On the Sub-Critical Bifurcation of Anti-Phase and In-Phase Synchronized Vortex Shedding Forms

Affiliation(s)

Department of Civil Engineering, National Chi Nan University, Puli 545, Taiwan, China..

Department of Civil Engineering, National Chi Nan University, Puli 545, Taiwan, China..

ABSTRACT

Transition of flows past a pair of side-by-side circular
cylinders are investigated by numerical simulations and the bifurcation
analysis of the numerical results. Various flow patterns behind the
cylinder-pair have been identified by the gap ratio (*G*) and Reynolds number (*Re*).
This study focus on transition of in-phase and anti-phase vortex shedding
synchronized forms. A nested Cartesian-grid formulation, in combination with an
effective immersed boundary method and a two-step fractional-step procedure,
has been adopted to simulate the flows. Numerical results reveal that the
in-phase and anti-phase vortex shedding flows at *Re** *= 100 can
co-exist at 2.08 ≤G≤ 2.58. Hysteresis loop with increasing/decreasing *G* at constant Reynolds number *Re** *= 100 is reported.

Cite this paper

Y. F. Peng, "On the Sub-Critical Bifurcation of Anti-Phase and In-Phase Synchronized Vortex Shedding Forms,"*Journal of Modern Physics*, Vol. 4 No. 5B, 2013, pp. 89-95. doi: 10.4236/jmp.2013.45B015.

Y. F. Peng, "On the Sub-Critical Bifurcation of Anti-Phase and In-Phase Synchronized Vortex Shedding Forms,"

References

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[2] P. W. Bearman and A. J. Wadcock, “The Interference between A Pair of Circular Cylinders Normal to A Stream,” Journal of Fluid Mechanics, Vol. 61. No. 3, 1973, pp. 499-511. doi:10.1017/S0022112073000832

[3] M. M. Zdravkovich, “Review of Flow Interference Between Two Circular Cylinders in Various Arrangements,” ASME, Vol. 99, No. 4, 1977, pp. 618-633. doi:10.1115/1.3448871

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[5] H. J. Kim and P. A. Durbin, “Investigation of the Flow between a Pair of Circular Cylinders in the Flopping Regime,” Journal of Fluid Mechanics, Vol. 196, 1988, pp. 431-448. doi:10.1017/S0022112088002769

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[9] S. Kang, “Characteristics of Flow over Two Circular Cylinders in A Side-by-side Arrangement at Low Reynolds Numbers,” Physics of Fluids, Vol. 15, 2003.

[10] S. Kumar, B. Gonzalez and O. Probst, “Flow Past Two Rotating Cylinders,” Physics of Fluids, Vol. 23, No. 1, 2011, 01402. doi:10.1063/1.3528260

[11] Md Mahbub Alam, Y. Zhou, and X. W. Wang, “The Wake of Two Side-by-side Square Cylinders,” Journal of Fluid Mechanics, Vol. 669, 2011, pp. 432-471. doi:10.1017/S0022112010005288

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[13] J. Mizushima and Y. Ino, “Stability of Flows Past A Pair of Circular Cylinders in A Side-by-side Arrangement,” Journal of Fluid Mechanics, Vol. 595, 2008, pp. 491-507. doi:10.1017/S0022112007009433

[14] Y. F. Peng, A. Sau, R. R. Hwang, W. C. Yang and C. M. Hsieh, “Criticality of Flow Transition behind Two Side-by-side Elliptic Cylinders,” Physics of Fluids, Vol. 24. No. 3, 2012, 034102. doi:10.1063/1.3687450

[15] Y. F. Peng, R. Mittal, A. Sau and R. Hwang, “Ested Cartesian Grid Method in Incompressible Viscous Fluid Flow,”Journal of Computational Physics, Vol. 229, No.19, 2010, pp. 7072-7101. doi:10.1016/j.jcp.2010.05.041

[16] M. Provansal, C. Mathis and L. Boyer, “Be’nard-von Ka’rma’n Instability: Transient and Forced Regimes,” Journal of Fluid Mechanics, Vol. 182, 1987, pp. 1-22. doi:10.1017/S0022112087002222

[17] C. H. K. Wil-liamson, “Oblique and Parallel Modes of Vortex Shedding in the Wake of a Circular Cylinder at Low Reynolds Numbers,” Journal of Fluid Mechanics, Vol. 206, 1989, pp. 579-627. doi:10.1017/S0022112089002429

[18] C. Norberg, “An Experimental Investigation of the Flow around a Circular Cylinder: Influence of Aspect Ratio,” Journal of Fluid Mechanics, Vol. 258, 1994, pp. 287-316. doi:10.1017/S0022112094003332

[19] C. P. Jackson, “A Fi-nite-element Study of the Onset of Vortex Shedding in Flow past Variously Shaped Bodies,” J ournal of Fluid Mechanics, Vol. 182, 1987, pp. 23-45. doi:10.1017/S0022112087002234

[20] B, Kumar and S. Mittal, “Effect of Blockage on Critical Parameters for Flow past A Circular Cylinder,” International Journal for Numerical Methods in Fluids, Vol. 50. No. 8, 2006, pp. 987-1001. doi:10.1002/fld.1098

[21] J. Dusek, P. Le Gal and P. Fraunie, “A Numerical and Theoretical Study of the First Hopf Bifurcation in A Cylinder Wake,” Journal of Fluid Mechanics, Vol. 264, 1994, pp. 59-80. doi:10.1017/S0022112094000583

[1] H. M. Spivac, “Vortex Frequency and Flow Pattern in the Wake of Two Parallel Cylinders at Varied Spacing Normal to An Airstream,” Journal of the Aeronautical Sciences, Vol. 13, 1946, pp. 289-301.

[2] P. W. Bearman and A. J. Wadcock, “The Interference between A Pair of Circular Cylinders Normal to A Stream,” Journal of Fluid Mechanics, Vol. 61. No. 3, 1973, pp. 499-511. doi:10.1017/S0022112073000832

[3] M. M. Zdravkovich, “Review of Flow Interference Between Two Circular Cylinders in Various Arrangements,” ASME, Vol. 99, No. 4, 1977, pp. 618-633. doi:10.1115/1.3448871

[4] C. H. K. Williamson, “Evolution of a Single Wake behind A Pair of Bluff Bodies,” Journal of Fluid Mechanics, Vol. 159, 1985, pp. 1-18. doi:10.1017/S002211208500307X

[5] H. J. Kim and P. A. Durbin, “Investigation of the Flow between a Pair of Circular Cylinders in the Flopping Regime,” Journal of Fluid Mechanics, Vol. 196, 1988, pp. 431-448. doi:10.1017/S0022112088002769

[6] D. Sumner, S. S. T. Wong, S. J. Price and M. P. Päidoussis, “Fluid Behavior of Side-by-side Circular Cylinders in Steady Cross-flow,” Journal of Fluids and Structures, Vol. 13. No. 3, 1999, pp. 309-339. doi:10.1006/jfls.1999.0205

[7] Y. Zhou, H. J. Zhang and M. W. Yiu, “The Turbulent Wake of Two Side-by-side Circular Cylinders,” Journal of Fluid Mechanics，Vol. 458, 2002, pp. 303-332. doi:10.1017/S0022112002007887

[8] S. J. Xu, Y. Zhou and R. M. C. So, “Reynolds Number Effects on the Flow Structure behind Two Side-by-side Cylinders,” Physics of Fluids, Vol. 15. No. 5, 2003, pp. 1214-1219. doi:10.1063/1.1561614

[9] S. Kang, “Characteristics of Flow over Two Circular Cylinders in A Side-by-side Arrangement at Low Reynolds Numbers,” Physics of Fluids, Vol. 15, 2003.

[10] S. Kumar, B. Gonzalez and O. Probst, “Flow Past Two Rotating Cylinders,” Physics of Fluids, Vol. 23, No. 1, 2011, 01402. doi:10.1063/1.3528260

[11] Md Mahbub Alam, Y. Zhou, and X. W. Wang, “The Wake of Two Side-by-side Square Cylinders,” Journal of Fluid Mechanics, Vol. 669, 2011, pp. 432-471. doi:10.1017/S0022112010005288

[12] D. Sumner, “Two Circular Cylinders in Cross-flow: A Review,” Journal of Fluids and Structures, Vol. 26. No. 6, 2010, pp. 849-899. doi:10.1016/j.jfluidstructs.2010.07.001

[13] J. Mizushima and Y. Ino, “Stability of Flows Past A Pair of Circular Cylinders in A Side-by-side Arrangement,” Journal of Fluid Mechanics, Vol. 595, 2008, pp. 491-507. doi:10.1017/S0022112007009433

[14] Y. F. Peng, A. Sau, R. R. Hwang, W. C. Yang and C. M. Hsieh, “Criticality of Flow Transition behind Two Side-by-side Elliptic Cylinders,” Physics of Fluids, Vol. 24. No. 3, 2012, 034102. doi:10.1063/1.3687450

[15] Y. F. Peng, R. Mittal, A. Sau and R. Hwang, “Ested Cartesian Grid Method in Incompressible Viscous Fluid Flow,”Journal of Computational Physics, Vol. 229, No.19, 2010, pp. 7072-7101. doi:10.1016/j.jcp.2010.05.041

[16] M. Provansal, C. Mathis and L. Boyer, “Be’nard-von Ka’rma’n Instability: Transient and Forced Regimes,” Journal of Fluid Mechanics, Vol. 182, 1987, pp. 1-22. doi:10.1017/S0022112087002222

[17] C. H. K. Wil-liamson, “Oblique and Parallel Modes of Vortex Shedding in the Wake of a Circular Cylinder at Low Reynolds Numbers,” Journal of Fluid Mechanics, Vol. 206, 1989, pp. 579-627. doi:10.1017/S0022112089002429

[18] C. Norberg, “An Experimental Investigation of the Flow around a Circular Cylinder: Influence of Aspect Ratio,” Journal of Fluid Mechanics, Vol. 258, 1994, pp. 287-316. doi:10.1017/S0022112094003332

[19] C. P. Jackson, “A Fi-nite-element Study of the Onset of Vortex Shedding in Flow past Variously Shaped Bodies,” J ournal of Fluid Mechanics, Vol. 182, 1987, pp. 23-45. doi:10.1017/S0022112087002234

[20] B, Kumar and S. Mittal, “Effect of Blockage on Critical Parameters for Flow past A Circular Cylinder,” International Journal for Numerical Methods in Fluids, Vol. 50. No. 8, 2006, pp. 987-1001. doi:10.1002/fld.1098

[21] J. Dusek, P. Le Gal and P. Fraunie, “A Numerical and Theoretical Study of the First Hopf Bifurcation in A Cylinder Wake,” Journal of Fluid Mechanics, Vol. 264, 1994, pp. 59-80. doi:10.1017/S0022112094000583