3.1.1 Steady-state flow field

Initially, the fully wetted Case L-RFW is considered. Figure 6 shows the flow field after it is fully developed. The wall-normal velocity (figure 6 a) and the vorticity magnitude along with surface pressure (figure 6 b) are shown. A wall-normal velocity component into the flow is induced due to the surface asperities. Additional vorticity is generated due to the surface roughness, and large variations of pressure on the surface are evident due to the presence of peaks and valleys. The penetration effect of the surface roughness is illustrated in figure 7, where the percent change in instantaneous streamwise velocity $(u(y)-u_{o}(y))/u_{o}(y)$ is shown for the fully wetted rough case (Case L-RFW) compared to the smooth channel case (Case L-S) at four wall-parallel planes varying from $y=0.02H$ to $y=0.4H$ . The baseline streamwise velocity $u_{o}(y)$ represents Case L-S and $u(y)$ represents Case L-RFW. Notice that it is not until the location $y=0.4H$ that the change in velocity is less than $1\,\%$ , suggesting that the surface roughness effects can penetrate up to that distance.

Figure 6.

Laminar Couette flow (Case L-RFW) with (a) wall-normal velocity contours normalised by the maximum wall-normal velocity $v_{max}$ and (b) vorticity magnitude line contours normalised by the maximum vorticity $\unicode[STIX]{x1D714}_{max}$ with surface pressure (normalised by $p_{max}$ ) on the roughness for the range shown in the colour bar.

Figure 7.

Percent change in the streamwise velocity field for the fully wetted laminar Couette flow (Case L-RFW) compared to the baseline smooth wall (Case L-S) at wall-parallel planes: (a)  $0.02H$ , (b)  $0.08H$ , (c)  $0.16H$ and (d)  $0.40H$ from the bottom surface.

Figure 8.

Laminar Couette flow: drag reduction as a function of interface height normalised by the RMS roughness height $S_{q}$ .

3.1.2 Mean flow field properties

Simulations are performed for each of the interface heights $h$ varying from $S_{v}$ to $S_{p}$ . $DR$ is shown for all the interface heights in figure 8. The increase in $DR$ is not linear when $h$ is varied from $S_{v}$ to $S_{p}$ . Note that the fully wetted case has negative $DR$ indicating that the absence of the interface has increased drag due the exposed asperities. The presence of the interface produces nearly the same drag reduction for $h/S_{q}\leqslant -0.32$ which we will refer to as region I. This suggests that the value of $DR$ is insensitive to $h$ in region $I$ . This holds in general for any surface with negative skewness ( $S_{sk}=-0.32$ ) since that region $I$ holds most of the valleys. Figure 9 shows a 2-D slice of the surface roughness; note that the valleys dominate over the peaks about the reference line. As the interface fills up more of the cavities, the slip area becomes significant enough to cause drag reduction. In region $II$ , for $-0.32<h/S_{q}<2.15$ , the increase in drag reduction is rapid since the increase in interface height exposes fewer asperities to the outer flow. The slip is enhanced due to a much larger area of air–water interface. In region $III$ beyond $h/S_{q}\geqslant 2.15$ , $DR$ hits a plateau and becomes insensitive to the interface height since it covers most of the asperities. $DR$ is therefore sensitive to $h$ in the vicinity of mean roughness $S_{q}$ . It is therefore evident from figure 8 that we can classify the interface cases into three distinct regions:

One can extract a representative case from each of the three regions shown above. The baseline smooth wall is denoted as Case L-S. For the rough wall, one can extract a representative case from each of the three regions shown above. The fully wetted case in region $I$ is denoted as Case L-RFW, the interface at the mean elevation height of the roughness $h=0$ denoted by Case L-RI1, the interface at $h=S_{q}$ in region $II$ is represented by Case L-RI2 and the interface at $h=S_{p}$ in region $III$ is represented by Case L-RI3. The velocity profiles are extracted and compared in figure 10. Case L-RFW exhibits an increase in velocity gradient when compared to Case L-S indicating an increase in drag. Once the interface is introduced, the effect is reversed and the velocity gradient decreases for Cases L-RI1, L-RI2 and L-RI3. The effect is more pronounced in Case L-RI3 since it corresponds to the interface being at the highest peak where most of the asperities are covered and the largest slip effect is achieved.

Figure 9.

A 2-D slice of the surface roughness (solid blue line) to highlight negative skewness about the reference line (dashed red line).

Figure 10.

Mean streamwise velocity $u$ as a function of the wall-normal distance $y$ , where $u$ and $h$ are normalised with $U_{\infty }$ and $H$ respectively.

Figure 11.

Gas fraction $\unicode[STIX]{x1D719}_{g}$ as a function of interface height $h$ normalised with the RMS roughness height $S_{q}$ . The red symbols represent the data and the solid blue line represents the nonlinear fit.

The increase in interface height reduces the amount of rough surface area exposed to the flow. As a result, the flow is subjected to an increase in slip area. The asperities exposed to the outer flow can be represented by a solid fraction $\unicode[STIX]{x1D719}_{s}$ , which is found by calculating the area of the rough surface above the interface normalised by the projected area of the bottom wall. It is evident that there exists a relationship between the interface height $h$ and gas fraction $\unicode[STIX]{x1D719}_{g}$ defined by $\unicode[STIX]{x1D719}_{g}=1-\unicode[STIX]{x1D719}_{s}$ . This is useful since $\unicode[STIX]{x1D719}_{g}$ is not known a priori and $h$ is prescribed as an initial condition. A simple nonlinear fit relates $\unicode[STIX]{x1D719}_{g}$ to $h/S_{q}$ of the form of $c[1+\tanh (ax+b)]$ and is described by the following equation:

This equation can be applied to any general rough surface: the coefficients do change for different surfaces, but the overall fit is general since any surface roughness can be represented by a bearing area curve (BAC). Region $I$ represents the index of the deepest valleys where the interface is retained, region $II$ represents the core interface retention index where the maximum amount of air is trapped within the cavities, and region $III$ the upper zone index related to the largest asperities that contribute to drag. Figure 11 shows a comparison between the actual data and (3.2) for $\unicode[STIX]{x1D719}_{g}$ as a function of $h/S_{q}$ . The negative skewness $S_{sk}=-0.32$ coincides with the transition between regions $I$ and $II$ .

Figure 12.

Laminar Couette flow: drag reduction $DR$ as a function of gas fraction $\unicode[STIX]{x1D719}_{g}$ .

Alternatively, $DR$ can be represented as a function of slip area instead of interface height by using $\unicode[STIX]{x1D719}_{g}$ as shown in figure 12. Based on the definition of $DR$ in (2.7), it can be shown that $DR$ is related to the slip length $b_{eff}$ by the following equation:

therefore,

Philip (Reference Philip1972a ,Reference Philip b ) obtained an analytic solution for the normalised slip lengths $b_{eff}/H$ for periodic grooves oriented parallel and perpendicular to the flow respectively:

Ybert et al. (Reference Ybert, Barentin, Cottin-Bizonne, Joseph and Bocquet2007) showed that for a post geometry, $b_{eff}$ scales with the solid fraction as

where $\unicode[STIX]{x1D6FC}$ is a prefactor that depends on the geometry. Davis & Lauga (Reference Davis and Lauga2010) were able to use superposition of point sources, where the infinite series is interpreted as a Riemann sum to obtain an analytical solution that agrees with Ybert et al. (Reference Ybert, Barentin, Cottin-Bizonne, Joseph and Bocquet2007) in the asymptotic limit of small surface coverage. A linear regression is performed on the numerically obtained data using the scaling given by (3.7) to obtain the following expression:

Figure 13.

Effective slip length $b_{eff}$ normalised with the channel height $H$ as a function of (a) the gas fraction $\unicode[STIX]{x1D719}_{g}$ and (b) the square root of gas fraction area $\unicode[STIX]{x1D719}_{g}$ . In (a), the solid black line is for longitudinal grooves, the solid dashed line for transverse grooves, the thick solid blue line for post geometry and the symbols for the random roughness. In (b), the solid blue line is the power-law fit and the symbols represent the random roughness.

Figure 13(a) shows a comparison between the different solutions obtained for the longitudinal and transverse grooves and posts to that of random roughness. The solution for post geometry best approximates the data for a random rough geometry for $\unicode[STIX]{x1D719}_{g}<0.85$ but starts to diverge as gas fraction increases. In the limit of large $\unicode[STIX]{x1D719}_{g}$ , the transverse groove solution captures the slip effect in random roughness more accurately. We present a power-law fit to the current data to obtain an expression for $b_{eff}$ as a function of $\unicode[STIX]{x1D719}_{g}$ :

This formula provides a simple expression for slip length over the rough surface given the amount of gas fraction present when $\unicode[STIX]{x1D719}_{g}>0$ and is shown in figure 13(b). The fully wetted roughness $b_{eff}$ remains an outlier.

3.2.1 Mean velocity profiles

Mean velocity profiles and Reynolds stresses are computed for five cases (T-S, T-RFW, T-RI1, T-RI2 and T-RI3). Figure 14(a) shows the mean velocity profile $U$ normalised by the average friction velocity $u_{\unicode[STIX]{x1D70F}}$ of each corresponding case, as a function of the wall-normal distance $y$ normalised by the channel half-height $\unicode[STIX]{x1D6FF}=L_{y}/2$ . The smooth channel (Case T-S) is also shown for reference. The presence of roughness (Case T-RFW) causes a small slip effect: it shows an insignificant decrease of 0.6 % in peak value of $U/u_{\unicode[STIX]{x1D70F}}$ and about a 3 % shift in its centreline location away from the rough wall. The presence of an interface (Case T-RI1) shows a further increase in slip, a slight increase in the centreline peak value of $U/u_{\unicode[STIX]{x1D70F}}$ and a 1 % shift in its location towards the SHS wall. The presence of an interface for Case T-RI1 adds just enough slip to offset the effect of roughness. The slip effect is more pronounced for Case T-RI2 and is largest for Case T-RI3 when the interface location covers all the roughness. The mean peak velocity $U/u_{\unicode[STIX]{x1D70F}}$ increases and the mean profile shifts towards the SHS wall by 5 % and 6 % for Cases T-RI2 and T-RI3 respectively when compared to Case T-S.

Figure 14.

Mean profile of (a) velocity normalised with the average friction velocity $u_{\unicode[STIX]{x1D70F}}$ as a function of wall-normal distance normalised by the channel half-height $\unicode[STIX]{x1D6FF}$ and (b) close-up near the wall region of the velocity normalised by the bottom wall friction velocity $u_{\unicode[STIX]{x1D70F}}^{B}$ as a function of wall-normal distance $y^{+}(u_{\unicode[STIX]{x1D70F}}^{B})$ . Symbols for each case are: Case T-S (▫), Case T-RFW (▵), Case T-RI1 (○), Case T-RI2 (♢), Case T-RI3 (▿). The symbols are not representative of the grid resolution.

Figure 14(b) shows a close-up view near the SHS wall of the mean velocity profile $U$ normalised by the bottom wall friction velocity $u_{\unicode[STIX]{x1D70F}}^{B}$ as a function of the wall-normal distance $y^{+}(u_{\unicode[STIX]{x1D70F}}^{B})$ . The scaling with $u_{\unicode[STIX]{x1D70F}}^{B}$ describes a more accurate picture in terms of slip and drag reduction. Case T-RFW shows a slip velocity at the wall due to the presence of roughness. The mean velocity profile is 5.8 % lower than the baseline case in the viscous wall region ( $y^{+}<50$ ), indicating an overall increase in drag. This effect is not apparent when the mean velocity profile is scaled with the average $u_{\unicode[STIX]{x1D70F}}$ . Case T-RI1 sees a further increase in slip at the wall due to the presence of an interface: the mean velocity profile is around 14 % higher within the viscous sublayer and extends into the buffer layer where the two velocity profiles of Case T-S and Case T-RI1 intersect at $y^{+}\sim 22$ . Cases T-RI2 and T-RI3 exhibit the largest slip at the wall: the two profiles intersect at $y^{+}\sim 28$ . The intersection of the velocity profiles is due to asymmetry caused by the slip effect which shifts the profile towards the SHS. This trend has been observed in Martell et al. (Reference Martell, Perot and Rothstein2009, Reference Martell, Rothstein and Perot2010) for longitudinal grooves and post geometries. In terms of scaling laws, the law of the wall $u^{+}=y^{+}+C$ still holds ( $C$ is the constant shift that represents the normalised slip velocity), with the exception of Case T-RFW and Case T-RI3. Case T-RFW shows a deviation from the law of the wall at $y^{+}<3$ , where the Reynolds stress is negligible compared to the viscous stress, but follows it for $3<y^{+}<10$ . This implies that shifting the profile by the normalised slip velocity $u_{s}^{+}$ will cause the profile to move further below the baseline case. This behaviour in the profile for the fully wetted case has been demonstrated in the literature (Yuan & Piomelli Reference Yuan and Piomelli2014). Case T-RI3 shows a complete deviation from the law of the wall due to the large amount of slip which modifies the slope of the velocity profile such that $u^{+}=\unicode[STIX]{x1D6FC}y^{+}+C$ where $\unicode[STIX]{x1D6FC}<1$ .

Figure 15.

Semi-log plot of the mean profile for (a) velocity normalised with the bottom wall friction velocity $u_{\unicode[STIX]{x1D70F}}^{B}$ of each respective case and (b) velocity shifted by the corresponding slip velocity all normalised by the bottom wall friction velocity $u_{\unicode[STIX]{x1D70F}}^{B}$ as a function of the wall-normal distance $y^{+}(u_{\unicode[STIX]{x1D70F}}^{B})$ . Symbols for each case are: Case T-S (▫), Case T-RFW (▵), Case T-RI1 (○), Case T-RI2 (♢), Case T-RI3 (▿). The symbols are not representative of the grid resolution.

A semi-log plot is shown in figure 15(a) where, unlike figure 14(a), the mean velocity profile $U$ is normalised with the bottom wall friction velocity $u_{\unicode[STIX]{x1D70F}}^{B}$ . The slip effect is more pronounced in the near-wall region and the difference in peaks are more apparent. The mean velocity profile of Case T-RFW shows a positive slip effect as mentioned earlier: the mean velocity profile is lower than Case T-S over all regions ranging from the viscous sublayer through the log law. The roughness reduces the overall mass flux, indicating an increase in drag. The profile shifts down and away from the wall by roughly 6 % from the centreline location of the baseline. The peak velocity at the centreline is lower than Case T-S by 2 %. Case T-RI1 shows a further increase in slip and the velocity profile is shifted further up indicating a drag reduction. The presence of an interface at that specific height is not enough to overcome the effect of roughness. This is evident from the velocity profile which is still 1 % below the baseline case. The profile shifts back towards the SHS by around 2 % from the centreline location of Case T-RFW. The near-wall slip seems to affect the velocity profile only within the viscous wall region for Cases T-RFW and T-RI1 when compared to Case T-S; however the log-law region shows a collapse in the data where the difference is within 2 %–3 %. Cases T-RI2 and T-RI3 show a large increase in slip near the wall with Case T-RI3 having the largest slip given that the roughness is fully covered by an interface. The peaks for those cases are 8 % and 10 % higher than Case T-S respectively. The profiles for both Cases T-RI2 and T-RI3 shift closer to the SHS wall such that their centrelines are 8.75 % and 11 % away from Case T-S respectively. Overall, the largest difference is clearly seen in the viscous wall region but the log-law region shows a collapse for Cases T-RI2 and T-RI3. This collapse holds until $y^{+}=40$ and the deviation in the slope of the log-law region in Case T-RI3 becomes more apparent for $y^{+}>50$ when compared to Case T-RI2. Overall, fully wetted roughness exhibits a decrease in mass flux whereas the presence of an interface increases mass flux. This is evident by the downward shift in the log-law region for Case T-RFW and an upward shift for Cases T-RI1, T-RI2 and T-RI3. We can conclude that the trend for an increase in mass flux (more fluid mass moving) directly correlates with drag reduction and vice versa. This is due to the fact that for all the cases, $Re_{\unicode[STIX]{x1D70F}}$ and pressure gradient are held constant.

It has been shown that for structured geometries (grooves and posts) not only does the gas fraction $\unicode[STIX]{x1D719}_{g}$ matter, but also the gap spacing (Ou & Rothstein Reference Ou and Rothstein2005; Daniello et al. Reference Daniello, Waterhouse and Rothstein2009; Martell et al. Reference Martell, Perot and Rothstein2009). For a random rough geometry with varying interface heights, the gas fraction is indeed increasing, but the gap spacing is also altered since more pockets are being filled with air. Notice that between Cases T-RI2 and T-RI3 there was not much increase in peak centreline velocity aside from the large increase in slip near the wall which does not substantially alter the total mass flux. However a significant increase in mass flux is observed between Cases T-RI1 and T-RI2. This can be attributed to the change in gap spacing. Although the gas fraction increases with increasing interface height, more gaps and surface valleys are covered up, since the surface roughness is dominated by valleys due to negative skewness. As the interface covers nearly all the valleys then a sharp increase in drag reduction is observed. Case T-RI1 is dominated by small gap size features which may be ineffective in reducing drag; this is in agreement with the literature on longitudinal grooves and posts.

The velocity profiles can be corrected by offsetting them with the slip velocity as shown in figure 15(b). Close to the wall we see a good collapse in the viscous sublayer with an early departure $y^{+}>1$ from the $u^{+}=y^{+}$ . With the roughness fully covered by the interface in Case T-RI3, a large deviation from the law of the wall is observed when compared to other cases. This implies that we should expect the structures of the wall-normal turbulence to remain intact for Cases T-RFW, T-RI1 and T-RI2, and Case T-RI3 to be fundamentally different. Away from the near wall, the relative velocity $(U-u_{s})/u_{\unicode[STIX]{x1D70F}}^{B}$ decreases with roughness and increasing interface height (equivalently with increasing $\unicode[STIX]{x1D719}_{g}$ ). This trend has also been observed in Türk et al. (Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014) for structured geometries. The profile in the log-law region is given by

where $\unicode[STIX]{x1D705}$ is the von Kármán constant and $B$ is the intercept. The value of $\unicode[STIX]{x1D705}$ decreases from $0.41$ to $0.38$ for Case T-RI3 but remains the same for the other cases. The value of $B$ decreases with increasing $\unicode[STIX]{x1D719}_{g}$ , going from $B\approx 5.5$ to $B\approx 2$ for Case T-RI2 and $B\approx -4$ for Case T-RI3. Similar trends have been observed in the literature (Busse & Sandham Reference Busse and Sandham2012; Yuan & Piomelli Reference Yuan and Piomelli2014; Busse, Thakkar & Sandham Reference Busse, Thakkar and Sandham2017). The decrease in $B$ implies an increase in friction as discussed in the literature through surface manipulation (Luchini, Manzo & Pozzi Reference Luchini, Manzo and Pozzi1991; Jiménez Reference Jiménez1994; Garcia-Mayoral & Jiménez Reference Garcia-Mayoral and Jiménez2011). Therefore a decrease in $B$ is associated with an increase in friction due to roughness and a decrease in friction due to SHS, and in order to differentiate the two, we consider the following argument. This increase in friction that is typically associated with surface roughness is offset by the drag reducing slip velocity $u_{s}$ due to the presence of an interface in SHS. Roughness induces slip, but the velocity profile in the viscous wall region does not necessarily follow the law of the wall. If we were to compare it to a smooth channel, the profile of the rough channel would intersect the profile of the smooth channel somewhere in the viscous sublayer ( $y^{+}<5$ ) to merge into the log region, which is shifted below the baseline indicating an increase in drag. If we offset the profile by the slip velocity, the near-wall region does not collapse but moves further below. If we take the SHS with structured geometries, the interface is flush with the top location of the roughness. The slip effect is also present but the velocity profile in the viscous wall region obeys the law of the wall with some offset $u^{+}=y^{+}+C$ ; therefore if the velocity profile is shifted by the slip velocity, then a collapse in the near-wall region is observed. What we see in our simulation is somewhere in between. This is simply due to the fact that even when we have an interface, some roughness protrudes. Our analysis shows a combination of both behaviours where the law of the wall holds to a certain extent in the viscous sublayer before any appreciable deviation is observed. Also, with increasing interface height, the profile in the viscous wall region tends to move closer to the baseline case and away from the fully wetted roughness, with the exception of Case T-RI3 in which the log-law region seems to extend to the vicinity of the wall, disrupting the near-wall cycle. In general, rough surfaces tend to shift both the law of the wall and the logarithmic region away from baseline while SHS tend to shift the law of the wall closer and simultaneously moving the logarithmic region away from baseline.

3.2.2 Scaling laws

Figure 16(a) shows the slip effect as a function of $\unicode[STIX]{x1D719}_{g}$ . The slip velocity $u_{s}$ is normalised by the mean bulk velocity $U_{b}$ for each corresponding case. As $\unicode[STIX]{x1D719}_{g}$ increases, the slip effect is more pronounced: $u_{s}$ exhibits a steady increase. Slip velocities can reach as much as 68 % of the bulk velocities which can obscure some of the effects due to roughness. This is another reason why it would be important to plot the mean profile offset by the slip as shown in figure 15(b). $DR$ (%) as a function of $u_{s}$ is not an accurate predictor of drag reduction since fully wetted roughness increases drag while achieving positive slip velocities as shown in figure 16(a). The slip length $b_{s}$ normalised by the average roughness gap width $L$ is reported in figure 16(b) as a function of $\unicode[STIX]{x1D719}_{g}$ . The average roughness gap $L$ is not a straightforward property to obtain since there is no periodicity (typically associated with structured geometry such as grooves and posts). We use a two-step process to calculate $L$ . First we find the profile peak count HSC, the number of profile peaks that exceed a pre-selected threshold (e.g. arithmetic mean elevation of the roughness), calculated over the entire streamwise length for all the spanwise slices. Second we calculate the mean peak spacing $S_{m}$ , the mean spacing between profile peaks, averaged over all the spanwise slices to obtain $L$ . We refer the reader to figure 28 in appendix B for an example of profile peak count. There is a good agreement between our data and a fit based on the scaling presented by Ybert et al. (Reference Ybert, Barentin, Cottin-Bizonne, Joseph and Bocquet2007) for posts given by the following equation:

Figure 16.

Mean flow field properties as a function of $\unicode[STIX]{x1D719}_{g}$ . (a) Percentage slip velocity $u_{s}$ normalised by the mean bulk velocity $U_{b}$ for each corresponding case, (b) slip length $b_{s}$ normalised by the average gap width of the roughness $L$ where the solid blue line represents the scaling law of Ybert et al. (Reference Ybert, Barentin, Cottin-Bizonne, Joseph and Bocquet2007) and the dashed black line that of Sbragaglia & Prosperetti (Reference Sbragaglia and Prosperetti2007a ), (c) percentage change in bulk velocity $\unicode[STIX]{x0394}U_{b}$ normalised by the mean bulk velocity $U_{b,0}$ of the baseline case, (d) percentage shear stress reduction $\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}_{w}$ based on the ratio of the bottom wall $\unicode[STIX]{x1D70F}_{w}^{B}$ to the average channel $\unicode[STIX]{x1D70F}_{w}$ and (e) correlation between the percentage shear stress reduction and percentage change in bulk velocity compared to a linear fit (solid blue line).

The solution by Sbragaglia & Prosperetti (Reference Sbragaglia and Prosperetti2007a ) which does not require a fit gives good agreement for $\unicode[STIX]{x1D719}_{g}<0.7$ . Slip length is not a good indicator of $DR$ since both fully wetted roughness and SHS produce a positive $b_{s}$ . As mentioned earlier during the discussion of the mean velocity profiles, overall mass flux appears to be a good indicator for drag reduction given that our channels are run at a constant $Re_{\unicode[STIX]{x1D70F}}$ and pressure gradient. Drag reduction would imply that more fluid mass is moving for the same conditions, i.e. a larger mass flux and a larger change in bulk velocity; the opposite is also true. Figure 16(c) shows the percent change in mean bulk velocity $\unicode[STIX]{x0394}U_{b}$ normalised by $U_{b,0}$ of the baseline Case T-S. A negative change in $\unicode[STIX]{x0394}U_{b}$ indicates an increase in drag, i.e. a lower mass flux. Once an interface is present (Case T-RI1) drag is reduced but is not enough to offset the effect roughness since $\unicode[STIX]{x0394}U_{b}$ is still negative. As the interface height increases, $\unicode[STIX]{x0394}U_{b}$ increases to become a positive value indicating an increase in mass flux and a larger $DR$ effect. This description is exactly what we see when compared to the analysis of the mean velocity profile $U/u_{\unicode[STIX]{x1D70F}}^{B}$ done earlier. $\unicode[STIX]{x0394}U_{b}$ does not exhibit a steady increase and plateaus for large values of $\unicode[STIX]{x1D719}_{g}$ ; this was also observed by Türk et al. (Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014). The reduction in the shear stress on the bottom wall $\unicode[STIX]{x1D70F}_{w}^{B}$ compared to the average shear stress of the channel $\unicode[STIX]{x1D70F}_{w}$ provides a straightforward result for $DR$ as shown in figure 16(d). Note the increase in drag by around 2.5 % for Case T-RFW. The roughness has $S_{q}^{+}\approx 1.6$ which is in the hydrodynamically smooth regime. Busse et al. (Reference Busse, Thakkar and Sandham2017) reported that the surface property $S_{z5\times 5}^{+}$ is a more suitable measure of the sand-grain roughness $k^{+}$ where $S_{z5\times 5}^{+}\approx 11$ for our surface. This explains why we see a drag increase for the current configuration. A reduction of 1 % in drag is obtained once an interface is introduced in Case T-RI1. A 7 % $DR$ is achieved for Case T-RI2 and peaks at 8 % for Case T-RI3. The results are in agreement with the experimental results of Ling et al. (Reference Ling, Srinivasan, Golovin, McKinley, Tuteja and Katz2016) obtained for turbulent boundary layers over SHS at higher $Re_{\unicode[STIX]{x1D70F}}$ , where $k_{rms}^{+}\approx 0.45$ – $0.75$ and $DR$ ranged from 9 % to 12 %. One can therefore reasonably assume that the change in shear stress and the change in bulk velocity correlate well with each other. Figure 16(e) shows $\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}_{w}$ as a function of $\unicode[STIX]{x0394}U_{b}$ compared to a linear fit given by

The roughness function $\unicode[STIX]{x0394}U^{+}$ depends on the three-dimensional topography of a surface where Bradshaw (Reference Bradshaw2000) suggested a scaling given by $\unicode[STIX]{x0394}U^{+}\sim (k_{rms}^{+})^{\unicode[STIX]{x1D6FC}}$ . Since the interface height essentially modifies $k_{rms}^{+}$ , then a similar scaling argument can be made for $\unicode[STIX]{x0394}U^{+}$ as a function of $b_{s}^{+}$ . Figure 17(a) shows a good agreement with a power-law behaviour where the roughness function goes to zero as the slip length diminishes. The power-law formula obtained from fitting the data is given by

A semi-analytical formula by Ybert et al. (Reference Ybert, Barentin, Cottin-Bizonne, Joseph and Bocquet2007) relates the slip velocity to the cavity width, in terms of wall units $u_{s}^{+}=C_{Y}L^{+}$ . We simply use the slip length $b_{s}^{+}$ instead since slip is a direct consequence of $L^{+}$ . Figure 17(b) shows the linear approximation near the wall for $C_{Y}=0.65$ where

The linear scaling deviates for $b_{s}^{+}>10$ which was also shown for $L^{+}>10$ in Seo et al. (Reference Seo, Garcia-Mayoral and Mani2015) for a value of $C_{Y}=0.535$ . This is expected since the linear relationship is based on Stokes’ flow which becomes less accurate as slip increases. It is worth noting that by definition $u_{s}^{+}=b_{s}^{+}$ for the cases where the air–water interface is aligned with the top of the roughness and where the total stress is equal to the viscous stress. However, as discussed in Ling et al. (Reference Ling, Srinivasan, Golovin, McKinley, Tuteja and Katz2016), $u_{s}^{+}<b_{s}^{+}$ for the cases when $-(u^{\prime }v^{\prime })^{+}>0$ or when the total stress $\unicode[STIX]{x1D70F}_{t}^{+}>\unicode[STIX]{x1D707}(\text{d}U/\text{d}y)^{+}$ due the random nature of the roughness and the variable interface heights.

Figure 17.

Roughness function $\unicode[STIX]{x0394}U^{+}$ as a function of slip length $b_{s}^{+}$ is shown in (a) and is compared to a power-law fit (solid blue line) and (b) shows the slip velocity scaling $u_{s}^{+}$ as a function of $b_{s}^{+}$ with a comparison to a linear approximation (solid black line).

3.2.3 Reynolds stresses

The Reynolds stress profiles are shown for the different cases as a function of the wall-normal distance. Figure 18(a) shows the streamwise component of the Reynolds stress normalised by $u_{\unicode[STIX]{x1D70F}}$ . Case T-RFW seems to slightly increase the peak of $u^{\prime }u^{\prime }$ while simultaneously shifting it away from the SHS wall. The presence of an interface seems to dampen that effect where Case T-RI1 shows a decrease in the peak while shifting it towards the SHS wall. This effect is further amplified in Case T-RI2 where the peak is clearly damped and the profile moves closer to the wall. Case T-RI3 exhibits the largest slip and hence is closest to the SHS wall where there exists a sharp rise in the streamwise component; however the peak is larger than the other cases, breaking the symmetry of the profile completely. The overall trend in damping the peak of $u^{\prime }u^{\prime }$ while shifting it closer to the slip wall has been observed in the literature for longitudinal grooves and posts. For reference, $u^{\prime }u^{\prime }$ is scaled by $u_{\unicode[STIX]{x1D70F}}^{B}$ in figure 18(b). We get a reasonable collapse in the profiles near the SHS wall with the exception of Case T-RI3 indicating a change in the near-wall behaviour. Figure 18(c) shows a $\log$ plot of $u^{\prime }u^{\prime }$ as a function of the wall-normal distance $y^{+}(u_{\unicode[STIX]{x1D70F}}^{B})$ in wall units scaled by $u_{\unicode[STIX]{x1D70F}}^{B}$ . Large differences are observed in the near-wall region due to the slip effect. The shift away from the SHS wall for Case T-RFW and the shift towards the SHS wall for Cases T-RI1 and T-RI2 are more evident.

Figure 18.

Streamwise component of the normal Reynolds stress normalised by (a) the average friction velocity $u_{\unicode[STIX]{x1D70F}}^{2}$ as a function of $y/\unicode[STIX]{x1D6FF}$ , (b) the bottom wall friction velocity $(u_{\unicode[STIX]{x1D70F}}^{B})^{2}$ as a function of $y/\unicode[STIX]{x1D6FF}$ and (c) the bottom wall friction velocity $(u_{\unicode[STIX]{x1D70F}}^{B})^{2}$ as a function of $y^{+}(u_{\unicode[STIX]{x1D70F}}^{B})$ on a $\log$ scale. Symbols for each case are: Case T-S (▫), Case T-RFW (▵), Case T-RI1 (○), Case T-RI2 (♢), Case T-RI3 (▿). The symbols are not representative of grid resolution.

Figure 19.

Wall-normal component of the normal Reynolds stress normalised by (a) the average friction velocity $u_{\unicode[STIX]{x1D70F}}^{2}$ as a function of $y/\unicode[STIX]{x1D6FF}$ , (b) the bottom wall friction velocity $(u_{\unicode[STIX]{x1D70F}}^{B})^{2}$ as a function of $y/\unicode[STIX]{x1D6FF}$ and (c) the bottom wall friction velocity $(u_{\unicode[STIX]{x1D70F}}^{B})^{2}$ as a function of $y^{+}(u_{\unicode[STIX]{x1D70F}}^{B})$ on a $\log$ scale. Symbols for each case are: Case T-S (▫), Case T-RFW (▵), Case T-RI1 (○), Case T-RI2 (♢), Case T-RI3 (▿). The symbols are not representative of grid resolution.

Figure 20.

Spanwise component of the normal Reynolds stress normalised by (a) the average friction velocity $u_{\unicode[STIX]{x1D70F}}^{2}$ as a function of $y/\unicode[STIX]{x1D6FF}$ , (b) the bottom wall friction velocity $(u_{\unicode[STIX]{x1D70F}}^{B})^{2}$ as a function of $y/\unicode[STIX]{x1D6FF}$ and (c) the bottom wall friction velocity $(u_{\unicode[STIX]{x1D70F}}^{B})^{2}$ as a function of $y^{+}(u_{\unicode[STIX]{x1D70F}}^{B})$ on a $\log$ scale. Symbols for each case are: Case T-S (▫), Case T-RFW (▵), Case T-RI1 (○), Case T-RI2 (♢), Case T-RI3 (▿). The symbols are not representative of grid resolution.

The wall-normal Reynolds stress component $v^{\prime }v^{\prime }$ is shown in figure 19. Initially $v^{\prime }v^{\prime }$ is normalised by $u_{\unicode[STIX]{x1D70F}}$ as shown in figure 19(a). Case T-RFW amplifies the peak stress and shifts it away from the slip wall. Comparing it to Case T-RI2, it is evident that the interface has a damping effect. It is interesting to note that once the interface was introduced initially, Case T-RI1 showed a further amplification in the peak from Case T-RFW and not the opposite. This is likely to be due to the fact that the interface height barely covers any of the roughness, and therefore the inhomogeneity between slip and no-slip due to the interface and random rough patches cause larger fluctuations. This is of course damped out once the interface covers more of the surface and larger slip areas are present. Case T-RI3 shows the largest shift towards the SHS wall as expected but the peak stress does not not seem to follow any further damping effect with increasing slip area. Figure 19(b) shows $v^{\prime }v^{\prime }$ scaled by $u_{\unicode[STIX]{x1D70F}}^{B}$ and no collapse in the data is observed. The $\log$ plot in figure 19(c) shows the velocities going to zero since at the interface an infinite surface tension is assumed. Therefore the wall-normal velocity $v_{N}=0$ at the interface and a no-slip boundary condition is applied elsewhere over the rough surface. Note that the profiles of Case T-RI1 and Case T-RI2 intersect around $y^{+}\approx 28$ where Case T-RI2 exhibits larger wall-normal stresses for $y^{+}<28$ and then Case T-RI1 tends to become larger for the region above that. Case T-RFW does not exhibit an observable shift away from the wall while it remains clear that when an interface is present, the shift towards the SHS wall remains considerable.

The streamwise component of Reynolds stress is shown in figure 20, which follows a similar trend in the behaviour to that described above. Case T-RFW shows an increase in $w^{\prime }w^{\prime }$ when scaled by $u_{\unicode[STIX]{x1D70F}}$ as shown in figure 20(a). The addition of an interface (Case T-RI1) further increases the peak in $w^{\prime }w^{\prime }$ . The same reasoning applies to the spanwise component as described earlier for the wall-normal component since the roughness has no preferential direction. Therefore the interplay between slip and no-slip due to the interface and the protruding roughness holds here too. Similarly as soon as the interface covers a large portion of the roughness as in Case T-RI2, the peak is damped out. Case T-RI3 is highly skewed towards the rough wall with a peak velocity that is much larger than the rest of the cases. Similar to what was observed in the wall-normal stresses, Cases T-RI1 and T-RI2 intersect each other as shown in figure 20(c). The location however is smaller, where $y^{+}\approx 13$ . Anywhere below that, Case T-RI2 exhibits higher stresses than Case T-RI1 and the opposite holds true when $y^{+}>13$ . The spanwise slip is evident due to the presence of the interface where Case T-RI3 still exhibits the largest slip effect.

The Reynolds shear stress is an important quantity to examine since its behaviour is closely related to turbulence levels and the structure of the near-wall turbulence. Figure 21(a) shows how the presence of an interface (Case T-RI2) reduces shear while the presence of roughness enhances it. Note that since the wall-normal shear stress tends to increase when an interface is introduced (Case T-RI1), that effect translates here where we see a further enhancement in mixing instead of a reduction in the peak. Similar to the discussion above, it is not until the interface covers a large portion of the roughness that the damping effect takes place. Therefore it is evident that there are competing effects between the interface suppressing vertical velocity fluctuations and the asperities doing the opposite by enhancing them. Case T-RI3 exhibits a large gradient near the SHS wall since the wall-normal velocity is zero yet the streamwise component sees a large slip effect as was shown in figure 18(a). As we move away from the wall, the shear stress has to balance out with the top wall given that we are running a constant pressure gradient, which explains why the profiles are parallel in that region. Figure 21(b) shows the $u^{\prime }v^{\prime }$ component normalised by $u_{\unicode[STIX]{x1D70F}}^{B}$ , and a good collapse of the data is observed with the exception of Case T-RI3. This indicates that, overall, the near-wall turbulence is not fundamentally changed for Cases T-RFW, T-RI1 and T-RI2 whereas the turbulent structures of Case T-RI3 are different. Figure 21(c) shows $-u^{\prime }v^{\prime }$ plotted on a $\log$ scale in wall units (normalised by $u_{\unicode[STIX]{x1D70F}}^{B}$ ) along with the viscous stress $\unicode[STIX]{x1D707}(\text{d}U/\text{d}y)$ and total stress $\unicode[STIX]{x1D70F}_{t}$ which shows the total shear stress budget. We can see the cases where $-u^{\prime }v^{\prime }>0$ and where $\unicode[STIX]{x1D70F}_{t}>\unicode[STIX]{x1D707}(\text{d}U/\text{d}y)$ leading to $u_{s}^{+}<b_{s}^{+}$ as discussed earlier. Case RFW seems to collapse onto the baseline while Cases T-RI1 and T-RI2 shift towards the slip wall. The shear stresses go to zero near the wall due to the infinite surface tension that keeps the interface flat.

Figure 21.

Reynolds shear stress normalised by (a) the average friction velocity $u_{\unicode[STIX]{x1D70F}}^{2}$ as a function of $y/\unicode[STIX]{x1D6FF}$ , (b) the bottom wall friction velocity $(u_{\unicode[STIX]{x1D70F}}^{B})^{2}$ as a function of $y/\unicode[STIX]{x1D6FF}$ . (c) The viscous stress, Reynolds shear stress and total stress normalised by the bottom wall friction velocity $(u_{\unicode[STIX]{x1D70F}}^{B})^{2}$ as a function of $y^{+}(u_{\unicode[STIX]{x1D70F}}^{B})$ on a $\log$ scale. Symbols for each case are: Case T-S (▫), Case T-RFW (▵), Case T-RI1 (○), Case T-RI2 (♢), Case T-RI3 (▿). In (c) the solid line represents total stress $\unicode[STIX]{x1D70F}_{t}^{+}$ , dashed lines the viscous stress $\unicode[STIX]{x1D707}(\text{d}U/\text{d}y)^{+}$ and dashed-dotted lines the Reynolds shear stress $-(u^{\prime }v^{\prime })^{+}$ . The symbols are not representative of grid resolution.

Earlier we discussed how $u_{s}$ is not an accurate predictor of $DR$ since both fully wetted roughness and SHS result in a positive slip velocity. $\unicode[STIX]{x0394}U_{b}$ on the other hand correlates with $\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}_{w}$ . One can show that the Reynolds shear stress is tied to the change in bulk velocity which is manifested in additional turbulent losses. This is done by applying a triple integration to the averaged transport equation for the streamwise momentum equation. This was demonstrated in Hasegawa, Frohnapfel & Kasagi (Reference Hasegawa, Frohnapfel and Kasagi2011) and Türk et al. (Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014) where the following identities are obtained:

For the baseline case (T-S) where we have no-slip walls:

This leads to the final form given by the following:

where $u_{0}^{\prime }v_{0}^{\prime }$ denotes the shear stress of the baseline case. Take Case T-RFW as an example. The fully wetted roughness enhances vertical velocity fluctuations and so does having an interface at a small height location (Case T-RI1). There $-u^{\prime }v^{\prime }>-u_{0}^{\prime }v_{0}^{\prime }$ and therefore $\int _{0}^{\unicode[STIX]{x1D6FF}}(1-y/\unicode[STIX]{x1D6FF})(-u^{\prime }v^{\prime }+u_{0}^{\prime }v_{0}^{\prime })>0$ which happens also to be larger than the $u_{s}$ caused by the roughness or the presence of the interface. This gives $\unicode[STIX]{x0394}U_{b}<0$ , indicating an increase in drag. For Cases T-RI2 and T-RI3, $u^{\prime }v^{\prime }$ is damped when compared to baseline. Therefore $-u^{\prime }v^{\prime }<-u_{0}^{\prime }v_{0}^{\prime }$ and the integral term ends up coming out to be negative which results in $\unicode[STIX]{x0394}U_{b}>0$ and hence drag reduction. One implication of this result is that a superhydrophobic surface might fail in the sense of reducing drag at high pressure in spite of the interface itself being stable.

3.2.4 Flow structures

The near-wall shear stress $u^{\prime }v^{\prime }$ collapsed when scaled with the bottom wall shear velocity indicating that the near-wall behaviour remains unchanged with the exception of Case T-RI3. In this section we look more closely at the near-wall region. Figure 22(a–d) shows instantaneous pressure fluctuations in wall units where $p^{+}=p/(u_{\unicode[STIX]{x1D70F}}^{B})^{2}$ . Figure 22(a) shows the pressure contours along the rough wall while figures 22(b)–22(d) are taken at the interface location (the reference plane) for each case. The axes are shown in wall units scaled by $u_{\unicode[STIX]{x1D70F}}^{B}/\unicode[STIX]{x1D708}$ . The contours change noticeably from figures 22(a) to 22(d). The instantaneous behaviour in pressure for figures 22(a) and 22(d) shows similar large length scales for fully wetted and fully covered roughness on the order of $100\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$ . Larger pressure intensities are visible in figure 22(d) due to slip at the interface. Local contribution due to the presence of protruding roughness elements is observed in figures 22(b) and 22(c). Asperities cause stagnation in front of them as observed by the high pressure values. Low pressure values are seen in their wake. This effect is reduced as more asperities are covered by the interface as shown in figure 22(c). This behaviour was also observed in Seo et al. (Reference Seo, Garcia-Mayoral and Mani2015).

Figure 22.

Instantaneous contours of $p^{+}$ for (a) Case T-RFW on the roughness, (b) Case T-RI1, (c) Case T-RI2 and (d) Case T-RI3 on the interface location. Range of the contours is from $-5$ to $5$ in wall units.

Figure 23.

Time-averaged contours of $p^{+}$ at $y^{+}=15$ for (a) Case T-RFW, (b) Case T-RI1, (c) Case T-RI2 and (d) Case T-RI3. Range of contours is from $-2$ to $2$ in wall units.

Figure 24.

Instantaneous greyscale contours of $u/U_{b}$ at $y^{+}=15$ for (a) Case T-RFW, (b) Case T-RI1, (c) Case T-RI2 and (d) Case T-RI3. Range of contours is from $0.2$ to $1$ .

We extract slices in the wall-normal plane such that $y^{+}=15$ and examine the time-averaged pressure distribution. Figure 23(a–d) shows striking visual difference between Case T-RFW, Cases T-RI1 and T-RI2, and Case T-RI3. The mean pressure is zero for a smooth channel; this is only observed in figure 23(d) when the interface fully covers the roughness. Figure 23(a) shows a variation in the mean pressure due to the signature of the rough wall. The contribution of stagnation pressure at the interface is overlaid with the contribution of asperities. Large variations in mean pressure interspersed across the domain are observed in figures 23(b) and 23(c) due to the interaction between the flow at the interface and around the asperities.

Since the pressure footprint can be related to near-wall velocity, we plot the instantaneous velocity $u$ normalised by $U_{b}$ in greyscale on the same $y$ -slice. The presence of an interface affects the spanwise streak motion, and the distance between streaks can be visually seen to be around $100$ wall units in figure 24(a). Figure 24(b) shows regions with larger streamwise velocities where the separation distance between streaks becomes larger than $100$ wall units in the spanwise direction. The slightly larger distance between the streaks is maintained in Case T-RI2 but that coherent structure is completely destroyed in Case T-RI3 as shown in figure 24(d). This behaviour for Case T-RI3 is likely to be due to the lower near-wall shear yielding lower values of $Sq^{2}/\unicode[STIX]{x1D716}$ where $S$ , $q^{2}$ and $\unicode[STIX]{x1D716}$ are the mean shear, twice the turbulence kinetic energy and turbulent dissipation respectively. As shown by Rogers & Moin (Reference Rogers and Moin1987) and Lee, Kim & Moin (Reference Lee, Kim and Moin1990) for homogeneous shear flow, only high values of $Sq^{2}/\unicode[STIX]{x1D716}$ as encountered in the near-wall region produce streaks.

Figure 25.

Instantaneous contours of $\unicode[STIX]{x1D714}_{y}^{+}$ at $y^{+}=15$ for (a) Case T-RFW, (b) Case T-RI1, (c) Case T-RI2 and (d) Case T-RI3. Range of contours is from $-0.5$ to $0.5$ in wall units.

Figure 26.

Illustration of the near-wall vortex structure. The asymmetry in slip due to the random roughness is illustrated in (a) whereas (b) shows the effect of having an interface cover most of the roughness and its effect on slip and induced secondary flow interaction.

The instantaneous wall-normal vorticity $\unicode[STIX]{x1D714}_{y}^{+}$ is shown in figure 25(a–d) to illustrate pairs of counter-rotating vortices in regions of low- and high-momentum fluid streaks. The $\unicode[STIX]{x1D714}_{y}^{+}$ contours show similar behaviour to the streamwise velocity with varying interface height. A striking feature in the $\unicode[STIX]{x1D714}_{y}^{+}$ plots is the slight asymmetry of counter-rotating vortex pairs. Figure 26(a) schematically shows how, due to the random nature of the slip no-slip behaviour over rough SHS, each vortex in a counter-rotating vortex pair experiences different slip areas, producing an inhomogeneous spanwise slip $w_{s}$ . This causes asymmetric velocity profiles to interact in the wall-normal direction. As the interface height increases, the slip area increases and less of the solid protrudes, and therefore $w_{s}$ becomes more homogeneous. The slip effect is however more amplified in this case (figure 26 b) and the spanwise velocity is much larger, penetrating further up in the wall-normal direction, with tertiary vortices set up above the pair of counter-rotating vortices. This could possibly explain why in figures 24(d) and 25(d) we observe less coherence and more violent mixing of those structures. While such mixing is associated with drag increase, the slip velocity in the streamwise direction is more dominant which offsets this deleterious effect to give a net positive $DR$ . This is explained by the change in bulk velocity as a function of slip velocity and the integral difference of shear stresses given by (3.17).