圆柱尾迹三维转捩特性研究
|
摘要
本文以三维不可压缩粘性流体的Navier-Stokes方程为数学模型对圆柱绕流问题作直接数值模拟，研究圆柱尾迹三维转捩区间转捩特征和动力学过程。Navier-Stokes方程的时间离散采用基于混合刚性稳定格式的三阶分裂算法，空间离散采用Fourier谱－谱元法。
文中对GLL配置点下的求导误差进行了分析，提出的双精度方法可以将求导误差从O((N4)减小到O((N2)，其中(为机器精度，N为单元内插值多项式阶数。利用N-S方程的精确解，以及前人研究二维静止圆柱绕流、旋转圆柱绕流等问题的实验和计算结果的比较，对算法和程序作了严格的校验。在此基础上对Re = 200、250和300下的流动进行了三维数值模拟。
Re = 200是刚超过模式A的临界雷诺数，计算结果表明，此时近尾迹流场处于三维准周期层流状态。占优展向模态的波长及流场的整体特性均受计算域的展向特征长度的影响。尤其值得注意的是发现在特定的展向特征长度下线性稳定的展向模态会取代模式A而在尾迹中占优并决定卡门涡脱落的展向相位差。
Re = 250时模式B处于亚临界状态。在本文的计算中，模式A首先在流场中自发地发展起来，并通过各展向模态间的相互作用激发了模式B的出现，使得尾迹中模式A和B共存，这与前人的结果是一致的。研究进一步发现流向涡结构在下游尾迹中演化为新的模式-“双涡对模式”，以及导致时间信号不规则性的独立低频fm的出现。
Re = 300时模式A和B均线性不稳定，模式B由于增长率较高，首先取代二维尾迹，而后随着模式A的增长尾迹中模式A和B共存。从圆柱表面到旋涡形成区、到近尾迹再到下游尾迹流向涡结构从模式A转换到模式B、双涡对模式、再恢复为模式A。
本文初步研究了三角波形的展向非均匀来流速度剖面对圆柱尾迹三维转捩的影响，考察不同雷诺数下不同波长和幅值的情况。结果表明，在流动发展相当长的一段时间内，非均匀来流速度剖面的波长起主导作用，尾迹中相同波长及其谐波的展向模态受到激发，而其他模态则被抑制。尾迹中流向涡结构与模式A的相似性支持了模式A的产生源于来流绕过圆柱时三维性的观点。
The flow past a uniform circular cylinder is studied in detail by direct numerical simulations of three-dimensional incompressible Navier-Stokes equations. The features and dynamics for various Reynolds numbers in the three-dimensional transition regime of the cylinder wake are investigated. The 3rd-order splitting algorithm based on the mixed stiffly stable scheme is employed in the temporal discretization of the N-S equations and the mixed Fourier-spectral-spectral-element method in the spatial discretization.
The errors in calculating derivatives for the GLL collocation points are evaluated, which can be alleviated from O((N4) to O((N2) by the double-precision method proposed in the present paper, where ( denotes the machine precision and N the order of the interpolation polynomials in the elements. The exact solutions of the N-S equations are employed to verify the algorithm and the program. The two-dimensional flows past both a still cylinder and a rotating cylinder are computed. Our numerical results show good conformance with that from the previous experimental and numerical studies. After the verification, the three-dimensional numerical simulations of the cylinder wake for Re = 200, 250 and 300 are performed.
Re = 200 is just beyond the critical Reynolds number of mode A. The numerical results of the present paper indicate that the near wake at this supercritical Reynolds number is in three-dimensional quasi-periodic laminar state with transitional behaviors. The spanwise characteristic length determines the transition features and global properties of the wake. Especially for the specific spanwise characteristic length linear stable mode can dominate the wake in place of mode A and determine the spanwise phase difference of the primary vortices shedding.
At Re = 250, mode B is subcritical. The present studies suggest that mode A spontaneously emerges in the wake preceding all the other spanwise modes. Then it excites the linear stable mode B through the nonlinear interactions among the various spanwise modes. Eventually mode A and B coexist in the wake, which confirms the previous studies. Besides, the present paper finds that downstream the streamwise
vortices evolve into a new type of mode - "dual vortex pair mode". An independent low frequency fm, which result in the irregularity of the temporal signals, other than the vortex shedding frequency is also identified.
When Re = 300, mode A and B are both unstable. Due to its higher growth rate mode B replaces the 2-D wake first. Then with the growth of mode A, the two modes coexist in the wake. From the surface of the cylinder to the formation region of the vortices then to the wake downstream, the structure of the streamwise vortices change from mode A to mode B, dual vortex pair mode then mode A again.
Preliminary studies are performed on the effect of the non-uniform stream with cosine form velocity profile in spanwise direction on the three-dimensional transition of the cylinder wake. The parameters taken into consideration include the Reynolds number, the wavelength and the amplitude of the stream profile. The numerical results indicate that in a quite long period of evolution the wavelength of the stream profile determines the flow. In the wake the spanwise modes with the identical wavelength and the harmonic waves are excited, while the other modes are strongly suppressed. The similarity between the streamwise vortex structure in the wake and mode A for the uniform stream supports the opinion that mode A derives from the three-dimensionality in the bypassing process of the flow around the cylinder.
文中对GLL配置点下的求导误差进行了分析，提出的双精度方法可以将求导误差从O((N4)减小到O((N2)，其中(为机器精度，N为单元内插值多项式阶数。利用N-S方程的精确解，以及前人研究二维静止圆柱绕流、旋转圆柱绕流等问题的实验和计算结果的比较，对算法和程序作了严格的校验。在此基础上对Re = 200、250和300下的流动进行了三维数值模拟。
Re = 200是刚超过模式A的临界雷诺数，计算结果表明，此时近尾迹流场处于三维准周期层流状态。占优展向模态的波长及流场的整体特性均受计算域的展向特征长度的影响。尤其值得注意的是发现在特定的展向特征长度下线性稳定的展向模态会取代模式A而在尾迹中占优并决定卡门涡脱落的展向相位差。
Re = 250时模式B处于亚临界状态。在本文的计算中，模式A首先在流场中自发地发展起来，并通过各展向模态间的相互作用激发了模式B的出现，使得尾迹中模式A和B共存，这与前人的结果是一致的。研究进一步发现流向涡结构在下游尾迹中演化为新的模式-“双涡对模式”，以及导致时间信号不规则性的独立低频fm的出现。
Re = 300时模式A和B均线性不稳定，模式B由于增长率较高，首先取代二维尾迹，而后随着模式A的增长尾迹中模式A和B共存。从圆柱表面到旋涡形成区、到近尾迹再到下游尾迹流向涡结构从模式A转换到模式B、双涡对模式、再恢复为模式A。
本文初步研究了三角波形的展向非均匀来流速度剖面对圆柱尾迹三维转捩的影响，考察不同雷诺数下不同波长和幅值的情况。结果表明，在流动发展相当长的一段时间内，非均匀来流速度剖面的波长起主导作用，尾迹中相同波长及其谐波的展向模态受到激发，而其他模态则被抑制。尾迹中流向涡结构与模式A的相似性支持了模式A的产生源于来流绕过圆柱时三维性的观点。
The flow past a uniform circular cylinder is studied in detail by direct numerical simulations of three-dimensional incompressible Navier-Stokes equations. The features and dynamics for various Reynolds numbers in the three-dimensional transition regime of the cylinder wake are investigated. The 3rd-order splitting algorithm based on the mixed stiffly stable scheme is employed in the temporal discretization of the N-S equations and the mixed Fourier-spectral-spectral-element method in the spatial discretization.
The errors in calculating derivatives for the GLL collocation points are evaluated, which can be alleviated from O((N4) to O((N2) by the double-precision method proposed in the present paper, where ( denotes the machine precision and N the order of the interpolation polynomials in the elements. The exact solutions of the N-S equations are employed to verify the algorithm and the program. The two-dimensional flows past both a still cylinder and a rotating cylinder are computed. Our numerical results show good conformance with that from the previous experimental and numerical studies. After the verification, the three-dimensional numerical simulations of the cylinder wake for Re = 200, 250 and 300 are performed.
Re = 200 is just beyond the critical Reynolds number of mode A. The numerical results of the present paper indicate that the near wake at this supercritical Reynolds number is in three-dimensional quasi-periodic laminar state with transitional behaviors. The spanwise characteristic length determines the transition features and global properties of the wake. Especially for the specific spanwise characteristic length linear stable mode can dominate the wake in place of mode A and determine the spanwise phase difference of the primary vortices shedding.
At Re = 250, mode B is subcritical. The present studies suggest that mode A spontaneously emerges in the wake preceding all the other spanwise modes. Then it excites the linear stable mode B through the nonlinear interactions among the various spanwise modes. Eventually mode A and B coexist in the wake, which confirms the previous studies. Besides, the present paper finds that downstream the streamwise
vortices evolve into a new type of mode - "dual vortex pair mode". An independent low frequency fm, which result in the irregularity of the temporal signals, other than the vortex shedding frequency is also identified.
When Re = 300, mode A and B are both unstable. Due to its higher growth rate mode B replaces the 2-D wake first. Then with the growth of mode A, the two modes coexist in the wake. From the surface of the cylinder to the formation region of the vortices then to the wake downstream, the structure of the streamwise vortices change from mode A to mode B, dual vortex pair mode then mode A again.
Preliminary studies are performed on the effect of the non-uniform stream with cosine form velocity profile in spanwise direction on the three-dimensional transition of the cylinder wake. The parameters taken into consideration include the Reynolds number, the wavelength and the amplitude of the stream profile. The numerical results indicate that in a quite long period of evolution the wavelength of the stream profile determines the flow. In the wake the spanwise modes with the identical wavelength and the harmonic waves are excited, while the other modes are strongly suppressed. The similarity between the streamwise vortex structure in the wake and mode A for the uniform stream supports the opinion that mode A derives from the three-dimensionality in the bypassing process of the flow around the cylinder.
引文
[1] Albarède P & Monkewitz P A. A model for the formation of oblique shedding and "chevron" patterns in cylinder wakes. Phys. Fluids A, 1992, 4(4): 744~756
[2] Balachandar S, Mittal R & Najjar F M. Properties of the mean recirculation region in the wakes of two-dimensional bluff bodies. J. Fluid Mech., 351: 167~199
[3] Baltensperger R. Improving the accuracy of the matrix differentiation method for arbitrary collocation points. Appl. Numer. Math., 2000, 33: 143~149
[4] Barkley D & Henderson R D. Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech., 1996, 322: 215~241
[5] Barnes F H. Vortex shedding in the wake of a rotating circular cylinder at low Reynolds numbers. J. Phys. D: Appl. Phys., 2000, 33: L141~L144
[6] Bearman P W. Investigation of the flow behind a two-dimensional model with a blunt trailing edge and fitted with splitter plates. J. Fluid Mech., 1965, 21: 241~255
[7] Bearman P W & Tombazis N. The effect of three-dimensional imposed disturbances on bluff body near wake flows. J. Wind Engng Indust. Aero., 1993, 49: 339~350
[8] Bernal L P & Roshko A. Streamwise vortex structure in plane mixing layers. J. Fluid Mech., 1986, 170: 499~525
[9] Bloor M S. The transition to turbulence in the wake of a circular cylinder. J. Fluid Mech., 1964, 19: 290~304
[10] Braza M, Faghani D & Persillon H. Successive stages and the role of natural vortex dislocations in three-dimensional wake transition. J. Fluid Mech., 2001, 439: 1~41
[11] Brede M, Eckelmann H & Rockwell D. On secondary vortices in the cylinder wake. Phys. Fluids, 1996, 8(8): 2117~2124
[12] Breuer K S & Everson R M. On the errors incurred calculating derivatives using Chebyshev polynomials. J. Comput. Phys., 1992, 99: 56~67
[13] Canuto C, Hussaini M Y & Quarteroni A, et al. Spectral Methods in Fluid Dynamics. New York: Springer-Verlag, 1988
[14] Cao W & Guo B. Preconditioning on element interfaces for the p-version finite element method and spectral element method. SIAM J. Sci. Comput., 1999, 21(2): 522~551
[15] Casarin M A. Diagonal edge preconditioners in p-version and spectral element methods. SIAM J. Sci. Comput., 1997, 18(2): 610~620
[16] Don W S & Solomonoff A. Accuracy and speed in computing the Chebyshev collocation derivative. SIAM J. Sci. Comput., 1995, 16(6): 1253~1268
[17] Eisenlohr H & Eckelmann H. Vortex splitting and its consequences in the vortex street of circular cylinders. Phys. Fluids A, 1989, 1(2): 189~192
[18] Gerrad J H. The wakes of cylindrical bluff bodies at low Reynolds number. Philos. Trans. R. Soc. London Ser. A, 1978, 288: 351
[19] Golub G H & Van Loan C F. Matrix Computations, Third Edition. Baltimore and London: Johns Hopkins University Press, 1996
[20] Hama F R. Three-dimensional vortex pattern behind a circular cylinder. J. Aeronaut. Sci., 1957, 24: 156
[21] Hammache M & Gharib M. An experimental study of the parallel and oblique vortex shedding from circular cylinders. J. Fluid Mech., 1991, 232: 567~590
[22] Henderson R D. Details of the drag curve near the onset of vortex shedding. Phys. Fluids, 1995, 7: 2102~2121
[23] Henderson R D. Nonlinear dynamics and pattern formation in turbulent wake transition. J. Fluid Mech., 1997, 352: 65~112
[24] Henderson R D. Dynamic refinement algorithms for spectral element methods. Computer Methods in Applied Mechanics and Engineering, 1999, 175(3): 395~411
[25] Henderson R D & Karniadakis G E. Unstructured spectral element methods for simulation of turbulent flows. J. Comput. Phys., 1995, 122: 191~217
[26] Hu G H, Sun D J & Yin X Y, et al. Hopf bifurcation in wakes behind a rotating and translating circular cylinder. Phys. Fluids, 1996, 8(7): 1972~1974
[27] Kang S, Choi H & Lee S. Laminar flow past a rotating circular cylinder. Phys. Fluids, 1999, 11(11): 3312~3321
[28] Karniadakis G E, Israeli M & Orszag S A. High-order splitting methods for incompressible Navier-Stokes equations. J. Comput. Phys., 1991, 97: 414~443
[29] Karniadakis G E, Orszag S A & Ronquist E M, et al. Spectral element and lattice gas methods for incompressible fluid dynamics. In: Gunzburger M D & Nicalaides R A eds. Incompressible Fluid Dynamics Trends and Advances. Cambridge University Press, 1991
[30] Karniadakis G E & Sherwin S J. Spectral/hp Element Methods for CFD. New York, Oxford: Oxford University Press, 1999
[31] Karniadakis G E & Triantafyllou G S. Three-dimensional dynamics and transition to turbulence in the wake of bluff objects. J. Fluid Mech., 1992, 238: 1~30
[32] Kleiser L, Hartel C & Wintergerste T. There is no error in the Kleiser-Schumann influence matrix method. J. Comput. Phys., 1998, 141: 85~87
[33] K?nig M & Eckelmann H. An experimental study of the three-dimensional structure of the wake of circular cylinders in the laminar and transitional Reynolds number range. In: Eckelmann H, Graham J M R, Huerre P, et al. eds. Proceedings of the IUTAM Symposium on Bluff-body Wakes and Instabilities. Berlin: Springer-Verlag, 1993, 341~344
[34] Korczak K Z & Patera A T. An isoparametric spectral element method for solution of the Navier-Stokes equations in complex geometry. J. Comput. Phys., 1986, 62: 361~382
[35] Kovasznay L I G. Laminar flow behind a two-dimensional grid. In: Proc. Combridge Philos. Soc., 1948, 44
[36] Kravchenko A G, Moin P & Shariff K. B-spline method and zonal grids for simulations of complex turbulent flows. J. Comput. Phys., 1999, 151: 757~789
[37] Landau L D & Lifshitz E M. Fluid Mechanics, translated from the Russian by Sykes and Reid, London: Pergamon, 1959
[38] Lee T & Budwig R. A study of the effect of aspect ratio on vortex shedding behind circular cylinders. Phys. Fluids A, 1991, 3(2): 309~315
[39] Leweke T & Provansal M. The flow behind rings: bluff body wakes without end effects. J. Fluid Mech., 1995, 288: 265~310
[40] Maday Y & Patera A T. Spectral element methods for the incompressible Navier-Stokes equations. In: Noor A K eds. State of the Art Surveys in Computational Mechanics. New York, 1989. 71
[41] Made M M M, Beauwens R & Warzee G. Preconditioning of discrete Helmholtz operators perturbed by a diagonal complex matrix. Commun. Numer. Meth. Engng, 2000, 16(11): 801~817
[42] Magere E & Deville M O. Simulation of the Taylor-Couette flow in a finite geometry by spectral element method. Applied Numerical Mathematics, 2000, 33: 241~249
[43] Meiburg E & Lasheras J C. Experimental and numerical investigation of the three- dimensional transition in plane wakes. J. Fluid Mech., 1988, 190: 1~37
[44] Norberg C. An experimental investigation of the flow around a circular cylinder: influence of aspect ratio. J. Fluid Mech., 1994, 258: 287~316
[45] Patera A T. A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys., 1984, 54: 468~488
[46] Pavarino L F. Preconditioned mixed spectral element methods for elasticity and Stokes problems. SIAM J. Sci. Comput., 1998, 19(6): 1941~1957
[47] Pavarino L F & Widlund O B. Iterative substructuring methods for spectral elements: problems in three dimensions based on numerical quadrature. Computers Math. Applic., 1997, 33(1/2): 193~209
[48] Persillon H & Braza M. Physical analysis of the transition to turbulence in the wake of a circular cylinder by three-dimensional Navier-Stokes simulation. J. Fluid Mech., 1998, 365: 23~88
[49] Pierrehumbert R & Widnall S. The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech., 1982, 114: 59~82
[50] Posdziech O & Grundmann R. Numerical simulation of the flow around an infinitely long circular cylinder in the transition regime. Theoret. Comput. Fluid Dynamics, 2001, 15: 121~141
[51] Provansal M, Mathis C & Boyer L. Bénard-von Kármán instability: transient and forced regimes. J. Fluid Mech., 1987, 182: 1~22
[52] Ronquist E M. A domain decomposition solver for the steady Navier-Stokes equation. In: Ilin A V & Scott L R eds. Proceedings of the 3rd International Conference on Spectral and Higher Order Methods. Houston, 1996. 469
[53] Roshko A. On the development of turbulent wakes from vortex streets. NACA Rep. 1191, 1954
[54] Roshko A. On the wake and drag of bluff bodies. J. Aeronaut. Sci., 1955, 22: 124
[55] Schatz M F, Barkley D & Swinney H L. Instability in a spatially periodic open flow. Phys. Fluids, 1995, 7(2): 344~358
[56] Thompson M, Hourigan K & Sheridan J. Three-dimensional instabilities in the wake of a circular cylinder. Exp. Therm. Fluid Sci., 1996, 12(2): 190~196
[57] Tombazis N & Bearman P W. A study of three-dimensional aspects of vortex shedding from a bluff body with a mild geometric disturbance. J. Fluid Mech., 1997, 330: 85~112
[58] Wilhelm D & Kleiser L. Stability analysis for different formulations of the nonlinear term in PN-PN(2 spectral element discretizations of the Navier-Stokes equations. J. Comput. Phys., 2001, 174: 306~326
[59] Williamson C H K. Defining a universal and continuous Strouhal-Reynolds number relationship for the laminar vortex shedding of a circular cylinder. Phys. Fluids, 1988a, 31(10): 2742~2744
[60] Williamson C H K. The existence of two stages in the transition to three dimensionality of a cylinder wake. Phys. Fluids, 1988b, 31(11): 3165~3167
[61] Williamson C H K. Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech., 1989, 206: 579~627
[62] Williamson C H K. The natural and forced formation of spot-like dislocations in the transition of a wake. J. Fluid Mech., 1992, 243: 393~441
[63] Williamson C H K. Three-dimensional wake transition. J. Fluid Mech., 1996a, 328: 345~407
[64] Williamson C H K. Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech., 1996b, 28: 477~539
[65] Williamson C H K. Three-dimensional vortex dynamics in bluff body wakes. Exp. Therm. Fluid Sci., 1996c, 12: 150~168.
[66] Zhang H Q, Fey U & Noack B R, et al. On the transition of the cylinder wake. Phys. Fluids, 1995, 7: 779~794
[67] 德拉津, 雷德. 流体动力稳定性. 周祖巍, 顾德炜 译. 北京: 宇航出版社, 1990
[68] 郁春伟. 圆柱尾迹三维转捩特征研究: [博士学位论文]. 北京: 中国科学院力学研究所, 1998
[69] 章本照. 流体力学中的有限元方法. 机械工业出版社, 1986
[2] Balachandar S, Mittal R & Najjar F M. Properties of the mean recirculation region in the wakes of two-dimensional bluff bodies. J. Fluid Mech., 351: 167~199
[3] Baltensperger R. Improving the accuracy of the matrix differentiation method for arbitrary collocation points. Appl. Numer. Math., 2000, 33: 143~149
[4] Barkley D & Henderson R D. Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech., 1996, 322: 215~241
[5] Barnes F H. Vortex shedding in the wake of a rotating circular cylinder at low Reynolds numbers. J. Phys. D: Appl. Phys., 2000, 33: L141~L144
[6] Bearman P W. Investigation of the flow behind a two-dimensional model with a blunt trailing edge and fitted with splitter plates. J. Fluid Mech., 1965, 21: 241~255
[7] Bearman P W & Tombazis N. The effect of three-dimensional imposed disturbances on bluff body near wake flows. J. Wind Engng Indust. Aero., 1993, 49: 339~350
[8] Bernal L P & Roshko A. Streamwise vortex structure in plane mixing layers. J. Fluid Mech., 1986, 170: 499~525
[9] Bloor M S. The transition to turbulence in the wake of a circular cylinder. J. Fluid Mech., 1964, 19: 290~304
[10] Braza M, Faghani D & Persillon H. Successive stages and the role of natural vortex dislocations in three-dimensional wake transition. J. Fluid Mech., 2001, 439: 1~41
[11] Brede M, Eckelmann H & Rockwell D. On secondary vortices in the cylinder wake. Phys. Fluids, 1996, 8(8): 2117~2124
[12] Breuer K S & Everson R M. On the errors incurred calculating derivatives using Chebyshev polynomials. J. Comput. Phys., 1992, 99: 56~67
[13] Canuto C, Hussaini M Y & Quarteroni A, et al. Spectral Methods in Fluid Dynamics. New York: Springer-Verlag, 1988
[14] Cao W & Guo B. Preconditioning on element interfaces for the p-version finite element method and spectral element method. SIAM J. Sci. Comput., 1999, 21(2): 522~551
[15] Casarin M A. Diagonal edge preconditioners in p-version and spectral element methods. SIAM J. Sci. Comput., 1997, 18(2): 610~620
[16] Don W S & Solomonoff A. Accuracy and speed in computing the Chebyshev collocation derivative. SIAM J. Sci. Comput., 1995, 16(6): 1253~1268
[17] Eisenlohr H & Eckelmann H. Vortex splitting and its consequences in the vortex street of circular cylinders. Phys. Fluids A, 1989, 1(2): 189~192
[18] Gerrad J H. The wakes of cylindrical bluff bodies at low Reynolds number. Philos. Trans. R. Soc. London Ser. A, 1978, 288: 351
[19] Golub G H & Van Loan C F. Matrix Computations, Third Edition. Baltimore and London: Johns Hopkins University Press, 1996
[20] Hama F R. Three-dimensional vortex pattern behind a circular cylinder. J. Aeronaut. Sci., 1957, 24: 156
[21] Hammache M & Gharib M. An experimental study of the parallel and oblique vortex shedding from circular cylinders. J. Fluid Mech., 1991, 232: 567~590
[22] Henderson R D. Details of the drag curve near the onset of vortex shedding. Phys. Fluids, 1995, 7: 2102~2121
[23] Henderson R D. Nonlinear dynamics and pattern formation in turbulent wake transition. J. Fluid Mech., 1997, 352: 65~112
[24] Henderson R D. Dynamic refinement algorithms for spectral element methods. Computer Methods in Applied Mechanics and Engineering, 1999, 175(3): 395~411
[25] Henderson R D & Karniadakis G E. Unstructured spectral element methods for simulation of turbulent flows. J. Comput. Phys., 1995, 122: 191~217
[26] Hu G H, Sun D J & Yin X Y, et al. Hopf bifurcation in wakes behind a rotating and translating circular cylinder. Phys. Fluids, 1996, 8(7): 1972~1974
[27] Kang S, Choi H & Lee S. Laminar flow past a rotating circular cylinder. Phys. Fluids, 1999, 11(11): 3312~3321
[28] Karniadakis G E, Israeli M & Orszag S A. High-order splitting methods for incompressible Navier-Stokes equations. J. Comput. Phys., 1991, 97: 414~443
[29] Karniadakis G E, Orszag S A & Ronquist E M, et al. Spectral element and lattice gas methods for incompressible fluid dynamics. In: Gunzburger M D & Nicalaides R A eds. Incompressible Fluid Dynamics Trends and Advances. Cambridge University Press, 1991
[30] Karniadakis G E & Sherwin S J. Spectral/hp Element Methods for CFD. New York, Oxford: Oxford University Press, 1999
[31] Karniadakis G E & Triantafyllou G S. Three-dimensional dynamics and transition to turbulence in the wake of bluff objects. J. Fluid Mech., 1992, 238: 1~30
[32] Kleiser L, Hartel C & Wintergerste T. There is no error in the Kleiser-Schumann influence matrix method. J. Comput. Phys., 1998, 141: 85~87
[33] K?nig M & Eckelmann H. An experimental study of the three-dimensional structure of the wake of circular cylinders in the laminar and transitional Reynolds number range. In: Eckelmann H, Graham J M R, Huerre P, et al. eds. Proceedings of the IUTAM Symposium on Bluff-body Wakes and Instabilities. Berlin: Springer-Verlag, 1993, 341~344
[34] Korczak K Z & Patera A T. An isoparametric spectral element method for solution of the Navier-Stokes equations in complex geometry. J. Comput. Phys., 1986, 62: 361~382
[35] Kovasznay L I G. Laminar flow behind a two-dimensional grid. In: Proc. Combridge Philos. Soc., 1948, 44
[36] Kravchenko A G, Moin P & Shariff K. B-spline method and zonal grids for simulations of complex turbulent flows. J. Comput. Phys., 1999, 151: 757~789
[37] Landau L D & Lifshitz E M. Fluid Mechanics, translated from the Russian by Sykes and Reid, London: Pergamon, 1959
[38] Lee T & Budwig R. A study of the effect of aspect ratio on vortex shedding behind circular cylinders. Phys. Fluids A, 1991, 3(2): 309~315
[39] Leweke T & Provansal M. The flow behind rings: bluff body wakes without end effects. J. Fluid Mech., 1995, 288: 265~310
[40] Maday Y & Patera A T. Spectral element methods for the incompressible Navier-Stokes equations. In: Noor A K eds. State of the Art Surveys in Computational Mechanics. New York, 1989. 71
[41] Made M M M, Beauwens R & Warzee G. Preconditioning of discrete Helmholtz operators perturbed by a diagonal complex matrix. Commun. Numer. Meth. Engng, 2000, 16(11): 801~817
[42] Magere E & Deville M O. Simulation of the Taylor-Couette flow in a finite geometry by spectral element method. Applied Numerical Mathematics, 2000, 33: 241~249
[43] Meiburg E & Lasheras J C. Experimental and numerical investigation of the three- dimensional transition in plane wakes. J. Fluid Mech., 1988, 190: 1~37
[44] Norberg C. An experimental investigation of the flow around a circular cylinder: influence of aspect ratio. J. Fluid Mech., 1994, 258: 287~316
[45] Patera A T. A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys., 1984, 54: 468~488
[46] Pavarino L F. Preconditioned mixed spectral element methods for elasticity and Stokes problems. SIAM J. Sci. Comput., 1998, 19(6): 1941~1957
[47] Pavarino L F & Widlund O B. Iterative substructuring methods for spectral elements: problems in three dimensions based on numerical quadrature. Computers Math. Applic., 1997, 33(1/2): 193~209
[48] Persillon H & Braza M. Physical analysis of the transition to turbulence in the wake of a circular cylinder by three-dimensional Navier-Stokes simulation. J. Fluid Mech., 1998, 365: 23~88
[49] Pierrehumbert R & Widnall S. The two- and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech., 1982, 114: 59~82
[50] Posdziech O & Grundmann R. Numerical simulation of the flow around an infinitely long circular cylinder in the transition regime. Theoret. Comput. Fluid Dynamics, 2001, 15: 121~141
[51] Provansal M, Mathis C & Boyer L. Bénard-von Kármán instability: transient and forced regimes. J. Fluid Mech., 1987, 182: 1~22
[52] Ronquist E M. A domain decomposition solver for the steady Navier-Stokes equation. In: Ilin A V & Scott L R eds. Proceedings of the 3rd International Conference on Spectral and Higher Order Methods. Houston, 1996. 469
[53] Roshko A. On the development of turbulent wakes from vortex streets. NACA Rep. 1191, 1954
[54] Roshko A. On the wake and drag of bluff bodies. J. Aeronaut. Sci., 1955, 22: 124
[55] Schatz M F, Barkley D & Swinney H L. Instability in a spatially periodic open flow. Phys. Fluids, 1995, 7(2): 344~358
[56] Thompson M, Hourigan K & Sheridan J. Three-dimensional instabilities in the wake of a circular cylinder. Exp. Therm. Fluid Sci., 1996, 12(2): 190~196
[57] Tombazis N & Bearman P W. A study of three-dimensional aspects of vortex shedding from a bluff body with a mild geometric disturbance. J. Fluid Mech., 1997, 330: 85~112
[58] Wilhelm D & Kleiser L. Stability analysis for different formulations of the nonlinear term in PN-PN(2 spectral element discretizations of the Navier-Stokes equations. J. Comput. Phys., 2001, 174: 306~326
[59] Williamson C H K. Defining a universal and continuous Strouhal-Reynolds number relationship for the laminar vortex shedding of a circular cylinder. Phys. Fluids, 1988a, 31(10): 2742~2744
[60] Williamson C H K. The existence of two stages in the transition to three dimensionality of a cylinder wake. Phys. Fluids, 1988b, 31(11): 3165~3167
[61] Williamson C H K. Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. J. Fluid Mech., 1989, 206: 579~627
[62] Williamson C H K. The natural and forced formation of spot-like dislocations in the transition of a wake. J. Fluid Mech., 1992, 243: 393~441
[63] Williamson C H K. Three-dimensional wake transition. J. Fluid Mech., 1996a, 328: 345~407
[64] Williamson C H K. Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech., 1996b, 28: 477~539
[65] Williamson C H K. Three-dimensional vortex dynamics in bluff body wakes. Exp. Therm. Fluid Sci., 1996c, 12: 150~168.
[66] Zhang H Q, Fey U & Noack B R, et al. On the transition of the cylinder wake. Phys. Fluids, 1995, 7: 779~794
[67] 德拉津, 雷德. 流体动力稳定性. 周祖巍, 顾德炜 译. 北京: 宇航出版社, 1990
[68] 郁春伟. 圆柱尾迹三维转捩特征研究: [博士学位论文]. 北京: 中国科学院力学研究所, 1998
[69] 章本照. 流体力学中的有限元方法. 机械工业出版社, 1986
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |