三种抛物线形渠道共轭水深的显式计算公式
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摘要
为了得到半立方、平方、立方抛物线形渠道共轭水深的显式计算公式,对这3种抛物线形渠道的水跃方程进行恒等变形,利用临界水深介于跃前水深和跃后水深之间的性质,得到了无量纲跃前水深x和无量纲跃后水深y之间的关系式,进一步分别得到其迭代公式。在工程常用范围内,利用excel拟合得到其迭代初值,提出了一套抛物线类渠道共轭水深的显式计算公式。最后,实例及误差分析表明:半立方、平方、立方抛物线形断面无量纲跃前水深x、无量纲跃后水深y最大相对误差分别为0.25%,-0.23%;0.17%,-0.29%;0.31%,0.39%。公式物理概念清晰,计算简捷,精度高,适用范围广。
In order to get the explicit calculation formula of conjugate depth for semi-cubic,square,and cubic parabolic-shaped channels,the jump equations of the three parabola-shaped channels were transformed identically and the relationships between the dimensionless water depth x before jump and the dimensionless water depth y after jump were obtained according to the property that the critical depth was between the pre-jump depth and the postjump depth. Their iterative formulas were further obtained respectively. Hence,a set of explicit calculation formulas of conjugate depth for semi-cubic,square,and cubic parabolic-shaped channels were obtained by fitting the iterative initial value through excel in common engineering scope. Finally,example and error analysis shows that the absolute value of maximum relative error of dimensionless water depth x before jump was 0. 25%,0. 17%,and0. 31% respectively for the semi-cubic,square,and cubic parabolic-shaped channel,and that of dimensionless water depth y after jump was respectively- 0. 23%,- 0. 29%,and 0. 39%. The formulas were convenient and highly accurate with clear physical meaning and wide application scope.
In order to get the explicit calculation formula of conjugate depth for semi-cubic,square,and cubic parabolic-shaped channels,the jump equations of the three parabola-shaped channels were transformed identically and the relationships between the dimensionless water depth x before jump and the dimensionless water depth y after jump were obtained according to the property that the critical depth was between the pre-jump depth and the postjump depth. Their iterative formulas were further obtained respectively. Hence,a set of explicit calculation formulas of conjugate depth for semi-cubic,square,and cubic parabolic-shaped channels were obtained by fitting the iterative initial value through excel in common engineering scope. Finally,example and error analysis shows that the absolute value of maximum relative error of dimensionless water depth x before jump was 0. 25%,0. 17%,and0. 31% respectively for the semi-cubic,square,and cubic parabolic-shaped channel,and that of dimensionless water depth y after jump was respectively- 0. 23%,- 0. 29%,and 0. 39%. The formulas were convenient and highly accurate with clear physical meaning and wide application scope.
引文
[1]刘计良,王正中,杨晓松,等.梯形渠道水跃共轭水深理论计算方法初探[J].水力发电学报,2010,29(5):216-219.(LIU Ji-liang,WANG Zheng-zhong,YANG Xiao-song,et al.Preliminary Study on the Theoretical Method for Calculating Conjugate Depth of Trapezoidal Channel[J].Journal of Hydroelectric Engineering,2010,29(5):216-219.(in Chinese))
[2]马吉明,谢省宗,梁元博.城门洞形及马蹄形输入隧洞内的水跃[J].水利学报,2000,(7):20-24.(MA Jiming,XIE Sheng-zong,LIANG Yuan-bo.Hydraulic Jumps in Rectangular Conduit with Circular Upper Wall and Horseshoe Tunnel[J].Shuili Xuebao,2000,(7):20-24.(in Chinese))
[3]冯家涛.梯形渠道水跃共轭水深直接计算公式[J].力学与实践,1998,20(5):50-51.(FENG Jia-tao.A Direct Calculation Formula for Conjugate Water Depth of Water Jump in a Trapezoid Channel[J].Mechanics in Engineering,1998,20(5):50-51.(in Chinese))
[4]李蕊,王正中,张宽地,等.梯形明渠共轭水深计算方法[J].长江科学院院报,2012,29(11):33-36.(LI Rui,WANG Zheng-zhong,ZHANG Kuan-di,et al.Calculation Method for Conjugate Water Depth in Open Trapezoidal Channel[J].Journal of Yangtze River Scientific Research Institute,2012,29(11):33-36.(in Chinese))
[5]郭振家,刘伟哲,马玉,等.圆形断面水跃共轭水深的迭代计算方法[J].吉林水利,2013,(1):7-9.(GUO Zhen-jia,LIU Wei-zhe,MA Yu,et al.Iterative Method of Conjugate Depth for Channels with Cross-section[J].Jilin Water Resources,2013,(1):7-9.(in Chinese))
[6]张新燕,吕宏兴.抛物线形断面渠道正常水深的显示计算[J].农业工程学报,2012,28(21):121-125.(ZHANG Xin-yan,LV Hong-xing.Explicit Solution for Normal Depth in Parabolic-shape Channel[J].Transactions of the CSAE,2012,28(21):121-125.(in Chinese))
[7]赵延风,王正中,方兴,等.半立方抛物线形渠道正常水深算法[J].排灌机械工程学报,2011,29(3):242-245.(ZHAO Yan-feng,WANG Zheng-zhong,FANG Xing,et al.Calculation Method for Normal Depth of Semi-cubic Parabolic Channels[J].Journal of Drainage and Irrigation Machinery Engineering,2011,29(3):242-245.(in Chinese))
[8]藤凯.半立方抛物线形渠道正常水深的近似计算公式[J].长江科学院院报,2012,29(12):30-33.(TENG Kai.The Approximate Formula for Normal Water Depth of Semi-cubic Parabolic Channels[J].Journal of Yangtze River Scientific Research Institute,2012,29(12):30-33.(in Chinese))
[9]文辉,李风玲.立方抛物线断面渠道收缩水深的直接计算方法[J].人民长江,2009,40(13):38-59.(WEN Hui,LI Feng-ling.The Direct Calculation Method of Contraction Depth for the Cubic Parabolic-shaped Channel[J].Yangtze River,2009,40(13):38-59.(in Chinese))
[10]文辉,李风玲.抛物线形断面渠道收缩水深的解析解[J].长江科学院院报,2009,26(9):32-34.(WEN Hui,LI Feng-ling.Analytical Solution of Water Depth in Parabolic-Shaped Channel with Contracted Section[J].Journal of Yangtze River Scientific Research Institute,2009,26(9):32-34.(in Chinese))
[11]赵延风,宋松柏,孟秦倩.抛物线形断面渠道收缩水深的直接计算方法[J].水利水电技术,2008,39(3):36-41.(ZHAO Yan-feng,SONG Song-bai,MENG Qin-qian.A Direct Calculation Method for Water Depth in Parabolic-shaped Channel with Contracted Section[J].Water Resources and Hydropower Engineering,2008,39(3):36-41.(in Chinese))
[12]冷畅俭,王正中.三次抛物线形渠道断面收缩水深的计算公式[J].长江科学院院报,2011,28(4):29-35.(LENG Chang-jian,WANG Zheng-zhong.Formula for Calculating Contracted Water Depth of Channel with Cubic Parabola Cross Section[J].Journal of Yangtze River Scientific Research Institute,2011,28(4):29-35.(in Chinese))
[13]张志昌,刘亚菲,刘松舰.抛物线形渠道水力最优断面的计算[J].西安理工大学学报,2002,18(3):235-237.(ZHANG Zhi-chang,LIU Ya-fei,LIU Song-jian.The Calculation of Optimal Hydraulic Cross Section in Parabola-Shape Canal[J].Journal of Xi’an University of Technology,2002,18(3):235-237.(in Chinese))
[14]魏文礼,杨国丽.立方抛物线形渠道水力最优断面的计算[J].武汉大学学报(工学版),2006,39(3):49-51.(WEI Wen-li,YANG Guo-li.Calculation of Optimal Hydraulic Cross-section for Cubic Parabola-shape Channel[J].Engineering Journal of Wuhan University,2006,39(3):49-51.(in Chinese))
[15]马子普,张根广,赵春龙,等.立方抛物线形渠道水跃共轭水深的迭代算法[J].人民长江,2013,44(1):90-92.(MA Zi-pu,ZHANG Gen-guang,ZHAO Chun-long,et al.The Iterative Formula of the Cubic Parabolicshaped Channel’s Conjugate Depth[J].Yangtze River,2013,44(1):90-92.(in Chinese))
[16]马子普,张根广,冯雪,等.半立方抛物线形明渠共轭水深的迭代算法[J].西北农林科技大学学报(自然科学版),2012,40(11):211-215.(MA Zi-pu,ZHANG Gen-guang,FENG Xue,et al.Iterative Algorithm of Conjugate Depth for Semi-cubic Parabolic Open Channels[J].Journal of Northwest A&F University(Natural Science Edition),2012,40(11):211-215.(in Chinese))
[17]冷畅俭,王弈,王正中.抛物线形断面渠道共轭水深的直接计算公式[J].排灌机械工程学报,2013,31(2):132-136,141.(LENG Chang-jian,WANG Yi,WANG Zheng-zhong.Explicit Calculation Equations for Conjugate Depths of Channels in Parabolic-shaped Crosssection[J].Journal of Drainage and Irrigation Machinery Engineering,2013,31(2):132-136,141.(in Chinese))
[18]汪清,曹青,马子普.二次抛物线形渠道临界水深与共轭水深简化算法[J].水电能源学,2012,30(6):111-113.(WANG Qing,CAO Qing,MA Zi-pu.Simplified Calculation Method of Critical Depth and Conjugate Depth for Second-degree Parabolic Channels[J].Water Resources and Power,2012,30(6):111-113.(in Chinese))
[19]吕宏兴,裴国霞,杨玲霞.水力学[M].北京:中国农业出版社,2002.(LV Hong-xing,PEI Guo-xia,YANG Ling-xia.Hydraulics[M].Beijing:China Agricultural Press,2002.(in Chinese))
[20]李庆扬,王能超,易大义.数值分析[M].北京:清华大学出版社,2008.(LI Qing-yang,WANG Neng-chao,YI Da-yi.Numerical Analysis[M].Beijing:Tsinghua University Press,2008.(in Chinese))
[21]郑阿奇,曹弋.MATLAB实用教程[M].北京:电子工业出版社,2007.(ZHENG A-qi,CAO Yi.Practical Tutorial of MATLAB[M].Beijing:Publishing House of Electronics Industry,2007.(in Chinese))
[2]马吉明,谢省宗,梁元博.城门洞形及马蹄形输入隧洞内的水跃[J].水利学报,2000,(7):20-24.(MA Jiming,XIE Sheng-zong,LIANG Yuan-bo.Hydraulic Jumps in Rectangular Conduit with Circular Upper Wall and Horseshoe Tunnel[J].Shuili Xuebao,2000,(7):20-24.(in Chinese))
[3]冯家涛.梯形渠道水跃共轭水深直接计算公式[J].力学与实践,1998,20(5):50-51.(FENG Jia-tao.A Direct Calculation Formula for Conjugate Water Depth of Water Jump in a Trapezoid Channel[J].Mechanics in Engineering,1998,20(5):50-51.(in Chinese))
[4]李蕊,王正中,张宽地,等.梯形明渠共轭水深计算方法[J].长江科学院院报,2012,29(11):33-36.(LI Rui,WANG Zheng-zhong,ZHANG Kuan-di,et al.Calculation Method for Conjugate Water Depth in Open Trapezoidal Channel[J].Journal of Yangtze River Scientific Research Institute,2012,29(11):33-36.(in Chinese))
[5]郭振家,刘伟哲,马玉,等.圆形断面水跃共轭水深的迭代计算方法[J].吉林水利,2013,(1):7-9.(GUO Zhen-jia,LIU Wei-zhe,MA Yu,et al.Iterative Method of Conjugate Depth for Channels with Cross-section[J].Jilin Water Resources,2013,(1):7-9.(in Chinese))
[6]张新燕,吕宏兴.抛物线形断面渠道正常水深的显示计算[J].农业工程学报,2012,28(21):121-125.(ZHANG Xin-yan,LV Hong-xing.Explicit Solution for Normal Depth in Parabolic-shape Channel[J].Transactions of the CSAE,2012,28(21):121-125.(in Chinese))
[7]赵延风,王正中,方兴,等.半立方抛物线形渠道正常水深算法[J].排灌机械工程学报,2011,29(3):242-245.(ZHAO Yan-feng,WANG Zheng-zhong,FANG Xing,et al.Calculation Method for Normal Depth of Semi-cubic Parabolic Channels[J].Journal of Drainage and Irrigation Machinery Engineering,2011,29(3):242-245.(in Chinese))
[8]藤凯.半立方抛物线形渠道正常水深的近似计算公式[J].长江科学院院报,2012,29(12):30-33.(TENG Kai.The Approximate Formula for Normal Water Depth of Semi-cubic Parabolic Channels[J].Journal of Yangtze River Scientific Research Institute,2012,29(12):30-33.(in Chinese))
[9]文辉,李风玲.立方抛物线断面渠道收缩水深的直接计算方法[J].人民长江,2009,40(13):38-59.(WEN Hui,LI Feng-ling.The Direct Calculation Method of Contraction Depth for the Cubic Parabolic-shaped Channel[J].Yangtze River,2009,40(13):38-59.(in Chinese))
[10]文辉,李风玲.抛物线形断面渠道收缩水深的解析解[J].长江科学院院报,2009,26(9):32-34.(WEN Hui,LI Feng-ling.Analytical Solution of Water Depth in Parabolic-Shaped Channel with Contracted Section[J].Journal of Yangtze River Scientific Research Institute,2009,26(9):32-34.(in Chinese))
[11]赵延风,宋松柏,孟秦倩.抛物线形断面渠道收缩水深的直接计算方法[J].水利水电技术,2008,39(3):36-41.(ZHAO Yan-feng,SONG Song-bai,MENG Qin-qian.A Direct Calculation Method for Water Depth in Parabolic-shaped Channel with Contracted Section[J].Water Resources and Hydropower Engineering,2008,39(3):36-41.(in Chinese))
[12]冷畅俭,王正中.三次抛物线形渠道断面收缩水深的计算公式[J].长江科学院院报,2011,28(4):29-35.(LENG Chang-jian,WANG Zheng-zhong.Formula for Calculating Contracted Water Depth of Channel with Cubic Parabola Cross Section[J].Journal of Yangtze River Scientific Research Institute,2011,28(4):29-35.(in Chinese))
[13]张志昌,刘亚菲,刘松舰.抛物线形渠道水力最优断面的计算[J].西安理工大学学报,2002,18(3):235-237.(ZHANG Zhi-chang,LIU Ya-fei,LIU Song-jian.The Calculation of Optimal Hydraulic Cross Section in Parabola-Shape Canal[J].Journal of Xi’an University of Technology,2002,18(3):235-237.(in Chinese))
[14]魏文礼,杨国丽.立方抛物线形渠道水力最优断面的计算[J].武汉大学学报(工学版),2006,39(3):49-51.(WEI Wen-li,YANG Guo-li.Calculation of Optimal Hydraulic Cross-section for Cubic Parabola-shape Channel[J].Engineering Journal of Wuhan University,2006,39(3):49-51.(in Chinese))
[15]马子普,张根广,赵春龙,等.立方抛物线形渠道水跃共轭水深的迭代算法[J].人民长江,2013,44(1):90-92.(MA Zi-pu,ZHANG Gen-guang,ZHAO Chun-long,et al.The Iterative Formula of the Cubic Parabolicshaped Channel’s Conjugate Depth[J].Yangtze River,2013,44(1):90-92.(in Chinese))
[16]马子普,张根广,冯雪,等.半立方抛物线形明渠共轭水深的迭代算法[J].西北农林科技大学学报(自然科学版),2012,40(11):211-215.(MA Zi-pu,ZHANG Gen-guang,FENG Xue,et al.Iterative Algorithm of Conjugate Depth for Semi-cubic Parabolic Open Channels[J].Journal of Northwest A&F University(Natural Science Edition),2012,40(11):211-215.(in Chinese))
[17]冷畅俭,王弈,王正中.抛物线形断面渠道共轭水深的直接计算公式[J].排灌机械工程学报,2013,31(2):132-136,141.(LENG Chang-jian,WANG Yi,WANG Zheng-zhong.Explicit Calculation Equations for Conjugate Depths of Channels in Parabolic-shaped Crosssection[J].Journal of Drainage and Irrigation Machinery Engineering,2013,31(2):132-136,141.(in Chinese))
[18]汪清,曹青,马子普.二次抛物线形渠道临界水深与共轭水深简化算法[J].水电能源学,2012,30(6):111-113.(WANG Qing,CAO Qing,MA Zi-pu.Simplified Calculation Method of Critical Depth and Conjugate Depth for Second-degree Parabolic Channels[J].Water Resources and Power,2012,30(6):111-113.(in Chinese))
[19]吕宏兴,裴国霞,杨玲霞.水力学[M].北京:中国农业出版社,2002.(LV Hong-xing,PEI Guo-xia,YANG Ling-xia.Hydraulics[M].Beijing:China Agricultural Press,2002.(in Chinese))
[20]李庆扬,王能超,易大义.数值分析[M].北京:清华大学出版社,2008.(LI Qing-yang,WANG Neng-chao,YI Da-yi.Numerical Analysis[M].Beijing:Tsinghua University Press,2008.(in Chinese))
[21]郑阿奇,曹弋.MATLAB实用教程[M].北京:电子工业出版社,2007.(ZHENG A-qi,CAO Yi.Practical Tutorial of MATLAB[M].Beijing:Publishing House of Electronics Industry,2007.(in Chinese))