实际地形溃坝水流的数值模拟
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摘要
水工建筑物会因自然因素或人为因素遭受破坏，一旦遭到破坏，其对下游人民生命财产的影响远远大于洪水，而水库和堤坝的安全又是水工建筑物设计和管理的核心问题，因此，溃坝计算对于水工建筑物失事后的影响评估起着重要的作用，也可为后期的环境与生态评估系统的开发提供水流模型基础。本论文在有限体积无结构网格上建立两类高性能格式——TVD-MacCormack格式和MUSCL格式，计算出溃坝坝址流量、水位及向下游洪水演进得出沿程各处的流量、水位、流速和波前。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，克服了许多格式在间断处不是过分耗散就是数值振荡的缺点。对复杂地形和边界进行了有效处理并在地形起伏较大处采用水面梯度法(SurfaceGradient Method,SGM)，得到满意的结果。显示出此类格式的守恒性、逆风性、TVD(Total Variation Diminishing)性及对间断的高分辨率等优良特性。
The hydraulic structure could be damaged due to natural or human factors. Once it happens, it would destroy lots of the properties and lives downstream of the structure. Therefore, safety of the hydraulic structure is the most important factor of the design and management. Simulation of the dam-break wave plays a key role hi assessing the effect of hydraulic structure damaging and provides the basis for exploring system assessing environmental and ecological consequences. Two schemes, TVD-MacCormack and TVD-MUSCL were constructed based on Finite Volume Method (FVM) with high performance. Two schemes can be used to compute discharge, water level at collapsed dam-toe and each section along the propagating route, the sharp front of the wave. Two models were tested by analytical solution of one-dimensional dam-break wave. The results showed that the models could capture the dam-break wave front correctly and eliminate the false numerical oscillations. The models overcome either too much dissipation or numerical
oscillation, which many other schemes do. This thesis also developed a technique treating complicated geometry. The results from the simulation of the dam-break wave propagating on a real topography prove that the technique developed in this thesis is useful in practice.
The hydraulic structure could be damaged due to natural or human factors. Once it happens, it would destroy lots of the properties and lives downstream of the structure. Therefore, safety of the hydraulic structure is the most important factor of the design and management. Simulation of the dam-break wave plays a key role hi assessing the effect of hydraulic structure damaging and provides the basis for exploring system assessing environmental and ecological consequences. Two schemes, TVD-MacCormack and TVD-MUSCL were constructed based on Finite Volume Method (FVM) with high performance. Two schemes can be used to compute discharge, water level at collapsed dam-toe and each section along the propagating route, the sharp front of the wave. Two models were tested by analytical solution of one-dimensional dam-break wave. The results showed that the models could capture the dam-break wave front correctly and eliminate the false numerical oscillations. The models overcome either too much dissipation or numerical
oscillation, which many other schemes do. This thesis also developed a technique treating complicated geometry. The results from the simulation of the dam-break wave propagating on a real topography prove that the technique developed in this thesis is useful in practice.
引文
1. Van Leer B. Towards the Ultimate Conservative Difference scheme Ⅱ. Journal of Computational Physics. 14: 363～389, 1974
2. Osher S. Shock Modeling in transonic and Supersonic Flow. Recent Advances in Numerical Methods in Fluids. Vol. 4, W. G. Habashi Ed
3. Yee H C, Warming R F and Harten A. Implicit Total Variation Diminishing (TVD) Schemes for Steaty-state Calculation. AIAA Paper 83～1902, 1983
4. Harten A. High resolution schemes for hyperbolic systems of conservation laws. Journal of Computational Physics. 49:357～393, 1983
5. Harten A, Osher S. Uniformly high order accurate essentially non-oscillatory schemes Ⅰ.SIAM J. on Num. Anal. 24:279～309, 1987
6. Harten A, Osher S, Engquist B, Chakravarthy S R. Uniformly high order accurate essentially non-oscillatory schemes Ⅲ. Journal of Computational Physics. 71:231～303, 1987
7. Van Leer B. Towards the Ultimate Conservative Difference Scheme Ⅴ: A Second Order Sequel to Godunov's Method. Journal of Computational Physics. 32:101～136, 1979
8．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．水动力学研究与进展，15(1)：1～7，2000
9. Wang J S, Ni H Ca, He Y S. Finite-difference TVD scheme for computation of dam-break problems. Journal of Hydraulic Engineering. 126(4):253～262, 2000
10. Aureli F, Mignosa P, Tomirotti M. Numerical simulation and experimental verification of dam-break flows with shocks. Journal of Hydraulic Research. 38(3): 197～206, 2000
11．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．水利水电技术．Vol．30，1999(增刊)
12．王国安．坝体瞬间全溃最大流量通用公式推导．华北水利水电学院学报．2001，22(3)：23～30
13．谢任之．溃坝坝址流量计算．水利水运科学研究．1982(1)
14．汪德爟．计算水力学理论与应用．南京：河海大学出版社，1989
15. Harten A. On the Symmetric form of Systems of Conservation Laws With Entwpy. Journal of Computational Physics. 49:151～164, 1983
16. Harten A, Lax P D and Van Leer B. On Upstream Differencing and Godunov-Type Schemes
for Hyperbolic Conservation Laws. SIAM Rev. 25:35～61, 1983
17．傅德薰．流体力学数值模拟．北京：国防工业出版社，1993
18．刘顺隆，郑群．计算流体力学．哈尔滨工程大学出版社，1998
19. Lax P D and Wendroff B. Difference Schemes for Hyperbolic Equations With High Order of accuracy. Comm. Pure Appl. Math. 17: 381～398, 1964
20．谭维炎，胡四一，钱塘江口涌潮的二维数值模拟．水科学进展．6(2)：83～93，1995
21．谭维炎．计算浅水动力学—有限体积法的应用．北京：清华大学出版社，1998，9
22. Spekreijse, S. P. Multigrid solution of the steady Euler equations. CWI Tract 46, Amsterdam, The Netherlands, 1988
23. Desiseri D and Jameson A. Multigrid solution of the Euler equations on unstructured and adaptive meshes. AIAA Paper 87-0353, 1987
24. Pan D and Cheng J. incompressible flow solution on unstructured triangular meshes. Int. J. Numer. Methods Fluids. 16:1079～1098, 1993
25. Zhao D H et al. Finite-volume two-dimensional unsteady-flow model for fiver basins. ASCE J. Hydraul. Eng. 120:864～883, 1994
26. Anastasiou K and Chan C T. Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes. Inc J. Numer. Methods Fluids.24:1225～1245, 1997
27．朱自强等．应用计算流体力学．北京：北京航空航天大学出版社，1998，8
28．王嘉松等．用TVD显格式模拟一维溃坝洪水波的演进与反射．水利学报，5：7～11，1998
29. Sweby P K. High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Num. Anal. 21:995～1011, 1984
30. Roe P L. Generalized formulation of TVD Lax-Wendroff schemes. ICASE Report No. 84-53,1984
31. Garcia-Navarro P et al. One-dimensional open-channel flow simulation using TVD-MacCormack scheme. Journal of Computational Physics. 118(10), 1992
32. Garcia R, Kahawita R A. Numerical solution of the ST. Venant equations with the MacCormack finite-difference scheme. Int. Num. Meth. Fluids, 6(5):259～274, 1986
33. Fennema R J, Chaudhry M H. Simulation of 1D dam-break flows. J. Hydr. Res. 25(1):41～51,1987
34. Savic L J, Holly F M. Dam-break flood waves computed by modified Godunov method. J. Hydr. Res., Delft, The Netherlands, 31(2): 187～204, 1993
35．汪迎春．溃坝水流二维演进模型．河海大学硕士学位论文．2001，3
36. Garcia P et al. MacCormack's method for the numerical solution of 1D discontinuous unsteady open-channel flow. Journal of Hydraulics Research. 1992, 30(1)
37. Tseng M H. Two-dimensional shallow water flows simulation using TVD-MacCormack Scheme Journal of Hydraulics Research. 38(2): 123～131, 2000
38. Roe P L. Approximate Riemann Solvers. Parameter Vector, and Difference Schemes. J. Com.Phys. 43:357～372, 1981
39. Tseng M H. Explicit finite-volume ENO/TVD schemes for 2D transient free-surface flows. Proc. 4th Conference on Computational Fluid Dynamics, Taiwan, R.O.C., 327～334, 1997
40. Hirsch C. Numerical Computation of Internal and External Flows: Vol. 2: Computational Methods for Inviscid and Viscous Flows, John Wiley & Sons, New York, 1988
41. Harten A and Hyman P. Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. Journal of Computational Physics. 50:235～269, 1983
42. Yang H Q and Przekwas A J. A Comparative Study Advanced Shock-Capturing Schemes Applied to Burgers' Equation. Journal of Computational Physics. 102:139～159, 1992
43. Jeng Y N and Payne U J. An Adaptive TVD Limiter. Journal of Computational Physics.118:229～241, 1995
44．于恒，张慧生．非线性双曲型守恒律的两步二阶TVD差分格式．计算力学学报．18(4)：414～418，2001
45. Van Leer B. On the relation between the upwind-differencing schemes of Godunov,Enguist-Osher and Roe. SIAM Journal of Science and Statistical Computations. 5(1): 1～20,1985
46. Alcrudo F and Garcia-Navarro P. A high-resolution Godunov-type scheme in finite-volumes for the 2D shallow-water equations. Int. J. Numer. Methods Fluids. 16:489～505, 1993
47．谭维炎，胡四一，二维浅水明流的一种二阶高性能算法．水科学进展．3(2)：89～95，1992
48. Mingham C G and Causon D M. High-resolution finite-volume method for shallow water flows. J. Hydr. Engineer. 6:605～414, 1998
49. Courant R, Friedrich K O, Lewy H. On the partial differential equation of mathematical physics. IBM Journal. 11(2): 215～234, 1967
50. Toro EF. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag: Berlin, Heidlberg, 1997
51．谭维炎，胡四一，二维浅水流动的一种普适的高性能格式．水科学进展．2(3)：154～161，1991
52．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．水科学进展．5(4)：262～269，1994
53. Zhao D H et al. Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling. J. Hydraulic Engineering, ASCE, 122(12): 692～702, 1996
54. Vincent A and Caltagirone J P. Numerical modeling of bore propagation and run-up on sloping beaches using a MacCormack TVD scheme. J. Hydr. Res. 39(1): 41～49, 2001
55．谭维炎，胡四一，计算浅水动力学的新方向．水科学进展．3(4)：310～318，1992
56. Zhou J G et al. The surface gradient method for the treatment of source terms in the shallow water equations. Journal of Computational Physics. 168: 1～25, 2001
57. Zhou J G et al. Numerical solutions of the shallow water equations with discontinuous bed topography. Int. Numer. Meth. Fluids. 38: 769～788, 2002
58. Hu K, Mingham C G, Causon D M. Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations. Coastal Engineering 41: 443～465,2000
59．韩志平，殷兴良，改进的MacCormack有限体积格式及其应用．系统工程与电子技术．24(7)：46～50，2002
60．汪继文，刘儒勋，间断解问题的有限体积法．计算物理．18(2)：97～103，2001
61. Fraccrollo L, Toro E F. Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems. J. Hydr. Res. 33(6):843～864, 1995
62．王嘉松，倪汉根，金生，二维溃坝问题的高分辨率数值模拟．上海交通大学学报．33(10)：1213～1216，1999
63．谢任之，溃坝水力学．山东：山东科学技术出版社，1993，5
64. Fennema, R J and Chaudhry, M H. Explicit methods for 2D transient free-surface flows. J. Hydr. Engrg.,ASCE, 116(11): 1013～1034, 1990
2. Osher S. Shock Modeling in transonic and Supersonic Flow. Recent Advances in Numerical Methods in Fluids. Vol. 4, W. G. Habashi Ed
3. Yee H C, Warming R F and Harten A. Implicit Total Variation Diminishing (TVD) Schemes for Steaty-state Calculation. AIAA Paper 83～1902, 1983
4. Harten A. High resolution schemes for hyperbolic systems of conservation laws. Journal of Computational Physics. 49:357～393, 1983
5. Harten A, Osher S. Uniformly high order accurate essentially non-oscillatory schemes Ⅰ.SIAM J. on Num. Anal. 24:279～309, 1987
6. Harten A, Osher S, Engquist B, Chakravarthy S R. Uniformly high order accurate essentially non-oscillatory schemes Ⅲ. Journal of Computational Physics. 71:231～303, 1987
7. Van Leer B. Towards the Ultimate Conservative Difference Scheme Ⅴ: A Second Order Sequel to Godunov's Method. Journal of Computational Physics. 32:101～136, 1979
8．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．水动力学研究与进展，15(1)：1～7，2000
9. Wang J S, Ni H Ca, He Y S. Finite-difference TVD scheme for computation of dam-break problems. Journal of Hydraulic Engineering. 126(4):253～262, 2000
10. Aureli F, Mignosa P, Tomirotti M. Numerical simulation and experimental verification of dam-break flows with shocks. Journal of Hydraulic Research. 38(3): 197～206, 2000
11．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．水利水电技术．Vol．30，1999(增刊)
12．王国安．坝体瞬间全溃最大流量通用公式推导．华北水利水电学院学报．2001，22(3)：23～30
13．谢任之．溃坝坝址流量计算．水利水运科学研究．1982(1)
14．汪德爟．计算水力学理论与应用．南京：河海大学出版社，1989
15. Harten A. On the Symmetric form of Systems of Conservation Laws With Entwpy. Journal of Computational Physics. 49:151～164, 1983
16. Harten A, Lax P D and Van Leer B. On Upstream Differencing and Godunov-Type Schemes
for Hyperbolic Conservation Laws. SIAM Rev. 25:35～61, 1983
17．傅德薰．流体力学数值模拟．北京：国防工业出版社，1993
18．刘顺隆，郑群．计算流体力学．哈尔滨工程大学出版社，1998
19. Lax P D and Wendroff B. Difference Schemes for Hyperbolic Equations With High Order of accuracy. Comm. Pure Appl. Math. 17: 381～398, 1964
20．谭维炎，胡四一，钱塘江口涌潮的二维数值模拟．水科学进展．6(2)：83～93，1995
21．谭维炎．计算浅水动力学—有限体积法的应用．北京：清华大学出版社，1998，9
22. Spekreijse, S. P. Multigrid solution of the steady Euler equations. CWI Tract 46, Amsterdam, The Netherlands, 1988
23. Desiseri D and Jameson A. Multigrid solution of the Euler equations on unstructured and adaptive meshes. AIAA Paper 87-0353, 1987
24. Pan D and Cheng J. incompressible flow solution on unstructured triangular meshes. Int. J. Numer. Methods Fluids. 16:1079～1098, 1993
25. Zhao D H et al. Finite-volume two-dimensional unsteady-flow model for fiver basins. ASCE J. Hydraul. Eng. 120:864～883, 1994
26. Anastasiou K and Chan C T. Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes. Inc J. Numer. Methods Fluids.24:1225～1245, 1997
27．朱自强等．应用计算流体力学．北京：北京航空航天大学出版社，1998，8
28．王嘉松等．用TVD显格式模拟一维溃坝洪水波的演进与反射．水利学报，5：7～11，1998
29. Sweby P K. High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Num. Anal. 21:995～1011, 1984
30. Roe P L. Generalized formulation of TVD Lax-Wendroff schemes. ICASE Report No. 84-53,1984
31. Garcia-Navarro P et al. One-dimensional open-channel flow simulation using TVD-MacCormack scheme. Journal of Computational Physics. 118(10), 1992
32. Garcia R, Kahawita R A. Numerical solution of the ST. Venant equations with the MacCormack finite-difference scheme. Int. Num. Meth. Fluids, 6(5):259～274, 1986
33. Fennema R J, Chaudhry M H. Simulation of 1D dam-break flows. J. Hydr. Res. 25(1):41～51,1987
34. Savic L J, Holly F M. Dam-break flood waves computed by modified Godunov method. J. Hydr. Res., Delft, The Netherlands, 31(2): 187～204, 1993
35．汪迎春．溃坝水流二维演进模型．河海大学硕士学位论文．2001，3
36. Garcia P et al. MacCormack's method for the numerical solution of 1D discontinuous unsteady open-channel flow. Journal of Hydraulics Research. 1992, 30(1)
37. Tseng M H. Two-dimensional shallow water flows simulation using TVD-MacCormack Scheme Journal of Hydraulics Research. 38(2): 123～131, 2000
38. Roe P L. Approximate Riemann Solvers. Parameter Vector, and Difference Schemes. J. Com.Phys. 43:357～372, 1981
39. Tseng M H. Explicit finite-volume ENO/TVD schemes for 2D transient free-surface flows. Proc. 4th Conference on Computational Fluid Dynamics, Taiwan, R.O.C., 327～334, 1997
40. Hirsch C. Numerical Computation of Internal and External Flows: Vol. 2: Computational Methods for Inviscid and Viscous Flows, John Wiley & Sons, New York, 1988
41. Harten A and Hyman P. Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. Journal of Computational Physics. 50:235～269, 1983
42. Yang H Q and Przekwas A J. A Comparative Study Advanced Shock-Capturing Schemes Applied to Burgers' Equation. Journal of Computational Physics. 102:139～159, 1992
43. Jeng Y N and Payne U J. An Adaptive TVD Limiter. Journal of Computational Physics.118:229～241, 1995
44．于恒，张慧生．非线性双曲型守恒律的两步二阶TVD差分格式．计算力学学报．18(4)：414～418，2001
45. Van Leer B. On the relation between the upwind-differencing schemes of Godunov,Enguist-Osher and Roe. SIAM Journal of Science and Statistical Computations. 5(1): 1～20,1985
46. Alcrudo F and Garcia-Navarro P. A high-resolution Godunov-type scheme in finite-volumes for the 2D shallow-water equations. Int. J. Numer. Methods Fluids. 16:489～505, 1993
47．谭维炎，胡四一，二维浅水明流的一种二阶高性能算法．水科学进展．3(2)：89～95，1992
48. Mingham C G and Causon D M. High-resolution finite-volume method for shallow water flows. J. Hydr. Engineer. 6:605～414, 1998
49. Courant R, Friedrich K O, Lewy H. On the partial differential equation of mathematical physics. IBM Journal. 11(2): 215～234, 1967
50. Toro EF. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag: Berlin, Heidlberg, 1997
51．谭维炎，胡四一，二维浅水流动的一种普适的高性能格式．水科学进展．2(3)：154～161，1991
52．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．水科学进展．5(4)：262～269，1994
53. Zhao D H et al. Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling. J. Hydraulic Engineering, ASCE, 122(12): 692～702, 1996
54. Vincent A and Caltagirone J P. Numerical modeling of bore propagation and run-up on sloping beaches using a MacCormack TVD scheme. J. Hydr. Res. 39(1): 41～49, 2001
55．谭维炎，胡四一，计算浅水动力学的新方向．水科学进展．3(4)：310～318，1992
56. Zhou J G et al. The surface gradient method for the treatment of source terms in the shallow water equations. Journal of Computational Physics. 168: 1～25, 2001
57. Zhou J G et al. Numerical solutions of the shallow water equations with discontinuous bed topography. Int. Numer. Meth. Fluids. 38: 769～788, 2002
58. Hu K, Mingham C G, Causon D M. Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations. Coastal Engineering 41: 443～465,2000
59．韩志平，殷兴良，改进的MacCormack有限体积格式及其应用．系统工程与电子技术．24(7)：46～50，2002
60．汪继文，刘儒勋，间断解问题的有限体积法．计算物理．18(2)：97～103，2001
61. Fraccrollo L, Toro E F. Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems. J. Hydr. Res. 33(6):843～864, 1995
62．王嘉松，倪汉根，金生，二维溃坝问题的高分辨率数值模拟．上海交通大学学报．33(10)：1213～1216，1999
63．谢任之，溃坝水力学．山东：山东科学技术出版社，1993，5
64. Fennema, R J and Chaudhry, M H. Explicit methods for 2D transient free-surface flows. J. Hydr. Engrg.,ASCE, 116(11): 1013～1034, 1990