贴体曲线坐标系下浅水数学模型的研究与应用
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摘要
近年来,我国水利水电建设事业得到了蓬勃发展。在水动力学的研究中,越来越多的学者将三维的天然河道水流模型转化为二维的浅水数学模型进行求解。在对天然河道和沿海水流的计算中,这一处理方法表现出了很强的优越性,在理论研究和生产实践中发挥着日益重要的作用。本文在广泛研究现有各种浅水数学模型的基础上,针对天然河道弯曲复杂的边界条件给计算网格生成带来的困难,将笛卡尔坐标系下的浅水模型引入到贴体曲线坐标系下,构建了基于同位网格系统的贴体曲线坐标系下的浅水数学模型,具体研究内容如下：
(1)针对天然河道边界弯曲复杂的特点,提出了一种可控制网格内部正交的贴体网格生成方法。在调节因子P,Q的选取上,根据流函数与势函数绝对正交的原则,推导了计算平面上一点(ξ,η)与物理平面上一点(x,y)对应关系的方程,运用TDMA算法对网格生成方程进行求解,最终生成贴体坐标网格,整个流程运用FORTRAN 95编程实现。通过程序计算得到的网格光滑性和正交性较好,不失为一种理想的计算网格。
(2)在曲线坐标下的浅水控制方程的基础上,针对天然河流大部分河段的弯曲性较大,岸线曲折,地形复杂,引起弯道环流,水流具有明显的三维特性,而且弯道环流的作用对水流泥沙运动影响较大,使水流凹岸流速增大等特点,在控制方程里增加了弯道环流项和科氏力项,以求能更好的描述浅水模型。在方程的离散上,运用了同位网格,对对流项采用乘方格式进行离散。在运用SIMPLE算法进行求解时,采用水位修正法对流速进行修正。在对连续弯道和天然河段进行求解中,通过计算值与实测值的对比,表明了计算模型与计算方法的可行性,也一定程度上反映出了弯道水流的水力特点。
(3)在求解三维水流时,平面上采用贴体坐标变换,而垂向上采用了σ坐标变换。在处理压力项时,采用了静水压强假定,这从对三维弯道水流的计算结果可以看出是相符的。针对三维弯道水流因受到离心力而产生的螺旋状流动,选取了7个横断面,对断面内的水深进行了分析；计算结果与实测值较为吻合,从而验证了该模型的可行性与准确性。
本论文的研究内容及其成果对于研究天然河道及近海水流具有一定的指导意义。
Recently,with the vigorous development of hydroelectric construction in our country,more and more experts have worked out many methods with which three-demensional natural river flow model can be converted into two-demensional mathematical model of shallow water and be solved.These method have showed strong superiority on calculations of natural rivers and coastal water and have played a more and more important role in theoretical research and production practice.On the basis of the wide research to the existing kinds of mathematical model of shallow water,for the difficulties of grid generation because of complex boundary conditions of natural river bend, mathematical model of shallow water based on cartesian coordinates is translated into a new one based on body-fitted coordinates.Based on the collocated grid system,a new mathematical model of shallow water based on body-fitted coordinates is built in this thesis.The specific contents studyed in this paper are as follows:
(1)When generating body-fitted grid, for complex boundary conditions of natural river bend,a method of generating body-fitted grid which can makes interior grid lines of computational zone is brought up. While choosing proper regulator,according to a principle that stream function is absolutely orthogonal with potential function, the equation corresponding to a node between computational plane and physical plane has been deducted.A algorithm called TDMA is used to solve the equation with which body-fitted grid is obtained.The whole process is achieved by programming with a senior language.The grid obtained by programmed here is proved to have high lubricity and orthogonality and to be a perfect computational grid.
(2)On the basis of mathematical model of shallow water based on body-fitted coordinates,for the characteristics of natural river such as bending, twists and turns shoreline,complex terrain,obvious characteristics of three demension, an important influence of turn circulation on flow and sediment movement which makes the velocity nearby concave bank larger than the velocity nearby convex bank,turn circulation and coriolis force are added to the mathematical model of shallow water so as to more exactly describe the actual situation.Convection item is discreted with a new scheme called power-law scheme based on collacted grid.A method that velocity is amended with water level is used to solve the control equations with SIMPLE algorithm.A continuous curve and a natural river are used to verified the computational model.Comperisons between computational results and experimental results show that the computational modes and algorithm have efficient feasibility.In the meanwhile,the computational results also show the characteristics of turnning flow.
(3)When solving the equations of three demensional flow,computational grid is gained by choosing body-fitted coordinate transformation horizontally andσcoordinate transformation vertically.A assumption of hydrostatic pressure is applied to deal with the pressure item. Comparisons between the calculating results and experimental results show good reasonableness and raliability.The computational depth of seven cross sections are used to verified to describe the spiral flow which gives birth to because of the centrifugal force.Obviously,comparisons between the computational results and experimental results show good consistency.It shown that this calculating model has good practicality and accuracy.
The contents of study and achievement obtained in this paper could provide a good guildline to study flow of natural river and offshore.
(1)针对天然河道边界弯曲复杂的特点,提出了一种可控制网格内部正交的贴体网格生成方法。在调节因子P,Q的选取上,根据流函数与势函数绝对正交的原则,推导了计算平面上一点(ξ,η)与物理平面上一点(x,y)对应关系的方程,运用TDMA算法对网格生成方程进行求解,最终生成贴体坐标网格,整个流程运用FORTRAN 95编程实现。通过程序计算得到的网格光滑性和正交性较好,不失为一种理想的计算网格。
(2)在曲线坐标下的浅水控制方程的基础上,针对天然河流大部分河段的弯曲性较大,岸线曲折,地形复杂,引起弯道环流,水流具有明显的三维特性,而且弯道环流的作用对水流泥沙运动影响较大,使水流凹岸流速增大等特点,在控制方程里增加了弯道环流项和科氏力项,以求能更好的描述浅水模型。在方程的离散上,运用了同位网格,对对流项采用乘方格式进行离散。在运用SIMPLE算法进行求解时,采用水位修正法对流速进行修正。在对连续弯道和天然河段进行求解中,通过计算值与实测值的对比,表明了计算模型与计算方法的可行性,也一定程度上反映出了弯道水流的水力特点。
(3)在求解三维水流时,平面上采用贴体坐标变换,而垂向上采用了σ坐标变换。在处理压力项时,采用了静水压强假定,这从对三维弯道水流的计算结果可以看出是相符的。针对三维弯道水流因受到离心力而产生的螺旋状流动,选取了7个横断面,对断面内的水深进行了分析；计算结果与实测值较为吻合,从而验证了该模型的可行性与准确性。
本论文的研究内容及其成果对于研究天然河道及近海水流具有一定的指导意义。
Recently,with the vigorous development of hydroelectric construction in our country,more and more experts have worked out many methods with which three-demensional natural river flow model can be converted into two-demensional mathematical model of shallow water and be solved.These method have showed strong superiority on calculations of natural rivers and coastal water and have played a more and more important role in theoretical research and production practice.On the basis of the wide research to the existing kinds of mathematical model of shallow water,for the difficulties of grid generation because of complex boundary conditions of natural river bend, mathematical model of shallow water based on cartesian coordinates is translated into a new one based on body-fitted coordinates.Based on the collocated grid system,a new mathematical model of shallow water based on body-fitted coordinates is built in this thesis.The specific contents studyed in this paper are as follows:
(1)When generating body-fitted grid, for complex boundary conditions of natural river bend,a method of generating body-fitted grid which can makes interior grid lines of computational zone is brought up. While choosing proper regulator,according to a principle that stream function is absolutely orthogonal with potential function, the equation corresponding to a node between computational plane and physical plane has been deducted.A algorithm called TDMA is used to solve the equation with which body-fitted grid is obtained.The whole process is achieved by programming with a senior language.The grid obtained by programmed here is proved to have high lubricity and orthogonality and to be a perfect computational grid.
(2)On the basis of mathematical model of shallow water based on body-fitted coordinates,for the characteristics of natural river such as bending, twists and turns shoreline,complex terrain,obvious characteristics of three demension, an important influence of turn circulation on flow and sediment movement which makes the velocity nearby concave bank larger than the velocity nearby convex bank,turn circulation and coriolis force are added to the mathematical model of shallow water so as to more exactly describe the actual situation.Convection item is discreted with a new scheme called power-law scheme based on collacted grid.A method that velocity is amended with water level is used to solve the control equations with SIMPLE algorithm.A continuous curve and a natural river are used to verified the computational model.Comperisons between computational results and experimental results show that the computational modes and algorithm have efficient feasibility.In the meanwhile,the computational results also show the characteristics of turnning flow.
(3)When solving the equations of three demensional flow,computational grid is gained by choosing body-fitted coordinate transformation horizontally andσcoordinate transformation vertically.A assumption of hydrostatic pressure is applied to deal with the pressure item. Comparisons between the calculating results and experimental results show good reasonableness and raliability.The computational depth of seven cross sections are used to verified to describe the spiral flow which gives birth to because of the centrifugal force.Obviously,comparisons between the computational results and experimental results show good consistency.It shown that this calculating model has good practicality and accuracy.
The contents of study and achievement obtained in this paper could provide a good guildline to study flow of natural river and offshore.
引文
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[7]潘存鸿.三角形网格下求解二维浅水方程的和谐Godunov格式.水科学进展[J],2007,2006,Vol.18,No.2:204-207
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[3]刘哲.复杂边界二维水流数值模拟研究与应用：[硕士学位论文][D].西安：西安理工大学,2005
[4]金忠青.N-S方程的数值解和紊流模型[J].南京：河海大学出版社,1989
[5]陈建忠,史思科,胡彦梅.二维浅水方程的高阶松弛格式求解.水动力学研究与进展[J],2007,Vol.22,No.3：306-308
[6]窦红,汪继文.求解二维浅水方程的一种高分辨率有限体积法.应用数学与计算数学学报[J],2006, Vol.20,No.2:84-86
[7]潘存鸿.三角形网格下求解二维浅水方程的和谐Godunov格式.水科学进展[J],2007,2006,Vol.18,No.2:204-207
[8]朱良生.近岸二维海流数值计算方法若干问题的研究和应用.热带海洋[J],1995, Vol.14,No.1: 31-36
[9]王立辉,胡四一,龚春生.二维浅水方程的非结构网格数值解.水利水运工程学报[J],2006,Vol.1,No.1:8-12
[10]孔俊,宋志尧,张金善.一种新的浅水方程计算模式及其应用.第十二届中国海岸工程学术讨论会论文集[J],2003,339-344
[11]杜彦良,褚君达.部分水深平均二维计算模型的研究.河海大学学报[J],2001, Vol.29,No.3:13-15
[12]胡四一,谭维炎.无结构网格上二维浅水流动的数值模拟.水科学进展[J],1995,6(1),1-9
[13]刘晓东,华祖林,赵玉萍.基于四叉树网格的Godunov型二维水流数值计算模式.河海大学学报[J],2002,30(6),7-10
[14]魏文礼,张宗孝,刘玉玲.分型算法在二维非恒定流的数值模拟中的应用.西北水资源与水工程[J],1999,10(4),41-43
[15]吴修广,沈永明,郑永红,王平义.非正交曲线坐标下二维水流计算的SIMPLEC算法.水利学报[J],2003,10(2),25-30
[16]谢鉴衡.河流模拟[M].北京：中国水利电力出版社,1993
[17]罗奇PJ.计算流体动力学[M].北京：科学出版社,1983,47-56,82-87
[18]张修忠,王光谦.浅水流动及输运三维数学模型研究进展.水利学报[J],2002,3(7),1-7
[19]魏文礼,王德意.计算水力学理论及应用[M].西安：陕西科学技术出版社,2001
[20]陶文铨.数值传热学[M].西安：西安交通大学出版社,2001
[21]魏文礼,戴会超.紊流模型理论及工程应用[M].西安：陕西科学技术出版社,2006
[22]饶盛文贺成才赵丰.有限差分法在低渗油藏数值模拟中的应用.石油工业计算机应用[J],2008,2(3),21-23
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