Critical Depth, Velocity Profiles, and Averaging

Open-channel hydraulics is a subset of shallow-water theory that, in turn, is a subset of hydrodynamics. The distinguishing feature of shallow-water theory is the assumption of hydrostatic pressure. One-dimensional, open-channel hydraulics goes further in that it averages the velocity in both the vertical and the horizontal directions. The derivation of the open-channel equations from the general hydrodynamics relationships displays the approximations. The hydrostatic approximation is well documented in the literature, but some confusion persists in the averaging process, a confusion that leads to erroneous definitions of critical depth, Froude number, and even Bernoulli's equation. Critical depth must be defined so that it displays the singularities in the unsteady and steady equations of motion. Bernoulli's equation stems from conservation of momentum and should not carry an energy correction factor for nonuniform velocity distribution. In most circumstances, the error made by neglecting momentum and energy correction factors is tertiary, smaller than errors of erroneous friction and nonhydrostatic pressure.