For turbulent boundary-layer flow under a uniform freestream speed $U_{\infty }$ over a plate of length $L$, covered with uniform roughness of nominal sand-grain scale $k_s$, the physical behaviors underlying two distinguished limits at large ${\text{Re}}_L\equiv U_{\infty }L/\nu$ are explored: the fully rough wall flow where $k_s/L$ is fixed and the long-plate limit where ${\text{Re}}_k\equiv U_{\infty }{k}_s/\nu$ is fixed. For the fully rough limit it is shown that not only is the drag coefficient $C_D$ independent of ${\text{Re}}_L$ but that a universal skin-friction coefficient $C_f$ and normalized boundary-layer thickness $\delta /k_s$ can be found that depends only on $k_s/x$, where $x$ is the downstream distance. In the long-plate limit, it is shown that the flow becomes asymptotically smooth at huge ${\text{Re}}_L$ at a rate that depends on ${\text{Re}}_k$. Comparisons with wind-tunnel and field data are made.

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• Roughness
• Structure & turbulence of boundary layers
• Turbulence