Explicit Solution for Flow Depth in Open Channels of Trapezoidal Cross-Sectional Area: Classic Problem of Interest

Abstract

This study presents an explicit solution for the flow depth in open channels of trapezoidal cross-sectional area. Four depths are encountered in most classical open-channel hydraulics problems, namely normal depth, critical depth, alternate depths from the specific energy equation, and sequent depths from the specific force equation. The study proposes new equations derived from dimensional and regression analysis to estimate directly those flow depths. The novelty of this work lies in the proposed regression equations for the prediction of the alternate and sequent flow depths, which can be regarded as a new contribution to the literature of classical open-channel flow because currently only analytical solutions of very complex formulations exist for those depths. The range of used data makes the proposed regression equations applicable to laboratory flumes, irrigation channels, and large rivers as well. The equations are examined over a wide range of flow and geometric conditions, providing satisfactory predictions when compared to the exact solution obtained from the governing hydraulic equations. Maximum and average errors in the flow-depth predictions are under 6 and 2.5%, respectively, which are acceptable in most hydraulic engineering practice. The proposed equations are coupled with fixed-point iterative equations to provide more-accurate predictions of the flow depths when desired. A sensitivity analysis showed that 6% uncertainty in the estimates of hydraulic parameters such as channel longitudinal bed slope and Manning’s coefficient provides errors in the flow-depth prediction comparable to the errors encountered in the proposed regression equations. Thus, the accuracy of the proposed regression equations for the prediction of flow depths can be regarded as satisfactory considering the errors that come from other sources of uncertainty. The proposed equations are simple and would be useful in hydraulic engineering practice when quick and accurate estimates are needed for those depths. They can also be used to find the initial values for the flow depth to be utilized in the proposed fixed-point iterative equations or any other alternative numerical scheme if the exact solution is required.