Modeling rough-wall turbulent channel flow with principal component analysis enhanced amplitude modulation

Direct numerical simulation of turbulent channel flows over rough surfaces, formed from hexagonally-packed arrays of hemispheres on both walls, were performed at friction Reynolds numbers $Re_\tau = 200$, $400$, and $600$. The inner normalized roughness height $k^+=20$ was maintained for all Reynolds numbers while the spacing between hemispheres was varied from $d/k=2-4$. The interactions between the near-wall small-scale fluctuations and outer layer large-scale turbulence were studied by amplitude modulation (AM) analysis that has been modified to include principal component analysis (PCA). Based on these interactions, a PCA-adapted predictive inner--outer model was developed to address the modeling of anisotropic effects near the roughness and effectively predict the near-wall statistics up to $4^{th}$ order moments of all velocity fluctuations, including cross terms. The predictions based on the PCA-adapted model were shown to agree excellently with the original statistics from the DNS with better predictions of the statistics of $v$ compared to model without the PCA.