Turbulent drag reduction by permeable coatings

We present an assessment of permeable coatings as a form of passive drag reduction, proposing a simplified model to quantify the effect of the coating thickness and permeability. To reduce skin friction, the porous layer must be preferentially permeable in the streamwise direction, so that a slip effect is produced. For small permeability, the controlling parameter is the difference between streamwise and spanwise permeability lengths, scaled in viscous units, $\sqrt{K_x^+}-\sqrt{K_z^+}$. In this regime, the reduction in drag is proportional to that difference. However, the proportional performance eventually breaks down for larger permeabilities. A degradation mechanism is investigated, common to other obstructed surfaces in general and permeable substrates in particular, which depends critically on the geometric mean of the streamwise and wall-normal permeabilities, $\sqrt{K_x^+ K_y^+}$. For a streamwise-to-cross-plane permeability ratio of order $K_x^+/K_y^+=K_x^+/K_z^+\sim10$-$100$, the model predicts a maximum drag reduction of order 15-25\%.