Explicit solutions for critical and normal depths in trapezoidal and parabolic open channels

Normal and critical depths are important parameters in the design of open channels and analysis of gradually varied flow. In trapezoidal and parabolic channels, the governing equations are highly nonlinear in the normal and critical flow depths and thus solution of the implicit equations involves numerical methods (except for critical depth in parabolic channels). In current research explicit solutions have been obtained using the non-dimensional forms of the governing equations. For the trapezoidal cross section, the maximum error of critical flow depth is less than 6 × 10−6% (near exact solution) and the maximum error of normal depth is less than 0.25% (very accurate solution). The maximum error of normal flow depth for parabolic cross section is also less than 8 × 10−3% (near exact solution). Proposed explicit equations have definite physical concept, high accuracy, easy calculation, and wide application range compared with the existing direct equations.