水中超空泡流及航行体弹道特性研究
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摘要
超空泡技术是主要在头部空化器的作用下，通过低压汽化和人工通气的共同作用，在航行体周围形成一个气泡并将其包裹，降低摩擦阻力，从而提高水中航行体速度的一种水中减阻技术。本文主要针对超空泡流动中相变、湍流等流体中复杂流动现象的问题，开展了超空泡流的相变模型、可压缩效应的数值计算及超空泡航行体弹道特性的研究。
研究了考虑混合相压缩性和不考虑混合相的压缩性条件下的空化源项问题。从R-P方程出发，建立了考虑混合相压缩性和不考虑混合相压缩性条件下的与蒸汽相输运方程量纲一致的自然空化源项公式及通气空化源项公式。利用CFD手段模拟了稳态空泡流动、空泡的非稳态发展特性及空泡流动的三维效应，并与文献试验数据进行对比分析。计算结果验证了数值计算公式及算法的有效性。
基于细长体理论研究了水中亚超音速可压缩空泡流动空泡形状及空化阻力的特征。研究表明，在亚音速区域，随着马赫数的增加，空泡长细比及空化阻力呈现上升的趋势；在超音速区域，随着马赫数的增加，空泡长细比及空化阻力呈现下降的趋势；同时马赫数对空泡长细比的影响没有空化数对空泡长细比的影响大。数值模拟了考虑水和蒸汽可压缩效应的水中亚音速超空泡流。数值计算结果表明，随着马赫数的增加，航行体头部的弓形区域内水的密度及流场压力明显增大。
联合考虑空泡中气体平衡方程、超空泡截面表达式、航行体与空泡之间的相互关系、水动力计算方法和航行体运动的动力学方程。研究航行体带空泡入水垂直平面弹道特性、加速段垂直平面弹道特性、巡航段无控和有控制方案垂直平面弹道特性、巡航段水平面弹道特性及空间6-DOF的弹道特性。研究表明，考虑卷入空气的弹性会引起入水空泡的不稳定，并会在空泡壁面出现壁面波；采取控制方案以后，质心位置基本保持在定深位置很少的偏差范围内变化；当同时采取重力和V型超空泡水翼稳定方案时，航行体水平面姿态将呈现出保持在较小的幅值以较大频率有阻尼振荡。
Supercavitating technology is a kind of underwater drag reduction technology which canreduce the friction drag through forming a cavity and enwrapping the vehicle with the actionof cavitator、 low pressure vaporization and artificial ventilation. The studied ofsupercavitating flow numerical simulation with considering the phase transition model,turbulence and compressibility effects and the trajectory characteristics of supercavitationvehicle were carried out.
The cavitation source term was studied under the condition of with and withoutconsidering the compressibility of mixed multiphase. The nature and ventilated cavitationsource term which dimension is consistent with the transport equation of vapor phase wereestablished based on the R-P equation under the condition of with and without considering thecompressibility of mixed multiphase. The steady cavitation flows, the cavity developmentcharacteristics of unsteady cavitation flows and the3D effects of cavitation flows werestudied and compared with experimental data of references. The validity of the formulas andalgorithms were verified based on the simulation results.
The cavity shape and cavitation drag characteristics of sub and supersonic cavitationflow in water were studied based on the slender body theory. The results shows that the valueof cavity slenderness ratio and cavitation drag is increases and decreases with the Machnumber increases in subsonic and supersonic region respectively and the influence of Machnumber to the cavity slenderness ratio is smaller than the cavitation number’s. The subsonicwith considering the compressibility effect of water and vapor were numerical simulated inwater. The results shows that the value of pressure and water density are increased evidentlyin bow region of the vehicle head in the subsonic supercavitation flows in water with theMach number increases.
The gas balance equations in cavity、equations of cavity section expression、positionrelationship between the vehicle and cavity、hydrodynamic calculation methods and vehicledynamic equations were considered. The trajectory characteristics of vehicle high-speed waterentry with supercavitation in vehicle vertical plane were studied. The trajectory characteristicsof vehicle with and without control scheme in ventilation accelerated motion stage（VAMS）and cruise stage were studied. The vehicle trajectory characteristics of horizontalmovement and6-DOF movement were studied. The results show that the water entry cavityunsteady phenomena will be arosed with considering the gas elasticity, and the wall-wavephenomena will be appeared; the vehicle mass centre remains in the fixed deep positionwithin a small deviation with considering the control scheme; the vehicle gesture in horizontalplane are damped oscillation with small amplitude and big frequency with considering thegravity and V-shape supercavitating hydrofoil stabilization schemes.
研究了考虑混合相压缩性和不考虑混合相的压缩性条件下的空化源项问题。从R-P方程出发，建立了考虑混合相压缩性和不考虑混合相压缩性条件下的与蒸汽相输运方程量纲一致的自然空化源项公式及通气空化源项公式。利用CFD手段模拟了稳态空泡流动、空泡的非稳态发展特性及空泡流动的三维效应，并与文献试验数据进行对比分析。计算结果验证了数值计算公式及算法的有效性。
基于细长体理论研究了水中亚超音速可压缩空泡流动空泡形状及空化阻力的特征。研究表明，在亚音速区域，随着马赫数的增加，空泡长细比及空化阻力呈现上升的趋势；在超音速区域，随着马赫数的增加，空泡长细比及空化阻力呈现下降的趋势；同时马赫数对空泡长细比的影响没有空化数对空泡长细比的影响大。数值模拟了考虑水和蒸汽可压缩效应的水中亚音速超空泡流。数值计算结果表明，随着马赫数的增加，航行体头部的弓形区域内水的密度及流场压力明显增大。
联合考虑空泡中气体平衡方程、超空泡截面表达式、航行体与空泡之间的相互关系、水动力计算方法和航行体运动的动力学方程。研究航行体带空泡入水垂直平面弹道特性、加速段垂直平面弹道特性、巡航段无控和有控制方案垂直平面弹道特性、巡航段水平面弹道特性及空间6-DOF的弹道特性。研究表明，考虑卷入空气的弹性会引起入水空泡的不稳定，并会在空泡壁面出现壁面波；采取控制方案以后，质心位置基本保持在定深位置很少的偏差范围内变化；当同时采取重力和V型超空泡水翼稳定方案时，航行体水平面姿态将呈现出保持在较小的幅值以较大频率有阻尼振荡。
Supercavitating technology is a kind of underwater drag reduction technology which canreduce the friction drag through forming a cavity and enwrapping the vehicle with the actionof cavitator、 low pressure vaporization and artificial ventilation. The studied ofsupercavitating flow numerical simulation with considering the phase transition model,turbulence and compressibility effects and the trajectory characteristics of supercavitationvehicle were carried out.
The cavitation source term was studied under the condition of with and withoutconsidering the compressibility of mixed multiphase. The nature and ventilated cavitationsource term which dimension is consistent with the transport equation of vapor phase wereestablished based on the R-P equation under the condition of with and without considering thecompressibility of mixed multiphase. The steady cavitation flows, the cavity developmentcharacteristics of unsteady cavitation flows and the3D effects of cavitation flows werestudied and compared with experimental data of references. The validity of the formulas andalgorithms were verified based on the simulation results.
The cavity shape and cavitation drag characteristics of sub and supersonic cavitationflow in water were studied based on the slender body theory. The results shows that the valueof cavity slenderness ratio and cavitation drag is increases and decreases with the Machnumber increases in subsonic and supersonic region respectively and the influence of Machnumber to the cavity slenderness ratio is smaller than the cavitation number’s. The subsonicwith considering the compressibility effect of water and vapor were numerical simulated inwater. The results shows that the value of pressure and water density are increased evidentlyin bow region of the vehicle head in the subsonic supercavitation flows in water with theMach number increases.
The gas balance equations in cavity、equations of cavity section expression、positionrelationship between the vehicle and cavity、hydrodynamic calculation methods and vehicledynamic equations were considered. The trajectory characteristics of vehicle high-speed waterentry with supercavitation in vehicle vertical plane were studied. The trajectory characteristicsof vehicle with and without control scheme in ventilation accelerated motion stage（VAMS）and cruise stage were studied. The vehicle trajectory characteristics of horizontalmovement and6-DOF movement were studied. The results show that the water entry cavityunsteady phenomena will be arosed with considering the gas elasticity, and the wall-wavephenomena will be appeared; the vehicle mass centre remains in the fixed deep positionwithin a small deviation with considering the control scheme; the vehicle gesture in horizontalplane are damped oscillation with small amplitude and big frequency with considering thegravity and V-shape supercavitating hydrofoil stabilization schemes.
引文
[1] J.P.Franc, J.M.Michel. Fundamentals of Cavitation[M]. Kluwer Academic Publishers,2004.
[2] J.M.Michel. Introduction to Cavitation and Supercavitation[C]. Paper presented at theRTO AVT Lecture Series on “Supercavitating Flows”, held at the von Kármán Institute（VKI） in Brussels, Belgium, and publicshed in RTO （PP RTO AVT LS “SF”VKI） EN-01,12-16February2001.
[3] G.M.Fridman, A.S. Achkinadze. Review of Theoretical Approaches to NonlinearSupercavitating Flows[C]. PP RTO AVT LS “SF” VKI EN-10,12-16February2001.
[4] K.V.Rozhdestvensky. Supercavitating Flows: Small Perturbation Theory and MatchedAsymptotics[C]. PP RTO AVT LS “SF” VKI EN-18,12-16February2001.
[5] K.V.Rozhdestvensky, G.M.Fridman. Supercavitating Nonlinear Flow Problems:Matched Asymptotics[C]. PP RTO AVT LS “SF” VKI EN-19,12-16February2001.
[6] S.A.Kinnas. Supercavitating2-D Hydrofoils: Prediction of Performance and Design[C].PP RTO AVT LS “SF” VKI EN-21,12-16February2001.
[7] S.A.Kinnas. Supercavitating3-D Hydrofoils and Propellers: Prediction of Performanceand Design[C]. PP RTO AVT LS “SF” VKI2001EN-23.
[8] A.S.Achkinadze. Supercavitating Propellers[C]. PP RTO AVT LS “SF” VKI EN-22,12-16February2001.
[9] P.Garabedian. Calculation of axially symmetric cavities and jets[J]. Pac. Math.,1956,（4）:611-684.
[10] M.Tulin. Supercavitating flows-small perturbation theory[J]. J. Ship Res.,1964,（7）:3,16-37.
[11] G.V.Logvinivich. Hydrodynamics of Flows with Free Boundaries[M]. Englishtranslation, New York: Halsted Press,1973.
[12] E.V.Paryshev. Mathematical Modeling of Unsteady Cavity Flows [C]. FifthInternational Symposium on Cavitation （CAV2003） Osaka, Japan, November1-4,2003.
[13] E.V.Paryshev. On unsteady planing of body over liquid curvilinear surface[J]. Secondinternational summer scientific school, High Speed Hydrodynamics, June2004,Cheboksary, Russia.
[14] E.V.Paryshev. Approximate mathematical models in high-speed hydrodynamics[J].Journal of Engineering Mathematics,2006,55:41-64.
[15] V.N.Semenenko. Artificial Supercavitation. Physics and Calculation[C]. PP RTO AVTLS “SF” VKI EN-11,12-16February2001.
[16] V.N.Semenenko. Dynamic Processes of Supercavitation and Computer Simulation [C].PP RTO AVT LS “SF” VKI EN-12,12-16February2001.
[17] Yu.N.Savchenko, V.N.Semenenko. The gas absorption into supercavity from liquid-gas bubble mixture[C]. Proc. Third International Symp. on Cavitation, Grenoble（France）,1998,2,49-53.
[18] V.N.Semenenko. Instability and oscillation of gas-filled supercavities[C]. Proc. ThirdInternational Symp. on Cavitation, Grenoble （France）,1998,2,25-30.
[19] V.N.Semenenko. Unsteady flow calculations past ventilated hydrofoils[C]. Proc.Seventh International Conf. on Numerical Ship Hydrodynamics. Nantes （France）,1999.
[20] V.N.Semenenko. Prediction of the2-d unsteady supercavity shapes[C] Presented atCAV2001: Fourth International Symposium on Cavitation. California Institute ofTechnology, Pasadena, CA USA. June20-23,2001.
[21] V.N.Semenenko. Notes of two-dimensional unsteady supercavity calculations[C]. Proc.International Summer Scientific School "High Speed Hydrodynamics". Cheboksary,Russia:2002, pp.167-176.
[22] V.N.Semenenko, T.N. Semenenko. Prediction of the unsteady supercavities incascades[C]. Presented at CAV2003: Fifth International Symposium on Cavitation.Osaka （Japan）, Nov1-4,2003.
[23] Yu.N.Savchenko. Perspectives of the supercavitation flow applications [C].International Conference on innovative approaches to further increase speed of fastmarine vehicles, moving above, under and in water surface, SuperFAST’2008, Saint-Petersburg, Russia July2-4,2008.
[24] Yu.N.Savchenko. Control of Supercavitation Flow and Stability of SupercavitatingMotion of Bodies[C]. PP RTO AVT LS “SF” VKI EN-14,12-16February2001.
[25] Yu.N.Savchenko. Supercavitating Object Propulsion [C]. PP RTO AVT LS “SF” VKIEN-17,12-16February2001.
[26] B.I.Romanovsky. Symmetric free-streamline flow over contours with a water intake[J].Trudy TsAGI,1980,（2060）:31-53（in Russian）.
[27] M.Yu.Tseytlin, V.M.Lapin. Free-streamline flow over a polygonal contour with awater intake having a flange for lift creation[J]. Trudy TsAGI,1993,（2496）:12-27（in Russian）.
[28] I.Nesteruk. Drag Reduction Tools in High-Speed Hydrodynamics: Supercavitation orUnseparated Shapes[C]. Sixth International Symposium on Cavitation CAV2006,Wageningen, The Netherlands, September,2006.
[29] I.Nesteruk. The Slender Axisymmetric Cavity Form Calculations Based on theIntegral-Differential Equation[J]. Izv.AN SSSR, MZhG,1985,5：83~90（inRussian）.
[30] I.Nesteruk. Some Problems of Slender Axisymmetric Cavitator Dynamics[J].Transactions of Boundary Problems Seminar in Kazan University,1990,24：187~197P（in Russian）.
[31] G.A.Aleve. Separating flow of circle cone by transonic flow of water[J]. Trans. of ASUSSR, MFG,1983,（1）:152-154.
[32] G.A.Aleve.3-D problem of disc entering in compressible fluid[J]. Trans. of AS USSR,1988, MFG,（1）:17-20.
[33] V.A.Frolov. High-speed Compressible Flows about Axisymmetric Bodies[C]. Cav03-OS-7-004, Fifth International Symposium on Cavitation （CAV2003） Osaka, Japan,November1-4,2003.
[34] N.M.Alexander. Planing for the Sub–and Supersonic Speeds[C]. Cav03-OS-7-009,Fifth International Symposium on Cavitation （CAV2003） Osaka, Japan, November1-4,2003.
[35] L.M.Zigangareva, O.M.Kiselev. Subsonic cavitational flow by compressible fluid forsmall Cavitation Numbers[J]. Trans RAN,1998,（4）:27-34.
[36] L.M.Zigangareva, O.M.Kiselev. Calculation of the Compressible Subsonic CavitationFlow past a Circular Cone[J]. Prikl. Mat. Mekh.,1994,58（4）:93-107（reprinted inthe USA）.
[37] A.D.Vasin. Thin Axisymmetric Cavities in Subsonic Compressible Flow[J]. zv. Akad.Nauk SSSR, Mekh. Zhidk. Gaza,1987,（5）:174-177（reprinted in the USA）.
[38] V.V.Serebryakov. Asymptotic Solutions of Axisymmetrical Problems of Subsonic andSupersonic Separated Water Flows with Zero Cavitation Numbers[J]. Dokl. Akad.Nauk Ukraine,1992,（9）:66-71（in Russian）.
[39] I.Nesteruk. Influence of the Flow Unsteadiness, Compressibility and Capillarity onLong Axisynnetric Cavitues[C].Cav03-GS-6-004,Fifth International Symposium onCavitation （CAV2003）Osaka, Japan, November1-4,2003.
[40] V.V.Serebryakov. Some models of prediction of supercavitation flows based onslender body approximation[C]. Fourth International Symposium on Cavitation,California Institute of Technology, Pasadena, CA USA, June20-23,2001.
[41] V.V.Serebryakov. Models of the Supercavitation Prediction for High-Speed Motion inWater[C]. Proceedings of International scientific school "High speed Hydrodynamics,2002, Jun.16-23, Chebocsary, Russia, pp.71-92（invited lecture）.
[42] V.V.Serebryakov.Guenter Schnerr. Some Problems of Hydrodynamics for Sub-, Trans-Supersonic Motion in Water with Supercavitation[C]. Fifth International Symposiumon Cavitation （CAV2003） Osaka, Japan, Cav03-OS7-015November1-4,2003.
[43] V.V.Serebryakov, G.H. Schnerr. On the Theory for Subsonic, Transonic andSupersonic Flows in Water With Supercavitation[C]. FM5S_12066, XXI InternationalCongress of Theoretical and Applied Mechanics Warsaw, Poland, August15–21,2004.
[44] V.V.Serebryakov. On methods for analysis of supercavitation in a high-speed motionin water: incompressible approximation[J]. International Journal of Fluid MechanicsResearch,2006,33（2）:186-210.
[45] A.D.Vasin. Thin Axisymmetric Cavities in Subsonic Compressible Flow[J]. zv. Akad.Nauk SSSR, Mekh. Zhidk. Gaza,1987,（5）:174-177（reprinted in the USA）.
[46] A.D.Vasin. Application of the Slender Body Theory to Investigate DevelopedAxisymmetric Cavitation Flows in a Subsonic Stream of Compressible Fluid[J].Applied hydromechanics, Kiev,2000,2（74）:17-25（in Russian）.
[47] A.D.Vasin. Supercavities in Compressible Fluid[C]. PP RTO AVT LS “SF” VKI EN-016,12-16February2001.
[48] A.D.Vasin. The Principle of Independence of the Cavity Sections Expansion（Logvinovich’s Principle） as the Basis for Investigation on Cavitation Flows[C]. PPRTO AVT LS “SF” VKI EN-08,12-16February2001.
[49] A.D.Vasin. Shocks and Conical Flows in a Supersonic Water Stream[J] Izv. Rus. Akad.Nauk, Mekh. Zhidk. Gaza,1998,（5）:196-199（reprinted in the USA）.
[50] A.D.Vasin. Problems of Hydrodynamics and Hydroelasticity at High Speed BodyMotion in Water[D].1999, Dissertation for a doctor.s degree, TsAGI, Moscow,（inRussian）.
[51] A.D.Vasin. High-Speed Body Motion in Compressible Fluid[C]. In Proceedings of theScientific Meeting on High-Speed Hydrodynamics and Supercavitation. Laboratoiredes Ecoulements Geophysiques et Industriels, Grenoble, France,2000.
[52] A.D.Vasin. Some problems of supersonic cavitation flows [C]. The4th InternationalSymposium on Cavitation, Pasadena, California,20-23June,2001.
[53] Massimo Ruzzene. Dynamic buckling of periodically stiffened shells: application tosupercavitating vehicles[J]. International Journal of Solids and Structures,2004（41）:1039-1059.
[54] Massimo Ruzzene. Non-axisymmetric buckling of stiffened supercavitating shells:static and dynamic analysis[J]. Computers&Structures,2004（82）:257-269.
[55] M.Ruzzene, F.Soranna. Impact Dynamics of Elastic Stiffened Supercavitatingunderwater Vehicles[J]. Journal of Vibration and Control,2004（10）:243-267.
[56] S.S.Ahn, M.Ruzzene. Optimal design of cylindrical shells for enhanced bucklingstability: Application to supercavitating underwater vehicles[J]. Finite Elements inanalysis and design,2006（42）:967-976.
[57] J.Y.Choi, M.Ruzzene. Stability analysis of supercavitating underwater vehicles withadaptive cavitator[J]. International Journal of Mechanical Sciences.
[58] Edward Alyanak et al. Structural response and optimization of a supercavitatingtorpedo[J]. Finite Elements Analysis,2005（6）:563~582.
[59] Edward Alyanak et al. Structural response of a supercavitating torpedo shell[A].45thAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics&MaterialsConference,19-22April2004, Palm Springs,California.
[60] Edward Alyanak et al. Variable shape cavitator design for a supercavitating torpedo[A].10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference,30August-1September2004, Albany, New York.
[61] Edward Alyanak et al. Optimum design of a supercavitating torpedo consideringoverall size, shape, and structural configuration[J]. International Journal of Solids andStructures,2006（43）:642-657.
[62] J.H. Choi, R.C. Penmetsa and R.V. Grandhi, Shape optimization of the cavitator for asupercavitating torpedo[J]. Struct Multidisc Optim,2005（29）:159~167.
[63]董世汤.有厚度水翼局部空泡流的理论解[J].中国造船,1983, pp.15-27.
[64]刘桦,李家春,何友声等.“十一五”水动力学发展规划的建议[J].力学进展,2007,37（1）:142-147.
[65]冯光，颜开.超空泡航行体水下弹道的数值计算[J].船舶力学,2005,9（2）:1-8.
[66]安伟光,蒋运华,安海.运动体高速入水非定常过程研究[J].工程力学,2011,28（3）:251-256.
[67]顾永维,安伟光,安海.水下高速航行体的抗弯稳定可靠性分析[J].哈尔滨工程大学学报,2008,29（7）:683-686.
[68]安伟光,顾永维,安海.初偏心影响下的水下高速航行体的抗弯稳定性及可靠性分析[J].兵工学报,2008,29（7）:824-828.
[69]周凌，安伟光，安海.超空泡航行体强度与稳定性的非概率可靠性分析[J].哈尔滨工程大学学报，2009，30（4）：362-367.
[70] An Wei-guang, Zhou Ling, An Hai. Structure buckling and non-probabilistic reliabilityanalysis of supercavitating vehicles[J]. Jou rnal of Harbin Institu te of Technology（New Series）,2009,16（4）:561-569.
[71]杨传武,王安稳.超空泡水下航行体振动特性分析[J].海军工程大学学报,2008,20（4）:12-14.
[72]杨传武,刘刚，王安稳.超空泡体结构响应问题的有限元分析[J].海军工程大学学报,2008,20（2）:101-104.
[73]杨传武,王安稳.动态轴向载荷对超空泡航行体振动特性的影响[J].华中科技大学学报,2008,36（12）:71-74.
[74]杨传武,王安稳.冲击载荷作用下超空泡水下航行体的结构响应[J].华中科技大学学报,2008,36（7）:129-132.
[75] O.Boulon, G.L.Chahine. Numerical simulation of unsteady cavitation on3Dhydrofoil[C]. Third International Symposium on Cavitation, Grenoble, France, April,1998.
[76] L.Dieval, M.Arnaud, R.Marcer. Numerical modeling of unsteady cavitating flows by aVOF method[C]. Third International Symposium on Cavitation, Grenoble, France,April,1998.
[77] F.Salvatore, P.G.Esposito. An improved boundary element analysis of cavitating three-dimensional hydrofoils[C]. Fourth International Symposium on Cavitation, California,USA, June,2001.
[78] S.A.Kinnas, H.Sun, H.Lee Numerical analysis of flow around the cavitating CAV2003hydrofoil[C]. Fifth International Symposium on Cavitation, Osaka, Japan, November,2003.
[79] I.N. Kirschner, Neal E. Fine, James S. Uhlman, et al. Numerical Modeling ofSupercavitating Flows[C]. PP RTO AVT LS “SF” VKI EN-09,12-16February2001.
[80] Cheng Xiao-jun, Lu Chuan-jing. On the partially cavitating flow around two-dimensional hydrofoils[J]. Applied Mathematics and Mechanics,2000,21（2）:1450-1459.
[81]程晓俊,鲁传敬.二维水翼的局部空泡流研究[J].应用数学和力学,2002,21（12）:1310-1318.
[82]冷海军,鲁传敬.轴对称体的局部空泡流研究[J].上海交通大学学报,2002,36（3）:395-398.
[83]戚定满,鲁传敬.空泡噪声的数值研究[J].水动力学研究与进展, A辑,2001,16（1）:9-17.
[84] A.Kubota, H.Kato, H.Yamaguchi. A new modeling of cavitating flows: a numericalstudy of unsteady cavitation on a hydrofoil section[J]. J. Fluid Mech.,1992, Vol.240,pp.59-96.
[85] Y.Delannoy, J.L.Kueny. Two-phase flow approach in unsteady cavitation modeling[C].ASME Cavitation and Multiphase Flow Forum, Toronto, June,1990.
[86] C.Song, J.He. Numerical simulation of cavitating flows by single-phase flowapproach[C]. Third International Symposium on Cavitation, Grenoble, France, April,1998, pp.295–300.
[87] C.L.Merkle, J.Z.Feng, P.E.O.Buelow. Computational modeling of the dynamics ofsheet cavitation[C]. Third International Symposium on Cavitation, Grenoble, France,April,1998, pp307–313.
[88] J.C.Reboud, Y.Delannoy.Two-phase flow modeling of unsteady cavitation[C]. SecondInternational Symposium on Cavitation, Tokyo, Japan, April,1994, pp.39-44.
[89] R.F.Kunz, T.S.Chyczewski, D.A.Boger. Multiphase CFD analysis of natural andventilated cavitation about submerged bodies[C]. Third ASME/JSME Joint FluidsEngineering Conference, San Francisco, USA,1999.
[90] R.F.Kunz, D.A.Boger, D.R.Stinebring. A Preconditioned Navier-Stokes method fortwo-phase flows with application to cavitation prediction[J]. Computers&Fluids,2000,Vol.29, pp.849-875.
[91] Y.Chen, S.Heister. Two-phase modeling of cavitating flows[J]. Computers and fluids,1995,24（7）:799-809.
[92] Y.Ventikos, G.Tzabiras. A numerical method for the simulation of steady and unsteadycavitating flows[J]. Computer&Fluids,2000, Vol.29, pp.63-88.
[93] A.Hosangadi, V.Ahuja, S.Arunajatesan. A generalized compressible cavitationmodel[C]. Fourth International Symposium on Cavitation, California, USA, June,2001.
[94] W. X.Yuan, J.Sauer, G.H.Schnerr. Modeling and computation of unsteady cavitationflows in injection nozzles[J]. Mec. Ind.,2001, No.2,383-394.
[95] I.Senocak, W.Shyy. A pressure-based method for turbulent cavitating flowcomputations[J]. J. Comp. Physics,2002, Vol.176, pp.363-383.
[96] A.K.Singhal, M.M.Athavale, H.Y.Li, et al. Mathematical basis and validation of thefull cavitation model[J]. J. Fluids Eng.,2002, Vol.124, pp.617-624.
[97] Y.Saito, I.Nakamori, T.Ikohagi. Numerical analysis of unsteady vaporous cavitatingflow around hydrofoil[C]. Fifth International Symposium on Cavitation, Osaka, Japan,November,2003.
[98] O.Coutier-Delgosha, R.Fortes-Patella, J.L.Reboud.Evaluation of the turbulence modelinfluence on the numerical simulations of unsteady cavitation[J]. J. Fluids Eng.,2003,125（1）:38-45.
[99] Q.Qin, C.C.S.Song, R.E. A.Arndt. A numerical study of an unsteady turbulent wakebehind a cavitating hydrofoil[C]. Fifth International Symposium on Cavitation, Osaka,Japan, November,2003.
[100] R.Vaidyanathan, I.Senocak, J.Y.Wu. Sensitivity evaluation of a transport-basedturbulent cavitation model[J]. J. Fluids Eng.,2003,125（5）:447-458.
[101] M.P.Kinzel, J.W.Lindau, E.G.Paterson, et al. The effects of a free-surface on computedturbulent, super and partially cavitating flows[C]. Sixth International Symposium onCavitation,W ageningen, The Netherlands, September,2006.
[102] S.H.akai, T.Ohta, T.Kajishima. Direct numerical simulation of vortex cavitation inturbulent shear layer[C]. Sixth International Symposium on Cavitation, Wageningen,The Netherlands, September,2006.
[103]熊永亮,郜冶,王革.水下超空泡航行体减阻能力的数值研究[J].弹道学报，2007，19（1）:51-54.
[104] Y.L.Xiong, Y.Gao, W.G.An. Comparisons of turbulence models in predicting unsteadysupercavitating flow[C]. Sixth International Symposium on Cavitation, Wageningen,The Netherlands, September,2006.
[105]袁绪龙,张宇文,杨武刚.高速超空化航行体典型空化器多相流CFD分析[J].弹箭与制导学报.2005,25（1）:53-59.
[106]傅慧萍,鲁传敬,吴磊.回转体空泡流特性研究[J].水动力学研究与进A,2005,20（1）:84-89.
[107]王海斌,张嘉钟,魏英杰等.空泡形态与典型空化器参数关系的研究---小空泡数下的发展空泡形态.水动力学研究与进展A,2005,25（2）:251-257.
[108]汤继斌，钟诚文.空化、超空化流动的数值模拟方法研究[J].力学学报,2005,37（5）:640-644.
[109]贾力平,王聪,于开平等.空化器参数对通气超空泡形态影响的实验研究[J].工程力学,2007,24（3）:159-163.
[110]陈鑫.通气空泡流研究[D].上海：上海交通大学,2006,12.
[111] Robert F. Kunz， Jules W. Lindau， Michael L. Billet， David R. Stinebring.Multiphase CFD Modeling of Developed and Supercavitating Flows[C]. PP RTO AVTLS “SF” VKI EN-013,12-16February.
[112] J.W.Lindau, R.F.Kunz, S.Venkateswaran, et al. Application of a Preconditioned,Multiple-Species, Compressible, Navier-Stokes Model to Cavitating Flows[C]. The4thInternational Symposium on Cavitation, Pasadena, California,20-23June,2001.
[113] J.W.Lindau, S.Venkateswaran, R.F.Kunz, et al. Development of a Fully-CompressibleMulti-Phase Reynolds-Averaged Navier-Stokes Model[C]. Submitted to the15thAIAA CFD Conference, Anaheim, California,11-14June,2001.
[114] Sankaran Venkateswaran, Jules W. Lindau, Robert F. Kunz, et al. Computation ofMultiphase Mixture Flows with Compressibility Effects[J]. Journal of ComputationalPhysics2002,180,54–77.
[115] Jules W. Lindau, Sankaran Venkateswaran, Robert F. Kunz, et al. Computation ofCompressble Multiphase Flows[C]. AIAA2003-1285,41st Aerospace SciencesMeeting and Exhibit6-9January2003, Reno, Nevada.
[116] Jules W. Lindau, Robert F. Kunz, Venkateswaran Sankaran, et al. HomogeneousMultiphase CFD Modeling of Large Scale Cavities [C]. European Congress onComputational Methods in Applied Sciences and Engineering ECCOMAS2004.
[117] A.D.Vasin. Calculation of Axisymmetric Cavities Downstream of a Disk in SubsonicCompressible FluidFlow[J]. Izv. Rus. Akad. Nauk, Mekh. Zhidk. Gaza,1996,（2）:94-103（reprinted in the USA）.
[118] A.D.Vasin Calculation of Axisymmetric Cavities Downstream of a Disk in SupersonicFlow[J]. Izv. Rus. Akad. Nauk, Mekh. Zhidk. Gaza,1997,（4）:54-62（reprinted inthe USA）.
[119] M. Schaffar, H.J. Pfeifer. Comparison of Two Computational Methods for High-Velocity Cavitating Flows around Conical Projectiles: OTi-HULL Hydrocode andTwo-Phase Flow Method [C]. The4th International Symposium on Cavitation,Pasadena, California,20-23June,2001.
[120] F.M.Owis, A.H.Nayfeh, Computation of the compressible multiphase flow over thecavitating high-speed torpedo[J]. J. Fluids Eng.,2003,125（5）:459-468.
[121] Duncan R. van der Heul, C. Vuik, P. Wesseling. Efficient Computation of Flow withCavitation by Compressible Pressure Correction[C]. European Congress onComputational Methods in Applied Sciences and Engineering ECCOMAS2000,Barcelona,11-14September2000.
[122] Vineet Ahuja, Ashvin Hosangadi, Srinivasan Arunajatesan. Simulations of CavitatingFlows Using Hybrid Unstructured Meshes [J]. Journal of Fluids Engineering,2001,123,331-340.
[123] V.Ahuja, A.Hosangadi, R.Ungewitter, et al. A Hybrid Unstructured Mesh Solver forMutli-Fluid Mixtures[J].1999, AIAA Paper99-3330.
[124] R.Saurel, J.P.Cocchi, P.B.Butler. Numerical Study of Cavitation in the Wake of aHypervelocity Underwater Projectile[J] AIAA Journal of Propulsion and Power,1999,15（4）:513-520.
[125] Gunter H. Schnerr, Steffen Schmidt, Ismail Sezal, Matthias Thalhamer. Shock andwave dynamics of compressible liquid flows with special emphasis on unsteady loadon hydrofoils and on cavitation in injection nozzles[C]. Sixth International Symposiumon Cavitation CAV2006, Wageningen, The Netherlands, September,2006.
[126]鲁传敬，陈瑛.空泡流数值模拟方法研究[C].第九届全国水动力学学术会议暨第二十二届全国水动力学研讨会文集，中国四川成都，2009,8.
[127]张鹏,傅慧萍.跨超音速射弹的超空泡数值模拟[J].弹箭与制导学报，2009,29（5）:166-169.
[128]贾力平.空化器诱导超空泡特性的数值仿真与试验研究[D].哈尔滨工业大学,2007,6.
[129]易文俊,李月洁,王中原等.小攻角下水下高速射弹的空泡形态特性[J].南京理工大学学报（自然科学版）,2008,32（4）:464-467.
[130]易文俊,王中原,熊天红等.水下高速射弹超空泡减阻特性研究[J].弹道学报,2008,20（4）:1-4.
[131]熊天红,李铁鹏,易文俊等.水下高速射弹超空泡形态与阻力特性研究[J].弹道学报,2009,21（2）:100-102.
[132] R.J.Etter. State of the art-cavitation test facilities and experimental methods[C].NCT’50International Conference on Propeller Cavitation,3-5April, University ofNewcastle, Newcastle uponTyne, United Kingdom,2000.
[133] H.Reichardt. The Laws if Cavitation Bubbles at Axially Symmetrical Bodies in a Flow
[Z]. Ministry of Aircraft Production Volkenrode, MAP-VC, Report andTranslations766, ONR,1946.
[134] J.P.Onerll. Flow Around Bodies with Attached Open Cavities [Z]. CIT Hyd. Rept. E-24:7,1954.
[135] R.N.Cox, J.W.Maccoll. Recent Contributions to Basic Hydroballistics [C]. NavalSymp.1956, pp215-239.
[136] R.H.Oversmith. Some Observations on Cavitating Flows [Z].Convair Eng. Dept. Rept.ZR-659-015.1959.
[137] J.Klose, A.J.Acosta. Some New Measurements on the Drag of Cavitating Disks [J].J.Ship Res:9, pp102-104.1965.
[138] R.David. Stinebring Michael L.Billet.Jules W.Lindau.Robert F.Kunz. DevelopedCavitation Cavity Dynamics[C]. Applied Research Laboratory P.O. Box30StateCollege,PA16804USA.
[139] J.M.Michel. Ventilated cavities: a contribution to the study of pulsation mechanism[C].Symposium IUTAM on rapid, non steady liquid flows, Leningrad,1971.
[140] J.M.Michel. Some features of water flows with ventilated cavities[J]. J. of FluidsEngineering,1984,106（3）:319-326.
[141] J.M.Michel. Oscillations of Ventilated Cavities Experimental Aspects[C]. PP RTOAVT LS “SF” VKI EN-02,12-16February2001.
[142] D.P.Hart, C.E.Brennen, A.J.Acosta. Observations of cavitation on a three-dimensionaloscillating hydrofoil[C]. ASME Cavitation and Multiphase Flow Forum,1990, Vol.98,pp.49-52.
[143] L.Wilczynski. The experimental investigation of the cavitation inception in the flowaround NACA16-018hydrofoil[C]. Third International Symposium on Cavitation:Vol. I, Grenoble, France, April,1998, pp.167-172.
[144] Y.Kawanami, H.Kato, H.Yamaguchi. Three-dimensional characteristics of the cavitiesformed on a two-dimensional hydrofoil[C]. Third International Symposium onCavitation: Vol. I, Grenoble, France, April,1998, pp.191-196.
[145] L.K.R.aberteaux, S.L.Ceccio. Flow in the closure region of closed partial attachedcavitation[C]. Third International Symposium on Cavitation: Vol. I, Grenoble, France,April,1998, pp.197-202.
[146] B.Stutz, J. L.Reboud. Experiments on unsteady cavitation[J]. Experiments in Fluids,1997, No.22, pp.191-198.
[147] B.Stutz, J.L.Reboud. Measurements within unsteady cavitation[J]. Experiments inFluids,2000, Vol.29, pp.545-552.
[148] H.Lohrberg, B.Stoffel, R.Fortes-Patella, et al. Numerical and Experimentalinvestigations on the cavitating flow in a cascade of hydrofoils[J]. Experiments inFluids,2002, No.33, pp.578-586.
[149] S.Takahashi, S.Washio, K.Uemura, et al. Experimental study on cavitation starting atand flow characteristics close to the point of separation[C]. Fifth InternationalSymposium on Cavitation （Cav2003）, Osaka, Japan, No. Cav03-OS-3-003, pp.1-8,2003.
[150] S.Washio, T S.akahashi, Y.Uda, et al. Study on cavitation inception in hydraulic oilflow through a long twodimensional constriction[J]. Proc Instn Mech Engrs,2001,215（Part J）:373-386.
[151] S.Washio, S.Takahashi, S.Kawahara, et al. Study on the cavitation mechanism inhydraulic oil flow using a needle projection[J]. Proc Instn Mech Engrs,2002,216（Part J）:27-34.
[152] S.Washio, S.Takahashi, S.Yoshimori. Study on cavitation starting at the point ofseparation on a smooth wall in hydraulic oil flow[J]. Proc Instn Mech Engrs,2003,217（Part C）:619-630.
[153] Yu.N. Savchenko, Experimental Investigation of Supercavitating Motion of Bodies[C].PP RTO AVT LS “SF” VKI EN-04,12-16February2001.
[154] V.T.Savchenko. Hydrodynamic Characteristics of Rudders Operating in Air-SeaInterface[C].Paper presented at the RTO AVT Symposium on “Fluid DynamicsProblems of Vehicles Operating near or in the Air-Sea Interface”, held in Amsterdam,The Netherlands,5-8October1998, and published in RTO MP-15.
[155] Robert Kuklinski, Charles Henoch, John Castano, Experimental Study of VentilatedCavities on Dynamic Test Model[C]. Presented at CAV2001: Fourth InternationalSymposium on Cavitation. California Institute of Technology, Pasadena, CA USA.June20-23,2001.
[156] R.E.A.Arndt. From Wageningen to Minnesota and Back: Perspectives on CavitationResearch Keynote Lecture[C]. Proceedings Sixth International Symposium onCavitation, CAV2006, Wageningen, The Netherlands, September2006.
[157] E.Amromin, J.Kopriva, R.E.A.Arndt. Hydrofoil Drag Reduction by PartialCavitation[J]. J. Fluids Engineering,128, Sept.2006.
[158] B.Vanek, J.Bokor, G.J.Balas, et al. Longitudinal motion control for a high-speedsupercavitation vehicle[J]. Journal of Vibration and Control, Sept.2006.
[159] G.J.Balas, B J.okor, B.Vanek, et al. Control of high-speed underwater vehicles. Controlof Uncertain Systems: Modelling, Approximation, and Design[J]. B. Francis, M. Smithand J. Willems, eds. Berlin: Springer-Verlag.
[160] R.E.A.Arndt, G. Balas, M.Wosnik. Control of cavitating flows: A perspective[J]. JapanSoc. Mech. Eng. Series B,48（2）:334-341.
[161] Yuriy D.Vlasenko. Experimental Investigation of Supercavitation Flow Regimes atSubsonic and Transonic Speeds[C]，Cav03-GS-6-006Fifth International Symposiumon Cavitation （cav2003） Osaka, Japan, November1-4,2003.
[162] I.N. Kirschner. Results of Selected Experiments Involving Supercavitating Flows.[C].PP RTO AVT LS “SF” VKI EN-013,12-16February.
[163]何友声,刘桦,赵岗.二维空泡流的脉动性态研究[J].力学学报,1997,29（1）:1-7.
[164]刘桦,何友声,赵岗,轴对称空泡流的脉动性态研究[J].上海力学,1997,18（2）:99-105.
[165]顾魏.非稳定空泡流瞬态和周期现象实验研究[D].上海:上海交通大学,1998.
[166]谢正桐,何友声.小攻角下轴对称细长体的充气肩空泡试验研究[J].试验力学.1999,14（3）:279-287.
[167]谢正桐,何友声,朱世权.小攻角带空泡细长体的试验研究[J].水动力学研究与进展,A辑,2001,16（3）:374-381.
[168] Feng Xue-mei, Lu Chuan-jing, Hu Tian-qun, et al., The fluctuation characteristics ofnatural and ventilated cavities on an axisymmetric body[J]. Journal of Hydrodynamics,Ser. B,2005,17（1）:87-91.
[169]顾建农,张志宏,高永琪,郑学龄.充气头型对超空泡轴对称体阻力特性影响的试验研究[J].兵工学报.2004,25（6）:766-769.
[170]袁绪龙,张宇文,刘乐华等.水下航行体通气超空泡形态试验研究[J].应用力学学报.2004,21（3）:33-37.
[171]袁绪龙,张宇文,王育才等.水下航行体通气超空泡非对称性研究[J].力学学报.2004,36（2）:146-150.
[172]邓飞,张宇文,陈伟政等.头形对细长体超空泡生成与外形影响的实验研究[J].西北工业大学学报.2004年6月第22卷第3期.
[173]王海斌,张嘉钟,魏英杰等.水下航行体通气超空泡减阻特性的实验研究[J].船舶工程.2006,28（3）:20-23.
[174]熊天红,易文俊.高速射弹超空泡减阻试验研究与数值模拟分析[J].工程力学,2009,26（8）:174-178.
[175] P.J. Zwart, A.G. Gerber, T. Belamri. A Two-Phase Flow Model for PredictingCavitation Dynamics[C]. Fifth International Conference on Multiphase Flow,Yokohama, Japan,2004.
[176] G.H. Schnerr, J. Sauer. Physical and Numerical Modeling of Unsteady CavitationDynamics[J]. Fourth International Conference on Multiphase Flow, New Orleans, USA,2001.
[177] A.K.Singhal, N.Vaidya, A.D.Leonard. Multidimensional simulation of cavitating flowsusing a PDF model for phase change[C]. ASME Fluids Engineering Division SummerMeeting, ASME Paper FEDSM97-3272.
[178] C.L.Merkle, J.Feng, P.O.Buelow. Computational Modeling of the Dynamics of SheetCavitation[C]. Proc.3rd International Symposium on Cavitation, Grenoble,France,1998.
[179] Chien-Chou Tseng, Wei Shyy. Modeling for isothermal and cryogenic cavitation[J].International Journal of Heat and Mass Transfer,2010（53）:513–525.
[180] H. Rouse, J. S. McNown. Cavitation and Pressure Distribution Head Forms at ZeroAngle of Yaw[J]. Iowa Institute of Hydraulic Research, State University of Iowa, IowaCity,1948.
[181] O.Coutier-Delgosha, R.Fortes-Patella, J.L.Reboud. Evaluation of the turbulence modelinfluence on the numerical simulations of unsteady cavitation[J]. J. Fluids Eng.,2003,Vol.125, No.1, pp.38-45.
[182]苏玉民,庞永杰编.潜艇原理[M].哈尔滨:哈尔滨工程大学出版社,2005.
[2] J.M.Michel. Introduction to Cavitation and Supercavitation[C]. Paper presented at theRTO AVT Lecture Series on “Supercavitating Flows”, held at the von Kármán Institute（VKI） in Brussels, Belgium, and publicshed in RTO （PP RTO AVT LS “SF”VKI） EN-01,12-16February2001.
[3] G.M.Fridman, A.S. Achkinadze. Review of Theoretical Approaches to NonlinearSupercavitating Flows[C]. PP RTO AVT LS “SF” VKI EN-10,12-16February2001.
[4] K.V.Rozhdestvensky. Supercavitating Flows: Small Perturbation Theory and MatchedAsymptotics[C]. PP RTO AVT LS “SF” VKI EN-18,12-16February2001.
[5] K.V.Rozhdestvensky, G.M.Fridman. Supercavitating Nonlinear Flow Problems:Matched Asymptotics[C]. PP RTO AVT LS “SF” VKI EN-19,12-16February2001.
[6] S.A.Kinnas. Supercavitating2-D Hydrofoils: Prediction of Performance and Design[C].PP RTO AVT LS “SF” VKI EN-21,12-16February2001.
[7] S.A.Kinnas. Supercavitating3-D Hydrofoils and Propellers: Prediction of Performanceand Design[C]. PP RTO AVT LS “SF” VKI2001EN-23.
[8] A.S.Achkinadze. Supercavitating Propellers[C]. PP RTO AVT LS “SF” VKI EN-22,12-16February2001.
[9] P.Garabedian. Calculation of axially symmetric cavities and jets[J]. Pac. Math.,1956,（4）:611-684.
[10] M.Tulin. Supercavitating flows-small perturbation theory[J]. J. Ship Res.,1964,（7）:3,16-37.
[11] G.V.Logvinivich. Hydrodynamics of Flows with Free Boundaries[M]. Englishtranslation, New York: Halsted Press,1973.
[12] E.V.Paryshev. Mathematical Modeling of Unsteady Cavity Flows [C]. FifthInternational Symposium on Cavitation （CAV2003） Osaka, Japan, November1-4,2003.
[13] E.V.Paryshev. On unsteady planing of body over liquid curvilinear surface[J]. Secondinternational summer scientific school, High Speed Hydrodynamics, June2004,Cheboksary, Russia.
[14] E.V.Paryshev. Approximate mathematical models in high-speed hydrodynamics[J].Journal of Engineering Mathematics,2006,55:41-64.
[15] V.N.Semenenko. Artificial Supercavitation. Physics and Calculation[C]. PP RTO AVTLS “SF” VKI EN-11,12-16February2001.
[16] V.N.Semenenko. Dynamic Processes of Supercavitation and Computer Simulation [C].PP RTO AVT LS “SF” VKI EN-12,12-16February2001.
[17] Yu.N.Savchenko, V.N.Semenenko. The gas absorption into supercavity from liquid-gas bubble mixture[C]. Proc. Third International Symp. on Cavitation, Grenoble（France）,1998,2,49-53.
[18] V.N.Semenenko. Instability and oscillation of gas-filled supercavities[C]. Proc. ThirdInternational Symp. on Cavitation, Grenoble （France）,1998,2,25-30.
[19] V.N.Semenenko. Unsteady flow calculations past ventilated hydrofoils[C]. Proc.Seventh International Conf. on Numerical Ship Hydrodynamics. Nantes （France）,1999.
[20] V.N.Semenenko. Prediction of the2-d unsteady supercavity shapes[C] Presented atCAV2001: Fourth International Symposium on Cavitation. California Institute ofTechnology, Pasadena, CA USA. June20-23,2001.
[21] V.N.Semenenko. Notes of two-dimensional unsteady supercavity calculations[C]. Proc.International Summer Scientific School "High Speed Hydrodynamics". Cheboksary,Russia:2002, pp.167-176.
[22] V.N.Semenenko, T.N. Semenenko. Prediction of the unsteady supercavities incascades[C]. Presented at CAV2003: Fifth International Symposium on Cavitation.Osaka （Japan）, Nov1-4,2003.
[23] Yu.N.Savchenko. Perspectives of the supercavitation flow applications [C].International Conference on innovative approaches to further increase speed of fastmarine vehicles, moving above, under and in water surface, SuperFAST’2008, Saint-Petersburg, Russia July2-4,2008.
[24] Yu.N.Savchenko. Control of Supercavitation Flow and Stability of SupercavitatingMotion of Bodies[C]. PP RTO AVT LS “SF” VKI EN-14,12-16February2001.
[25] Yu.N.Savchenko. Supercavitating Object Propulsion [C]. PP RTO AVT LS “SF” VKIEN-17,12-16February2001.
[26] B.I.Romanovsky. Symmetric free-streamline flow over contours with a water intake[J].Trudy TsAGI,1980,（2060）:31-53（in Russian）.
[27] M.Yu.Tseytlin, V.M.Lapin. Free-streamline flow over a polygonal contour with awater intake having a flange for lift creation[J]. Trudy TsAGI,1993,（2496）:12-27（in Russian）.
[28] I.Nesteruk. Drag Reduction Tools in High-Speed Hydrodynamics: Supercavitation orUnseparated Shapes[C]. Sixth International Symposium on Cavitation CAV2006,Wageningen, The Netherlands, September,2006.
[29] I.Nesteruk. The Slender Axisymmetric Cavity Form Calculations Based on theIntegral-Differential Equation[J]. Izv.AN SSSR, MZhG,1985,5：83~90（inRussian）.
[30] I.Nesteruk. Some Problems of Slender Axisymmetric Cavitator Dynamics[J].Transactions of Boundary Problems Seminar in Kazan University,1990,24：187~197P（in Russian）.
[31] G.A.Aleve. Separating flow of circle cone by transonic flow of water[J]. Trans. of ASUSSR, MFG,1983,（1）:152-154.
[32] G.A.Aleve.3-D problem of disc entering in compressible fluid[J]. Trans. of AS USSR,1988, MFG,（1）:17-20.
[33] V.A.Frolov. High-speed Compressible Flows about Axisymmetric Bodies[C]. Cav03-OS-7-004, Fifth International Symposium on Cavitation （CAV2003） Osaka, Japan,November1-4,2003.
[34] N.M.Alexander. Planing for the Sub–and Supersonic Speeds[C]. Cav03-OS-7-009,Fifth International Symposium on Cavitation （CAV2003） Osaka, Japan, November1-4,2003.
[35] L.M.Zigangareva, O.M.Kiselev. Subsonic cavitational flow by compressible fluid forsmall Cavitation Numbers[J]. Trans RAN,1998,（4）:27-34.
[36] L.M.Zigangareva, O.M.Kiselev. Calculation of the Compressible Subsonic CavitationFlow past a Circular Cone[J]. Prikl. Mat. Mekh.,1994,58（4）:93-107（reprinted inthe USA）.
[37] A.D.Vasin. Thin Axisymmetric Cavities in Subsonic Compressible Flow[J]. zv. Akad.Nauk SSSR, Mekh. Zhidk. Gaza,1987,（5）:174-177（reprinted in the USA）.
[38] V.V.Serebryakov. Asymptotic Solutions of Axisymmetrical Problems of Subsonic andSupersonic Separated Water Flows with Zero Cavitation Numbers[J]. Dokl. Akad.Nauk Ukraine,1992,（9）:66-71（in Russian）.
[39] I.Nesteruk. Influence of the Flow Unsteadiness, Compressibility and Capillarity onLong Axisynnetric Cavitues[C].Cav03-GS-6-004,Fifth International Symposium onCavitation （CAV2003）Osaka, Japan, November1-4,2003.
[40] V.V.Serebryakov. Some models of prediction of supercavitation flows based onslender body approximation[C]. Fourth International Symposium on Cavitation,California Institute of Technology, Pasadena, CA USA, June20-23,2001.
[41] V.V.Serebryakov. Models of the Supercavitation Prediction for High-Speed Motion inWater[C]. Proceedings of International scientific school "High speed Hydrodynamics,2002, Jun.16-23, Chebocsary, Russia, pp.71-92（invited lecture）.
[42] V.V.Serebryakov.Guenter Schnerr. Some Problems of Hydrodynamics for Sub-, Trans-Supersonic Motion in Water with Supercavitation[C]. Fifth International Symposiumon Cavitation （CAV2003） Osaka, Japan, Cav03-OS7-015November1-4,2003.
[43] V.V.Serebryakov, G.H. Schnerr. On the Theory for Subsonic, Transonic andSupersonic Flows in Water With Supercavitation[C]. FM5S_12066, XXI InternationalCongress of Theoretical and Applied Mechanics Warsaw, Poland, August15–21,2004.
[44] V.V.Serebryakov. On methods for analysis of supercavitation in a high-speed motionin water: incompressible approximation[J]. International Journal of Fluid MechanicsResearch,2006,33（2）:186-210.
[45] A.D.Vasin. Thin Axisymmetric Cavities in Subsonic Compressible Flow[J]. zv. Akad.Nauk SSSR, Mekh. Zhidk. Gaza,1987,（5）:174-177（reprinted in the USA）.
[46] A.D.Vasin. Application of the Slender Body Theory to Investigate DevelopedAxisymmetric Cavitation Flows in a Subsonic Stream of Compressible Fluid[J].Applied hydromechanics, Kiev,2000,2（74）:17-25（in Russian）.
[47] A.D.Vasin. Supercavities in Compressible Fluid[C]. PP RTO AVT LS “SF” VKI EN-016,12-16February2001.
[48] A.D.Vasin. The Principle of Independence of the Cavity Sections Expansion（Logvinovich’s Principle） as the Basis for Investigation on Cavitation Flows[C]. PPRTO AVT LS “SF” VKI EN-08,12-16February2001.
[49] A.D.Vasin. Shocks and Conical Flows in a Supersonic Water Stream[J] Izv. Rus. Akad.Nauk, Mekh. Zhidk. Gaza,1998,（5）:196-199（reprinted in the USA）.
[50] A.D.Vasin. Problems of Hydrodynamics and Hydroelasticity at High Speed BodyMotion in Water[D].1999, Dissertation for a doctor.s degree, TsAGI, Moscow,（inRussian）.
[51] A.D.Vasin. High-Speed Body Motion in Compressible Fluid[C]. In Proceedings of theScientific Meeting on High-Speed Hydrodynamics and Supercavitation. Laboratoiredes Ecoulements Geophysiques et Industriels, Grenoble, France,2000.
[52] A.D.Vasin. Some problems of supersonic cavitation flows [C]. The4th InternationalSymposium on Cavitation, Pasadena, California,20-23June,2001.
[53] Massimo Ruzzene. Dynamic buckling of periodically stiffened shells: application tosupercavitating vehicles[J]. International Journal of Solids and Structures,2004（41）:1039-1059.
[54] Massimo Ruzzene. Non-axisymmetric buckling of stiffened supercavitating shells:static and dynamic analysis[J]. Computers&Structures,2004（82）:257-269.
[55] M.Ruzzene, F.Soranna. Impact Dynamics of Elastic Stiffened Supercavitatingunderwater Vehicles[J]. Journal of Vibration and Control,2004（10）:243-267.
[56] S.S.Ahn, M.Ruzzene. Optimal design of cylindrical shells for enhanced bucklingstability: Application to supercavitating underwater vehicles[J]. Finite Elements inanalysis and design,2006（42）:967-976.
[57] J.Y.Choi, M.Ruzzene. Stability analysis of supercavitating underwater vehicles withadaptive cavitator[J]. International Journal of Mechanical Sciences.
[58] Edward Alyanak et al. Structural response and optimization of a supercavitatingtorpedo[J]. Finite Elements Analysis,2005（6）:563~582.
[59] Edward Alyanak et al. Structural response of a supercavitating torpedo shell[A].45thAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics&MaterialsConference,19-22April2004, Palm Springs,California.
[60] Edward Alyanak et al. Variable shape cavitator design for a supercavitating torpedo[A].10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference,30August-1September2004, Albany, New York.
[61] Edward Alyanak et al. Optimum design of a supercavitating torpedo consideringoverall size, shape, and structural configuration[J]. International Journal of Solids andStructures,2006（43）:642-657.
[62] J.H. Choi, R.C. Penmetsa and R.V. Grandhi, Shape optimization of the cavitator for asupercavitating torpedo[J]. Struct Multidisc Optim,2005（29）:159~167.
[63]董世汤.有厚度水翼局部空泡流的理论解[J].中国造船,1983, pp.15-27.
[64]刘桦,李家春,何友声等.“十一五”水动力学发展规划的建议[J].力学进展,2007,37（1）:142-147.
[65]冯光，颜开.超空泡航行体水下弹道的数值计算[J].船舶力学,2005,9（2）:1-8.
[66]安伟光,蒋运华,安海.运动体高速入水非定常过程研究[J].工程力学,2011,28（3）:251-256.
[67]顾永维,安伟光,安海.水下高速航行体的抗弯稳定可靠性分析[J].哈尔滨工程大学学报,2008,29（7）:683-686.
[68]安伟光,顾永维,安海.初偏心影响下的水下高速航行体的抗弯稳定性及可靠性分析[J].兵工学报,2008,29（7）:824-828.
[69]周凌，安伟光，安海.超空泡航行体强度与稳定性的非概率可靠性分析[J].哈尔滨工程大学学报，2009，30（4）：362-367.
[70] An Wei-guang, Zhou Ling, An Hai. Structure buckling and non-probabilistic reliabilityanalysis of supercavitating vehicles[J]. Jou rnal of Harbin Institu te of Technology（New Series）,2009,16（4）:561-569.
[71]杨传武,王安稳.超空泡水下航行体振动特性分析[J].海军工程大学学报,2008,20（4）:12-14.
[72]杨传武,刘刚，王安稳.超空泡体结构响应问题的有限元分析[J].海军工程大学学报,2008,20（2）:101-104.
[73]杨传武,王安稳.动态轴向载荷对超空泡航行体振动特性的影响[J].华中科技大学学报,2008,36（12）:71-74.
[74]杨传武,王安稳.冲击载荷作用下超空泡水下航行体的结构响应[J].华中科技大学学报,2008,36（7）:129-132.
[75] O.Boulon, G.L.Chahine. Numerical simulation of unsteady cavitation on3Dhydrofoil[C]. Third International Symposium on Cavitation, Grenoble, France, April,1998.
[76] L.Dieval, M.Arnaud, R.Marcer. Numerical modeling of unsteady cavitating flows by aVOF method[C]. Third International Symposium on Cavitation, Grenoble, France,April,1998.
[77] F.Salvatore, P.G.Esposito. An improved boundary element analysis of cavitating three-dimensional hydrofoils[C]. Fourth International Symposium on Cavitation, California,USA, June,2001.
[78] S.A.Kinnas, H.Sun, H.Lee Numerical analysis of flow around the cavitating CAV2003hydrofoil[C]. Fifth International Symposium on Cavitation, Osaka, Japan, November,2003.
[79] I.N. Kirschner, Neal E. Fine, James S. Uhlman, et al. Numerical Modeling ofSupercavitating Flows[C]. PP RTO AVT LS “SF” VKI EN-09,12-16February2001.
[80] Cheng Xiao-jun, Lu Chuan-jing. On the partially cavitating flow around two-dimensional hydrofoils[J]. Applied Mathematics and Mechanics,2000,21（2）:1450-1459.
[81]程晓俊,鲁传敬.二维水翼的局部空泡流研究[J].应用数学和力学,2002,21（12）:1310-1318.
[82]冷海军,鲁传敬.轴对称体的局部空泡流研究[J].上海交通大学学报,2002,36（3）:395-398.
[83]戚定满,鲁传敬.空泡噪声的数值研究[J].水动力学研究与进展, A辑,2001,16（1）:9-17.
[84] A.Kubota, H.Kato, H.Yamaguchi. A new modeling of cavitating flows: a numericalstudy of unsteady cavitation on a hydrofoil section[J]. J. Fluid Mech.,1992, Vol.240,pp.59-96.
[85] Y.Delannoy, J.L.Kueny. Two-phase flow approach in unsteady cavitation modeling[C].ASME Cavitation and Multiphase Flow Forum, Toronto, June,1990.
[86] C.Song, J.He. Numerical simulation of cavitating flows by single-phase flowapproach[C]. Third International Symposium on Cavitation, Grenoble, France, April,1998, pp.295–300.
[87] C.L.Merkle, J.Z.Feng, P.E.O.Buelow. Computational modeling of the dynamics ofsheet cavitation[C]. Third International Symposium on Cavitation, Grenoble, France,April,1998, pp307–313.
[88] J.C.Reboud, Y.Delannoy.Two-phase flow modeling of unsteady cavitation[C]. SecondInternational Symposium on Cavitation, Tokyo, Japan, April,1994, pp.39-44.
[89] R.F.Kunz, T.S.Chyczewski, D.A.Boger. Multiphase CFD analysis of natural andventilated cavitation about submerged bodies[C]. Third ASME/JSME Joint FluidsEngineering Conference, San Francisco, USA,1999.
[90] R.F.Kunz, D.A.Boger, D.R.Stinebring. A Preconditioned Navier-Stokes method fortwo-phase flows with application to cavitation prediction[J]. Computers&Fluids,2000,Vol.29, pp.849-875.
[91] Y.Chen, S.Heister. Two-phase modeling of cavitating flows[J]. Computers and fluids,1995,24（7）:799-809.
[92] Y.Ventikos, G.Tzabiras. A numerical method for the simulation of steady and unsteadycavitating flows[J]. Computer&Fluids,2000, Vol.29, pp.63-88.
[93] A.Hosangadi, V.Ahuja, S.Arunajatesan. A generalized compressible cavitationmodel[C]. Fourth International Symposium on Cavitation, California, USA, June,2001.
[94] W. X.Yuan, J.Sauer, G.H.Schnerr. Modeling and computation of unsteady cavitationflows in injection nozzles[J]. Mec. Ind.,2001, No.2,383-394.
[95] I.Senocak, W.Shyy. A pressure-based method for turbulent cavitating flowcomputations[J]. J. Comp. Physics,2002, Vol.176, pp.363-383.
[96] A.K.Singhal, M.M.Athavale, H.Y.Li, et al. Mathematical basis and validation of thefull cavitation model[J]. J. Fluids Eng.,2002, Vol.124, pp.617-624.
[97] Y.Saito, I.Nakamori, T.Ikohagi. Numerical analysis of unsteady vaporous cavitatingflow around hydrofoil[C]. Fifth International Symposium on Cavitation, Osaka, Japan,November,2003.
[98] O.Coutier-Delgosha, R.Fortes-Patella, J.L.Reboud.Evaluation of the turbulence modelinfluence on the numerical simulations of unsteady cavitation[J]. J. Fluids Eng.,2003,125（1）:38-45.
[99] Q.Qin, C.C.S.Song, R.E. A.Arndt. A numerical study of an unsteady turbulent wakebehind a cavitating hydrofoil[C]. Fifth International Symposium on Cavitation, Osaka,Japan, November,2003.
[100] R.Vaidyanathan, I.Senocak, J.Y.Wu. Sensitivity evaluation of a transport-basedturbulent cavitation model[J]. J. Fluids Eng.,2003,125（5）:447-458.
[101] M.P.Kinzel, J.W.Lindau, E.G.Paterson, et al. The effects of a free-surface on computedturbulent, super and partially cavitating flows[C]. Sixth International Symposium onCavitation,W ageningen, The Netherlands, September,2006.
[102] S.H.akai, T.Ohta, T.Kajishima. Direct numerical simulation of vortex cavitation inturbulent shear layer[C]. Sixth International Symposium on Cavitation, Wageningen,The Netherlands, September,2006.
[103]熊永亮,郜冶,王革.水下超空泡航行体减阻能力的数值研究[J].弹道学报，2007，19（1）:51-54.
[104] Y.L.Xiong, Y.Gao, W.G.An. Comparisons of turbulence models in predicting unsteadysupercavitating flow[C]. Sixth International Symposium on Cavitation, Wageningen,The Netherlands, September,2006.
[105]袁绪龙,张宇文,杨武刚.高速超空化航行体典型空化器多相流CFD分析[J].弹箭与制导学报.2005,25（1）:53-59.
[106]傅慧萍,鲁传敬,吴磊.回转体空泡流特性研究[J].水动力学研究与进A,2005,20（1）:84-89.
[107]王海斌,张嘉钟,魏英杰等.空泡形态与典型空化器参数关系的研究---小空泡数下的发展空泡形态.水动力学研究与进展A,2005,25（2）:251-257.
[108]汤继斌，钟诚文.空化、超空化流动的数值模拟方法研究[J].力学学报,2005,37（5）:640-644.
[109]贾力平,王聪,于开平等.空化器参数对通气超空泡形态影响的实验研究[J].工程力学,2007,24（3）:159-163.
[110]陈鑫.通气空泡流研究[D].上海：上海交通大学,2006,12.
[111] Robert F. Kunz， Jules W. Lindau， Michael L. Billet， David R. Stinebring.Multiphase CFD Modeling of Developed and Supercavitating Flows[C]. PP RTO AVTLS “SF” VKI EN-013,12-16February.
[112] J.W.Lindau, R.F.Kunz, S.Venkateswaran, et al. Application of a Preconditioned,Multiple-Species, Compressible, Navier-Stokes Model to Cavitating Flows[C]. The4thInternational Symposium on Cavitation, Pasadena, California,20-23June,2001.
[113] J.W.Lindau, S.Venkateswaran, R.F.Kunz, et al. Development of a Fully-CompressibleMulti-Phase Reynolds-Averaged Navier-Stokes Model[C]. Submitted to the15thAIAA CFD Conference, Anaheim, California,11-14June,2001.
[114] Sankaran Venkateswaran, Jules W. Lindau, Robert F. Kunz, et al. Computation ofMultiphase Mixture Flows with Compressibility Effects[J]. Journal of ComputationalPhysics2002,180,54–77.
[115] Jules W. Lindau, Sankaran Venkateswaran, Robert F. Kunz, et al. Computation ofCompressble Multiphase Flows[C]. AIAA2003-1285,41st Aerospace SciencesMeeting and Exhibit6-9January2003, Reno, Nevada.
[116] Jules W. Lindau, Robert F. Kunz, Venkateswaran Sankaran, et al. HomogeneousMultiphase CFD Modeling of Large Scale Cavities [C]. European Congress onComputational Methods in Applied Sciences and Engineering ECCOMAS2004.
[117] A.D.Vasin. Calculation of Axisymmetric Cavities Downstream of a Disk in SubsonicCompressible FluidFlow[J]. Izv. Rus. Akad. Nauk, Mekh. Zhidk. Gaza,1996,（2）:94-103（reprinted in the USA）.
[118] A.D.Vasin Calculation of Axisymmetric Cavities Downstream of a Disk in SupersonicFlow[J]. Izv. Rus. Akad. Nauk, Mekh. Zhidk. Gaza,1997,（4）:54-62（reprinted inthe USA）.
[119] M. Schaffar, H.J. Pfeifer. Comparison of Two Computational Methods for High-Velocity Cavitating Flows around Conical Projectiles: OTi-HULL Hydrocode andTwo-Phase Flow Method [C]. The4th International Symposium on Cavitation,Pasadena, California,20-23June,2001.
[120] F.M.Owis, A.H.Nayfeh, Computation of the compressible multiphase flow over thecavitating high-speed torpedo[J]. J. Fluids Eng.,2003,125（5）:459-468.
[121] Duncan R. van der Heul, C. Vuik, P. Wesseling. Efficient Computation of Flow withCavitation by Compressible Pressure Correction[C]. European Congress onComputational Methods in Applied Sciences and Engineering ECCOMAS2000,Barcelona,11-14September2000.
[122] Vineet Ahuja, Ashvin Hosangadi, Srinivasan Arunajatesan. Simulations of CavitatingFlows Using Hybrid Unstructured Meshes [J]. Journal of Fluids Engineering,2001,123,331-340.
[123] V.Ahuja, A.Hosangadi, R.Ungewitter, et al. A Hybrid Unstructured Mesh Solver forMutli-Fluid Mixtures[J].1999, AIAA Paper99-3330.
[124] R.Saurel, J.P.Cocchi, P.B.Butler. Numerical Study of Cavitation in the Wake of aHypervelocity Underwater Projectile[J] AIAA Journal of Propulsion and Power,1999,15（4）:513-520.
[125] Gunter H. Schnerr, Steffen Schmidt, Ismail Sezal, Matthias Thalhamer. Shock andwave dynamics of compressible liquid flows with special emphasis on unsteady loadon hydrofoils and on cavitation in injection nozzles[C]. Sixth International Symposiumon Cavitation CAV2006, Wageningen, The Netherlands, September,2006.
[126]鲁传敬，陈瑛.空泡流数值模拟方法研究[C].第九届全国水动力学学术会议暨第二十二届全国水动力学研讨会文集，中国四川成都，2009,8.
[127]张鹏,傅慧萍.跨超音速射弹的超空泡数值模拟[J].弹箭与制导学报，2009,29（5）:166-169.
[128]贾力平.空化器诱导超空泡特性的数值仿真与试验研究[D].哈尔滨工业大学,2007,6.
[129]易文俊,李月洁,王中原等.小攻角下水下高速射弹的空泡形态特性[J].南京理工大学学报（自然科学版）,2008,32（4）:464-467.
[130]易文俊,王中原,熊天红等.水下高速射弹超空泡减阻特性研究[J].弹道学报,2008,20（4）:1-4.
[131]熊天红,李铁鹏,易文俊等.水下高速射弹超空泡形态与阻力特性研究[J].弹道学报,2009,21（2）:100-102.
[132] R.J.Etter. State of the art-cavitation test facilities and experimental methods[C].NCT’50International Conference on Propeller Cavitation,3-5April, University ofNewcastle, Newcastle uponTyne, United Kingdom,2000.
[133] H.Reichardt. The Laws if Cavitation Bubbles at Axially Symmetrical Bodies in a Flow
[Z]. Ministry of Aircraft Production Volkenrode, MAP-VC, Report andTranslations766, ONR,1946.
[134] J.P.Onerll. Flow Around Bodies with Attached Open Cavities [Z]. CIT Hyd. Rept. E-24:7,1954.
[135] R.N.Cox, J.W.Maccoll. Recent Contributions to Basic Hydroballistics [C]. NavalSymp.1956, pp215-239.
[136] R.H.Oversmith. Some Observations on Cavitating Flows [Z].Convair Eng. Dept. Rept.ZR-659-015.1959.
[137] J.Klose, A.J.Acosta. Some New Measurements on the Drag of Cavitating Disks [J].J.Ship Res:9, pp102-104.1965.
[138] R.David. Stinebring Michael L.Billet.Jules W.Lindau.Robert F.Kunz. DevelopedCavitation Cavity Dynamics[C]. Applied Research Laboratory P.O. Box30StateCollege,PA16804USA.
[139] J.M.Michel. Ventilated cavities: a contribution to the study of pulsation mechanism[C].Symposium IUTAM on rapid, non steady liquid flows, Leningrad,1971.
[140] J.M.Michel. Some features of water flows with ventilated cavities[J]. J. of FluidsEngineering,1984,106（3）:319-326.
[141] J.M.Michel. Oscillations of Ventilated Cavities Experimental Aspects[C]. PP RTOAVT LS “SF” VKI EN-02,12-16February2001.
[142] D.P.Hart, C.E.Brennen, A.J.Acosta. Observations of cavitation on a three-dimensionaloscillating hydrofoil[C]. ASME Cavitation and Multiphase Flow Forum,1990, Vol.98,pp.49-52.
[143] L.Wilczynski. The experimental investigation of the cavitation inception in the flowaround NACA16-018hydrofoil[C]. Third International Symposium on Cavitation:Vol. I, Grenoble, France, April,1998, pp.167-172.
[144] Y.Kawanami, H.Kato, H.Yamaguchi. Three-dimensional characteristics of the cavitiesformed on a two-dimensional hydrofoil[C]. Third International Symposium onCavitation: Vol. I, Grenoble, France, April,1998, pp.191-196.
[145] L.K.R.aberteaux, S.L.Ceccio. Flow in the closure region of closed partial attachedcavitation[C]. Third International Symposium on Cavitation: Vol. I, Grenoble, France,April,1998, pp.197-202.
[146] B.Stutz, J. L.Reboud. Experiments on unsteady cavitation[J]. Experiments in Fluids,1997, No.22, pp.191-198.
[147] B.Stutz, J.L.Reboud. Measurements within unsteady cavitation[J]. Experiments inFluids,2000, Vol.29, pp.545-552.
[148] H.Lohrberg, B.Stoffel, R.Fortes-Patella, et al. Numerical and Experimentalinvestigations on the cavitating flow in a cascade of hydrofoils[J]. Experiments inFluids,2002, No.33, pp.578-586.
[149] S.Takahashi, S.Washio, K.Uemura, et al. Experimental study on cavitation starting atand flow characteristics close to the point of separation[C]. Fifth InternationalSymposium on Cavitation （Cav2003）, Osaka, Japan, No. Cav03-OS-3-003, pp.1-8,2003.
[150] S.Washio, T S.akahashi, Y.Uda, et al. Study on cavitation inception in hydraulic oilflow through a long twodimensional constriction[J]. Proc Instn Mech Engrs,2001,215（Part J）:373-386.
[151] S.Washio, S.Takahashi, S.Kawahara, et al. Study on the cavitation mechanism inhydraulic oil flow using a needle projection[J]. Proc Instn Mech Engrs,2002,216（Part J）:27-34.
[152] S.Washio, S.Takahashi, S.Yoshimori. Study on cavitation starting at the point ofseparation on a smooth wall in hydraulic oil flow[J]. Proc Instn Mech Engrs,2003,217（Part C）:619-630.
[153] Yu.N. Savchenko, Experimental Investigation of Supercavitating Motion of Bodies[C].PP RTO AVT LS “SF” VKI EN-04,12-16February2001.
[154] V.T.Savchenko. Hydrodynamic Characteristics of Rudders Operating in Air-SeaInterface[C].Paper presented at the RTO AVT Symposium on “Fluid DynamicsProblems of Vehicles Operating near or in the Air-Sea Interface”, held in Amsterdam,The Netherlands,5-8October1998, and published in RTO MP-15.
[155] Robert Kuklinski, Charles Henoch, John Castano, Experimental Study of VentilatedCavities on Dynamic Test Model[C]. Presented at CAV2001: Fourth InternationalSymposium on Cavitation. California Institute of Technology, Pasadena, CA USA.June20-23,2001.
[156] R.E.A.Arndt. From Wageningen to Minnesota and Back: Perspectives on CavitationResearch Keynote Lecture[C]. Proceedings Sixth International Symposium onCavitation, CAV2006, Wageningen, The Netherlands, September2006.
[157] E.Amromin, J.Kopriva, R.E.A.Arndt. Hydrofoil Drag Reduction by PartialCavitation[J]. J. Fluids Engineering,128, Sept.2006.
[158] B.Vanek, J.Bokor, G.J.Balas, et al. Longitudinal motion control for a high-speedsupercavitation vehicle[J]. Journal of Vibration and Control, Sept.2006.
[159] G.J.Balas, B J.okor, B.Vanek, et al. Control of high-speed underwater vehicles. Controlof Uncertain Systems: Modelling, Approximation, and Design[J]. B. Francis, M. Smithand J. Willems, eds. Berlin: Springer-Verlag.
[160] R.E.A.Arndt, G. Balas, M.Wosnik. Control of cavitating flows: A perspective[J]. JapanSoc. Mech. Eng. Series B,48（2）:334-341.
[161] Yuriy D.Vlasenko. Experimental Investigation of Supercavitation Flow Regimes atSubsonic and Transonic Speeds[C]，Cav03-GS-6-006Fifth International Symposiumon Cavitation （cav2003） Osaka, Japan, November1-4,2003.
[162] I.N. Kirschner. Results of Selected Experiments Involving Supercavitating Flows.[C].PP RTO AVT LS “SF” VKI EN-013,12-16February.
[163]何友声,刘桦,赵岗.二维空泡流的脉动性态研究[J].力学学报,1997,29（1）:1-7.
[164]刘桦,何友声,赵岗,轴对称空泡流的脉动性态研究[J].上海力学,1997,18（2）:99-105.
[165]顾魏.非稳定空泡流瞬态和周期现象实验研究[D].上海:上海交通大学,1998.
[166]谢正桐,何友声.小攻角下轴对称细长体的充气肩空泡试验研究[J].试验力学.1999,14（3）:279-287.
[167]谢正桐,何友声,朱世权.小攻角带空泡细长体的试验研究[J].水动力学研究与进展,A辑,2001,16（3）:374-381.
[168] Feng Xue-mei, Lu Chuan-jing, Hu Tian-qun, et al., The fluctuation characteristics ofnatural and ventilated cavities on an axisymmetric body[J]. Journal of Hydrodynamics,Ser. B,2005,17（1）:87-91.
[169]顾建农,张志宏,高永琪,郑学龄.充气头型对超空泡轴对称体阻力特性影响的试验研究[J].兵工学报.2004,25（6）:766-769.
[170]袁绪龙,张宇文,刘乐华等.水下航行体通气超空泡形态试验研究[J].应用力学学报.2004,21（3）:33-37.
[171]袁绪龙,张宇文,王育才等.水下航行体通气超空泡非对称性研究[J].力学学报.2004,36（2）:146-150.
[172]邓飞,张宇文,陈伟政等.头形对细长体超空泡生成与外形影响的实验研究[J].西北工业大学学报.2004年6月第22卷第3期.
[173]王海斌,张嘉钟,魏英杰等.水下航行体通气超空泡减阻特性的实验研究[J].船舶工程.2006,28（3）:20-23.
[174]熊天红,易文俊.高速射弹超空泡减阻试验研究与数值模拟分析[J].工程力学,2009,26（8）:174-178.
[175] P.J. Zwart, A.G. Gerber, T. Belamri. A Two-Phase Flow Model for PredictingCavitation Dynamics[C]. Fifth International Conference on Multiphase Flow,Yokohama, Japan,2004.
[176] G.H. Schnerr, J. Sauer. Physical and Numerical Modeling of Unsteady CavitationDynamics[J]. Fourth International Conference on Multiphase Flow, New Orleans, USA,2001.
[177] A.K.Singhal, N.Vaidya, A.D.Leonard. Multidimensional simulation of cavitating flowsusing a PDF model for phase change[C]. ASME Fluids Engineering Division SummerMeeting, ASME Paper FEDSM97-3272.
[178] C.L.Merkle, J.Feng, P.O.Buelow. Computational Modeling of the Dynamics of SheetCavitation[C]. Proc.3rd International Symposium on Cavitation, Grenoble,France,1998.
[179] Chien-Chou Tseng, Wei Shyy. Modeling for isothermal and cryogenic cavitation[J].International Journal of Heat and Mass Transfer,2010（53）:513–525.
[180] H. Rouse, J. S. McNown. Cavitation and Pressure Distribution Head Forms at ZeroAngle of Yaw[J]. Iowa Institute of Hydraulic Research, State University of Iowa, IowaCity,1948.
[181] O.Coutier-Delgosha, R.Fortes-Patella, J.L.Reboud. Evaluation of the turbulence modelinfluence on the numerical simulations of unsteady cavitation[J]. J. Fluids Eng.,2003,Vol.125, No.1, pp.38-45.
[182]苏玉民,庞永杰编.潜艇原理[M].哈尔滨:哈尔滨工程大学出版社,2005.