摘要
水污染的治理关键是污染源的控制,造成水环境污染的主要原因之一就是城市污水的无序排放。由于污水江河湖海处置的工程造价和运行费用相对较低,污水江河湖海处置已经成为解决城市水污染优先考虑的工程措施。科学的设计污水排放方式和科学的选择排放水域都有赖于对排放到环境水体中污水的动力特性以及污染物扩散、输移特性等环境水力学问题的深刻认识。污水通常以射流的形式排放到江河湖海中,系统开展动水环境中射流特性的基础性研究不仅可以加深对湍流的认识,而且能够为射流在生产实践中的应用提供更加完善的理论基础,有着重要的科学价值和实际意义。
本文从物理实验和数值模拟两方面对动水环境中射流特性,包括射流的流态、射流扩展率以及污染物浓度场分布特性等进行了较为系统的研究。在实验方面,利用波浪水槽结合粒子图像测速(PIV)系统对波浪环境中圆形垂向射流流场的时均特性和紊动特性进行了研究;在数值模拟方面,以Reynolds时均方程为基本控制方程,利用应力代数紊流模型和VOF方法建立了可模拟带自由表面的紊动射流二维数学模型和三维数学模型,对动水环境中窄缝射流和圆形射流进行了数值模拟,分析了横流环境中射流喷角对射流特性的影响,开展了波浪环境中窄缝射流和圆形射流基本特性的系统研究。
本文基于不同波要素波浪环境中射流流场实验结果,分析、总结了波浪周期对射流核长度、横剖面上射流垂向平均速度分布以及射流扩展率的影响规律。根据对波浪环境中射流紊动特性实验结果的分析,依据射流的紊动特性,将波浪环境中紊动射流的发展分为三个阶段,分析、总结了波浪环境中射流的紊动量在横剖面上的分布规律,及其在紊动射流三个不同发展阶段随波高的变化规律,并指出波浪环境下射流的紊动扩散量值大小约是对流的1/8~1/3,紊动扩散的作用不可忽略。
本文基于对横流环境中窄缝射流的数值模拟结果,分析、总结了横流环境中不同射流比下各种喷角对涡心点、分离点位置以及稀释度的影响规律。提出“面积湍动能”的概念,并通过分析喷角对“面积湍动能”的影响规律,得出喷角为90度时最有利于射流水体与环境水体间掺混的结论。
本文基于对波浪环境中窄缝射流的数值模拟结果,详细描述了波浪环境中的窄缝射流二维即时流场以及浓度场的特点,指出波浪环境中窄缝射流污染物的传播扩散过程分为三个阶段。通过对波浪环境中窄缝射流周期平均流场的分析,指出喷口轴线上周期平均的垂向速度衰减分为四个阶段,并总结了波高和波周期对波浪环境下窄缝射流的射流核长度、横剖面上射流垂向平均速度分布以及射流扩展率的影响规律。指出波浪影响射流特性的直接因素包括两个:第一,波浪水质点的最大水平速度;第二,波浪的摆动。波浪水质点的水平运动半径是影响垂向射流特性的根本因素,波浪水质点水平运动半径越大,波浪对射流特性的影响越显著。
本文基于波浪环境中圆形射流的数值模拟结果,详细描述了波浪环境中的圆形射流三维即时流场以及浓度场的特点,着重对比了过喷口轴线顺浪向和垂直浪向的两个垂面上的即时流场、浓度场的差异,以及波浪环境中圆形射流过喷口轴线顺浪向垂面上浓度场与窄缝射流浓度场的差异。另外,通过对不同波要素、不同水平面上即时涡流场的分析,指出波浪环境中圆形射流水平面上出现的各旋涡结构中,肾形漩涡对出现频率最高。通过对波浪环境中圆形射流周期平均流场的分析,总结了波要素对三维圆形射流扩展率的影响规律,并着重对比了过喷口轴线顺浪向和垂直浪向的两个垂面上射流扩展率的差异,以及波浪环境中三维圆形射流与二维窄缝射流扩展率受波要素影响规律的差异。
The key problem for renovating water environment is the control on the origin of pollution. One of the factors polluting water environment is just the out-of-order letting of cities' polluted water. Because the cost of the waste water treated project is correspondingly cheap by letting the polluted water into rivers or sea, it has been the preferentially measure for dealing with polluted water of cities. Scientifically designing the mode of waste water letting and choosing the region of waste water letting all depend on the profoundly understanding for the polluted water's hydrodynamic characteristics and the contamination's diffusion, transport characteristics. Polluted water usually flows into rivers and sea by the mode of jet. Systemically making the basic study of the jet's characteristics in the flowing ambient fluid not only proves the understanding on the turbulent flow, but also supplies more perfect academic base for the jet's application in the production, so it has important scientific value and factual significance.
This paper makes more systemically study on the jet's characteristics in the flowing ambient fluid by the experiment and numerical simulation, including the jet's flow state, expansion rate, and the distribution of concentration. In the experiment the time-averaged and turbulent characteristics of the vertical round jet in wave environment is studied by the wave flume and the system of particle image velocimetry (PIV); in the numerical simulation based on Reynolds time-averaged equations the turbulent jet's 2D and 3D numerical modes are established by the stress-algebraic turbulent model and VOF method. Numerically simulate the plan jet and round jet in the flowing ambient environment, analyze the jet angle influence on the jet's characteristics, and makes the systemically study on the plan and round jet's characteristics in wave environment.
Based the experimental results of the jet's velocity field this paper summarizes the rules of the wave period influence on the length of potential core, the distribution of vertical velocity on the cross-profiles, and the jet's expansion rate. By analyzing the jet's turbulent characteristics the process of the turbulent jet in wave environment is divided into three phases. Summarize the rules of the distribution of turbulent terms on the cross-profiles and the change of turbulent terms with the wave height, and indicate that the value of the jet's turbulent diffusion is about the 1/8-1/3 of the value of convection in wave environment, so the function of the diffusion can't be ignored.
Based the numerical simulated results of the jet in the cross flow this paper summarizes the rules of the jet angle influence on the position of the vortex center, separation point and dilution. The concept of "area turbulent kinetic energy" is proposed and by using this concept the relationship between mixing intensity and jet angles is analyzed. The results show that 90 degree jet angle is the most favorite condition for jet water mixing with environment water.
Based the numerical simulated results of the plan jet in wave environment this paper describes the characteristics of the instantaneous velocity and concentration field, and indicates that the progress of the contamination's spreading is divided into three phases. By analyzing the period-averaged velocity field of the plan jet in wave environment this paper divides the progress of the vertical period-averaged velocity falling on the axis of the spout into four phases, and summarizes the rules of the wave height and wave period influence on the length of potential core, the distribution of vertical velocity on the cross-profiles, and the plan jet's expansion rate. The factors of the wave influence on the characteristics of the jet include: the first one is the horizontal maximum velocity of wave particle; the second one is the oscillatory wave motion. Based on the two reasons it can be conclude that the horizontal radius of wave particle motion is the essential factor affecting the characteristics of vertical jet, and the horizontal radius of wave particle motion is larger, the effect on the characteristics of vertical jet is more remarkable.
Based the numerical simulated results of the round jet in wave environment this paper describes the 3D characteristics of the instantaneous velocity and concentration field, and emphasizes the contrast between the instantaneous velocities, concentration fields in the two vertical plans through the axis of the spout. In addition the conclusion is obtained that the pairs of kidney vortex has the highest frequency in all the vortexes in the horizontal plan by analyzing the vortex and velocity fields in the different cases. Summarize the rules of the wave terms influence on the rate of the 3D jet expansion by analyzing the period-averaged velocity field of the round jet in wave environment, and emphasizes the contrast between the rules of the wave terms influence on the rate of the 3D round jet expansion and the 2D plan jet expansion.
引文
[1] 张永良,阎鸿邦.污水海洋处置技术指南.北京:中国环境科学出版社,1996年.
[2] 王超.污水处置理论及技术.南京:河海大学出版社,1998.
[3] 熊鳌魁,詹德新等.近水面淹没喷嘴射流流场特性的实验研究.水动力学研究与进展.1995,A辑10(4):429-433.
[4] Anthony, D. G. and Willmarth, W. W. Turbulence measurements in a round jet beneath a free surface. Journal of Fluid Mechanics, 1992,243:699-720.
[5] Wallace, R.B. and S.J. Wright. Spreading layer of two-dimensional buoyant jet. Journal of Hydraulic Engineering, 1984,110(6):813-828.
[6] Lee, J.H.W. and V.W.L. Cheung. Inclined olane buoyant jet in stratified fluid. Journal of Hydraulic Engineering, 1986,112(7):580-589.
[7] Sobey, R.J. and R.D. Keane. Horizontal round buoyant jet in shallow water. Journal of Hydraulic Engineering, 1988,114(8):910-929.
[8] Guo, Z.-R. and J.J. Sharp, Characteristics of radial jets and mixing under buoyant conditions. Journal of Hydraulic Engineering, 1996,122(9):495-502.
[9] Frick, W.E. Non-empirical closure of the plume equations. Atmospheric Environment, 1984,18(4):653-662.
[10] Chiang, H.-C. and B.L. Sill. Entrainment models and their application to jets in a turbulent cross flow. Atmospheric Environment, 1985,19(9):1425-1438.
[11] Lee, J.H. V and V Cheung, Generalized Lagrangian model for buoyant jets in current. Journal of Environmental Engineering, 1990, 116(6): 1085-1106.
[12] Cheung, S.K.B., et al., VISJET-A computer ocean outfall modeling system. IEEE Computer Graphics International, 2000: 75-80.
[13] 周连伟.三维非恒定流场中圆形浮射流的数值模拟研究.:(硕士论文).大连:大连理工大学,2004.
[14] Lee, J. and W.I. Seo. Numerical simulation of advected thermal using Gaussian-vortex model. Journal of Engineering Mechanics, 2000,26(10): 1098-1106.
[15] Wang, H. and A.W.-K.Law. Second-order integral model for a round turbulent buoyantjet. Journal of Fluid Mechanics, 2002, (459):397-428.
[16] Mukhtasor, L.M. Lye, and J.J. Sharp. A new approach to modelling initial dilution of a buoyancy-dominated jet in moving water. Journal of Environmental Engineering and Science, 2002,1:101-111.
[17] 金忠青.用标准k-ε模型求解二维紊动淹没射流.河海大学学报,1987,15(6):15-26.
[18] 杨志峰,周雪漪,许协庆,潮汐环境中垂向湍射流流场的数值模拟.水科学进展,1993,4(1):23—29.
[19] 槐文信,李炜.静止分层环境中圆形浮力射流全场特性的数值模拟.水动力学研究与进展,1993增刊:595—600.
[20] 槐文信,李炜,彭文启.横流中单圆孔紊动射流计算与特性分析.水利学报,1998,(4):7—14.
[21] 曾玉红,槐文信.横流中三维圆形垂直浮力射流特性数值分析.应用基础与工程科学学报,2005,13(2):120—128.
[22] Chang Y. R., then K. S. Measurements of opposing heated line jets discharged at an angle to a confined crossflow. Int. J. Heat Mass Transfer. 1994,37(12): 2935-2946.
[23] Chang Y. R., Chen K. S. Prediction of opposing turbulent line jets discharged laterally into a confined crossflow. Int. J. Heat Mass Transfer. 1995,38(9): 1693-1703.
[24] 张长高.一个新的紊流模式.河海大学学报,1994,22(4):39—47.
[25] 袁丽蓉.波流环境中垂向紊动射流的数值模拟研究:(博士论文).大连:大连理工大学,2006.
[26] 陈景仁.湍流模型及有限分析法.上海:上海交通大学出版社,1998.
[27] 李炜,槐文信.平面射流与浮力射流的数学模型及计算方法.水动力学研究与进展,1989,4(4):41—49.
[28] 姜春波,许协庆.各向异性湍浮力射流的三维有限元计算.水利学报,1997,(7):8-18.
[29] Dai H C, Wang L L. Numerical study of submerged vertical plane jets under progressive water surface waves. China Ocean Engineering, 2005, 19 (3):433-442.
[30] Hahn S, Choi H. Unsteady simulation of jets in a cross flow. Journal of Computational Physics, 1997,134:342-356.
[31] Fischer, H.B., et al. Mixing in inland and coastal waters. California: Academic Press, 1979.
[32] Wygnanski, 1., and Fiedler, H. Some measurements in the self-preserving jet. J. Fluid Mech, 1969,38:577-612.
[33] Gutmark, E., et al. Mean and turbulent structure of noncircular jets. in AIAA Shear Flow Control Conference. Boulder, CO, USA: AIAA, New York, NY, USA. 1985.
[34] Kotsovinos, N.E., Study of the entrainment and turbulence in a plane buoyant jet. California Institute of Technology; W. M. Keck Laboratory of Hydraulics and Water Resources, Report .n KH-R-32,1975:324.
[35] Lam, K.M. Penetration of a submerged round jet into a counter-flowing current. Envir. Hydr.: Proc., Int. Symp. on Envir. Hydr. eds. J. H. W. Lee and Y. K. Cheung, A. A. Balkema Publishers, Rotterdam, The Netherlands, 1991:115-120.
[36] Chan, C.H.C., and Lam, K.M. The velocity field of a circular jet in a counterflow. Envir. Hydr.; In: Proceeding of Environmental Hydraulics, eds., Lee, Jayawardena and Wang. Balkema, Rotterdam, 1999:223-228.
[37] Chan, C.H.C., and Lam, K.M. and Bemero, S. On the penetration of around jet into a counterflow at different velocity ratios. Envir. Hydr.: In: Proceeding of Environmental Hydraulics,. eds., Lee, Jayawardena and Wang. Balkema, Rotterdam, 1999:229-234.
[38] Ramaprian, B.R. and M.S. Chandrasekhara. LDA measurements in plane turbulent jets. Journal of Fluids Engineering, Transactions of the ASME, 1985,107(2): 264-271.
[39] Lau, J.C., Whiffen, M.C., Fisher, M.J., and Smith, D.M. Anote on turbulence measurements with a laser velocimeter. J Fluid Mech.,1981,102: 353-366.
[40] Lee, J.H.W. and V H. Chu, Turbulent jets and Plumes: A Lagranglan Approach. Kluwer Academic Publishers, 2003.
[41] Nickels, T. B., and Perry, A.E. An experimental and theoretical study of the turbulent coflowing jet. J. Fluid Mech, 1996,309:157182
[42] Weisgraber, T. H.,and Liepmam, D. Turbulent structure during transition to self-similarity in a round jet. Exp. Fluids. 1998,24: 210-224.
[43] Haven, B.A. and M. Kurosaka. Kidney and anti-kidney vortices in crossflow jets. Journal of Fluid Mechanics, 1997,352:27-64.
[44] Law, A.W.-K. and H.J. Wang. Measurement of mixing processes with combined digital particle image velocimetry and planar laser induced fluorescence. Experimental Thermal and Fluid Science, 2000,22:213-229.
[45] Ryu, Y., Chang, H.-A., and Mori N. Dispersion of neutrally buoyant horizontal round jet in wave environment. J. Hydraul. Eng.,2005,131(12):1088-1097.
[46] 姜国强,张晓元,李炜.PⅣ在横流中的湍射流实验研究中的应用.水科学进展,2002,13(5):588-593.
[47] 姜国强,李炜.横流中有限宽窄缝射流的旋涡结构.水利学报,2004,(5):52-57.
[48] 姜国强,李炜,陶建华.动水环境中有限宽窄缝湍射流的水力特性研究.2004,(5):51-55。
[49] 张明亮,陈刚,许联锋,杨敦敏.PⅣ技术在水垫塘实验模型淹没射流中的应用.实验流体力学,2005,19(3):79-83.
[50] 张燕,樊靖郁,王道增.横流冲击射流尾迹涡结构的实验研究.力学季刊,2005,26(4):539—543。
[51] 肖洋.横向流动条件下多孔水平动量射流掺混特性研究:(博士论文).南京:河海大学,2005.
[52] 黄真理.浅水域中垂向纯射流的流动特征.水利学报,1992,9:31-37.
[53] 槐文信,李炜,彭文官.横流中单圆孔紊动射流计算与特性分析.水利学报,1998,4:6-14.
[54] 陈道毅.非恒定环境中紊动射流的研究:(博士学位论文).北京:清华大学,1988.
[55] 陈昭泉.数字图像和PLIF技术研究潮流底部多孔射流浓度场及分形:(博士学位论文).北京:清华大学,1997.
[56] 李玉梁,陈朝泉,陈嘉范.潮流底部多孔垂向排放污染物影响区的计算.清华大学学报(自然科学版),1999,(11):35-37.
[57] 周雪漪,阳坤,李玉梁,陈永灿.各向异性紊浮力模型在潮流底部排放计算中的应用.清华大学学报(自然科学版),1998,(11):91-94.
[58] 岳钧堂,敖柏川,孙淑卿.温排水在潮流中的运动特性及工程应用.力学与实践,1996,(3):52-54.
[59] 杨志峰.潮汐流动中垂向排放近区数值模拟和优化差分方法研究:(博士学位论文).北京:清华大 学,1989.
[60] Paoanicolaous P N, List E J. Investigation of roud vertical turbulent buoyant jets. Journal of Fluid Mechanics, 1988,195:341-391.
[61] Pryputniewicz R J, Bowley W W. An experimental study of vertical buoyant jets discharged into water of finite depth. Journal of Heat rranfer, 1975, 5:274-281.
[62] Jirka G H, Harleman D R F. Stability and mixing in a vertical plane buoyant jet in confined depth, Part2. Journal of Fluid Mechanics, 1979, 92:2 75-304.
[63] Lee J H W, Jirka G H. Vertical round buoyant jet in shallow jet in shallow water. Journal of Hydraulics Division, 1981,109(12):1651-1675.
[64] Sobey R J, Johnston A J, Keane R D. Horizontal round buoyant jet in shallow water. Journal of Hydraulic Engineering, 1988, 114(8):910-929.
[65] Johnston A J, Volker R E. Round buoyant jet entering shallow water. Journal of Hydraulic Research, 1993,31(1):121-138.
[66] Johnston A J, Nguyen N, Volker R E. Round buoyant jet entering shallow water in motion. Journal of Hydraulic Engineering, 1993,119(12):1364-1382.
[67] 槐文信,那宁彤,黄纪忠等.浅水环境中垂直圆形射流的试验研究.水科学进展,2002,13(1):26-30.
[68] 槐文信,那宇彤,童汉毅等.静止浅水环境中铅垂紊动射流的试验研究.水利学报,2002,(9):32-36.
[69] 曾玉红.静止浅水环境中浮力射流稳定性与混合特性研究:(博士论文).武汉:武汉大学,2005.
[70] Antonia, R.A. and R. W Bilger. An experimental investigation of an axisymmetric jet in a co-flowing air stream. J. Fluid Mech.,1973,138:91-127.
[71] Antonia, R.A. and R.W Bilger. Prediction of the axisyrnmetric turbulent jet issuing into a Co-Flowing Stream. Aeronautical Quarterly, 1974,25:69-80.
[72] Chu, P.C.K., J.H. Lee, and V.M. Chu. Spreading of turbulent round jet in coflow. Journal ofHydraulic Engineering, 1999,125(2):193-204.
[73] 肖洋,唐洪武,华明,王志良.同向圆射流混合特性实验研究.水科学进展,2005,17(4):512-517.
[74] Yoda, M. and H.E. Fiedler, Round jet in a uniform counterflow: Flow visualization and mean concentration measurements. Experiments in Fluids,,1996,21(6): 427-436.
[75] Lam, K.M. and H. C. Chan. Round jet in ambient counterflowing stream. Journal of Hydraulic Engineering, 1997,123(10):895-904.
[76] Subramanya K, Porey P D. Trajectory of a turbulent cross jet. Journal of Hydraulic research, 1984,22:343-354.
[77] List H B, Imberger J. Turbulent entrainment in buoyant jets. Journal of Hydraulics Division, Proc. of. ASCE.,1973,99:1467-1474.
[78] Rajaratnam N, Langat J K Mixing region of circular turbulent wall jets in cross flows. Journal of Hydraulic Engineering, 1995,121(10):694-698.
[79] Tian X D, Roberts P J W. A 3D LIF system for turbulent buoyant jet flows. Experiments in Fluids, 2003,35(6):636-647.
[80] Patankar S V, Basu b K, Alpay S A. Prediction of the three-dimensional velocity field of a deflected turbulent jet. Journal of Fluid Mechanics, 1977,99:758-762.
[81] Sykes R I, Lewellen W S, Parker S F. On the vorticity dynamics of a turbulent jet in a crossflow. Journal of Fluid Mechanics, 1986,168:393-413.
[82] Wood I R. Asymptotic solutions and behavior of outfall plumes. Journal of Hydraulic Engineering, 1993,119:555-580.
[83] Lee J H W, Li L, Cheung V. Semianalytical self-similar solution of bent-over jet in cross-flow. Journal of Engineering Mechanics, 1999,125(7):733-746.
[84] Kuang C P, Lee J H W. Effect of downstream control on stability and mixing of a vertical plane buoyant jet in confined depth. Journal of Hydraulic Research, 2001,39(4):375-391.
[85] Kuang C P, Lee J H W. Stability and mixing of a vertical axisymmetric buoyant jet in shallow water. Environmental Fluid Mechanics, 2006,6(2):153-180.
[86] 赵明登,郑邦民.横流中底部排污混合区的分析计算.武汉水利水电大学学报,1995,28(2):149-155.
[87] 梁爱国,槐文信.浅水横流中底部水平热水射流近区的初始稀释—(I)数学模型及其验证.四川大学学报(程科学版),2005,37(3):1-4.
[88] Anwar H. O., Otkins R. Turbulence measurements in simulated tidal flow. J. Hydr. Div., ASCE, 1980,106(8):1273-1289.
[89] 王立新.潮汐流动中的排放特性与数字图像处理技术的应用:(博士学位论文).北京:清华大学,1990.
[90] 陈朝泉,数字幽像和PL1F技术研究潮流底部多孔射流浓度场及其分形:(博士学位论文).北京:清华大学,1997.
[91] 丁剡,周雪漪,李玉梁,余常昭.完全深度平均紊流模型及在潮流侧向排污计算中的应用.水利学报,1994,11:70-76.
[92] 夏丽萍,林杰明,杨志峰,王金生.圆射流在非恒定横向流中的掺混扩散.水利学报,2001,(5):82—88.
[93] Chin D A. Model of buoyant-surface-wave interaction. Journal of Waterway, Port, Coastal and Ocean Engineering, 1988,114(3):331-345.
[94] Koole R, Swan C. Measurements of a 2-d non-buoyant jet in a wave environment. Coastal Engineering, 1994,24(2):151-169.
[95] Chyan J M, Hwung H H. On the interaction of a turbulent jet with waves, Journal of Hydraulic Research, 1993,31(6):791-810.
[96] Mossa M. Behavior of nonbuoyant jets in a wave environment. Journal of Hydraulic Engineering, 2004,130(7):704-717.
[97] Mori n, Chang K A. Experimental study of a horizontal jet in a wavy environment. Journal of Engineering Mechanics, 2003,129(10):1149-1155.
[98] Abdel-Rahman A A, Eleshaky M E. Diffusion characteristics of a plane jet discharged in awavy crossflowing stream. Proceedings of The Institution Of Mechanical Engineers Part C:Journal Of Mechanical Engineering Science, 2004, 218(4):411-423.
[99] 范洁川.近代流动显示技术.北京:国防工业出版社,2002.
[100] 张燕.横流冲击射流涡旋结构的实验和数值研究:(博士论文).上海:上海大学,2005.
[101] 吴龙华,严忠民,唐洪武.DPⅣ相关分析中相关窗口大小的确定.水科学进展.2002,13(5):594—598.
[102] Hussain, A. K. M. F. and Reynolds, W. C.. The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech.. 1970, 41:295-300.
[103] Hwung, H. H., Wang, S. C. and Lin, C.. The investigation of turbulent characteristics in the surf zone by LDV. Proc. 10th Conf. Ocean Eng., Taiwan, 1988: 17-34.
[104] Pope, S. B.. Turbulent flows. U.K.: Cambridge University Press, 2000.
[105] 陶文铨.数值传热学(第2版).西安:西安交通大学出版社,2001.
[106] 倪浩清,王能家,周力行.应力代数模型在各向异性湍浮力回流中的应用.力学学报.1989,21(1):26-33.
[107] 倪浩清,贺益英,王能家等.浅水渠道突扩湍浮力回流数学模拟.力学学报.1987,19(1):45—51.
[108] 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.
[109] 邹志利.水波理论及其应用.北京:科学出版社,2005.
[110] Launder, B.E.,Splanding, D.B, The numerical computation of turbulent flow[J].Comp. Meth,. In Appl. Mech.,1974,3:269-289.
[111] Celik I, Rodi W. Simulation of free-surface effects in turbulent channel flows. Phys Chem Hydr. 1984, 5(3):217-227.
[112] 徐高田,韦鹤平.上海市污水治理二期工程白龙岗排放口潜没多孔排放污水近区稀释扩散效果研究.海洋环境科学.2000,19(4):41-45.
[113] 李爱华,槐文信.流动环境中二维铅锤射流的试验研究及数值模拟.水利学报.2002,(12):49-55.
[114] Mcguirk J J., Rodi W. A depth-averaged mathematical model for the near field of side discharges into open channel flows. J Fluid Mech. 1978, 86:761-81.
[115] 梁书秀,沈永明,孙昭晨.深度平均的应力-通量代数全场模型及其验证.海洋环境科学.1999,18(3):33-38.
[116] 刘儒勋,王志峰.数值模拟方法和运动界面追踪.合肥:中国科学技术大学出版社,2001.