Apparent slip and drag reduction for the flow over superhydrophobic and lubricant-impregnated surfaces

The motion of a liquid above textured surfaces, impregnated with either a gas or a lubricant oil, is considered by homogenizing the microscopic near-wall flow to obtain protrusion heights (or Navier slip lengths), which can then be used in (macroscopic) direct numerical simulations of turbulence. The microstructure of the wall consists in longitudinally aligned ridges; this renders the small-scale Stokes problem amenable to decoupling into two problems, solved by the boundary element method with full account of the deformation of the liquid-lubricant interface, finally yielding two different protrusion heights, a longitudinal and a transverse one, respectively $h_{\left|\right|}$ and $h_{\perp }$. Such heights are then employed in direct numerical simulations of the turbulent flow in a channel bounded by superhydrophobic or lubricant-impregnated walls, demonstrating the reduction in skin-friction drag which can be achieved with the rise in protrusion height difference, $\mathrm{\Delta }h=h_{\left|\right|}-h_{\perp }$. The results are in good agreement with an analytical approximation, provided $\mathrm{\Delta }{h}^{+}$ ($\mathrm{\Delta }h$ scaled in wall units) remains below 2 and the protrusion heights small in magnitude.

Figure 1

Geometry, governing equations and boundary conditions for the transverse (bottom left) and the longitudinal (bottom right) problems.

Figure 2

Wall geometry for a depressing ($\mathrm{\Phi }<1$) or protruding ($\mathrm{\Phi }>1$) interface $\mathsf{\text{I}}$.

Figure 3

Comparison in transverse protrusion heights (scaled by $b$) between the analytical model by Davis and Lauga [19] and the present numerical simulations. The solid lines correspond to the analytical model by Davis and Lauga for $c=0.30$ (lower line), $c=0.50$ (intermediate line), and $c=0.70$ (upper line). Filled circles denote results of the present simulations, for the same values of $c$, with $\lambda =0.018$ at $\text{Ca}=0.1$.

Figure 4

Comparison in longitudinal protrusion heights (scaled by $b$) between the analytical model by Crowdy [20] (dashed lines), the numerical results by Teo and Khoo [23] (solid lines), and the present numerical simulations (filled circles) for different values of $c$ at $\lambda =0.018$.

Figure 5

Computed transverse (left column) and longitudinal protrusion heights for increasing values of $c$ and a wide range of $\lambda$, increasing according to the direction of the arrow, at $\text{Ca}=0.1$. The value of the interface span $c$ is increased from top to bottom, namely, $c=0.30$ in the top row, $c=0.50$ in the middle row, and $c=0.70$ in the bottom row. The dashed lines illustrate the values of the protrusion heights when $\lambda \to \infty$, i.e., when the interface is solid. The dots are the values of the protrusion heights calculated using the analytical formulas by Schönecker et al. [27] for a flat interface, and excellent agreement is found with our results in that limit.

Figure 6

Computed difference between transverse and longitudinal protrusion heights for increasing values of $c$ and a wide range of $\lambda$ at $\text{Ca}=0.1$. The value of the interface span $c$ is increased from top to bottom, from $c=0.30$ to $c=0.70$.

Figure 7

Flow field developing into the computational domain for two representative cases with a depressing (left, $\mathrm{\Phi }=0.90$) and protruding (right, $\mathrm{\Phi }=1.10$) interface at $\lambda =20$ and $\text{Ca}=0.1$. (a), (b) longitudinal velocity component; (c), (d) wall normal velocity component; (e), (f) transverse velocity component.

Figure 8

(a) Estimated limits of validity of relation (27) (region enclosed by dashed lines) together with the present simulations (red circles), those by Min and Kim [32] (green squares), and those by Busse and Sandham [36] (magenta triangles). The isocontours with labels show the percentage drag increase or reduction at ${\text{Re}}_{\tau }=180$ as computed by Busse and Sandham [36]. (b) Friction factors as function of the difference in protrusion heights in wall units.