Theory for flow resistance caused by submerged roughness elements

Herein, a phenomenological theory that unifies the flow resistance caused by a variety of submerged bluff bodies, including cylindrical vegetation, two-dimensional dunes and the roughness elements studied in Schlichting's 1936 experiments, is presented. For the different flow obstructions, scaling relations that relate the equivalent roughness length to the obstructions' spatial dimensions are derived, by adopting the concepts of the turbulent energy cascade and the momentum thickness in the wake of a bluff body. In the derivation, it is hypothesized that the equivalent roughness height represents the volumetric average of the prevailing turbulence length scale that is present in wakes of the roughness elements. A comparison with experimental literature data indicates that the derived scaling relations capture the dominant dependencies of the equivalent roughness length. These expressions have important engineering applications in large-scale flows where employment of more detailed flow-modelling techniques is not feasible.