Laminar-turbulent patterns with rough walls

Oblique large-scale laminar-turbulent patterns are found near the onset of turbulence in subcritical planar shear flows. Their robustness to the introduction of wall roughness is investigated numerically in plane Couette flow as a function of the Reynolds number and the roughness height. The effect of roughness is considered on either one or two walls and is modeled numerically using a parametric model suggested recently [Busse and Sandham, J. Fluid Mech. 712, 169 (2012)JFLSA70022-112010.1017/jfm.2012.408]. In the case of two rough walls, the patterns are robust for a mean roughness height up to 10% of the wall gap, but the flow shows larger turbulent fractions with increasing roughness height. In the case of one rough wall only, the trend is similar, but the onset Reynolds number decreases faster with increasing roughness height. Roughness height levels above 15% of the wall gap give rise to new coherent structures, including turbulent bands with nonoblique interfaces. The energetic efficiency of the various regimes is investigated by monitoring the friction factor versus the friction Reynolds number. The mechanisms allowing for streamwise localization of the stripe patterns are discussed, with or without roughness, in the light of various low-order models.