Published online by Cambridge University Press: 22 January 2021
A surface roughness from a recently cleaned and painted ship's hull was scanned, scaled and replicated for laboratory testing to systematically investigate the influence of the ratio of in-plane roughness wavelength, $\lambda$, with respect to the boundary layer thickness
$\delta$. The experiments were performed by geometrically scaling the surface which maintains a constant effective slope
$ES_x$ and solidity
$\varLambda$, while the ratio of
$\lambda /\delta$ is varied. Here we scale the scanned roughness topography by a factor of 2.5 and 15, and measure the mean velocity profiles in the turbulent boundary layers developing over these surfaces at a range of free stream velocities and streamwise measurement locations. The results show that the
$2.5\times$ scaled roughness, which has
$\lambda /\delta \ll 1$, behaves in the expected
$k$-type manner, with a roughness function
${\rm \Delta} U^+$ that is proportional to the viscous-scaled roughness height. The
$15\times$ surface, however, which has
$\lambda /\delta \approx 1$, exhibits very different non-
$k$-type behaviour. This larger surface does not approach the fully rough asymptote and also exhibits a drag penalty that is comparable to the
$2.5\times$ case despite the sixfold increase in the roughness height. Measurements on a spanwise–wall-normal plane reveal that the
$15\times$ surface has introduced a large-scale spanwise variation in mean streamwise velocity (dispersive stresses) that extend far beyond the logarithmic region. Together this evidence suggests that a demarcation between
$k$-type and non-
$k$-type behaviour can occur in situations where the in-plane roughness wavelength approaches the boundary layer thickness. This finding has important implications to how we scale small-scale roughness from high Reynolds number (Re) large-scale applications for testing in low Re small-scale laboratory facilities or simulations.