Dependence of the drag over superhydrophobic and liquid infused surfaces on the asperities of the substrate

Direct numerical simulations (DNSs) of two superposed fluids in a turbulent channel flow with a textured surface made of pinnacles of random height have been performed at ${\text{Re}}_{\tau }\approx 180$ and ${\text{Re}}_{\tau }\approx 395$. The texture reproduces an etched sandblasted aluminum. The viscosity ratio between the two fluids is either $m=0.02$, mimicking the viscosity ratio of a superhydrophobic surface (water over air), or $m=0.40$ (water over heptane) resembling a liquid infused surface. A parametric study has been carried out varying the position of the interface between the two fluids to assess the contribution to the drag of the portion of the texture emerging above the interface and to provide guidelines for design. Simulations with a deformable interface, at Weber numbers ${\text{We}}^{+}\approx {10}^{-2}$ and ${10}^{-3}$ have been performed. Results have been compared to those obtained assuming the ideal case of a flat interface which is slippery in the streamwise and spanwise direction (corresponding to a Weber number $\text{We}=0$). The time-averaged position of the interface has been correlated to the pressure field induced by the random-height pinnacles while the instantaneous deformation is due to the passage of the near-wall coherent structures. We attempted to reconcile the effect of superhydrophobic and liquid infused surfaces on the coherent structures by calculating how the shear rate parameter depends on the amount of drag reduction.

Figure 1

(a) Sketch of the surface and definitions of $k$, $k_{\max }$, and $h$. (b) A portion of the computational domain showing the substrate, the interface between the two fluids, and the definition of $H$.

Figure 2

Mean velocity profiles in wall units for (a), (c) $m=0.02$ and (b), (d) $m=0.4$ at (a), (b) ${\text{Re}}_{\tau }=180$ and (c), (d) ${Re}_{\tau }\approx 395$ (the ${\text{Re}}_{\tau }$ of each case is provided in Table 2) for $\text{We}=0$: $k/k_{\max }=0$ ($\mathbf{—}$), $k/k_{\max }=0.2$ ($\mathbf{—}$), $k/k_{\max }=0.4$ ($\mathbf{—}$), $k/k_{\max }=0.6$ ($\mathbf{—}$), $k/k_{\max }=0.8$ ($\mathbf{—}$), $k/k_{\max }=1$ ($\mathbf{—}$); for ${\text{We}}^{+}\approx {10}^{-3}$: $\overline{k}/k_{\max }=0.755$ (- - -) and for ${\text{We}}^{+}\approx {10}^{-2}$: $\overline{k}/k_{\max }=0.637$ (—). Velocity profiles of a smooth channel flow with the corresponding ${\text{Re}}_{\tau }$ from Moser et al. [25] ($◯$) and over SHS etched sandblasted aluminum (SB-AL) at ${\text{Re}}_{\tau }\approx 700$ from Ling et al. [26] ($\square$) are included as a reference. The interface position is indicated with $•$. For the cases $k/k_{\max }<0.6$ the interface is not shown in the figure because it is below ${\left({y-h}_{\mathrm{mean}}\right)}^{+}=0.1$.

Figure 3

Slip velocity in wall units as a function of the mean interface position: $m=0.02$ (SHS, $◯$); $m=0.4$ (LIS, $\square$); open symbols, ${\text{Re}}_{\tau }\approx 180$; solid symbols, ${\text{Re}}_{\tau }\approx 395$; red represents $\text{We}=0$, green ${\text{We}}^{+}\approx {10}^{-2}$, and blue ${\text{We}}^{+}\approx {10}^{-3}$.

Figure 4

Color contours of time-averaged pressure: (a)–(d) $\text{Re}=2800$, $m=0.02$ (SHS), and $\text{We}=0$; (e)–(g) $\text{Re}=2800$, $\text{We}=40$, and $\overline{k}/k_{\max }=0.75$. For $\text{We}=0$, two interface positions are shown: (a), (b) $k/k_{\max }=0.6$ and (c), (d) $k/k_{\max }=0.8$. Vertical sections are shown at (b), (d) $z/H=0.56$ for $\text{We}=0$, and (f), (g) $z/H=0.56$ and 1.25 for $\text{We}=40$.

Figure 5

Color contours of time-averaged interface position $\langle k\rangle /k_{\max }$ for $\text{Re}=2800$: (a), (b) $m=0.02$ (SHS), (c), (d) $m=0.40$ (LIS), (a), (c) $\text{We}=400$, (b), (d) $\text{We}=40$.

Figure 6

Maximum and minimum time-averaged displacement with respect to the closest asperity as a function of the equivalent width of the valleys: (a), (b) $\text{Re}=2800$, (c), (d) $\text{Re}=6900$; (a), (c) $\text{We}=400$, (b), (d) $\text{We}=40;◯$, $m=0.02$ (SHS), and $\square$, $m=0.4$ (LIS).

Figure 7

Drag reduction as a function of the slip length $b^{+0}$ in wall units: $m=0.02$ (SHS, $◯$); $m=0.4$ (LIS, $\square$); open symbols, ${\text{Re}}_{\tau }\approx 180$; solid symbols, ${\text{Re}}_{\tau }\approx 395$; ($---$) analytic result with ${\text{Re}}_{\tau }=180$ and (....) with ${\text{Re}}_{\tau }=395$ from Ref. [10]; red, $\text{We}=0$; green, ${\text{We}}^{+}\approx {10}^{-2}$; and blue, ${\text{We}}^{+}\approx {10}^{-3}$.

Figure 8

Drag reduction (a) as a function of the mean interface position, $\overline{k}$, and (b) as a function of the area fraction: $m=0.02$ (SHS, $◯$); $m=0.4$ (LIS, $\square$); open symbols, ${\text{Re}}_{\tau }\approx 180$; solid symbols, ${\text{Re}}_{\tau }\approx 395$; red, $\text{We}=0$; green, ${\text{We}}^{+}\approx {10}^{-2}$; and blue, ${\text{We}}^{+}\approx {10}^{-3}$.

Figure 9

(a) Form drag and (b) its contribution with respect to the total shear at the wall as function of the position of the interface: $m=0.02$ (SHS, $◯$); $m=0.4$ (LIS, $\square$); open symbols, ${\text{Re}}_{\tau }\approx 180$; solid symbols, ${\text{Re}}_{\tau }\approx 395$; red, $\text{We}=0$, green, ${\text{We}}^{+}\approx {10}^{-2}$; and blue, ${\text{We}}^{+}\approx {10}^{-3}$.

Figure 10

Drag reduction as a function (a) of the mean surface above the interface normalized in wall units and (b) of its rms for $\text{We}=0$: $m=0.02$ (SHS, $◯$); $m=0.4$ (LIS, $\square$); open symbols, ${\text{Re}}_{\tau }\approx 180$; solid symbols, ${\text{Re}}_{\tau }\approx 395$.

Figure 11

Color contours of the variance of (a), (d) streamwise velocity fluctuations $\langle uu\rangle$, (b), (e) wall-normal velocity fluctuations $\langle vv\rangle$, and (c), (f) covariance of streamwise and wall-normal velocity fluctuations $\langle uv\rangle$ in a vertical section at $z/H=0.56$: the interface positions (a)–(c) $k/k_{\max }=0.6$ and (d)–(f) $k/k_{\max }=0.8$; in both cases ${\text{Re}}_{\tau }\approx 180$ and $m=0.02$.

Figure 12

(a) Variance of streamwise velocity fluctuations, (b) variance of wall-normal velocity fluctuations, and (c) Reynolds shear stress for SHS with ${\text{Re}}_{\tau }\approx 180$ (symbols and lines as in Fig. 2).

Figure 13

Isocontours of streamwise velocity fluctuations at $\text{Re}=2800$ and $\text{We}=0$: (a), (c) $k=0.6$ and (b), (d) $k=0.8$. For (e), (g) $\text{We}={10}^{-2}$ and (f), (h) ${\text{We}}^{+}\simeq {10}^{-3}$, the isocontours of streamwise velocity fluctuations are superposed to the displacement of the interface with respect to its time-averaged position (e), (g) $\overline{k}=0.6154$, (f), (h) $\overline{k}=0.76$. Two viscosity ratios are shown, (a), (b), (e), (f) $m=0.02$ and (c), (d), (g), (h) $m=0.4$, which mimic idealized SHSs and LISs, respectively. A sketch of streamwise vortices, low and high speed streaks, and deformed interface (dotted line) with respect to the mean interface position $\overline{k}$ is included as a reference.

Figure 14

Two-point streamwise velocity correlation in the $x–y$ plane ($r_{uu}({\mathbf{x}}_{\mathbf{0}},\mathbf{x})$) at ($x_0=0.7236H,z_0=0.9652H$). The value of $y_0$ is taken 10 wall units above the interface. Contours from 0.4 to 1 with increments of 0.1: (a), (e) smooth wall, (b), (f) $k/k_{\max }=0.6$, (c), (g) $k/k_{\max }=0.8$, (d), (h) $k/k_{\max }=1$ with $\text{We}=0$, ${\text{Re}}_{\tau }=395$, and $m=0.02$ (SHS).

Figure 15

Shear rate parameter ($Sq/\epsilon$) for (a) $m=0.02$ (SHS) and (b) $m=0.4$ (LIS) with ${\text{Re}}_{\tau }\approx 395$ and $\text{We}=0$ for different interface positions (symbols and lines as in Fig. 2).

Figure 16

Maximum of shear rate parameter $Sq/\epsilon$ as a function of drag reduction for $\text{We}=0$: $m=0.02$ (SHS, $◯$); $m=0.4$ (LIS, $\square$); open symbols, ${\text{Re}}_{\tau }\approx 180$; solid symbols, ${\text{Re}}_{\tau }\approx 395$; the black dot is the smooth wall.

Figure 17

(a), (c) Eddy turnover timescale, $q/\epsilon$, and (b), (d) mean shear flow timescale, ($1/S$), for ${\text{Re}}_{\tau }\approx 395$: (a), (b) $m=0.02$, (c), (d) $m=0.4$ (symbols and lines as in Fig. 2).