Direct calculation of critical depth of horseshoe section tunnel

Horseshoe section tunnel is a common type in the free-flow tunnel. There is no analytical expression to solve the critical depth of horseshoe section tunnel under free flow. The traditional solving methods, such as trial method and chart method, are very complicated with low precision. In this study, by mathematical transformation of critical flow equation of horseshoe section tunnel, the relationship between the dimensionless parameters and relative critical depth was analyzed and simulated. Using the gradual optimization and fitting principle, a direct formula for calculating the critical depths of standard Ⅰ-type and Ⅱ-type horseshoe section tunnel was established. Results of the example calculation and error analysis show that the maximum relative error of critical depth is 0.8%, in agreement with the application scope of engineering. The characteristics of this formula is exact and simple, which supplies a reference to manual compilation of engineering design and hydraulic design.