水工自由水面及掺气问题的数值模拟研究
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摘要
自由液面问题在工程实践中大量存在，对自由液面问题的研究具有十分重要的工程价值和学术意义，数值模拟是研究和解决实际工程问题主要的方法之一，由于自由液面的特殊性质，对自由液面的处理在数值模拟中显得尤为困难和重要。本文主要对水利工程中一些涉及自由水面及掺气的流动的数值模拟问题进行研究。
首先回顾了在数值模拟中自由液面问题的处理方法和存在的问题，并对数值计算方法进行了简要的综述，根据自由液面的形状，将自由液面分为相对稳定的自由液面和起伏较大的自由液面。对含有相对稳定的和起伏较大的自由水面的水流问题进行了数值模拟。给出了流动的控制方程，利用刚盖假定法对相对稳定的自由水面进行了简化近似处理，用弹性盖法对起伏较大的自由水面进行了简化近似处理，对控制方程进行了封闭、离散和数值求解。以小孤山水电站枢纽冲砂闸闸门上下游流场为实际算例，对计算结果进行了分析。从计算结果来看，对相对稳定的自由液面采用刚盖假定法处理是适合的，计算得到的速度场、压力场与实际比较吻合，根据数值模拟的结果，对闸门胸墙的体型进行了优化；但对闸后起伏较大的自由水面问题采用弹性盖法处理则存在不满意的地方。
对含有自由水面的水气两相流，采用两流体模型，简要给出了水气两相流的控制方程，并考虑相间作用力的影响，在同位网格下，对方程进行离散，并结合明渠掺气水流进行了数值模拟，对明渠水流的自由水面采用改进的刚盖假定法处理并获
西安理工大学硕士学位论文
得了较好的结果。根据计算结果，对水气两相流中气泡尺寸对掺气浓度预测结果的
影响进行了分析，得到沿程气泡不断上逸，气泡粒径越大，其上逸速度越大，则留
在渠底附近的气泡越少的气泡运动规律，提出了掺气槽的设计，除满足掺气量外还
应满足掺气粒径要求的工程建议。
用VOF法对自由水面问题进行了初步研究。给出了VOF法的控制方程，利用VOF
法对明渠水流、闸后水流进行了数值模拟，计算得到了明渠水流和闸后水流水气界
面的形状、含气率的分布，流场的速度场、压力场、紊动能及紊动耗散率分布场，
计算结果显示了VOF法对无论是相对稳定的自由水面还是起伏较大的自由水面问题
的处理，都显现了其优势。
通过本文对数值模拟中的自由液面问题的研究表明，对闸前相对稳定的自由液
面问题可以采用相对简单的方法进行处理，但对于闸后起伏较大的自由液面，则需
要采用更精细的自由液面处理方法。
Many free fluid surface problems exist in engineering practice, so there is very importance engineering value and academic meaning to make scientific researches on free fluid surface. Numerical simulation is one of the primary method to study and solve engineering problems. Because of special quality of free surface, the disposal of free surface become more difficult and important in numerical simulation. Numerical simulation problems about free water surface and air entrained flows in hydraulic engineering are studied in this paper.
First, the solving methods and exiting problems of free surface were reviewed and the numerical simulation methods were also brief summarized. Based on the shapes of free surface, the free surface is divided into a relative stable one and a rather fluctuant one. Water current problems including relative steady and rather fluctuant free surface were simulating numerically. Flow control equations were given, relative steady and rather fluctuant free surface were treated approximately with using rigidity cover assume, at the same time, control equations were closed and discretized and numerically solved. Flow field of upstream and downstream of Xiao gushan
water power station hinge discharge sluice were practical calculated examples. The results were analyzed and showed that rigidity cover assume was fitted to relative steady free surface flow upstream sluice. The velocity field and pressure field from the calculation accord with the fact. Based the results of numerical simulation, the bodily form of sluice parapet is optimized. But the elasticity assume existed dissatisfactory aspects to rather fluctuant free surface flow downstream sluice.
To air-water two-phase flow including free surface, the control equations of two-fluid model were briefly listed, and the effect of the interfacial momentum transfer term was considered, then the equations were discretized on collocated grid. The open-channel aerated water flow was numerically simulated and the free surface was treated with improved rigidity cover assume. Based on simulation results which were satisfied, the effect of bubble diameter on entrainment concentration forcast result was analyzed. The bubble moving rule was found which includs the bubble is escaping upwards along the way, the diameter of the bubble is bigger, the velocity upwards is larger and the bladder remaining near channel bottom is smaller. To make best use of the function of the aerator, the demands of enough air entrainment quantity and suitable air bubble diameter are needed to be satisfied.
The problems of free surface were dealed with by VOF which is more advanced method. The governing equations were listed and the open-channel flow and sluice upstream flow are simulated by using the method of VOF. From the simulation, the shape of air-water surface, the distributing of gas ratio, the velocity field, pressure field, turbulence momentum and turbulence moment dissipation ratio were gained. The results showed that the advantage was displayed by using method of VOF not only for relative steady water surface but for rather fluctuant one.
The study on free surface showed the relative simple methods can be used for relative steady free water surface flows upstream the sluice, but more advanced methods must be adopted to rather fluctuant one downstream the sluice.
首先回顾了在数值模拟中自由液面问题的处理方法和存在的问题，并对数值计算方法进行了简要的综述，根据自由液面的形状，将自由液面分为相对稳定的自由液面和起伏较大的自由液面。对含有相对稳定的和起伏较大的自由水面的水流问题进行了数值模拟。给出了流动的控制方程，利用刚盖假定法对相对稳定的自由水面进行了简化近似处理，用弹性盖法对起伏较大的自由水面进行了简化近似处理，对控制方程进行了封闭、离散和数值求解。以小孤山水电站枢纽冲砂闸闸门上下游流场为实际算例，对计算结果进行了分析。从计算结果来看，对相对稳定的自由液面采用刚盖假定法处理是适合的，计算得到的速度场、压力场与实际比较吻合，根据数值模拟的结果，对闸门胸墙的体型进行了优化；但对闸后起伏较大的自由水面问题采用弹性盖法处理则存在不满意的地方。
对含有自由水面的水气两相流，采用两流体模型，简要给出了水气两相流的控制方程，并考虑相间作用力的影响，在同位网格下，对方程进行离散，并结合明渠掺气水流进行了数值模拟，对明渠水流的自由水面采用改进的刚盖假定法处理并获
西安理工大学硕士学位论文
得了较好的结果。根据计算结果，对水气两相流中气泡尺寸对掺气浓度预测结果的
影响进行了分析，得到沿程气泡不断上逸，气泡粒径越大，其上逸速度越大，则留
在渠底附近的气泡越少的气泡运动规律，提出了掺气槽的设计，除满足掺气量外还
应满足掺气粒径要求的工程建议。
用VOF法对自由水面问题进行了初步研究。给出了VOF法的控制方程，利用VOF
法对明渠水流、闸后水流进行了数值模拟，计算得到了明渠水流和闸后水流水气界
面的形状、含气率的分布，流场的速度场、压力场、紊动能及紊动耗散率分布场，
计算结果显示了VOF法对无论是相对稳定的自由水面还是起伏较大的自由水面问题
的处理，都显现了其优势。
通过本文对数值模拟中的自由液面问题的研究表明，对闸前相对稳定的自由液
面问题可以采用相对简单的方法进行处理，但对于闸后起伏较大的自由液面，则需
要采用更精细的自由液面处理方法。
Many free fluid surface problems exist in engineering practice, so there is very importance engineering value and academic meaning to make scientific researches on free fluid surface. Numerical simulation is one of the primary method to study and solve engineering problems. Because of special quality of free surface, the disposal of free surface become more difficult and important in numerical simulation. Numerical simulation problems about free water surface and air entrained flows in hydraulic engineering are studied in this paper.
First, the solving methods and exiting problems of free surface were reviewed and the numerical simulation methods were also brief summarized. Based on the shapes of free surface, the free surface is divided into a relative stable one and a rather fluctuant one. Water current problems including relative steady and rather fluctuant free surface were simulating numerically. Flow control equations were given, relative steady and rather fluctuant free surface were treated approximately with using rigidity cover assume, at the same time, control equations were closed and discretized and numerically solved. Flow field of upstream and downstream of Xiao gushan
water power station hinge discharge sluice were practical calculated examples. The results were analyzed and showed that rigidity cover assume was fitted to relative steady free surface flow upstream sluice. The velocity field and pressure field from the calculation accord with the fact. Based the results of numerical simulation, the bodily form of sluice parapet is optimized. But the elasticity assume existed dissatisfactory aspects to rather fluctuant free surface flow downstream sluice.
To air-water two-phase flow including free surface, the control equations of two-fluid model were briefly listed, and the effect of the interfacial momentum transfer term was considered, then the equations were discretized on collocated grid. The open-channel aerated water flow was numerically simulated and the free surface was treated with improved rigidity cover assume. Based on simulation results which were satisfied, the effect of bubble diameter on entrainment concentration forcast result was analyzed. The bubble moving rule was found which includs the bubble is escaping upwards along the way, the diameter of the bubble is bigger, the velocity upwards is larger and the bladder remaining near channel bottom is smaller. To make best use of the function of the aerator, the demands of enough air entrainment quantity and suitable air bubble diameter are needed to be satisfied.
The problems of free surface were dealed with by VOF which is more advanced method. The governing equations were listed and the open-channel flow and sluice upstream flow are simulated by using the method of VOF. From the simulation, the shape of air-water surface, the distributing of gas ratio, the velocity field, pressure field, turbulence momentum and turbulence moment dissipation ratio were gained. The results showed that the advantage was displayed by using method of VOF not only for relative steady water surface but for rather fluctuant one.
The study on free surface showed the relative simple methods can be used for relative steady free water surface flows upstream the sluice, but more advanced methods must be adopted to rather fluctuant one downstream the sluice.
引文
【1】王建军，陆明万，张雄，李其汉，朱梓根．自由液面流体大晃动有限元方法．水动力学研究与进展，16(3)，2001：390—395．
【2】孙江龙，叶恒奎．ALE有限元法解二维自由面流体大晃动问题。华中科技大学学报(自然科学版)，30(11)，2002：80—82．
【3】李廷秋，缪国平，刘应中．三维船体自由面贴体网格的数值生成．上海交通大学学报，28(4)，1994：5—11．
【4】ZHANG Zhi-rong, ZHAO Feng, Li Bai-qi. Numerical Calculation of Viscous Free-Surface Flow About Ship Hull. Journal of Ship Mechanics, 6(6), 2002:10—17.
【5】CHANG Huai-xin, LIU Ying-zhong, Mao Guo-ping. Numerical Simulation of Three-dimensional Viscous Flow with Free Surface about a Ship. Journal of Ship Mechanics, 7(3), 2003:33—39.
【6】许唯临，杨永全，吴持恭．具有起伏表面的紊流数值模拟．水力学报，(10)，1990：16—22．
【7】牛争鸣，孙静，张鸣远．水平旋转内消能泄水道环流空腔直径的变化规律．水利学报，(2)，2003：31—37．
【8】孙静，牛争鸣，章晋雄．竖井进流水平旋转内消能泄水道流速分布的试验研究．18(2)，2003：240—247．
【9】牛争鸣，孙静，张明远．竖井进流水平旋转内消能泄水道的壁面压力分布规律．水力发电学报，(4)，2002：71—80．
【10】金忠青．N-S方程的数值解与紊流模型．河海大学出版社，1989．
【11】Hirt.C W.and Nichols B.D.Volume of Fluid(VOF)Method for the Dynamics of Free Boundary.J.Comp.Phys,1981.
【12】袁新明，毛根海，吴寿荣，唐锦春．闸后水跃水流的数值模拟．水力学报，(5)，1999：44—48．
【13】杨小亭．二维溃坝水波MAC方法数值模拟．武汉水利电力大学学报，
30(2)，1997：54—58．
【14】陈群，戴光清．鱼背山水库岸边阶梯溢洪道流场的三维数值模拟．水力发电学报，(3)，2002：62—72．
【15】朱仁庆，杨松林，王志东．闸门开启过程中水体流动的数值模拟．华东船舶工业学院学报，12(3)，1998：18—21．
【16】陈黎，刁明军．表孔泄洪的水气二相流数值模拟．四川水力发电，21(1)，2002：91—93．
【17】何枫，叶宏开．水池中自由液面波动的数值模拟 (二维)．山东烟台，第三界全国环境流体力学会议，1994．
【18】韩标，何枫，叶宏开，王学芳．工程中液面波动的数值模拟．水动力学研究与进展，14(2)，1999：142—147．
【19】李治勤，陈刚，李建中．用紊流模型通过座标变换预测过坝水流流场．水力学报，(8)，1996：25—32．
【20】谭立新，许唯临，杨永全．明渠水气两相流数值模拟．四川联合大学学报 (工程科学版)，3(1)，1999：93—97．
【21】谭立新，许唯临，杨永全，李建中，魏文礼．明渠气水二相流的双流体模型．应用力学学报，18(4)，2001：28—32．
【22】许唯临，王韦，谭立新，杨永全．水工水气两相流的数值模拟．水动力学研究与进展，16(2)，2001：225—229．
【23】张宏伟．掺气水流双流体模型数值模拟研究．西安理工大学硕士学位论文．2002．4
【24】李国栋，陈刚，李建中．明流泄洪洞流场数值模拟．水动力学研究与进展，11(6)，1996：633—639．
【25】许联锋，李建中，陈刚．泄水建筑物体型优选的数值模拟研究．水动力学研究与进展，17(6)，2002：701—708。
【26】廖华胜，汝树勋，吴持恭．阶梯溢流坝流场的数值模拟．成都科技大学学报，(5)，1995：27—33．
【27】许联锋，陈刚，李建中，邵建斌．气液两相流中气泡运动速度场的PIV分析与研究．试验力学，17(4)，2002：458—463．
【28】许联锋，陈刚，李建中，邵建斌．粒子图像测速技术研究进展．力学进展，33(4)，2003：533—540．
【29】陶文铨．数值传热学(第2版)．西安交通大学出版社，2001．
【30】ShihTH, Liou W W, Shabbir A,Yang Z G, Zhu J. A new k-ε eddy viscosity model for high Reynolds number turbulent flows. Comput Fluids, 24(3), 1995:227—238.
【31】陶文铨．计算传热学的近代进展．科学出版社，2000。
【32】吴持恭主编．水力学(上册)．北京：高等教育出版社，1982：73—142．
【33】张绍芳．堰闸水力设计．北京：水利电力出版社，1987：214—260．
【34】李建中．水力学．陕西：陕西科学技术出版社，2002：169—183．
【35】刘大有．二相流体动力学．高教出版社，1993：534—549．
【36】Drew D A, Segel.L.A. Averaged equations for two-phase flow. Studies in Applied Math,50(3),1971.
【37】Delhaye J M. Space/time and time/space-averaged equations. Two-phase flow and heat transfer, vol.1, Hemisphere, Washington,D.C.,1977.
【38】D.A.Drew and Lahey, R T, Jr. In particulate two-phase flow,1990:509—566, Butterworth-Heinemann, Boston.
【39】M.Ishii. Yhermo-fluid dynamic theory of two-phase flow. Eyrolles Paris, 1975.
【40】H.Enwald, E.Peirano, A.E.Almstedt. Eulerian two-phase flow theory applied to fouidization. Int.J.Multiphase Flow,vol:22,1996:21-66.
【41】Lance M, Lopez de B M. Phase distribution phenoma and effects in bubbly two-phase flow.Multiphase Science and Technology, vol:8.
【42】Ishii M, Mishima K.Two-fluid modeling and hydrodynamic constitutive relation.Nucl.Eng.Des.vol:82, 1984:107-126.
[43] M.Ishii. Two-fluid model for two-phase flow.Science and technology of multi-phase,vol:5, 1990.
[44] L.Schiller and A.Nauman. Vdi Zeits. 1933:318.
[45] M.Ishii and N.Zuber. AICHEJ.Vol:25,Iss:5,1979:843.
[46] F.Ihme, H.Schmidt-Traub, H.Brauer. Chemic-Ing-Tech, Vol:44,Iss:4,1972:306.
[47] P.G.Suffman, The lift on a small sphere in a slow shear flow, J.Fluid Mech,22(3),1965.
[48] D.A.Drew and Lahey, R T, Jr. (1990) .Some suppleymental analysis concerning the virtual mass and lift force on a sphere in a rotating and straining flow. Int.J.Multiphase Flow, 16(6).
[49] Rhie C M and Chow W L. A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation. AIAAJ,vol:21,1983:1525—1532.
[50] Peric M. Kessler R and Scheuerer G.. Comparison of finite-volume numerical method with staggered and collocated grids.Computers and fluid, 16(4),1988:389-40.
[51] Majumdar S. Role of underrelaxation in a partially divided enclosure.Int.J.Heat Mass Transfer.vol:10,1988:635-640.
[52] Miller T F and Schmidt F W. Use of a pressure-weighted interpolation method for solution of the incompressible navier-stokes equations on a nonstaggered grid system. Numerical Heat Transfer,vol:14,1988:213-233.
[53] Rodi W Majumdar S and Schonung B. Finite-volum methods for two-dimensional incompressible flows with complex boundaries. Compute.Meth.Appl.Meth.eng.vol:14,1989:213-233.
[54] Thiart G.D. Finite-difference scheme for the numerical solution of fluid flow and heat transfer problems on nonstaggered grid. Numerical Heat
Transfer.vol: 17B, 1990:43-62.
[55] Thiart G.D. Improved finite-difference scheme for the solution of convective-diffusion problem with the SIMPLEN algorithm. Numerical Heat Transfer.vol:18B,1990:81-95.
[56] Date A.W.C. Solution of incompressible navier-stokes equations on a nonstaggered grid. Int.J.Heat Mass Transfer.vol:36,1993:1913-1922.
[57] Date A.W.C. Complete pressure correction algorithm for solution of incompressible navier-stokes equations on a nonstaggered grid. Int.J.Heat Mass Transfer.vol:29B, 1996:441-458.
[58] 杨建明，吴建华．淹没水跃的数值模拟．水动力学研究与进展，16(4)，2001：405—411．
[59] A. R. Zarrati, Mathematical modeling of air-water mixtures in open channels. J Hydraulic Research, Vo1.32, 1994, No. 5.
[60] 张宏伟，谭立新，陈刚，李建中，杨永全，等．水气二相流双流体模型研究现状．陕西水力发电．第17卷，第四期，2001．12．
[61] 滕爱国，沈永明，郑永红，邱大洪．波浪变形数值模拟．大连理工大学学报，41(3)，2001：359—362．
[62] 邓军，许唯临，曲景学，董建伟．明流泄洪隧洞中自掺气水流浓度分布及其计算．水利学报，2002：64—68．
[63] 中国电力企业联合会标准化部编．电力工业标准汇编(水电卷)[S]．北京：水利电力出版社，1995，3：285-286．
[64] 李建中，宁利中．高速水力学．陕西：西北工业大学出版社，1994：93—155．
【2】孙江龙，叶恒奎．ALE有限元法解二维自由面流体大晃动问题。华中科技大学学报(自然科学版)，30(11)，2002：80—82．
【3】李廷秋，缪国平，刘应中．三维船体自由面贴体网格的数值生成．上海交通大学学报，28(4)，1994：5—11．
【4】ZHANG Zhi-rong, ZHAO Feng, Li Bai-qi. Numerical Calculation of Viscous Free-Surface Flow About Ship Hull. Journal of Ship Mechanics, 6(6), 2002:10—17.
【5】CHANG Huai-xin, LIU Ying-zhong, Mao Guo-ping. Numerical Simulation of Three-dimensional Viscous Flow with Free Surface about a Ship. Journal of Ship Mechanics, 7(3), 2003:33—39.
【6】许唯临，杨永全，吴持恭．具有起伏表面的紊流数值模拟．水力学报，(10)，1990：16—22．
【7】牛争鸣，孙静，张鸣远．水平旋转内消能泄水道环流空腔直径的变化规律．水利学报，(2)，2003：31—37．
【8】孙静，牛争鸣，章晋雄．竖井进流水平旋转内消能泄水道流速分布的试验研究．18(2)，2003：240—247．
【9】牛争鸣，孙静，张明远．竖井进流水平旋转内消能泄水道的壁面压力分布规律．水力发电学报，(4)，2002：71—80．
【10】金忠青．N-S方程的数值解与紊流模型．河海大学出版社，1989．
【11】Hirt.C W.and Nichols B.D.Volume of Fluid(VOF)Method for the Dynamics of Free Boundary.J.Comp.Phys,1981.
【12】袁新明，毛根海，吴寿荣，唐锦春．闸后水跃水流的数值模拟．水力学报，(5)，1999：44—48．
【13】杨小亭．二维溃坝水波MAC方法数值模拟．武汉水利电力大学学报，
30(2)，1997：54—58．
【14】陈群，戴光清．鱼背山水库岸边阶梯溢洪道流场的三维数值模拟．水力发电学报，(3)，2002：62—72．
【15】朱仁庆，杨松林，王志东．闸门开启过程中水体流动的数值模拟．华东船舶工业学院学报，12(3)，1998：18—21．
【16】陈黎，刁明军．表孔泄洪的水气二相流数值模拟．四川水力发电，21(1)，2002：91—93．
【17】何枫，叶宏开．水池中自由液面波动的数值模拟 (二维)．山东烟台，第三界全国环境流体力学会议，1994．
【18】韩标，何枫，叶宏开，王学芳．工程中液面波动的数值模拟．水动力学研究与进展，14(2)，1999：142—147．
【19】李治勤，陈刚，李建中．用紊流模型通过座标变换预测过坝水流流场．水力学报，(8)，1996：25—32．
【20】谭立新，许唯临，杨永全．明渠水气两相流数值模拟．四川联合大学学报 (工程科学版)，3(1)，1999：93—97．
【21】谭立新，许唯临，杨永全，李建中，魏文礼．明渠气水二相流的双流体模型．应用力学学报，18(4)，2001：28—32．
【22】许唯临，王韦，谭立新，杨永全．水工水气两相流的数值模拟．水动力学研究与进展，16(2)，2001：225—229．
【23】张宏伟．掺气水流双流体模型数值模拟研究．西安理工大学硕士学位论文．2002．4
【24】李国栋，陈刚，李建中．明流泄洪洞流场数值模拟．水动力学研究与进展，11(6)，1996：633—639．
【25】许联锋，李建中，陈刚．泄水建筑物体型优选的数值模拟研究．水动力学研究与进展，17(6)，2002：701—708。
【26】廖华胜，汝树勋，吴持恭．阶梯溢流坝流场的数值模拟．成都科技大学学报，(5)，1995：27—33．
【27】许联锋，陈刚，李建中，邵建斌．气液两相流中气泡运动速度场的PIV分析与研究．试验力学，17(4)，2002：458—463．
【28】许联锋，陈刚，李建中，邵建斌．粒子图像测速技术研究进展．力学进展，33(4)，2003：533—540．
【29】陶文铨．数值传热学(第2版)．西安交通大学出版社，2001．
【30】ShihTH, Liou W W, Shabbir A,Yang Z G, Zhu J. A new k-ε eddy viscosity model for high Reynolds number turbulent flows. Comput Fluids, 24(3), 1995:227—238.
【31】陶文铨．计算传热学的近代进展．科学出版社，2000。
【32】吴持恭主编．水力学(上册)．北京：高等教育出版社，1982：73—142．
【33】张绍芳．堰闸水力设计．北京：水利电力出版社，1987：214—260．
【34】李建中．水力学．陕西：陕西科学技术出版社，2002：169—183．
【35】刘大有．二相流体动力学．高教出版社，1993：534—549．
【36】Drew D A, Segel.L.A. Averaged equations for two-phase flow. Studies in Applied Math,50(3),1971.
【37】Delhaye J M. Space/time and time/space-averaged equations. Two-phase flow and heat transfer, vol.1, Hemisphere, Washington,D.C.,1977.
【38】D.A.Drew and Lahey, R T, Jr. In particulate two-phase flow,1990:509—566, Butterworth-Heinemann, Boston.
【39】M.Ishii. Yhermo-fluid dynamic theory of two-phase flow. Eyrolles Paris, 1975.
【40】H.Enwald, E.Peirano, A.E.Almstedt. Eulerian two-phase flow theory applied to fouidization. Int.J.Multiphase Flow,vol:22,1996:21-66.
【41】Lance M, Lopez de B M. Phase distribution phenoma and effects in bubbly two-phase flow.Multiphase Science and Technology, vol:8.
【42】Ishii M, Mishima K.Two-fluid modeling and hydrodynamic constitutive relation.Nucl.Eng.Des.vol:82, 1984:107-126.
[43] M.Ishii. Two-fluid model for two-phase flow.Science and technology of multi-phase,vol:5, 1990.
[44] L.Schiller and A.Nauman. Vdi Zeits. 1933:318.
[45] M.Ishii and N.Zuber. AICHEJ.Vol:25,Iss:5,1979:843.
[46] F.Ihme, H.Schmidt-Traub, H.Brauer. Chemic-Ing-Tech, Vol:44,Iss:4,1972:306.
[47] P.G.Suffman, The lift on a small sphere in a slow shear flow, J.Fluid Mech,22(3),1965.
[48] D.A.Drew and Lahey, R T, Jr. (1990) .Some suppleymental analysis concerning the virtual mass and lift force on a sphere in a rotating and straining flow. Int.J.Multiphase Flow, 16(6).
[49] Rhie C M and Chow W L. A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation. AIAAJ,vol:21,1983:1525—1532.
[50] Peric M. Kessler R and Scheuerer G.. Comparison of finite-volume numerical method with staggered and collocated grids.Computers and fluid, 16(4),1988:389-40.
[51] Majumdar S. Role of underrelaxation in a partially divided enclosure.Int.J.Heat Mass Transfer.vol:10,1988:635-640.
[52] Miller T F and Schmidt F W. Use of a pressure-weighted interpolation method for solution of the incompressible navier-stokes equations on a nonstaggered grid system. Numerical Heat Transfer,vol:14,1988:213-233.
[53] Rodi W Majumdar S and Schonung B. Finite-volum methods for two-dimensional incompressible flows with complex boundaries. Compute.Meth.Appl.Meth.eng.vol:14,1989:213-233.
[54] Thiart G.D. Finite-difference scheme for the numerical solution of fluid flow and heat transfer problems on nonstaggered grid. Numerical Heat
Transfer.vol: 17B, 1990:43-62.
[55] Thiart G.D. Improved finite-difference scheme for the solution of convective-diffusion problem with the SIMPLEN algorithm. Numerical Heat Transfer.vol:18B,1990:81-95.
[56] Date A.W.C. Solution of incompressible navier-stokes equations on a nonstaggered grid. Int.J.Heat Mass Transfer.vol:36,1993:1913-1922.
[57] Date A.W.C. Complete pressure correction algorithm for solution of incompressible navier-stokes equations on a nonstaggered grid. Int.J.Heat Mass Transfer.vol:29B, 1996:441-458.
[58] 杨建明，吴建华．淹没水跃的数值模拟．水动力学研究与进展，16(4)，2001：405—411．
[59] A. R. Zarrati, Mathematical modeling of air-water mixtures in open channels. J Hydraulic Research, Vo1.32, 1994, No. 5.
[60] 张宏伟，谭立新，陈刚，李建中，杨永全，等．水气二相流双流体模型研究现状．陕西水力发电．第17卷，第四期，2001．12．
[61] 滕爱国，沈永明，郑永红，邱大洪．波浪变形数值模拟．大连理工大学学报，41(3)，2001：359—362．
[62] 邓军，许唯临，曲景学，董建伟．明流泄洪隧洞中自掺气水流浓度分布及其计算．水利学报，2002：64—68．
[63] 中国电力企业联合会标准化部编．电力工业标准汇编(水电卷)[S]．北京：水利电力出版社，1995，3：285-286．
[64] 李建中，宁利中．高速水力学．陕西：西北工业大学出版社，1994：93—155．