浅水流动及其挟沙状态下的数值模拟
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摘要
河道和近海周围一直是人类的聚集之地,因此河道的演变对人类活动有很大影响,从而需要研究水流的运动规律来为人类服务,而数值模拟则是有效的研究方法之一。
’在浅水动力学的研究中,出现很多较好的实用算法,例如一维的LAX-Wendroff显格式,Preissomann隐格式及特征差分格式,并通过算子分裂法或者ADI法将其应用到二维问题,这些方法可以应用到一定的场合,但对于实际的工程问题却不能完全尽人意。
本文基于数值模拟方法中最有前景的有限体积法对二维浅水方程进行离散求解,建立了二维浅水模型,在模型的求解过程中,对数值通量的求解用Roe格式的近似黎曼解,使其能够处理强间断问题,求解中引入矩阵理论来处理系数矩阵；对源项的处理用迎风格式的特征方向法,使其在处理非平底问题时保证了界面通量上的平衡,并使格式具有高精度和稳定性。在挟沙水流问题的分析中,基于扩散理论用张洪武的考虑泥沙影响的流速沿垂线分布公式建立含沙量沿垂线分布的规律,克服应用比较广泛的罗斯公式中含沙量在水面处为0的弊端,分布状态更符合实际。另外在二维浅水方程的基础上增加了泥沙扩散方程和河床变形方程,对其耦合求解,从而模拟挟沙水流的运动规律及河床的冲淤变形情况。
对比之前浅水动力学的方法,本文所建立的模型能够应用到工程实际中,在本文的最后一章用第二章所建立的二维浅水模型验证了实验室中两个有代表性的溃坝实验：方形非对称溃坝问题及带有三角形障碍物的溃坝问题,通过与实验室内的实测资料对比,其结果吻合良好,从而验证了二维浅水模型的准确性。用二维挟沙水流模型在不考虑冲淤的条件下模拟了温州海域内潮位场,流速场及含沙量变化过程,并与各个测站上实测资料的进行对比验证；对长江上沙市至监利段的水流运动规律进行数值模拟,在考虑冲淤及河床变形时,对其在一定时间段内的河床冲淤变形进行模拟验证。结果证明了模型在实际中的可应用性。
The river's surrounding and offshore have been the gathering place for human beings,so the evolution of the river will take a lot of affects for human being. It is necessary to research the law of water movement, and the numerical simulation is one of the effective ways within these methods.
In the study of the shallow water, there are a lot of good practical algorithms, such as one-dimensional LAX-Wendroff、Preissomann format and characteristics of implicit difference scheme, and be applied to two-dimensional problem by operator splitting method or the ADI method. These methods can be applied to certain situations, but they are not fully intended the need of people for practical engineering problems.
This article use the finite volume method to scatter the shallow water equation, and create the two-dimensional shallow water model, at the solving process of the model, pull in the approximate Riemann solution of Roe Format for obtaining numerical flux, to enable them to deal with discontinuous problems; at the question of deal with the source term, proposed the characteristic direction method of upwind format, to ensure the balance on the interface flux when deal with non-flat bottom, in the problem of the sediment analysis, based on diffusion theory, suspended sediment concentration variation along the vertical line which based on HongWu Zhang velocity distribution, this formula overcoming the shortcomings of 0 at the water surface, and the regular pattern is line with the practice. Then at the foundation of shallow water equation, add the diffusion equation and the bed deformation equation, simulating the movement of sediment and erosion deformation of the riverbed.
Testing and verifying two experiment of dam break which did by some researchers in laboratory with the shallow water model:the dam break model of asymmetric square and the dam break model with triangular obstacle, by the contrast with the data of laboratory, proved the accurateness of the model.
Simulated the sea of WenZhou with the model of sediment-laden flow, and contrast the tidemark、viocity and the sediment concentration with the data through measurement, then simulated the law of water movement and the erosion deformation of the riverbed between ShaShi and JianLi at Yangtze River. The result certificated the accurateness of the model.
’在浅水动力学的研究中,出现很多较好的实用算法,例如一维的LAX-Wendroff显格式,Preissomann隐格式及特征差分格式,并通过算子分裂法或者ADI法将其应用到二维问题,这些方法可以应用到一定的场合,但对于实际的工程问题却不能完全尽人意。
本文基于数值模拟方法中最有前景的有限体积法对二维浅水方程进行离散求解,建立了二维浅水模型,在模型的求解过程中,对数值通量的求解用Roe格式的近似黎曼解,使其能够处理强间断问题,求解中引入矩阵理论来处理系数矩阵；对源项的处理用迎风格式的特征方向法,使其在处理非平底问题时保证了界面通量上的平衡,并使格式具有高精度和稳定性。在挟沙水流问题的分析中,基于扩散理论用张洪武的考虑泥沙影响的流速沿垂线分布公式建立含沙量沿垂线分布的规律,克服应用比较广泛的罗斯公式中含沙量在水面处为0的弊端,分布状态更符合实际。另外在二维浅水方程的基础上增加了泥沙扩散方程和河床变形方程,对其耦合求解,从而模拟挟沙水流的运动规律及河床的冲淤变形情况。
对比之前浅水动力学的方法,本文所建立的模型能够应用到工程实际中,在本文的最后一章用第二章所建立的二维浅水模型验证了实验室中两个有代表性的溃坝实验：方形非对称溃坝问题及带有三角形障碍物的溃坝问题,通过与实验室内的实测资料对比,其结果吻合良好,从而验证了二维浅水模型的准确性。用二维挟沙水流模型在不考虑冲淤的条件下模拟了温州海域内潮位场,流速场及含沙量变化过程,并与各个测站上实测资料的进行对比验证；对长江上沙市至监利段的水流运动规律进行数值模拟,在考虑冲淤及河床变形时,对其在一定时间段内的河床冲淤变形进行模拟验证。结果证明了模型在实际中的可应用性。
The river's surrounding and offshore have been the gathering place for human beings,so the evolution of the river will take a lot of affects for human being. It is necessary to research the law of water movement, and the numerical simulation is one of the effective ways within these methods.
In the study of the shallow water, there are a lot of good practical algorithms, such as one-dimensional LAX-Wendroff、Preissomann format and characteristics of implicit difference scheme, and be applied to two-dimensional problem by operator splitting method or the ADI method. These methods can be applied to certain situations, but they are not fully intended the need of people for practical engineering problems.
This article use the finite volume method to scatter the shallow water equation, and create the two-dimensional shallow water model, at the solving process of the model, pull in the approximate Riemann solution of Roe Format for obtaining numerical flux, to enable them to deal with discontinuous problems; at the question of deal with the source term, proposed the characteristic direction method of upwind format, to ensure the balance on the interface flux when deal with non-flat bottom, in the problem of the sediment analysis, based on diffusion theory, suspended sediment concentration variation along the vertical line which based on HongWu Zhang velocity distribution, this formula overcoming the shortcomings of 0 at the water surface, and the regular pattern is line with the practice. Then at the foundation of shallow water equation, add the diffusion equation and the bed deformation equation, simulating the movement of sediment and erosion deformation of the riverbed.
Testing and verifying two experiment of dam break which did by some researchers in laboratory with the shallow water model:the dam break model of asymmetric square and the dam break model with triangular obstacle, by the contrast with the data of laboratory, proved the accurateness of the model.
Simulated the sea of WenZhou with the model of sediment-laden flow, and contrast the tidemark、viocity and the sediment concentration with the data through measurement, then simulated the law of water movement and the erosion deformation of the riverbed between ShaShi and JianLi at Yangtze River. The result certificated the accurateness of the model.
引文
[1]谭维炎.浅水动力学的回顾和当代前沿问题[J],水科学进展.1999[10].296-302
[2]王福军.计算流体动力学分析-CFD软件原理及应用[M].北京：清华大学出版社,2004.9.
[3]白玉川,顾元棪,邢焕政.水流泥沙水质数学模型理论及应用[M].天津：天津大学出版社,2005.10.
[4]Duboys,P.,"Etudes du Regime du Rhone et 1'Aotion Exercee par les Laux sur un Lit a Fond de Graviers Undefiniment Affouillable",Annales des Ponts et Chausses,Ser,5, Vol.18,1879,pp.141-195
[5]李人宪.有限体积法基础[M].北京：国防工业出版社,2008.3.
[6]丛文相.数值网格划分方法[J].黑龙江大学自然科学学报.1998[15].23-25.
[7]苏铭德,朱芳林.二维非结构网格生成及自动加密技术[J].计算物理.1998[15].6-10.[8]周正贵,计算流体力学-基础理论与实际应用.[M].南京：东南大学出版社,2008.3.
[9]谭维炎.计算浅水动力学-有限体积法的应用[M].北京：清华大学出版社,1998.9.
[10]H.欧特尔,朱自强(译),钱翼稷(译),李宗瑞(译),普朗特流体力学基础[M].北京：科学出版社,2008.6.
[11]槐文信,赵明登,童汉毅.河道及近海水流的数值模拟[M].北京：科学出版社,2005.5.
[12]吕彪,金生,艾丛芳.非结构网格下求解二维浅水方程的和谐Roe格式[J].水里水运工程学报.2010[2],39-43.
[13]王志力.基于Godunov和Semi-Lagranian法的二三维浅水方程的非结构化网格离散研究[D].大连：大连理工土木水利学院,2005.
[14]杨明远,严以新.孔俊,宋志尧,珠江口水流泥沙运动模拟研究[M].北京：海洋出版社,2008.11.
[15]钱宁,万兆惠.泥沙运动力学[M].北京：科学出版社,1983.12.
[]6] 中国水力学会泥沙专业委员会,泥沙手册[M].科学出版社.1989.
[17]王昌杰,河流动力学[M].北京：人民交通出版社,2004.6
[]8]悬移质含沙量沿垂线分布的探讨[J].武汉人学学报,2002.4,2,22-24.
[19]熊治平.悬移质泥沙级配沿垂线分布公式探求[J].武汉水利电力大学学报,1999.
[20]张小峰,谈广鸣.非均匀沙含沙量沿垂线分布特征[J].水利学报,1992,10,48-52.
[2]] 张洪武.挟沙水流流速的垂线分布公式[J].泥沙研究,1995.6.,2,1-10.
[22]Einstein, H. A. and Ning Chien(钱宁), " Effects of Sendiment Concentration Near t he Bed on the Velocity and Sediment Distribution", M. R. D. Sediment Series No.8, Missouri River Div., Corps Engrs.,1955, p.76
[23]陈丕翔.基于有限体积法的二维水流水质模拟[D].南京：河海大学水文水资源学院.2007.
[24]陈界仁,沙捞·巴里,陈国祥.二维水库水流泥沙数值模拟[J].河海大学学报.2000[2].11-14.
[25]张细兵,殷瑞兰.平面二维水流泥沙数值模拟[J].水科学进展.2002[13],665-668.
[2]王福军.计算流体动力学分析-CFD软件原理及应用[M].北京：清华大学出版社,2004.9.
[3]白玉川,顾元棪,邢焕政.水流泥沙水质数学模型理论及应用[M].天津：天津大学出版社,2005.10.
[4]Duboys,P.,"Etudes du Regime du Rhone et 1'Aotion Exercee par les Laux sur un Lit a Fond de Graviers Undefiniment Affouillable",Annales des Ponts et Chausses,Ser,5, Vol.18,1879,pp.141-195
[5]李人宪.有限体积法基础[M].北京：国防工业出版社,2008.3.
[6]丛文相.数值网格划分方法[J].黑龙江大学自然科学学报.1998[15].23-25.
[7]苏铭德,朱芳林.二维非结构网格生成及自动加密技术[J].计算物理.1998[15].6-10.[8]周正贵,计算流体力学-基础理论与实际应用.[M].南京：东南大学出版社,2008.3.
[9]谭维炎.计算浅水动力学-有限体积法的应用[M].北京：清华大学出版社,1998.9.
[10]H.欧特尔,朱自强(译),钱翼稷(译),李宗瑞(译),普朗特流体力学基础[M].北京：科学出版社,2008.6.
[11]槐文信,赵明登,童汉毅.河道及近海水流的数值模拟[M].北京：科学出版社,2005.5.
[12]吕彪,金生,艾丛芳.非结构网格下求解二维浅水方程的和谐Roe格式[J].水里水运工程学报.2010[2],39-43.
[13]王志力.基于Godunov和Semi-Lagranian法的二三维浅水方程的非结构化网格离散研究[D].大连：大连理工土木水利学院,2005.
[14]杨明远,严以新.孔俊,宋志尧,珠江口水流泥沙运动模拟研究[M].北京：海洋出版社,2008.11.
[15]钱宁,万兆惠.泥沙运动力学[M].北京：科学出版社,1983.12.
[]6] 中国水力学会泥沙专业委员会,泥沙手册[M].科学出版社.1989.
[17]王昌杰,河流动力学[M].北京：人民交通出版社,2004.6
[]8]悬移质含沙量沿垂线分布的探讨[J].武汉人学学报,2002.4,2,22-24.
[19]熊治平.悬移质泥沙级配沿垂线分布公式探求[J].武汉水利电力大学学报,1999.
[20]张小峰,谈广鸣.非均匀沙含沙量沿垂线分布特征[J].水利学报,1992,10,48-52.
[2]] 张洪武.挟沙水流流速的垂线分布公式[J].泥沙研究,1995.6.,2,1-10.
[22]Einstein, H. A. and Ning Chien(钱宁), " Effects of Sendiment Concentration Near t he Bed on the Velocity and Sediment Distribution", M. R. D. Sediment Series No.8, Missouri River Div., Corps Engrs.,1955, p.76
[23]陈丕翔.基于有限体积法的二维水流水质模拟[D].南京：河海大学水文水资源学院.2007.
[24]陈界仁,沙捞·巴里,陈国祥.二维水库水流泥沙数值模拟[J].河海大学学报.2000[2].11-14.
[25]张细兵,殷瑞兰.平面二维水流泥沙数值模拟[J].水科学进展.2002[13],665-668.