We show that Manning's empirical formula for the mean velocity of turbulent flows in channels represents the power-law asymptotic behavior of a flow of incomplete similarity in the relative roughness. We then derive the formula based on the phenomenological theory of turbulence. Our derivation yields the correct similarity exponent; it justifies Manning's use of a single parameter, the hydraulic radius, to characterize the geometry of the cross section; and it affords insight into the mechanism of momentum transfer.
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Gioia et al. (2001) studied this question.