Direct calculation for normal depth and critical depth of circular section tunnel under free flow

There is no analytical expression to solve the normal depth and critical depth of circular section tunnel under free flow. The traditional solving methods, such as trial method and chart method, are very complicated and low precision. In this paper, Newton iteration formula for computing normal depth and critical depth was put forward by mathematical transformation of uniform flow equation and critical flow equation of circular section tunnel. Then, the relationship between the corresponding central angle and the introduced parameters was analyzed. An approximate formula for angle was obtained according to optimal uniform approximation principle. Using this approximate formula as initial value of iteration formula, a direct formula for calculating normal depth and critical depth of circular section tunnel was established after its first iterative. Results of the example calculation and error analysis showed that the maximum relative error of normal depth and critical depth was 0.32% and 0.0049%, respectively with the application scope of engineering. The precision will be 103 and 105 times higher than the original one after using the iterative formula once again.