近壁方柱绕流湍流场非定常特性的PIV实验研究
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摘要
本文研究工作受国家自然科学基金(50606024)“后缘尾涡脱落非定常流场涡声相关的实验研究”和NSFC(中国自然科学基金会)-KOSEF(韩国科学与工程基金会)合作研究项目“湍流场相干结构涡声相关实验和理论研究”资助。本文利用PIV(Particle Image Velocimetry)对近壁方柱绕流的非定常特性进行了细致的研究,揭示了壁面对近壁柱体绕流流动的影响;同时,通过对流动控制方程的分析,提出了基于PIV速度场测试结果的流场压力预测方法。
首先,作者根据实验要求设计了用于湍流机理实验的低速循环水槽,并建立了用于获取方柱绕流平均速度场数据的高分辨率PIV系统,以及用于流场非定常测量的时序PIV(Time-Resolved PIV,TR-PIV)系统。通过对PIV测量得到的速度场进行统计分析,获得了流场流线图、剪切力分布、回流间歇因子分布等。对四种近壁结构(G/D=0.8、0.4、0.2、0.1,G为方柱与壁面间距,D为方柱截面边长)的流场对比分析后,揭示了方柱靠近壁面时,壁面对方柱绕流流动的影响。通过对时序PIV结果的频谱分析、相关分析及瞬态流场的分析,阐述了方柱与壁面间距不同时,方柱后旋涡脱落的频率和形态变化,以及旋涡脱落后在方柱尾部的运动状态和发展演化过程。
此外,本文对基于PIV速度场测试结果的流场压力预测方法进行了探索研究。作者分别在层流和湍流两种情况下,对流动控制方程进行分析并简化,得到压力计算公式。在对压力计算公式中的微分项处理时,本文给出了两种离散方法并比较了其优越性。在对本文所提出的算法进行验证后,对结果和产生误差的原因进行了分析。
Because of the practical importance and the wealth of interesting fluid flow phenomena, much progress has been made in understanding of the flow around a square cylinder placed near a wall. However, due to its physical complexity, the experimental results of unsteady characteristics and a complete and conclusive explanation of the physics are still lacking. The use of advanced techniques in experiments and data post-processing methodologies has made it possible to enhance our knowledge of the phenomenon.
In the present work, detailed results of time averaged flow properties as well as unsteady flow characteristics in the near wake of a square cylinder with side length D placed near a wall are measured in a low-speed water tunnel. Particle image velocimetry(PIV) and time-resolved particle image velocimetry(TR-PIV) measurements are conducted for cylinder to wall gap heights, G, from G/D=0.1 to G/D=1.0.The PIV measurements obtain some time-mean results such as velocity field, recirculation length and shear stress distribution. According to the comparison of these results in different kinds of G/D, the influence of wall proximity on vortex shedding and flow characteristics in the wake is clarified, as well as a critical gap height of 0.3D is deduced. For 0.3 The approach to determine pressure field from PIV velocity data is also discussed in the present work. The method relies upon the application of control volume approach in combination with the discretization of Navier-Stokes equation. By considering laminar and turbulent flow conditions, differential governing equations are simplified and discretized by finite difference method and circulation method respectively. Then the reliability of pressure prediction approach is verified by CFD method. Finally, sensitivity of the pressure results to the resolution level and to the discretization scheme of differential terms is discussed.
首先,作者根据实验要求设计了用于湍流机理实验的低速循环水槽,并建立了用于获取方柱绕流平均速度场数据的高分辨率PIV系统,以及用于流场非定常测量的时序PIV(Time-Resolved PIV,TR-PIV)系统。通过对PIV测量得到的速度场进行统计分析,获得了流场流线图、剪切力分布、回流间歇因子分布等。对四种近壁结构(G/D=0.8、0.4、0.2、0.1,G为方柱与壁面间距,D为方柱截面边长)的流场对比分析后,揭示了方柱靠近壁面时,壁面对方柱绕流流动的影响。通过对时序PIV结果的频谱分析、相关分析及瞬态流场的分析,阐述了方柱与壁面间距不同时,方柱后旋涡脱落的频率和形态变化,以及旋涡脱落后在方柱尾部的运动状态和发展演化过程。
此外,本文对基于PIV速度场测试结果的流场压力预测方法进行了探索研究。作者分别在层流和湍流两种情况下,对流动控制方程进行分析并简化,得到压力计算公式。在对压力计算公式中的微分项处理时,本文给出了两种离散方法并比较了其优越性。在对本文所提出的算法进行验证后,对结果和产生误差的原因进行了分析。
Because of the practical importance and the wealth of interesting fluid flow phenomena, much progress has been made in understanding of the flow around a square cylinder placed near a wall. However, due to its physical complexity, the experimental results of unsteady characteristics and a complete and conclusive explanation of the physics are still lacking. The use of advanced techniques in experiments and data post-processing methodologies has made it possible to enhance our knowledge of the phenomenon.
In the present work, detailed results of time averaged flow properties as well as unsteady flow characteristics in the near wake of a square cylinder with side length D placed near a wall are measured in a low-speed water tunnel. Particle image velocimetry(PIV) and time-resolved particle image velocimetry(TR-PIV) measurements are conducted for cylinder to wall gap heights, G, from G/D=0.1 to G/D=1.0.The PIV measurements obtain some time-mean results such as velocity field, recirculation length and shear stress distribution. According to the comparison of these results in different kinds of G/D, the influence of wall proximity on vortex shedding and flow characteristics in the wake is clarified, as well as a critical gap height of 0.3D is deduced. For 0.3
引文
[1]刘宇,苏中地,不同雷诺数下方柱绕流的数值模拟.中国计量学院学报,2006,17(1):40-43.
[2] Gerrard J.H., The mechanics of the formation region of vortices behind bluff bodies. Journal of Fluid Mechanics,1966,25:401-413.
[3] Cantwell B.,Coles D., An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. Journal of Fluid Mechanics,1983,136: 321-374.
[4] Lyn D.A.,Rodi W., The flapping shear layer formed by flow separation from the forward corner of a square cylinder. Journal of Fluid Mechanics,1994,267:353- 376.
[5] Zdravkovich M.M., Flow around circular cylinders, Vol.1:Fundamentals.Oxford University Press,1997,London.
[6] Bearman P. W.,Zdravkovich M.M., Flow around a circular cylinder near a plane boundary. Journal of Fluid Mechanics,1978,89:33-47.
[7] Taniguchi S.,Miyakoshi K.,Dohada S., Interference between plane wall and two-dimensional rectangular cylinder. Trans. JSME,1983,49(477):2522-2529.
[8] Taneda S.,Experimental investigation of vortex streets. Journal of the Physical Society of Japan,1965,20:1714-1721.
[9] Roshko A.,Steinolfson A.,Chattoorgoon V., Flow forces on a cylinder near a wall or near another cylinder. In Proceedings of the 2nd National Conference on wind Engineering Research,1975,Colorado State University, paper IV-15.
[10] Bailey S.C.C.,Kopp G.A.,Martinuzzi R.J., Vortex shedding from a square cylinder near a wall[J],Journal of Turbulence,Volume 3,Art.No.N3
[11] Buresti G.,Lanciotti A., Vortex shedding from smooth and roughened cylinders in cross-flow near a plane surface.The Aeronautical Quarterly,1979,30:305-321.
[12] Angrilli F.,Bergamaschi S.,Cossalter V., Investigation of wall induced modifications to vortex shedding from a circular cylinder. ASME Journal of Fluids Engineering,1982,104:518-522.
[13] Cheng M.,Tsuei H.E.,Chow K.L., Experimental study on flow interference phenomena of cylinder/cylinder and cylinder/plane arrangements. ASME J. FlowInduced Vibration,1994,273:173-184.
[14] Grass A.J.,Raven P.W.J.,Stuart R.J.,Bray J.A., The influence of boundary layer velocity gradients and bed proximity on vortex shedding from free spanning pipelines. Journal of Energy Resources Technology 1984,106:70-78.
[15] Taniguchi S.,Miyakoshi K., Fluctuating fluid forces acting on a circular cylinder and interference with a plane wall. Experiments in Fliuds,1990, 9(4):197-204.
[16] Fuh Min Fang,J.C. Chen,Y.T.Hong, Experimental and analytical evaluation of flow in a square-to-square wind tunnel contraction. Journal of Wind Engineering and Industrial Aerodynamics,2001,89(3-4):247-262.
[17] Ronald J.Adrian, Particle-imaging techniques for experimental fluid mechanics. Annual Review of Fluid Mechanics,1991,23:261-304.
[18] Wolfgang Merzkirch, Flow visualization(Second Edition), Academic Press Inc., New York,1987.
[19]代钦,魏润杰,黄湛等.超音速喷流DPIV瞬时速度场实验测量,北京航空航天大学学报,2001,27(6):666-669.
[20]赵宇,PIV测试中示踪粒子性能的研究[硕士论文].大连理工大学,2004年6月.
[21] R.J.Adrian,Twenty years of particle image velocimetry. Experiments in Fluids, 2005, 39(2): 159-169.
[22] Dimotakis P.E.,Debussy F.D.,Koochesfahani M.M., Particle streak velocity field measurements in a two-dimensional mixing layer. Physics of Fluids, 1981, 24: 995-999.
[23] Khalighi B.,Study of the intake swirl process in an engine using flow visualization and particle tracking velocimetry.ASME-FED.1989,85:37-41.
[24] Simpkins P.G..,Dudderar T.D., Laser speckle measurement of transient Benard convection. Journal of Fluid Mechanics,1978,89:665-71.
[25] Kawahashi M.,Yamamoto K., Speckle method using beam scanning techniques. Proceedings of The International Workshop on PIV’95,Fukui,1995,155~158.
[26] Willert C.E.,Gharib M., Digital particle image velocimetry. Experiments in Fliuds,1991, 10(4): 181-193.
[27] Hart D.P.,PIV error correction. Experiments in Fliuds,2000,29(1):13-22.
[28] Westerweel J., Analysis of PIV interrogation with low-pixel resolution. OpticalDiagnostics in Fluid and Thermal Flow,1993,7:14-16.
[29] Westerweel J., Fundamentals of digital particle image velocimetry. Measurement Science and Technology,1997,8(12):1379-1392.
[30] Westerweel J., Theoretical analysis of the measurement precision in particle image velocimetry. Experiments in Fliuds,2000,29(7):3-12.
[31]段俐,康琦,申功炘，PIV技术的粒子图像处理方法.北京航空航天大学学报,2000,26(1):79-82.
[32]魏润杰,申功炘，丁汉泉，数字全息罻油枷癫馑偌际跹芯浚本┖娇蘸教齑?学学报,2004,30(q5):456-460.
[33]张伟,葛耀君,杨詠昕,粒子图像测速技术互相关算法研究进展.力学进展,2007,3(37):443-452.
[34] LIU Ying-zheng,CAO Zhao-min, Recent progress on particle image velocimetry in China.J.Hydrodynamics,Ser.B,2006,18(1):11-19.
[35] Keane R.D.,Adrian R.J., Optimization of particle image velocimetry:II. Multiple pulsed system. Measurement Science and Technology,1991,2(10):963-974.
[36] Keane R.D,Adrian R.J., Theory of cross-correlation analysis of PIV images. Applied Scientific Research,1992,49(3):191-215.
[37]翁文国,廖光煊,范维澄,小波变换应用于DPIV图像去噪.中国图像图形学报,2000,5(3):211-215.
[38] Raffel M.,Willert C.E.,Kompenhans J., Particle image velocimetry, a practical guide. Springer,Berlin,1998.
[39] Nogueira J.,Lecuona A.,Rodriguez P.A., Local field correction PIV:on the increase of accuracy of digital PIV systems. Experiments in Fliuds,1999,27(2):107-116.
[40] Huang H.T.,Fiedler H.E.,Wang J.J., Limitation and improvement of PIV.I. Limitation of conventional techniques due to deformation of particle image patterns. Experiments in Fliuds,1993,15(4):168-174.
[41] Huang H.T.,Fiedler H.E.,Wang J.J., Limitation and improvement of PIV. II. Particle image distortion . Experiments in Fliuds,1993,15(5):263-273.
[42] Scarano F.,Riethmuller M.L., Advances in iterative multigrid PIV image processing. Experiments in Fliuds,2000,29(S1):51-60.
[43] Scarano F.,Riethmuller M.L., Iterative multigrid approach in PIV image processing with discrete window offset . Experiments in Fliuds,1999,26(6):513- 523.
[44] Nogueira J.,Lecuona A.,Rodriguez P.A., Limits on the resolution of correlation PIV iterative methods, Fundamentals. Experiments in Fliuds,2005,39(2):305-313.
[45] Scarano F., Iterative image deformation methods in PIV. Measurement Science and Technology,2002,13(1):R1-R19.
[46] Adrian R.J.,Christensen K.T.,Liu Z.C., Analysis and interpretation of instantaneous turbulent velocity fields. Experiments in Fliuds,2000,29(3):275-290.
[47]康文,基于相分离的PIV-PTV两相流测量技术研究及动态测量系统[硕士论文].上海大学,2007年6月.
[48] Bailey,S.C.C.,Kopp,G.A.and Martinuzzi,R.J., Vortex shedding from a square cylinder near a wall[J],Journal of Turbulence, Volume 3,Art.No.N3
[49] Martinuzzi R.J.,Bailey S.C.C.,Kopp G.A., Influence of wall proximity on vortex shedding from a square cylinder. Experiments in Fluids,2003,34(5):585-596.
[50] Durao D.F.G.,Gouveia P.S.T.,Pereira .J.C.F., Velocity characteristics of the flow around a square cross section cylinder placed near a channel wall. Experiments in Fluids,1991,11(6):341-350.
[51] Unal M.F.,Lin J.C.,Rockwell D.,Force prediction by PIV imaging: a momentum-based approach. Journal of Fluids and Structures,1997,11(8):965-971.
[52] Gurka R.,Liberzon A.,Hefetz D.,Rubinstein D.,Shavit U., Computation of pressure distribution using PIV velocity data, Proceedings of the Third International Workshop on PIV,Santa Barbara,Sep.1999,671-682.
[53] Jakobsen M.L.,Dewhirst T.P.,Greated C.A., Particle image velocimetry for predictions of acceleration fields and force within fluid flows. Measurement Science and Technology,1997,8(12):1502-1516
[54] Jensen A.,Pedersen G.K., Optimization of acceleration measurements using PIV.Measurement Science and Technology,2004,15(11):2275-2283.
[55] Fujisawa N.,Tanahashi S.,Srinivas K., Evaluation of pressure field and fluid forces on a circular cylinder with and without rotational oscillating using velocity data from PIV measurement. Measurement Science and Technology, 2005, 16(4): 989- 996.
[56] Liu X.,Katz J., Instantaneous pressure and material acceleration measurements using a four-exposure PIV system. Experiments in Fluids,2006,41(2):227-240.
[57]张兆顺,崔桂香,许春晓,湍流理论与模拟.清华大学出版社,2005.
[58]章梓雄,董曾南,粘性流体力学.清华大学出版社,1996.
[59]陶文铨,数值传热学(第二版).西安交通大学出版社,2001.
[60] Raffel M.,Willert C.E.,Kompenhans J., Particle image velocimetry,a practical guide(Second Edition). Springer,Berlin,2007.
[61] Van O.B.W.,Scarano F.,Roosenboom E.W.M.,Casimiri E.W.F.,Souverein L. J., Evaluation of integral forces and pressure fields from planar velocimetry data for incompressible and compressible flows. Experiments in Fluids, 2007, 43(2-3): 153-162.
[62] Murai Y.,Nakada T.,Suzuki T.,Yamamoto F., Particle tracking velocimetry applied to estimate the pressure field around a Savonius turbine. Measurement Science and Technology,2007,18(8):2491-2503.
[2] Gerrard J.H., The mechanics of the formation region of vortices behind bluff bodies. Journal of Fluid Mechanics,1966,25:401-413.
[3] Cantwell B.,Coles D., An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. Journal of Fluid Mechanics,1983,136: 321-374.
[4] Lyn D.A.,Rodi W., The flapping shear layer formed by flow separation from the forward corner of a square cylinder. Journal of Fluid Mechanics,1994,267:353- 376.
[5] Zdravkovich M.M., Flow around circular cylinders, Vol.1:Fundamentals.Oxford University Press,1997,London.
[6] Bearman P. W.,Zdravkovich M.M., Flow around a circular cylinder near a plane boundary. Journal of Fluid Mechanics,1978,89:33-47.
[7] Taniguchi S.,Miyakoshi K.,Dohada S., Interference between plane wall and two-dimensional rectangular cylinder. Trans. JSME,1983,49(477):2522-2529.
[8] Taneda S.,Experimental investigation of vortex streets. Journal of the Physical Society of Japan,1965,20:1714-1721.
[9] Roshko A.,Steinolfson A.,Chattoorgoon V., Flow forces on a cylinder near a wall or near another cylinder. In Proceedings of the 2nd National Conference on wind Engineering Research,1975,Colorado State University, paper IV-15.
[10] Bailey S.C.C.,Kopp G.A.,Martinuzzi R.J., Vortex shedding from a square cylinder near a wall[J],Journal of Turbulence,Volume 3,Art.No.N3
[11] Buresti G.,Lanciotti A., Vortex shedding from smooth and roughened cylinders in cross-flow near a plane surface.The Aeronautical Quarterly,1979,30:305-321.
[12] Angrilli F.,Bergamaschi S.,Cossalter V., Investigation of wall induced modifications to vortex shedding from a circular cylinder. ASME Journal of Fluids Engineering,1982,104:518-522.
[13] Cheng M.,Tsuei H.E.,Chow K.L., Experimental study on flow interference phenomena of cylinder/cylinder and cylinder/plane arrangements. ASME J. FlowInduced Vibration,1994,273:173-184.
[14] Grass A.J.,Raven P.W.J.,Stuart R.J.,Bray J.A., The influence of boundary layer velocity gradients and bed proximity on vortex shedding from free spanning pipelines. Journal of Energy Resources Technology 1984,106:70-78.
[15] Taniguchi S.,Miyakoshi K., Fluctuating fluid forces acting on a circular cylinder and interference with a plane wall. Experiments in Fliuds,1990, 9(4):197-204.
[16] Fuh Min Fang,J.C. Chen,Y.T.Hong, Experimental and analytical evaluation of flow in a square-to-square wind tunnel contraction. Journal of Wind Engineering and Industrial Aerodynamics,2001,89(3-4):247-262.
[17] Ronald J.Adrian, Particle-imaging techniques for experimental fluid mechanics. Annual Review of Fluid Mechanics,1991,23:261-304.
[18] Wolfgang Merzkirch, Flow visualization(Second Edition), Academic Press Inc., New York,1987.
[19]代钦,魏润杰,黄湛等.超音速喷流DPIV瞬时速度场实验测量,北京航空航天大学学报,2001,27(6):666-669.
[20]赵宇,PIV测试中示踪粒子性能的研究[硕士论文].大连理工大学,2004年6月.
[21] R.J.Adrian,Twenty years of particle image velocimetry. Experiments in Fluids, 2005, 39(2): 159-169.
[22] Dimotakis P.E.,Debussy F.D.,Koochesfahani M.M., Particle streak velocity field measurements in a two-dimensional mixing layer. Physics of Fluids, 1981, 24: 995-999.
[23] Khalighi B.,Study of the intake swirl process in an engine using flow visualization and particle tracking velocimetry.ASME-FED.1989,85:37-41.
[24] Simpkins P.G..,Dudderar T.D., Laser speckle measurement of transient Benard convection. Journal of Fluid Mechanics,1978,89:665-71.
[25] Kawahashi M.,Yamamoto K., Speckle method using beam scanning techniques. Proceedings of The International Workshop on PIV’95,Fukui,1995,155~158.
[26] Willert C.E.,Gharib M., Digital particle image velocimetry. Experiments in Fliuds,1991, 10(4): 181-193.
[27] Hart D.P.,PIV error correction. Experiments in Fliuds,2000,29(1):13-22.
[28] Westerweel J., Analysis of PIV interrogation with low-pixel resolution. OpticalDiagnostics in Fluid and Thermal Flow,1993,7:14-16.
[29] Westerweel J., Fundamentals of digital particle image velocimetry. Measurement Science and Technology,1997,8(12):1379-1392.
[30] Westerweel J., Theoretical analysis of the measurement precision in particle image velocimetry. Experiments in Fliuds,2000,29(7):3-12.
[31]段俐,康琦,申功炘，PIV技术的粒子图像处理方法.北京航空航天大学学报,2000,26(1):79-82.
[32]魏润杰,申功炘，丁汉泉，数字全息罻油枷癫馑偌际跹芯浚本┖娇蘸教齑?学学报,2004,30(q5):456-460.
[33]张伟,葛耀君,杨詠昕,粒子图像测速技术互相关算法研究进展.力学进展,2007,3(37):443-452.
[34] LIU Ying-zheng,CAO Zhao-min, Recent progress on particle image velocimetry in China.J.Hydrodynamics,Ser.B,2006,18(1):11-19.
[35] Keane R.D.,Adrian R.J., Optimization of particle image velocimetry:II. Multiple pulsed system. Measurement Science and Technology,1991,2(10):963-974.
[36] Keane R.D,Adrian R.J., Theory of cross-correlation analysis of PIV images. Applied Scientific Research,1992,49(3):191-215.
[37]翁文国,廖光煊,范维澄,小波变换应用于DPIV图像去噪.中国图像图形学报,2000,5(3):211-215.
[38] Raffel M.,Willert C.E.,Kompenhans J., Particle image velocimetry, a practical guide. Springer,Berlin,1998.
[39] Nogueira J.,Lecuona A.,Rodriguez P.A., Local field correction PIV:on the increase of accuracy of digital PIV systems. Experiments in Fliuds,1999,27(2):107-116.
[40] Huang H.T.,Fiedler H.E.,Wang J.J., Limitation and improvement of PIV.I. Limitation of conventional techniques due to deformation of particle image patterns. Experiments in Fliuds,1993,15(4):168-174.
[41] Huang H.T.,Fiedler H.E.,Wang J.J., Limitation and improvement of PIV. II. Particle image distortion . Experiments in Fliuds,1993,15(5):263-273.
[42] Scarano F.,Riethmuller M.L., Advances in iterative multigrid PIV image processing. Experiments in Fliuds,2000,29(S1):51-60.
[43] Scarano F.,Riethmuller M.L., Iterative multigrid approach in PIV image processing with discrete window offset . Experiments in Fliuds,1999,26(6):513- 523.
[44] Nogueira J.,Lecuona A.,Rodriguez P.A., Limits on the resolution of correlation PIV iterative methods, Fundamentals. Experiments in Fliuds,2005,39(2):305-313.
[45] Scarano F., Iterative image deformation methods in PIV. Measurement Science and Technology,2002,13(1):R1-R19.
[46] Adrian R.J.,Christensen K.T.,Liu Z.C., Analysis and interpretation of instantaneous turbulent velocity fields. Experiments in Fliuds,2000,29(3):275-290.
[47]康文,基于相分离的PIV-PTV两相流测量技术研究及动态测量系统[硕士论文].上海大学,2007年6月.
[48] Bailey,S.C.C.,Kopp,G.A.and Martinuzzi,R.J., Vortex shedding from a square cylinder near a wall[J],Journal of Turbulence, Volume 3,Art.No.N3
[49] Martinuzzi R.J.,Bailey S.C.C.,Kopp G.A., Influence of wall proximity on vortex shedding from a square cylinder. Experiments in Fluids,2003,34(5):585-596.
[50] Durao D.F.G.,Gouveia P.S.T.,Pereira .J.C.F., Velocity characteristics of the flow around a square cross section cylinder placed near a channel wall. Experiments in Fluids,1991,11(6):341-350.
[51] Unal M.F.,Lin J.C.,Rockwell D.,Force prediction by PIV imaging: a momentum-based approach. Journal of Fluids and Structures,1997,11(8):965-971.
[52] Gurka R.,Liberzon A.,Hefetz D.,Rubinstein D.,Shavit U., Computation of pressure distribution using PIV velocity data, Proceedings of the Third International Workshop on PIV,Santa Barbara,Sep.1999,671-682.
[53] Jakobsen M.L.,Dewhirst T.P.,Greated C.A., Particle image velocimetry for predictions of acceleration fields and force within fluid flows. Measurement Science and Technology,1997,8(12):1502-1516
[54] Jensen A.,Pedersen G.K., Optimization of acceleration measurements using PIV.Measurement Science and Technology,2004,15(11):2275-2283.
[55] Fujisawa N.,Tanahashi S.,Srinivas K., Evaluation of pressure field and fluid forces on a circular cylinder with and without rotational oscillating using velocity data from PIV measurement. Measurement Science and Technology, 2005, 16(4): 989- 996.
[56] Liu X.,Katz J., Instantaneous pressure and material acceleration measurements using a four-exposure PIV system. Experiments in Fluids,2006,41(2):227-240.
[57]张兆顺,崔桂香,许春晓,湍流理论与模拟.清华大学出版社,2005.
[58]章梓雄,董曾南,粘性流体力学.清华大学出版社,1996.
[59]陶文铨,数值传热学(第二版).西安交通大学出版社,2001.
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