Time of Concentration Formula for Sheet Flow of Varying Flow Regime

For an overland plane with rainfall as the lateral inflow, the flow regime is complicated by the varying flow depth and velocity along the plane. The flow regime becomes variable. For a plane that is sufficiently long, from the upstream to the downstream end of the plane, the flow regime may change from laminar through transitional to turbulent. In this paper, by coupling the kinematic wave equations with the Darcy-Weisbach friction formula, a time of concentration formula for a portion of a plane subject to uniform rainfall excess and a constant upstream inflow is derived. This formula is applicable to a portion of plane with a single flow regime only. For practical application, this formula is further developed for a plane of uniform slope subject to uniform rainfall excess, with flow regime varying from laminar through transitional to turbulent. An application of the formula for varying flow regime shows that its time of concentration estimate is longer than those based on a single flow regime covering the entire plane.