Direct solutions for normal depth in parabolic and rectangular open channels using asymptotic matching technique

• An asymptotic matching solution for normal depth in a parabolic channel.
• An asymptotic matching solution for normal depth in a rectangular channel.
• Improved asymptotic matching solution for normal depth in rectangular channels.
• A clear graphical representation of the solution method.
• Full range and near-exact solutions for normal depth (parabolic and rectangular).

Normal flow depth is an important parameter in design of open channels and analysis of gradually varied flow. In open channels with parabolic and rectangular cross-sections, the governing equations are nonlinear in terms of the normal depth and thus solution of the implicit equations involves numerical methods. In current research explicit solutions for these channels have been obtained using asymptote matching technique. For the parabolic channel, the maximum error of proposed equation for normal depth is less than 0.07% (near exact solution). But, in rectangular channels, the maximum error of proposed equation for normal depth is less than 1.94% which is not very accurate. The efficiency of the asymptote matching technique can be considerably improved by adding a power-law function between two asymptotes. For rectangular channel a new solution for normal flow depth is developed using the improved asymptote matching technique proposed in this research. The maximum error of this full range solution is less than 0.12%. The results showed that the improvement in proposed solution is substantial. Proposed full range solutions have definite physical concept, high accuracy and easy calculation and are well-suited for manual calculations and computer programming.