde {v} \tilde {v} \rangle ^+$ is higher than that of $\langle \tilde {w} \tilde {w} \rangle ^+$ in figure 14$(b)$.

The pressure perturbations at the same locations are shown in figures 15$(g)$ and 15$(h)$. High $\tilde {p}$ (red region) is found to occur in front of the roughness asperities where the flow stagnates, while low $\tilde {p}$ (blue region) occurs at the crests and behind the roughness elements. The form drag is therefore mainly produced by the peaks of the rough surface.

The form-induced quantities for Case R600 were also examined but are not shown here in the interests of brevity. The main features remain the same, while their magnitudes increase at the higher $Re_{\tau }$.

Figure 16 takes a closer look at how roughness protrusions contribute to mean wall-normal fluxes. Two representative regions labelled $A$ and $B$ in figure 15$(b)$ are considered. Region $A$ represents a dense and peaky region where the flow is dominated by negative $\tilde {u}$, while $B$ is a relatively flat and smooth region. Both figures 16$(a)$ and 16$(b)$ show that negative $\tilde {u}$ is induced at the back of the protrusions, and the largest crest yields the strongest negative $\tilde {u}$. The upward $\tilde {v}$ induced in front of the crests accompanies the negative $\tilde {u}$, resulting in ‘ejection’ motions, borrowing terminology from the boundary layer literature (Bailey & Stoll Reference Bailey and Stoll2013). Similar strong ‘ejection’ motions are present in front of the crests while ‘sweeping’ motions occur within the troughs. Both ‘ejection’ and ‘sweeping’ motions are responsible for the negative dispersive shear stress shown in figure 14$(b)$. A clockwise ‘roll-up’ motion is shown at the roughness crests in figure 16$(b)$; such motions occur when the surface topography is less rough and the flow is not dominated by negative $\tilde {u}$. Thus roughness geometry influences wall-normal momentum transfer by influencing the ‘ejection’ and ‘roll-up’ motions.

Figure 16.

Vectors of ($\tilde {u}$,$\tilde {v}$) with background contours of $\tilde {p}^+$ for representative roughness sections, corresponding to the regions labelled ($a$) $A$ and ($b$) $B$ in figure 15$(b)$.

Figure 17 shows the joint p.d.f. of ($\tilde {u}$, $\tilde {p}$) and ($\tilde {v}$, $\tilde {p}$) near the roughness mean height at $y^+=3$ for Case R400. The results for Case R600 are not shown since the main conclusions remain the same. Stronger correlations between $\tilde {u}$ and $\tilde {p}$ are observed in the third quadrant $Q3$ compared with the smooth channel (Lenaers et al. Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012), which corresponds to the ‘roll-up’ motions induced by the roughness crests. The regions with these motions account for the negative $\tilde {u}$ and correlate with the negative $\tilde {p}$, as shown in figure 16$(b)$. Lenaers et al. (Reference Lenaers, Li, Brethouwer, Schlatter and Örlü2012) found a correlation between negative $v$ and positive $p'$ near smooth walls, indicating splatting events. In the rough case, however, the correlation in the second quadrant $Q2$ is weak since positive $p'$ is more due to stagnation on the roughness elements. Meanwhile, the correlation in $Q1$ and $Q3$ is stronger since the upward $\tilde {v}$ is always induced in front of the crests where positive $\tilde {p}$ occurs, while the downward $\tilde {v}$ appears in the wake regions where negative $\tilde {p}$ occurs.

Figure 17.

Joint p.d.f. of $(a)$ ($\tilde {u}^+$, $\tilde {p}^+$) and $(b)$ ($\tilde {v}^+$, $\tilde {p}^+$) for Case R400 at the wall-normal location $(y-y_0)/\delta =0.0075$ ($y^+=3$). Contour levels are defined by the common logarithm of joint p.d.f. and the interval between successive contour levels is 0.5.

4.5. Mean momentum balance

The mean momentum balance reveals the different contributions to momentum conservation in the roughness layer. The streamwise DA momentum equation for a fully developed channel flow (Raupach & Shaw Reference Raupach and Shaw1982; Yuan & Jouybari Reference Yuan and Jouybari2018) is

(4.4)\begin{equation} \underbrace{\frac{\partial \langle \tilde{u} \tilde{v} \rangle}{\partial y}}_{\text{A}} = -\underbrace{\frac{\partial \langle \bar{p} \rangle}{\partial x}}_{\text{B}} + \underbrace{\nu \frac{\partial^2 \langle \bar{u} \rangle}{\partial y^2}}_{\text{C}} -\underbrace{ \left \langle \frac{\partial \tilde{p} }{\partial x} \right \rangle}_{\text{D}} + \underbrace{\nu \left \langle \frac{\partial^2 \tilde{u} }{\partial x_j^2}\right \rangle}_{\text{E}} - \underbrace{\frac{\partial \langle \overline{u'v'} \rangle}{\partial y}}_{\text{F}} + \langle K \rangle, \end{equation}

where the subscript $j$ denotes streamwise, wall-normal and spanwise components for position vectors and $K$ is the constant pressure gradient (divided by density). In this equation, term A is the dispersive shear stress gradient, term B represents the mean pressure gradient which is zero in our simulations, term C shows the viscous force (viscous stress gradient), terms D and E account for the pressure drag and viscous drag caused by the roughness, respectively, and term F is the net effect of turbulent inertia (Reynolds stress gradient).

Figure 18 shows the budget of terms in the mean momentum equation for the bottom half-channel. Since terms A, D and E are zero for Case SW, the balance is among the constant pressure gradient, the viscous and Reynolds stress gradients. For Case R400, the Reynolds stress gradient is slightly increased at the wall and the peak but follows a similar variation as Case SW in general. As $y^+$ increases, the viscous stress gradient increases and reaches a maximum value at $y^+=3$, then decreases to negative values and collapses with Case SW above the roughness layer. The pressure and viscous drag are induced by the roughness and only exist in the roughness layer. Compared with other terms, the contribution of the dispersive term is quite small. Overall, the roughness transfers mean momentum downward through the viscous and Reynolds stress gradients to balance the pressure and viscous drag. The observation for Case R600 is the same except that the magnitude of each term is larger than for Case R400.

Figure 18.

Budget of the streamwise DA momentum equation for Case SW (symbols) and Case R400 (lines). The vertical dotted line denotes the top of the roughness layer $k_c^+$. The summation of all terms is denoted by the short-dashed line.

4.6. Pressure fluctuations

4.6.1. Mean-square pressure fluctuations

Figure 19$(a)$ shows the mean-square pressure fluctuations $\langle \overline {p'p'} \rangle ^+$ in outer coordinates for cases R400 and R600 over the bottom half of the channel. Also shown are Case SW and other smooth channel flows from Panton et al. (Reference Panton, Lee and Moser2017). Note that away from the wall, both smooth- and rough-wall curves collapse on to the logarithmic correlation described by Panton et al. (Reference Panton, Lee and Moser2017):

(4.5)\begin{equation} \langle \overline{p'p'} \rangle^+_{cp}(y/\delta) = -2.5625\ln(y/\delta) + 0.2703. \end{equation}

As $Re_{\tau }$ increases, $\langle \overline {p'p'} \rangle ^+$ increases near the wall, peaks closer to the wall and collapses onto the logarithmic correlation closer to the wall. Roughness increases the peak levels of $\langle \overline {p'p'} \rangle ^+$ near the wall.

Figure 19.

$(a)$ Profiles of mean-square pressure fluctuations $\langle \overline {p'p'} \rangle$ normalized by $u_{\tau }^b$ and $(b)$ inner correlation $\langle \phi \rangle (y^+)$. Symbols denote: Case SW (solid), Case R400 (circle), Case R600 (delta), Panton et al. (Reference Panton, Lee and Moser2017), $Re_{\tau }=182$ (dashed), $Re_{\tau }=544$ (dash-dotted), $Re_{\tau }=1001$ (dash-dot-dot). The vertical dashed lines denote the roughness layers.

Panton et al. (Reference Panton, Lee and Moser2017) derived an inner correlation to account for the $Re_{\tau }$ dependence for smooth channel flows:

(4.6)\begin{equation} \langle \phi \rangle(y^+) = \langle \overline{p'p'} \rangle^+(y^+,Re_{\tau}) - 2.5625\ln(Re_{\tau}) - 0.2703. \end{equation}

Figure 19$(b)$ shows $\langle \phi \rangle (y^+)$ computed for both smooth and rough cases. The smooth curves show acceptable collapse both near to and away from the wall for all $Re_{\tau }$. The rough curves, however, collapse only beyond $y^+=50$. The increased level of pressure fluctuations below $y^+=50$ indicates that the roughness increases the pressure fluctuations in the inner layer and the enhancement is much stronger for higher $Re_{\tau }$.

The vertical lines in figure 19$(b)$ illustrate the top of the roughness layer. Note that $\langle \overline {p'p'} \rangle ^+$ peaks outside the roughness layer. Comparing Case R400 with Case SW, the level of pressure fluctuations within the roughness layer increases by $20.2\,\%$, and the peak value by $7.8\,\%$. For Case R600, the enhancement of pressure fluctuations in the roughness layer is even higher.

The spatial variation of $\overline {p'p'}$ is examined at a representative location $(y-y_0)/\delta =0.0075$ and $z/\delta =1.57$ in figure 20. At both Reynolds numbers, high pressure fluctuations are observed in front of the roughness asperities, while the regions behind the roughness elements have lower levels. The unsteady stagnation and wake regions induced by the roughness elements result in higher pressure fluctuations. Smooth-wall flows do not display this behaviour, and hence have lower levels of pressure fluctuations for the same $Re_{\tau }$.

Figure 20.

Mean-square pressure fluctuations $\overline {p'p'}$ normalized by $U_b$ for Case R400 at ($a$) $(y-y_0)/\delta =0.0075$ (corresponding to $y^+=3$), ($b$) $z/\delta =1.57$, and Case R600 at ($c$) $(y-y_0)/\delta =0.0075$ (corresponding to $y^+=4.5$) and ($d$) $z/\delta =1.57$.

4.6.2. Poisson equation source terms

The source terms in the pressure Poisson equation provide further insight into how roughness influences pressure fluctuations. The Poisson equation for the fluctuating pressure is given by

(4.7)\begin{equation} \frac{\partial^2 p'}{\partial x_i \partial x_i} = - 2 \frac{\partial {\langle \overline{u_i} \rangle}}{\partial {x_j}}\frac{\partial u_j'}{\partial x_i} - \frac{\partial^2}{\partial x_i \partial x_j}\left(u_i'u_j' - \overline{u_i'u_j'}\right). \end{equation}

The first term on the right-hand side is called the mean-shear or rapid source term (MS) and the second term is known as the turbulence-turbulence or slow source term (TT). The dominant component of the MS term is

(4.8)\begin{equation} MS_{12} = - 2 \frac{\partial {\langle \bar{u} \rangle}}{\partial {y}}\frac{\partial v'}{\partial x}. \end{equation}

Since this term involves the streamwise mean velocity gradient, it is large near the wall (Johansson, Her & Haritonidis Reference Johansson, Her and Haritonidis1987). For the TT term, it has been shown that the component

(4.9)\begin{equation} TT_{23}=\frac{\partial {v'}}{\partial {z}}\frac{\partial w'}{\partial y} - \frac{\partial ^2 \overline{v'w'}}{\partial y \partial z} \end{equation}

is the dominant term, located in the buffer region whose peak location is correlated with the average position of quasi-streamwise vortices (Chang III et al. Reference Chang III, Piomelli and Blake1999). The dominant terms $MS_{12}$ and $TT_{23}$ for the random rough channel flows are discussed below.

Figure 21 shows how roughness impacts ${\partial {\langle \bar {u} \rangle }}/{\partial {y}}, {\partial v'}/{\partial x}$ and $MS_{12}$ individually. The profiles are normalized by the local friction velocity at the two walls. Roughness is seen to affect both the mean and fluctuating velocity gradients. Figure 21$(a)$ shows that for the smooth half of the channel, the mean velocity gradient is largest at the wall and gradually decreases with increasing $y$. On the other hand, at the rough wall, ${\partial {\langle \bar {u} \rangle }}/{\partial {y}}$ peaks at $y^+=6$ which lies within the roughness layer, and collapses onto the smooth-wall profile for larger $y$. Figure 21$(b)$ shows $\partial {v'} /\partial x$. Note that the r.m.s. of $\partial {v'} /\partial x$ significantly increases within the roughness layer, and collapses with the smooth-wall profile above the roughness. Both terms combine to increase $MS_{12}$ as shown in figure 21$(c)$. The peak values are within the roughness layer at $y^+=6$, consistent with the peak locations observed in figure 21$(a)$. Both Reynolds numbers show the same behaviour except that Case R600 has higher magnitudes.

Figure 21.

$(a)$ Streamwise mean velocity gradient, $(b)$ $\partial {v'} /\partial x$, $(c)$ $MS_{12}$ and $(d)$ $TT_{23}$. Here $u_{\tau }^l$ and $\delta$ are used for normalization. The vertical dash-dotted line denotes the mean roughness height $y=y_0$, and the vertical dashed lines denote the top of the roughness layer $k_c^+$.

Figures 22$(a)$ and 22$(b)$ show the local variation of r.m.s. $MS_{12}$ in the near-wall region for Case R400. The intense high-magnitude spots (red regions) are produced in front of the roughness protrusions and on the crests. The time-averaged spanwise vorticity in figures 22$(c)$ and 22$(d)$ shows the local mean shear layers that form at the same locations. These shear layers combine to yield the mean velocity gradient at a wall-normal location. The higher levels of $\partial {v'} /\partial x$ within the roughness layer are due to the upward velocity in front of the protrusions and the ‘roll-up’ motions on the crests observed in § 4.4.

Figure 22.

Spatial variation of r.m.s. $MS_{12}$ for Case R400 at $(a)$ $y^+=3$ and $(b)$ $z^+=628$. Time-averaged spanwise vorticity $\overline {\omega _z}$ normalized by $u_{\tau }/\delta$ at $(c)$ $y^+=3$ and $(d)$ $z^+=628$. The dashed line in $(a)$ denotes the $x^+$–$y^+$ plane observed in $(b)$ and $(d)$, while the dashed line in $(b)$ denotes the edge of the roughness layer.

The high-magnitude regions of r.m.s. $MS_{12}$ and $\overline {\omega _z}^+$ are consistent with the regions of high pressure fluctuations in figure 20. The more intense pressure fluctuations in front of and above the roughness elements can be attributed to the attached shear layers formed upstream of the obstacles. The low level of r.m.s. $MS_{12}$ in the wake regions is because the flow is more quiescent behind the roughness elements.

The r.m.s. of $TT_{23}$ is examined in figure 21$(d)$. Note that for both cases R400 and R600, the level of $TT_{23}$ increases in the roughness layer, the peak location remains at $y^+=18$ and the profile above the roughness layer collapses with that of the smooth wall. The spatial distribution of r.m.s. $TT_{23}$ at $y^+=3$ and $x^+=688$ for Case R400 is shown in figures 23$(a)$ and 23$(b)$, respectively. The high-magnitude regions (red) are observed as the streaky structures in the troughs as well as a large band at $y^+=18$. To specifically investigate how the r.m.s. of $TT_{23}$ is increased in the roughness layer, two typical high-magnitude streaks are identified in figure 23$(a)$ and the time-averaged streamwise vorticity in the corresponding $z$–$y$ planes are examined in figures 23$(c)$ and 23$(d)$. Two main features are identified. For location A, the level of $TT_{23}$ is increased in the immediate vicinity upstream of the roughness protrusion. Meanwhile, a pair of streamwise vortices is observed at the same location which is induced by the upward velocity in front of the protrusions. A pair of secondary vortices is observed below the induced streamwise vortices, closely attaching to the bottom wall. For location B, the high-magnitude streak is located in the valley in the same region where the streamwise vortices occur. These observations show that $TT_{23}$ is correlated with the quasi-streamwise vortices. Since quasi-streamwise vortices are present in the troughs (Jouybari, Brereton & Yuan Reference Jouybari, Brereton and Yuan2019) and induced in front of the protrusions, the r.m.s. of $TT_{23}$ is accordingly increased.

Figure 23.

Spatial variation of r.m.s. $TT_{23}$ for Case R400 at $(a)$ $y^+=3$ and $(b)$ $x^+=xu_{\tau }/\nu =688$. Time-averaged streamwise vorticity $\overline {\omega _x}$ normalized by $u_{\tau }/\delta$ corresponding to $(c)$ the square region A at $x^+=160$ and $(d)$ the square region B at $x^+=688$. The dashed lines in $(a)$ denote the $z^+$–$y^+$ planes observed in $(b\text {--}d)$, while the dashed line in $(b)$ denotes the edge of the roughness layer. Note that for the $z^+$–$y^+$ plane, the mean flow direction points out of the page.

The flow characteristics related to the contributions of $MS_{12}$ and $TT_{23}$ to the pressure fluctuations are summarized in figure 24. The shear layer along with the ‘roll-up’ motion indicated by the dashed ellipse is generated at the upstream stagnation point of the protrusion in figure 24$(a)$, and is a primary source of the larger pressure fluctuations. Figure 24$(b)$ illustrates the streamwise vortices that occur in front of the protrusion and also in the troughs, generating secondary vortical structures below the main vortices.

Figure 24.

Illustration of the typical flow features contributing to $(a)$ $MS_{12}$ and $(b)$ $TT_{23}$. The dashed ellipse in $(a)$ denotes the shear layer formed on the protuberance. The red arrows in $(b)$ represent a pair of streamwise vortices and secondary vortices beneath them.

4.7. Wall shear stress fluctuations

The effects of roughness on wall shear stress fluctuations are examined in this section. The wall shear stress components $\tau _{yx}$ and $\tau _{yz}$ are calculated at the rough surface. Data are collected for a time period $T=32\delta /u_{\tau }$ with sampling time interval $t=2.5\times 10^{-3}\delta /u_{\tau }$. The wall shear stress fluctuations are denoted by $\tau _{yx}'$ and $\tau _{yz}'$, the time-averaged values are expressed as $\overline {\tau _{yx}}$ and $\overline {\tau _{yz}}$ and the r.m.s. values are $\tau _{yx,rms}$ and $\tau _{yz,rms}$.

4.7.1. Local variation of wall shear stress

Shear stress $\overline {\tau _{yx}}$ and the roughness height for a portion of the surface are shown in figure 25. The results of Case R400f are compared with those of Case R400 and found to be in acceptable agreement.

Figure 25.

Roughness height and time-averaged streamwise wall shear stress for a portion of the rough surface. The red circles denote locations A–E where wall shear stress is probed.

Figure 25 shows that the variation of $\overline {\tau _{yx}}$ in the rough case is highly correlated with the roughness height. Parameter $\overline {\tau _{yx}}$ has large positive values at the roughness protrusions, and is nearly zero or has small negative values in the valleys.

Five different locations (see table 5) on the roughness profile in figure 25 are probed to show the dependence of wall shear on the surface topography. Point A is located at a trough, where $\overline {\tau _{yx}}$ is negative, due to the reverse flow that occurs in the trough. Points B, C and D have a similar $y^+$. However, B is at a trough where $\overline {\tau _{yx}}$ is close to zero. Point C is located at the ascent of the protrusion, which shows a higher $\overline {\tau _{yx}}$ and $\tau _{yx,rms}$. In contrast to points B and C, point D is located at a crest, leading to even larger mean and r.m.s. of wall shear stress. The wall shear stress at the crest for an even higher $y^+$ is examined at point E. The mean and r.m.s. values are much higher than those of other probed locations.

Table 5.

Locations and time-averaged and r.m.s. values of $\tau _{yx}$ for the probed points in figure 25.

4.7.2. Probability distribution function of wall shear stress

The effect of roughness on the p.d.f. of wall shear stress is examined. The p.d.f.s of the streamwise shear stress component $\tau _{yx}$, spanwise shear stress component $\tau _{yz}$ and the shear stress yaw angle $\phi _{\tau } = \tan ^{-1}(\tau _{yz}(t)/\tau _{yx}(t))$ are shown in figure 26, and tabulated in table 6. Case SW is the baseline, whose statistics of wall shear stress fluctuations show good agreement with the results for a smooth-wall turbulent boundary layer (Diaz-Daniel et al. Reference Diaz-Daniel, Laizet and Vassilicos2017). Motivated by the influence of topography in § 4.7.1, the shear stress is probed separately in two regions: the ‘peak region’ (above the mean height location) and the ‘valley region’ (below the mean height location). A schematic of the decomposition is shown in figure 27.

Figure 26.

The p.d.f.s of $(a)$ streamwise wall shear stress, $(b)$ spanwise wall shear stress and $(c)$ angle between shear stress vector and streamwise direction. Cases SW (solid black), R400 peak (dashed red), R400 valley (dash-dotted red), R600 peak (long-dashed blue) and R600 valley (dash-dot-dot blue).

Figure 27.

Illustration of how wall shear stress is divided into peak regions (above the mean height location) and valley regions (below the mean height location).

Table 6.

Statistics of the wall shear stress components $\tau _{yx}$, $\tau _{yz}$ and yaw angle $\phi _{\tau }$: mean $\mu (\cdot )$, standard deviation $\sigma (\cdot )$, skewness $Sk(\cdot )$ and kurtosis $Ku(\cdot )$.

The r.m.s. of the streamwise shear stress for Case SW (table 6) is in good agreement with the correlation $\tau _{yx,rms}^{+}=\tau _{yx,rms}/\tau _{w}=0.298+0.018\ln Re_{\tau }$, proposed by Örlü & Schlatter (Reference Örlü and Schlatter2011). Comparison of the rough walls with Case SW at the same $Re_{\tau }$ reveals the following features. A significant increase in the probability of events with positive $\tau _{yx}$ is observed for the peak region in figure 26$(a)$ since $\tau _{yx}$ is associated with the streamwise velocity which generally increases as $y$ increases. The probability of events with negative $\tau _{yx}$ is higher in the valleys, corresponding to figure 28 that shows how reverse flows occur mainly in the valleys. The p.d.f. for rough surfaces has longer tails (larger kurtosis), since intermittent events in the flow can interact randomly with peaks or valleys in the rough surface, resulting in more extreme events.

Figure 28.

Contours of time-averaged streamwise velocity and vectors of ($\bar {u}$,$\bar {v}$) in the vicinity of the rough surface for Case R400.

The p.d.f. of spanwise wall shear stress fluctuations $\tau _{yz}$, shown in figure 26$(b)$, has zero mean and zero skewness for both smooth and rough cases. The normalization by the r.m.s. values collapses the peak of the p.d.f.; however, the tails are longer for the rough walls. This behaviour is more pronounced at the higher $Re_{\tau }$.

The probability distribution of $\phi _{\tau }$ in figure 26$(c)$ examines the correlation between $\tau _{yx}$ and $\tau _{yz}$. Jeon et al. (Reference Jeon, Choi, Yoo and Moin1999) found that the probability for events with $|\phi _{\tau }|>45^{\circ }$ is very small in smooth channel flows; i.e. the flow is dominated by events where large positive $\tau _{yx}$ is associated with small $\tau _{yz}$. The standard deviation of $\phi _{\tau }$ is much higher for the rough wall, compared with Case SW. The probability of events with $|\phi _{\tau }| >45^{\circ }$ is enhanced by the roughness, suggesting that the events with comparable magnitudes of $\tau _{yx}$ and $\tau _{yz}$ occur more frequently. This behaviour can be explained by the three-dimensionalization of the vortical motions in the roughness layer. The roughness elements break up the directional bias of the near-wall streaks and generate more isotropic vorticity fields (Mehdi, Klewicki & White Reference Mehdi, Klewicki and White2010; Jouybari et al. Reference Jouybari, Brereton and Yuan2019). In the valley region, events with $90^{\circ } < |\phi _{\tau }| <180^{\circ }$ (corresponding to negative $\tau _{yx}$) are more probable, consistent with reverse flow in the valleys. The p.d.f. of $|\phi _{\tau }|$ tends to be more evenly distributed for Case R600, implying that the reverse flow is enhanced as $Re_{\tau }$ increases.