2 Numerical methods and simulation parameters

The DNS studies were performed in turbulent channel flows using standard D3Q19 (three-dimensional, 19 discrete velocity), single-relaxation-time lattice Boltzmann methods (Succi Reference Succi2001) with grid embedding (Lagrava et al. Reference Lagrava, Malaspinas, Latt and Chopard2012), of grid ratio $4:1$ , in the near-wall regions to better resolve the flow features near the SH walls. The resulting grid spacings were $\unicode[STIX]{x1D6E5}_{f}^{+0}\approx 0.5$ for $z^{+0}\lesssim 30$ , and $\unicode[STIX]{x1D6E5}_{c}^{+0}\approx 2$ for $z^{+0}\gtrsim 30$ in all three directions in all the simulations, where $\unicode[STIX]{x1D6E5}_{f}$ and $\unicode[STIX]{x1D6E5}_{c}$ denote the grid spacings on the fine grid and the coarse grid, respectively, and the $+0$ superscript denotes normalization with respect to the kinematic viscosity, $\unicode[STIX]{x1D708}$ , and the wall-friction velocity, $u_{\unicode[STIX]{x1D70F}_{0}}$ , of a ‘base’ turbulent channel flow with smooth, no-slip walls at the same bulk Reynolds number as the channel flow with SH walls. To preserve the order of accuracy of the lattice Boltzmann method, and ensure smoothness and continuity of the variables at the transition between the coarse and the fine grids, the two grids were overlapped for one cell width of the coarse grid, as suggested by Lagrava et al. (Reference Lagrava, Malaspinas, Latt and Chopard2012), and the interpolations between the two grids were performed using third-order bi-cubic Hermite splines in space, and second-order Hermite polynomials in time. The details of the numerical methods and verification studies performed to ensure the accuracy of the numerical methods and adequacy of the domain sizes and grid resolutions are described in the appendix of Rastegari & Akhavan (Reference Rastegari and Akhavan2018a ).

The liquid/gas interfaces on SH walls were modelled as stationary, curved or flat shear-free boundaries, with the shape of the meniscus determined from an analytical solution of the Young–Laplace equation (Rastegari & Akhavan Reference Rastegari and Akhavan2018a ). Two interface protrusion angles of $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ and $\unicode[STIX]{x1D703}_{p}=-30^{\circ }$ were investigated, corresponding to liquid/gas interfaces which were either flat or at maximum advancing contact angle ( $\unicode[STIX]{x1D703}_{c}=\unicode[STIX]{x1D703}_{F,adv}=120^{\circ }$ ), respectively.

The simulations were performed in channels of size $5h\times 2.5h\times 2h$ in the streamwise ( $x$ ), spanwise ( $y$ ) and wall-normal ( $z$ ) directions, respectively, as shown in figure 1. A constant flow rate was maintained in the channel during the course of all simulations, corresponding to bulk Reynolds numbers of $Re_{b}\equiv q/2\unicode[STIX]{x1D708}=3600$ or $Re_{b}=7860$ , where $q$ denotes the flow rate per unit spanwise width in the channel. For reference, DNS studies were also performed in base turbulent channel flows with smooth, no-slip walls at the same $Re_{b}$ as the SH channels. The corresponding friction Reynolds numbers in the base turbulent channel flows were $Re_{\unicode[STIX]{x1D70F}_{0}}\equiv u_{\unicode[STIX]{x1D70F}_{0}}h/\unicode[STIX]{x1D708}\approx 222$ at $Re_{b}=3600$ , and $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ at $Re_{b}=7860$ .

Figure 1. Schematic of the channels with SH walls, the coordinate system and the computational grid: (a) SH longitudinal microgrooves; (b) SH aligned microposts; (c) computational grid and protrusion angle.

A total of 53 simulations were performed with SH longitudinal microgrooves, covering the range of microgroove widths $4\lesssim g^{+0}\lesssim 128$ , solid fractions of $1/64\leqslant \unicode[STIX]{x1D719}_{s}\leqslant 1/2$ , protrusion angles of $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ and $\unicode[STIX]{x1D703}_{p}=-30^{\circ }$ and bulk Reynolds numbers of $Re_{b}=3600$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ ) and $Re_{b}=7860$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ ), as shown in table 1. In addition, as shown in table 1, four simulations were also performed with SH aligned square microposts with $g^{+0}\approx 28,56$ , $\unicode[STIX]{x1D719}_{s}=1/64$ and $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ at $Re_{b}=3600$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ ) and $Re_{b}=7860$ ( $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ ).

Table 1. Summary of simulation parameters and resulting drag reductions, $DR$ , in the present study.

3 The scaling of drag reduction with surface microtexture and flow parameters

It has been suggested that in drag reduction with microtextured surfaces, be they riblets, SH surfaces or liquid-infused (LI) surfaces, the magnitude of drag reduction depends not only on the geometry and size of the surface microtexture in wall units, but also the Reynolds number of the base flow (Bechert et al. Reference Bechert, Bruse, Hage, Van Der Hoeven and Hoppe1997; Spalart & McLean Reference Spalart and McLean2011; García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2012; Rastegari & Akhavan Reference Rastegari and Akhavan2018a ). It has also been suggested (Bechert et al. Reference Bechert, Bruse, Hage, Van Der Hoeven and Hoppe1997; Spalart & McLean Reference Spalart and McLean2011) that a Reynolds number independent measure of drag reduction can be constructed by representing the normalized mean velocity profiles in the base flow and in the flow with microtextured walls as logarithmic profiles, $\langle \overline{U}\rangle /u_{\unicode[STIX]{x1D70F}_{0}}=(1/\unicode[STIX]{x1D705})\ln z^{+0}+B_{0}$ and $\langle \overline{U}\rangle /u_{\unicode[STIX]{x1D70F}}=(1/\unicode[STIX]{x1D705})\ln z^{+}+B$ , throughout the cross-section of the flow, and expressing the magnitude of drag reduction in terms of the shift, $(B-B_{0})$ , in the intercepts of these logarithmic-law representations of the mean velocity profiles, as shown in the figure 2. Here, and throughout this study, an overbar denotes Reynolds averaging in time and any homogeneous flow directions, and brackets $\langle \,\rangle$ denote averaging in the wall-parallel directions.

Using this formulation, the magnitude of drag reduction with microtextured surfaces can be expressed as (Rastegari & Akhavan Reference Rastegari and Akhavan2018a )

(3.1) $$\begin{eqnarray}\left(1-DR\right)=\left\{1+\left[\frac{1}{2\unicode[STIX]{x1D705}}\ln (1-DR)+(B-B_{0})\right]\sqrt{\frac{C_{f_{0}}}{2}}\right\}^{-2},\end{eqnarray}$$

where $DR\equiv 1-C_{f}/C_{f_{0}}$ is the magnitude of drag reduction, $C_{f}$ and $C_{f_{0}}$ are the skin-friction coefficients in the flow with microtextured walls and in the base flow with smooth no-slip walls, defined as $C_{f}\equiv 2u_{\unicode[STIX]{x1D70F}}^{2}/U_{b}^{2}$ and $C_{f_{0}}\equiv 2u_{\unicode[STIX]{x1D70F}_{0}}^{2}/U_{b_{0}}^{2}$ in internal flow and $C_{f}\equiv 2u_{\unicode[STIX]{x1D70F}}^{2}/U_{\infty }^{2}$ and $C_{f_{0}}\equiv 2u_{\unicode[STIX]{x1D70F}_{0}}^{2}/U_{\infty }^{2}$ in external flow, respectively, $\unicode[STIX]{x1D705}$ is the von Kármán constant, and $(B-B_{0})$ is Reynolds number independent and only a function of the geometry and size of the surface microtexture in wall units. The appearance of $C_{f_{0}}$ in (3.1) leads to a degradation in the magnitude of drag reduction with increasing Reynolds number of the base flow for a given $(B-B_{0})$ .

Figure 2. Schematic representation of $(B-B_{0})$ and $U_{s}^{+}$ in turbulent flow with microtextured walls: —— (black), –  $\cdot$   $\cdot$  – (turquoise), mean velocity profiles in the base turbulent flow and the flow with microtextured walls, respectively; – – – (black), –  $\cdot$  – (turquoise), logarithmic representations of the mean velocity profiles in the base turbulent flow and the flow with microtextured walls, respectively.

It was brought to our attention by one of the anonymous referees that (3.1) can be recast into an explicit expression for $DR$ , given by

(3.2) $$\begin{eqnarray}DR=1-\unicode[STIX]{x1D705}^{2}\left(\frac{2}{C_{f_{0}}}\right)\left\{W\left(\unicode[STIX]{x1D705}\sqrt{\frac{2}{C_{f_{0}}}}\exp \left(\unicode[STIX]{x1D705}\left[(B-B_{0})+\sqrt{\frac{2}{C_{f_{0}}}}\right]\right)\right)\right\}^{-2},\end{eqnarray}$$

where $W$ denotes the Lambert $W$ function.

Figure 3. The magnitude of drag reduction, $DR$ , as a function of $(B-B_{0})$ and the friction Reynolds number of the base flow, $Re_{\unicode[STIX]{x1D70F}_{0}}$ . Present DNS studies in SH turbulent channel flow: 

, 

,

, 

, 

 (navy), SH longitudinal microgrooves, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ , $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/2$ , $1/8$ , $1/16$ , $1/32$ , $1/64$ ; 

, 

, 

, 

, 

 (navy), SH longitudinal microgrooves, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ , $\unicode[STIX]{x1D703}_{p}=-30^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/2$ , $1/8$ , $1/16$ , $1/32$ , $1/64$ ; 

, 

,  

, 

 (blue), SH longitudinal microgrooves, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ , $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/8$ , $1/16$ , $1/32$ , $1/64$ ; 

, 

, 

, 

 (blue), SH longitudinal microgrooves, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ , $\unicode[STIX]{x1D703}_{p}=-30^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/8$ , $1/16$ , $1/32$ , $1/64$ ; 

(navy), SH aligned microposts, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ , $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/64$ ; 

 (blue), SH aligned microposts, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ , $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/64$ ; 

 (lavender), DNS of turbulent channel flow with LI longitudinal microgrooves, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 180$ , $\unicode[STIX]{x1D703}_{p}=0^{o}$ , $\unicode[STIX]{x1D719}_{s}=1/2$ , $N=0.1,1,2.5,10,20,100$ (Fu et al. Reference Fu, Arenas, Leonardi and Hultmark2017); ▪ (red), experiments in turbulent boundary layer flow with SH random microposts, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 460$ and $560$ (Zhang et al. Reference Zhang, Tian, Yao, Hao and Jiang2015); ♦ (green), experiments in turbulent Taylor–Couette flow with SH random microposts, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 1193$ and $3810$ (Srinivasan et al. Reference Srinivasan, Kleingartner, Gilbert, Cohen, Milne and McKinley2015); ● (orange), experiments in turbulent boundary layer flow with SH longitudinal microgrooves or SH random microposts, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 863,1408$ and $4287$ (Ling et al. Reference Ling, Srinivasan, Golovin, McKinley, Tuteja and Katz2016). The black lines show the predictions of (3.1) in turbulent channel flow for: ——, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ ( $Re_{b}=3600$ ); — —, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ ( $Re_{b}=7860$ ); – – –, $Re_{\unicode[STIX]{x1D70F}_{0}}=10^{3}$ ( $Re_{b}\approx 2\times 10^{4}$ ); –  $\cdot$  –,  $Re_{\unicode[STIX]{x1D70F}_{0}}=10^{4}$ ( $Re_{b}\approx 2.6\times 10^{5}$ ); –  $\cdot$   $\cdot$  –,  $Re_{\unicode[STIX]{x1D70F}_{0}}=10^{5}$ ( $Re_{b}\approx 3.2\times 10^{6}$ );  $\cdots \cdots$ , $Re_{\unicode[STIX]{x1D70F}_{0}}=10^{6}$ ( $Re_{b}\approx 3.8\times 10^{7}$ ).

Figure 3 shows the predictions of (3.1) in turbulent channel flow for $222\lesssim Re_{\unicode[STIX]{x1D70F}_{0}}\lesssim 10^{6}$ compared to results from present DNS studies in turbulent channel flows with SH longitudinal microgrooves or SH aligned microposts at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ and $442$ , as summarized in table 1. For $Re_{\unicode[STIX]{x1D70F}_{0}}\geqslant 1000$ , the values of $C_{f_{0}}$ in (3.1) were obtained from the logarithmic skin-friction correlation suggested by Zanoun, Nagib & Durst (Reference Zanoun, Nagib and Durst2009), which has been shown to give improved agreement with experimental data at high Reynolds numbers compared to the classical Dean’s correlation (Dean Reference Dean1978). In agreement with the predictions of (3.1), for a given value of $(B-B_{0})$ , the DNS data show small degradations of ${\approx}1.0{-}2.5\,\%$ in drag reduction as the friction Reynolds number of the base flow increases from $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ to $442$ . Also shown in figure 3 are the experimental data of Zhang et al. (Reference Zhang, Tian, Yao, Hao and Jiang2015), obtained in turbulent boundary layer flow with SH random microposts at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 460$ and $560$ , the experimental data of Ling et al. (Reference Ling, Srinivasan, Golovin, McKinley, Tuteja and Katz2016), obtained in turbulent boundary layer flow with SH longitudinal microgrooves at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 863$ or with SH random microposts at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 863$ , $1408$ and $4287$ , and the experimental data of Srinivasan et al. (Reference Srinivasan, Kleingartner, Gilbert, Cohen, Milne and McKinley2015), obtained in turbulent Taylor–Couette flow with SH random microposts at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 1193$ and $3810$ . In plotting these experimental data, the values of $(B-B_{0})$ were approximated by the values of $\unicode[STIX]{x0394}U^{+}$ reported in the experiments. Good agreement can be seen between the predictions of (3.1) and the experimental data, including the case where the SH surface gives a negative drag reduction of $-10\,\%$ in turbulent boundary layer flow at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 4287$ (Ling et al. Reference Ling, Srinivasan, Golovin, McKinley, Tuteja and Katz2016). Evidence of the degradation of drag reduction with increasing Reynolds number of the base flow, for a given value of $(B-B_{0})$ , can also be seen in these experimental data. For example, in turbulent boundary layer flow with SH random microposts, a $(B-B_{0})=1.1$ is reported to give rise to $12\,\%$ drag reduction at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 863$ , but only $11\,\%$ drag reduction at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 1408$ (Ling et al. Reference Ling, Srinivasan, Golovin, McKinley, Tuteja and Katz2016). Similarly, a $(B-B_{0})=3.2$ is found to give rise to $24.7\,\%$ drag reduction in turbulent boundary layer flow with SH random microposts at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 560$ (Zhang et al. Reference Zhang, Tian, Yao, Hao and Jiang2015), but only $22\,\%$ drag reduction in turbulent Taylor–Couette flow with SH random microposts at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 3810$ (Srinivasan et al. Reference Srinivasan, Kleingartner, Gilbert, Cohen, Milne and McKinley2015). While these degradations in drag reduction are small at moderate Reynolds number, they become far more significant at higher Reynolds numbers, requiring ever higher values of $(B-B_{0})$ to achieve a given level of drag reduction, as shown in figure 3. Nevertheless, with a judicious choice of $(B-B_{0})$ , drag reductions of up to ${\sim}50\,\%$ should still be feasible at $Re_{\unicode[STIX]{x1D70F}_{0}}\sim 10^{5}{-}10^{6}$ of practical interest. Finally, it should be noted that the drag reduction relation given by (3.1) is equally valid for all microtexture surfaces, including LI surfaces. This can be seen in figure 3 by a comparison of the predictions of (3.1) with the DNS results of Fu et al. (Reference Fu, Arenas, Leonardi and Hultmark2017), obtained in turbulent channel flows with LI longitudinal microgrooves at viscosity ratios of $0.1\leqslant N\equiv \unicode[STIX]{x1D707}_{o}/\unicode[STIX]{x1D707}_{i}\leqslant 100$ with $\unicode[STIX]{x1D719}_{s}=1/2$ , $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ and $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 180$ .

Designing optimal SH surfaces for drag reduction in high Reynolds number turbulent flows requires, in addition to (3.1), an understanding of the scaling of $(B-B_{0})$ with the geometrical parameters of the surface microtexture. Progress can be made by noting that the shift, $(B-B_{0})$ , arises primarily because of the presence of an effective streamwise slip velocity, $U_{s}^{+}$ , at the wall, as shown in figure 2 (Min & Kim Reference Min and Kim2004; Seo & Mani Reference Seo and Mani2016; Rastegari & Akhavan Reference Rastegari and Akhavan2018a ). However, the magnitude of $(B-B_{0})$ is always smaller than $U_{s}^{+}$ due to concurrent presence of spanwise slip. As such, $(B-B_{0})$ and $U_{s}^{+}$ can be expected to have similar scalings with surface microtexture geometrical parameters, but with a smaller multiplicative coefficient in the scaling law for $(B-B_{0})$ compared to $U_{s}^{+}$ .

Figure 4. Contours of instantaneous streamwise velocity, $U^{+}(x,y,z)$ , at $z=0$ for: (a) SH longitudinal microgrooves, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ , $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/64$ , $g^{+0}\approx 63$ ; (b) SH longitudinal microgrooves, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ , $\unicode[STIX]{x1D703}_{p}=-30^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/64$ , $g^{+0}\approx 63$ ; (c) SH aligned microposts, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ , $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/64$ , $g^{+0}\approx 56$ .

Figure 5. Production and dissipation of turbulence kinetic energy in turbulence channel flow with SH longitudinal microgrooves at (a,c,d,e) $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ , $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/64$ , $g^{+0}\approx 63$ and (b,f,g,h) $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ , $\unicode[STIX]{x1D703}_{p}=-30^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/64$ , $g^{+0}\approx 63$ : (a,b) ——, — —, $\cdots \cdots$ , –  $\cdot$   $\cdot$  – (turquoise), total production $\langle {\mathcal{P}}\rangle ^{+}=-\langle \overline{u_{i}u_{j}}\overline{S}_{ij}\rangle ^{+}$ , production arising from $-\langle \overline{uw}\unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}z\rangle ^{+}$ , production arising from $-\langle \overline{uv}\unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}y\rangle ^{+}$ , dissipation $\langle \unicode[STIX]{x1D700}\rangle ^{+}=\langle 2\unicode[STIX]{x1D708}\overline{s_{ij}s_{ij}}\rangle ^{+}$ , in turbulent channel flow with SH walls; – – –, –  $\cdot$  – (black), production arising from $-\langle \overline{uw}\unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}z\rangle ^{+}$ and dissipation, $\langle \unicode[STIX]{x1D700}\rangle ^{+}$ , in base turbulent channel flow with smooth, no-slip walls; (c,d,e; f,g,h) spanwise variation of the mean streamwise velocity, $\bar{U}^{+}$ , total production, ${\mathcal{P}}^{+}$ , and dissipation, $\unicode[STIX]{x1D700}^{+}$ , at different $z^{+}$ in turbulent channel flow with SH longitudinal microgrooves at $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ and $\unicode[STIX]{x1D703}_{p}=-30^{\circ }$ , respectively. The blue bars on the lower horizontal axes of (c–h) denote the locations of the no-slip surfaces. $L=g+w$ denotes the pitch of the microgrooves.

Figure 6. Production and dissipation of turbulence kinetic energy in turbulent channel flow with SH aligned microposts at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ , $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ , $\unicode[STIX]{x1D719}_{s}=1/64$ , $g^{+0}\approx 56$ : (a) total production $\langle {\mathcal{P}}\rangle ^{+}=-\langle \overline{u_{i}u_{j}}\overline{S}_{ij}\rangle ^{+}$ , production arising from $-\langle \overline{uw}\unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}z\rangle ^{+}$ , production arising from $-\langle \overline{uv}\unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}y\rangle ^{+}$ , dissipation $\langle \unicode[STIX]{x1D700}\rangle ^{+}=\langle 2\unicode[STIX]{x1D708}\overline{s_{ij}s_{ij}}\rangle ^{+}$ , in turbulent channel flow with SH walls, compared to base turbulent channel flow with smooth, no-slip walls; (b) schematic representation of sections A-A and B-B where turbulence statistics are shown in (c–e) and (f–h); (c–e; f–h) spanwise variation of the mean streamwise velocity, $\bar{U}^{+}$ , total production, ${\mathcal{P}}^{+}$ , dissipation, $\unicode[STIX]{x1D700}^{+}$ , at different $z^{+}$ of sections A-A and B-B, respectively. Line types in (a) as in figure 5. The blue bars on the lower horizontal axes of (c–h) denote the locations of the no-slip surfaces. $L=g+w$ denotes the pitch of the microposts.

The scaling of $U_{s}^{+}$ with surface microtexture parameters can be clarified by considering the dynamics of turbulence kinetic energy (TKE) within the ‘surface layer’. This is a layer of thickness ${\sim}g$ above the microtextured walls, in which the flow transitions from the slip/no-slip patterns at the tip of the wall microtexture, at $z=0$ , to a homogeneous turbulent flow in the wall-parallel directions at $z\sim g$ (Rastegari & Akhavan Reference Rastegari and Akhavan2015). For surface microtextures where the flow can find an unobstructed passageway through the surface micro-features, such as with longitudinal microgrooves or aligned microposts, sharp spanwise gradients of the mean streamwise velocity develop between the slip and no-slip regions of the boundary, giving rise to strong shear layers within the surface layer above these regions, as shown in figures 4(a–c), 5(c,f) and 6(c,f). These shear layers are the strongest with longitudinal microgrooves, but are also present with aligned microposts, where entire stripes aligned with the row of microposts can act as low-slip regions, as shown in figures 4(c) and 6(c,f). The presence of interface curvature somewhat mitigates the strength of these shear layers, but their mechanism is still at work, as can be seen by a comparison of figures 4(a) and 4(b) or figures 5(c) and 5(f).

These shear layers are the source of additional production of TKE within the surface layer through the $-\overline{uv}~\unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}y$ term, as shown in figures 5(a,b) and 6(a). This additional production of TKE, which is above and beyond the normal production of TKE in wall-bounded flows through the $-\overline{uw}~\unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}z$ term, dominates the overall production of TKE within the surface layer, as shown in figures 5(a,b,d,g) and 6(a,d,g). The additional production of TKE through these shear layers is dissipated by the turbulent eddies above the no-slip regions, as shown in figures 5(e,h) and 6(e,h), such that

(3.3) $$\begin{eqnarray}\{-\overline{uv}~\unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}y\}_{shear\,layers}\sim \{\unicode[STIX]{x1D708}\overline{s_{ij}s_{ij}}\}_{no\text{-}slip}.\end{eqnarray}$$

An order of magnitude estimate of the terms in (3.3) leads to the scaling law for $U_{s}^{+}$ . Specifically, within the surface layer, the spanwise gradients of the mean streamwise velocity can be estimated as $\unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}y\sim U_{ss}/g$ , where $U_{ss}=U_{s}/(1-\unicode[STIX]{x1D719}_{s})$ is the average slip velocity on the slip portions of the microtextured walls. Furthermore, the fluctuating strain rate above the no-slip regions can be estimated as $\{{s_{ij}\}}_{no-slip}\sim \sqrt{u_{\unicode[STIX]{x1D70F}_{n}}u_{\unicode[STIX]{x1D70F}}}/(\unicode[STIX]{x1D708}/\sqrt{u_{\unicode[STIX]{x1D70F}_{n}}u_{\unicode[STIX]{x1D70F}}})$ , where $\sqrt{u_{\unicode[STIX]{x1D70F}_{n}}u_{\unicode[STIX]{x1D70F}}}$ is the characteristic velocity of the turbulent eddies above the no-slip regions within the surface layer, estimated as the geometric average of the wall friction velocities $u_{\unicode[STIX]{x1D70F}_{n}}=u_{\unicode[STIX]{x1D70F}}/\sqrt{\unicode[STIX]{x1D719}_{s}}$ and $u_{\unicode[STIX]{x1D70F}}$ above the no-slip regions at $z=0$ and $z\sim g$ , respectively, and $\unicode[STIX]{x1D708}/\sqrt{u_{\unicode[STIX]{x1D70F}_{n}}u_{\unicode[STIX]{x1D70F}}}$ is the associated inner length scale. Finally, the Reynolds shear stress $-\overline{uv}$ can be estimated using a mixing-length model as $-\overline{uv}\sim \unicode[STIX]{x1D708}_{t}\unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}y\sim u^{\prime }\ell ~\unicode[STIX]{x2202}\bar{U}/\unicode[STIX]{x2202}y$ , where $u^{\prime }\sim \sqrt{u_{\unicode[STIX]{x1D70F}_{n}}u_{\unicode[STIX]{x1D70F}}}$ is the characteristic velocity of the largest eddies within the surface layer, and $\ell \sim \sqrt{g\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}}$ is their characteristic size, approximated as the geometric average of $g$ and $\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$ , representing the size of the largest eddies at $z\sim g$ and $z\sim 0$ , respectively. Substitution of these expressions into (3.3) gives the scaling law for $U_{s}^{+}$ as

(3.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D719}_{s}^{3/8}}{(1-\unicode[STIX]{x1D719}_{s})}\,U_{s}^{+}\sim \{{g^{+}\}}^{3/4}.\end{eqnarray}$$

Given that the dominant contribution to $(B-B_{0})$ arises from $U_{s}^{+}$ , a similar scaling can also be assumed for $(B-B_{0})$ :

(3.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D719}_{s}^{3/8}}{(1-\unicode[STIX]{x1D719}_{s})}\,(B-B_{0})\sim \{{g^{+}\}}^{3/4},\end{eqnarray}$$

with the understanding that the multiplicative coefficient in the expression for $(B-B_{0})$ would be smaller than that for $U_{s}^{+}$ .

Figure 7. The scaling of $U_{s}^{+}$ and $(B-B_{0})$ with surface microtexture parameters in turbulent flow with SH or LI walls: (a–d) present DNS data in turbulent channel flow with SH longitudinal microgrooves or SH aligned microposts at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ and $442$ , $1/64\leqslant \unicode[STIX]{x1D719}_{s}\leqslant 1/2$ , $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ and $-30^{\circ }$ ; (e,f) DNS data from other investigators in turbulent channel flow with SH longitudinal microgrooves, LI longitudinal microgrooves or SH aligned microposts at $180\lesssim Re_{\unicode[STIX]{x1D70F}_{0}}\lesssim 590$ , $1/36\leqslant \unicode[STIX]{x1D719}_{s}\leqslant 1/2$ . Symbols as in figure 3. Other symbols in (e,f):

, 

, 

 (brown), $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 180$ , $\unicode[STIX]{x1D719}_{s}=1/8$ , $1/4$ , $1/2$ , 

, 

, 

, 

 (green),  $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 395$ , $\unicode[STIX]{x1D719}_{s}=1/16$ , $1/8$ , $1/4$ , $1/2$ , 

, 

, 

, 

 (red),  $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 590$ , $\unicode[STIX]{x1D719}_{s}=1/16$ , $1/8$ , $1/4$ , $1/2$ , SH longitudinal microgrooves, $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ (Park et al. Reference Park, Park and Kim2013); 

, 

 (brown),  $Re_{\unicode[STIX]{x1D70F}}\approx 195$ , $\unicode[STIX]{x1D719}_{s}=1/64$ , $1/9$ , 

, 

, 

 (green),  $Re_{\unicode[STIX]{x1D70F}}\approx 400$ , $\unicode[STIX]{x1D719}_{s}=1/36$ , $1/16$ , $1/9$ , SH aligned microposts, $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ (Seo & Mani Reference Seo and Mani2016).

Figure 7(a–d) shows $U_{s}^{+}$ and $(B-B_{0})$ from the DNS studies summarized in table 1, compared to (3.4) and (3.5). At both $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 222$ and $442$ , and for both $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ and $\unicode[STIX]{x1D703}_{p}=-30^{\circ }$ , $U_{s}^{+}$ and $(B-B_{0})$ scale as $\{{g^{+}\}}^{3/4}$ , while normalization with $\unicode[STIX]{x1D719}_{s}^{3/8}/(1-\unicode[STIX]{x1D719}_{s})$ collapses the data from different $\unicode[STIX]{x1D719}_{s}$ . Both $U_{s}^{+}$ and $(B-B_{0})$ are Reynolds number independent and only functions of the geometrical parameters of the surface microtexture in wall units, in agreement with the predictions of (3.4) and (3.5).

A best fit to the DNS data gives the multiplicative coefficients in the expressions for $U_{s}^{+}$ and $(B-B_{0})$ as

(3.6) $$\begin{eqnarray}\displaystyle U_{s}^{+} & = & \displaystyle 0.52\{(1-\unicode[STIX]{x1D719}_{s})\unicode[STIX]{x1D719}_{s}^{-3/8}\}\{{g^{+}\}}^{3/4},\end{eqnarray}$$

(3.7) $$\begin{eqnarray}\displaystyle (B-B_{0}) & = & \displaystyle 0.41\{(1-\unicode[STIX]{x1D719}_{s})\unicode[STIX]{x1D719}_{s}^{-3/8}\}\{{g^{+}\}}^{3/4},\end{eqnarray}$$

with $p$ -values of 0.98, obtained from the Kolmogorov–Smirnov goodness of fit test.

Figure 7(e,f) shows the comparison of (3.6) and (3.7) with available DNS data from other investigators, obtained in turbulent channel flows using SH longitudinal microgrooves (Park et al. Reference Park, Park and Kim2013), SH aligned microposts (Seo & Mani Reference Seo and Mani2016) or LI longitudinal microgrooves at viscosity ratios of $N\equiv \unicode[STIX]{x1D707}_{o}/\unicode[STIX]{x1D707}_{i}=0.1,1,10,20,100$ (Fu et al. Reference Fu, Arenas, Leonardi and Hultmark2017). Good agreement can be seen between the predictions of (3.6) and (3.7) and available DNS data from other investigators for both SH longitudinal microgrooves and SH aligned microposts as long as $g^{+}<100$ . For $g^{+}>100$ , both $U_{s}^{+}$ and $(B-B_{0})$ saturate and branch off from the predictions of (3.6) and (3.7). This saturation has been attributed to the width of the microgrooves, $g^{+}$ , approaching the mean streak spacing of $\unicode[STIX]{x1D706}^{+}\sim 100$ in turbulent wall flows (Park et al. Reference Park, Park and Kim2013). Presumably, for $g^{+}\gtrsim 100$ , the dynamics of turbulence above the slip regions becomes independent of the dynamics of turbulence above the no-slip regions, and both $U_{s}^{+}$ and $(B-B_{0})$ approach their limiting values.

Good agreement can also be seen in figure 7(e,f) between the predictions of (3.6) and (3.7) and DNS results in turbulent channel flows with LI longitudinal microgrooves (Fu et al. Reference Fu, Arenas, Leonardi and Hultmark2017) when the viscosity ratio, $N$ , is greater than ${\sim}10$ . These results suggest that for high enough viscosity ratios, of $N\gtrsim 10$ , the drag reduction of LI surfaces begins to approach that of SH surfaces. These findings are consistent with analytical solutions in laminar shear flow with LI surfaces obtained by Schönecker, Baier & Hardt (Reference Schönecker, Baier and Hardt2014), and reproduced in figure 8, which show that for LI longitudinal microgrooves of depth $d=g$ , $\unicode[STIX]{x1D719}_{s}=0.5$ and viscosity ratio of $N=10$ , the effective slip length is ${\approx}86\,\%$ of the effective slip length predicted by Philip’s solution (Philip Reference Philip1972) for the same surface microtexture with ‘idealized’, shear-free, SH interfaces. The LI results shown in figure 7(e,f) are also consistent with experimental measurements in turbulent Taylor–Couette flow with LI longitudinal microgrooves of depth $d=g$ , $\unicode[STIX]{x1D719}_{s}=0.5$ and $0.7\leqslant N\leqslant 3.0$ (Van Buren & Smits Reference Van Buren and Smits2017), which show the drag reduction of the LI surface at $N\approx 2.0{-}3.0$ to be ${\approx}50{-}60\,\%$ of the drag reduction of a SH surface with the same surface microtexture.

In the derivation of (3.6) and (3.7), the multiplicative coefficients, of 0.52 and 0.41, were obtained based on DNS data which assumed ‘idealized’, shear-free boundaries at the air/water interfaces and, as such, did not consider the flow of air in the SH microgrooves. This raises a concern that the magnitudes of $U_{s}^{+}$ and $(B-B_{0})$ would be over-predicted by (3.6) and (3.7). An estimate of the degree of this over-prediction can be obtained by comparing analytical solutions for the effective slip length, $L_{s,N}$ , obtained with a finite viscosity ratio, $N$ , in laminar shear flow with LI longitudinal microgrooves (Schönecker et al. Reference Schönecker, Baier and Hardt2014) to the effective slip length, $L_{s,\infty }$ , which would be predicted in the same flow based on the assumption of shear-free liquid/air interfaces (Philip Reference Philip1972). Figure 8 shows $L_{s,N}/L_{s,\infty }$ as a function of $N$ for longitudinal microgrooves of depth $d=g$ and the range of solid fractions, $1/64\leqslant \unicode[STIX]{x1D719}_{s}\leqslant 1/2$ , investigated in the present study. It can be seen that for $N\approx 56$ , corresponding to air/water interfaces, the ratio $L_{s,N}/L_{s,\infty }$ ranges from $0.92\leqslant L_{s,N}/L_{s,\infty }\leqslant 0.97$ for $1/64\leqslant \unicode[STIX]{x1D719}_{s}\leqslant 1/2$ . These results suggest that for the range of $1/64\leqslant \unicode[STIX]{x1D719}_{s}\leqslant 1/2$ , considered in the present study, the errors in (3.6) and (3.7) due to the assumption of shear-free interfaces are no more than 10 %. These errors can become larger for solid fractions of $\unicode[STIX]{x1D719}_{s}\lesssim 1/64$ or for shallow grooves of $d\lesssim g/2$ (Schönecker et al. Reference Schönecker, Baier and Hardt2014). However, such configurations of microgrooves are impractical to build, either because the microgroove blades become too thin to be structurally sustainable, or because the depth of the grooves becomes so shallow that the air/water interfaces can hit the bottom of the grooves during their deformation, thus expediting their collapse.

Figure 8. Ratio of the effective slip length, $L_{s,N}$ , obtained with a finite viscosity ratio, $N$ , based on the analytical solution of Schönecker et al. (Reference Schönecker, Baier and Hardt2014), to the effective slip length, $L_{s,\infty }$ , obtained with the assumption of shear-free liquid/air interfaces (Philip Reference Philip1972), in laminar shear flow over LI longitudinal microgrooves for groove depth $d=g$ and $1/64\leqslant \unicode[STIX]{x1D719}_{s}\leqslant 1/2$ : $\cdots \cdots$ , $\unicode[STIX]{x1D719}_{s}=1/2$ ; — ⋅ ⋅ —, $\unicode[STIX]{x1D719}_{s}=1/8$ ; — ⋅ —, $\unicode[STIX]{x1D719}_{s}=1/16$ ; – – –, $\unicode[STIX]{x1D719}_{s}=1/32$ ; ——, $\unicode[STIX]{x1D719}_{s}=1/64$ .

Another concern is the effect of dynamic deformation of the interface on (3.6) and (3.7), as the multiplicative coefficients in (3.6) and (3.7) were obtained from DNS studies which were performed with ‘stationary’, flat or curved, shear-free interfaces. The effect of dynamic deformation of the interface can be assessed by examining available DNS results obtained with deformable interfaces, but pinned contact lines, in turbulent channel flows with LI longitudinal microgrooves at a viscosity ratio of $N=2.5$ for $We_{\unicode[STIX]{x1D70F}}=0$ and $We_{\unicode[STIX]{x1D70F}}=0.024$ (Arenas-Navarro Reference Arenas-Navarro2017), where $We_{\unicode[STIX]{x1D70F}}\equiv \unicode[STIX]{x1D70C}_{o}u_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D708}_{o}/\unicode[STIX]{x1D70E}$ is the Weber number, $\unicode[STIX]{x1D70C}_{o}$ and $\unicode[STIX]{x1D708}_{o}$ are the density and kinematic viscosity of the outer fluid, respectively, and $\unicode[STIX]{x1D70E}$ is the surface tension.

Figure 9 shows $\{\unicode[STIX]{x1D719}_{s}^{3/8}/(1-\unicode[STIX]{x1D719}_{s})\}\,U_{s}^{+}$ and $\{\unicode[STIX]{x1D719}_{s}^{3/8}/(1-\unicode[STIX]{x1D719}_{s})\}\,(B-B_{0})$ computed from these DNS databases. Both $\{\unicode[STIX]{x1D719}_{s}^{3/8}/(1-\unicode[STIX]{x1D719}_{s})\}\,U_{s}^{+}$ and $\{\unicode[STIX]{x1D719}_{s}^{3/8}/(1-\unicode[STIX]{x1D719}_{s})\}\,(B-B_{0})$ can be seen to scale as ${\sim}{g^{+}\{}^{3/4}\}$ , in agreement with (3.6) and (3.7). However, the multiplicative coefficients in the best fit lines to the LI data are smaller than those in (3.6) and (3.7) by a factor of $0.6$ for $U_{s}^{+}$ and a factor of $0.4$ for $(B-B_{0})$ . These factors are comparable to the ratio $L_{s,N}/L_{s,\infty }=0.6$ observed for $N=2.5$ and $\unicode[STIX]{x1D719}_{s}=1/2$ in figure 8.

The dynamic deformation of the interface is seen to have little effect on the slip velocity, with the values for $U_{s}^{+}$ at the two Weber numbers being nearly identical, as seen in figure 8(a). However, the surface ripples formed by the dynamic deformation of the interface can act as ‘surface roughness’ and reduce the magnitude of $(B-B_{0})$ by $2.5-25.5\,\%$ at $We_{\unicode[STIX]{x1D70F}}=0.024$ compared to $We_{\unicode[STIX]{x1D70F}}=0$ , as seen in figure 8(b). The results shown in figure 9 were obtained with LI longitudinal microgrooves at $N=2.5$ and pinned contact lines. As will be shown in § 4 (figure 12), for SH longitudinal microgrooves, a $We_{\unicode[STIX]{x1D70F}}=0.024$ is very large and will lead to interface instability for $g^{+}>6.5$ at $Re_{\unicode[STIX]{x1D70F}}\approx 200$ . Stable SH longitudinal microgrooves require much smaller values of $We_{\unicode[STIX]{x1D70F}}$ , for which the effect of dynamic deformation of the interface on $(B-B_{0})$ is much smaller than the results shown in figure 8(b). Nevertheless, these effects can be accounted for by constructing a $\unicode[STIX]{x0394}B_{k}$ associated with the ‘roughness’ of the liquid/gas interface, and subtracting this $\unicode[STIX]{x0394}B_{k}$ from the $(B-B_{0})$ given by (3.7) to get the drag reduction. More recent DNS studies (Rastegari & Akhavan Reference Rastegari and Akhavan2018b ) suggest that the motion of the contact line may be a more important source of roughness effects. A similar procedure can also be used to account for the roughness effects arising from the motion of the contact line on SH or LI longitudinal microgrooves or aligned microposts, assuming the interface has not fully collapsed.

Figure 9. Effect of dynamic deformation of the interface on (a) $U_{s}^{+}$ and (b) $(B-B_{0})$ , based on DNS results with deformable interfaces, but pinned contact lines, in turbulent channel flows with LI longitudinal microgrooves at $Re_{\unicode[STIX]{x1D70F}}\approx 180$ , $\unicode[STIX]{x1D719}_{s}=1/2$ , $N=2.5$ (Arenas-Navarro Reference Arenas-Navarro2017): 

 (lavender),  $We_{\unicode[STIX]{x1D70F}}=0$ ; 

 (dark violet), $We_{\unicode[STIX]{x1D70F}}=0.024$ .

Finally, it should be noted that, given that the derivations of (3.6) and (3.7) require an unobstructed passageway through the surface microtexture, these scalings are not expected to apply to staggered microposts or transverse microgrooves.

4 Sustainability bounds of superhydrophobic surfaces

Equations (3.1) and (3.7) provide a tool for a priori prediction of the magnitude of drag reduction with any pattern of SH longitudinal microgrooves or aligned microposts in turbulent flow at any Reynolds number. However, design of optimal SH surfaces also requires consideration of the sustainability bounds of such surfaces, as all SH surfaces are susceptible to a wetting transition from the Cassie–Baxter state to a Wenzel state. This wetting transition can occur because of (i) depinning of the contact line and/or sagging of the liquid–gas interface under high local pressures (Zheng et al. Reference Zheng, Yu and Zhao2005; Checco et al. Reference Checco, Ocko, Rahman, Black, Tasinkevych, Giacomello and Dietrich2014) or (ii) diffusion or entrainment of the gas layer into the working liquid under high local shear rates (Samaha et al. Reference Samaha, Tafreshi and Gad-el Hak2012; Karatay et al. Reference Karatay, Tsai and Lammertink2013; Wexler et al. Reference Wexler, Jacobi and Stone2015; Ling et al. Reference Ling, Katz, Fu and Hultmark2017) of turbulent flow. Of the two mechanisms, the wetting transition due to high local pressures is generally the first mode of failure, and of the different SH surface microtextures, longitudinal microgrooves are the most stable under this mode of failure, as impingement of the flow on the back of microposts creates large stagnation pressures which can expedite the wetting transition (Seo, García-Mayoral & Mani Reference Seo, García-Mayoral and Mani2015). Consequently, in this section we will focus only on the pressure stability bounds of SH longitudinal microgrooves.

For SH longitudinal microgrooves with stable interfaces, the contact line is pinned and the shape of the interface is governed by the Young–Laplace equation

(4.1) $$\begin{eqnarray}\unicode[STIX]{x0394}P=-\frac{\unicode[STIX]{x1D70E}}{R}=-\frac{2\unicode[STIX]{x1D70E}\cos (\unicode[STIX]{x1D703}_{c})}{g},\end{eqnarray}$$

where $\unicode[STIX]{x0394}P$ is the pressure difference across the interface, $\unicode[STIX]{x1D70E}$ is the surface tension, $R$ is the radius of curvature of the interface, $\unicode[STIX]{x1D703}_{c}$ is the contact angle and $g$ is the microgroove width. With increasing $\unicode[STIX]{x0394}P$ , the contact angle increases until it reaches the maximum advancing contact angle, $\unicode[STIX]{x1D703}_{F,adv}$ , at which point the contact line depins (Checco et al. Reference Checco, Ocko, Rahman, Black, Tasinkevych, Giacomello and Dietrich2014). Assuming a maximum advancing contact angle of $\unicode[STIX]{x1D703}_{F,adv}=120^{\circ }$ (Nishino et al. Reference Nishino, Meguro, Nakamae, Matsushita and Ueda1999), the criterion for stability of SH interfaces becomes $\unicode[STIX]{x0394}Pg/\unicode[STIX]{x1D70E}\leqslant 1$ , or

(4.2) $$\begin{eqnarray}(\unicode[STIX]{x0394}P^{+})(g^{+})(We_{\unicode[STIX]{x1D70F}})\leqslant 1,\end{eqnarray}$$

where $\unicode[STIX]{x0394}P^{+}\equiv \unicode[STIX]{x0394}P/(\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}^{2})$ is the non-dimensional pressure, and $We_{\unicode[STIX]{x1D70F}}\equiv \unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D708}/\unicode[STIX]{x1D70E}$ is the Weber number.

Figure 10. Probability density function and cumulative distribution function of the wall pressure fluctuations in turbulent channel flow with SH longitudinal microgrooves, compared to turbulent channel flow with smooth, no-slip walls: (a) probability density function, $f_{p}(p_{w}^{+}/\{\langle \overline{p^{2}}\rangle _{w}^{+}\!\}^{1/2})$ ; (b) cumulative distribution function, $F_{p}(p_{w}^{+}/\{\langle \overline{p^{2}}\rangle _{w}^{+}\}^{1/2})$ ; – – –, –  $\cdot$  –, SH longitudinal microgrooves, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ , $\unicode[STIX]{x1D719}_{s}=1/64$ , $g^{+0}\approx 63$ , $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ and $-30^{\circ }$ , respectively; $\cdots \cdots$ , base turbulent channel flow with smooth, no-slip walls, $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ ; —— (orange), Gaussian distribution.

Assessing the stability bounds of SH surfaces in turbulent flow from (4.2) requires a knowledge of $\unicode[STIX]{x0394}P^{+}$ . In turbulent flow, $\unicode[STIX]{x0394}P^{+}$ arises from the instantaneous pressure fluctuations on the SH walls. The range of values assumed by the instantaneous pressure fluctuations on the SH walls can be assessed by examining the probability density function (p.d.f.) of the wall pressure fluctuations, $f_{p}(p_{w}^{+}/\{\langle \overline{p^{2}}\rangle _{w}^{+}\!\}^{1/2})$ . Figure 10(a) shows the p.d.f. of wall pressure fluctuations in turbulent channel flow with SH longitudinal microgrooves at $Re_{\unicode[STIX]{x1D70F}_{0}}\approx 442$ , $\unicode[STIX]{x1D719}_{s}=1/64$ , $g^{+0}\approx 63$ and $\unicode[STIX]{x1D703}_{p}=0^{\circ }$ and $\unicode[STIX]{x1D703}_{p}=-30^{\circ }$ , compared to turbulent channel flow with smooth, no-slip walls. The p.d.f. of instantaneous wall pressure fluctuations in turbulent channel flow with SH walls is found to be similar to that for smooth, no-slip walls. Both p.d.f.s follow a non-Gaussian distribution with exponential tails. Similar non-Gaussian distributions have also been reported experimentally in high Reynolds number turbulent boundary layers (Tsuji et al. Reference Tsuji, Fransson, Alfredsson and Johansson2007). The probability that the instantaneous pressure fluctuations fall within a given range of values, $a\leqslant p_{w}^{+}/\{\langle \overline{p^{2}}\rangle _{w}^{+}\!\}^{1/2}<b$ , can be estimated from $F_{p}(b)-F_{p}(a)=\int _{a}^{b}f_{p}(\unicode[STIX]{x1D702})\,\text{d}\unicode[STIX]{x1D702}$ , where $F_{p}$ is the cumulative distribution function. Figure 10(b) shows the cumulative distribution functions, $F_{p}(p_{w}^{+}/\{\langle \overline{p^{2}}\rangle _{w}^{+}\!\}^{1/2})$ , for the cases shown in figure 10(a). Examination of these cumulative distribution functions shows that the probability of the instantaneous wall pressure fluctuations falling within $\pm 4\{\langle \overline{p^{2}}\rangle _{w}^{+}\!\}^{1/2}$ is $99.75\,\%$ . These results are consistent with experimental measurements in turbulent boundary layer flows (Schewe Reference Schewe1983), which show a probability of $99.7\,\%$ for the wall pressure fluctuations never exceeding $4\{\overline{p^{2}}_{w}^{+}\!\}^{1/2}$ . Based on these results, an upper estimate for the magnitude of $\unicode[STIX]{x0394}P^{+}$ can be obtained as $\unicode[STIX]{x0394}P^{+}\approx 4\{\langle \overline{p^{2}}\rangle _{w,SH}^{+}\!\}^{1/2}$ , with $99.75\,\%$ confidence that the instantaneous wall pressure fluctuations will not exceed this value.

Figure 11. Mean-squared turbulent wall pressure fluctuations in turbulent channel flow with SH longitudinal microgrooves: (a) $\langle \bar{p^{2}}\rangle _{w,SH}^{+}$ as a function of $Re_{\unicode[STIX]{x1D70F}}$ ; (b) $\langle \bar{p^{2}}\rangle _{w,SH}^{+}-\unicode[STIX]{x1D6E4}(g^{+})$ as a function of $Re_{\unicode[STIX]{x1D70F}}$ , where $\unicode[STIX]{x1D6E4}(g^{+})=2.32\ln (\{g^{+}\}^{3/4})-4.5$ . Symbols as in figure 3. $\cdots \cdots$ , the fit $\langle \bar{p^{2}}\rangle _{w,smooth}^{+}=2.31\ln (Re_{\unicode[STIX]{x1D70F}})-9.5$ , suggested by Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013) for turbulent channel flows with smooth, no-slip walls.

Figure 11(a) shows $\langle \overline{p^{2}}\rangle _{w,SH}^{+}$ as a function of $Re_{\unicode[STIX]{x1D70F}}$ from present DNS studies of turbulent channel flow with SH longitudinal microgrooves at fixed protrusion angle of $\unicode[STIX]{x1D703}_{p}=-30^{\circ }$ ( $\unicode[STIX]{x1D703}_{c}=120^{\circ }$ ). For reference, the fit $\langle \overline{p^{2}}\rangle _{w,smooth}^{+}=2.31\ln (Re_{\unicode[STIX]{x1D70F}})-9.5$ , suggested by Sillero et al. (Reference Sillero, Jiménez and Moser2013), for turbulent channel flow with smooth no-slip walls is also shown. Similar to turbulent flow with smooth no-slip walls, the mean-squared wall pressure fluctuations in turbulent flow with SH walls is strongly Reynolds number dependent. However, due to the presence of spanwise slip, the magnitude of $\langle \bar{p^{2}}\rangle _{w,SH}^{+}$ on SH walls is higher than that on smooth no-slip walls at comparable $Re_{\unicode[STIX]{x1D70F}}$ . Analysis of DNS data suggests that the effect of this spanwise slip can be captured in a function $\unicode[STIX]{x1D6E4}(g^{+})$ , such that

(4.3) $$\begin{eqnarray}\langle \overline{p^{2}}\rangle _{w,SH}^{+}\approx \unicode[STIX]{x1D6E4}(g^{+})+\langle \overline{p^{2}}\rangle _{w,smooth}^{+}\approx \unicode[STIX]{x1D6E4}(g^{+})+2.31\ln (Re_{\unicode[STIX]{x1D70F}})-9.5.\end{eqnarray}$$

A best fit to the DNS data suggests the correlation

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}(g^{+})\approx \left\{\begin{array}{@{}ll@{}}2.32\ln (\{g^{+}\}^{3/4})-4.5,\quad & g^{+}\gtrsim 5\\ 0,\quad & g^{+}\lesssim 5.\end{array}\right.\end{eqnarray}$$

For $g^{+}\lesssim 5$ , the microgrooves become too narrow to generate any significant spanwise slip (Rastegari & Akhavan Reference Rastegari and Akhavan2018a ) and $\unicode[STIX]{x1D6E4}(g^{+})\approx 0$ . The correlations (4.3)–(4.4) fit the DNS data with a $p$ -value of $0.97$ , based on the Kolmogorov–Smirnov goodness of fit test. Figure 11(b) shows the comparison of these correlations to the DNS data.

Figure 12. The maximum permissible $We^{+}$ and $g^{+}$ at a given $Re_{\unicode[STIX]{x1D70F}}$ to maintain the stability of SH longitudinal microgrooves in turbulent channel flow. Lines show the predictions of (4.2) for: ——, $Re_{\unicode[STIX]{x1D70F}}\approx 200$ ; — —, $Re_{\unicode[STIX]{x1D70F}}\approx 400$ ; – – –, $Re_{\unicode[STIX]{x1D70F}}\approx 10^{3}$ ; — ⋅ —, $Re_{\unicode[STIX]{x1D70F}}\approx 10^{4}$ ; –  $\cdot$   $\cdot$  –,  $Re_{\unicode[STIX]{x1D70F}}\approx 10^{5}$ ; $\cdots \cdots$ ,  $Re_{\unicode[STIX]{x1D70F}}\approx 10^{6}$ .

Figure 12 shows the pressure stability bounds of SH longitudinal microgrooves obtained from (4.2), with $\unicode[STIX]{x0394}P^{+}$ approximated as $\unicode[STIX]{x0394}P^{+}\approx 4\{\langle \overline{p^{2}}\rangle _{w,SH}^{+}\!\}^{1/2}$ and $\langle \overline{p^{2}}\rangle _{w,SH}^{+}$ computed from (4.3) to (4.4). Due to the strong Reynolds number dependence of $\langle \overline{p^{2}}\rangle _{w,SH}^{+}$ , the pressure stability bounds of SH surfaces become significantly curtailed with increasing Reynolds number of the flow. Specifically, for the range of $10^{4}\lesssim Re_{\unicode[STIX]{x1D70F}_{0}}\lesssim 10^{5}$ and $We_{\unicode[STIX]{x1D70F}_{0}}\approx 5\times 10^{-3}$ , typical of practical applications, figure 3 shows that achieving ${\sim}50\,\%$ drag reduction requires a $(B-B_{0})\approx 12{-}14$ . At the same time, figure 12 shows that the microgroove widths should be kept to $g^{+}\lesssim 17{-}20$ to maintain the stability of the interface. The only way such large values of $(B-B_{0})$ can be achieved with such small values of $g^{+}$ is by using ‘blade’ or ‘scalloped’ longitudinal microgrooves with $\unicode[STIX]{x1D719}_{s}\lesssim 0.02{-}0.05$ , per (3.7), where the nomenclature for ‘blade’ or ‘scalloped’ longitudinal microgrooves is borrowed from the riblet literature (Bechert et al. Reference Bechert, Bruse, Hage, Van Der Hoeven and Hoppe1997).

The combination of (3.1), (3.7), (4.2) and (4.3)–(4.4) provides a set of design tools for a priori prediction of the magnitude of drag reduction and the sustainability bounds of SH surfaces in high Reynolds number turbulent flows. Given the Reynolds number, $Re_{\unicode[STIX]{x1D70F}_{0}}$ , and Weber number, $We_{\unicode[STIX]{x1D70F}_{0}}$ , of the base flow and a desired level of drag reduction, (3.1) provides an estimate of the required $(B-B_{0})$ , while (4.2) and (4.3)–(4.4) provide an estimate of the maximum groove width $g^{+}$ for stability. These $(B-B_{0})$ and $g^{+}$ values can then be used in (3.7) to determine the maximum allowable solid fraction, $\unicode[STIX]{x1D719}_{s}$ .

5 Summary and conclusions

Using scaling arguments and analysis of results from DNS, we present scaling laws which allow for a priori prediction of the magnitude of skin-friction drag reduction with SH or LI surfaces and the pressure stability bounds of SH surfaces in turbulent flow. It is shown that the magnitude of drag reduction can be parameterized in terms of the friction coefficient of the base flow and the shift, $(B-B_{0})$ , in the intercept of logarithmic-law representations of the mean velocity profiles in the flow with SH or LI walls compared to the base flow, where $(B-B_{0})$ is Reynolds number independent. The parameterization of $(B-B_{0})$ in terms of the geometrical parameters of the SH or LI surface, and the pressure stability bounds of SH longitudinal microgrooves in turbulent flow are presented. It is shown that, for a given geometry and size of the surface microtexture in wall units, both the magnitude of drag reduction and the sustainability bounds of SH surfaces degrade with increasing Reynolds number of the base flow. Using these scalings, the narrow range of surface microtexture geometrical parameters which can yield large drag reduction while maintaining the stability of SH interfaces in high Reynolds number turbulent flows have been identified.