溢洪道掺气水流的三维数值模拟
|
摘要
随着水利水电工程建设的发展,泄水建筑物的泄流量越来越大,单宽流量进一步增加,与之相关的脉动、空化空蚀等高速水流问题日益突出。向高速水流中掺气,是减免建筑物发生空蚀破坏的主要工程措施。因此,开展溢洪道水力特性,特别是掺气水流特性的研究对泄水建筑物的安全运行具有重要的现实意义。
鉴于数值计算具有易操作、低成本和短周期等优越性,在建筑物的选型和优化设计工作中,数值模拟方法能发挥较大的作用甚至部分代替试验,缩短周期。但是由于涉及掺气的理论还很不成熟,对其进行数值模拟还有许多研究工作需要去做。本文对巴贡水利工程中长溢洪道整体三维流场进行了数值模拟计算,以期深入了解其水力特性,为溢洪道的合理设计提供依据。采用多相流模型,对掺气坎处的掺气水流进行了模拟,通过对不同气泡特征直径,不同相间作用力模型计算结果的对比分析,探索改善掺气水流预测水平的途径,。
本文选用标准κ-ε紊流模型,有限体积法离散控制方程,VOF方法模拟自由表面,对不规则边界采用非结构网格进行处理。通过对该工程溢洪道紊动流场进行的三维数值模拟,得到了泄流量,速度场、压力场、水面线等的分布规律。数值模拟和试验结果的对比分析,两者吻合较好,表明模拟结果能满足工程需要。
对于掺气坎附近水流及掺气的模拟,本文采用双流体理论的欧拉模型,考虑水气间的相互作用,并以某个特征直径气泡代替气相对其进行模拟。计算结果表明,采用欧拉模型能较好的模拟出通气孔处的风速和通气量,不同的特征直径气泡对通气量的影响较为明显,但总体上来说,模拟值较实测值偏大。跌坎下游的掺气浓度无论实验数据和模拟结果,均显示水体中心部位掺气浓度沿程衰减,但模拟值的衰减率明显小于实测值,空腔负压较实测值偏大,导致模拟得到的掺气空腔不完整。在考虑充分通气的情况下,对比两种不同相间作用力模型对下游掺气浓度的影响,采用对称模型时特征气泡直径对水面气泡上浮影响较大,但掺气沿程衰减不明显的现象依然没有得到很好的改善。
由于紊动、水气两相流以及自由面问题的复杂性,两相流机理还处在发展阶段,并且由于气泡大小分布和变化规律尚不明确,在掺气模拟方面依然还有很多问题有待进一步探索。
With the development of hydropower engineering industry, the discharge of outlet structure is getting greater and greater,unit discharge is further increasing and problems brought by high velocity flow such as pulsating movement,cavitation and erosion become more prominent day by day. The main engineering measure to prevent cavitation damage of the stucture is to add gas into high velocity flow.Therefore,study on the hydraulic characteristics of spillway,especially on aerated flow,becomes more and more important for safe operation of outlet structure.
As numerical simulation has the advantages of handleability, low cost and short cycle and so on,it can play a larger role and even replace apart of test in the construction selection and optimization,the period can be shorted.Because the theory of aerification is not very mature,there is mass of numerical simulation research ahout it to be done.In order to understand the hydraulic characteristics of spillway and provide basis for rational design,the numerical simulation of flow field of spillway in BaGong Project is done in this paper. The simulation of aerated flow at aerator is done with multiphase model, A new avenues to better improve the forecast level of aerated flow will be explored by comparisons of the caculation results in the case of characteristic diameter of different bubbles and different forces between phases.
Standardκ-εmodel combining with finite volume method are applied to this paper,free surface was trated with VOF method and irregular boundaries were dealed with unstructured grid.Compared with the data of model test of engineering project, a three-dimensional numerical simulation of turbulence of the spillway in this project was done,the discharge, velocity field, pressure field and water surface profile were all obtained. The comparison analysis shows that numerical simulation was in good accordance with experimental results,which satisfied the engineering requirements.
In order to exactly study the flow near the aerator, the two-fluid of Eulerian Model is used in this paper.Considering the interaction between gas and liquid.the simulaiton is done with a acharacteristic diameter bubble instead of gas.The results show that the ventilation volume and wind speed by the vent are obtained with Euler model, the different characteristics diameter of bubble has a notable impact on ventilation volume,yet on the whole,the values of simulation are a little greater than the experimental ones. The experimental datas and simulation results show that,air concentration of simulation results is smaller than experiment and negative pressure and cavity is greater than it which lead to the cavity is incomplete.In view of the condiotn of sufficient aeration,the impacts of two different interphase forces model on air concentration of downstream are compared,the diameter of characteristic bubble has a great effect on bubble floating on the water surface with symmetric model in compution,but the phenomenon that on-way attenuation of aerification is inapparent has not been improved better.
Because of the complexity of turbulent flow, water-air two-phase flow and free surface, two-phase flow mechanism is still in development stage, because the bubble size and the variation is not clear, the numerical simulation of aerated flow still has many questions to be further explore
鉴于数值计算具有易操作、低成本和短周期等优越性,在建筑物的选型和优化设计工作中,数值模拟方法能发挥较大的作用甚至部分代替试验,缩短周期。但是由于涉及掺气的理论还很不成熟,对其进行数值模拟还有许多研究工作需要去做。本文对巴贡水利工程中长溢洪道整体三维流场进行了数值模拟计算,以期深入了解其水力特性,为溢洪道的合理设计提供依据。采用多相流模型,对掺气坎处的掺气水流进行了模拟,通过对不同气泡特征直径,不同相间作用力模型计算结果的对比分析,探索改善掺气水流预测水平的途径,。
本文选用标准κ-ε紊流模型,有限体积法离散控制方程,VOF方法模拟自由表面,对不规则边界采用非结构网格进行处理。通过对该工程溢洪道紊动流场进行的三维数值模拟,得到了泄流量,速度场、压力场、水面线等的分布规律。数值模拟和试验结果的对比分析,两者吻合较好,表明模拟结果能满足工程需要。
对于掺气坎附近水流及掺气的模拟,本文采用双流体理论的欧拉模型,考虑水气间的相互作用,并以某个特征直径气泡代替气相对其进行模拟。计算结果表明,采用欧拉模型能较好的模拟出通气孔处的风速和通气量,不同的特征直径气泡对通气量的影响较为明显,但总体上来说,模拟值较实测值偏大。跌坎下游的掺气浓度无论实验数据和模拟结果,均显示水体中心部位掺气浓度沿程衰减,但模拟值的衰减率明显小于实测值,空腔负压较实测值偏大,导致模拟得到的掺气空腔不完整。在考虑充分通气的情况下,对比两种不同相间作用力模型对下游掺气浓度的影响,采用对称模型时特征气泡直径对水面气泡上浮影响较大,但掺气沿程衰减不明显的现象依然没有得到很好的改善。
由于紊动、水气两相流以及自由面问题的复杂性,两相流机理还处在发展阶段,并且由于气泡大小分布和变化规律尚不明确,在掺气模拟方面依然还有很多问题有待进一步探索。
With the development of hydropower engineering industry, the discharge of outlet structure is getting greater and greater,unit discharge is further increasing and problems brought by high velocity flow such as pulsating movement,cavitation and erosion become more prominent day by day. The main engineering measure to prevent cavitation damage of the stucture is to add gas into high velocity flow.Therefore,study on the hydraulic characteristics of spillway,especially on aerated flow,becomes more and more important for safe operation of outlet structure.
As numerical simulation has the advantages of handleability, low cost and short cycle and so on,it can play a larger role and even replace apart of test in the construction selection and optimization,the period can be shorted.Because the theory of aerification is not very mature,there is mass of numerical simulation research ahout it to be done.In order to understand the hydraulic characteristics of spillway and provide basis for rational design,the numerical simulation of flow field of spillway in BaGong Project is done in this paper. The simulation of aerated flow at aerator is done with multiphase model, A new avenues to better improve the forecast level of aerated flow will be explored by comparisons of the caculation results in the case of characteristic diameter of different bubbles and different forces between phases.
Standardκ-εmodel combining with finite volume method are applied to this paper,free surface was trated with VOF method and irregular boundaries were dealed with unstructured grid.Compared with the data of model test of engineering project, a three-dimensional numerical simulation of turbulence of the spillway in this project was done,the discharge, velocity field, pressure field and water surface profile were all obtained. The comparison analysis shows that numerical simulation was in good accordance with experimental results,which satisfied the engineering requirements.
In order to exactly study the flow near the aerator, the two-fluid of Eulerian Model is used in this paper.Considering the interaction between gas and liquid.the simulaiton is done with a acharacteristic diameter bubble instead of gas.The results show that the ventilation volume and wind speed by the vent are obtained with Euler model, the different characteristics diameter of bubble has a notable impact on ventilation volume,yet on the whole,the values of simulation are a little greater than the experimental ones. The experimental datas and simulation results show that,air concentration of simulation results is smaller than experiment and negative pressure and cavity is greater than it which lead to the cavity is incomplete.In view of the condiotn of sufficient aeration,the impacts of two different interphase forces model on air concentration of downstream are compared,the diameter of characteristic bubble has a great effect on bubble floating on the water surface with symmetric model in compution,but the phenomenon that on-way attenuation of aerification is inapparent has not been improved better.
Because of the complexity of turbulent flow, water-air two-phase flow and free surface, two-phase flow mechanism is still in development stage, because the bubble size and the variation is not clear, the numerical simulation of aerated flow still has many questions to be further explore
引文
[1]李建中.高速水力学[M].西安：西北工业大学出版社,118-149
[2]Rasmuissen R.E.H.Some experiment on cavitation erosion in water mixed with air.-<>, London, 1956
[3]J.A.彼得卡,掺气对气蚀的影响,高速水流论文译丛,第一辑,第一册,PP.499～513,科学出版社,1958
[4]Rasmussen,R.E.H, Experiments on Flow with cavitation in water mixed with air, Transactions of the Danish Academy of Technical Sciences, A.T.S., No.1, Copenhagen, Danmark,1949
[5]Pinto, N.L.S, Aeration at High velocity Flows, water Power&Dam construction, No.2, No.3, 1982
[6]水利部长江水利委员会：长江三峡水利枢纽表孔和深孔体型优化专题报告[R].1996年7月.18-19
[7]R.依伦伯格,陡槽水流掺气,高速水流论文译丛,第1辑第1册,科学出版社,1958年
[8]L.S.霍尔,高速明渠水流的掺气,高速水流论文译丛,第1辑第1册,科学出版社,1958年
[9]п.A.伏依诺维奇,A.и.舒华兹,掺气水流的均匀运动,高速水流论文译丛,第1辑第1册,科学出版社,1958年
[10]Lane, E.W., Entrainment on spillway faces,Civil Engineering, Vol.9,1939年
[11]G.哈尔勃朗,陡槽中的掺气,高速水流论文译丛,第1辑第1册,科学出版社,1958年
[12]Michels, V. and Lovely,M.,Some\prototype Observation of Air Entrained Flows, Proceedings of Minnesota Conference,IAHR,1953
[13]Bauer,W.J., Turbulent Boundary Layer on Steep slopes, ASCE,1954
[14]Kanpp, Daily, Hammitt, Cavitaton, McGraw-Hill,1970
[15]刘家峡水电厂.刘家峡水电站泄水道空蚀、磨损破坏与修补[C].第三届水工混凝土建筑物修补技术交流会论文集,1992,6：1-100
[16]W.S.汉密尔顿,水工建筑物气蚀破坏与防护[J],中南水电,1989年第3期
[17]Rasmuissen R.E.H.Some experiment on cavitation erosion in water mixed with air. Cavitation in Hydrodynamics[M], London,1956
[18]吴持恭.水力学下册(第二版)[M]．北京：高等教育出版社,1996：20-30
[19]祁庆和.水工建筑物[M].北京：中国水利水电出版社,1981(9)
[20]B.г.沙科洛夫,在溢流式水电站模型上的水流掺气的研究,高速水流论文译丛,第1辑,第1册,科学出版社,1958年
[21]刘韩生、张勇.论掺气减蚀[J].西北水资源与水工程,1996,7(1)
[22]吴持恭,明渠水气二相流[M].成都科技大学出版社,1989年
[23]张丽,解伟,黄晓玲.高水头泄水建筑物的掺气减蚀试验[J].华北水利水电学报, 1998.3.
[24]黄继汤.空化与空蚀的原理[M]．北京：清华大学出版社,1991(2)
[25]P.C.加尔彼凌登.水工建筑物的空蚀[M].水利出版社,1981：149-180第一章绪论
[26]刘家峡水电厂.刘家峡水电站泄水道空蚀、磨损破坏与修补[C].第三届水工混凝土建筑物修补技术交流会论文集,1992,6：1-100
[27]《水工隧洞设计经验选编》编写组,水工隧洞设计经验选编,水利出版社,1981年
[28]李建中,张书长,二龙山水库低空闸室的空蚀问题,人民长江,1984年第3期
[29]肖兴斌,王才欢,王业红,弧形闸门突扩跌坎式通气减蚀研究进展综述[J],长江科学院院报,2000年10月
[30]SL253-2000溢洪道设计规范[s],1-5
[31]丁道扬,吴时强,泄水建筑物水力数值模型应用,水利水电泄水工程与高速水流信息网、水利部东北勘侧设计研究院科学研究院编,吉林科学技术出版社,1922年10月。
[32]金忠青编著,N一方程的数值解和紊流模型,河海大学出版社,1989年6月
[33]李国栋,陈刚,李建中．明流泄洪洞流场数值模拟[J].水动力学研究与进展,1996(12)：634—639
[34]丁道扬,吴时强.龙抬头式泄洪洞水流数学模型及其应用[J].水利水运科学研究,1995(9)：246260
[35]邓军,许唯临,等.高水头岸边泄洪洞水力特性的数值模拟[J].水利学报,2005(10)：1209—1218
[36]沙海飞,吴时强.泄洪洞有压段体型优化的数值模拟[J]水动力学研究与进展,2004(12)：942—946
[37]许联锋,李建中,陈刚.泄水建筑物体型优选的数值模拟研究[J].水动力学研究与进展,2002(12)：701—708
[38]魏文礼等,泄水建筑物曲面边壁三维定常紊流数值模拟[J].2000.12：105—110
[39]刁明军.高坝大流量泄洪消能数值模拟及实验研究[J].四川大学博士论文,2004.
[40]李玲,陈永灿等.泄洪隧洞防空蚀体型数值研究[J]水利学报,2003(7)：12—15
[41]林宗虎.两相流与沸腾传热.西安：西安交通大学出版社,2002
[42]刘大有.二相流动力学.北京：高等教育出版社,1993
[43]鲁钟琪.两相流与沸腾传热.北京：清华大学出版社,2002
[44]陈学俊,陈立勋,周芳德.气-液两相流与传热基础.北京：科学出版社,1995
[45]刘儒勋,王志峰.运动界面追踪数值模拟方法.合肥：中国科学技术大学出版社,2001
[46]徐济鋆.沸腾传热和气液两相流.北京：科学出版社,2001
[47]Brackbill J U, Kothe D B, Zemach C. A Continuum Method for Modeling Surface Tension.J Computational Physics,1992,100:335-354
[48]陶文铨.计算传热学的近代发展.北京：科学出版社,2001
[49]郭照立,郑楚光,李青,王能超.流体动力学的个子——Bolzmann方法.武汉：湖北科学技术出版社,2002
[50]刘全,张晓轶,水鸿寿.数值模拟界面流方法进展.力学进展,2002.32(2)：259-274
[51]Kolev N I.three Fluid Modeling with Dynamic Fragmentation and Coalescence:Fiction or Daily Practice.Proceedings of 4th world Conference on Experimental Heat Transfer,Fluid Mechanics and Thermodynamics,ExHFT-4,Brussels,June2-6,1997
[52]Marzio Piller,Enrico Nobile,J.Thoms,DNS study of turbulent tranaport at low Prandtl numbers in a channel flow.Journal of Fluid Mechanics,(458):491-441,2002
[53]J.G. Wissink. DNS of separating low Reynolds number flow in a turbine cascade with incoming wakes.International Journal of Heat and Fluid Flow,24(4):626-635,2003
[54]M.B.Abbott, D.R.Basco,Computational Fluid Dynamics-A Introduction for Engineers. Longman Scientific& Technical, Harlow, England,1989
[55]Fluent Inc., FLUENT User's Guide. Fluent Inc.,2003
[56]J.O.Hinze,Turbulence. McGraw-Hill, New York,1975
[57]金忠青.N-S方程的数值解与紊流模拟[M].武汉：河海大学出版社1989,6
[58]陶文铨.数值传热学(第二版)[M].西安：交通大学出版社,1995(4)
[59]C.J.Chen,S.Y.Jaw.Fundamentals of Turbulence Modeling.Taylor&Francis,Wsahington,1998
[60]H.K.Versteeg,W.Malalasekera,An Introduction to Computational Fluid Dynamics:The Finite Volume Method. Wiley,New York,1995
[61]戴会超,槐文信,吴玉林,许唯临等.水利水电工程水流精细模拟理论与应用[M].北京：科学出版社,2006.4
[62]Hirt C W,Nichols B D.Volume of fluid(VOF) method for the dynamics of ree boundaries.J Comp Phy,1981.39:201～225
[63]张云勇,百斤重,敬万钧.虚拟实现系统的建立.计算机应用,2000,(20)：229～231
[64]S.V.Patanker,D.B.Spalding,A calculation processure for heat,mass and momentum transfer in three-dimensional parabolic flows.Int J Heat Mass Transfer,15:1787-1806,1972
[65]王海云,高水头龙抬头泄洪洞掺气减蚀实验研究及数值模拟.四川大学
[66]韩占忠,王敬,兰小平.Fluent流体工程仿真计算实例与应用.北京：北京理工大学出版社,2004
[2]Rasmuissen R.E.H.Some experiment on cavitation erosion in water mixed with air.-<
[3]J.A.彼得卡,掺气对气蚀的影响,高速水流论文译丛,第一辑,第一册,PP.499～513,科学出版社,1958
[4]Rasmussen,R.E.H, Experiments on Flow with cavitation in water mixed with air, Transactions of the Danish Academy of Technical Sciences, A.T.S., No.1, Copenhagen, Danmark,1949
[5]Pinto, N.L.S, Aeration at High velocity Flows, water Power&Dam construction, No.2, No.3, 1982
[6]水利部长江水利委员会：长江三峡水利枢纽表孔和深孔体型优化专题报告[R].1996年7月.18-19
[7]R.依伦伯格,陡槽水流掺气,高速水流论文译丛,第1辑第1册,科学出版社,1958年
[8]L.S.霍尔,高速明渠水流的掺气,高速水流论文译丛,第1辑第1册,科学出版社,1958年
[9]п.A.伏依诺维奇,A.и.舒华兹,掺气水流的均匀运动,高速水流论文译丛,第1辑第1册,科学出版社,1958年
[10]Lane, E.W., Entrainment on spillway faces,Civil Engineering, Vol.9,1939年
[11]G.哈尔勃朗,陡槽中的掺气,高速水流论文译丛,第1辑第1册,科学出版社,1958年
[12]Michels, V. and Lovely,M.,Some\prototype Observation of Air Entrained Flows, Proceedings of Minnesota Conference,IAHR,1953
[13]Bauer,W.J., Turbulent Boundary Layer on Steep slopes, ASCE,1954
[14]Kanpp, Daily, Hammitt, Cavitaton, McGraw-Hill,1970
[15]刘家峡水电厂.刘家峡水电站泄水道空蚀、磨损破坏与修补[C].第三届水工混凝土建筑物修补技术交流会论文集,1992,6：1-100
[16]W.S.汉密尔顿,水工建筑物气蚀破坏与防护[J],中南水电,1989年第3期
[17]Rasmuissen R.E.H.Some experiment on cavitation erosion in water mixed with air. Cavitation in Hydrodynamics[M], London,1956
[18]吴持恭.水力学下册(第二版)[M]．北京：高等教育出版社,1996：20-30
[19]祁庆和.水工建筑物[M].北京：中国水利水电出版社,1981(9)
[20]B.г.沙科洛夫,在溢流式水电站模型上的水流掺气的研究,高速水流论文译丛,第1辑,第1册,科学出版社,1958年
[21]刘韩生、张勇.论掺气减蚀[J].西北水资源与水工程,1996,7(1)
[22]吴持恭,明渠水气二相流[M].成都科技大学出版社,1989年
[23]张丽,解伟,黄晓玲.高水头泄水建筑物的掺气减蚀试验[J].华北水利水电学报, 1998.3.
[24]黄继汤.空化与空蚀的原理[M]．北京：清华大学出版社,1991(2)
[25]P.C.加尔彼凌登.水工建筑物的空蚀[M].水利出版社,1981：149-180第一章绪论
[26]刘家峡水电厂.刘家峡水电站泄水道空蚀、磨损破坏与修补[C].第三届水工混凝土建筑物修补技术交流会论文集,1992,6：1-100
[27]《水工隧洞设计经验选编》编写组,水工隧洞设计经验选编,水利出版社,1981年
[28]李建中,张书长,二龙山水库低空闸室的空蚀问题,人民长江,1984年第3期
[29]肖兴斌,王才欢,王业红,弧形闸门突扩跌坎式通气减蚀研究进展综述[J],长江科学院院报,2000年10月
[30]SL253-2000溢洪道设计规范[s],1-5
[31]丁道扬,吴时强,泄水建筑物水力数值模型应用,水利水电泄水工程与高速水流信息网、水利部东北勘侧设计研究院科学研究院编,吉林科学技术出版社,1922年10月。
[32]金忠青编著,N一方程的数值解和紊流模型,河海大学出版社,1989年6月
[33]李国栋,陈刚,李建中．明流泄洪洞流场数值模拟[J].水动力学研究与进展,1996(12)：634—639
[34]丁道扬,吴时强.龙抬头式泄洪洞水流数学模型及其应用[J].水利水运科学研究,1995(9)：246260
[35]邓军,许唯临,等.高水头岸边泄洪洞水力特性的数值模拟[J].水利学报,2005(10)：1209—1218
[36]沙海飞,吴时强.泄洪洞有压段体型优化的数值模拟[J]水动力学研究与进展,2004(12)：942—946
[37]许联锋,李建中,陈刚.泄水建筑物体型优选的数值模拟研究[J].水动力学研究与进展,2002(12)：701—708
[38]魏文礼等,泄水建筑物曲面边壁三维定常紊流数值模拟[J].2000.12：105—110
[39]刁明军.高坝大流量泄洪消能数值模拟及实验研究[J].四川大学博士论文,2004.
[40]李玲,陈永灿等.泄洪隧洞防空蚀体型数值研究[J]水利学报,2003(7)：12—15
[41]林宗虎.两相流与沸腾传热.西安：西安交通大学出版社,2002
[42]刘大有.二相流动力学.北京：高等教育出版社,1993
[43]鲁钟琪.两相流与沸腾传热.北京：清华大学出版社,2002
[44]陈学俊,陈立勋,周芳德.气-液两相流与传热基础.北京：科学出版社,1995
[45]刘儒勋,王志峰.运动界面追踪数值模拟方法.合肥：中国科学技术大学出版社,2001
[46]徐济鋆.沸腾传热和气液两相流.北京：科学出版社,2001
[47]Brackbill J U, Kothe D B, Zemach C. A Continuum Method for Modeling Surface Tension.J Computational Physics,1992,100:335-354
[48]陶文铨.计算传热学的近代发展.北京：科学出版社,2001
[49]郭照立,郑楚光,李青,王能超.流体动力学的个子——Bolzmann方法.武汉：湖北科学技术出版社,2002
[50]刘全,张晓轶,水鸿寿.数值模拟界面流方法进展.力学进展,2002.32(2)：259-274
[51]Kolev N I.three Fluid Modeling with Dynamic Fragmentation and Coalescence:Fiction or Daily Practice.Proceedings of 4th world Conference on Experimental Heat Transfer,Fluid Mechanics and Thermodynamics,ExHFT-4,Brussels,June2-6,1997
[52]Marzio Piller,Enrico Nobile,J.Thoms,DNS study of turbulent tranaport at low Prandtl numbers in a channel flow.Journal of Fluid Mechanics,(458):491-441,2002
[53]J.G. Wissink. DNS of separating low Reynolds number flow in a turbine cascade with incoming wakes.International Journal of Heat and Fluid Flow,24(4):626-635,2003
[54]M.B.Abbott, D.R.Basco,Computational Fluid Dynamics-A Introduction for Engineers. Longman Scientific& Technical, Harlow, England,1989
[55]Fluent Inc., FLUENT User's Guide. Fluent Inc.,2003
[56]J.O.Hinze,Turbulence. McGraw-Hill, New York,1975
[57]金忠青.N-S方程的数值解与紊流模拟[M].武汉：河海大学出版社1989,6
[58]陶文铨.数值传热学(第二版)[M].西安：交通大学出版社,1995(4)
[59]C.J.Chen,S.Y.Jaw.Fundamentals of Turbulence Modeling.Taylor&Francis,Wsahington,1998
[60]H.K.Versteeg,W.Malalasekera,An Introduction to Computational Fluid Dynamics:The Finite Volume Method. Wiley,New York,1995
[61]戴会超,槐文信,吴玉林,许唯临等.水利水电工程水流精细模拟理论与应用[M].北京：科学出版社,2006.4
[62]Hirt C W,Nichols B D.Volume of fluid(VOF) method for the dynamics of ree boundaries.J Comp Phy,1981.39:201～225
[63]张云勇,百斤重,敬万钧.虚拟实现系统的建立.计算机应用,2000,(20)：229～231
[64]S.V.Patanker,D.B.Spalding,A calculation processure for heat,mass and momentum transfer in three-dimensional parabolic flows.Int J Heat Mass Transfer,15:1787-1806,1972
[65]王海云,高水头龙抬头泄洪洞掺气减蚀实验研究及数值模拟.四川大学
[66]韩占忠,王敬,兰小平.Fluent流体工程仿真计算实例与应用.北京：北京理工大学出版社,2004