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⁢𝛱∗ becomes almost independent of 𝑅𝑒_𝑏 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⁢𝛱∗ at low 𝑅𝑒_𝑏 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 𝑅𝑒_𝑏 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⁢𝛱∗ 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.