2.1 Dimensionless parameters

In this problem we consider three key dimensionless parameters on scales relevant to turbulence, superhydrophobic surfaces, and surface tension of the gas–liquid interface. The first dimensionless parameter is the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}=u_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D708}$ , which measures the separation of length scales from the boundary layer thickness, $\unicode[STIX]{x1D6FF}$ , to the viscous unit length $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$ , where $u_{\unicode[STIX]{x1D70F}}$ is the friction velocity defined by $\sqrt{\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D70C}}$ and $\unicode[STIX]{x1D70F}_{w}$ is wall shear stress. $\unicode[STIX]{x1D70F}_{w}$ is the mean shear averaged over the entire interface area (including solid zones and shear-free air zones). This mean shear can be computed from the mean pressure gradient, using a momentum balance leading to $2\unicode[STIX]{x1D70F}_{w}=(\overline{\text{d}p}/\text{d}x)(2\unicode[STIX]{x1D6FF})$ . In this work we run our simulations at $Re_{\unicode[STIX]{x1D70F}}\approx 200$ –400. These Reynolds numbers are much lower than realizable numbers in practical applications for naval applications, typically $Re_{\unicode[STIX]{x1D70F}}\gtrsim 4000$ ; for example, a free-stream velocity of ${\sim}5~\text{m}~\text{s}^{-1}$ over a plate ${\sim}1$  m long.

However, since superhydrophobic surfaces only modify the inner region of turbulent wall-bounded flows, their effects can be studied using low $Re_{\unicode[STIX]{x1D70F}}\approx 180$ –200 as long as the dimensionless quantities based on the inner scale match the application of interest (Martell et al. Reference Martell, Rothstein and Perot2010; Seo et al. Reference Seo, García-Mayoral and Mani2015). Martell et al. (Reference Martell, Rothstein and Perot2010) first showed that the mean velocity profiles with two different $Re_{\unicode[STIX]{x1D70F}}$ are collapsed when $L^{+}$ is fixed. Seo et al. (Reference Seo, García-Mayoral and Mani2015) showed that the effects of superhydrophobic surfaces on turbulent flows are confined to the near-wall region, $y^{+}\lessapprox 80$ , as long as the superhydrophobic surface texture scale is smaller than the outer scale of the flow. They concluded that this modification to the inner region of the flow can be captured insensitive to the Reynolds number down to $Re_{\unicode[STIX]{x1D70F}}\approx 200$ . Increasing Reynolds number only modifies the outer flow in the same fashion as in the boundary layer over a smooth wall. In other words, the Reynolds number effects can be captured well by extending the log-layer in turbulent wall-bounded flows.

Another important dimensionless parameter is the size of the texture in viscous units $L^{+}=L/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ , a measure of how large the texture is compared to the near-wall eddies. Many computational (Martell et al. Reference Martell, Rothstein and Perot2010; Park et al. Reference Park, Park and Kim2013; Türk et al. Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014; Lee et al. Reference Lee, Jelly and Zaki2015; Rastegari & Akhavan Reference Rastegari and Akhavan2015; Seo et al. Reference Seo, García-Mayoral and Mani2015; Seo & Mani Reference Seo and Mani2016) and experimental (Daniello et al. Reference Daniello, Waterhouse and Rothstein2009; Park et al. Reference Park, Sun and Kim2014) investigations show that the slip length, and thus drag reduction, increase with larger $L^{+}$ when the solid fraction is fixed. Seo & Mani (Reference Seo and Mani2016) showed that the slip length followed a linear scaling of texture size, and matched with the analytical solution from Stokes flow (Ybert, Barentin & Cottin-Bizonne Reference Ybert, Barentin and Cottin-Bizonne2007; Davis & Lauga Reference Davis and Lauga2010) that obtained from the same geometry, when texture size in wall units is smaller than approximately 10 for isotropic posts. When the texture size becomes larger, the slip length in wall units increases nonlinearly with a scaling exponent less than 1 (Park et al. Reference Park, Park and Kim2013; Türk et al. Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014; Rastegari & Akhavan Reference Rastegari and Akhavan2015; Seo & Mani Reference Seo and Mani2016). For small $L^{+}$ , flows with superhydrophobic surfaces preserve the behaviour of canonical smooth-wall flows, while for large $L^{+}$ the effect of superhydrophobic surfaces completely disrupts the buffer layer, similar to the behaviour of flows over rough walls. Recent DNS (Türk et al. Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014; Rastegari & Akhavan Reference Rastegari and Akhavan2015; Seo et al. Reference Seo, García-Mayoral and Mani2015) reached $L^{+}$ down to the size relevant to practical applications $L^{+}\approx 6$ –8, comparable to typical experimental values of the texture period $L^{+}\approx 0.5$ –5 (Daniello et al. Reference Daniello, Waterhouse and Rothstein2009; Woolford et al. Reference Woolford, Prince, Maynes and Webb2009; Park et al. Reference Park, Sun and Kim2014). In the present work, we have investigated textures with sizes $L^{+}\approx 13$ –155, where the smallest texture size is close to the realistic texture size. Texture size on the order of hundreds is not considered in our study since the gas pockets would be destroyed due to high momentum of turbulent shear and slip flows.

Another texture-related dimensionless parameter is the solid fraction, which is equal to $\unicode[STIX]{x1D719}_{s}=w^{2}/L^{2}$ in the case of isotropic posts. This parameter is typically in the range 10 %–20 % and does not vary by an order of magnitude. In the present study we considered $\unicode[STIX]{x1D719}_{s}=1/9$ , which is in the range of practical scenarios. We considered one case for a streamwise ridge with a solid fraction of $\unicode[STIX]{x1D719}_{s}=w/L=1/3$ .

The last dimensionless number is the Weber number, which measures the relative importance of the surface tension to the momentum. Using inner scalings, Weber number could be defined as $We^{+}=\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}^{2}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}/\unicode[STIX]{x1D70E}$ . Noting $u_{\unicode[STIX]{x1D70F}}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D708}$ , this parameter can be also presented as a Capillary number $We^{+}=Ca^{+}=\unicode[STIX]{x1D707}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D70E}$ . When discussing the DNS results, we will use an alternative Weber number defined based on the texture size, $We_{L}=\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}^{2}L/\unicode[STIX]{x1D70E}=We^{+}L^{+}$ . In our simulations we use $We_{L}=10^{-3}{-}8\times 10^{-3}$ . Another Weber number can be defined based on the slip velocity, $U_{s}$ , as $We_{s}=\unicode[STIX]{x1D70C}U_{s}^{2}L/\unicode[STIX]{x1D70E}$ . We will show that this definition leads to collapse of data related to the flow-induced capillary waves. However, since $We_{s}$ is not known a priori, we prefer to keep $We_{L}$ (or $We^{+}$ ) as an input dimensionless parameter, and to provide explicit relations leading to $We_{s}$ after investigating DNS data.

In § 7, we will discuss the parameter space and regions of stable design. In this case, it is beneficial to use $We^{+}$ as the dimensionless parameter instead of $We_{L}$ . This is because $We^{+}$ is solely dependent on the imposed flow, and independent of superhydrophobic surface design choices (e.g., $L^{+}$ and advancing and receding contact angles). For channel flows with a prespecified pressure gradient, $u_{\unicode[STIX]{x1D70F}}$ is known a priori and so is $We^{+}$ . For boundary layers, $U_{\infty }$ is known instead of $u_{\unicode[STIX]{x1D70F}}$ , and thus $We^{+}$ is not exactly independent of texture parameters. As a useful rough approximation, however, one can write $We^{+}$ in terms of the free-stream velocity as $We^{+}\simeq \unicode[STIX]{x1D707}U_{\infty }/(25\unicode[STIX]{x1D70E})$ . This approximation can be justified given that the ratio $U_{\infty }/u_{\unicode[STIX]{x1D70F}}$ is in the range 22–30 in boundary layers over practical ranges of Reynolds numbers, $Re_{\unicode[STIX]{x1D70F}}\approx 1000{-}20\,000$ , with a weak logarithmic dependence on the Reynolds number. In situations with slip velocity, one can use the shifted-TBL model (Seo & Mani Reference Seo and Mani2016) and find that the same approximation holds as long as drag reduction is much less than the drag itself. In other words, in a design problem, a reasonable approximation of $We^{+}$ is available prior to the decision on the design details, while one can easily improve on these approximations a posteriori via algebraic relations provided for the shifted-TBL model and slip length in Seo & Mani (Reference Seo and Mani2016). These small corrections are expected to converge very rapidly with a small number of iterations.

In § 7 we will discuss additional parameters relating to design, including the advancing contact angle, which is related to the surface chemistry and determines the onset of failure. Another design consideration is the use of barriers in between the texture grooves to keep air regions in the form of isolated pockets. However, for the majority of the paper, where the physics of flow-induced capillary waves are discussed, we only need $L^{+}$ and $We^{+}$ as the key controlling parameters.

4.1 Effects of interface deformation

4.2 Reynolds number dependence

Next, we examine the dependence of the upstream-travelling capillary waves on system parameters in wall units. We first change the friction Reynolds number from $Re_{\unicode[STIX]{x1D70F}}\approx 200$ to $Re_{\unicode[STIX]{x1D70F}}\approx 400$ and investigate the effect of $Re_{\unicode[STIX]{x1D70F}}$ . In figure 9, turbulence statistics show that the near-wall behaviour of superhydrophobic surfaces with a deformable interface is essentially independent of Reynolds number when the other parameters are fixed in wall units. The mean quantities at the wall, for instance the slip velocity or the wall pressure, are well collapsed to the same values at two Reynolds numbers. The statistics away from the wall are identical to the outer region of smooth-wall channel flows and consistent with previous studies. The investigation of wall pressure in figure 10(a) qualitatively confirms that the structure of the flow-induced capillary waves remains essentially unmodified. Space–time correlation of the pressure signal in figure 10(b) further confirms that the convection velocity of higher Reynolds number, $U_{c_{2}}^{+}\approx -38$ , is unaltered from the lower Reynolds number simulation. Therefore, we believe our current results near the gas–liquid interface can be extended to higher Reynolds numbers, and the characteristics of pressure fluctuations are mainly governed by $L^{+}$ and $We_{L}$ .

Figure 11. Instantaneous wall pressure, $p^{+}$ , contours for cases with $Re_{\unicode[STIX]{x1D70F}}\approx 200$ and $L^{+}\approx 77$ . (a) $We_{L}=10^{-3}$ , (b) $We_{L}=2\times 10^{-3}$ , (c) $We_{L}=4\times 10^{-3}$ .

4.3 Effect of surface tension

In DNS, we systematically change $We_{L}$ and investigate how the dynamic characteristics of pressure changes. In figure 11, we plot instantaneous realizations of wall pressure with three different Weber numbers for $L^{+}\approx 77$ . As $We_{L}$ increases, the spanwise-coherent pressure becomes dominant and its magnitude intensifies. In addition, it is visually shown that the wavelength $\unicode[STIX]{x1D706}_{x}^{+}$ changes with $We_{L}$ . For instance, wavelength becomes shortened approximately from $\unicode[STIX]{x1D706}_{x}^{+}\approx 3L^{+}$ to $\unicode[STIX]{x1D706}_{x}^{+}\approx 2L^{+}$ as $We_{L}$ increases from $2\times 10^{-3}$ to $4\times 10^{-3}$ .

The consequent interface fluctuations in figure 12 intensify with larger $We_{L}$ in response to the increased spanwise-coherent pressure. The large deformation of the interface implies that the gas–liquid interface becomes less stable when $We_{L}$ increases.

Figure 12. Instantaneous interface deformation, $\unicode[STIX]{x1D702}^{+}$ , for cases with $Re_{\unicode[STIX]{x1D70F}}\approx 200$ and $L^{+}\approx 77$ . (a) $We_{L}=10^{-3}$ , (b) $We_{L}=2\times 10^{-3}$ , (c) $We_{L}=4\times 10^{-3}$ . The dashed lines represent the baseline wall location $y^{+}=0$ . Snapshots are taken at the same instance in which the pressure snapshots in figure 11 are taken. The dashed lines are the baseline wall location $y^{+}=0$ . The axis for $\unicode[STIX]{x1D702}^{+}$ is 300 times magnified for visualization purposes.

Figure 13. Comparison of space–time correlations of dynamic wall pressure signals, $p^{+}$ , for cases with $Re_{\unicode[STIX]{x1D70F}}\approx 200$ and $L^{+}\approx 77$ . (a) $We_{L}=10^{-3}$ , – – – –: $U_{c_{2}}^{+}\approx -23$ ; (b) $We_{L}=2\times 10^{-3}$ , – – – –: $U_{c_{2}}^{+}\approx -42$ ; (c) $We_{L}=4\times 10^{-3}$ , – – – –: $U_{c_{2}}^{+}\approx -78$ . ——: $U_{c_{1}}^{+}\approx 21$ for all cases.

In figure 13, the space–time correlations of the pressure signal show that for the larger $We_{L}$ , $\unicode[STIX]{x1D706}_{x}^{+}$ becomes shorter, the wave period, $T^{+}$ becomes longer, and thus  $U_{c}^{+}$ becomes slower. The advection of turbulence remains unchanged, $U_{c_{1}}^{+}\approx 21$ , regardless of the change of $We_{L}$ since slip velocity is the same for same $L^{+}$ . That is, the advection of turbulence is not affected by pressure fluctuations imposed by deformability of the interface while the upstream-travelling wave is strongly dependent on the Weber number.

4.4 Dependence on the texture size

Next, we investigate the effect of texture size by reducing the texture size down to $L^{+}\approx 13$ . In figure 14, we portray wall pressure snapshots with texture size spanning $L^{+}\approx 26$ –77 at fixed $We_{L}=4\times 10^{-3}$ and $Re_{\unicode[STIX]{x1D70F}}\approx 200$ . At texture size $L^{+}\approx 26$ , the stagnation pressure and the footprint from overlying turbulence, which scales ${\sim}100\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ , are dominant components of the total pressure fluctuations, and the pressure due to the capillary wave is hard to recognize. The weakness of the capillary pressure holds for smaller texture size $L^{+}\approx 13$ , and in that case the capillary pressure is completely overwhelmed by other fluctuations so that only the downstream-propagating component remains in the space–time correlation analysis. These results indicate that the deleterious effect of spanwise-coherent waves on the stability of gas pockets becomes significant when $L^{+}$ becomes large.

Figure 14. Instantaneous wall pressure contours, $p^{+}$ with (a) $L^{+}\approx 26$ , (b) $L^{+}\approx 38$ , (c)  $L^{+}\approx 77$ at $We_{L}=4\times 10^{-3}$ and $Re_{\unicode[STIX]{x1D70F}}\approx 200$ . From blue to yellow, the fluctuations range between $-10$ and 10 wall units.

Figure 15. Statistics of interfacial quantities scaled in viscous units versus $We_{L}$ . (a) Wavelength of upstream-travelling wave in the streamwise direction, $\unicode[STIX]{x1D706}_{x}^{+}$ ; (b) time period of the upstream-travelling wave, $T^{+}$ ; (c) convection velocity of the upstream-travelling wave, $U_{c}^{+}$ . ♦: $L^{+}\approx 26$ , ▪: $L^{+}\approx 38$ , ▴: $L^{+}\approx 77$ , ●: $L^{+}\approx 155$ . Symbols are computed by averaging at least four uncorrelated signals, and error bars plotted on top of the symbols.

The phase velocity, wavelength, and time scale of the capillary wave covering a wide range of $L^{+}$ and $We_{L}$ are summarized in figure 15. The wavelength and time period are calculated from space–time correlations of the capillary pressure. The averaged data from at least four different uncorrelated time windows are plotted with error bars. The results show that $\unicode[STIX]{x1D706}_{x}^{+}$ is comparable to several times $L^{+}$ , and becomes longer for increasing $L^{+}$ under the same $We_{L}$ . With larger $L^{+}$ for fixed $We_{L}$ , the time scale of the capillary waves increases, and the convection speed decreases. In the present form, the data do not show collapse on any of the measured quantities.

4.5 Data collapse with $We_{s}$

So far we have reported the capillary wave speed, wavelength, and period in wall units, i.e. $u_{\unicode[STIX]{x1D70F}},\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ , for length and velocity scales. The friction Reynolds number independence shown in § 4.2 indicates that the interfacial phenomena would be not directly connected to the overlying outer turbulence scales. Since the interface deformation is small and its effect is confined to the near-wall region, we hypothesize that the interfacial phenomena are separable from the scales associated with the overlying turbulence. In this case, a more suitable set of reference dimensions would be the texture size, $L$ , and the flow slip velocity, $U_{s}$ .

In figure 16, we rescale the wavelength, time frequency, and convection velocity with $L$ and $U_{s}$ . As a result, the dimensionless time $T^{\ast }=TU_{s}/L$ and velocity $U_{c}^{\ast }=U_{c}/U_{s}$ show excellent collapse when scaled with $We_{s}=\unicode[STIX]{x1D70C}U_{s}^{2}L/\unicode[STIX]{x1D70E}$ . The dimensionless wavelength $\unicode[STIX]{x1D706}^{\ast }=\unicode[STIX]{x1D706}_{x}/L$ shows reasonable collapse. This data collapse reveals that the main parameter governing the interfacial flow is $We_{s}$ . This result supports the hypothesis that the observed coherent pressure waves are capillary waves that develop as modes of oscillation of the interface as a membrane.

5.1 Model formulation

We consider an inviscid flow of density $\unicode[STIX]{x1D70C}$ slipping over a superhydrophobic surface with posts separated by a distance $L$ , with uniform mean velocity $U$ , and with small fluctuations $(u,v,w;p)$ on the mean flow induced by a deformable interface with finite surface tension $\unicode[STIX]{x1D70E}$ . This problem is analogous to the wave propagation in liquid films (Squire Reference Squire1953; Taylor Reference Taylor1959), except for the presence of the solid posts, which make the film discontinuous. The base state involves a uniform streamwise velocity, with no interface deformation, which trivially satisfies the Euler equation, continuity, and the no-penetration condition on the interface. The fluctuating velocity field can be described in terms of a potential $\unicode[STIX]{x1D713}(x,y,z)$ , which satisfies

Using Fourier decomposition along the wall-parallel directions ( $x$ and $z$ ), the above Poisson equation adopts the form $(-k_{x}^{2}-k_{z}^{2}+\unicode[STIX]{x2202}_{y}^{2})\,\widehat{\unicode[STIX]{x1D713}}=0$ for each $(k_{x},k_{z})$ mode, where $\widehat{\unicode[STIX]{x1D713}}$ means Fourier coefficient of $\unicode[STIX]{x1D713}$ . For vanishing $\widehat{\unicode[STIX]{x1D713}}$ at $y\rightarrow \infty$ , the solution is $\widehat{\unicode[STIX]{x1D713}}(y)=\widehat{\unicode[STIX]{x1D713}}_{y=0}\exp (-\sqrt{k_{x}^{2}+k_{z}^{2}}\,y)$ . This allows us to establish a relationship at $y=0$ between the potential and its $y$ -derivative,

where $\unicode[STIX]{x1D641}$ and $\unicode[STIX]{x1D641}^{-1}$ denote discrete direct and inverse discrete Fourier transform operators, and $\unicode[STIX]{x1D646}$ is the diagonal matrix formed by the derivative eigenvalues $-\sqrt{k_{x}^{2}+k_{z}^{2}}$ . Note that, by the definition of the potential, $\unicode[STIX]{x2202}_{y}\unicode[STIX]{x1D713}=v$ . Therefore, we are interested only in the restriction of (5.2) that also satisfies $(\unicode[STIX]{x2202}_{y}\unicode[STIX]{x1D713})_{y=0}=v_{y=0}=0$ over the solid posts.

Figure 16. Statistics of interfacial quantities scaled with $L$ and $U_{s}$ versus $We_{s}$ . (a) wavelength of the upstream-travelling wave scaled by the texture size, $\unicode[STIX]{x1D706}^{\ast }=\unicode[STIX]{x1D706}_{x}/L$ ; (b) time frequency of the upstream-travelling wave, $T^{\ast }=TU_{s}/L$ ; (c) convection velocity of the upstream-travelling wave scaled by the slip velocity, $U_{c}^{\ast }=U_{c}/U_{s}$ . ♦: $L^{+}\approx 26$ , ▪: $L^{+}\approx 38$ , ▴: $L^{+}\approx 77$ , ●: $L^{+}\approx 155$ .

The potential $\unicode[STIX]{x1D713}$ can also be related to the fluctuating pressure through the linearized inviscid momentum equation,

which can be particularized at $y=0$ . Let us also note that the fluctuating deformation of the interface satisfies the Young–Laplace equation

where $\unicode[STIX]{x1D702}$ is connected to the wall-normal velocity through the material derivative,

We are interested in wave solutions of the fully coupled system of equations (5.2)–(5.5). Using the corresponding transformation $\unicode[STIX]{x2202}_{t}=\text{i}\unicode[STIX]{x1D714}$ , the problem can be written in the frequency- $\unicode[STIX]{x1D714}$ domain,

where the dependence of $\unicode[STIX]{x1D63C}$ on $\unicode[STIX]{x1D714}$ is quadratic. The problem then reduces to finding values of $\unicode[STIX]{x1D714}$ for which the determinant of $\unicode[STIX]{x1D63C}$ is zero, and the corresponding eigen-solutions in the null subspace. The above system of equations can be normalized using reference scales of length, $L_{r}=L$ , velocity $U_{r}=U$ , which lead to subsequent scales of time, $T_{r}=L/U$ , pressure, $p_{r}=\unicode[STIX]{x1D70C}U^{2}$ , and surface tension $\unicode[STIX]{x1D70E}_{r}=\unicode[STIX]{x1D70C}U^{2}L$ . In the end, the dimensionless system of equation requires only one physical input parameter – that is, the Weber number, $We=\unicode[STIX]{x1D70C}U^{2}L/\unicode[STIX]{x1D70E}$ . Solutions exist only for a discrete set of frequencies $\unicode[STIX]{x1D714}^{\ast }=\unicode[STIX]{x1D714}L/U$ , and take the form of either upstream- or downstream-travelling waves with a phase velocity $U_{c}$ that can be several times larger than the flow velocity  $U$ .

5.4 A dispersion relation of semianalytical capillary waves

In order to develop some intuition about the prediction of the semianalytical model, in this section we present a comparison between the results of this model and phenomenological scaling laws of the frequency versus wavelength relations. We note, however, this comparison is made in simplifying limits and regimes that do not necessarily represent the considered turbulent flow scenarios. We first recall that in the small-capillary-wavelength limit, $\unicode[STIX]{x1D706}^{\ast }=\unicode[STIX]{x1D706}/L\ll 1$ , the texture would not affect the capillary waves, and the standard theory would be capable of predicting the dispersion relation.

This scaling can be obtained by combining a system of equations from (5.3) to (5.5), providing $\unicode[STIX]{x1D714}\unicode[STIX]{x1D713}\sim P/\unicode[STIX]{x1D70C}$ , $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D706}^{2}\sim P/\unicode[STIX]{x1D70E}$ , and $\unicode[STIX]{x1D714}\unicode[STIX]{x1D702}\sim \unicode[STIX]{x1D713}/\unicode[STIX]{x1D706}$ , therefore $\unicode[STIX]{x1D714}\sim \sqrt{\unicode[STIX]{x1D70E}/(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D706}^{3})}$ . When scaled with $L$ and $U_{s}$ , the time frequency associated with this wave, $\unicode[STIX]{x1D714}^{\ast }\equiv \unicode[STIX]{x1D714}L/U_{s}$ , would be

In the large-wavelength-limit $\unicode[STIX]{x1D706}\gg L$ , however, this scaling should be modified. When the wavelength is sufficiently larger than texture size, one can assume that dependence of pressure on the amplitude would scale as $p\sim \unicode[STIX]{x1D70E}\unicode[STIX]{x1D702}/L^{2}$ . This is because the dominant spatial variation of the capillary wave would depend on $L$ . This relation combined with $\unicode[STIX]{x1D714}\unicode[STIX]{x1D713}\sim P/\unicode[STIX]{x1D70C}$ , and $\unicode[STIX]{x1D714}\unicode[STIX]{x1D702}\sim \unicode[STIX]{x1D713}/\unicode[STIX]{x1D706}$ , results in $\unicode[STIX]{x1D714}\sim \sqrt{\unicode[STIX]{x1D70E}/(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D706}L^{2})}$ . The consequent dimensionless frequency of the wave is

In this analysis, in addition to the large-wavelength limit, we also assume that the capillary speed, $U_{c}\sim \sqrt{\unicode[STIX]{x1D70E}\unicode[STIX]{x1D706}/(\unicode[STIX]{x1D70C}L^{2})}$ , is much larger than the slip velocity, $U_{s}$ , and thus the effect of the background advection can be ignored (i.e., small $We_{s}$ ). In figure 18, we show log–log plots of time frequency of the interface from a semianalytical solution for the range of $0.1\leqslant We_{s}\leqslant 2.0$ and $2^{2}\leqslant \unicode[STIX]{x1D706}/L\leqslant 2^{4}$ . In figure 18(a), the semianalytical model shows the expected inverse square root scaling for $\unicode[STIX]{x1D706}^{\ast }$ for the whole range of $We_{s}$ . In figure 18(b), the frequency is rescaled with $w^{\ast }\sqrt{\unicode[STIX]{x1D706}^{\ast }}$ . As $\unicode[STIX]{x1D706}^{\ast }$ becomes large, $\unicode[STIX]{x1D706}^{\ast }\gg 1$ , the whole range of $We_{s}$ exhibits the inverse square root scaling with respect to $We_{s}$ , as expected from (5.8).

Figure 18. Dimensionless time frequency solutions, $\unicode[STIX]{x1D714}^{\ast }$ , from the semianalytical model. (a)  $\unicode[STIX]{x1D714}^{\ast }$ versus $\unicode[STIX]{x1D706}^{\ast }$ with an arrow with increasing $We_{s}=0.1{-}2.0$ with successive interval 0.1. (b)  $\unicode[STIX]{x1D714}^{\ast }\sqrt{\unicode[STIX]{x1D706}^{\ast }}$ versus $We_{s}$ with an arrow with increasing $\unicode[STIX]{x1D706}^{\ast }=2^{2},2^{3},2^{4}$ .

While this phenomenological scaling provides an understanding of the dispersion relations in the induced capillary waves, the DNS data does not necessarily fall in the assumed simplified regimes considered here. Therefore, the prediction of dispersion relations in the DNS results requires comparison with the full semianalytical model. Another shortcoming is that the provided scalings do not predict which capillary wavelength(s) are most energized by the overlying turbulent flow, neither do they predict the amplitude of these waves in the statistically stationary condition. In the next section we use DNS data directly to infer scaling laws predicting the capillary wavelength and amplitude in terms of input conditions. In the discussion section we suggest possible methods for a physics-based explanation of these observations.

6.1 Amplitude of capillary pressure and interface deformation

To assess the magnitude of flow-induced capillary pressure fluctuations, we decompose the spatio-temporal pressure fields. We have identified three separate phenomena contributing to pressure fluctuations: the overlaying turbulent flow, stagnation phenomena, and capillary waves. First, we exclude the effect of stagnation pressure by a decomposition of the total signal to obtain an unsteady signal, $p^{\prime \prime }$ , so that

where $\bar{p}(y)$ is averaged pressure over wall-parallel domain and time, and $\tilde{x}=\text{modulo}(x,L_{x})$ and $\tilde{z}=\text{modulo}(z,L_{z})$ are the periodic streamwise and spanwise coordinates within each pattern unit. The stagnation component $\tilde{p}$ is averaged over time and over the number of periodic units. The above decomposition method, first introduced by Reynolds & Hussain (Reference Reynolds and Hussain1972), was used to analyse turbulent flows over stagnation wall modifications (Choi, Moin & Kim Reference Choi, Moin and Kim1993; García-Mayoral & Jiménez Reference García-Mayoral and Jiménez2011; Jelly et al. Reference Jelly, Jung and Zaki2014; Türk et al. Reference Türk, Daschiel, Stroh, Hasegawa and Frohnapfel2014; Seo et al. Reference Seo, García-Mayoral and Mani2015). The resulting pressure fluctuation $p^{\prime \prime }$ is a sum of the two time-dependent effects from overlying turbulence $p_{t}^{\prime \prime }$ , and flow-induced capillary pressure $p_{c}^{\prime \prime }$ . We assume the capillary pressure is statistically uncorrelated from the stagnation pressure and overlying turbulence. Therefore, r.m.s. fluctuation of capillary pressure is obtained by

where $p_{t,rms}^{\prime \prime }$ is the wall pressure fluctuation from overlying turbulence when the interface is flat. $p_{t,rms}^{\prime \prime }$ is calculated from the DNS data by Seo et al. (Reference Seo, García-Mayoral and Mani2015). The total wall pressure fluctuation $p_{rms}^{\prime +}$ , and each pressure component from stagnation pressure $\tilde{p}_{rms}^{+}$ , capillary pressure $p_{c,rms}^{\prime \prime +}$ , and turbulence pressure $p_{t,rms}^{\prime \prime +}$ are plotted in figure 19.

We do not apply this decomposition method to extract the capillary contribution from the $\unicode[STIX]{x1D702}^{+}$ fluctuation, since decomposition of the turbulence contribution to interface deformation requires solution of the Young–Laplace equation over a smooth wall, in a one-way coupled fashion. This is a more cumbersome postprocessing step than decomposing the pressure data.

Figure 19. Decomposition of wall pressure fluctuations for $L^{+}\approx 155$ and $We_{L}=4\times 10^{-3}$ . (a) total r.m.s. pressure fluctuation, $p_{rms}^{\prime +}$ , (b) r.m.s. stagnation pressure fluctuation, $\tilde{p}_{rms}^{+}$ , (c) r.m.s. capillary pressure $p_{c,rms}^{\prime \prime +}$ , (d) r.m.s. turbulence pressure, $p_{t,rms}^{\prime \prime +}$ .

Figure 20 presents $\unicode[STIX]{x1D702}_{rms}^{+}$ and $p_{c,rms}^{\prime \prime +}$ against the input Weber number $We_{L}$ for different texture sizes. In figure 20(a), $\unicode[STIX]{x1D702}_{rms}^{+}$ includes all interfacial responses to pressure fluctuations from stagnation, capillary waves, and turbulence, while in figure 20(b), $p_{c,rms}^{\prime \prime +}$ includes only the flow-induced capillary pressure fluctuation. The error bars in figure 20(b) are obtained from standard error of mean (SEM) of each $p_{rms}^{\prime \prime }$ and $p_{t,rms}^{\prime \prime }$ sampling mean values in subdivided time intervals. Both interface deformations and capillary pressure fluctuations increase with increasing $We_{L}$ for a fixed $L^{+}$ ; however, they do not collapse for different texture sizes.

Figure 20. Statistics of fluctuations of interface deformation and capillary pressure versus $We_{L}$ . (a) r.m.s. interface deformation in wall units, $\unicode[STIX]{x1D702}_{rms}^{+}$ , (b) r.m.s. capillary pressure $p_{c}^{\prime \prime +}$ in wall units, ▾: $L^{+}\approx 13$ , ♦: $L^{+}\approx 26$ , ▪: $L^{+}\approx 38$ , ▴: $L^{+}\approx 77$ , ●: $L^{+}\approx 155$ .

Similar to § 4.5, we reconsider what is the most relevant physical quantities for the capillary wave. Again, $We_{s}$ should be used instead of $We_{L}$ . Considering input variables $We_{s}$ , $L^{+}$ , and $U_{s}^{+}$ , we report the best collapsed fit of data to seek the power law for each input variables. We use a linear least squares fit to find the coefficients, $a_{i}$ , of $\log (p_{c}^{\prime \prime +})=a_{0}+a_{1}\log (We_{s})+a_{2}\log (L^{+})+a_{3}\log (U_{s}^{+})$ , in which $We_{s}$ , $L^{+}$ , and $U_{s}^{+}$ values are used for the corresponding $p_{c}^{\prime \prime +}$ data. As a result, a fitting law for the capillary wave is

as shown in figure 21. We note that the prefactor 0.06 is specifically obtained for $\unicode[STIX]{x1D719}_{s}=1/9$ . For superhydrophobic surface design purposes, using $We^{+}$ is more suitable than using $We_{s}$ , since it is dependent only on the imposed flow and independent of texture size. Noting that $We_{s}$ is simply a combination of the non-dimensional parameters of the system, $We_{s}=We^{+}L^{+}{U_{s}^{+}}^{2}$ , the equation (6.3) can be rewritten in terms of wall units as

Equation (6.4) is one of the key results of this study. This equation shows that the capillary pressure has a strong dependency on slip velocity. For cases that the working flow and the flow conditions are prespecified, $We^{+}$ would be fixed. However, if the texture size increases, the slip velocity increases accordingly, and the capillary pressure rapidly increases corresponding to the slip velocity.

Figure 21. Statistics of fluctuations of dimensionless r.m.s. capillary pressure, $p_{c}^{\prime \prime +}$ , versus $We_{s}$ . ▾: $L^{+}\approx 13$ , ♦: $L^{+}\approx 26$ , ▪: $L^{+}\approx 38$ , ▴: $L^{+}\approx 77$ , ●: $L^{+}\approx 155$ . – – – –: $p_{c,rms}^{\prime \prime +}\approx 0.06We_{s}^{0.57}{U_{s}^{+}}^{2.8}{L^{+}}^{-0.74}$ .

Figure 22. Scaling of interfacial quantities scaled in $L$ and $U_{s}$ versus $We_{s}$ . (a) Wavelength of capillary wave in streamwise direction, $\unicode[STIX]{x1D706}^{\ast }$ ; (b) convection velocity of capillary wave, $U_{c}^{\ast }$ . ♦: $L^{+}\approx 26$ , ▪: $L^{+}\approx 38$ , ▴: $L^{+}\approx 77$ , ●: $L^{+}\approx 155$ . – – – –: $\unicode[STIX]{x1D706}^{\ast }\approx 1.9{We_{s}}^{-0.5}$ , — ⋅ —: $U_{c}^{\ast }\approx 1.2{We_{s}}^{-1}$ .

6.2 Scaling of wavelength and phase speed

We report scalings as well as fitting coefficients for the estimation of expected behaviour of capillary waves. The log–log scale of data presented in figure 22 reveals that $\unicode[STIX]{x1D706}^{\ast }$ scales with $We_{s}^{-0.5}$ , and $U_{c}^{\ast }$ scales with $We_{s}^{-1}$ . The best fit for $\unicode[STIX]{x1D706}^{\ast }$ is

The best fit for phase speed $U_{c}^{\ast }$ measured from DNS data is

Both (6.5) and (6.3) are plotted in figure 22.

While $We_{s}$ is directly measured from DNS in the presented plots, we note that $We_{s}$ can be predicted a priori, given the texture parameters, flow conditions, and fluid properties, from the relation introduced by Seo & Mani (Reference Seo and Mani2016), that is $U_{s}^{+}=U_{s}^{+}(L^{+},\unicode[STIX]{x1D719}_{s})$ . Specifically they predicted $U_{s}^{+}\sim L^{+}$ in the small-texture-size limit, and $U_{s}^{+}\sim {L^{+}}^{1/3}$ for the large-texture-size limit.

7.1 Onset of failure by capillary pressure

We compare the pressure fluctuations from the stagnation pressure versus capillary pressure as a function of slip velocity in wall units in figure 23(a). While previous analysis by Seo et al. (Reference Seo, García-Mayoral and Mani2015) suggested a linear scaling of the stagnation pressure with respect to slip velocity, $\tilde{p}_{0,rms}^{+}\approx 0.28U_{s}^{+}+1.26$ , the pressure load imposed by flow-induced capillary waves is $p_{c,rms}^{\prime \prime +}\approx 0.06{We^{+}}^{0.57}{U_{s}^{+}}^{3.9}{L^{+}}^{-0.17}$ . Therefore, the capillary pressure becomes dominant over stagnation pressure shortly after $U_{s}^{+}\gtrsim 5$ , when we consider the surface tension in the practical application regime, $We^{+}=10^{-3}$ – $10^{-2}$ .

Figure 23. (a) Scalings of wall pressure fluctuations, $p_{0,rms}^{+}$ . $\tilde{p}_{s}^{+}$ is wall pressure fluctuation by stagnation pressure, $\tilde{p}_{s,rms}^{+}\approx 0.28U_{s}^{+}+1.26$ (Seo et al. Reference Seo, García-Mayoral and Mani2015).  $p_{c}^{\prime \prime +}$ is wall pressure fluctuation by capillary pressure, $p_{c,rms}^{\prime \prime +}\approx 0.06{We^{+}}^{0.57}{U_{s}^{+}}^{3.9}{L^{+}}^{-0.17}$ . (b) Schematic of example calculation showing interface deformation due to an overlying pressure field. When the slope of the interface at the edge of the post form an angle, $\unicode[STIX]{x1D703}_{L}$ , larger than the advancing contact angle, the interface starts to move.

Similar to Seo et al. (Reference Seo, García-Mayoral and Mani2015), we define the onset of failure as the conditions resulting in microscopic contact angles leading to the onset of motion for contact lines. A proper dimensional analysis of the Young–Laplace equation leads to the following scaling law for this failure criterion for interface breakage (Seo et al. Reference Seo, García-Mayoral and Mani2015),

The order one coefficient on the right-hand side of the equation is determined by the failure criteria that the contact angle of gas–liquid interface, $\unicode[STIX]{x1D703}_{L}=\unicode[STIX]{x03C0}/2+\tan ^{-1}(\text{d}\unicode[STIX]{x1D702}/\text{d}x|_{L})$ , exceeds the advancing contact angle, $\unicode[STIX]{x1D703}_{adv}$ , where $\text{d}\unicode[STIX]{x1D702}/\text{d}x|_{L}$ is the slope of the gas–liquid interface at the leading edge of the post as shown in figure 23(b). Seo et al. (Reference Seo, García-Mayoral and Mani2015) used the self-similar stagnation pressure fields to obtain the $O(1)$ constant on the right-hand side of (7.1) for $\unicode[STIX]{x1D703}_{adv}=100^{\circ }$ and $120^{\circ }$ . For instance, the resulting relationship for stagnation pressure is $0.28U_{s}^{+}We^{+}L^{+}=f(\unicode[STIX]{x1D703}_{adv})$ , where $f$ is computed to be equal to 1.7 for $\unicode[STIX]{x1D703}_{adv}=120^{\circ }$ and equal to 0.5 for $\unicode[STIX]{x1D703}_{adv}=100^{\circ }$ . $U_{s}^{+}$ is a known function of $L^{+}$ obtained by the scaling introduced in Seo & Mani (Reference Seo and Mani2016) as:

We repeated the same procedure for the capillary waves. Given that the capillary waves observed in DNS have wavelengths larger than the texture size, $L$ , we used a uniform pressure on the textured interface to approximate $f(\unicode[STIX]{x1D703}_{adv})=O(1)$ constant on the right-hand side of (7.1) for this mechanism. The $f$ constant is computed by measuring the uniform $p^{+}$ that leads to deformation angle equal to advancing contact angle. With this simplification the right-hand-side constant is computed to be equal to 0.6 and 0.2, respectively, for $\unicode[STIX]{x1D703}_{adv}=120^{\circ }$ and $100^{\circ }$ , considering square patterns with $\unicode[STIX]{x1D719}_{s}=1/9$ . In order to consider the worst case scenario we used the peak of the capillary pressure estimated as $P_{peak}\simeq \sqrt{2}p_{c,rms}^{\prime \prime }$ . The resulting equation for capillary waves with $\unicode[STIX]{x1D703}_{adv}=120^{\circ }$ is

where the coefficient 0.06 is subject to change with $\unicode[STIX]{x1D719}_{s}$ , and $U_{s}^{+}$ can be estimated from (7.2). In appendix B, we discuss the validity of linearization of the Young–Laplace equation to obtain the coefficient on the right-hand side.

7.2 Boundary map for stable superhydrophobic surface design

Using the failure mode described in § 7.1, we provide boundary maps for stable drag reduction of superhydrophobic surfaces. The boundary map consists of two independent design parameters: the texture size in wall units, $L_{c}^{+}$ , and Weber number in wall units, $We^{+}=\unicode[STIX]{x1D707}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D70E}$ . We will consider multiple contributions to the failure mode, so thus using $L^{+}$ and $We^{+}$ in design space is suitable while $We_{s}$ is the controlling parameter only for the capillary pressure. In figure 24(a), both criteria from stagnation pressure and capillary pressure fluctuations are represented together. In most of the practical applications of hydrodynamic flows, $We^{+}=10^{-3}{-}10^{-2}$ , and the texture size regime of interest for noticeable drag reduction is $L^{+}>1$ . In this regime, the capillary pressure sets a more restrictive boundary than the criterion imposed by the stagnation pressure. The change of slope in each plot indicates transition from $U_{s}^{+}\sim L^{+}$ to $U_{s}^{+}\sim {L^{+}}^{1/3}$ , respectively, for small and large texture sizes, as discussed by Seo & Mani (Reference Seo and Mani2016) and indicated in (7.2). The error bar, indicated by the dashed dotted line in figure 24(a), is computed from the error of the fit in figure 21. The 95 % confidence interval of the fit in figure 21 is computed from $1.96\sqrt{\text{mean}[(0.06{We_{s}}^{0.57}{U_{s}^{+}}^{2.8}{L^{+}}^{-0.74}-p_{c}^{\prime \prime +})^{2}+\text{SEM}(p_{c}^{\prime \prime +})^{2} )]}$ . This error, translated in the stability diagram, leads to 45 % error in the $L_{c}^{+}$ for $We^{+}=10^{-4}$ and 35 % error in the $L_{c}^{+}$ for $We^{+}=10$ .

Figure 24. Design map for the stable drag reduction of superhydrophobic surfaces in turbulent flows. $L_{c}^{+}$ is the maximum allowable texture size for stable drag reduction. $We^{+}=\unicode[STIX]{x1D707}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D70E}$ is Weber number based on inner scaling. The grey area is the unstable region and the white area is the stable region. (a) Shows the stability criteria imposed by stagnation pressure and capillary pressure. The dashed-dot lines are error bounds of the estimation of stability boundary for capillary pressure. (b) Shows the effect of texture randomness on the total stability region. Dashed lines are stability criteria imposed by stagnation pressure and capillary pressure for aligned posts. (c) Shows the effect of turbulence pressure on the total stability region. Dashed line is the solid curve in panel (b). Three different lines are plotted from different confidence intervals for pressure variation: 95 % (c), 99 % (b), 99.99 % (a). The area below the dotted lines indicates the region where $DR$ is less than 1 %. All analyses use the advancing contact angle $\unicode[STIX]{x1D703}_{adv}=120^{\circ }$ with the solid fraction of $\unicode[STIX]{x1D719}_{s}=0.11$ .

Next, we include more practical considerations to the presented analysis to make it more relevant to realistic applications. Our recent finding on the effect of texture randomness (Seo & Mani Reference Seo and Mani2017) indicates that the maximum deformation angle of the randomly distributed textured superhydrophobic surface is approximately twice that of the perfectly aligned posts considered in this study. For randomly distributed textured superhydrophobic surfaces, produced by spray coating or an etched process, this geometric randomness will push the boundaries imposed by our analysis from aligned, periodic textured superhydrophobic surfaces as shown in figure 24(b).

Moreover, the stability region will be further shrunk when we consider unsteady, intermittent turbulence pressure fluctuations in addition to the stagnation pressure and capillary pressure. The worst scenario for the interface stability can occur instantaneously when some of travelling turbulence pressure with high-intensity encounters an interface. An instantaneous local turbulence pressure fluctuation can be estimated by multiplication of 1.96 on $p_{t,rms}^{\prime \prime +}$ with 95 % confidence. We consider the multiplier 2.58 for 99 % confidence and 4.0 for 99.99 % confidence and the effect of using different confidence intervals is shown in figure 24(c). Here we avoid a higher number of digits in the confidence interval since the extremely rare high-pressure events are local in space and time (unlike capillary waves that have large features). Therefore, although they can cause instant deformation of the interface, it is unclear whether they lead to failure, as they may not persist long enough over a significant portion of a texture. In this analysis we use $p_{t,rms}^{\prime \prime +}\approx 3$ , which has been observed in DNS of turbulent channel flow up to $Re_{\unicode[STIX]{x1D70F}}\approx 5000$ (Lee & Moser Reference Lee and Moser2015). Our previous analysis has shown that the turbulence pressure fluctuation from overlying turbulent flows in the presence of texture, $p_{t,rms}^{\prime \prime +}$ , is marginally modified from a conventional smooth channel flow (Seo et al. Reference Seo, García-Mayoral and Mani2015). Higher Reynolds number involves slightly higher $p_{t,rms}^{\prime \prime +}$ with a logarithmic dependence on $Re_{\unicode[STIX]{x1D70F}}$ . As an example, a case with extremely large $Re_{\unicode[STIX]{x1D70F}}=5\times 10^{5}$ would lead to $p_{t,rms}^{\prime \prime +}\approx 4.5$ (Farabee & Casarella Reference Farabee and Casarella1991). This modification is smaller than the uncertainty associated with the discussed precision of the confidence interval. With these considerations, the boundary map, between stable and unstable designs, in figure 24(c) shows the stable design spaces, considering all of the aforementioned effects.

Figure 25. Total design map for the stable drag reduction of superhydrophobic surfaces in turbulent flows. $L_{c}^{+}$ is the maximum allowable texture size for stable drag reduction. $We^{+}=\unicode[STIX]{x1D707}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D70E}$ is Weber number based on inner scaling. Arrows indicate the variability of stability of gas pockets when increasing velocity fixing fluid and design parameters. The upper solid line uses $\unicode[STIX]{x1D703}_{adv}=120^{\circ }$ , and the bottom solid line uses $\unicode[STIX]{x1D703}_{adv}=100^{\circ }$ in the analysis of the failure mechanisms. Dashed lines are the failure boundaries imposed by stagnation and capillary pressure fluctuations. The dotted line is a boundary below which the drag reduction is expected to be under 1 %. The boundary map is obtained for the solid fraction $\unicode[STIX]{x1D719}_{s}=0.11$ and 95 % confidence level for the effect of turbulence pressure.

In summary, we recap the design criteria for stable drag reduction in figure 25. If the speed of overlying flow, $U$ , increases for fixed design parameters and fluid properties $(L,\unicode[STIX]{x1D719}_{s},\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707})$ , the viscous length, $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ , decreases, leading to a proportional increase in both $L^{+}$ and $We^{+}$ , and thus the state in the design map would move along the $45^{\circ }$ line up and to the right, as indicated by the arrows in figure 25. The maximum allowable drag reduction can be estimated by the shifted-TBL model and a phenomenological model for slip length suggested by Seo & Mani (Reference Seo and Mani2016) for the critical texture size in figure 25. We note that drag reduction is not only a function of slip length, but subject to change with Reynolds number. In this particular case we consider a scenario when the solid fraction is $\unicode[STIX]{x1D719}_{s}=1/9$ and $Re_{\unicode[STIX]{x1D70F}}\approx 5000$ for the estimation of drag reduction. The critical texture length in figure 25 shows that the maximum allowable drag reduction in a realistic superhydrophobic surface with flow condition $We^{+}=10^{-3}{-}10^{-2}$ would be approximately from 15 % to 45 % for the best chemically coated superhydrophobic surfaces with the maximum microscopic contact angle of $\unicode[STIX]{x1D703}_{adv}=120^{\circ }$ . If a superhydrophobic surface has a different chemistry which lowers the advancing contact angle, e.g. $\unicode[STIX]{x1D703}_{adv}=100^{\circ }$ , the boundary would be further pushed downwards, limiting the maximum allowable drag reduction from ${\approx}45\,\%$ to 30 % at $We^{+}\approx 10^{-3}$ . The overall prediction of the maximum stable drag reduction by our analysis is consistent with current experimental observations, in which most successful drag reductions were limited to be approximately less than 30 % (Bidkar et al. Reference Bidkar, Leblanc, Kulkarni, Bahadur, Ceccio and Perlin2014; Haibao et al. Reference Haibao, Peng, Feng, Dong and Yang2015; Srinivasan et al. Reference Srinivasan, Kleingartner, Gilbert, Cohen, Milne and McKinley2015; Zhang et al. Reference Zhang, Tian, Yao, Hao and Jiang2015).

Figure 26. Schematic representations of barriers to prevent the shear-driven drainage and stability criteria. $L$ is pattern period of post arrays, $L_{\infty }$ is period of barriers, and $N_{p}$ is the number of periods in a barrier, $N_{p}=L_{\infty }/L$ . (a) Posts enclosed by barriers with $N_{p}=4$ , (b) posts enclosed by barriers with $N_{p}=2$ . (c) Change of stability criteria according to the relative importance of shear-driven failure mechanism to stagnation pressure failure mechanism. $\unicode[STIX]{x1D707}_{a}$ is viscosity of air, $\unicode[STIX]{x1D707}_{w}$ is viscosity of water, $L^{+}$ is pattern period in wall units. Dashed lines are predicted stability boundaries imposed by shear-driven failure depending on the ratio between stagnation and shear-driven pressure.

7.3 Considerations of failure due to shear-driven drainage

An additional failure mechanism for superhydrophobic surfaces is shear-driven drainage (Wexler et al. Reference Wexler, Jacobi and Stone2015b ; Liu et al. Reference Liu, Wexler, Schönecker and Stone2016). Wexler et al. (Reference Wexler, Grosskopf, Chow, Fan, Jacobi and Stone2015a ) proposes a remedy for the shear-driven drainage, by installing barriers with finite periodicity which should be smaller than or equal to a threshold length, $L_{\infty }$ , that can retain the enclosed fluid. Two examples of such barriers with different barrier periodicity defined by $N_{p}=L_{\infty }/L$ , are portrayed in figure 26(a,b). According to Wexler et al. (Reference Wexler, Jacobi and Stone2015b ), the threshold length for shear-driven drainage is ${\sim}O(1)\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FE}h/\unicode[STIX]{x1D70F}_{yx}$ , where $h$ is the height of the texture, $\unicode[STIX]{x1D70F}_{yx}$ is the imposed shear on top of the cavity, and $\unicode[STIX]{x1D6FE}$ is the curvature of the interface. Assuming the texture height is ${\sim}L$ , the shear-driven pressure that can balance the capillary pressure, $\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FE}$ , would then be $p_{shear}\sim O(1)\unicode[STIX]{x1D707}_{air}U_{s}N_{p}/L$ . Considering the stagnation pressure scales as $p_{stagnation}\sim 0.28\unicode[STIX]{x1D707}_{water}U_{s}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$ (Seo et al. Reference Seo, García-Mayoral and Mani2015), the ratio between the shear-driven pressure to the stagnation is

For superhydrophobic surfaces, due to the low viscosity ratio of air to water, $\unicode[STIX]{x1D707}_{air}/\unicode[STIX]{x1D707}_{water}\approx 2\,\%$ , this ratio remains small if $N_{p}$ is on the order of 10 or smaller. When $N_{p}$ is comparable to the viscosity ratio, the shear-driven pressure will dominate the stagnation pressure.

In figure 26(c), we show the rough estimate of the change of the stability boundary according to the ratio between the shear-driven pressure and the stagnation pressure. When the number of periods within barriers, $N_{p}$ , becomes large, the shear-driven failure will be the dominant mechanism for the interface breakage. The optimal choice of $N_{p}$ is required since small $N_{p}$ ensures stable gas pockets from shear-driven drainage, but it also suppresses the drag reduction due to additional solid–liquid contact area. For example, an extremely small $N_{p}=1$ formulates shear-free holes with periodicity, $L$ , which separately contain air pockets. While this configuration may consist a highly robust design for superhydrophobic surfaces, the drag reduction is significantly impacted. Our preliminary results show isolated holes can lead to half the drag reduction compared to posts with the same $\unicode[STIX]{x1D719}_{s}=1/9$ . Based on the scaling analysis of (7.4) we estimate that $N_{p}\sim O(10)$ provides a good compromise between robustness and drag reduction for superhydrophobic surfaces subject to turbulent flows.