3.1 Mean slip length and skin-friction reduction

The mean slip length is obtained from $\bar{b}=\bar{u}_{s}/(\partial \bar{u}_{l}/\partial y)_{s}$ , where $\bar{u}_{s}$ is the mean slip velocity and $(\partial \bar{u}_{l}/\partial y)_{s}$ is the mean velocity gradient of the water at the interface. Figure 2 shows the variation of the mean slip length with the air-layer thickness. For both cases 1 and 2, the mean slip lengths linearly increase with the air-layer thickness and their slopes are ${\it\mu}_{R}$ and ${\it\mu}_{R}/4$ respectively, where ${\it\mu}_{R}$ is the viscosity ratio ( ${\it\mu}_{R}={\it\mu}_{l}/{\it\mu}_{g}$ ). At a given air-layer thickness, more slip is induced by the shear-driven air flow (case 1) than by the recirculating air flow (case 2). It is interesting to note that the mean slip length of turbulent flow is the same as that of laminar flow. This counterintuitive result can be explained by the definition of the mean slip length. The mean slip length is rewritten as $\bar{b}={\it\mu}_{R}\bar{u}_{s}/(\partial \bar{u}_{g}/\partial y)_{s}$ by matching the shear stresses at the air–water interface. When the velocity profiles inside the air layer are linear (case 1) and quadratic (case 2) functions, the $(\partial \bar{u}_{g}/\partial y)_{s}$ values are $\bar{u}_{s}/D$ and $4\bar{u}_{s}/D$ for cases 1 and 2 respectively. Then, the slip length depends only on ${\it\mu}_{R}$ and $D$ , independent of the flow condition in water. The mean velocity profiles of the air layer are indeed linear and quadratic for cases 1 and 2 respectively (see below). The slip length in the viscous wall unit ( $b^{+_{o}}=\bar{b}u_{{\it\tau}_{o}}/{\it\nu}_{l}$ ) in figure 2 shows that relaminarization starts when $b^{+_{o}}\geqslant 100$ in both cases. This is consistent with the observation by Busse & Sandham (Reference Busse and Sandham2012).

Figure 2.

Variation of the mean slip length with the air-layer thickness: symbols, turbulent flow; lines, laminar flow.

Figure 3.

Variations of the DR rate (▪, ●) and the mean slip velocity (▫, ○) with the mean slip length normalized by (a) $u_{{\it\tau}_{o}}$ and ${\it\nu}_{l}$ and (b) $u_{{\it\tau}}$ (shear velocity at the interface) and ${\it\nu}_{l}$ . In (a), the solid black, green and blue lines are the DR rates by empirical formulae: lines with triangles, Fukagata et al. (Reference Fukagata, Kasagi and Koumoutsakos2006); lines with inverse triangles, Busse & Sandham (Reference Busse and Sandham2012). The red open rectangles (case 1) and circles (case 2) denote the DR rates from DNS with the present slip-velocity model, (3.1) and (3.2), without solving the air layer. The dashed line in (a) is the DR rate and normalized slip velocity of laminar flow. In (b), the open triangles are the DNS results from various idealized SHSs (Park Reference Park2015).

Figure 3(a) shows the variation of the DR rate and the mean slip velocity with the mean slip length normalized by $u_{{\it\tau}_{o}}$ and ${\it\nu}_{l}$ , where the DR rate is defined as $DR=({\it\Pi}_{o}-{\it\Pi}_{l})/{\it\Pi}_{o}$ ; ${\it\Pi}_{o}$ is the mean pressure gradient with the no-slip wall. As the mean slip length increases (i.e. as $D$ increases), the DR rate and slip velocity also increase. Although the mean slip length remains the same for laminar and turbulent flows, the DR rates in turbulent flows are larger than that in laminar flow and they are even larger than the normalized mean slip velocity. This is because the DR of turbulent flow is determined by the combination of the pure slip effect and the weakening of near-wall turbulence (Park et al. Reference Park, Park and Kim2013; Rastegari & Akhavan Reference Rastegari and Akhavan2015). The present DR rate falls in between the empirical formulae for the streamwise slip only ( $b_{z}=0$ , Fukagata et al. Reference Fukagata, Kasagi and Koumoutsakos2006) and isotropic slip ( $b_{x}=b_{z}$ , Fukagata et al. Reference Fukagata, Kasagi and Koumoutsakos2006, Busse & Sandham Reference Busse and Sandham2012), indicating (see below) that the present idealized SHS with an air layer generates an anisotropic slip ( $b_{x}\neq b_{z}$ ), and more slip in the streamwise direction than in the spanwise one ( $b_{x}>b_{z}$ ). Figure 3(b) shows the variation of the DR rate with the mean slip length normalized by the shear velocity at the air–water interface, $u_{{\it\tau}}$ , together with the results from Park (Reference Park2015). As indicated by Park et al. (Reference Park, Park and Kim2013), the DR rate correlates well with the mean slip length in the viscous wall unit, irrespective of the flow type inside the air layer and the SHS geometry.

3.2 Anisotropic slip at the air–water interface

Figure 4 shows the modified mean velocity profiles in the viscous wall unit, $\bar{u}^{+}-\bar{u}_{s}^{+}$ , except inside the air layer, where $y^{+}=0$ is the location of the air–water interface. As the air-layer thickness increases, the modified mean velocity shifts downward in the buffer and log layers (note that the mean velocity itself shifts upwards with increasing air-layer thickness). Min & Kim (Reference Min and Kim2004) reported that these mean velocities fall on one curve for streamwise slip only, but show downward shifts for combined streamwise and spanwise slip, indicating that the present interface has non-negligible spanwise slip. Hahn et al. (Reference Hahn, Je and Choi2002) showed for turbulent flow over a permeable wall that the downward shift in the modified mean velocity in the buffer and log layers is the result of the decrease in the viscous sublayer thickness.

Figure 4.

Modified mean velocity ( $\bar{u}^{+}-\bar{u}_{s}^{+}$ ) profiles in the viscous wall unit: (a) case 1; (b) case 2.

Figure 5.

Joint probability density function of $u_{s}$ versus $(\partial u_{l}/\partial y)_{s}$ : (a–c) case 1; (d–f) case 2; (a) $D=0.0002H$ , (b) $D=0.0005H$ , (c) $D=0.001H$ , (d) $D=0.001H$ , (e) $D=0.002H$ , (f) $D=0.005H$ . The contour levels are from 0.005 to 0.045 in increments of 0.005. The solid and dashed lines have slopes of 1 and $1/4$ respectively. The solid circles denote the mean values of the slip velocity and streamwise velocity gradient.

Figure 6.

Joint probability density function of $w_{s}$ versus $(\partial w_{l}/\partial y)_{s}$ : (a–c) case 1; (d–f) case 2; (a) $D=0.0002H$ , (b) $D=0.0005H$ , (c) $D=0.001H$ , (d) $D=0.001H$ , (e) $D=0.002H$ , (f) $D=0.005H$ . The contour levels are from 0.05 to 0.45 in increments of 0.05. The solid and dashed lines have slopes of 1 and $1/4$ respectively.

To further investigate the nature of the slip at the interface, we examine the relation between the slip velocity and the velocity gradient at the interface. Figure 5 shows the joint probability density functions (PDFs) between the instantaneous streamwise slip velocity and the velocity gradient at the interface for both cases. The slope between these two (normalized) variables is near 1 for both cases 1 and 2, although this slope becomes lower for larger $D/H$ . These results mean that $u_{s}\sim {\it\mu}_{R}D(\partial u_{l}/\partial y)_{s}$ or $b_{x}\sim {\it\mu}_{R}D$ for both cases. This result is surprising, because the mean slip lengths for cases 1 and 2 are $\bar{b}={\it\mu}_{R}D$ and $(1/4){\it\mu}_{R}D$ (or $\bar{u}_{s}={\it\mu}_{R}D(\partial \bar{u}_{l}/\partial y)_{s}$ and $\bar{u}_{s}=(1/4){\it\mu}_{R}D(\partial \bar{u}_{l}/\partial y)_{s}$ ) respectively (see figure 2). As shown in figure 5 (solid circles), the slip lengths between the mean slip velocities and the mean velocity gradients hold for the present two cases in terms of joint PDF. The present results clearly indicate that the slip length used for the mean slip velocity and interface velocity gradient should not be applied to instantaneous ones (or fluctuating quantities). Rather, for the two cases considered here, the slip length of $b_{x}={\it\mu}_{R}D$ should be applied to the fluctuating components. Thus, the instantaneous streamwise slip velocity may be modelled as (unless the DR rate is very large)

Figure 6 shows the joint PDFs between the instantaneous spanwise slip velocity and the velocity gradient at the interface for both cases, where the mean spanwise velocity is zero. From this result, the instantaneous spanwise slip velocity is modelled as

and $b_{z}={\it\mu}_{R}D/4$ . Thus, the slip effect in the spanwise direction is much weaker than that in the streamwise direction. The reason why the slip lengths for the streamwise and spanwise fluctuating velocities are different from each other is examined in § 3.3.

The present slip-velocity model, (3.1) and (3.2), provides the anisotropic slip-length relation of $b_{z}=b_{x}/4$ . The DR rates versus the slip length with $b_{z}=b_{x}/4$ (using the formulae by Fukagata et al. (Reference Fukagata, Kasagi and Koumoutsakos2006) and Busse & Sandham (Reference Busse and Sandham2012)) are plotted in figure 3(a), showing excellent agreement with the present simulation results of case 1 but slight overprediction for case 2. We also perform separate DNS with the present model as the slip boundary conditions (without solving the air layer) and provide the results in figure 3(a). As shown, the present slip-velocity model successfully predicts the drag changes.

3.3 Flow structures in air and water

Figure 7 shows the mean velocity profiles in air and water for cases 1 and 2. As mentioned before, the mean velocity profiles inside the air layer are linear (case 1) and quadratic (case 2). With increasing air-layer thickness, the mean velocity in water increases near the interface due to the increased slip velocity but decreases near the centreline to maintain the same mass flow rate inside the channel. Figure 8 shows the r.m.s. velocity fluctuations normalized by $u_{{\it\tau}_{o}}$ . The r.m.s. velocity fluctuations are significantly reduced in the water flow for both cases as the air-layer thickness increases. At the air–water interface, for case 1, the streamwise velocity fluctuations increase but the spanwise velocity fluctuations first increase and then decrease with increasing air-layer thickness, whereas for case 2, both the streamwise and spanwise velocity fluctuations first increase and then decrease.

Figure 7.

Mean velocity profiles in air and water: (a) case 1; (b) case 2.

Figure 8.

Profiles of the r.m.s. velocity fluctuations normalized by $u_{{\it\tau}_{o}}$ : (a) case 1; (b) case 2. Black lines, $u_{rms}$ ; red lines, $v_{rms}$ ; blue lines, $w_{rms}$ .

The results shown in figures 5 and 6, $b_{x}={\it\mu}_{R}D$ and $b_{z}={\it\mu}_{R}D/4$ , indicate that the profiles of the streamwise and spanwise velocity fluctuations inside the air layer are more like linear and quadratic respectively, regardless of the cases considered here. In figure 9, we plot some of the instantaneous streamwise and spanwise velocity fluctuations inside the air layer from the present simulations. This figure clearly shows that the profiles of the streamwise velocity fluctuations are very different from those of the spanwise velocity fluctuations, and the first are more like linear but the latter are quadratic. This is because the streamwise velocity fluctuations at the interface have dominant low-frequency components originated from the near-wall streak, but the spanwise ones do not (see the solutions of Stokes’ second problem between two parallel plates for low- and high-frequency oscillations; Sir Lamb (Reference Lamb1916)).

Figure 9.

Profiles of the instantaneous velocity fluctuations ( $u^{\prime }$ and $w^{\prime }$ ) inside the air layer at every eighth grid point: (a) case 1 ( $D/H=0.0005$ ); (b) case 2 ( $D/H=0.002$ ).

Table 1 shows the Reynolds numbers based on the shear velocities at the interface in air and water respectively, where the Reynolds numbers in air flow are defined as $Re_{{\it\tau}}=u_{{\it\tau}}D/{\it\nu}_{g}$ (case 1) and $u_{{\it\tau}}(D/2)/{\it\nu}_{g}$ (case 2) respectively. It is notable that the Reynolds numbers in air flow are very small for all of the cases considered. Therefore, turbulence in air flow hardly survives and dissipates although it is continuously supplied at the interface from water flow.

Table 1.

Reynolds numbers based on the shear velocities in air and water.