航道整治河段流动特性的三维数值模拟
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摘要
丁坝是主要的航道整治建筑物之一。修建丁坝可束水归槽，保护岸堤，但也会带来分离、回流等一系列复杂的流动现象。研究丁坝对水流的影响规律以及丁坝附近水流的流动结构对于航道整治建筑物的设计具有十分重要的意义。本文通过模型实验和数值模拟相结合的研究方法，在整治河段一维水面曲线计算，单丁坝绕流数值模拟，紊流数学模型在航道整治工程中的应用以及丁坝群附近水流的三维流动特性等方面做了一些初步研究。
首先进行了单丁坝水槽实验，采用测压管和旋浆流速仪对实验水槽中水流的水位以及流速场进行了详细的测量。分析了丁坝附近水位在纵横向上的变化规律。实测水位资料表明水流纵横向比降大的区域位于丁坝上下游附近，在丁坝的上游出现壅水，下游存在有收缩区和恢复区，离丁坝越远纵横向水位的变幅越小。
然后根据天然河流横断面流速分布公式，推导了能量方程中动能修正系数α的计算公式。动能修正系数实际上反映了河道横断面流速分布的不均匀性，与断面的水力要素有关，可用谢才系数C来表示，水流所受的阻力越大，边界对其影响也就越大，流速分布愈不均匀，导致动能修正系数增大。
随后结合实测水位资料，通过采用调整局部水头损失系数、扣除回水面积以及壅水公式等不同的水面曲线计算方法对实验水槽一维水面曲线进行计算。计算结果表明，局部水头损失系数并不能完全反映出由于丁坝阻挡所带来的水头损失，而丁坝附近的有效过水面积难以准确地确定，这些导致前两种方法均不能很好地计算出丁坝上游水位的壅高值。而采用壅水公式则能很好地计算出
四川大学硕士学位论文
丁坝上下游的水位差，从而解决工程应用中丁坝上下游水位的衔接问题。
随后采用标准k一e紊流数学模型，依据水槽试验资料，数值模拟了绕坝和
漫坝两种情况下水流的流动过程。实测流速资料和数值模拟结果表明，所选用
的紊流数学模型可很好的模拟出由于丁坝阻隔水流所产生的分离、回流等流动
现象。丁坝绕流和漫流作用机理非常复杂。水流三维流动特性主要集中在丁坝
附近区域.丁坝对下游的影响区域远远大于对上游的影响区域。
最后将标准k一:紊流数学模型应用于实际航道整治工程应用中，引入相应
的边界条件，数值模拟了整治河段水流的三维流动过程。模拟结果表明，整治
河段水流的流速分布主要受制于河道的平面儿何形态，局部区域航道整治建筑
物起主要作用。在河道平面上水流流速基本呈中心流速大，两岸流速小的抛物
线型分布。群坝对水流的作用机理与单丁坝的作用机理不相同。整治河段中第
一根丁坝对水流的阻挡作用最大。受丁坝阻挡，坝田所在的边滩区域流速很小，
泥沙在此淤积:而坝头附近流速较大，通常在此区域形成冲刷，影响丁坝的稳
定性。
Spur dike is a kind of hydraulic structure that is widely used in channel regulation engineering for controlling the shape of the nature river. Some complex flow characteristics such as separation, circumfluence appears around spur dikes. It is of great importance for the design of channel regulation structure to explore the effect of spur dikes on flow, and to study the flow structure near the spur dike. Through flume experiment and numerical simulation, preliminary work on computation of water surface profile in regulation river reach, numerical simulation of flow with single spur dike in flume, and 3-D numerical solution of flow around spur dikes group are carried out in this paper.
The flume experiment with single spur dike is carried out. Water surface profile and velocity field of flow in flume are measured by manometer tube and propeller current meter respectively. Variations of water surface along longitudinal and horizontal direction are analyzed. The experiment data shows that the region with large surface gradients in longitudinal and horizontal direction locates at the vicinity of spur dike. Backwater appear upstream the spur dike, and there exist a recirculation area at the same side downstream the spur dike.
Based on the cross-sectional velocity distribution of flow in natural river, computation expression for correction coefficient of kinetic energy is derived. Actually correction coefficient of kinetic energy reflects the non-uniform characteristics of cross-sectional velocity in natural rivers. It is related with hydraulic parameters and can be expressed by Chezy's coefficient. The larger the resistance of boundary, the more non-uniform the distribution of cross-sectional velocity. The
value of correction coefficient of kinetic energy increases as a result.
The water surface profiles of flume experiments are computed by three different methods: adjusting the local head loss coefficient, deduction of backwater area, and application of backwater expression. Comparison of water level between measured and computation shows that the local head loss coefficient cannot give head loss by spur dike completely, and the effective area of passage is difficult to be ascertain. For this reason the two methods for water surface profile computation cannot predict the raise of water level upstream the spur dike accurately. Only the method for using backwater expression can predict the variation of water surface near the spur dike very well.
Numerical simulations of the flume experiment cases are carried out with the standard k-e turbulence model. Comparison between the numerical results and the measured data shows that it is adequate for the standard k ~ e turbulence model to simulate the complicated flow pattern with the existence of separation and circulation zones. Flow structure near the spur dike is very complicated, and areas with apparent three-dimensional characteristic exist in the vicinity of spur dike.
The standard k-e model is adopted to simulate the flow of practical channel regulation projects. The numerical results show that the velocity distribution of flow in the regulation project river reach are mainly controlled by the geometries of natural river. But in local region it is mainly controlled by the regulating structures. The planar velocity is of the parabola distribution. The velocity value in the middle of the river is larger than that of the two banks. The effects of spur dikes group to the flow are different to that of single spur dike. The first one among spur dikes group in regulation river reach has the most important holdback effect than the others. Velocities in regions between upstream and downstream spur dikes are very small, and are often less than the sediment deposit velocity. Silt will deposit there. Velocities around groin head are larger than the scour velocity and scour hole may develop in this area. This would affect the stability of spur dikes.
首先进行了单丁坝水槽实验，采用测压管和旋浆流速仪对实验水槽中水流的水位以及流速场进行了详细的测量。分析了丁坝附近水位在纵横向上的变化规律。实测水位资料表明水流纵横向比降大的区域位于丁坝上下游附近，在丁坝的上游出现壅水，下游存在有收缩区和恢复区，离丁坝越远纵横向水位的变幅越小。
然后根据天然河流横断面流速分布公式，推导了能量方程中动能修正系数α的计算公式。动能修正系数实际上反映了河道横断面流速分布的不均匀性，与断面的水力要素有关，可用谢才系数C来表示，水流所受的阻力越大，边界对其影响也就越大，流速分布愈不均匀，导致动能修正系数增大。
随后结合实测水位资料，通过采用调整局部水头损失系数、扣除回水面积以及壅水公式等不同的水面曲线计算方法对实验水槽一维水面曲线进行计算。计算结果表明，局部水头损失系数并不能完全反映出由于丁坝阻挡所带来的水头损失，而丁坝附近的有效过水面积难以准确地确定，这些导致前两种方法均不能很好地计算出丁坝上游水位的壅高值。而采用壅水公式则能很好地计算出
四川大学硕士学位论文
丁坝上下游的水位差，从而解决工程应用中丁坝上下游水位的衔接问题。
随后采用标准k一e紊流数学模型，依据水槽试验资料，数值模拟了绕坝和
漫坝两种情况下水流的流动过程。实测流速资料和数值模拟结果表明，所选用
的紊流数学模型可很好的模拟出由于丁坝阻隔水流所产生的分离、回流等流动
现象。丁坝绕流和漫流作用机理非常复杂。水流三维流动特性主要集中在丁坝
附近区域.丁坝对下游的影响区域远远大于对上游的影响区域。
最后将标准k一:紊流数学模型应用于实际航道整治工程应用中，引入相应
的边界条件，数值模拟了整治河段水流的三维流动过程。模拟结果表明，整治
河段水流的流速分布主要受制于河道的平面儿何形态，局部区域航道整治建筑
物起主要作用。在河道平面上水流流速基本呈中心流速大，两岸流速小的抛物
线型分布。群坝对水流的作用机理与单丁坝的作用机理不相同。整治河段中第
一根丁坝对水流的阻挡作用最大。受丁坝阻挡，坝田所在的边滩区域流速很小，
泥沙在此淤积:而坝头附近流速较大，通常在此区域形成冲刷，影响丁坝的稳
定性。
Spur dike is a kind of hydraulic structure that is widely used in channel regulation engineering for controlling the shape of the nature river. Some complex flow characteristics such as separation, circumfluence appears around spur dikes. It is of great importance for the design of channel regulation structure to explore the effect of spur dikes on flow, and to study the flow structure near the spur dike. Through flume experiment and numerical simulation, preliminary work on computation of water surface profile in regulation river reach, numerical simulation of flow with single spur dike in flume, and 3-D numerical solution of flow around spur dikes group are carried out in this paper.
The flume experiment with single spur dike is carried out. Water surface profile and velocity field of flow in flume are measured by manometer tube and propeller current meter respectively. Variations of water surface along longitudinal and horizontal direction are analyzed. The experiment data shows that the region with large surface gradients in longitudinal and horizontal direction locates at the vicinity of spur dike. Backwater appear upstream the spur dike, and there exist a recirculation area at the same side downstream the spur dike.
Based on the cross-sectional velocity distribution of flow in natural river, computation expression for correction coefficient of kinetic energy is derived. Actually correction coefficient of kinetic energy reflects the non-uniform characteristics of cross-sectional velocity in natural rivers. It is related with hydraulic parameters and can be expressed by Chezy's coefficient. The larger the resistance of boundary, the more non-uniform the distribution of cross-sectional velocity. The
value of correction coefficient of kinetic energy increases as a result.
The water surface profiles of flume experiments are computed by three different methods: adjusting the local head loss coefficient, deduction of backwater area, and application of backwater expression. Comparison of water level between measured and computation shows that the local head loss coefficient cannot give head loss by spur dike completely, and the effective area of passage is difficult to be ascertain. For this reason the two methods for water surface profile computation cannot predict the raise of water level upstream the spur dike accurately. Only the method for using backwater expression can predict the variation of water surface near the spur dike very well.
Numerical simulations of the flume experiment cases are carried out with the standard k-e turbulence model. Comparison between the numerical results and the measured data shows that it is adequate for the standard k ~ e turbulence model to simulate the complicated flow pattern with the existence of separation and circulation zones. Flow structure near the spur dike is very complicated, and areas with apparent three-dimensional characteristic exist in the vicinity of spur dike.
The standard k-e model is adopted to simulate the flow of practical channel regulation projects. The numerical results show that the velocity distribution of flow in the regulation project river reach are mainly controlled by the geometries of natural river. But in local region it is mainly controlled by the regulating structures. The planar velocity is of the parabola distribution. The velocity value in the middle of the river is larger than that of the two banks. The effects of spur dikes group to the flow are different to that of single spur dike. The first one among spur dikes group in regulation river reach has the most important holdback effect than the others. Velocities in regions between upstream and downstream spur dikes are very small, and are often less than the sediment deposit velocity. Silt will deposit there. Velocities around groin head are larger than the scour velocity and scour hole may develop in this area. This would affect the stability of spur dikes.
引文
1．杨尊伟等．中国21世纪内河航运论坛．北京：人民交通出版社，2001
2．尹畅安．我国与欧美国家内河航运的比较及思考．水运管理，2002(4)
3．张俊华等．河道整治及堤防管理．河南：黄河出版社，1998
4．许念曾．河道水力学．北京：中国建材工业出版社，1994
5．班久，梁川．日喀则地区的丁坝工程设计．四川水利发电，2000年第19卷增刊
6．刘树坤，李嘉等．中国水力学2000．成都：四川大学出版社，2000
7．余文韬．国外丁坝研究综述．人民长江，1979(3)
8．窦国仁．丁坝回流及其相似律的研究．水利水运科技情报，1978(3)
9．卢汉才．丁坝水力计算一些问题的讨论．山区航道整治论文集，1980
10．王益良等．航道整治建筑物——丁坝试验．天津水运科学研究所，1987
11．张俊兰．护岸工程附近的水流结构．长办水文处，1975
12．程年生，李昌华．有边坡丁坝回流试验研究．水利水运科学研究，1992(4)
13. Rajaratnam, N., et al. Flow Near Groine-Like Structucs. J. of Hydraulic Engineering, ASCE, 1983(109)
14. Francis, J., et al. Observations of Flow Patterns Around Some Simplified Groin Structure in channels. Proc. of the Institution of Civil Engineers, 1968(41)
15．孔祥柏，程年生．丁、潜坝局部水头损失的试验研究．水利水运科学研究，1992(4)
16. Gill, M. A.. Erosion of sand beds around spur dikes. J. H D, ASCE, 1972, 98(9)
17．孔祥柏等．整治建筑物作用下河床演变规律及其对洪水影响的实验研究．南京水利科学研究院，1990．
18．孔祥柏．丁坝绕流试验研究．第二届中日河工坝工技术讨论文集，1985
19．旺德胜．漫水丁坝局部冲刷的研究．水动力学研究与进展，1988，3(2)
20．张家善．群坝局部冲刷有关问题实验研究．合肥工业大学学报，1998，21(3)
21. Tingsanchali, T., et al. 2-D Depth-Averaged Flow Computation Near Groin. J. of Hydraulic Engineering, ASCE, 1990,116(1)
22．陆永军，周耀庭．勾透丁坝下游流场初探．水道港口，1989(1)
23．陆永军．透水丁坝下游流场初探．河海大学学报，1987，15(2)
24．张小梅，陈志轩．丁坝下游流场初步研究．水科学青年论文集(一)，1990
25．陆永军，周耀庭．丁坝下游恢复区流场初探．水动力学研究与进展，1989，5(2)
26．陆永军，周耀庭．指导建筑物绕流机理及其下游流场的探讨．水动力学研究与进展，1999，5(2)
27．陆永军，周耀庭．Flow Mechanism and Velocity Field Near Groin-Like Structure.J.of Hydraulic Engineering,ASCE,1983(19)
28．林秀维，陈阳．Prandtl混合长紊流模型模拟丁坝绕流．水道港口，1998(2)
29．程年生，李昌华．丁坝绕流的k～ε紊流数值解．水利水运科学研究，1989(3)
30．徐成伟，陆永军．丁坝绕流的数学模型．水道港口，1989(3)
31．贾云峰等．用水深平均k～ε紊流模型计算淹没丁坝流场．水利水运科学研究，1993(2)
32．王万群．丁坝绕流问题的紊流数学模型．交通部上海船舶运输科学研究所学报，1991，14(2)
33．蒋昌波等．丁坝绕流的二维大涡数值模拟．长沙交通学院学报，1999，15(3)
34．吴持恭．水力学(上、下册)．北京：高等教育出版社，1983
35．格拉夫，阿廷拉卡等．河川水力学．成都：成都科技大学出版社，1997
36．李然．气液两相流理论在明渠水气界面计算中的应用．水动力学研究与进展，2002，17(1)
37．韩标，何枫等．工程中液面波动的数值模拟．水动力学研究与进展，1999，14(2)
38．孙大鹏，李玉成．三维数值波动水槽波浪变形计算的0-1混合型边界元法．水动力学研究与进展．1999，14(4)
39．H．史里希廷．边界层理论．北京：科学出版社，1991
40．梁在潮．工程湍流．武汉：华中理工大学出版社，1999
41．林建忠．湍流的拟序结构．机械出版社，1995
42. Markatos, N.C.. The Mathematical in Turbulence Modeling of Turbulent Flows. Appl. Math. Modeling, Vol.10, June 1986
43. Rodi, W.. Recent Developments in Turbulence Modeling, Proc.3rd Int. Syp. On Refined Flow Modeling and Turbulence Measurements, Tokyo, July, 26-28, 1998
44. Handler R.A., Hendricks E.W. and Leighton R.L.. Low Reynolds number calculation of turbulent channel flow: a general discuss NRL Memorandum Report 6410. Naval Research Lab., Washington D.C., 1989
45．是勋刚．湍流直接数值模拟的进展前景．水动力学研究与进展，1992(1)
46．余常昭．环境流体力学导论．北京：清华大学出版社，1998
47. Chou, P.Y.. On the velocity correlations and the solution of the equation turbulent fluctuation. Quart. J., Appl. Math. 3.1, 1945
48. Dabidov, B.I.. On the statistical dynamics of an incompressible turbulent fluid. Dokl. ANSSSR, 1961, 49:127
49. Harlow, F.H., and Nakayama P.I.. Transport of turbulence energy decay rate. Los Alamos Sci. lab. Univ. California Rep. LA-3854,1968
50. Jones, W.P., and Launder B.E.. The prediction of Laminarisation with a Z-equation model of turbulence. Int. J. Heat mass Transfer, 1972,15
51. Spalding, D.B.. The prediction of two-dimensional steady turbulent flows. Imperial College, Heat Transfer Section Rep. EF/TN/A/16,1969
52. Kolavadin, B.A., and Vatutin, I.A.. On the statistical theory of non-uniform turbulence. Int. Seminar on Heat and Mass transfer, Herceg Novi Yugoslavia, 1969
53. Rotta, J.. Statistical theories nichthomogener turbulent. Zeitsch, fur Phydik, 1951,129
54. Launder, B. E., et al. Progress in the development of a Reynolds stress turbulence closure. J. Fluid Mech., 1975,68
55. Reecc, G.J.. A generalized Reynolds-Stress model of turbulence. PH.D. Thesis Univ. of London, 1977
56. Wolfgang Rodi. Turbulent models and their applications in hydraulics IAHR Publication. Delft, The Netherlands, June 1984
57．金忠青．N-S方程的数值解和紊流模型．南京：河海大学出版社，1989
58．钱壬章等．传热分析与计算．北京：高等教育出版社，1987
59. Chen, C.J., and chen, H.C.. Development of finite analytic method for Unsteady Two dimensional convection transport equation. J. of Computational Physics, 1983
60．李义天等．河道平面二维水沙数学模型．北京：中国水利水电出版社，2002
61．陶君安等．求解非定常对流扩散方程的混合有限分析六点格式．武汉水电学院学报，有限分析法专辑，1990
62．何子干，倪汉根．三维N-S方程的分层计算．第四届全国计算流体力学学术会议论文集，上海复旦大学，1988
63. Patankar, S.V.. Numerical Transfer and Fluid Flow. Hemisphere Publishing Corporation and
McGraw Hill Book Company, 1980
64. Splading, D.B.. Mathematical Modeling of fluid-Mechanics Heat Transfer and Chemical Reaction Process. Imperial College, London, HTS/80 11 1980
65. Patanker, S.V., and Splading, D.B.. A calculation Proceeding for Heat Mass Momentum Transfer in Three-Dimensional Parabilic Flows. Inter. J. Heat and Mass transfer, Vol. 15 1972
66. Shih, T. M., and Ren, A. L.. Primitive-Variable Formulations Using Nonstaggered Grids. Numer. Heat Transfer, vol. 7 1984
67. Rhie, C. M., and Chow, W. L.. A Numerical Study of Turbulent Flow Past an Isolated Airfoil with Trailing Edge Separation. AIAA J., vol. 21, 1983
68. Majumdar, S.. Role of Underrelaxation on Momentum Interpolation for Calculation of Flow with Nonstaggered Grids. Number. Heat Transfer, vol. 13,1988
69．陶文铨．数值传热学．西安：西安交通大学出版社，1988
2．尹畅安．我国与欧美国家内河航运的比较及思考．水运管理，2002(4)
3．张俊华等．河道整治及堤防管理．河南：黄河出版社，1998
4．许念曾．河道水力学．北京：中国建材工业出版社，1994
5．班久，梁川．日喀则地区的丁坝工程设计．四川水利发电，2000年第19卷增刊
6．刘树坤，李嘉等．中国水力学2000．成都：四川大学出版社，2000
7．余文韬．国外丁坝研究综述．人民长江，1979(3)
8．窦国仁．丁坝回流及其相似律的研究．水利水运科技情报，1978(3)
9．卢汉才．丁坝水力计算一些问题的讨论．山区航道整治论文集，1980
10．王益良等．航道整治建筑物——丁坝试验．天津水运科学研究所，1987
11．张俊兰．护岸工程附近的水流结构．长办水文处，1975
12．程年生，李昌华．有边坡丁坝回流试验研究．水利水运科学研究，1992(4)
13. Rajaratnam, N., et al. Flow Near Groine-Like Structucs. J. of Hydraulic Engineering, ASCE, 1983(109)
14. Francis, J., et al. Observations of Flow Patterns Around Some Simplified Groin Structure in channels. Proc. of the Institution of Civil Engineers, 1968(41)
15．孔祥柏，程年生．丁、潜坝局部水头损失的试验研究．水利水运科学研究，1992(4)
16. Gill, M. A.. Erosion of sand beds around spur dikes. J. H D, ASCE, 1972, 98(9)
17．孔祥柏等．整治建筑物作用下河床演变规律及其对洪水影响的实验研究．南京水利科学研究院，1990．
18．孔祥柏．丁坝绕流试验研究．第二届中日河工坝工技术讨论文集，1985
19．旺德胜．漫水丁坝局部冲刷的研究．水动力学研究与进展，1988，3(2)
20．张家善．群坝局部冲刷有关问题实验研究．合肥工业大学学报，1998，21(3)
21. Tingsanchali, T., et al. 2-D Depth-Averaged Flow Computation Near Groin. J. of Hydraulic Engineering, ASCE, 1990,116(1)
22．陆永军，周耀庭．勾透丁坝下游流场初探．水道港口，1989(1)
23．陆永军．透水丁坝下游流场初探．河海大学学报，1987，15(2)
24．张小梅，陈志轩．丁坝下游流场初步研究．水科学青年论文集(一)，1990
25．陆永军，周耀庭．丁坝下游恢复区流场初探．水动力学研究与进展，1989，5(2)
26．陆永军，周耀庭．指导建筑物绕流机理及其下游流场的探讨．水动力学研究与进展，1999，5(2)
27．陆永军，周耀庭．Flow Mechanism and Velocity Field Near Groin-Like Structure.J.of Hydraulic Engineering,ASCE,1983(19)
28．林秀维，陈阳．Prandtl混合长紊流模型模拟丁坝绕流．水道港口，1998(2)
29．程年生，李昌华．丁坝绕流的k～ε紊流数值解．水利水运科学研究，1989(3)
30．徐成伟，陆永军．丁坝绕流的数学模型．水道港口，1989(3)
31．贾云峰等．用水深平均k～ε紊流模型计算淹没丁坝流场．水利水运科学研究，1993(2)
32．王万群．丁坝绕流问题的紊流数学模型．交通部上海船舶运输科学研究所学报，1991，14(2)
33．蒋昌波等．丁坝绕流的二维大涡数值模拟．长沙交通学院学报，1999，15(3)
34．吴持恭．水力学(上、下册)．北京：高等教育出版社，1983
35．格拉夫，阿廷拉卡等．河川水力学．成都：成都科技大学出版社，1997
36．李然．气液两相流理论在明渠水气界面计算中的应用．水动力学研究与进展，2002，17(1)
37．韩标，何枫等．工程中液面波动的数值模拟．水动力学研究与进展，1999，14(2)
38．孙大鹏，李玉成．三维数值波动水槽波浪变形计算的0-1混合型边界元法．水动力学研究与进展．1999，14(4)
39．H．史里希廷．边界层理论．北京：科学出版社，1991
40．梁在潮．工程湍流．武汉：华中理工大学出版社，1999
41．林建忠．湍流的拟序结构．机械出版社，1995
42. Markatos, N.C.. The Mathematical in Turbulence Modeling of Turbulent Flows. Appl. Math. Modeling, Vol.10, June 1986
43. Rodi, W.. Recent Developments in Turbulence Modeling, Proc.3rd Int. Syp. On Refined Flow Modeling and Turbulence Measurements, Tokyo, July, 26-28, 1998
44. Handler R.A., Hendricks E.W. and Leighton R.L.. Low Reynolds number calculation of turbulent channel flow: a general discuss NRL Memorandum Report 6410. Naval Research Lab., Washington D.C., 1989
45．是勋刚．湍流直接数值模拟的进展前景．水动力学研究与进展，1992(1)
46．余常昭．环境流体力学导论．北京：清华大学出版社，1998
47. Chou, P.Y.. On the velocity correlations and the solution of the equation turbulent fluctuation. Quart. J., Appl. Math. 3.1, 1945
48. Dabidov, B.I.. On the statistical dynamics of an incompressible turbulent fluid. Dokl. ANSSSR, 1961, 49:127
49. Harlow, F.H., and Nakayama P.I.. Transport of turbulence energy decay rate. Los Alamos Sci. lab. Univ. California Rep. LA-3854,1968
50. Jones, W.P., and Launder B.E.. The prediction of Laminarisation with a Z-equation model of turbulence. Int. J. Heat mass Transfer, 1972,15
51. Spalding, D.B.. The prediction of two-dimensional steady turbulent flows. Imperial College, Heat Transfer Section Rep. EF/TN/A/16,1969
52. Kolavadin, B.A., and Vatutin, I.A.. On the statistical theory of non-uniform turbulence. Int. Seminar on Heat and Mass transfer, Herceg Novi Yugoslavia, 1969
53. Rotta, J.. Statistical theories nichthomogener turbulent. Zeitsch, fur Phydik, 1951,129
54. Launder, B. E., et al. Progress in the development of a Reynolds stress turbulence closure. J. Fluid Mech., 1975,68
55. Reecc, G.J.. A generalized Reynolds-Stress model of turbulence. PH.D. Thesis Univ. of London, 1977
56. Wolfgang Rodi. Turbulent models and their applications in hydraulics IAHR Publication. Delft, The Netherlands, June 1984
57．金忠青．N-S方程的数值解和紊流模型．南京：河海大学出版社，1989
58．钱壬章等．传热分析与计算．北京：高等教育出版社，1987
59. Chen, C.J., and chen, H.C.. Development of finite analytic method for Unsteady Two dimensional convection transport equation. J. of Computational Physics, 1983
60．李义天等．河道平面二维水沙数学模型．北京：中国水利水电出版社，2002
61．陶君安等．求解非定常对流扩散方程的混合有限分析六点格式．武汉水电学院学报，有限分析法专辑，1990
62．何子干，倪汉根．三维N-S方程的分层计算．第四届全国计算流体力学学术会议论文集，上海复旦大学，1988
63. Patankar, S.V.. Numerical Transfer and Fluid Flow. Hemisphere Publishing Corporation and
McGraw Hill Book Company, 1980
64. Splading, D.B.. Mathematical Modeling of fluid-Mechanics Heat Transfer and Chemical Reaction Process. Imperial College, London, HTS/80 11 1980
65. Patanker, S.V., and Splading, D.B.. A calculation Proceeding for Heat Mass Momentum Transfer in Three-Dimensional Parabilic Flows. Inter. J. Heat and Mass transfer, Vol. 15 1972
66. Shih, T. M., and Ren, A. L.. Primitive-Variable Formulations Using Nonstaggered Grids. Numer. Heat Transfer, vol. 7 1984
67. Rhie, C. M., and Chow, W. L.. A Numerical Study of Turbulent Flow Past an Isolated Airfoil with Trailing Edge Separation. AIAA J., vol. 21, 1983
68. Majumdar, S.. Role of Underrelaxation on Momentum Interpolation for Calculation of Flow with Nonstaggered Grids. Number. Heat Transfer, vol. 13,1988
69．陶文铨．数值传热学．西安：西安交通大学出版社，1988