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中文摘要：
马蹄形过水断面是无压隧洞较常采用的断面形式之一,水力计算中临界水深的求解无显函数公式,传统的试算法或查图法不仅计算过程繁琐复杂,而且计算精度不高。该文通过对马蹄形断面临界流方程的数学变换,并对本文引入的无量纲参数与相对临界水深的关系分析及数值计算,应用逐步优化拟合原理,得到标准Ⅰ、Ⅱ型马蹄形断面临界水深的直接计算式。实例计算及误差分析表明：在工程实用范围内该公式最大相对误差仅为0.8%,且该式物理概念清晰明确、公式形式简捷,为工程设计及水工设计手册的编制提供有益的参考。
英文摘要：
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.
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