Modelling inner--outer interactions of average wall-shear-stress in rough-wall turbulent channel flows using amplitude modulation

DNS of turbulent channel flows over hemispherical roughness were performed with friction Reynolds numbers $Re_\tau=200$, $400$ and $600$. The inner scaled roughness height $k^+=20$ was maintained for all Reynolds numbers while the spacing between hemispheres was varied from $d/k=2$--$4$. Wall-shear-stresses on both walls were extracted by integrating the stress tensor over the rough surfaces and averaged within arrays of cells of various sizes. Analysis of amplitude modulation was applied to the cell-averaged wall-shear-stress to investigate the inner--outer interactions between the outer large-scale structures and the wall-shear-stress. Pre-multiplied streamwise spectra of the universal wall-shear-stress signals (based on AM model) was investigated for all roughness and a scaling relation between the spectra and cell sizes was found, which collapses all spectra at large wavelengths between various roughness parameters. However, roughness shedding and spectral aliasing modify the scaling at small wavelengths. Spherical harmonics was used to extract the detailed power-spectrum of the wall-shear-stress to help quantify the observed behavior.