河道溃堤与溃堤波的一、二维耦合计算数值模拟
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摘要
河道溃堤后形成的溃堤波传播速度快、破坏性大,对人类生命、财产造成巨大危害,因此,河道溃堤和溃堤波的耦合计算为溃堤造成的圩区淹没过程的预测起着重要的作用。本文建立了一维河道溃堤与二维溃堤波耦合计算的水力模型。其中,河道的计算采用一维明渠非恒定流Saint-Venant方程组,方程采用稳定性好、精度高的Preissmann四点隐式差分格式离散,求解的过程与溃口的数值模拟耦合;溃坝按渐溃计算,将溃口模拟为梯形,随时间线性增大;溃堤波的计算采用在有限体积无结构网格上建立的TVD-MUSCL格式。模拟算例表明,此类格式能够自动俘获间断且能消除激波附近的虚假数值振荡。本文建立的河道溃堤与溃堤波的一、二维耦合计算水力模型实现了真正意义上的一维计算水域向二维计算水域的过渡,模拟了河道溃堤过程、溃口流速变化情况,以及溃堤波传播过程中的传播、绕射、反射及变形等运动特征,模拟比较接近真实溃堤过程。其研究成果能较好的预测圩区淹没过程和淹没范围,为建立河流洪水预警系统提供了可靠的理论基础。
The fast-spreading dike-break waves caused by river dike-burst can do great harm to the safe and properties of human being. Therefore, the coupled numerical simulation of river and dike-break waves is of great importance to the prediction of submerging process in polder. A hydraulic model of coupling 1-D river dike-breaking and 2-D dike-break waves is established in the paper. The Saint-Venant equations used in the simulation of river are discretized by the Preissmann implicit scheme, which shows good stability and high precision. The calculation is coupled with the numerical simulation of the breach. The dyke-break process is supposed to be gradual and the breach is simulated as an echelon growing wider with time. The TVD-MUSCL scheme based on FVM (Finite Volume Method) and unstructured grids is used to the computation of dike-break waves. The results show that this scheme is not only nonoscillatory, but also capable of treating hydraulic jump automatically. The hydraulic model of coupling 1-D river and 2-D dike-break waves realizes the transition from 1-D calculation water area to 2-D water area, and it shows the process of river dyke-breaking, the velocity variation of the breach flow and the flow characteristic of diffraction, reflection and deformation during the propagation of dike-break waves. The simulation can realistically show the burst process of dike. It is concluded that the model can predict the submerge process and area effectively, and it provides academic base for establishment of flood forecast system.
The fast-spreading dike-break waves caused by river dike-burst can do great harm to the safe and properties of human being. Therefore, the coupled numerical simulation of river and dike-break waves is of great importance to the prediction of submerging process in polder. A hydraulic model of coupling 1-D river dike-breaking and 2-D dike-break waves is established in the paper. The Saint-Venant equations used in the simulation of river are discretized by the Preissmann implicit scheme, which shows good stability and high precision. The calculation is coupled with the numerical simulation of the breach. The dyke-break process is supposed to be gradual and the breach is simulated as an echelon growing wider with time. The TVD-MUSCL scheme based on FVM (Finite Volume Method) and unstructured grids is used to the computation of dike-break waves. The results show that this scheme is not only nonoscillatory, but also capable of treating hydraulic jump automatically. The hydraulic model of coupling 1-D river and 2-D dike-break waves realizes the transition from 1-D calculation water area to 2-D water area, and it shows the process of river dyke-breaking, the velocity variation of the breach flow and the flow characteristic of diffraction, reflection and deformation during the propagation of dike-break waves. The simulation can realistically show the burst process of dike. It is concluded that the model can predict the submerge process and area effectively, and it provides academic base for establishment of flood forecast system.
引文
[1] 王嘉松,倪汉根,金生.瞬间全溃溃坝波的传播、反射和绕射的数值模拟[J].水动力学研究与进展,2000,15(1):1～7
[2] Wang J S, Ni H G, He Y S. Finite-difference TVD scheme for computation of dam-break problems[J]. Journal of Hydraulic Engineering, 2000, 126(4):253～262
[3] Aureli F, Mignosa P, Tomirotti M. Numerical simulation and experimental verification of dam-break flows with shocks[J]. Journal of Hydraulic Research, 2000, 38(3):197～206
[4] 柯礼丹等,我国大坝安全问题及展望[J].水利工程管理论文集,第1集,1981年全国大坝安全学术讨论会专辑,中国水利学会工程管理专业委员会,1984
[5] 徐玉英,宋榜科,韩晓东等.水库下游无水位资料时淹没水位的计算[J].水利水电技术,1999,30(1):77～78
[6] Ritter, A. Die Fortpflanzung der Wasserwel-len [The propagation of water waves] Ver. Deutsch Ingenieure Zeitschr[M]. Berlin, 1892, 36(33):947～954
[7] 谢任之.溃坝坝址流量计算[M].水利水运科学研究,1982(1)
[8] 汪德爟.计算水力学理论与应用[M].河海大学出版社,1989:140,158～165
[9] 徐正凡..水力学[M].北京:高等教育出版社,1986
[10] 吴持恭.水力学[M].北京.高等教育出版社,1982
[11] 齐鄂荣,罗昌.库区河道非恒定流糙率的选取及特性[J].武汉大学学报(工学版),2003,36(2):1～5
[12] 石国钰,叶敏,唐佩文.堤防分洪溃口变化特征初探[J].人民长江,1997,28(1):30～32
[13] 孔晋豫.溃坝洪水数学模型的初步应用[J].电力勘测,1995.9,(3):53～57
[14] 谢任之.溃坝水力学[M].山东科学技术出版社,1992.2
[15] 陶文铨.数值传热学[M].西安:西安交通大学出版社,1988
[16] Patankar S V, Spalding D B. A Calculation Procedures for Heat,Mass and Momentum Transfer in Three-Dimensional Parabolic Flows[J]. Int .J. Heat and Mass Transfer, 1972, 15: 1787-1806.
[17] Rhie C M, Chow W L. A Numerical Study of the Turbulent Flow Past an Isolated Airfoil With Trailing Edge Seperation[J]. AIAA J., 1983, 21:1525-1532.
[18] Peric M, Kessler R, Scheuerer G. Comparison of Finite-Volume Numerical Methods Wiih Staggered and Collocated Grids[J]. Computers and Fluids, 1988, 16(4):389～403.
[19] Miller T F, Sehmidt F W. Use of a Pressure-Weighted Interpolation Method for Solution of the Incompressible Navier-stokes Equations on a Nonstaggered Grid System[J]. Numerical HeatTransfer, 1988, 14:213～233.
[20] Rodi W, Majumdar S and Sehonung B. Finite-Volume Methods for Two-dimensional Incompressible Flows With complex Boundaries[J]. Comput. Meth. Appl. Meth. Eng., 1989, 75:369～392.
[21] Thiart G.D. Finite-Difference Scheme for the Numerical Solution of Fluid Flow and Heat Transfer Problems on Nonstaggered grid[J]. Numerical Heat Transfer, 1990, 17B,43～62.
[22] ZHAO D H, SHEN H W, TABIOS Ⅲ G Q, LAI J S. andTAN W Y. Finite-volume twodimensional unsteady-flow model for river basins[J]. Hydr. Engrg, ASCE,1994, 120(7):863～883.
[23] ZHAO D H, SHEN H W, LAI J S, et al. Approximate Riemann solvers in FVM for 2D hydraulic shock waves modeling[J]. Hydr. Engrg., ASCE, 1996, 122(12):693～702.
[24] 赵棣华,戚晨,庾维德,徐保华,裴中平.平面二维水流水质有限体积法及黎曼近似界模型[J].水科学进展,2000,11(4):368～373.
[25] ZHAO D H, et al. 2D-FVM model for modeling riverbed deformation due to cutoff [A]. Proceedings of International Conference on Hydraulic Engineering[C]. Iowa, US., 1999
[26] 赵棣华,沈福新,颜志俊,卢恭和.基于有限体积法及FDS格式的感潮河段二维泥沙冲淤模型[J].水动力学研究与进展A辑,2004,19.:98-103.
[27] J Douglas, Jr. Russell T F. Numerical Methods for Convection-dominated Diffusion ProblemsBased on Combing the Method of Characteristics with Finite Element or FiniteDifference Procedures[J]. SLAM J. Numer. Anal, 1982,19:871～885.
[28] Russell T F. Time Stepping along Characteristics with Incomplete Iteration for a Galerkin Approximation of Miscible Displacement in Porous Media[J]. SIAM J. Numer. Anal., 1985, 22:970～1013.
[29] Ewing R.E, Russell T F, Wheeler M F. Convergence Analysis of an Approximation fo Miscible Displacement in Porous Media by Mixed Finite Elements and a Modified Method of Characteristics[J]. Comp. Meth. In Appl. Mech. And Eng., 1984,14:73～92
[30] Gray W G. An efficient finite element scheme for two-dimensional surface computation[C]. In: Proceedings of the lst Intn. Conf. on Finite Elements in Water Resources,Peatech Press, 1977:433～449
[31] 于天常.二维浅水环流问题的一个有限元模型[J].海洋与湖沼,1984,15(1):37～45
[32] 史金松.计算二维非恒定流的守恒型有限元法[J].水利学报,1987,(10):41～48
[33] 谭维炎,赵棣华.二维非恒定浅水明流的三种算法[J].水利学报,1985,(9):1～9
[34] 李浩麟,易家豪.河口浅水方程的隐式和显式有限元解法[J].水利水运科学研究,1983,(10):15～26.
[35] 耿兆铨.二维非恒定流的显式迎流有限元模型[C].全国第一届水动力学学术会议论文集,北京:海洋出版社,1984
[36] 张存智,杨连武等.具有潮滩移动边界的浅海环流有限元模型[J].海洋学报,1990,(12):1～13
[37] Lax P D, Wendroff B. Difference Schemes for Hyperbolic Equations With High Order of accuracy[J]. Comm. Pure Appl. Math, 1964, 17:381～398
[38] 谭维炎,胡四一.钱塘江口涌潮的二维数值模拟[J].水科学进展,1995,6(2):83～93
[39] 谭维炎.计算浅水动力学一有限体积法的应用[M].北京:清华大学出版社,1998.9
[40] Spekreijse, S. P. Multigrid solution of the steady Euler equations. CWI Tract 46, Amsterdam, The Netherlands, 1988
[41] Wurbs R A. Dam-break flood wave models[J], Journal of Hydraulic Engineering, ASCE, 1987, 113(1): 29～46
[42] 朱自强等.应用计算流体力学[M].北京航空航天大学出版社,1997,7:66～70
[43] Lax P D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves[J]. SIAM Reg. Conf. Series Lectures in Appliced Math.,1972
[44] Harten A. High resoluteion schemes for hyperbolic systems of conservation laws[J]. J. Comp. Phys, 1983,49:357-393
[45] Jameson A, Lax P D. Conditions for the construction of multi-point total variation diminishing difference schemes[M]. ICASE Report 86-18, 1986
[46] Yee H C. Upwind and symmetric shock capturing schemes. NASA TM 89464, 1987
[47] 陆夕云,庄礼贤,童秉纲等.对称TVD差分格式的构造及其应用[J].计算物理,1994,11(1):45～50
[48] 王志刚.实际地形溃坝水流的数值模拟[M].河海大学硕士学位论文,2003:15～16
[49] Van Leer B. On the relation between the upwind-differencing schemes of Godunov, Enguist-Osher and Roe[J]. SIAM Journal of Science and Statistical Computations, 1985,5(1): 1～20
[50] Roe P L. Generalized formulation of TVD Lax-Wendroff schemes[J]. ICASE Report No.84-53, 1984
[51] Van Leer B, Thomas J L, Roe P L, Newasome T W. A comparison of numerical flux formulas for the Euler and Navier-Stokes equations[J]. AIAA 87-1104
[52] 水鸿寿.一维流体力学差分方法[M].国防工业出版社,1998.2
[53] 刘儒勋,王志峰等.数值模拟方法和运动界面追踪[M].中国科技大学出版社,2001.10
[54] Courant R, Friedrich K O, Lewy H. On the partial differential equation of mathematical physics[J]. IBM Journal, 1967, 11(2): 215～234
[55] 谭维炎,胡四一.二维浅水流动的一种普适的高性能格式[J].水科学进展,2000,2(3)
[56] Zhao D H et al. Finite-volume two-dimensional unsteady-flow model for river basins[J]. ASCE J. Hydraul. Eng, 1994, 120:864～883
[57] 谭维炎,胡四一.浅水流动计算中一阶有限体积法Osher格式的实现[J].水科学进展,,1994,5(4):262～269
[58] Zhao D Het al. Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling[J]. J. Hydraulic Engineering, ASCE, 1996, 122(12):692～702
[59] Garcia R, Kahawita R A. Numerical solution of the ST. Venant equations with the MacCormack finite-difference scheme[J]. Int. Num. Meth. Fluids, 1986, 6(5):259～274
[2] Wang J S, Ni H G, He Y S. Finite-difference TVD scheme for computation of dam-break problems[J]. Journal of Hydraulic Engineering, 2000, 126(4):253～262
[3] Aureli F, Mignosa P, Tomirotti M. Numerical simulation and experimental verification of dam-break flows with shocks[J]. Journal of Hydraulic Research, 2000, 38(3):197～206
[4] 柯礼丹等,我国大坝安全问题及展望[J].水利工程管理论文集,第1集,1981年全国大坝安全学术讨论会专辑,中国水利学会工程管理专业委员会,1984
[5] 徐玉英,宋榜科,韩晓东等.水库下游无水位资料时淹没水位的计算[J].水利水电技术,1999,30(1):77～78
[6] Ritter, A. Die Fortpflanzung der Wasserwel-len [The propagation of water waves] Ver. Deutsch Ingenieure Zeitschr[M]. Berlin, 1892, 36(33):947～954
[7] 谢任之.溃坝坝址流量计算[M].水利水运科学研究,1982(1)
[8] 汪德爟.计算水力学理论与应用[M].河海大学出版社,1989:140,158～165
[9] 徐正凡..水力学[M].北京:高等教育出版社,1986
[10] 吴持恭.水力学[M].北京.高等教育出版社,1982
[11] 齐鄂荣,罗昌.库区河道非恒定流糙率的选取及特性[J].武汉大学学报(工学版),2003,36(2):1～5
[12] 石国钰,叶敏,唐佩文.堤防分洪溃口变化特征初探[J].人民长江,1997,28(1):30～32
[13] 孔晋豫.溃坝洪水数学模型的初步应用[J].电力勘测,1995.9,(3):53～57
[14] 谢任之.溃坝水力学[M].山东科学技术出版社,1992.2
[15] 陶文铨.数值传热学[M].西安:西安交通大学出版社,1988
[16] Patankar S V, Spalding D B. A Calculation Procedures for Heat,Mass and Momentum Transfer in Three-Dimensional Parabolic Flows[J]. Int .J. Heat and Mass Transfer, 1972, 15: 1787-1806.
[17] Rhie C M, Chow W L. A Numerical Study of the Turbulent Flow Past an Isolated Airfoil With Trailing Edge Seperation[J]. AIAA J., 1983, 21:1525-1532.
[18] Peric M, Kessler R, Scheuerer G. Comparison of Finite-Volume Numerical Methods Wiih Staggered and Collocated Grids[J]. Computers and Fluids, 1988, 16(4):389～403.
[19] Miller T F, Sehmidt F W. Use of a Pressure-Weighted Interpolation Method for Solution of the Incompressible Navier-stokes Equations on a Nonstaggered Grid System[J]. Numerical HeatTransfer, 1988, 14:213～233.
[20] Rodi W, Majumdar S and Sehonung B. Finite-Volume Methods for Two-dimensional Incompressible Flows With complex Boundaries[J]. Comput. Meth. Appl. Meth. Eng., 1989, 75:369～392.
[21] Thiart G.D. Finite-Difference Scheme for the Numerical Solution of Fluid Flow and Heat Transfer Problems on Nonstaggered grid[J]. Numerical Heat Transfer, 1990, 17B,43～62.
[22] ZHAO D H, SHEN H W, TABIOS Ⅲ G Q, LAI J S. andTAN W Y. Finite-volume twodimensional unsteady-flow model for river basins[J]. Hydr. Engrg, ASCE,1994, 120(7):863～883.
[23] ZHAO D H, SHEN H W, LAI J S, et al. Approximate Riemann solvers in FVM for 2D hydraulic shock waves modeling[J]. Hydr. Engrg., ASCE, 1996, 122(12):693～702.
[24] 赵棣华,戚晨,庾维德,徐保华,裴中平.平面二维水流水质有限体积法及黎曼近似界模型[J].水科学进展,2000,11(4):368～373.
[25] ZHAO D H, et al. 2D-FVM model for modeling riverbed deformation due to cutoff [A]. Proceedings of International Conference on Hydraulic Engineering[C]. Iowa, US., 1999
[26] 赵棣华,沈福新,颜志俊,卢恭和.基于有限体积法及FDS格式的感潮河段二维泥沙冲淤模型[J].水动力学研究与进展A辑,2004,19.:98-103.
[27] J Douglas, Jr. Russell T F. Numerical Methods for Convection-dominated Diffusion ProblemsBased on Combing the Method of Characteristics with Finite Element or FiniteDifference Procedures[J]. SLAM J. Numer. Anal, 1982,19:871～885.
[28] Russell T F. Time Stepping along Characteristics with Incomplete Iteration for a Galerkin Approximation of Miscible Displacement in Porous Media[J]. SIAM J. Numer. Anal., 1985, 22:970～1013.
[29] Ewing R.E, Russell T F, Wheeler M F. Convergence Analysis of an Approximation fo Miscible Displacement in Porous Media by Mixed Finite Elements and a Modified Method of Characteristics[J]. Comp. Meth. In Appl. Mech. And Eng., 1984,14:73～92
[30] Gray W G. An efficient finite element scheme for two-dimensional surface computation[C]. In: Proceedings of the lst Intn. Conf. on Finite Elements in Water Resources,Peatech Press, 1977:433～449
[31] 于天常.二维浅水环流问题的一个有限元模型[J].海洋与湖沼,1984,15(1):37～45
[32] 史金松.计算二维非恒定流的守恒型有限元法[J].水利学报,1987,(10):41～48
[33] 谭维炎,赵棣华.二维非恒定浅水明流的三种算法[J].水利学报,1985,(9):1～9
[34] 李浩麟,易家豪.河口浅水方程的隐式和显式有限元解法[J].水利水运科学研究,1983,(10):15～26.
[35] 耿兆铨.二维非恒定流的显式迎流有限元模型[C].全国第一届水动力学学术会议论文集,北京:海洋出版社,1984
[36] 张存智,杨连武等.具有潮滩移动边界的浅海环流有限元模型[J].海洋学报,1990,(12):1～13
[37] Lax P D, Wendroff B. Difference Schemes for Hyperbolic Equations With High Order of accuracy[J]. Comm. Pure Appl. Math, 1964, 17:381～398
[38] 谭维炎,胡四一.钱塘江口涌潮的二维数值模拟[J].水科学进展,1995,6(2):83～93
[39] 谭维炎.计算浅水动力学一有限体积法的应用[M].北京:清华大学出版社,1998.9
[40] Spekreijse, S. P. Multigrid solution of the steady Euler equations. CWI Tract 46, Amsterdam, The Netherlands, 1988
[41] Wurbs R A. Dam-break flood wave models[J], Journal of Hydraulic Engineering, ASCE, 1987, 113(1): 29～46
[42] 朱自强等.应用计算流体力学[M].北京航空航天大学出版社,1997,7:66～70
[43] Lax P D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves[J]. SIAM Reg. Conf. Series Lectures in Appliced Math.,1972
[44] Harten A. High resoluteion schemes for hyperbolic systems of conservation laws[J]. J. Comp. Phys, 1983,49:357-393
[45] Jameson A, Lax P D. Conditions for the construction of multi-point total variation diminishing difference schemes[M]. ICASE Report 86-18, 1986
[46] Yee H C. Upwind and symmetric shock capturing schemes. NASA TM 89464, 1987
[47] 陆夕云,庄礼贤,童秉纲等.对称TVD差分格式的构造及其应用[J].计算物理,1994,11(1):45～50
[48] 王志刚.实际地形溃坝水流的数值模拟[M].河海大学硕士学位论文,2003:15～16
[49] Van Leer B. On the relation between the upwind-differencing schemes of Godunov, Enguist-Osher and Roe[J]. SIAM Journal of Science and Statistical Computations, 1985,5(1): 1～20
[50] Roe P L. Generalized formulation of TVD Lax-Wendroff schemes[J]. ICASE Report No.84-53, 1984
[51] Van Leer B, Thomas J L, Roe P L, Newasome T W. A comparison of numerical flux formulas for the Euler and Navier-Stokes equations[J]. AIAA 87-1104
[52] 水鸿寿.一维流体力学差分方法[M].国防工业出版社,1998.2
[53] 刘儒勋,王志峰等.数值模拟方法和运动界面追踪[M].中国科技大学出版社,2001.10
[54] Courant R, Friedrich K O, Lewy H. On the partial differential equation of mathematical physics[J]. IBM Journal, 1967, 11(2): 215～234
[55] 谭维炎,胡四一.二维浅水流动的一种普适的高性能格式[J].水科学进展,2000,2(3)
[56] Zhao D H et al. Finite-volume two-dimensional unsteady-flow model for river basins[J]. ASCE J. Hydraul. Eng, 1994, 120:864～883
[57] 谭维炎,胡四一.浅水流动计算中一阶有限体积法Osher格式的实现[J].水科学进展,,1994,5(4):262～269
[58] Zhao D Het al. Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling[J]. J. Hydraulic Engineering, ASCE, 1996, 122(12):692～702
[59] Garcia R, Kahawita R A. Numerical solution of the ST. Venant equations with the MacCormack finite-difference scheme[J]. Int. Num. Meth. Fluids, 1986, 6(5):259～274