Influence of irregular three-dimensional rough surfaces on the roughness function

The influence of irregular three-dimensional rough surfaces on the displacement of the logarithmic velocity profile relative to that of a smooth wall in turbulent flow, known as the roughness function, is studied using direct numerical simulations. Five different surface power spectral density (PSD) shapes were considered, and for each, several rough Gaussian surfaces were generated by varying the root mean square ( $k_{rms}$ ) of the surface heights. It is shown that the roughness function ( $\Delta U^{+}$ ) depends on both the PSD and $k_{rms}$ . For a given $k_{rms}$ , $\Delta U^{+}$ increases as the wavenumbers of the PSD expand to large values, but at a rate that decreases with the magnitude of the wavenumbers. Although $\Delta U^{+}$ generally does not scale with either $k_{rms}$ or the effective slope $ES$ when these variables are considered singularly, for PSDs with low wavenumbers, $\Delta U^{+}$ tends to scale with $ES$ , whereas as wavenumbers increase, $\Delta U^{+}$ tends to scale with $k_{rms}$ . An equivalent Nikuradse sand roughness of about eight times $k_{rms}$ is found, which is similar to that observed in previous studies for a regular three-dimensional roughness. Finally, it is shown that $k_{rms}$ and the effective slope are sufficient to describe the roughness function in the transitional rough regime.