Numerical integration of gradually varied flow in trapezoidal channel

Mathematical expressions are derived for the hydraulic exponents as functions of the gradually varied flow (GVF) depth using a trapezoidal channel cross-section. An approach is developed utilizing the continuously varying exponents to calculate the GVF length based on several numerical integration approaches, namely four and five points Gaussian quadratures and adaptive Simpson quadrature. Several applications for various channel bed slopes and GVF depths are presented. Comparison between the results obtained by the various integration methods using continously varying hydraulic exponents and the classical direct integration method using constant averaged hydraulic exponents, reveals no significant difference in the computed values of the GVF length. This conclusion justified the previous results obtained using the direct integration method. However, due to the availability of high speed computers, an exact approach to the problem is appropriate. An integrated computer program is developed using four and/or five points Gaussian and adaptive Simpson quadratures for the numerical integration. The computer model has the capability to calculate the total GVF length or detailed X-Y values for tabular and/or graphical representation of the GVF profile.