Turbulent boundary-layer flow over a long plate with a uniformly rough surface

We develop a semi-empirical model for a zero-pressure-gradient turbulent boundary layer flowing over a flat plate of length $L$ and covered with homogeneous, uniform roughness of equivalent sand-grain roughness $k_s$. Use is made of the log-wake model for the stream-wise mean velocity that includes a transitional-asymptotic roughness correction together with the K\'arm\'an integral relation. For $Re_L=U_\infty\,L/\nu$ very large, the velocity ratio $S=U_\infty/u_\tau$ at $x=L$, the plate drag coefficient $C_D$ and other mean-flow properties can be obtained for given $Re_L$ and $k_s/L$. Three distinct cases are discussed; the smooth-wall, fully-rough and long-plate limits. Of these, the most important is the fully-rough case where $k_s/L$ is fixed with $Re_L\to \infty$, giving that $C_D=f_1(k_s/L)$, $\delta_L/L=f_2(k_s/L)$ independent of $Re_L$. This agrees qualitatively with Granville (1958) although somewhat different $C_D(k_s/L)$ is obtained owing to the present use of a wake function. Thus for a given $k_s$ and $x = L$ location on a fully rough vehicle, the boundary layer thickness and the drag coefficient is invariant with unit Reynolds number $U_\infty/\nu$.