Turbulent flow over a long flat plate with uniform roughness

For turbulent boundary-layer flow under a uniform freestream speed U∞ over a plate of length L, covered with uniform roughness of nominal sand-grain scale ks , the physical behaviors underlying two distinguished limits at large ReL ≡ U∞L/ν are explored: the fully rough wall flow where ks/L is fixed and the long-plate limit where Rek ≡ U∞ks/ν is fixed. For the fully rough limit it is shown that not only is the drag coefficient CD independent of ReL but that a universal skin-friction coefficient Cf and normalized boundary-layer thickness δ/ks can be found that depends only on ks/x, where x is the downstream distance. In the long-plate limit, it is shown that the flow becomes asymptotically smooth at huge ReL at a rate that depends on Rek . Comparisons with wind-tunnel and field data are made.