4.1. The magnitude of DR in laminar and turbulent SH channel flow

Figure 2(a) shows the DRs obtained in the present study in laminar and turbulent SH channel flow with longitudinal MG, transverse MG and micro-posts, and $g/w=$ $1$ , $7$ , $15$ , as a function of the geometric parameter $(g+w)/2H$ . Also shown are the DNS data of Park et al. (Reference Park, Park and Kim2013), obtained in turbulent channel flow at $Re_{{\it\tau}_{0}}\approx 180$ , $395$ , $590$ with longitudinal MG and $g/w\approx$ $1$ , $3$ , $7$ , $16$ , and the DNS data of Türk et al. (Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014) in turbulent channel flow at $Re_{{\it\tau}_{0}}\approx 180$ with longitudinal MG and $g/w=1$ . The DRs for the DNS of Türk et al. (Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014) were estimated from the quantities reported in their paper using the procedures described in the supplementary data. For each category of SH surface micro-pattern, the DNS data show a trend for increase in DR with increasing $(g+w)/2H$ or $g/w$ , at a given $Re_{{\it\tau}_{0}}$ . The DNS data of Park et al. (Reference Park, Park and Kim2013) also show a trend for increase in DR with increasing $Re_{{\it\tau}_{0}}$ for longitudinal MG. For each category of SH surface micro-pattern, the DRs in laminar flow follow similar trends to those in turbulent flow. However, for a given SH surface micro-pattern, the magnitude of DR in turbulent flow is higher than that in laminar flow, by $143\,\%$ to $267\,\%$ with longitudinal MG, and by $10\,\%$ to $61\,\%$ with micro-posts, as shown in figure 2(a) and table 1. These trends are reversed with transverse MG, for which the DRs in turbulent flow are always lower than the corresponding cases in laminar flow, and in some cases even negative, indicating a drag increase (DI) in the presence of the SH surface. The significantly higher DRs observed in turbulent flow compared to laminar flow with longitudinal MG has led a number of investigators (Daniello et al. Reference Daniello, Waterhouse and Rothstein2009; Martel et al. Reference Martel, Rothstein and Perot2010; Rothstein Reference Rothstein2010; Park et al. Reference Park, Park and Kim2013) to conclude that the enhanced DR observed in turbulent flow must be due to mechanisms beyond the effective slip on the wall, such as modifications to the turbulence dynamics.

Figure 2.

Scaling of DR and $U_{s}/U_{b}$ in laminar and turbulent channel flow: (a) DR as a function of the geometric parameter $(g+w)/2H$ for different $g/w$ and $Re_{{\it\tau}_{0}}$ ; (b) DR as a function of the average slip velocity $U_{s}/U_{b}$ ; (c) average slip velocity on the slip surfaces, $(1+1/SFF)U_{s}/U_{b}$ , as a function of $g/h$ ; (d) DR as a function of the slip length $L_{s}^{+0}$ . ●, ▪, ◆ $\,$ , present study, $Re_{{\it\tau}_{0}}\approx 223$ , longitudinal MG, $g/w=1$ , $7$ , $15$ , respectively; ▶, ◀ $\,$ , present study, $Re_{{\it\tau}_{0}}\approx 223$ , transverse MG, $g/w=1$ , $7$ , respectively; ▲, ▼, present study, $Re_{{\it\tau}_{0}}\approx 223$ , micro-posts, $g/w=1$ , $7$ , respectively; ◯, ▫, ♢, present study, $Re_{b}=50$ , longitudinal MG, $g/w=1$ , $7$ , $15$ , respectively; ▷, ◁ $\,$ , present study, $Re_{b}=50$ , transverse MG, $g/w=1$ , $7$ , respectively; △, ▽, present study, $Re_{b}=50$ , micro-posts, $g/w=1$ , $7$ , respectively;

,  

,  

, DNS of Park et al. (Reference Park, Park and Kim2013),  $Re_{{\it\tau}_{0}}\approx 180$ , $g/w=1$ , 3, 7, respectively;  

,  

,  

,   

, DNS of Park et al. (Reference Park, Park and Kim2013),  $Re_{{\it\tau}_{0}}\approx 395$ , $g/w=1$ , 3, 7, 15, respectively;   

,  

,  

,  

DNS of Park et al. (Reference Park, Park and Kim2013),  $Re_{{\it\tau}_{0}}\approx 590$ , $g/w=1$ , 3, 7, 15, respectively;  

, DNS of Türk et al. (Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014),  $Re_{{\it\tau}_{0}}=180$ , $g/w=1$ ;  

, experiments of Daniello et al. (Reference Daniello, Waterhouse and Rothstein2009),  $2450\leqslant Re_{b}\leqslant 4000$ , $g=w=30~{\it\mu}\text{m}$ , MG on one wall; ——, $DR=U_{s}/U_{b}$ in (b), and $DR=L_{s}^{+0}/[1+(Re_{b}/Re_{{\it\tau}_{0}})]$ with $Re_{b}=3600$ , $Re_{{\it\tau}_{0}}\approx 223$ in (d); $\boldsymbol{-}\boldsymbol{\cdot }\boldsymbol{-}$ ,  $\boldsymbol{-}\boldsymbol{\cdot }\!\boldsymbol{\cdot }\,\boldsymbol{-}$ , analytical expressions of Fukagata et al. (Reference Fukagata, Kasagi and Koumoutsakos2006) for uniform streamwise slip and uniform combined slip, respectively, at $Re_{{\it\tau}_{0}}\approx 223$ .

However, when the same DR data is plotted as a function of $U_{s}/U_{b}$ , as shown in figure 2(b), the DRs from all DNS databases in laminar and turbulent flow, for all SH surface micro-patterns, nearly collapse and cluster around the line $DR=U_{s}/U_{b}$ . In laminar flow, the magnitude of DR exactly matches $U_{s}/U_{b}$ for all SH surface micro-patterns, as shown in table 1, confirming that DR in laminar flow is entirely due to the effective slip on the walls. In turbulent flow, the magnitude of DR either falls slightly above $U_{s}/U_{b}$ , as with longitudinal MG or micro-posts with high SFF, or slightly below $U_{s}/U_{b}$ , as with transverse MG or micro-posts with moderate SFF ratio. These reflect the effect of the $O({\it\varepsilon})$ term in (2.5), and indicate that changes to the structure and dynamics of turbulence within the flow can lead to either a further DR or a DI beyond the DRs resulting from the effective slip on the walls. For the former, the modifications to the turbulence dynamics lead to a normalized Reynolds shear stress, $-\langle \overline{uw}\rangle ^{+}$ , in the SH channel which is lower than that in the base flow, thus making ${\it\varepsilon}$ positive. For the latter, the modifications to the turbulence dynamics lead to a normalized Reynolds shear stress, $-\langle \overline{uw}\rangle ^{+}$ , in the SH channel which is slightly higher than that in the base flow, thus making ${\it\varepsilon}$ negative. The SH surface micro-pattern in the latter case acts similarly to surface roughness, enhancing the Reynolds shear stress. However, these additional DRs or DIs, resulting from modifications to the turbulence dynamics, are in all cases small and contribute no more than 10 % to 15 % additional DR with longitudinal MG and micro-posts of high SFF, and no more than 10 % DI with transverse MG or micro-posts of moderate SFF. In all cases, 80 % to 100 % of the total DR arises from the effective slip on the walls.

The higher DRs observed in turbulent flow compared to laminar flow arise primarily from the higher normalized average slip velocities on the slip surfaces, $(1+1/SFF)U_{s}/U_{b}$ , which can be achieved in turbulent flow compared to laminar flow with a given SH surface micro-pattern, as shown in figure 2(c), not from mechanisms beyond the effective slip on the wall. The only exceptions are transverse MG and micro-posts with high SFF, for which the normalized average slip velocities in laminar and turbulent flow become comparable, either because the orientation of the MG relative to flow direction makes the flow regime immaterial, as with transverse MG, or because the nature of the flow is predominantly inviscid, as with micro-posts of high SFF.

Using the relations $U_{s}^{+}=L_{s}^{+}$ and $U_{s}/U_{b}=$ $L_{s}^{+0}(1-DR)Re_{{\it\tau}_{0}}/Re_{b}$ , (2.5) can be rearranged to give an expression for the magnitude of DR in terms of the slip length,

(4.1) $$\begin{eqnarray}DR=\displaystyle \frac{L_{s}^{+0}}{L_{s}^{+0}+\displaystyle \frac{Re_{b}}{Re_{{\it\tau}_{0}}}}+O({\it\varepsilon}).\end{eqnarray}$$

Similar to (2.5), the scaling given by (4.1) remains valid in both laminar and turbulent SH channel flow. However, unlike earlier similar expressions (Fukagata et al. Reference Fukagata, Kasagi and Koumoutsakos2006), no assumptions about the shape of the mean velocity profile or the SH surface micro-pattern is required in the derivation of (4.1). Good agreement is seen in figure 2(d) between (4.1) and all available DNS data. Reasonable agreement is also seen in figure 2(b,d) between (2.5) and (4.1) and experimental data (Daniello et al. Reference Daniello, Waterhouse and Rothstein2009) for which the slip velocities were reported.

Figure 3.

Spanwise variation of the mean streamwise velocity, and vector plots of the secondary mean flow, in SH turbulent channel flow with longitudinal MG: (a,d) $g^{+0}=w^{+0}\approx 32$ ; (b,e) $g^{+0}\approx 56,w^{+0}\approx 8$ ; (c,f) $g^{+0}\approx 120,w^{+0}\approx 8$ ;   

, (blue), location of the no-slip surfaces on the SH wall.

4.2. Turbulence statistics and energy budget

The roles of the effective slip on the wall and modifications to the turbulence dynamics within the flow in the mechanism of turbulent DR with SH surfaces can be further clarified by examining the one-point turbulence statistics and TKE budget in the SH channel. In view of their superior DR performance in turbulent flow, the following discussion will focus on longitudinal MG. However, the general conclusions remain valid for other SH micro-patterns, as well.

The presence of SH longitudinal MG on the channel walls gives rise to spanwise variations in all Reynolds-averaged quantities in turbulent flow, within a ‘surface layer’ of thickness $z\approx g$ . Figure 3(a–c) show examples of these spanwise variations in the Reynolds-averaged streamwise velocity. Similar spanwise variations exist in all Reynolds-averaged turbulence statistics, including the Reynolds stresses. The latter gives rise to the development of a secondary mean flow, in the form of pairs of counter-rotating vortices (Perkins Reference Perkins1970; Rastegari & Akhavan Reference Rastegari and Akhavan2013; Jelly et al. Reference Jelly, Jung and Zaki2014; Türk et al. Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014), within the ‘surface layer’ $(z\lesssim g)$ , as shown in figure 3(d–f). Both this secondary mean flow and the spanwise variations in the Reynolds-averaged turbulence statistics need to be properly accounted for to obtain accurate turbulence statistics in the SH channel. To facilitate comparison of these statistics to those in turbulent channel flows with no-slip walls, all Reynolds-averaged statistics in the SH channel are reported here after performing an additional spanwise average, $\langle \,\rangle$ , over the pattern of slip and no-slip surfaces on the SH walls.

Figure 4(a–c) show the normalized profiles of the spanwise-averaged mean streamwise velocity, $\langle \bar{U}\rangle ^{+}$ , turbulence intensities, $\{\langle \bar{u_{i}^{2}}\rangle ^{+}\}^{1/2}$ , and shear stresses, $\langle {\it\tau}_{xz}\rangle ^{+}$ , in the SH channel, compared to a base flow with no-slip walls. With the average slip velocity subtracted, the mean streamwise velocity profiles, $\langle \bar{U}\rangle ^{+}-U_{s}^{+}$ , in the SH channel display a progressive thinning of the buffer layer, along with a downward shift of the logarithmic layer, with increasing DR. These features are in stark contrast to other examples of drag-reduced turbulent flow, for which DR is generally accompanied by a thickening of the buffer layer and an upward shift of the logarithmic layer (Bushnell & Hefner Reference Bushnell and Hefner1990). The difference lies in the mechanism of DR in the present flow, which is primarily the effective slip on the walls. The presence of this effective slip leads to weaker gradients of the mean streamwise velocity over the slip surfaces up to distances of $z\approx g$ , which when mixed and averaged with the mean streamwise velocities over the no-slip surfaces, leads to weaker gradients in $\langle \bar{U}\rangle ^{+}$ within the ‘surface layer’, and manifests as a progressive thinning of the buffer layer and a downward shift of the logarithmic layer in the profiles of $\langle \bar{U}\rangle ^{+}-U_{s}^{+}$ .

Within this ‘surface layer’ ( $z\lesssim g$ ), the normalized streamwise and spanwise turbulence intensities are enhanced compared to the base flow, as shown in figure 4(b). Outside of the ‘surface layer’, however, the normalized turbulence intensities collapse to those in the base flow, indicating that the normalized dynamics of turbulence has reverted to that of turbulent flows with no-slip walls.

Figure 4.

Turbulence statistics and TKE budget in SH channel flow with longitudinal MG: (a) mean streamwise velocity with the average slip velocity subtracted; (b) turbulence intensities; (c) Reynolds shear stress $\langle {\it\tau}_{R,xz}\rangle ^{+}$ , viscous shear stress $\langle {\it\tau}_{v,xz}\rangle ^{+}$ , convective stress $\langle {\it\tau}_{c,xz}\rangle ^{+}$ , and their sum $\langle {\it\tau}_{t}\rangle ^{+}$ ; (d) bulk production $\langle P\rangle$ and bulk dissipation $\langle {\it\varepsilon}\rangle$ ; (e) breakdown of $\langle P\rangle$ into $-\langle \overline{uw}\partial \bar{U}/\partial z\rangle$ and $-\langle \overline{uv}\partial \bar{U}/\partial y\rangle$ ; (f) total bulk transport $\langle t^{({\it\Sigma})}\rangle$ ; (g) production $\langle P\rangle ^{+}$ and dissipation $\langle {\it\varepsilon}\rangle ^{+}$ ; (h) breakdown of $\langle P\rangle ^{+}$ into $-\langle \overline{uw}\partial \bar{U}/\partial z\rangle ^{+}$ and $-\langle \overline{uv}\partial \bar{U}/\partial y\rangle ^{+}$ ; (i) total transport $\langle t^{({\it\Sigma})}\rangle ^{+}$ .  

,  $g^{+0}=w^{+0}\approx 4$ ;  

,  $g^{+0}=w^{+0}\approx 8$ ;  

,  $g^{+0}=w^{+0}\approx 16$ ;  

,  $g^{+0}=w^{+0}\approx 32$ ;  

,  $g^{+0}=w^{+0}\approx 64$ ;   

,  $g^{+0}=w^{+0}\approx 128$ ;   

,  $g^{+0}\approx 28,w^{+0}\approx 4$ ;   

,  $g^{+0}\approx 56,w^{+0}\approx 8$ ;   

,  $g^{+0}\approx 120,w^{+0}\approx 8$ ; $\boldsymbol{\cdot }\boldsymbol{\cdot }\boldsymbol{\cdot }\,\boldsymbol{\cdot }$ , no-slip channel flow at $Re_{b}=3600$ $(Re_{{\it\tau}_{0}}\approx 223)$ ;  - - - -, no-slip channel flow at $Re_{{\it\tau}_{0}}\approx 100$ . Arrows indicate increasing DR.

Figure 4(c) shows the normalized profiles of the spanwise-averaged Reynolds shear stress, $\langle {\it\tau}_{R,xz}\rangle ^{+}\equiv -\langle \overline{uw}\rangle ^{+}$ , viscous shear stress, $\langle {\it\tau}_{v,xz}\rangle ^{+}\equiv \langle \partial \overline{U}/\partial y\rangle ^{+}$ , mean convective stress, $\langle {\it\tau}_{c,xz}\rangle ^{+}\equiv -\langle \bar{U}\bar{W}\rangle ^{+}$ , and total stress, $\langle {\it\tau}_{t}\rangle ^{+}\equiv \langle {\it\tau}_{R,xz}\rangle ^{+}+\langle {\it\tau}_{v,xz}\rangle ^{+}+\langle {\it\tau}_{c,xz}\rangle ^{+}$ , in the SH channel, compared to the base flow. The profiles of normalized Reynolds shear stress display a progressive drop in their peak magnitudes compared to the base flow, with increasing DR. However, the magnitudes of these drops are nowhere as large as those observed at comparable DR with other means of skin-friction DR. Indeed, the profiles of normalized Reynolds shear stress remain comparable to those which would be observed in a turbulent channel flow with no-slip walls at the $Re_{{\it\tau}}$ of the drag-reduced SH channel flow, as shown in figure 4(c). This keeps the differences between the Reynolds shear stress in the SH flow and that in the base flow small, thus keeping the $O({\it\varepsilon})$ term in (2.5) small. Similarly, for cases where the $O({\it\varepsilon})$ term leads to a DI (not shown), the magnitude of the Reynolds shear stress in the SH channel becomes slightly larger than that in the base flow. In computing the total balance of shear stresses, in addition to the Reynolds and viscous shear stresses, a small contribution is also observed from the mean convective stress, $-\langle \bar{U}\bar{W}\rangle ^{+}$ , arising from the secondary mean flow in the channel. With this contribution, the total stress shows good agreement with the theoretical line $\langle {\it\tau}_{t}\rangle ^{+}=(1-z/h)$ , as required by (2.2). These secondary flows are found to make a negligible contribution to the $O({\it\varepsilon})$ term in (2.5).

Figure 4(d–i) show the balance of the terms in the TKE budget in the SH channel, compared to the base flow, where $\langle P\rangle \equiv -\langle \overline{u_{i}u_{j}}\partial \bar{U}_{j}/\partial x_{i}\rangle$ is the rate of TKE production, $\langle {\it\varepsilon}\rangle \equiv 2{\it\nu}\langle \overline{\unicode[STIX]{x1D634}_{ij}\partial u_{j}/\partial x_{i}}\rangle$ is the rate of viscous dissipation, $\langle t^{({\it\Sigma})}\rangle \equiv -(1/2)\langle \partial \overline{u_{j}^{2}u_{i}}/\partial x_{i}\rangle -(1/{\it\rho})$ $\langle \partial \overline{pu_{i}}/\partial x_{i}\rangle +2{\it\nu}\langle \partial \overline{u_{j}\unicode[STIX]{x1D634}_{ij}}/\partial x_{i}\rangle$ is the total TKE transport, and $\unicode[STIX]{x1D634}_{ij}=1/2(\partial u_{i}/\partial x_{j}+\partial u_{j}/\partial x_{i})$ is the fluctuating strain-rate tensor (Hinze Reference Hinze1975). When normalized with the bulk variables, $h$ and $U_{b}$ , the production, dissipation and transport of TKE in the SH channel display significant drops compared to the base flow, as shown in figure 4(d–f), consistent with the presence of DR and the lower $Re_{{\it\tau}}$ in the SH channel. When normalized in wall units, however, the profiles of TKE production, $\langle P\rangle ^{+}$ , and dissipation, $\langle {\it\varepsilon}\rangle ^{+}$ , collapse to those in the base flow outside of the ‘surface layer’, while within the ‘surface layer’ there is additional production and dissipation of TKE compared to the base flow, as shown in figure 4(g). A breakdown of TKE production into its non-zero components, $-\langle \overline{uv}\partial \bar{U}/\partial y\rangle ^{+}$ and $-\langle \overline{uw}\partial \bar{U}/\partial z\rangle ^{+}$ , reveals that TKE production through the $-\langle \overline{uw}\partial \bar{U}/\partial z\rangle ^{+}$ term in the SH channel remains virtually unchanged from that in the base flow, as shown in figure 4(h). The additional production of TKE within the ‘surface layer’ arises from the $-\langle \overline{uv}\partial \bar{U}/\partial y\rangle ^{+}$ term, resulting from the spanwise variations in the mean streamwise velocity within the ‘surface layer’. In the low-DR regime, this enhanced production of TKE is balanced locally by viscous dissipation within the ‘surface layer’, as shown in figure 4(g–i). In the high-DR regime, the additional production of TKE within the ‘surface layer’ cannot be fully dissipated locally, and is transported to the interior of the flow, where it is dissipated. This leads to a reversal in the sign of the transport terms within the ‘surface layer’, compared to the base flow, as shown in figure 4(i). The main effect of the SH surface on the dynamics of turbulence is thus additional production of TKE within a ‘surface layer’ of thickness of the order of the width of SH MG. This effect, however, is not drag reducing, but turbulence enhancing. Outside of this ‘surface layer’, the normalized dynamics of turbulence in the SH channel proceeds as in channel flows with no-slip walls at the $Re_{{\it\tau}}$ of the SH channel flow. Collectively, these results indicate that the effect of the SH surface on the internal dynamics of turbulence is small, and the dominant mechanism of turbulent DR with SH surfaces remains the effective slip on the wall.