Analysis of anisotropically permeable surfaces for turbulent drag reduction

The present work proposes the use of anisotropically permeable substrates as a means to reduce turbulent skin friction. We conduct an a priori analysis to assess the potential of these surfaces, based on the effect of small-scale surface manipulations on near-wall turbulence. The analysis, valid for small permeability, predicts a monotonic decrease in friction as the streamwise permeability increases. Empirical results suggest that the drag-reducing mechanism is however bound to fail beyond a certain permeability. We investigate the development of Kelvin-Helmholtz-like rollers at the surface as a potential mechanism for this failure. These rollers, which are a typical feature of turbulent flows over permeable walls, are known to increase drag and their appearance is known to limit the drag-reducing effect. We propose a model, based on linear stability analysis, that predicts the onset of these rollers for sufficiently large permeability and allows us to bound the maximum drag reduction that these surfaces can achieve.

Figure 1

Drag reduction $\mathit{DR}=-\mathrm{\Delta }\tau /{\tau }_0$ of the seal fur surface from Itoh et al. [5]: $▵$, seal fur; $\blacksquare$, trapezoidal flat-peak riblets, also from Itoh et al. [5]. For the seal fur, the wavelength identified by Itoh et al. by analogy with the riblet pitch $s^{+}$ has been used to express their results in terms of a viscous length scale.

Figure 2

Schematic representation of a permeable layer for drag reduction.

Figure 3

Sketch of (a) streamwise and (b) spanwise slip lengths. In (b), the top arrow represents an overlying spanwise perturbation, which is perceived locally as quasisteady and quasistationary under the assumption of vanishingly small surface texture.

Figure 4

Growth rate ${\sigma }_I=\text{Im}\left(\sigma \right)$ of the most amplified modes given by (21). (a) Isotropic case with ${\mathrm{\Phi }}_{xy}=1$ and $h/H=1$. Curves are shown for $(\sqrt{K_x{K}_y}/H^2)(UH/\nu )={10}^{[-2(0.4\left)6\right]}$. (b) Anisotropic case obtained using all possible combinations of $h/H={10}^{[-1,0,1]}$ and ${\mathrm{\Phi }}_{xy}={10}^{[-3(1\left)3\right]}$. Curves are shown for $\tilde{\mathcal{K}}={10}^{[-2(1\left)6\right]}$.

Figure 5

(a) Growth rate ${\sigma }^{+}={\alpha }^{+}\text{Im}\left(c^{+}\right)$ of the most amplified mode as a function of the longitudinal wavelength ${\lambda }_x^{+}$: $-·-, {\text{Re}}_{\tau }=180$; $--, {\text{Re}}_{\tau }=550$; $---, {\text{Re}}_{\tau }=1000$. For the isotropic case ${\mathrm{\Phi }}_{xy}=1, K^{+}={10}^{[-0.66(0.66\left)2.66\right]}, h^{+}=100$. (b) Stream-function contours of the mode with the highest growth rate at ${\text{Re}}_{\tau }=550$ for fully developed instability ${\tilde{K}}^{+}={10}^4$. Solid and dashed lines correspond to clockwise and counterclockwise rotation, respectively.

Figure 6

(a) Growth rate ${\sigma }^{+}={\alpha }^{+}\text{Im}\left(c^{+}\right)$ of the most amplified mode as a function of the longitudinal wavelength ${\lambda }_x$: $---, {\mathrm{\Phi }}_x={10}^{-3}$ and $h^{+}=10$; $--, {\mathrm{\Phi }}_x={10}^3$ and $h^{+}=10$; $▵, {\mathrm{\Phi }}_x=1$ and $h^{+}=1$; $\circ , {\mathrm{\Phi }}_x=1$ and $h^{+}=100$; $\square , {\mathrm{\Phi }}_x=1$ and $h^{+}=10$. Here ${\tilde{K}}^{+}={10}^{[0.36,0.82,1.28,2.20]}$ at ${\text{Re}}_{\tau }=550$. (b) Maximum growth rate ${\sigma }^{+}$ as a function of the permeability ${\tilde{K}}^{+}$ at ${\text{Re}}_{\tau }=550$: $---, {\mathrm{\Phi }}_x={10}^{-3}$ and $h^{+}=1$; $--, {\mathrm{\Phi }}_x={10}^3$ and $h^{+}=100$; $-·-, {\mathrm{\Phi }}_x=1$ and $h^{+}=10$.

Figure 7

Schematic summarizing the initial linear behavior of $\mathit{DR}=\mathrm{\Delta }\tau /{\tau }_0$ for low permeability. The hatched area represents the region $\tilde{K}>{\tilde{K}}_{\text{KH}}$, where the drag reducing effect is destroyed by the appearance of Kelvin-Helmholtz rollers.

Figure 8

Limit for the maximum drag reduction achievable as a function of the anisotropy ratio of the permeable layer ${\mathrm{\Phi }}_{xy}$. The dashed lines denote Eq. (26) for ${\tilde{K}}_{\text{KH}}^{+}=5$ and 10, and $\xi =1$. The light gray shaded region indicates the range for which Kelvin-Helmholtz rollers are expected to appear. The dark gray shaded region indicates the range for which the Kelvin-Helmholtz rollers would be fully developed.