Simplified calculation formulas for critical water depth of horseshoe cross section

The channel with standard horseshoe cross section was commonly used in diversion tunnel engineering. To get the critical depth in the section, a transcendental equation has to be solved. In order to solve the problem that the calculation formulas of critical depth in the section were not expressed by explicit function, direct calculation formulas with simple form and high efficiency were presented in the current work. The quasi-linear function, which was treated as the approximation function, was introduced. The formulas for the critical depth of the two standard horseshoe cross sections were approximated by the quasi-linear function in engineering scope, when the dimensionless critical depth ranged from 0.01 to 1.80. The expressions of the quasi-linear function for the critical depth of the two sections were also obtained, and the error analysis and evaluation were conducted. The results showed that the maximum error was less than 0.6%, indicating that the direct formulas of the quasi-linear function were much simpler, precise and wider than previous ones in applications.