Influence of Reynolds number on the pressure gradient of turbulent channel flow with heterogeneous roughness

Pressure-gradient measurements are reported for turbulent channel flows over six heterogeneous rough surfaces with $50\,%$ roughness coverage, composed of P60 sandpaper patches arranged as streamwise-aligned strips, spanwise-aligned strips, or a checkerboard pattern of square patches. A new metric quantifies the relative pressure-gradient increase of heterogeneous surfaces compared with a homogeneous rough ( $100\,%$ roughness coverage) surface. Above a surface-dependent critical Reynolds number, this metric Δ ^* becomes almost independent of Reb for all investigated surfaces, whereas below this threshold a clear dependence is observed. Complementary hot-wire measurements provide insight into the corresponding flow fields. According to the flow field data, the surfaces exhibiting the smallest Δ ^* at low Reb appear almost homogeneous to the flow, a feature that is not present at higher Reynolds number. Based on these observations the concept of a hydraulically heterogeneous surface is introduced. Surfaces with sandpaper patch dimensions of the order of the channel half-height can be perceived by the flow as homogeneous when the Reynolds number is low. As Reb rises, surface heterogeneity translates into flow heterogeneity, which first intensifies in the transitionally heterogeneous regime, then approaches an almost self-similar state in the fully heterogeneous regime where Δ ^* is approximately constant. In this regime, comparable pressure gradients for surfaces that generate markedly different mean flow fields indicate that turbulent secondary flows induced by streamwise-aligned roughness strips have little effect on overall drag. Remarkably, the pressure gradient in this regime is captured well by a simple predictive model.