混合对流绕方柱涡旋脱落及对传热影响的数值研究
|
摘要
非定常绕方柱混合对流研究,是一个涉及到绕尖缘矩形截面钝物体流动分离和传热两方面相结合的课题,在电子元器件的冷却、燃烧技术及其优化、建筑结构抗振设计等工程领域有着广泛的应用。因此,在过去的几十年里,该问题引起了众多的应用数学家、流体动力学家和传热学研究人员的极大关注。本文在对国内外关于绕矩形截面柱体流动分离和传热研究现状及发展进行综述的基础上,系统地论述了作者采用原始变量时间推进法对非定常绕方柱流动与传热的研究成果。
首先,本文在SIMPLE算法系列处理速度与压力耦合问题的思路和均匀网格下改进的关于流函数涡量方程的数值求解方法的基础上,将非定常原始变量Navier-Stokes方程的求解推广至非等距交错网格剖分,形成了具有本文特色的原始变量时间推进方法,其中对流项和扩散项中各变量的各阶导数均采用二阶精度公式,包括温度在内的离散方程组采用ADI迭代方法求解。原始变量时间推进法采用外节点法布置计算区域的网格节点,固体壁面上节点的压力和壁面外垂直壁面的速度分量采用动量方程在固体壁面简化形式的边界条件进行特别处理,对更为特殊的节点,即数学上存在间断点的地方如本文中方柱四个角点处的压力则采用了交替方向扫描平均法进行处理。应用原始变量时间推进法对方腔顶盖驱动流、封闭空腔内自然对流等流动与传热经典问题进行了数值模拟,通过本文计算结果与基准解的对比,验证了该算法对内流场流动与传热问题的有效性。其中根据计算结果,本文在前人工作的基础上总结出了封闭腔内层流自然对流换热的变化规律,比较准确地提出了导热占主导地位的层流流动和导热与对流共同作用的层流流动的分界点为瑞利数Ra=5×104。
其次,本文采用原始变量时间推进法,研究了低雷诺数条件下冷态场二维方柱绕流问题,讨论了出口条件对减小有效计算区域的作用规律,并分别研究了进口距离Xu、出口距离Xd、阻塞参数B、侧边界条件和出口条件等因素对绕方柱非定常流动的斯特鲁哈尔数St和阻力系数CD的影响,给出了各种人工侧边界条件下可以忽略侧边界影响的阻塞参数范围以及对不同人工侧边界条件进行比较的结果,在此基础上提出了一套适用于本文原始变量时间推进算法研究绕方柱流动与传热问题的优化计算域参数和人工边界条件。在此基础上,本文对绕方柱定常流动和非定常流动进行了详细的数值模拟,首先,根据计算结果总结出了方柱绕流定常流动过渡到非定常流动的临
界雷诺数Rec的范围为50至55之间;其次,本文成功地模拟了雷诺数Re=100方柱绕流非定常涡旋形成与脱落的完整过程,得到了严格的Kármán涡街,从而验证了该算法对冷态场非定常绕钝体流动问题(即冷态外流场问题)的有效性。
最后,本文在冷态场方柱绕流数值计算的基础上,引入能量方程并考虑热对方柱尾迹的影响,研究了低雷诺数Re层流绕方柱强迫对流和混合对流问题。根据本文划分方柱绕流定常流动和非定常流动的临界雷诺数Re区间(50 绕方柱定常(Re<50)和非定常(55In the field of computational heat transfer, the subject of unsteady mixed convection about a square cylinder is of relevance to the flow and heat transfer around a sharp-edged rectangular cross-sectional cylinder. For several decades, the problem has been a subject of great attention among applied mathematicians, fluid dynamicists and heat transfer analysts owing to its numerous engineering applications such as the cooling of electronic components, energy conservation, structural design and acoustic emissions. Based on a review of its current condition and development home and abroad, the thesis systematically expounds some achievements in the numerical arithmetic and the methodology that facilitate the study of the flow around a square cylinder and heat transfer of unsteady mixed convection by using the primitive-variable-time-dependent approach.
Firstly, based on the method of pressure-velocity coupling introduced by SIMPLE algorithm and a revised method of solving the unsteady equation of stream function and vorticity on uniform grid, a special algorithm called primitive-variable-time-dependent approach is formed in the present study. In this algorithm the Navier-Stokes equation in primitive variable form is solved by using the time-dependent approach on non-uniform staggered grid system. The second-order accuracy finite difference is used for the first and second partial derivatives of all variables of the convective terms and diffusion terms. The finite difference equations are solved with ADI method. Outer-node method is used to form the computational grid. The pressure on the wall and the nearest velocity normal to the wall are computed from the momentum equations in special form, and for the corners of the wall, the pressure is averaged by the value gained from the alternating direction calculation of the momentum equations. To validate the algorithm, two classical test problems are studied: namely, the driven cavity flow and the laminar natural convection flow in a square cavity. The calculation results are consistent with the results of the other numerical methods. It is valuable that a new law of laminar natural convection flow and heat transfer is summarized on the basis of previous scholar’s works and Ra=5×104 is accurately proposed as the flow transition point between the laminar flow predominated by the heat conduct and the laminar
flow cooperated by the convection and the heat conduct.
Secondly, the primitive-variable-time-dependent approach is used to compute the steady and unsteady two-dimensional flow around a square cylinder at different Reynolds numbers. The appropriate length of the calculation domain is selected by comparing the results calculated by using some different outlet boundary condition. Several computational parameters, such as Xu(distance from body to inlet), Xd(distance from body to outlet), blockage parameter B, lateral and outlet boundary conditions, are taken into consideration to analyze their effects on vortex shedding frequency scaled by the Strouhal number St and drag coefficient CD. Ranges of blockage parameters are studied for the different lateral boundary conditions to demonstrate where its effects can be ignored numerically, and then the effects of the lateral boundaries are compared. A set of optimized computational domain and artificial boundary conditions are selected to study the fluid flow and heat transfer around a square cylinder by using the primitive-variable-time-dependent approach. Numerical calculations of the steady flow around a square cylinder have been reported, and the onset of vortex shedding is predicted in between Re=50~55. To validate the algorithm, the unsteady process of vortex formation and shedding from the square cylinder for Re=100 is simulated successfully and the strict Kármán vortex street is also obtained.
Finally, based on the simulation of the flow around a square cylinder, the energy equation is taken into consideration and the vortex shedding from a square cylinder and its effects on the heat transfer of laminar forced c
首先,本文在SIMPLE算法系列处理速度与压力耦合问题的思路和均匀网格下改进的关于流函数涡量方程的数值求解方法的基础上,将非定常原始变量Navier-Stokes方程的求解推广至非等距交错网格剖分,形成了具有本文特色的原始变量时间推进方法,其中对流项和扩散项中各变量的各阶导数均采用二阶精度公式,包括温度在内的离散方程组采用ADI迭代方法求解。原始变量时间推进法采用外节点法布置计算区域的网格节点,固体壁面上节点的压力和壁面外垂直壁面的速度分量采用动量方程在固体壁面简化形式的边界条件进行特别处理,对更为特殊的节点,即数学上存在间断点的地方如本文中方柱四个角点处的压力则采用了交替方向扫描平均法进行处理。应用原始变量时间推进法对方腔顶盖驱动流、封闭空腔内自然对流等流动与传热经典问题进行了数值模拟,通过本文计算结果与基准解的对比,验证了该算法对内流场流动与传热问题的有效性。其中根据计算结果,本文在前人工作的基础上总结出了封闭腔内层流自然对流换热的变化规律,比较准确地提出了导热占主导地位的层流流动和导热与对流共同作用的层流流动的分界点为瑞利数Ra=5×104。
其次,本文采用原始变量时间推进法,研究了低雷诺数条件下冷态场二维方柱绕流问题,讨论了出口条件对减小有效计算区域的作用规律,并分别研究了进口距离Xu、出口距离Xd、阻塞参数B、侧边界条件和出口条件等因素对绕方柱非定常流动的斯特鲁哈尔数St和阻力系数CD的影响,给出了各种人工侧边界条件下可以忽略侧边界影响的阻塞参数范围以及对不同人工侧边界条件进行比较的结果,在此基础上提出了一套适用于本文原始变量时间推进算法研究绕方柱流动与传热问题的优化计算域参数和人工边界条件。在此基础上,本文对绕方柱定常流动和非定常流动进行了详细的数值模拟,首先,根据计算结果总结出了方柱绕流定常流动过渡到非定常流动的临
界雷诺数Rec的范围为50至55之间;其次,本文成功地模拟了雷诺数Re=100方柱绕流非定常涡旋形成与脱落的完整过程,得到了严格的Kármán涡街,从而验证了该算法对冷态场非定常绕钝体流动问题(即冷态外流场问题)的有效性。
最后,本文在冷态场方柱绕流数值计算的基础上,引入能量方程并考虑热对方柱尾迹的影响,研究了低雷诺数Re层流绕方柱强迫对流和混合对流问题。根据本文划分方柱绕流定常流动和非定常流动的临界雷诺数Re区间(50
Firstly, based on the method of pressure-velocity coupling introduced by SIMPLE algorithm and a revised method of solving the unsteady equation of stream function and vorticity on uniform grid, a special algorithm called primitive-variable-time-dependent approach is formed in the present study. In this algorithm the Navier-Stokes equation in primitive variable form is solved by using the time-dependent approach on non-uniform staggered grid system. The second-order accuracy finite difference is used for the first and second partial derivatives of all variables of the convective terms and diffusion terms. The finite difference equations are solved with ADI method. Outer-node method is used to form the computational grid. The pressure on the wall and the nearest velocity normal to the wall are computed from the momentum equations in special form, and for the corners of the wall, the pressure is averaged by the value gained from the alternating direction calculation of the momentum equations. To validate the algorithm, two classical test problems are studied: namely, the driven cavity flow and the laminar natural convection flow in a square cavity. The calculation results are consistent with the results of the other numerical methods. It is valuable that a new law of laminar natural convection flow and heat transfer is summarized on the basis of previous scholar’s works and Ra=5×104 is accurately proposed as the flow transition point between the laminar flow predominated by the heat conduct and the laminar
flow cooperated by the convection and the heat conduct.
Secondly, the primitive-variable-time-dependent approach is used to compute the steady and unsteady two-dimensional flow around a square cylinder at different Reynolds numbers. The appropriate length of the calculation domain is selected by comparing the results calculated by using some different outlet boundary condition. Several computational parameters, such as Xu(distance from body to inlet), Xd(distance from body to outlet), blockage parameter B, lateral and outlet boundary conditions, are taken into consideration to analyze their effects on vortex shedding frequency scaled by the Strouhal number St and drag coefficient CD. Ranges of blockage parameters are studied for the different lateral boundary conditions to demonstrate where its effects can be ignored numerically, and then the effects of the lateral boundaries are compared. A set of optimized computational domain and artificial boundary conditions are selected to study the fluid flow and heat transfer around a square cylinder by using the primitive-variable-time-dependent approach. Numerical calculations of the steady flow around a square cylinder have been reported, and the onset of vortex shedding is predicted in between Re=50~55. To validate the algorithm, the unsteady process of vortex formation and shedding from the square cylinder for Re=100 is simulated successfully and the strict Kármán vortex street is also obtained.
Finally, based on the simulation of the flow around a square cylinder, the energy equation is taken into consideration and the vortex shedding from a square cylinder and its effects on the heat transfer of laminar forced c
引文
[1] 童秉纲, 张丙暄, 崔尔杰. 非定常流与涡运动. 北京: 国防工业出版社, 1993
[2] Braza M., Chassaing P., Minh H. H. Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech, 1986, 165: 79-130
[3] Williamson C. H. K. Vortex dynamics in the cylinder wake. Ann. Rev. Fluid Mech, 1996, 28: 477-539
[4] Zdravkovich M. M. Flow around circular cylinders. Vol 1: Fundamentals. Oxford University Press, New York, 1997
[5] 李光正. 绕圆柱非定常周期性涡旋脱落的数值研究. 水动力学研究与进展, A辑, 2000, 15(4): 493-504
[6] Yoshiaki K., Michitoshi T., Kenji O. Unsteady CFD analysys around a circular cylinder at various Reynolds numbers. 9th international symposium on flow visualization, 2000
[7] Johe W. W. Direct numerical simulation of flow over circular cylinders for Large-eddy simulation modeling. Ph.D. Thesis. University of Illinois at Urbana-Champaign, 2001
[8] Rhie C., Chow W. Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal, 1983, 21: 1525-1532
[9] Hebbark S. Mean and turbulence measurements in the boundary layer and wake of a symmetric airfoil. Exp., Fluids, 1986, 4: 214-222
[10] Davidson L. Prediction of the flow around an airfoil using a Reynolds stress transport model. ASME: Journal of fluids engineering, 1995, 117: 50-57
[11] Davidson L., Cokljat D. LESFOIL-Large eddy simulation of flow around a high-lift airfoil. Springer, 2002
[12] Voke P. R. Flow past a square cylinder test case LES2. Direct and Large-eddy ⅡFrance, 1996: 335-373
[13] Yu D. H., Kareen A. Numerical simulation of flow around rectangular prism. J Wind
Ind Aerodyn, 1997, 67: 195-208
[14] Bouris D., Bergeles G. 2D LES of vortex shedding from a square cylinder. J Wind Ind Aerodyn, 1999, 80: 31-46
[15] Murakami S., Muchida A. On turbulent vortex shedding flow past 2D square cylinder predicted by CFD. J. Wind Ind. Aerodyn, 1995, 54: 191-211
[16] Davidson L., Norberg C. Application of different subgrid scale model in Large Eddy Simulation of Flow around a square cylinder. Technical Report, 1998, Chalmer University of Technology, Sweden
[17] 童兵, 祝兵, 周本宽. 方柱绕流的数值模拟. 力学季刊, 2002, 23(1): 77-82
[18] Kim D. H., Yang K. S., Senda M. Large eddy simulation of turbulent flow past a square cylinder confined in a channel. Computers Fluids, 2004, 33: 81-96
[19] Badr H. M. Laminar combined convection from a horizontal cylinder-parallel and contra flow regimes. Int. J. Heat Mass Transfer, 1984, 27(1): 15-27
[20] Chang K. S., Sa J. Y. The effect of buoyancy on vortex shedding in the near wake of a circular cylinder. J. Fluid Mech, 1990, 220: 253-266
[21] Nguyen H. D., et al. Unsteady mixed convection about a rotating circular cylinder with small fluctuations in the free-stream velocity. Int. J. Heat Mass Transfer, 1996, 39(3): 511-525
[22] 李光正. 混合对流绕圆柱涡旋脱落及对传热影响的数值研究. 水动力学研究与进展, A辑, 2000, 15(1): 49-57
[23] Strouhal V. über eine besondere Art der Tonerregung. Ann Phsik und Chemie, Neue Folge, 1878, 5:216-251
[24] Von Kármán T. über den Mechanismus des Widerstandes, den ein bewegter K?rper in einer Flussigkeit erzeugt. Nachr Ges Wiss G?ttingen Math-Phys, KlasseⅡ, 1991: 509-517
[25] Griffin O. M. A note on bluff body vortex formation. J. Fluid Mech, 1995, 258: 217-224
[26] Franke R. Numerische Berechnung der instation?ren Wirbelabl?sung hinter zylindrischen K?rpern. Ph.D. Thesis. University of Karlsruhe, 1991
[27] Kelkar K. M., Patankar S. V. Numerical prediction of vortex shedding behind a
square cylinder. Int. J. Num Methods in Fluids, 1992, 14: 327-341
[28] Sohankar A., Davidson L., Norberg C. Numerical simulation of unsteady flow around a square two-dimensional cylinder. In 12th Australasian Fluid Mechanics Conference, The University of Sydney, Australia, 1995: 517-520
[29] Jackson C. P. A finite-element study of the onset of vortex shedding in flow past variusly shaped bodies. J. Fluid Mech, 1987, 182: 23-45
[30] Provansal M., Mathis C., Boyer L. Bénard-von Kármán instability: transient and forced regimes. J. Fluid Mech, 1987, 182: 1-22
[31] Schumm M., Berger E., Monkewitz P. A. Self-excited oscillations in the wake of two-dimensional bluff bodies and their control. J. Fluid Mech, 1994, 253: 17-35
[32] Sa J. Y., Chang K. S. Shedding patterns of the near-wake vortices behind a circular cylinder. J. Num Methods Fluids, 1991, 12: 463-474
[33] Sohankar A., Norberg C., Davidson L. Numerical simulation of unsteady low-reynolds number flow around rectangular cylinders at incidence. J Wind Ind Aerodyn, 1997, 71: 189-201
[34] Saha A. K., Biswas G., Muralidhar K. Three-dimensional study of flow past a square cylinder at low Reynolds numbers. Int. J. Heat and Fluid Flow, 2003, 24: 54-66
[35] Versteeg H. K., Malalasekera W. An introduction to computational fluid dynamics, The finite volume method. Essex: Longman Scientific and Techical, 1995
[36] Patankar S. V., Spalding D. B. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat and Mass Transfer, 1972, 15: 1787-1806
[37] Knisely C. W. Strouhal number of rectangular cylinders at incidence: A review and new data. J. of Fluids and structures, 1990, 4: 371-393
[38] Vickery B. J. Fluctuating lift and drag on a long cylinder of square cross-section in a smooth and in a turbulent stream. J. Fluid Mech, 1966, 25: 481-494
[39] Nakaguchi H., Hashimoto K., Muto S. An experimental study on aerodynamic drag of rectangular cylinders. J. Japan Soc. Aero. Space Sci., 1968, 16: 1-5
[40] Bearman P. W., Trueman D. M. An investigation of the flow around rectangular cylinders. Aeronautical Quarterly, 1972, 23: 229-237
[41] Okajima A., Nagahisa T., Rokugoh A. A numerical analysis of flow around rectangular cylinders. JSME Int. J. SerⅡ,1990, 33: 702-711
[42] Ohya Y., Washizu K., Fujii K., et al. Wind tunnel experiments on aerodynamic forces and pressure distributions of rectangular cylinder in a uniform flow. J. Fluids Structures, 1980
[43] Lee B. E. The effect of turbulence on the surface pressure field of a square prism. J. Fluid Mech, 1975, 69: 263-282
[44] Okajima A. Strouhal numbers of rectangular cylinders. J. Fluid Mech, 1982, 123: 379-398
[45] Hasan M. A. Z. The near wake structure of a square cylinder. Int. J. Heat and Fluid Flow, 1989, 10: 339-348
[46] Norberg C. Flow around rectangular cylinders: Pressure forces and wake frequencies. J. Wind Eng. Ind. Aero., 1993, 49: 187-196
[47] Zaki T. G., Sen M., Gad-El-Hak M. Numerical and experimental investigation of flow past a freely rotationable square cylinder. J. Fluids and Structures, 1994, 8: 555-562
[48] Luo S. C., Yazdani Md G., Chew Y T., et al. Effects of incidence and afterbody shape on flow past bluff cylinders. J. Wind Eng. Ind. Aero., 1994, 53: 375-399
[49] Chen J. M., Liu C. H. Vortex shedding and surface pressure on a square cylinder at incidence to a uniform air stream. Int. J. Heat and Fluid Flow, 1999, 20: 592-597
[50] Parkinson G. V. Wind-induced instability of structures. Philosophical Transactions Of The Royal Society Of London Series A-Mathematical Physical And Engineering Sciences, 1971, 269: 395-409
[51] Novak J. Strouhal number of a quadrangular prism, angle iron, and two circular cylinders arranged in tandem. In Acta Technical CSAV, Czechoslovakia, 1974: 361-373
[52] Rockwell D. O. Organized fluctuations due to flow past a square cross section cylinder. ASME J.of Fluids Engineering, 1977, 99: 511-516
[53] Otsuki Y., Fujii K., Washizu K., et al. Wind tunnel experiments on aerodynamics forces and pressure distributions of rectangular in a uniform cylinders in a unform flow. In Proceeding of the Japan Symposium on Wind-Structure Interaction, 1978: 169-176
[54] Washizu K., Ohya A., Otsuki Y., et al. Aeroelastic instability of rectangular cylinders in a heaving mode. J.Sound and Vibration, 1978, 59: 481-494
[55] Grant I., Barnes F. H. The vortex shedding associated with structural angles. J. of Wind Eng. And Ind. Aero., 1981, 8: 115-122
[56] Modi V. J., Slater J. E. Unsteady aerodynamics and vortex induced aeroelastic instability of a structural angle section. J. of Wind Eng. And Ind. Aero., 1983, 11: 321-334
[57] Bokaian A. K., Geoola F. F. Hydrodynamics galloping of rectangular cylinders. In H. S. Stephens and G. B. Warren, editors, Int. Conf. Flow-Induced Vibrations In Fluids Engineerings, Reading, England, 1982: 105-129
[58] Bokaian A. K., Geoola F. F. On the cross flow response of cylindrical structures. In Proc. Inst. Civil Engineering, 1983, 75: 379-418
[59] Obasaju E. D. An investigation of the effects of incidence on the flow around a square section cylinder. Aeronautical Quarterly, 1983, 34: 243-259
[60] Parker P., Welsh M. C. Effects of sound on flow separation from blunt flat plates. Int. J. Heat and Fluid Flow, 1983, 4: 113-127
[61] Davis R. W., Moore E. F. A numerical study of vortex shedding from rectangles. J.Fluid Mech., 1982, 116: 475-506
[62] Okajima A. Numerical analysis of the flow around an oscillating cylinder. In 6th Int. Conf. Flow-Induced Vibration, London, U.K., 1995
[63] Oosthuizen P. H., Madan S. The effect of flow direction on combined convective heat transfer from cylinders to air, Trans. ASME C: Journal of Heat Transfer, 1971, 93: 240-241
[64] Yasuo M., et al. A fundamental study of symmetrical vortex generation behind a cylinder by wake heating or by splitter plate or mesh. Int. J. Heat mass transfer, 1986, 26(8): 1193-1201
[65] Sparrow E. M., Tien K. K. Forced convection heat transfer at an inclined and yawed square plate-application to solar collectors. Trans. ASME J. Heat Transfer, 1977, 99: 507-512
[66] Test F. L., Lessmann R. C. An experimental study of heat transfer during forced
convection over a rectangular body. Trans. ASME J. Heat Transfer, 1980, 102: 146-151
[67] Motwani D. G., Gaitonde U. N., Sukhatme S P. Heat transfer from rectangular plates inclined at different angles of attack and yaw to an air stream. Trans. ASME J. Heat Transfer, 1985: 307-312
[68] Igarashi T. Characteristics of the flow around a square prism. Bull. JSME, 1984, 27(231): 1858-1865
[69] Igarashi T. Characteristics of the flow around rectangular cylinders (the case of the angle of attack 0 deg). Bull. JSME, 1985(a), 28(242): 1690-1696
[70] Igarashi T. Heat transfer from a square prism to an air stream. Int. J. Heat Mass Transfer, 1985b, 28(1): 175-181
[71] Igarashi T. Fluid flow and heat transfer around rectangular cylinders (the case of a width/height ratio of a section of 2.0-4.0). Trans. Jpn. Soc. Mech. Engrs. 1986a, B52(480): 3011-3016
[72] Igarashi T. Local heat transfer from a square prism to an airstream. Int. J. Heat Mass Transfer, 1986b, 29(5): 777-784
[73] Igarashi T. Fluid flow and heat transfer around rectangular cylinders (the case of a width/height ratio of a section of 0.33-1.5). Int. J. Heat Mass Transfer, 1987, 30(5): 893-901
[74] Igarashi T., Mayumi Y. Fluid flow and heat transfer around a rectangular cylinder with small inclined angle (the case of a width/height ratio of a section of 5). Int. J. Heat and Fluid Flow, 2001, 22: 279-286
[75] Fromm J. E., Harlow F. H. Numerical solution of the problem of vortex street development. Physics of Fluids, 1963, 6: 975-982
[76] Leonard B. P. A stable and accurate convective modeling based on quadratic upstream interpolation. Comp. Meth. Appl. Mech. Eng., 1979, 19: 59-98
[77] Davis R. W., Moore E. F., Purtell L. P. A numerical-experimental study of confined flow around rectangular cylinders. Phys. Fluids, 1984, 27: 46-59
[78] Nagano S., Naito M., Takata H. A numerical analysis of two-dimensional flow past a rectangular prism by a discrite vortex model. Computers and Fluids, 1982, 10:
243-259
[79] Belotserkovsky S. M., Kotovskii V. N., Nisht M. I., et al. Two-dimensional separated flows. Technical report, 1993, CRC Press, Boca Roton, Florida
[80] Franke R., Rodi W., Sch?nung B. Numerical calculation of laminar vortex-shedding flow past cylinders. J. Wind Eng. Ind. Aero., 1990, 35: 237-257
[81] Tamura T., Kuwahara K. Numerical study of aerodynamics behavior of a square cylinder. J. of Wind Eng. And Ind. Aero., 1990, 33: 161-170
[82] Tamura T., Ohta I., Kuwahara K. On the reliability of two-dimensional simulation for unsteady flows around a cylinder-type structure. J. of Wind Eng. And Ind. Aero., 1990, 35: 275-288
[83] Okajima A. Numerical simulation of flow around rectangular cylinder. J. Wind Eng. Ind. Aero, 1990, 33: 171-180
[84] Arnal M. P., Goering D. J., Humphrey J. A. C. Vortex shedding from a bluff body adjacent to a plate sliding wall. J. Fluids Eng., Trans. ASME, 1991, 113: 384-398
[85] Okajima A., Ueno H., Sakai H. Numerical simulation of laminar and turbulent flows around rectangular cylinders. Int. J. for Numerical Methods in Fluids, 1992, 15: 999-1012
[86] Humphrey A. C., Li G. Numerical modeling of confined flow past a cylinder of square cross-section at various orientations. Int. J. for Numerical Methods in Fluids, 1995, 20: 1215-1236
[87] Sohankar A., Norberg C. Low-Reynolds flow around a square cylinder at incidence. Int. J. for Numerical Methods in Fluids, 1998, 26: 39-56
[88] 李光正. 非定常流函数涡量方程的一种数值解法研究. 力学学报, 1998, 31(1): 10-20
[89] 李光正. 绕流流场无穷远处流函数边值条件的数值研究. 水动力学研究与进展, A辑, 1998, 13(2): 189-198
[90] Patankar S. V. Numerical heat transfer and fluid flow. New York: McGraw-Hill, 1980
[91] Van Doormaal J. P., Raithby G. D. Enhancement of the SIMPLE method for predicting imcompressible fluid flows. Numer Heat Transfer, 1984, 7: 147-163
[92] 陶文铨. 计算传热学的近代进展. 北京: 科学出版社, 2000
[93] Raithby G. D., Schneider G. E. Elliptic systems: finite differnce methods Ⅱ. In: Minkowycz W. J., Sprrow E. W., Pletcher R. H., Schneider G. E., eds. Handbook of numerical heat transfer. New York: John Wiley ﹠ Sons, 1988: 241-289
[94] Sheng Y., Shoukri M., Sheng G., Wood P. A modification to the SIMPLE method for buoyancy-driven flows. Numer Heat Transfer, Part B,1998, 33: 65-78
[95] 吴望一. 流体力学. 北京: 北京大学出版社, 1982
[96] De Vahl D. G., Jones I. P. Natural convection in a square cavity: a comparison exercise. In: Lewis R. W., Morgan K., Schrefler B. A., eds. Numerical methods in thermal problems. Swansea: Prineridge Press, 1981: 552-572
[97] Chia U., Ghia K. N., Shin C. T. High-Re solutions for imcompressible flow using the navier-stokes equations and a multigrid method. J Comput. Phys, 1982, 48: 387-411
[98] Barakos G., Mitsulis E. Natural convection flow in a square cavity revisited: laminar and turbulent models with wall functions. Int J Num Methods Fluids, 1994, 18(7): 695-719
[99] Markatos N. C., Pericleous K. A. Laminar and turbulent natural convection in an enclosed cavity. Int. J. Heat Mass Transfer, 1984, 27: 775-792
[100] John M. House, Christoph Beckermann, Theodore F Smith. Effect of a centered conducting body on natural convection heat transfer in an enclosure. Numerical Heat Transfer, 1990, 18: 213-225
[101] Joshua Y. C., Schultz D. H. A stable high-order method for the heated cavity problem. Int. J. for Numerical Methods in Fluids, 1992, 15: 1313-1332
[102] Yoshida T., Watanabe T., Nakamura I. Numerical analysis of open boundary conditions for incompressible viscous flow past a square cylinder. Trans JSME, 1993, 59(2): 2799-2806
[103] Sohankar A., Norberg C., Davidson L. Low-Reynolds-Number flow around a square cylinder at incidence:study of blockage,onset of vortex shedding and outlet boundary condition. International Journal for Numerical Methods in Fluids, 1998, 2(6): 39-56
[104] Bao Weizhu. Artificial boundary conditions for two-dimensional incompressible viscous flows around an obstacle. Comput Methods Appl Mech Engng, 1997, 147(2): 263-273
[105] Stansby P. K., Slaouti A. Simulation of vortex shedding including blockage by the random-vortex and other methods. Int J Numer Meth Fluids, 1993, 17: 1003-1013
[106] Behr M., Hastretier D., Mittal S., et al. Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries. Comput Meth Appl Mech Engng, 1995, 123: 309-316
[107] Anagnostopoulos P., Illiadis G., Richardson S. Numerical study of the blockage effects on viscous flow past a circular cylinder. Int J Numer Meth Fluids, 1996, 22: 1061-1074
[108] 胡影影, 朱克勤, 席保树. 二维方柱绕流的人工侧边界数值研究, 清华大学学报, 2000, 40(4): 43-46
[109] Liehard, John H. A heat transfer textbook. Prentice-Hall, Inc., Englewood Cliffs, N. J. 07632, 1981
[110] Gray D. D., Giorgin A. The validity of the Boussinesq approximation for liquids and gases. Int. J. Heat Mass Transfer, 1976, 19(1): 545-551
[111] Thompson J. E., Warsi Z. U. A., Mastin C. W. Numerical grid generation, foundation and applications. North-Holland, 1985
[112] Kiya, M., Matsumura, M. Incoherent turbulence structure in the near wake of a normal plate. Journal of Fluid Mechanics, 1988, 190: 343-356
[2] Braza M., Chassaing P., Minh H. H. Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech, 1986, 165: 79-130
[3] Williamson C. H. K. Vortex dynamics in the cylinder wake. Ann. Rev. Fluid Mech, 1996, 28: 477-539
[4] Zdravkovich M. M. Flow around circular cylinders. Vol 1: Fundamentals. Oxford University Press, New York, 1997
[5] 李光正. 绕圆柱非定常周期性涡旋脱落的数值研究. 水动力学研究与进展, A辑, 2000, 15(4): 493-504
[6] Yoshiaki K., Michitoshi T., Kenji O. Unsteady CFD analysys around a circular cylinder at various Reynolds numbers. 9th international symposium on flow visualization, 2000
[7] Johe W. W. Direct numerical simulation of flow over circular cylinders for Large-eddy simulation modeling. Ph.D. Thesis. University of Illinois at Urbana-Champaign, 2001
[8] Rhie C., Chow W. Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal, 1983, 21: 1525-1532
[9] Hebbark S. Mean and turbulence measurements in the boundary layer and wake of a symmetric airfoil. Exp., Fluids, 1986, 4: 214-222
[10] Davidson L. Prediction of the flow around an airfoil using a Reynolds stress transport model. ASME: Journal of fluids engineering, 1995, 117: 50-57
[11] Davidson L., Cokljat D. LESFOIL-Large eddy simulation of flow around a high-lift airfoil. Springer, 2002
[12] Voke P. R. Flow past a square cylinder test case LES2. Direct and Large-eddy ⅡFrance, 1996: 335-373
[13] Yu D. H., Kareen A. Numerical simulation of flow around rectangular prism. J Wind
Ind Aerodyn, 1997, 67: 195-208
[14] Bouris D., Bergeles G. 2D LES of vortex shedding from a square cylinder. J Wind Ind Aerodyn, 1999, 80: 31-46
[15] Murakami S., Muchida A. On turbulent vortex shedding flow past 2D square cylinder predicted by CFD. J. Wind Ind. Aerodyn, 1995, 54: 191-211
[16] Davidson L., Norberg C. Application of different subgrid scale model in Large Eddy Simulation of Flow around a square cylinder. Technical Report, 1998, Chalmer University of Technology, Sweden
[17] 童兵, 祝兵, 周本宽. 方柱绕流的数值模拟. 力学季刊, 2002, 23(1): 77-82
[18] Kim D. H., Yang K. S., Senda M. Large eddy simulation of turbulent flow past a square cylinder confined in a channel. Computers Fluids, 2004, 33: 81-96
[19] Badr H. M. Laminar combined convection from a horizontal cylinder-parallel and contra flow regimes. Int. J. Heat Mass Transfer, 1984, 27(1): 15-27
[20] Chang K. S., Sa J. Y. The effect of buoyancy on vortex shedding in the near wake of a circular cylinder. J. Fluid Mech, 1990, 220: 253-266
[21] Nguyen H. D., et al. Unsteady mixed convection about a rotating circular cylinder with small fluctuations in the free-stream velocity. Int. J. Heat Mass Transfer, 1996, 39(3): 511-525
[22] 李光正. 混合对流绕圆柱涡旋脱落及对传热影响的数值研究. 水动力学研究与进展, A辑, 2000, 15(1): 49-57
[23] Strouhal V. über eine besondere Art der Tonerregung. Ann Phsik und Chemie, Neue Folge, 1878, 5:216-251
[24] Von Kármán T. über den Mechanismus des Widerstandes, den ein bewegter K?rper in einer Flussigkeit erzeugt. Nachr Ges Wiss G?ttingen Math-Phys, KlasseⅡ, 1991: 509-517
[25] Griffin O. M. A note on bluff body vortex formation. J. Fluid Mech, 1995, 258: 217-224
[26] Franke R. Numerische Berechnung der instation?ren Wirbelabl?sung hinter zylindrischen K?rpern. Ph.D. Thesis. University of Karlsruhe, 1991
[27] Kelkar K. M., Patankar S. V. Numerical prediction of vortex shedding behind a
square cylinder. Int. J. Num Methods in Fluids, 1992, 14: 327-341
[28] Sohankar A., Davidson L., Norberg C. Numerical simulation of unsteady flow around a square two-dimensional cylinder. In 12th Australasian Fluid Mechanics Conference, The University of Sydney, Australia, 1995: 517-520
[29] Jackson C. P. A finite-element study of the onset of vortex shedding in flow past variusly shaped bodies. J. Fluid Mech, 1987, 182: 23-45
[30] Provansal M., Mathis C., Boyer L. Bénard-von Kármán instability: transient and forced regimes. J. Fluid Mech, 1987, 182: 1-22
[31] Schumm M., Berger E., Monkewitz P. A. Self-excited oscillations in the wake of two-dimensional bluff bodies and their control. J. Fluid Mech, 1994, 253: 17-35
[32] Sa J. Y., Chang K. S. Shedding patterns of the near-wake vortices behind a circular cylinder. J. Num Methods Fluids, 1991, 12: 463-474
[33] Sohankar A., Norberg C., Davidson L. Numerical simulation of unsteady low-reynolds number flow around rectangular cylinders at incidence. J Wind Ind Aerodyn, 1997, 71: 189-201
[34] Saha A. K., Biswas G., Muralidhar K. Three-dimensional study of flow past a square cylinder at low Reynolds numbers. Int. J. Heat and Fluid Flow, 2003, 24: 54-66
[35] Versteeg H. K., Malalasekera W. An introduction to computational fluid dynamics, The finite volume method. Essex: Longman Scientific and Techical, 1995
[36] Patankar S. V., Spalding D. B. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat and Mass Transfer, 1972, 15: 1787-1806
[37] Knisely C. W. Strouhal number of rectangular cylinders at incidence: A review and new data. J. of Fluids and structures, 1990, 4: 371-393
[38] Vickery B. J. Fluctuating lift and drag on a long cylinder of square cross-section in a smooth and in a turbulent stream. J. Fluid Mech, 1966, 25: 481-494
[39] Nakaguchi H., Hashimoto K., Muto S. An experimental study on aerodynamic drag of rectangular cylinders. J. Japan Soc. Aero. Space Sci., 1968, 16: 1-5
[40] Bearman P. W., Trueman D. M. An investigation of the flow around rectangular cylinders. Aeronautical Quarterly, 1972, 23: 229-237
[41] Okajima A., Nagahisa T., Rokugoh A. A numerical analysis of flow around rectangular cylinders. JSME Int. J. SerⅡ,1990, 33: 702-711
[42] Ohya Y., Washizu K., Fujii K., et al. Wind tunnel experiments on aerodynamic forces and pressure distributions of rectangular cylinder in a uniform flow. J. Fluids Structures, 1980
[43] Lee B. E. The effect of turbulence on the surface pressure field of a square prism. J. Fluid Mech, 1975, 69: 263-282
[44] Okajima A. Strouhal numbers of rectangular cylinders. J. Fluid Mech, 1982, 123: 379-398
[45] Hasan M. A. Z. The near wake structure of a square cylinder. Int. J. Heat and Fluid Flow, 1989, 10: 339-348
[46] Norberg C. Flow around rectangular cylinders: Pressure forces and wake frequencies. J. Wind Eng. Ind. Aero., 1993, 49: 187-196
[47] Zaki T. G., Sen M., Gad-El-Hak M. Numerical and experimental investigation of flow past a freely rotationable square cylinder. J. Fluids and Structures, 1994, 8: 555-562
[48] Luo S. C., Yazdani Md G., Chew Y T., et al. Effects of incidence and afterbody shape on flow past bluff cylinders. J. Wind Eng. Ind. Aero., 1994, 53: 375-399
[49] Chen J. M., Liu C. H. Vortex shedding and surface pressure on a square cylinder at incidence to a uniform air stream. Int. J. Heat and Fluid Flow, 1999, 20: 592-597
[50] Parkinson G. V. Wind-induced instability of structures. Philosophical Transactions Of The Royal Society Of London Series A-Mathematical Physical And Engineering Sciences, 1971, 269: 395-409
[51] Novak J. Strouhal number of a quadrangular prism, angle iron, and two circular cylinders arranged in tandem. In Acta Technical CSAV, Czechoslovakia, 1974: 361-373
[52] Rockwell D. O. Organized fluctuations due to flow past a square cross section cylinder. ASME J.of Fluids Engineering, 1977, 99: 511-516
[53] Otsuki Y., Fujii K., Washizu K., et al. Wind tunnel experiments on aerodynamics forces and pressure distributions of rectangular in a uniform cylinders in a unform flow. In Proceeding of the Japan Symposium on Wind-Structure Interaction, 1978: 169-176
[54] Washizu K., Ohya A., Otsuki Y., et al. Aeroelastic instability of rectangular cylinders in a heaving mode. J.Sound and Vibration, 1978, 59: 481-494
[55] Grant I., Barnes F. H. The vortex shedding associated with structural angles. J. of Wind Eng. And Ind. Aero., 1981, 8: 115-122
[56] Modi V. J., Slater J. E. Unsteady aerodynamics and vortex induced aeroelastic instability of a structural angle section. J. of Wind Eng. And Ind. Aero., 1983, 11: 321-334
[57] Bokaian A. K., Geoola F. F. Hydrodynamics galloping of rectangular cylinders. In H. S. Stephens and G. B. Warren, editors, Int. Conf. Flow-Induced Vibrations In Fluids Engineerings, Reading, England, 1982: 105-129
[58] Bokaian A. K., Geoola F. F. On the cross flow response of cylindrical structures. In Proc. Inst. Civil Engineering, 1983, 75: 379-418
[59] Obasaju E. D. An investigation of the effects of incidence on the flow around a square section cylinder. Aeronautical Quarterly, 1983, 34: 243-259
[60] Parker P., Welsh M. C. Effects of sound on flow separation from blunt flat plates. Int. J. Heat and Fluid Flow, 1983, 4: 113-127
[61] Davis R. W., Moore E. F. A numerical study of vortex shedding from rectangles. J.Fluid Mech., 1982, 116: 475-506
[62] Okajima A. Numerical analysis of the flow around an oscillating cylinder. In 6th Int. Conf. Flow-Induced Vibration, London, U.K., 1995
[63] Oosthuizen P. H., Madan S. The effect of flow direction on combined convective heat transfer from cylinders to air, Trans. ASME C: Journal of Heat Transfer, 1971, 93: 240-241
[64] Yasuo M., et al. A fundamental study of symmetrical vortex generation behind a cylinder by wake heating or by splitter plate or mesh. Int. J. Heat mass transfer, 1986, 26(8): 1193-1201
[65] Sparrow E. M., Tien K. K. Forced convection heat transfer at an inclined and yawed square plate-application to solar collectors. Trans. ASME J. Heat Transfer, 1977, 99: 507-512
[66] Test F. L., Lessmann R. C. An experimental study of heat transfer during forced
convection over a rectangular body. Trans. ASME J. Heat Transfer, 1980, 102: 146-151
[67] Motwani D. G., Gaitonde U. N., Sukhatme S P. Heat transfer from rectangular plates inclined at different angles of attack and yaw to an air stream. Trans. ASME J. Heat Transfer, 1985: 307-312
[68] Igarashi T. Characteristics of the flow around a square prism. Bull. JSME, 1984, 27(231): 1858-1865
[69] Igarashi T. Characteristics of the flow around rectangular cylinders (the case of the angle of attack 0 deg). Bull. JSME, 1985(a), 28(242): 1690-1696
[70] Igarashi T. Heat transfer from a square prism to an air stream. Int. J. Heat Mass Transfer, 1985b, 28(1): 175-181
[71] Igarashi T. Fluid flow and heat transfer around rectangular cylinders (the case of a width/height ratio of a section of 2.0-4.0). Trans. Jpn. Soc. Mech. Engrs. 1986a, B52(480): 3011-3016
[72] Igarashi T. Local heat transfer from a square prism to an airstream. Int. J. Heat Mass Transfer, 1986b, 29(5): 777-784
[73] Igarashi T. Fluid flow and heat transfer around rectangular cylinders (the case of a width/height ratio of a section of 0.33-1.5). Int. J. Heat Mass Transfer, 1987, 30(5): 893-901
[74] Igarashi T., Mayumi Y. Fluid flow and heat transfer around a rectangular cylinder with small inclined angle (the case of a width/height ratio of a section of 5). Int. J. Heat and Fluid Flow, 2001, 22: 279-286
[75] Fromm J. E., Harlow F. H. Numerical solution of the problem of vortex street development. Physics of Fluids, 1963, 6: 975-982
[76] Leonard B. P. A stable and accurate convective modeling based on quadratic upstream interpolation. Comp. Meth. Appl. Mech. Eng., 1979, 19: 59-98
[77] Davis R. W., Moore E. F., Purtell L. P. A numerical-experimental study of confined flow around rectangular cylinders. Phys. Fluids, 1984, 27: 46-59
[78] Nagano S., Naito M., Takata H. A numerical analysis of two-dimensional flow past a rectangular prism by a discrite vortex model. Computers and Fluids, 1982, 10:
243-259
[79] Belotserkovsky S. M., Kotovskii V. N., Nisht M. I., et al. Two-dimensional separated flows. Technical report, 1993, CRC Press, Boca Roton, Florida
[80] Franke R., Rodi W., Sch?nung B. Numerical calculation of laminar vortex-shedding flow past cylinders. J. Wind Eng. Ind. Aero., 1990, 35: 237-257
[81] Tamura T., Kuwahara K. Numerical study of aerodynamics behavior of a square cylinder. J. of Wind Eng. And Ind. Aero., 1990, 33: 161-170
[82] Tamura T., Ohta I., Kuwahara K. On the reliability of two-dimensional simulation for unsteady flows around a cylinder-type structure. J. of Wind Eng. And Ind. Aero., 1990, 35: 275-288
[83] Okajima A. Numerical simulation of flow around rectangular cylinder. J. Wind Eng. Ind. Aero, 1990, 33: 171-180
[84] Arnal M. P., Goering D. J., Humphrey J. A. C. Vortex shedding from a bluff body adjacent to a plate sliding wall. J. Fluids Eng., Trans. ASME, 1991, 113: 384-398
[85] Okajima A., Ueno H., Sakai H. Numerical simulation of laminar and turbulent flows around rectangular cylinders. Int. J. for Numerical Methods in Fluids, 1992, 15: 999-1012
[86] Humphrey A. C., Li G. Numerical modeling of confined flow past a cylinder of square cross-section at various orientations. Int. J. for Numerical Methods in Fluids, 1995, 20: 1215-1236
[87] Sohankar A., Norberg C. Low-Reynolds flow around a square cylinder at incidence. Int. J. for Numerical Methods in Fluids, 1998, 26: 39-56
[88] 李光正. 非定常流函数涡量方程的一种数值解法研究. 力学学报, 1998, 31(1): 10-20
[89] 李光正. 绕流流场无穷远处流函数边值条件的数值研究. 水动力学研究与进展, A辑, 1998, 13(2): 189-198
[90] Patankar S. V. Numerical heat transfer and fluid flow. New York: McGraw-Hill, 1980
[91] Van Doormaal J. P., Raithby G. D. Enhancement of the SIMPLE method for predicting imcompressible fluid flows. Numer Heat Transfer, 1984, 7: 147-163
[92] 陶文铨. 计算传热学的近代进展. 北京: 科学出版社, 2000
[93] Raithby G. D., Schneider G. E. Elliptic systems: finite differnce methods Ⅱ. In: Minkowycz W. J., Sprrow E. W., Pletcher R. H., Schneider G. E., eds. Handbook of numerical heat transfer. New York: John Wiley ﹠ Sons, 1988: 241-289
[94] Sheng Y., Shoukri M., Sheng G., Wood P. A modification to the SIMPLE method for buoyancy-driven flows. Numer Heat Transfer, Part B,1998, 33: 65-78
[95] 吴望一. 流体力学. 北京: 北京大学出版社, 1982
[96] De Vahl D. G., Jones I. P. Natural convection in a square cavity: a comparison exercise. In: Lewis R. W., Morgan K., Schrefler B. A., eds. Numerical methods in thermal problems. Swansea: Prineridge Press, 1981: 552-572
[97] Chia U., Ghia K. N., Shin C. T. High-Re solutions for imcompressible flow using the navier-stokes equations and a multigrid method. J Comput. Phys, 1982, 48: 387-411
[98] Barakos G., Mitsulis E. Natural convection flow in a square cavity revisited: laminar and turbulent models with wall functions. Int J Num Methods Fluids, 1994, 18(7): 695-719
[99] Markatos N. C., Pericleous K. A. Laminar and turbulent natural convection in an enclosed cavity. Int. J. Heat Mass Transfer, 1984, 27: 775-792
[100] John M. House, Christoph Beckermann, Theodore F Smith. Effect of a centered conducting body on natural convection heat transfer in an enclosure. Numerical Heat Transfer, 1990, 18: 213-225
[101] Joshua Y. C., Schultz D. H. A stable high-order method for the heated cavity problem. Int. J. for Numerical Methods in Fluids, 1992, 15: 1313-1332
[102] Yoshida T., Watanabe T., Nakamura I. Numerical analysis of open boundary conditions for incompressible viscous flow past a square cylinder. Trans JSME, 1993, 59(2): 2799-2806
[103] Sohankar A., Norberg C., Davidson L. Low-Reynolds-Number flow around a square cylinder at incidence:study of blockage,onset of vortex shedding and outlet boundary condition. International Journal for Numerical Methods in Fluids, 1998, 2(6): 39-56
[104] Bao Weizhu. Artificial boundary conditions for two-dimensional incompressible viscous flows around an obstacle. Comput Methods Appl Mech Engng, 1997, 147(2): 263-273
[105] Stansby P. K., Slaouti A. Simulation of vortex shedding including blockage by the random-vortex and other methods. Int J Numer Meth Fluids, 1993, 17: 1003-1013
[106] Behr M., Hastretier D., Mittal S., et al. Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries. Comput Meth Appl Mech Engng, 1995, 123: 309-316
[107] Anagnostopoulos P., Illiadis G., Richardson S. Numerical study of the blockage effects on viscous flow past a circular cylinder. Int J Numer Meth Fluids, 1996, 22: 1061-1074
[108] 胡影影, 朱克勤, 席保树. 二维方柱绕流的人工侧边界数值研究, 清华大学学报, 2000, 40(4): 43-46
[109] Liehard, John H. A heat transfer textbook. Prentice-Hall, Inc., Englewood Cliffs, N. J. 07632, 1981
[110] Gray D. D., Giorgin A. The validity of the Boussinesq approximation for liquids and gases. Int. J. Heat Mass Transfer, 1976, 19(1): 545-551
[111] Thompson J. E., Warsi Z. U. A., Mastin C. W. Numerical grid generation, foundation and applications. North-Holland, 1985
[112] Kiya, M., Matsumura, M. Incoherent turbulence structure in the near wake of a normal plate. Journal of Fluid Mechanics, 1988, 190: 343-356