Effect of texture randomization on the slip and interfacial robustness in turbulent flows over superhydrophobic surfaces

Superhydrophobic surfaces demonstrate promising potential for skin friction reduction in naval and hydrodynamic applications. Recent developments of superhydrophobic surfaces aiming for scalable applications use random distribution of roughness, such as spray coating and etched process. However, most previous analyses of the interaction between flows and superhydrophobic surfaces studied periodic geometries that are economically feasible only in laboratory-scale experiments. In order to assess the drag reduction effectiveness as well as interfacial robustness of superhydrophobic surfaces with randomly distributed textures, we conduct direct numerical simulations of turbulent flows over randomly patterned interfaces considering a range of texture widths $w^{+}\approx 4\text{–}26$, and solid fractions ${\varphi }_s=11\%-25\%$. Slip and no-slip boundary conditions are implemented in a pattern, modeling the presence of gas-liquid interfaces and solid elements. Our results indicate that slip of randomly distributed textures under turbulent flows is about $30\%$ less than those of surfaces with aligned features of the same size. In the small texture size limit $w^{+}\approx 4$, the slip length of the randomly distributed textures in turbulent flows is well described by a previously introduced Stokes flow solution of randomly distributed shear-free holes. By comparing DNS results for patterned slip and no-slip boundary against the corresponding homogenized slip length boundary conditions, we show that turbulent flows over randomly distributed posts can be represented by an isotropic slip length in streamwise and spanwise direction. The average pressure fluctuation on a gas pocket is similar to that of the aligned features with the same texture size and gas fraction, but the maximum interface deformation at the leading edge of the roughness element is about twice as large when the textures are randomly distributed. The presented analyses provide insights on implications of texture randomness on drag reduction performance and robustness of superhydrophobic surfaces.

Figure 1

An illustration of a periodic turbulent channel with superhydrophobic walls. An instantaneous snapshot of streamwise velocity is plotted. A Voronoi decomposition of surface around textures is shown.

Figure 2

Turbulence statistics with different post arrangements with ${\varphi }_s=1/9$ at ${\text{Re}}_{\tau }\approx 200$. (a) Mean streamwise velocity profile; (b) $w^{+}\approx 4$, rms velocity, and pressure fluctuations; (c) $w^{+}\approx 26$, rms velocity, and pressure fluctuations. , $\mathrm{R}{04}_1$; , $\mathrm{R}{04}_2$; , $\mathrm{R}{26}_1$; , $\mathrm{R}{26}_2$; , $\mathrm{R}{26}_3$; , $\mathrm{R}{26}_4$.

Figure 3

(a) Slip length, $b^{+}$, versus texture size, $L_n^{+}=w^{+}/\sqrt{{\varphi }_s}$, with ${\varphi }_s=1/9$. Symbols are DNS solutions; $⚫$: randomly distributed posts, $\times$: aligned posts [28]. Lines are Stokes flow solutions; : Eq. (1) by Ref. [38]; $\_\hspace{0.17em}.\_\hspace{0.17em}.\_$: Ybert et al. [5]. (b) Slip length, $b^{+}$, normalized by texture width, $w^{+}$, versus solid fraction, ${\varphi }_s$, at $w^{+}\approx 4$. $⚫$: $\mathrm{R}{04}_1$, $▾$: $\mathrm{R}04{\varphi }_{s16}$, $\blacksquare$: $\mathrm{R}04{\varphi }_{s25}$, $\_\_\_\_$: Ref. [38].

Figure 4

Instantaneous streamwise velocity contours at $y^{+}=0$ for $L_n^{+}\approx 38.7$: (a) aligned posts, (b) randomly distributed posts.

Figure 5

Mean velocity profiles: (a) $U^{+}$ and (b) $U^{+}-b^{+}$ for ${\varphi }_s=1/9$; : patterned slip with randomly distributed posts, $\mathrm{R}{04}_1$; : patterned slip, $\mathrm{R}{26}_1$; : homogenized slip using the slip length from H04; : homogenized slip, H26; : smooth wall.

Figure 6

(a) Streamwise velocity, (c) wall-normal velocity, (e) spanwise velocity, (g) pressure rms fluctuations for R04 with $L_n^{+}\approx 12.9$. (b) $u_x$, (d) $u_y$, (f) $u_z$, (h) $p$, for R26 with $L_n^{+}\approx 77.4$. : $q_{rms}^{\prime }$; : $q_{rms}^{\prime \prime +}$; : ${\tilde{q}}_{rms}^{+}$; homogenized slip, : H04; : H26; : smooth wall.

Figure 7

Coherent secondary flow structure near the post with $w^{+}\approx 26$: (a) vector plots of ${\tilde{u}}_y$ and $U+{\tilde{u}}_x$ in the $x\text{−}y$ plane, (b) velocity contours of $U+{\tilde{u}}_x$ and ${\tilde{u}}_z$ in the $x\text{−}z$ plane, and (c) velocity contours of ${\tilde{u}}_y$ and ${\tilde{u}}_z$ in the $y\text{−}z$ plane. The location of the post is indicated by gray boxes.

Figure 8

Shear stress budget in wall normal direction. (a) $L_n^{+}\approx 12.9$; : randomly distributed posts, R04; : homogenized slip, HP04. (b) $L_n^{+}\approx 77.4$; : randomly distributed posts, R26; : homogenized slip, HP26; : smooth wall. ${\tau }_{xy,v}^{+}$ is viscous shear stress, $d{\overline{U}}^{+}/d{y}^{+},{\tau }_{xy,R}^{+}$ is Reynolds stress, $-{\overline{u^{″}{v}^{″}}}^{+},{\tau }_{xy,c}^{+}$ is Reynolds stress from coherent fluctuation, $-{\overline{\tilde{u}\tilde{v}}}^{+}$, and ${\tau }_{xy,\mathrm{tot}}^{+}$ is total shear stress, ${\tau }_{xy,\mathrm{tot}}^{+}={\tau }_{xy,R}^{+}+{\tau }_{xy,v}^{+}+{\tau }_{xy,c}^{+}$.

Figure 9

Instantaneous wall pressure snapshots, $p_0^{\prime +}=p_0^{\prime }/\left(\rho u_{\tau }^2\right)$, on superhydrophobic surfaces for (a) randomly distributed posts, $\mathrm{R}{04}_1,L_n^{+}\approx 13$; (b) randomly distributed posts, $\mathrm{R}{26}_1,L_n^{+}\approx 77$; and (c) aligned posts, $L_n^{+}\approx 77$. From blue to yellow, color shows pressure fluctuations from $-10$ to 10.

Figure 10

(a) Wall rms pressure fluctuation due to stagnation, ${\tilde{p}}_{rm{s}_0}^{+}$ against nominal texture size $L_n^{+}$; (b) ${\tilde{p}}_{rm{s}_0}^{+}$ versus slip velocity $U_s^{+}$. : randomly distributed posts; : aligned posts. (c) Self-similar profiles of wall pressure distribution due to stagnation on the gas-liquid interface. The plots show pressure profiles on the centerline going through the middle of post width. $L_n^{+}\approx 13$; : $L_n^{+}\approx 26$; : $L_n^{+}\approx 38$; : $L_n^{+}\approx 77$.

Figure 11

Interface deformation angle, $(d\eta /dx{|}_L)$, due to stagnation pressure for randomly distributed posts versus number of posts, $N_p$, as compared to aligned posts with Weber number ${\text{We}}^{+}=5\times {10}^{-3}$ for texture size: (a) $L_n^{+}\approx 77$; (b) $L_n^{+}\approx 26$. $\circ$: randomly distributed posts. Insets: Histogram of interface deformation angle due to stagnation pressure for randomly distributed posts versus number of posts. : mean of $(d\eta /dx{|}_L)$ of randomly distributed posts; : $(d\eta /dx{|}_L)$ of aligned posts.

Figure 12

(a) Comparison of maximum interface deformation slope, max ${(d\eta /dx)|}_L$, due to stagnation pressure for randomly distributed posts and aligned posts versus effective texture size, $L_n^{+}=w^{+}/\sqrt{{\varphi }_s}$, at Weber number ${\text{We}}^{+}=5\times {10}^{-3}$. : randomly distributed posts; : aligned posts [27]. For $L_n^{+}\approx 13$ and $L_n^{+}\approx 77$, error bars are plotted by taking account of different texture arrangements. (b) The critical texture size, $L_c^{+}$, is plotted against ${\text{We}}^{+}$ when ${\theta }_{adv}={120}^{\circ }$. : current study; : aligned posts.