平底悬链线形明渠水力最优断面求解
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摘要
断面设计是渠道设计的重要内容之一,适宜的渠道断面不仅能够增加过流能力,提高输水效率,减小输水损失,还能降低建造成本。该文提出了一种具有平底和悬链线形侧边的明渠断面。这种断面将平底和悬链线侧边平滑连接,既具有平底断面建造容易、灵活,管护方便,底部容易压实,侧边和平底可以用不同材料建造(以降低成本)等优点,也具有悬链线形断面过流能力大、无应力集中拐角、不宜渗漏、防冻胀能力强,耐久性好等优点,可广泛应用于大、中、小型渠道及寒区,具有良好的实用价值。推导了过流面积、湿周、水面宽度等水力断面特性计算公式。提出了一个更简单的正常水深的迭代算法。基于拉格朗日乘子法,推导出了平底悬链线形明渠的水力最优断面,结果表明其水力最优断面的底宽与水深比、水面宽与水深比、底宽与形状系数比、水面宽度与形状系数比、形状系数与水深比均为常数:宽深比等于0.405,形状系数与水深比等于0.474,水面宽与水深的比值为2.112,底宽与形状系数的比值为0.855。与现有平底断面(梯形、平底抛物线形、平底半立方抛物线形)进行了比较,结果表明,在过流面积或湿周一定的情况下,平底悬链线形断面的过流能力最大,相反,在流量一定的情况下,平底悬链线形断面的过流面积、湿周、水面宽度是最小的。与传统的悬链线形渠道进行了比较,增加平底后,在同等条件下,平底悬链线形渠道水力最优断面的过流能力不仅没有降低,反而增加了,意味着其经济性也优于传统的悬链线断面。研究为平底悬链线形渠道设计提供理论支撑。
A suitable channel section cannot only increase flow capacity of channel, improve efficiency of water resources, and reduce water leakage loss, but also decrease construction cost. This paper proposed a channel section with a horizontal bottom and catenary sides(HBC). The HBC section, on one hand, provided a larger flow capacity, lesser-sharp angles of stress concentration, less leakage, better slope stability and frost heave resistance than trapezoids and rectangles sections. On the other hand, it had lots of advantages of horizontal bottom sections, such as simpler construction, easier leveling and compaction of the foundation, and lesser construction cost. The most important advantage of this section was that the horizontal bottom and sides could be built with different materials or thickness for decreasing the construction cost or other purposes. The shape function for HBC was defined. The formulas for the flow area, wetted perimeter, and water surface width were presented. A simpler iterative algorithm for calculation of the normal depth was developed. The iterative convergence by this algorithm was evidenced. Comparisons showed that this simpler iterative algorithm was better than classic Newton iterative algorithm. The optimal model of the best hydraulic section of HBC was built. The general differential equations for all the sections having horizontal bottomed and curve sides were derived. The best hydraulic section of HBC channel was obtained according to Lagrange multiplier method and its characteristics were presented including shape factor, ratio of horizontal bottom width to shape factor etc. The results showed that the following optimum parameters were constant for the best hydraulic HBC section: bottom width to water depth, water surface width to depth, bottom width to shape factor, water surface width to shape factor, shape factor to water depth equals. The ratio of water surface width for catenary part to shape factor equaled 3.602, the ratio of bottom width to shape factor equaled 0.855, the ratio of bottom width to water depth equaled 0.405, and the ratio of shape factor to water depth equaled 0.474, and the ratio of total water surface width to water depth equaled 2.112. Various explicit formulae to calculate the normal depth, critical depth, shape factor, flow area, wetted perimeter and water surface width of the HBC section were derived for the best hydraulic section for HBC channel. These formulas should make the design of the HBC section easier and promote its practical applications. The optimum parameters of the best hydraulic section for existing horizontal bottom(HB) sections, such as trapezoidal, rectangle, horizontal bottomed parabolic, and horizontal bottomed semi-cubic parabolic were derived. The comparison results showed that the HBC section had larger discharge than those of existing horizontal bottom(HB) sections under the same conditions. In addition, the flow area, wetted perimeter, and water surface of the HBC section were the smallest, which means that earthwork cost, lining cost and land expropriation cost are all decreased, which means HBC section is more economical. Comparison with classic catenary section showed that the discharge of the HBC was larger than that of the classic catenary section under the same conditions. The flow area, wetted perimeter and water surface of the HBC were smaller than these of the classic catenary section, which means the HBC section has better hydraulic characteristics. Its economy was also superior to the traditional catenary section. The results were verified by examples. The proposed section should enrich existing types of open channel sections. The research provides a new practical and flexible channel section for channel design and theoretical support for horizontal-bottom catenary channel design and applications.
A suitable channel section cannot only increase flow capacity of channel, improve efficiency of water resources, and reduce water leakage loss, but also decrease construction cost. This paper proposed a channel section with a horizontal bottom and catenary sides(HBC). The HBC section, on one hand, provided a larger flow capacity, lesser-sharp angles of stress concentration, less leakage, better slope stability and frost heave resistance than trapezoids and rectangles sections. On the other hand, it had lots of advantages of horizontal bottom sections, such as simpler construction, easier leveling and compaction of the foundation, and lesser construction cost. The most important advantage of this section was that the horizontal bottom and sides could be built with different materials or thickness for decreasing the construction cost or other purposes. The shape function for HBC was defined. The formulas for the flow area, wetted perimeter, and water surface width were presented. A simpler iterative algorithm for calculation of the normal depth was developed. The iterative convergence by this algorithm was evidenced. Comparisons showed that this simpler iterative algorithm was better than classic Newton iterative algorithm. The optimal model of the best hydraulic section of HBC was built. The general differential equations for all the sections having horizontal bottomed and curve sides were derived. The best hydraulic section of HBC channel was obtained according to Lagrange multiplier method and its characteristics were presented including shape factor, ratio of horizontal bottom width to shape factor etc. The results showed that the following optimum parameters were constant for the best hydraulic HBC section: bottom width to water depth, water surface width to depth, bottom width to shape factor, water surface width to shape factor, shape factor to water depth equals. The ratio of water surface width for catenary part to shape factor equaled 3.602, the ratio of bottom width to shape factor equaled 0.855, the ratio of bottom width to water depth equaled 0.405, and the ratio of shape factor to water depth equaled 0.474, and the ratio of total water surface width to water depth equaled 2.112. Various explicit formulae to calculate the normal depth, critical depth, shape factor, flow area, wetted perimeter and water surface width of the HBC section were derived for the best hydraulic section for HBC channel. These formulas should make the design of the HBC section easier and promote its practical applications. The optimum parameters of the best hydraulic section for existing horizontal bottom(HB) sections, such as trapezoidal, rectangle, horizontal bottomed parabolic, and horizontal bottomed semi-cubic parabolic were derived. The comparison results showed that the HBC section had larger discharge than those of existing horizontal bottom(HB) sections under the same conditions. In addition, the flow area, wetted perimeter, and water surface of the HBC section were the smallest, which means that earthwork cost, lining cost and land expropriation cost are all decreased, which means HBC section is more economical. Comparison with classic catenary section showed that the discharge of the HBC was larger than that of the classic catenary section under the same conditions. The flow area, wetted perimeter and water surface of the HBC were smaller than these of the classic catenary section, which means the HBC section has better hydraulic characteristics. Its economy was also superior to the traditional catenary section. The results were verified by examples. The proposed section should enrich existing types of open channel sections. The research provides a new practical and flexible channel section for channel design and theoretical support for horizontal-bottom catenary channel design and applications.
引文
[1]Chow V T.Open Channel Hydraulics[M].New York:McGraw-Hill,1959.
[2]Jain A,Bhattacharjya R K,Sanaga S.Optimal design of composite channels using genetic algorithm[J].Journal of Irrigation&Drainage Engineering,2004,130(4):286-295.
[3]Han Yancheng,Gao Xueping,Xu Zhenghe.The best hydraulic section of horizontal-bottomed parabolic channel section[J].水动力学研究与进展B辑,2017,29(2):305-313.
[4]韩延成,徐征和,高学平,等.二分之五次方抛物线形明渠设计及提高水力特性效果[J].农业工程学报,2017,33(4):131-136.Han Yancheng,Xu Zhenghe,Gao Xueping,et al.Design of two and a half parabola-shaped canal and its effect in improving hydraulic property[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2017,33(4):131-136.(in Chinese with English abstract)
[5]罗金耀,王长德,夏富洲,等.大型输水渠道系统优化设计研究[J].水利学报,1996(6):24-28.Luo Jinyao,Wang Changde,Xia Fuzhou,et al.Optimization design of large-scale water conveyance channel system[J].Journal of Hydraulic Engineering,1996(6):24-28.(in Chinese with English abstract)
[6]Han Yancheng.Horizontal bottomed semi-cubic parabolic channel and best hydraulic section[J].Flow Measurement&Instrumentation,2015,45:56-61.
[7]Das A.Optimal design of channel having horizontal bottom and parabolic sides[J].Journal of Irrigation&Drainage Engineering,2007,133(2):192-197.
[8]Loganathan G V.Optimal design of parabolic canals[J].Journal of Irrigation&Drainage Engineering,1991,117(5):716-735.
[9]Han Yancheng,Easa S M.New and improved three and one-third parabolic channel and most efficient hydraulic section[J].Canadian Journal of Civil Engineering,2017,44(5):387-391.
[10]Han Yancheng,Easa S M.Superior cubic channel section and analytical solution of best hydraulic properties[J].Flow Measurement&Instrumentation,2016,50:169-177.
[11]Mironenko A P,Willardson L S,Jenab S A.Parabolic canal design and analysis[J].Journal of Irrigation&Drainage Engineering,1984,110(2):241-246.
[12]郑文锦,林琳.U形断面抗冻胀能力优越性分析[J].水利与建筑工程学报,2006,4(1):51-53.Zheng Wenjin,Lin Lin.Advantages of U-shaped section for frost heaving resistance[J].Journal of Water Resources and Architectural Engineering,2006,4(1):51-53.(in Chinese with English abstract)
[13]杨汉林.实用弧形坡脚梯形断面的设计[J].中国农村水利水电,2000(12):28-31.Yang Hanlin.Design of practical curved slope trapezoidal section[J].China Rural Water and Hydropower,2000(12):28-31.(in Chinese with English abstract)
[14]王正中,李甲林,陈涛,等.弧底梯形渠道砼衬砌冻胀破坏的力学模型研究[J].农业工程学报,2008,24(1):18-23.Wang Zhengzhong,Li Jialin,Chen Tao,et al.Mechanics models of frost-heaving damage of concrete lining trapezoidal canal with arc-bottom[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2008,24(1):18-23.(in Chinese with English abstract)
[15]宋清林.抛物线形混凝土衬砌渠道冻胀破坏力学模型及数值模拟[D].杨凌:西北农林科技大学,2015.Song Qinglin.Mechanical Model and Numerical Simulation of Frost Heaving Failure of Parabolic Concrete Lining Channel[D].Yangling:Northwest A&F University,2015.(in Chinese with English abstract)
[16]文辉,李风玲.数值积分法计算抛物线形渠道恒定渐变流水面线[J].农业工程学报,2014,30(24):82-86.Wen Hui,Li Fengling.Numerical integration method for calculating water surface profile of gradually varied steady flow in parabola shaped channel[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2014,30(24):82-86.(in Chinese with English abstract)
[17]张新燕,吕宏兴.抛物线形断面渠道正常水深的显式计算[J].农业工程学报,2012,28(21):121-125.Zhang Xinyan,LüHongxing.Explicit solution for normal depth in parabolic-shape channel[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2012,28(21):121-125.(in Chinese with English abstract)
[18]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,2006,39(3):49-51.Wei Wenli,Yang Guoli.Calculation of optimal hydraulic cross-section for cubic parabola-shape channel[J].Journal of Wuhan University,2006,39(3):49-51.(in Chinese with English abstract)
[19]张宽地,吕宏兴,赵延风.明流条件下圆形隧洞正常水深与临界水深的直接计算[J].农业工程学报,2009,25(3):1-5.Zhang Kuandi,LüHongxing,Zhao Yanfeng.Direct calculation for normal depth and critical depth of circular section tunnel under free flow[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2009,25(3):1-5.(in Chinese with English abstract)
[20]梁元博,马吉明,谢省宗.城门洞形及马蹄形输入隧洞内的水跃[J].水利学报,2000,31(7):20-24.Liang Yuanbo,Ma Jiming,Xie Shengzong.Hydraulic jumps in rectangular conduit with circular upper wall and horseshoe tunnel[J].Journal of Hydraulic Engineering,2000,31(7):20-24.(in Chinese with English abstract)
[21]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):36-37.Zhao Yanfeng,Song Songbai,Meng Qinqian.A direct calculation method for water depth in parabolic-shaped channel with contracted section[J].Water Resources and Hydropower Engineering,2008,39(3):36-37.(in Chinese with English abstract)
[22]张志昌,贾斌,李若冰,等.抛物线形渠道的水力特性[J].水利水运工程学报,2015(1):61-67.Zhang Zhichang,Jia Bin,Li Ruobing,et al.Hydraulic characteristics of parabolic channels[J].Journal of Water Resources and Hydropower Engineering,2015(1):61-67.(in Chinese with English abstract)
[23]汪艺义.弧形坡脚梯形渠道中弧形半径的确定[J].水利规划与设计,2017(1):115-117.Wang Yiyi.Determination of radius radius in trapezoidal channel of curved slope[J].Water Resources Planning and Design,2017(1):115-117.(in Chinese with English abstract)
[24]Abdulrahman A.Best hydraulic section of a composite channel[J].Journal of Hydraulic Engineering,2007,133(6):695-697.
[25]Babaeyan-Koopaei K,Valentine E M,Swailes D C J.Optimal design of parabolic-bottomed triangle canals[J].Journal of Irrigation and Drainage Engineering,2000,126(6):408-411.
[26]Vatankhah A R.Semi-regular polygon as the best hydraulic section in practice(Generalized solutions)[J].Flow Measurement&Instrumentation,2014,38(8):67-71.
[27]Easa S M.Improved channel cross section with two-segment parabolic sides and horizontal bottom[J].Journal of Irrigation&Drainage Engineering,2009,135(3):357-365.
[28]Easa S M.Versatile general elliptic open channel cross section[J].Ksce Journal of Civil Engineering,2016,20(4):1572-1581.
[29]Easa S M,Vatankhah A R.New open channel with elliptic sides and horizontal bottom[J].Ksce Journal of Civil Engineering,2014,18(4):1197-1204.
[30]吕宏兴,冯家涛.明渠水力最佳断面的比较[J].人民长江,1994(11):42-45.LüHongxing,Feng Jiatao.Comparison of the best hydraulic sections in open channels[J].People’s Yangtze River,1994(11):42-45.(in Chinese with English abstract)
[31]韩延成,初萍萍,梁梦媛,等.冰盖下梯形及抛物线形输水明渠正常水深显式迭代算法[J].农业工程学报,2018,34(14):101-106Han Yancheng,Chu Pingping,Liang Mengyuan,et al.Explicit iterative algorithm of normal water depth for trapezoid and parabolic open channels under ice cover[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2018,34(14):101-106.(in Chinese with English abstract)
[32]张新燕,吕宏兴,朱德兰.U形渠道正常水深的直接水力计算公式[J].农业工程学报,2013,29(14):115-119.Zhang Xinyan,LüHongxing,Zhu Delan.Direct calculation formula for normal depth of U-shaped channel[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2013,29(14):115-119.(in Chinese with English abstract)
[33]Bijankhan M,Kouchakzadeh S.Egg-shaped cross section:Uniform flow direct solution and stability identification[J].Flow Measurement&Instrumentation,2011,22(6):511-516.
[34]Li Yonghong,Gao Zhaoliang Explicit solution for normal depth of parabolic section of open channels[J].Flow Measurement&Instrumentation,2014,38(2):36-39.
[35]Zhang X Y,Wu L.Direct solutions for normal depths in curved irrigation canals[J].Flow Measurement&Instrumentation,2014,36(4):9-13.
[2]Jain A,Bhattacharjya R K,Sanaga S.Optimal design of composite channels using genetic algorithm[J].Journal of Irrigation&Drainage Engineering,2004,130(4):286-295.
[3]Han Yancheng,Gao Xueping,Xu Zhenghe.The best hydraulic section of horizontal-bottomed parabolic channel section[J].水动力学研究与进展B辑,2017,29(2):305-313.
[4]韩延成,徐征和,高学平,等.二分之五次方抛物线形明渠设计及提高水力特性效果[J].农业工程学报,2017,33(4):131-136.Han Yancheng,Xu Zhenghe,Gao Xueping,et al.Design of two and a half parabola-shaped canal and its effect in improving hydraulic property[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2017,33(4):131-136.(in Chinese with English abstract)
[5]罗金耀,王长德,夏富洲,等.大型输水渠道系统优化设计研究[J].水利学报,1996(6):24-28.Luo Jinyao,Wang Changde,Xia Fuzhou,et al.Optimization design of large-scale water conveyance channel system[J].Journal of Hydraulic Engineering,1996(6):24-28.(in Chinese with English abstract)
[6]Han Yancheng.Horizontal bottomed semi-cubic parabolic channel and best hydraulic section[J].Flow Measurement&Instrumentation,2015,45:56-61.
[7]Das A.Optimal design of channel having horizontal bottom and parabolic sides[J].Journal of Irrigation&Drainage Engineering,2007,133(2):192-197.
[8]Loganathan G V.Optimal design of parabolic canals[J].Journal of Irrigation&Drainage Engineering,1991,117(5):716-735.
[9]Han Yancheng,Easa S M.New and improved three and one-third parabolic channel and most efficient hydraulic section[J].Canadian Journal of Civil Engineering,2017,44(5):387-391.
[10]Han Yancheng,Easa S M.Superior cubic channel section and analytical solution of best hydraulic properties[J].Flow Measurement&Instrumentation,2016,50:169-177.
[11]Mironenko A P,Willardson L S,Jenab S A.Parabolic canal design and analysis[J].Journal of Irrigation&Drainage Engineering,1984,110(2):241-246.
[12]郑文锦,林琳.U形断面抗冻胀能力优越性分析[J].水利与建筑工程学报,2006,4(1):51-53.Zheng Wenjin,Lin Lin.Advantages of U-shaped section for frost heaving resistance[J].Journal of Water Resources and Architectural Engineering,2006,4(1):51-53.(in Chinese with English abstract)
[13]杨汉林.实用弧形坡脚梯形断面的设计[J].中国农村水利水电,2000(12):28-31.Yang Hanlin.Design of practical curved slope trapezoidal section[J].China Rural Water and Hydropower,2000(12):28-31.(in Chinese with English abstract)
[14]王正中,李甲林,陈涛,等.弧底梯形渠道砼衬砌冻胀破坏的力学模型研究[J].农业工程学报,2008,24(1):18-23.Wang Zhengzhong,Li Jialin,Chen Tao,et al.Mechanics models of frost-heaving damage of concrete lining trapezoidal canal with arc-bottom[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2008,24(1):18-23.(in Chinese with English abstract)
[15]宋清林.抛物线形混凝土衬砌渠道冻胀破坏力学模型及数值模拟[D].杨凌:西北农林科技大学,2015.Song Qinglin.Mechanical Model and Numerical Simulation of Frost Heaving Failure of Parabolic Concrete Lining Channel[D].Yangling:Northwest A&F University,2015.(in Chinese with English abstract)
[16]文辉,李风玲.数值积分法计算抛物线形渠道恒定渐变流水面线[J].农业工程学报,2014,30(24):82-86.Wen Hui,Li Fengling.Numerical integration method for calculating water surface profile of gradually varied steady flow in parabola shaped channel[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2014,30(24):82-86.(in Chinese with English abstract)
[17]张新燕,吕宏兴.抛物线形断面渠道正常水深的显式计算[J].农业工程学报,2012,28(21):121-125.Zhang Xinyan,LüHongxing.Explicit solution for normal depth in parabolic-shape channel[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2012,28(21):121-125.(in Chinese with English abstract)
[18]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,2006,39(3):49-51.Wei Wenli,Yang Guoli.Calculation of optimal hydraulic cross-section for cubic parabola-shape channel[J].Journal of Wuhan University,2006,39(3):49-51.(in Chinese with English abstract)
[19]张宽地,吕宏兴,赵延风.明流条件下圆形隧洞正常水深与临界水深的直接计算[J].农业工程学报,2009,25(3):1-5.Zhang Kuandi,LüHongxing,Zhao Yanfeng.Direct calculation for normal depth and critical depth of circular section tunnel under free flow[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2009,25(3):1-5.(in Chinese with English abstract)
[20]梁元博,马吉明,谢省宗.城门洞形及马蹄形输入隧洞内的水跃[J].水利学报,2000,31(7):20-24.Liang Yuanbo,Ma Jiming,Xie Shengzong.Hydraulic jumps in rectangular conduit with circular upper wall and horseshoe tunnel[J].Journal of Hydraulic Engineering,2000,31(7):20-24.(in Chinese with English abstract)
[21]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):36-37.Zhao Yanfeng,Song Songbai,Meng Qinqian.A direct calculation method for water depth in parabolic-shaped channel with contracted section[J].Water Resources and Hydropower Engineering,2008,39(3):36-37.(in Chinese with English abstract)
[22]张志昌,贾斌,李若冰,等.抛物线形渠道的水力特性[J].水利水运工程学报,2015(1):61-67.Zhang Zhichang,Jia Bin,Li Ruobing,et al.Hydraulic characteristics of parabolic channels[J].Journal of Water Resources and Hydropower Engineering,2015(1):61-67.(in Chinese with English abstract)
[23]汪艺义.弧形坡脚梯形渠道中弧形半径的确定[J].水利规划与设计,2017(1):115-117.Wang Yiyi.Determination of radius radius in trapezoidal channel of curved slope[J].Water Resources Planning and Design,2017(1):115-117.(in Chinese with English abstract)
[24]Abdulrahman A.Best hydraulic section of a composite channel[J].Journal of Hydraulic Engineering,2007,133(6):695-697.
[25]Babaeyan-Koopaei K,Valentine E M,Swailes D C J.Optimal design of parabolic-bottomed triangle canals[J].Journal of Irrigation and Drainage Engineering,2000,126(6):408-411.
[26]Vatankhah A R.Semi-regular polygon as the best hydraulic section in practice(Generalized solutions)[J].Flow Measurement&Instrumentation,2014,38(8):67-71.
[27]Easa S M.Improved channel cross section with two-segment parabolic sides and horizontal bottom[J].Journal of Irrigation&Drainage Engineering,2009,135(3):357-365.
[28]Easa S M.Versatile general elliptic open channel cross section[J].Ksce Journal of Civil Engineering,2016,20(4):1572-1581.
[29]Easa S M,Vatankhah A R.New open channel with elliptic sides and horizontal bottom[J].Ksce Journal of Civil Engineering,2014,18(4):1197-1204.
[30]吕宏兴,冯家涛.明渠水力最佳断面的比较[J].人民长江,1994(11):42-45.LüHongxing,Feng Jiatao.Comparison of the best hydraulic sections in open channels[J].People’s Yangtze River,1994(11):42-45.(in Chinese with English abstract)
[31]韩延成,初萍萍,梁梦媛,等.冰盖下梯形及抛物线形输水明渠正常水深显式迭代算法[J].农业工程学报,2018,34(14):101-106Han Yancheng,Chu Pingping,Liang Mengyuan,et al.Explicit iterative algorithm of normal water depth for trapezoid and parabolic open channels under ice cover[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2018,34(14):101-106.(in Chinese with English abstract)
[32]张新燕,吕宏兴,朱德兰.U形渠道正常水深的直接水力计算公式[J].农业工程学报,2013,29(14):115-119.Zhang Xinyan,LüHongxing,Zhu Delan.Direct calculation formula for normal depth of U-shaped channel[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2013,29(14):115-119.(in Chinese with English abstract)
[33]Bijankhan M,Kouchakzadeh S.Egg-shaped cross section:Uniform flow direct solution and stability identification[J].Flow Measurement&Instrumentation,2011,22(6):511-516.
[34]Li Yonghong,Gao Zhaoliang Explicit solution for normal depth of parabolic section of open channels[J].Flow Measurement&Instrumentation,2014,38(2):36-39.
[35]Zhang X Y,Wu L.Direct solutions for normal depths in curved irrigation canals[J].Flow Measurement&Instrumentation,2014,36(4):9-13.
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