Calculation of water normal depth in channels with common shapes
LIU Ji-Liang-1, WANG Zheng-Zhong-1, SU De-Hui-2, YANG Xiao-Song-1, LIU Quan-Hong-1
(1. Institute of Water Resources and Hydropower Engineering, Northwest A & F University, Yangling, Shaanxi 712100, China; 2. Montgomery & Barnes, Inc., Houston, Texas 77067, USA)
Abstract:Water normal depth calculation is an important task in the design of an open channel for drainage and irrigation. First, dimensionless variables including channel roughness coefficient, bed slope, geometric parameters, and discharge were introduced for each common section. In order to obtain general equations and evaluate their accuracy, the explicit equations available for water normal depth were expressed by those dimensionless variables. Then, the real ranges of those variables were specified for each common section in applications, the relative errors in the explicit equations that involve the variables were analyzed in the ranges. Also, the global relative error distribution diagrams were plotted to compare the maximum relative error and the global one. According to these analyses, the most appropriate equations, which can explicitly calculate the water normal depth in the channels with five types of common section, were spotted. Finally, based on the best approximations algorithm a new piecewise explicit equation was proposed for the water normal depth in a standard castlegate section. It was shown that the maximum relative errors in all the recommended equations are less than 1% for the water normal depth in six types of common section. The results in the paper have provided an effective method for designing channels with six types of common section and carrying out hydraulic calculations.
刘计良, 王正中, 苏德慧, 杨晓松, 刘铨鸿. 典型断面渠道正常水深计算[J]. 排灌机械工程学报, 2012, 30(3): 324-329.
Liu-Ji-Liang, WANG Zheng-Zhong, SU De-Hui, YANG Xiao-Song, LIU Quan-Hong. Calculation of water normal depth in channels with common shapes. Journal of Drainage and Irrigation Machinery Engin, 2012, 30(3): 324-329.
参考文献(References)[1]李家星,赵振兴. 水力学(上册)[M].2版.南京: 河海大学出版社, 2001.[2]刘计良,王正中,苏德慧,等. 典型断面渠道临界水深计算[J]. 排灌机械工程学报, 2012,30(2):181-187.Liu Jiliang, Wang Zhengzhong, Su Dehui, et al. Calculation of water critical depth in channels with common shapes[J].Journal of Drainage and Irrigation Machinery Engineering, 2012,30(2):181-187.(in Chinese)[3]Swamee P K. Normaldepth equations for irrigation canals[J]. Journal of Irrigation and Drainage Engineering, 1994, 120(5):942-948.[4]Swamee P K, Rathie P N. Exact solutions for normal depth problem[J]. Journal of Hydraulic Research, 2004, 42(5):541-547.[5]Vatankhah A R, Easa S M. Explicit solutions for critical and normal depths in channels with different shapes[J]. Flow Measurement and Instrumentation, 2011,22(1): 43-49.[6]王正中,席跟战,宋松柏,等. 梯形明渠正常水深直接计算公式[J]. 长江科学院院报, 1998,15(6):1-3.Wang Zhengzhong, Xi Genzhan, Song Songbai, et al. A direct calculation formula for normal depth in open trapezoidal channel[J]. Journal of Yangtze River Scientific Research Institute, 1998,15(6):1-3.(in Chinese)[7]赵延风,祝晗英,王正中. 一种新的圆形过水断面正常水深近似计算公式[J]. 河海大学学报:自然科学版, 2010, 38(1):68-71.Zhao Yanfeng, Zhu Hanying, Wang Zhengzhong. New approximate formula for circular cross section of normal depth[J]. Journal of Hohai University:Natural Science, 2010,38(1):68-71.(in Chinese)[8]王正中,宋松柏,王世民,等. 弧底梯形明渠正常水深的直接算法[J]. 长江科学院院报, 1999,16(4):31-34.Wang Zhengzhong, Song Songbai, Wang Shimin, et al. A direct formula calculating normal depth for open trapezoidal channel with spherical bed[J]. Journal of Yangtze River Scientific Research Institute, 1999,16(4):31-34.(in Chinese)[9]文辉,李风玲,欧军利,等. 城门洞形断面隧洞正常水深的近似算法[J]. 给水排水, 2007,33(7):19-21.Wen Hui, Li Fengling, Ou Junli, et al. Approximate solution of the normal depth inside tunnel with arch crosssection[J]. Water and Wastewater Engineering, 2007, 33(7):19-21.(in Chinese)[10]赵延风,刘军,梅淑霞,等. 普通城门洞形断面正常水深的近似计算方法[J]. 武汉大学学报:工学版, 2009,42(6):773-775.Zhao Yanfeng, Liu Jun, Mei Shuxia, et al. Approximate method for calculating normal water depth in common cityopening shaped crosssection[J]. Engineering Journal of Wuhan University, 2009,42(6):773-775.(in Chinese)[11]张宽地,吕宏兴,王光谦,等. 普通城门洞形隧洞正常水深的直接计算方法[J]. 农业工程学报, 2009,25(11): 8-12.Zhang Kuandi, Lü Hongxing, Wang Guangqian, et al. Direct calculating method for normal depth of arched section tunnel[J]. Transactions of the CSAE, 2009,25(11): 8-12.(in Chinese)[12]吕宏兴,辛全才,华立峰. 马蹄形过水断面正常水深的迭代计算[J]. 长江科学院院报, 2001,18(3):7-10.Lü Hongxing, Xin Quancai, Hua Lifeng. Calculation on normal depth of horseshoe cross section by iterative method[J]. Journal of Yangtze River Scientific Research Institute, 2001,18(3):7-10.(in Chinese)[13]王顺久,侯玉,张欣莉,等. 马蹄形过水断面正常水深计算的遗传算法[J]. 灌溉排水, 2002,21(3):43-45.Wang Shunjiu, Hou Yu, Zhang Xinli, et al. Genetic algorithm for calculating normal water depth of Horseshoe cross section[J]. Irrigation and Drainage, 2002,21(3):43-45.(in Chinese)[14]文辉,李凤玲,李霞. 标准I型马蹄形断面正常水深的近似计算[J]. 人民黄河, 2008,30(7):89-90.Wen Hui, Li Fengling, Li Xia. Approximate calculation of normal depth of the standard Itype horseshoe tunnel[J]. Yellow River, 2008,30(7):89-90.(in Chinese)[15]张宽地,王光谦,吕宏兴,等. 基于改进粒子群算法求解马蹄形断面正常水深[J]. 排灌机械工程学报, 2011,29(1):54-60.Zhang Kuandi, Wang Guangqian, Lü Hongxing, et al. Applying particle swam optimization algorithm on normal depth calculation of horseshoe section tunnel[J]. Journal of Drainage and Irrigation Machinery Engineering, 2011, 29(1):54-60.(in Chinese)[16]Liu Jiliang, Wang Zhengzhong, Fang Xing. Iterative formulas and estimation formulas for computing normal depth of horseshoe crosssection tunnel[J]. Journal of Irrigation and Drainage Engineering, 2010,136(11): 786-790.