平底马蹄形断面的水力计算
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摘要
平底Ⅱ型马蹄形断面是将标准Ⅱ型马蹄形断面的底拱改成平底而来,是水利水电工程较常用的断面形式之一,但其正常水深和临界水深的水力计算需求解超越方程,无解析解。传统的试算法或查表法计算过程繁琐且计算精度不高。本文通过对平底Ⅱ型马蹄形断面基本方程进行数学变换,并对引入的无量纲参数与无量纲水深的关系进行分析及计算,应用拟合优化原理得到了其水力计算的直接计算公式。该公式不需分段和进行判别选取,直接可得到答案。在工程的适用范围内(即水深与顶拱半径之比:0.05≤h/r≤1.41),计算正常水深的最大误差小于0.41%,计算临界水深的最大误差小于0.20%。其成果对该断面形式的隧洞设计有一定参考价值。
As one of the simplified type of the standard Ⅱ type horseshoe, the Ⅱ type horseshoe cross-section with flat bottom is one of the general used shape types for spillway tunnel and irrigation tunnel. However, the hydraulic calculation for this type tunnels has not been reported. Aiming at this problem, a new hydraulic calculation formula was proposed. The Ⅱ type horseshoe cross-section with flat bottom is consisted of four parts: one flat bottom, two side circular arc arch and one circular arc roof umbrella arch. This simplified horseshoe type is special suitable for the geological conditions with low foundation pressure. The characteristics of hydraulics and mechanics are between horseshoe-shaped and U-shaped arch. Due to the flat bottom, this type can keep a normal water flow regime, and a smoothly connection of the water flow before and behind the tunnel. For obtaining the calculation formula of the normal and critical water depth for Ⅱ type horseshoe tunnel with flat bottom, its geometrical design of hydraulic tunnel were analyzed. In order to guarantee the status without pressure in a free flow tunnel with changing water level, the free space above the water level was not less than 15 % of the whole cross-sectional areas of the tunnel. The upper limit of dimensionless water depth was 1.41. For most of the application engineering, the lower limit of dimensionless water depth was 0.05. After ascertained the utility range of the formula and based on fundamental equations of uniform flow and critical flow, the interrelation between dimensionless water depth and the dimensionless parameter were analyzed. We used the power function as the formula form, and coefficients of the formula were calculated based on the theory of optimization and regression. The calculation formula for normal water depth and critical water depth were obtained. Results showed that the maximum relative error of normal water depth and critical water depth were 0.41% and 0.20%, respectively. This research indicated that the hydraulic calculation formula could be widely used in engineering design and project management with high accurate and simple form.
As one of the simplified type of the standard Ⅱ type horseshoe, the Ⅱ type horseshoe cross-section with flat bottom is one of the general used shape types for spillway tunnel and irrigation tunnel. However, the hydraulic calculation for this type tunnels has not been reported. Aiming at this problem, a new hydraulic calculation formula was proposed. The Ⅱ type horseshoe cross-section with flat bottom is consisted of four parts: one flat bottom, two side circular arc arch and one circular arc roof umbrella arch. This simplified horseshoe type is special suitable for the geological conditions with low foundation pressure. The characteristics of hydraulics and mechanics are between horseshoe-shaped and U-shaped arch. Due to the flat bottom, this type can keep a normal water flow regime, and a smoothly connection of the water flow before and behind the tunnel. For obtaining the calculation formula of the normal and critical water depth for Ⅱ type horseshoe tunnel with flat bottom, its geometrical design of hydraulic tunnel were analyzed. In order to guarantee the status without pressure in a free flow tunnel with changing water level, the free space above the water level was not less than 15 % of the whole cross-sectional areas of the tunnel. The upper limit of dimensionless water depth was 1.41. For most of the application engineering, the lower limit of dimensionless water depth was 0.05. After ascertained the utility range of the formula and based on fundamental equations of uniform flow and critical flow, the interrelation between dimensionless water depth and the dimensionless parameter were analyzed. We used the power function as the formula form, and coefficients of the formula were calculated based on the theory of optimization and regression. The calculation formula for normal water depth and critical water depth were obtained. Results showed that the maximum relative error of normal water depth and critical water depth were 0.41% and 0.20%, respectively. This research indicated that the hydraulic calculation formula could be widely used in engineering design and project management with high accurate and simple form.
引文
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[13]张耀哲,王亚林,王文娥.含沙量对U型渠道水流流速横向分布律的影响[J].农业工程学报,2012,28(12):134-139.Zhang Yaozhe,Wang Yalin,Wang Wen'e.Effects of sediment concentration on lateral distribution of water velocity in U-channel[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2012,28(12):134-139.(in Chinese with English abstract)
[14]李学军,费良军,李改琴.大型U形混凝土衬砌渠道季节性冻融水热耦合模型研究[J].农业工程学报,2008,24(1):13-17.Li Xuejun,Fei Liangjun,Li Gaiqin.Model of coupled heat-fluid transport of U-shape canal lining with concrete during seasonal freezing and thawing stage[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2008,24(1):13-17.(in Chinese with English abstract)
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[17]马吉明,梁海波,梁元博.城门洞形及马蹄形过水隧洞的临界水流[J].清华大学学报,1999,39(11):32-34.Ma Jiming,Liang Haibo,Liang Yuanbo,et al.Critical flow in city-gate and horseshoe conduit[J].Tsinghua Univ(Sci and Tech),1999,39(11):32-34.(in Chinese with English abstract)
[18]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,2000,31(7):20-24.Ma Jiming,Xie Shengzong,Liang Yuanbo.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)
[19]王正中,陈涛,张新民,等.城门洞形断面隧洞临界水深度的近似算法[J].清华大学学报:自然科学版,2004,44(6):812-814.Wang Zhengzhong,Chen Tao,Zhang Xinming,et al.Approximate solution for the critical depth of a arched turnel[J].Tsinghua Univ:Sci and Tech,2004,44(6):812-814.(in Chinese with English abstract)
[20]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):18-21.Wen Hui,Li Fengling,Ou Junli,et al.Approximate solution of the normal depth inside tunnel with arch cross-section[J].Water and Wastewater Engineering,2007,33(7):18-21.(in Chinese with English abstract)
[21]文辉,李风玲.再论城门洞形断面隧洞正常水深的近似计算[J].给水排水,2008,34(11):42-43.Wen Hui,Li Fengling.Discussion again of the approximate calculation for the normal water depth of the city-opening Section Tunnel[J].Water and Wastewater Engineering,2008,34(11):42-43.(in Chinese with English abstract)
[22]王正中,陈涛,芦琴,等.马蹄形过水断面临界水深的直接计算[J].水力发电学报,2005,24(5):95-98.Wang Zhengzhong,Chen Tao,Lu Qin,et al.The direct solution on critical depth of horseshoe section tunnel[J].Journal of Hydroelectric Engineering,2005,24(5):95-98.(in Chinese with English abstract)
[23]张宽地,吕宏兴,陈俊英.马蹄形过水断面临界水深的直接计算法[J].农业工程学报,2009,25(4):15-18.Zhang Kuandi,LüHongxing,Chen Junying.Direct calculation of critical depth of horseshoe section tunnel[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2009,25(4):15-18.(in Chinese with English abstract)
[24]刘计良,王正中,赵延风.马蹄形隧洞断面收缩水深的迭代法计算[J].农业工程学报,2010,26(S2):167-171.Liu Jiliang,Wang Zhengzhong,Zhao Yanfeng.Iterative calculation for contracted depth of horseshoe tunnel cross-section[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2010,26(Supp.2):167-171.(in Chinese with English abstract)
[25]文辉,李风玲,李霞.标准I型马蹄形断面正常水深的近似算法[J].人民黄河,2008,30(7):89-90.Wen Hui,Li Fengling,Li Xia.Approximate solution on normal depth of standard I type horseshoe section tunnel[J].Yellow River,2008,30(7):89-90.(in Chinese with English abstract)
[26]赵延风,王正中,芦琴.马蹄形断面正常水深的直接计算公式[J].水力发电学报,2012,31(1):173-178.Zhao Yanfeng,Wang Zhengzhong,Lu Qin.Direct calculation formulae for normal depth of horseshoe section[J].Journal of Hydroelectric Engineering,2012,31(1):173-178.(in Chinese with English abstract)
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[2]郝树棠.梯形渠道临界水深的计算及讨论[J].水利学报,1994,25(8):48-52.Hao Shutang.Calculation and discussion of critical depthfor ladder-canal[J].Journal of Hydraulic Engineering,1994,25(8):48-52.(in Chinese with English abstract)
[3]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)
[4]文辉,李风玲.再论圆管明渠均匀流正常水深的直接计算公式[J].水利水电科技进展,2012,32(6):15-17.Wen Hui,Li Fengling:Further study on explicit formula for normal water depth of uniform flows in circular pipes[J].Advances in Science and Technology of Water Resources,2012,32(6):15-17.(in Chinese with English abstract)
[5]文辉,李风玲,彭波,等.圆管明渠临界水深的直接近似计算公式[J].人民黄河,2007,29(4):67-68.Wen Hui,Li Fengling,Peng Bo,et al.Direct approximate formula for cirtical depth of pipe-channel[J].Yellow River,2007,29(4):67-68.(in Chinese with English abstract)
[6]李风玲,文辉,彭波.圆形过水断面临界流的近似水力计算[J].人民长江,2008,39(11):77-78.Li Fengling,Wen Hui,Peng Bo,et al.Approximate hydraulic calculation for critical flow in circular cross-section[J].Yangtze River,2008,39(11):77-78.(in Chinese with English abstract)
[7]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)
[8]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):32-33.Wen Hui,Li Fengling.Analytical solution of water depthin parabolic-shaped channel with contracted section[J].Journal of Yangtze River Scientific Research Institute,2009,26(9):32-33.(in Chinese with English abstract)
[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):58-59.Wen Hui,Li Fengling.A direct calculation method of contracted depth in semi-cubic parabola-shaped channel[J].Yangtze River,2009,40(13):58-59.(in Chinese with English abstract)
[10]文辉,李风玲.立方抛物线形渠道水力计算的显式计算式[J].人民黄河,2010,32(1):75-76.Wen Hui,Li Fengling.Explicit formula of hydraulic calculation of semi-cubic parabola-shaped channel[J].Yellow River,2010,32(1):75-76.(in Chinese with English abstract)
[11]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 Wenli,Yang Guoli.Calculation of optimal hydraulic cross-section for semi-cubic parabola-shape channel[J].Engineering Journal of Wuhan University,2006,39(3):49-51.(in Chinese with English abstract)
[12]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):36-37,41.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):36-37,41.(in Chinese with English abstract)
[13]张耀哲,王亚林,王文娥.含沙量对U型渠道水流流速横向分布律的影响[J].农业工程学报,2012,28(12):134-139.Zhang Yaozhe,Wang Yalin,Wang Wen'e.Effects of sediment concentration on lateral distribution of water velocity in U-channel[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2012,28(12):134-139.(in Chinese with English abstract)
[14]李学军,费良军,李改琴.大型U形混凝土衬砌渠道季节性冻融水热耦合模型研究[J].农业工程学报,2008,24(1):13-17.Li Xuejun,Fei Liangjun,Li Gaiqin.Model of coupled heat-fluid transport of U-shape canal lining with concrete during seasonal freezing and thawing stage[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2008,24(1):13-17.(in Chinese with English abstract)
[15]李风玲,文辉,陈雄.U形渠道水力计算的显式计算式[J].水利水电科技进展,2010,30(1):65-67.Li Fengling,Wen Hui,Chen Xiong.Explicit formulae for hydraulic calculation of U-shaped channels[J].Advances in Science and Technology of Water Resources,2010,30(1):65-67.(in Chinese with English abstract)
[16]张宽地,吕宏兴,王光谦,等.普通城门洞形隧洞正常水深直接计算方法[J].农业工程学报,2009,25(11):8-12.Zhang Kuandi,LüHongxing,Wang Guangqian,et al.Direct calculation method for normal depth of arched section tunnel[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2009,25(11):8-12.(in Chinese with English abstract)
[17]马吉明,梁海波,梁元博.城门洞形及马蹄形过水隧洞的临界水流[J].清华大学学报,1999,39(11):32-34.Ma Jiming,Liang Haibo,Liang Yuanbo,et al.Critical flow in city-gate and horseshoe conduit[J].Tsinghua Univ(Sci and Tech),1999,39(11):32-34.(in Chinese with English abstract)
[18]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,2000,31(7):20-24.Ma Jiming,Xie Shengzong,Liang Yuanbo.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)
[19]王正中,陈涛,张新民,等.城门洞形断面隧洞临界水深度的近似算法[J].清华大学学报:自然科学版,2004,44(6):812-814.Wang Zhengzhong,Chen Tao,Zhang Xinming,et al.Approximate solution for the critical depth of a arched turnel[J].Tsinghua Univ:Sci and Tech,2004,44(6):812-814.(in Chinese with English abstract)
[20]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):18-21.Wen Hui,Li Fengling,Ou Junli,et al.Approximate solution of the normal depth inside tunnel with arch cross-section[J].Water and Wastewater Engineering,2007,33(7):18-21.(in Chinese with English abstract)
[21]文辉,李风玲.再论城门洞形断面隧洞正常水深的近似计算[J].给水排水,2008,34(11):42-43.Wen Hui,Li Fengling.Discussion again of the approximate calculation for the normal water depth of the city-opening Section Tunnel[J].Water and Wastewater Engineering,2008,34(11):42-43.(in Chinese with English abstract)
[22]王正中,陈涛,芦琴,等.马蹄形过水断面临界水深的直接计算[J].水力发电学报,2005,24(5):95-98.Wang Zhengzhong,Chen Tao,Lu Qin,et al.The direct solution on critical depth of horseshoe section tunnel[J].Journal of Hydroelectric Engineering,2005,24(5):95-98.(in Chinese with English abstract)
[23]张宽地,吕宏兴,陈俊英.马蹄形过水断面临界水深的直接计算法[J].农业工程学报,2009,25(4):15-18.Zhang Kuandi,LüHongxing,Chen Junying.Direct calculation of critical depth of horseshoe section tunnel[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2009,25(4):15-18.(in Chinese with English abstract)
[24]刘计良,王正中,赵延风.马蹄形隧洞断面收缩水深的迭代法计算[J].农业工程学报,2010,26(S2):167-171.Liu Jiliang,Wang Zhengzhong,Zhao Yanfeng.Iterative calculation for contracted depth of horseshoe tunnel cross-section[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2010,26(Supp.2):167-171.(in Chinese with English abstract)
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