Hydraulic exponents M and N for gravity flow pipes

Mathematical expressions are derived for the hydraulic exponents M and N as functions of the gradually varied flow (GVF) depth using a circular channel section. The derived expressions of M(y/d₀) and N(y/d₀) for continuously varying depth are compared to exponents M and N proposed by Chow [Chow, V.T., Integrating the equation of gradually varied flow. ASCE, Proc. 81, Vol. 11, 1955] using constant averaged GVF depths. The exponent N shows a remarkable difference (an opposite trend). The computation of the GVF length is calculated using the exponents M and N for varying and constant averaged GVF depths. Numerical integration approach based on the Simpson's rule method is used to calculate the GVF length. The results of the calculated GVF profile length using the derived exponents M(y/d₀) and N(y/d₀) are found to be closer to the GVF length calculated from the exact formulation of the GVF dynamic equation. The percentage difference ranges from 0.017 to 6.9% for various bed slopes and GVF depth limits. Using the Chow [ibid.] constant exponents M and N, the calculated GVF length resulted in a wider values with a percentage difference ranges from 1.2 to 130%. Hence, a remarkable improvement of the computation of GVF profile length is achieved using the derived M(y/d₀) and N(y/d₀) exponents.