Analytical solution of gradually varied flow equation in circular channels using variable Manning coefficient

• A semi-analytical solution for water depth profile along a circular channel.
• The proposed direct solution uses a single step for the profile computation.
• The proposed solution does not depend on mesh size as standard numerical methods.
• The solution can be simply implemented by Microsoft Excel spreadsheet.

The direct integration method is used to solve the equation of gradually varied flow (GVF) in open prismatic channels. The GVF is a non-uniform flow with gradual changes in flow depth. In circular channels, Manning roughness coefficient under the partially full flow condition varies with the flow depth, and thus a variable Manning coefficient should be used to calculate the water surface profile. It is accepted that the Manning coefficient varies with flow depth in accordance with Camp’s curve. Given that circular channels have an important application in sewer systems, this research presents a semi-analytical approach for establishing the gradually varied flow profiles in circular channels through application of a variable Manning coefficient. For this, using two approximation expressions, the integrand of the GVF equation is expanded into a finite set of partial fractions and then every term of them is integrated separately. The proposed semi-analytical solution uses a single step for the computation of water surface profiles and provides an accurate and simple way to compute GVF flow profiles.